DRAG COEFFICI ENTS FOR FLAT PLATE S , S PH ERES, AND CYLINDERS MOVING AT LOW REYNOLDS NUMBERS IN A VISCOUS FLUID by ALVA MERLE JONE S A THESIS submitte<i to OREGON STATE COLLEGE in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE June 1958
103
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DRAG COEFFICIENTS FOR FLAT PLATES SPHERES AND CYLINDERS MOVING AT LOW REYNOLDS
NUMBERS IN A VISCOUS FLUID
by
ALVA MERLE JONES
A THESIS
submittelti to
OREGON STATE COLLEGE
in partial fulfillment of the requirements for the
degree of
MASTER OF SCIENCE
June 1958
APFROVED T
Redacted for Privacy
In 0hrrg of laJar
Redacted for Privacy
Redacted for Privacy
Redacted for Privacy
Drtc tbrclr la prrrontr a h4ul^r-J-trlqql
lfypcd by ftrcdeetr Or Joncr
i
ACKNOWLEDGEMLNT
The author wishes to express his appreciation to
Dr J G Knudsen for helping with this investigation and
to the Do Chemical Company for aiding this work through
FIGURE e- PHOTOGRAPH OF SPHERES CYLINDERS AND PLATES
34
holes were drilled so that each plate could be used for
two geometric ratios by changing the wires (See for
example plates la and lb in Table I
35
EXPERI MENTA L PROCEDURE
Viscosity and Density Calibration
A calibrated hydrometer measuring to the nearest
0002 was used to measure the density Table VI shows that
the effect of temperature on density is practically negli shy
gible in the small temperature range used
A Brookfield Synchro-lectric viscometer was used to
measure the viscosity of both the light and heavy oil
Figures 18 and 19 show the effect of temperature on visshy
cosity In addition the viscosity of the light oil was
checke d using the falling ball method and the equation
D2--ltA (f s bull fl) g (34) l 8v
The viscometer was calibrated by the National Bureau of bull
Standards and was accurate to l tb
Velocity Measurements
The velocity of movement through the oil was measured
by determining the rate of rotation of the pulleys with a
stop watch Usually the time for 10 revolutions was
measured at the highe r ve locities and for 5 revolutions at
the low velocities From this information and the di
amaters of the pulleys the velocities ere calculated
36
The time was measured to the nearest tenth of a second
Since the measured time was usually between 20 and 40
aeconds 1 the error in ~easuring velocity was considered to
be less tha~ 0 5~
force Measurements
The object connected to the scale 1 was dropped to the
bottom of the oil drum The motor was started and the scale
was read as the object vms being pulled towards the top of
the drum Two or three readings were taken for each object
at each velocity In nearly all cases these readings were
the same
37
ti XPER I MENTAL RE STJLTS
The dra g coefficient and the Reynolds number were
calculated by the use of Equations (l or (15) for each of
the spheres cylinders and plates from the measured
quantities of force and velocity a~d the values of the vis shy
cosity and density corresponding to the temperature of the
oil It was necessary to ~ubtract from the measured force
the force on the wire The corrected force measurement was
then used to determine the drag coefficient The force on
the wire has been determined as being proportional to the
velocity A correction curve relating force on the wire
and ve l ocity is plo tted in Figure 9 for the li ght oil and
Fi gure 10 for the heavy oil
The calculated drag coefficients Reynolds numbers
and velocities along with the measured force for the spheres
cylinders flat plates - parallel flow and flat plates shy
perpendicular flow have been tabulated in Tables II III
I V and v respectively
The calculated drag coefficients have been plotted as
a function of the Reynolds number on logarithic graph paper
with geometric ratios as a parameter
Drag coefficients for the spheres are plo tted in
Figure 11 The data for the cylinders are plotted in
CD_ bull 0 G 0
03
Tshy02
01
10 20 30 410 50 60 70 80
VELOCITY- FTJSEC
DRAG FORCE ON THE WIRE-LIGHT OIL
FIGURE 9
I -shy I -middot -- -shy -1shy _i-i I --~ I I _ -middot- shy I i
_I_ - _ middot- LL I l l tmiddot - middot1middot ~- - - - -+i middotshy I - --+-cl - l
1 1 I I IV jc---- --r--middotmiddottmiddot r-middotmiddot--tmiddotmiddot---shy _____ _L __ --~- --1shy middotmiddotr-r-middott- 1 -f-f-T- _~ +-L--1---~- 1--l
~- - shy I-+---Rmiddot-- I I I l i ~~ i -~~ ~- -T f i rshy ~-- --shy i- ----~-- shy - middot1 shy
I i I i I I 1--- -middot - fshy middot i----1---+-shy - i-middot -~+-- --~- --~-- ---- -t+ I v-~~ -middot j
i I middot 1_ _ I tmiddot---+-+1-+--li~+middot -+--+-+-1-+-+-+-+--tc--1-+-t-11-shy - middot --t- 1---t- t----tmiddotshy --~-- -middot i-shy I 1i - ~ i I i v i middotmiddotmiddot
[~v +L~ + ~ - I~~j-+ r V I ~t--- -~-- I +---~-- I f-middot ---1-- ~ -- --- ) Li --+--+--+-+-+-+--1--+--+---t---4 -1--1--+-+--+-l-i
tl~ I I Q Y +l~~ii-+-++++-middotHH-++-+-+-+--H--++ -i t Imiddot i i 1 j _V I f1 r-t~-middot l--r-tshy -~ 7 middot 1 -shy middot middotmiddot I
DRAG FORCE ON THE WIRE- HEAVY OIL
FIGURE 10
40
+shy l i~ltgt ~ bull r-rshy I i t _l
1 lf-1-1 l+r+ fJ-Ct I+ t li 1~t rtH r+l rf-l It llil I I
l l~pound 11 1 ~middot ~~middott ~ It lqf L
t I+--= ~r 17 -Er I _ ~ _pound~- sect Imiddot I+
iU=ff=t 1 +~ t_ - ~ r 111= t h=
I middot
t= IE I 1 1
plusmn~ kplusmni - -STOKE S EQ
(~ l h+middot
ru HmiddotHti+H1 11
c lffii l t~ 4 ~ ~middot ~ff l ~ ~h i ltlri
1 yen~ middot I ~ I I T ~ gt l+t H+h l+ i j l tfl-l Imiddotmiddot ft+ ++ l f+ Imiddotmiddot I+ I+ middott bulli I 1middot1 I ftt-1shy middot I middot r 11 I IH Ij ~ ~ middotishy J F 1= 6= ~
=f l~iit rtti l lit~ I FS lf~ l=i-+
l-11ffi tt lr 1 ~1 -t =l=Rttl 1ft i- 1 ~ I+ I
~~ lflJ
t I lfl m ~~WFB Lt
41plusmn811 IF I Hir tt ft itttplusmn i I~
1-+++middot
I ~ I (~ ffitrHf1 Ittmiddot ~ l r i H-t-r r HHt m 11 H++ I
bull I I
1_ _ F bullmiddot Imiddotmiddot t-- 1-T h iT
f-t+ ftt I+ I lt + T Imiddot 1
1t _plusmn middot~~ ~- 11shy
=a~ 1~ - =itf lttti
H I
=
DATA FOR SPHERES
FIGURE II
41
I -1---1-1-+--+--Ti-+-------+----r--shy --r--- -shy + t----+shy ----4-~---+-f----f--+-f--l--1 I t--shy --t-- ---+-shy
J-+-~f--~~ -___l_ ~---
i 1 L~L~-~tr-l----H~4-----~-f------+------+-----+----+---+middot-t-middot-H5000
middot~ ~ m- ~ - ~t plusmn~ 3t i t~ -f--- bullbull - ~~ h middot-
01 0~ 10
Re
-
DATA FOR CYLINDERS - LD = 6 8 AND 12
FIGURE I 4
44
Figures 12 13 and 14 The data for LD values of 16 24
and 32 were nearly the same and have been plotted to gether
i n Figure 12 In addition the curves for the other LD
ratios determined fro m Fib~res 13 and 14 have been drawn
in Figure 12 so that the effect of the length-to-diameter
is clearly shown Figure 13 shows the data for LD values
of 2 and 4 and the curves determined from this data
Firure 14 shows the data for LD values of 6 8 and 12
and the curves determined from this data
The data for flat plates in parallel flow are plotted
in Fi gure 15 A correction factor for the edge effect has
beon used so that the width-to-length ratio is not a
parameter in this plot A portion of the data of Janour
(5 p 31) is also shown in the diagram
The data for fla t plates in perpendicular flow is
plotted in Figures 16 a nd 17 Figure 16 shows the data for
WL values of 2 Also the curves for the three WL ratios
1 2 and 4 have been drawn in the fi gure Figure 17 shows
the data for WL values of 1 and 4 The curves determined
from the data have also been dravm in the figure
45
10~ ~ ~--- -shy
t==Ff1TR=+ iJ+--_-_--r_-_---+-+---+--+-+--_---_-~r-=r~=~+--=---=---=---=--~=--=_~1=_--=_~_-middot~~--+-+-t~ 1 Ll~+--+-- ---jtshyl~t L--+ I
I
P------ _l -- --1---L i
20 ~-- I ~g I --- - ---+-- r t L_shy
~ ~B 1) I --o-o- JONES - () - - ~~ p f---j- -~-- e e JANOU R
c gt ~c ~ ------ JANSSEN I 0 0 ~ I
IO ~2=i~~~~~~a=~~f=j= ---- TOM OTIKA bulll= I
~~n ~~--~~~~~~o~~~~~--4- NDCIgttl o shy
-
~--~~~~~+--+~+--4-r-~1+-~-middot+1~ ~ --H--~-~~os I i i i-4 ---~T I I f-- t --- li-------~--+-_--+--t-----~~-~_+---_-_-_--+------+-+-__+-[- +_- ___ _______ __+---+-r-+--H----_+--r--------+shy
it I+ ~ bull t ~1 ri j t++t+t++tft bullm H--~+H-t+t-++H-f+t+~HtttH t bull~H-IrttI-H
iH-H u nH m
I
t H+t-~ 1-r f-tj
i it iT -t middotHt I I I I Ill
~middot __
r middotshy
i I r-
f H- jLj f r H rr t~
II
t f f-l -t+tt ~ ==_ =~middot irE
I I
I
I
f
I --
i
t
1 r bull - r
~- ltt++l=tUtt~S-t+t+++~-++U +HJJm~-fl~HHtt1 tttn ll+t-Tt-~- ~ r fH T --r -1 t ---t- -tshy w _+ _ I-shy middotI
-shy -r- + Hbull Hshy t-I --r++ -t iHr -1 H-e-- -t I 1IT 1
1 H-rf-I IJftJ Jf+i+ ~ L
=+shy - tjshy rtmiddotshy ~ -
+ H 1-Jt I tt o =tt ~-
~1 l +fill l plusmn~ fplusmn -shy + I t-
DATA FOR FLAT PLATES PERPENDICULAR FLOW- WL= I 4
FIGURE 17
48
DI SCUSS ION OF RESULTS
Correction and Accuracy of Measurements
After a few pre liminary force measurements with the
spheres and a check with Stokes law (Equation 2) it was
apparent that the drag force on the wire was appreciable
and needed to be considered It was decided to take a
series of measurements with the spheres and calculate the
difference between the measured force and the force calcushy
lated from Stokes law The difference in force could then
be attributed to the drag on the wire If Stokes law is
followed the force on the wire should be proportional to
the velocity
A series of twenty measurements of the force on the
spheres was taken for each oil and the difference between
the measured force and that calcula ted by Stokes 1 law was
determined For each oil this difference as plo tted vs
the velocity The points grouped fairly ell around a
strai ght line nearly passing through the origin The
method of least squares was used to determine the equation
of the line best fitting the da t a The equa tion of the
line for the li bht oil tas found to be
Fe bullbull05605v - oooa (35)
which was determined at about 62 7degF Since the intercept
49
of the line is very close to zero it is believed that the
line is a good indication of the drag on the wire The
equation of the line for the heavy oil was found to be
F - 19llv I oo2o1 (36 ) c shy
which was determined at about 64 2deg The intercept of this
line is also quite close to zero These lines plotted in
Fi poundures 9 and 10 were used throughout the investigation
for the correction factor of the drag on the wires For
the cylinders and flat plates in parallel flow which were
pulled by two wires the values determined from Equations
35) and (36) were doubled For the plates in perpendicular
flow pulled by four wires the correction force was multishy
plied by four
The spring scale had 12 ounce divisions but could be
read to the nearest sixth of an ounce Some of the measureshy
ments of force were under an ounce hence a considerable
spread of the measurements was noticed in the pre liminary
data and throughout the experiment However sufficient
points were obtained so that it was possible to draw a
reliable curve through the data in all casas An analysis
was made to determine the average deviation from Stokes
equation for the spheres It raa found that the average
deviation was 15 1 for the light oil 16 6 for the heavy
oil and 15 9 overall The maximum deviation was 89
50
Inspection of the other data shows that these deviations
are also representative of the cylinders and flat plates
The force measurement is the least accurate part of the
experiment Other insignificant errors are introduced by
a small variation in the temperature This variation was
held to about 10 from the temperature of the calibrated
correction curve The velocity measurements and the
dimensions of the cylinders spheres and pl~ tes are conshy
sidered go od enough so tha t no appreciable errors occur
In order to e l iminate the WL parameter for flat plates
in parallel f l ow an additional factor for the effect of
the edges was subtracted from the measured force Janour
(5 p 27) presented the foll owing equation for the edge
correction for one edge of a flat plate in parallel flow
F ~ lv~ bull (37 ) edge gc
In present work this equation as doubled because both
edges of the plates were submerged in fluid It is assumed
in appl ying this correction that the lowe r limit of a
Reynolds number of 10 proposed by Janour can be extended
close to 0 1
Analysis of Results
Forty of the points for the spheres were used to get
51
the correction factor for the wires The remaining thirty
points are well erouped about Stokes law
The data for cylinders for LD ratios of 16 24 and
32 did not seem to be se gregated therefore these data
were plotted together It would seem that in the low range
of Reyno l ds numbers an LD of 16 and greater can be con shy
sidered an ~nfini tely long cylinder The other LD ratios
of 2 4 6 a 12 provided fairly distinct and separate
lines The best straight lines were drawn through the data
for each of the LD ratios It was evident that in eaeh
case a slope of -1 on a lo g-log graph gave the best straight
line which would indicate that the force varies directly
as the velocity It was possible to develop an empirical
expression relating dra g coefficient Reynolds number and
LD The following equation was obtained from the straight
line plots of Re vs fd for the various LD ratios
(38 )
Equation (38) applies for Reyno l ds numbers from 01 to 10
and for LD ratios of 2 to 16 For LD ratios greater
than 16
10 re (39 )
The data for flat plates in parallel flow is plotted
in Figure 15 after the correction factor for tho edge
52
effect was subtracted When the edge correction is made
no effect of WL ratio is indicated This result would be
expected The data followed a straight line with a slope
of -1 up to a Reynolds number of 2 After that a curve was
dravm connecting the line to that obtained by Janour The
equation for the straight section of the curve is
f - 6 (40)- Re
which applies for Reynolds numbers of 0 1 to 2 0 Here
a gain the force is proportional to the velocity Vfuen
determining drag force for flat plates in parallel flow
the force is first calculated from Equations (40) and (15 )
then the edge correction is added
The effect of the geometric ratios is clearly shown in
the data for flat plates in perpendicul ar flow which are
plotted in Figures 16 and 17 As with the other data the
best straight line was drawn through the various points
for eaoh of the WL ratios Again the line had a slope of
-1 The equation relating fd Re and wL was found t o be
rd 37 (w) -o 3o (41)Irel
which applies for Reynolds numbers of about 05 to 2 0 and
WL ratios of 1 to 4 It is possible but it has not been
proved that Equation (41) is suitable for higher WL ratios
The exponent on WL in Equation 41) is very close to that
53
on L D i n Equation ( 38 )~ It i s possible t ha t these
exponents are t he same but this cannot be sho~~ depound1nitely
until more accura te da ta are available It would be exshy
pected that a s the Reynolds number approaches zero t he
effect of geometric ratios would be the same for cylinders
and fla t pla tes in perpendicula r flow
It is seen in the t a bles of data that occasionally a
ne gative force was obtained because the correction applie d
due to t he wire dra g was greater than the mea sured force
These points obviously are incorrect This occurred only
for the smallest plates in the heavy oil at t he highest
velocities However these knom bad points occur in less
tha n 5~ of the data
It is clearl y shown that for cylinders and plates the
fd increases as L D or W L decreases This is in direct
contrast to Wiesel aberger s investigation However his
work is for hi gher Reynolds numbers at which a turbulent
wake forms bull
Comparison of Results with Other Data and Theoretical So l utions
The data for sphere~ a grees of course with Stokes
l aw since that law was used to determine the correction
factor for the wire Liebster (9 Pbull 548 ) has
54
substantiated Stokes equation
There are no experimental data with which to compare
the results of the cylinders Wieselsbergers minimum
Reynolds number of 4 is above the ran ge covered in the preshy
sent investigation The da ta for the highest LD ratios
(16 24 and 32) does agree almost exactly wi t h the solution
of Allen and Southwell (1 P bull 141) (LD =00) in the range
of Reynolds numbers from 0 1 to 1 0 Allen and Southwells
solution a greed with the data of Wieselsberger (16 p 22)
However the present data is above the theoretical solutions
of Lamb (8 p 112-121) throughout the range of Reynolds
numbers from 0 01 to 1 0 and above the solutions of
Bairstow Cave and Lang (2 p 404) I mai (4 p 157) and
Tomotika and Aoi (15 p 302) for Reynolds numbers of 0 1
to 1 0 Allen and Southwells solution a grees dth both
Wieselsberger 1 s a nd the present data Their solution and
the present data represent the best means for predicting
drag coefficients for flow over long cylinders for Reynolds
numbers of 0 01 to 10 It should be remembered that the
o t her solutions should a gree with eac h other since they
were all essentially derived by linearizing the Na viershy
Stokes equation
The data for flat plates in parallel flow is
55
considerably above the theoretical solutions of Janssen
(6 p 183 ) and Tomotika and Aoi (15 Pbull 302) However
Fi f~re 15 shows that a smooth transition occurs bet een
the present work and the data of Janour (5 P bull 31) The
present data considerably extend the experimental inforshy
mation previously available for laminar flow paral lel to
flat plates In the re gion of Reynol ds numbers less than
2 the drag coefficient is shown to be inversely proportional
to the Reynolds number Janours data covers a range of
Reynolds numbers from 11 to 1000 The results of the
present investigation line up with Janours results which
in turn on extrapolation to higher Reyno l ds numbers
(greater than 1000) make a smooth transition into Blasius
curve represented by Equation (10) At Reyno l ds numbers
greater than 20 000 the drag coefficient is inversely proshy
portional to the square root of the Reynolds number
The data for flat plates in perpendicular flow is conshy
siderably above the solutions of Tomotika and Aoi
(15 p 302) and Imai (4 p 157 However their solutions
f or cylinders and plates in parallel flow are also below
the present data Also it should be remembered that their
solutions are for infinitely wide plates If a value of
WL of above 100 is used in Equation (41) then the present
data and the solutions of Tomotika and Aoi are fairly close
56
The present results indicate that Equation (41~ can be
used with an accuracy of 15 to 20 within the limitations
of the equation (WL 1 to 4 Re = 0 05 to 2)
57
SUM RY AND CONCLUSIONS
Only a small amount of work has been done in the past
on the study of laminar flow over immersed bodies There
are many areas in the chemical process industries and the
field of aeronautics where this information would be very
helpful The purpose of the present investi gation wa s to
study the almost totally unexplored range of Reynol ds
numbers from 0 01 to 10
Drag coefficients have been determined for spheres
cylinders and flat plates in paralle l and perpendicular
flow The drag coefficients have been plotted as a
function of the Reynolds number with dimension ratios as
a parameter on lo g-log graphs The best straight lines
have been drawn through the data In all cases these lines
had a slope of -1 hich shows that the dra g coefficient is
inversely proportional to the Reynolds number at very low
Reynolds numbers for all shapes and dimension ratios The
following equations have been determined from the data
For cylinders
fd - 27 L -0 36 (38 ) - Re ())
which applies for Reynolds numbers of 0 01 to 1 and LD of
2 to 16 For LD greater than 16 the equation is
58
(39)
For flat plates in parallel flow a correction factor has
been applied to account for the edge effect The equation
which applies for Reyno l ds numbers of 0 1 to 2 is
f 6Re
(40)
For flat plates in perpendicular flow
f d
- 37 - Re (w) t -
0 bull 30 (41)
wbieh applies for W L of 1 to 4 and Reynolds numbers of
0 05 to 2
It is concluded tha t Equations (38-41) give the best
values of drag coefficients within an accuracy of 20~ for
the range of Reynolds numbers that were considered Also
it is evident that the dimension ratios are a n important
factor in determining the drag coefficient for a given
Reynolds number Furthermore the drag coefficient inshy
creases with decreasing values of L D or W L for a constant
Reynolds number The da ta obtained in this investi gation
compare favorably with the other experimental data and with
some of the theoretical sol utions It should be remembered
that when comparing the experimental data with theoretical
solutions that practically all of the solutions are for an
infinitely long cylinder or an infinitely wide plate
It is recommended tha t the present apparatus be
59
modified so that a force of 001 pound can be measured
Also it would improve tho accuracy to set up a constant
temperature bath so that the temperature of the oil can not
vary over 02degF A few check points on the present data
is all that is necessary to confirm the validity of
Equations (38- 41) It is also r ecommended that only SAE 140
oil be used and that 2 inches should be the minimum plate
width and cylinder length to be studi3d These conditions
would help to maintain the accuracy of the correction force
for the wire
60
~WMENCIATURE
Symbol Dimensions
A area sq ft
D diameter ft
F force lb f
L length ft
M mas s lb m Re Reynolds number Dvf= -ltr w width ft
a area sq ft
b characteristic length ft
d diameter ft
f drag coefficientfd
gravitation constant l b mft gc 2= 32 17 l b _ rsec
1 length ft
m mass l b bullm
p pressure lbrsqft
r radius ft
t time see
u velocity ft sec
v velocity ft sec
w width ft
61
Symbol Dimensions
X xbullcoordinate ft
y y- coordinate ft
o( vorticity
time sec
viscosity lb m ft -sec
kinematic viscosity ft 2sec
circumference diameter = 3 1416
3density lb m ft
function
stream function
Laplacian operator
infinity
Subscripts
c corrected
f force
1 l iquid
m mass
p projected
s solid
w wetted
62
BI BLIOGRAPHY
1 Allan D N de G and R v Southwell Re laxation methods applied to determine the motion in two di shymensions of a viscous fluid past a fixed cylinder Quarterly Journal of Mechanics and Applied Mathe shymatics 8 129-145 1955
2 Bairstow L B M Cave and E D Lang The reshysistance of a cylinder moving in a viscous fluid Philosophical Transactions of the Royal Society of London ser A 223383- 432 1923
3 Goldstein Sidney The steady flow of viscous fluid past a fixed spherical obstacle at small Reyno l ds numbers Proceedings of the Royal Society of London ser A 123225-235 1929
4 Imai I A new method of solving Oseens equations and its application to the flow past an inclined elliptic cylinder Proceedings of the Royal Society of London ser A 224 141-160 1954
5 Janour Zbynek Resistance of a plate in paralle l flow at low Reyno lds numbers Washington Nov 1951 40 p National Advisory Committee for Aeronautics Te chnica l Memorandum 1316)
6 Janssen E An analog solution of the Navier-Stokes equation for the case of flow past a f l at plate at low Reynolds numbers In 1956 Heat Transfer and Fluid Mechanics Institute (Preprints of Papers) p 173-183
7 Knudsen James G and Donal d L Katz Fluid Dynamics a nd Heat Transfer Ann Arbor University of Michigan 1953 243 p (Michi gan University Engineering Research Bulletin no 37)
8 La~b Horace On the uniform motion of a spherethrough a viscous fluid Philosophical Magazine and Journal of Science s~r 6 21112-121 1911
9 Liebster H Uben den widerstrand von kugeln Annalen Der Physik ser 4 82 541- 562 1 927
63
10 McAdams William H Heat transmission 3d ed New York McGraw- Hill 1954 532 p
11 Pai Shih- I Viscous f l ow theory I Laminar flow Princeton D Van Nostrand 1956 384 p
12 Prandtlbull Ludwi g Es sentials of fluid dynamics London Blackie amp Son 1954 452 p
13 Relf i F Discussion of the results of measure shyments of the resistance of wires with some additionshyal tests of the resistance of wires of small diame shyters In Technical report of the Advisory Committee for Aeronautics London) March 1914 p 47 - 51 (Report and memoranda no 102 )
14 Stokes George Gabriel Mathematical and physical papers Vol 3 Cambridge University Press 1922 413 p
15 Tomotika s and T Aoi The steady flow of a viscous fluid past an elliptic cylinder and a flat plate at smal l Reynolds numbers Quarterly Journal of Me chanics and Applie d Ma thematics 6 290- 312 1953
16 Wieselsbergo r c Versuche Ube r der luftwiderstand gerundeter und kant iger korper Er gebnisse der Aeroshydynamischen Versucbsansta l t Vol 2 G~ttingen 1923 80 p
FIGURE e- PHOTOGRAPH OF SPHERES CYLINDERS AND PLATES
34
holes were drilled so that each plate could be used for
two geometric ratios by changing the wires (See for
example plates la and lb in Table I
35
EXPERI MENTA L PROCEDURE
Viscosity and Density Calibration
A calibrated hydrometer measuring to the nearest
0002 was used to measure the density Table VI shows that
the effect of temperature on density is practically negli shy
gible in the small temperature range used
A Brookfield Synchro-lectric viscometer was used to
measure the viscosity of both the light and heavy oil
Figures 18 and 19 show the effect of temperature on visshy
cosity In addition the viscosity of the light oil was
checke d using the falling ball method and the equation
D2--ltA (f s bull fl) g (34) l 8v
The viscometer was calibrated by the National Bureau of bull
Standards and was accurate to l tb
Velocity Measurements
The velocity of movement through the oil was measured
by determining the rate of rotation of the pulleys with a
stop watch Usually the time for 10 revolutions was
measured at the highe r ve locities and for 5 revolutions at
the low velocities From this information and the di
amaters of the pulleys the velocities ere calculated
36
The time was measured to the nearest tenth of a second
Since the measured time was usually between 20 and 40
aeconds 1 the error in ~easuring velocity was considered to
be less tha~ 0 5~
force Measurements
The object connected to the scale 1 was dropped to the
bottom of the oil drum The motor was started and the scale
was read as the object vms being pulled towards the top of
the drum Two or three readings were taken for each object
at each velocity In nearly all cases these readings were
the same
37
ti XPER I MENTAL RE STJLTS
The dra g coefficient and the Reynolds number were
calculated by the use of Equations (l or (15) for each of
the spheres cylinders and plates from the measured
quantities of force and velocity a~d the values of the vis shy
cosity and density corresponding to the temperature of the
oil It was necessary to ~ubtract from the measured force
the force on the wire The corrected force measurement was
then used to determine the drag coefficient The force on
the wire has been determined as being proportional to the
velocity A correction curve relating force on the wire
and ve l ocity is plo tted in Figure 9 for the li ght oil and
Fi gure 10 for the heavy oil
The calculated drag coefficients Reynolds numbers
and velocities along with the measured force for the spheres
cylinders flat plates - parallel flow and flat plates shy
perpendicular flow have been tabulated in Tables II III
I V and v respectively
The calculated drag coefficients have been plotted as
a function of the Reynolds number on logarithic graph paper
with geometric ratios as a parameter
Drag coefficients for the spheres are plo tted in
Figure 11 The data for the cylinders are plotted in
CD_ bull 0 G 0
03
Tshy02
01
10 20 30 410 50 60 70 80
VELOCITY- FTJSEC
DRAG FORCE ON THE WIRE-LIGHT OIL
FIGURE 9
I -shy I -middot -- -shy -1shy _i-i I --~ I I _ -middot- shy I i
_I_ - _ middot- LL I l l tmiddot - middot1middot ~- - - - -+i middotshy I - --+-cl - l
1 1 I I IV jc---- --r--middotmiddottmiddot r-middotmiddot--tmiddotmiddot---shy _____ _L __ --~- --1shy middotmiddotr-r-middott- 1 -f-f-T- _~ +-L--1---~- 1--l
~- - shy I-+---Rmiddot-- I I I l i ~~ i -~~ ~- -T f i rshy ~-- --shy i- ----~-- shy - middot1 shy
I i I i I I 1--- -middot - fshy middot i----1---+-shy - i-middot -~+-- --~- --~-- ---- -t+ I v-~~ -middot j
i I middot 1_ _ I tmiddot---+-+1-+--li~+middot -+--+-+-1-+-+-+-+--tc--1-+-t-11-shy - middot --t- 1---t- t----tmiddotshy --~-- -middot i-shy I 1i - ~ i I i v i middotmiddotmiddot
[~v +L~ + ~ - I~~j-+ r V I ~t--- -~-- I +---~-- I f-middot ---1-- ~ -- --- ) Li --+--+--+-+-+-+--1--+--+---t---4 -1--1--+-+--+-l-i
tl~ I I Q Y +l~~ii-+-++++-middotHH-++-+-+-+--H--++ -i t Imiddot i i 1 j _V I f1 r-t~-middot l--r-tshy -~ 7 middot 1 -shy middot middotmiddot I
DRAG FORCE ON THE WIRE- HEAVY OIL
FIGURE 10
40
+shy l i~ltgt ~ bull r-rshy I i t _l
1 lf-1-1 l+r+ fJ-Ct I+ t li 1~t rtH r+l rf-l It llil I I
l l~pound 11 1 ~middot ~~middott ~ It lqf L
t I+--= ~r 17 -Er I _ ~ _pound~- sect Imiddot I+
iU=ff=t 1 +~ t_ - ~ r 111= t h=
I middot
t= IE I 1 1
plusmn~ kplusmni - -STOKE S EQ
(~ l h+middot
ru HmiddotHti+H1 11
c lffii l t~ 4 ~ ~middot ~ff l ~ ~h i ltlri
1 yen~ middot I ~ I I T ~ gt l+t H+h l+ i j l tfl-l Imiddotmiddot ft+ ++ l f+ Imiddotmiddot I+ I+ middott bulli I 1middot1 I ftt-1shy middot I middot r 11 I IH Ij ~ ~ middotishy J F 1= 6= ~
=f l~iit rtti l lit~ I FS lf~ l=i-+
l-11ffi tt lr 1 ~1 -t =l=Rttl 1ft i- 1 ~ I+ I
~~ lflJ
t I lfl m ~~WFB Lt
41plusmn811 IF I Hir tt ft itttplusmn i I~
1-+++middot
I ~ I (~ ffitrHf1 Ittmiddot ~ l r i H-t-r r HHt m 11 H++ I
bull I I
1_ _ F bullmiddot Imiddotmiddot t-- 1-T h iT
f-t+ ftt I+ I lt + T Imiddot 1
1t _plusmn middot~~ ~- 11shy
=a~ 1~ - =itf lttti
H I
=
DATA FOR SPHERES
FIGURE II
41
I -1---1-1-+--+--Ti-+-------+----r--shy --r--- -shy + t----+shy ----4-~---+-f----f--+-f--l--1 I t--shy --t-- ---+-shy
J-+-~f--~~ -___l_ ~---
i 1 L~L~-~tr-l----H~4-----~-f------+------+-----+----+---+middot-t-middot-H5000
middot~ ~ m- ~ - ~t plusmn~ 3t i t~ -f--- bullbull - ~~ h middot-
01 0~ 10
Re
-
DATA FOR CYLINDERS - LD = 6 8 AND 12
FIGURE I 4
44
Figures 12 13 and 14 The data for LD values of 16 24
and 32 were nearly the same and have been plotted to gether
i n Figure 12 In addition the curves for the other LD
ratios determined fro m Fib~res 13 and 14 have been drawn
in Figure 12 so that the effect of the length-to-diameter
is clearly shown Figure 13 shows the data for LD values
of 2 and 4 and the curves determined from this data
Firure 14 shows the data for LD values of 6 8 and 12
and the curves determined from this data
The data for flat plates in parallel flow are plotted
in Fi gure 15 A correction factor for the edge effect has
beon used so that the width-to-length ratio is not a
parameter in this plot A portion of the data of Janour
(5 p 31) is also shown in the diagram
The data for fla t plates in perpendicular flow is
plotted in Figures 16 a nd 17 Figure 16 shows the data for
WL values of 2 Also the curves for the three WL ratios
1 2 and 4 have been drawn in the fi gure Figure 17 shows
the data for WL values of 1 and 4 The curves determined
from the data have also been dravm in the figure
45
10~ ~ ~--- -shy
t==Ff1TR=+ iJ+--_-_--r_-_---+-+---+--+-+--_---_-~r-=r~=~+--=---=---=---=--~=--=_~1=_--=_~_-middot~~--+-+-t~ 1 Ll~+--+-- ---jtshyl~t L--+ I
I
P------ _l -- --1---L i
20 ~-- I ~g I --- - ---+-- r t L_shy
~ ~B 1) I --o-o- JONES - () - - ~~ p f---j- -~-- e e JANOU R
c gt ~c ~ ------ JANSSEN I 0 0 ~ I
IO ~2=i~~~~~~a=~~f=j= ---- TOM OTIKA bulll= I
~~n ~~--~~~~~~o~~~~~--4- NDCIgttl o shy
-
~--~~~~~+--+~+--4-r-~1+-~-middot+1~ ~ --H--~-~~os I i i i-4 ---~T I I f-- t --- li-------~--+-_--+--t-----~~-~_+---_-_-_--+------+-+-__+-[- +_- ___ _______ __+---+-r-+--H----_+--r--------+shy
it I+ ~ bull t ~1 ri j t++t+t++tft bullm H--~+H-t+t-++H-f+t+~HtttH t bull~H-IrttI-H
iH-H u nH m
I
t H+t-~ 1-r f-tj
i it iT -t middotHt I I I I Ill
~middot __
r middotshy
i I r-
f H- jLj f r H rr t~
II
t f f-l -t+tt ~ ==_ =~middot irE
I I
I
I
f
I --
i
t
1 r bull - r
~- ltt++l=tUtt~S-t+t+++~-++U +HJJm~-fl~HHtt1 tttn ll+t-Tt-~- ~ r fH T --r -1 t ---t- -tshy w _+ _ I-shy middotI
-shy -r- + Hbull Hshy t-I --r++ -t iHr -1 H-e-- -t I 1IT 1
1 H-rf-I IJftJ Jf+i+ ~ L
=+shy - tjshy rtmiddotshy ~ -
+ H 1-Jt I tt o =tt ~-
~1 l +fill l plusmn~ fplusmn -shy + I t-
DATA FOR FLAT PLATES PERPENDICULAR FLOW- WL= I 4
FIGURE 17
48
DI SCUSS ION OF RESULTS
Correction and Accuracy of Measurements
After a few pre liminary force measurements with the
spheres and a check with Stokes law (Equation 2) it was
apparent that the drag force on the wire was appreciable
and needed to be considered It was decided to take a
series of measurements with the spheres and calculate the
difference between the measured force and the force calcushy
lated from Stokes law The difference in force could then
be attributed to the drag on the wire If Stokes law is
followed the force on the wire should be proportional to
the velocity
A series of twenty measurements of the force on the
spheres was taken for each oil and the difference between
the measured force and that calcula ted by Stokes 1 law was
determined For each oil this difference as plo tted vs
the velocity The points grouped fairly ell around a
strai ght line nearly passing through the origin The
method of least squares was used to determine the equation
of the line best fitting the da t a The equa tion of the
line for the li bht oil tas found to be
Fe bullbull05605v - oooa (35)
which was determined at about 62 7degF Since the intercept
49
of the line is very close to zero it is believed that the
line is a good indication of the drag on the wire The
equation of the line for the heavy oil was found to be
F - 19llv I oo2o1 (36 ) c shy
which was determined at about 64 2deg The intercept of this
line is also quite close to zero These lines plotted in
Fi poundures 9 and 10 were used throughout the investigation
for the correction factor of the drag on the wires For
the cylinders and flat plates in parallel flow which were
pulled by two wires the values determined from Equations
35) and (36) were doubled For the plates in perpendicular
flow pulled by four wires the correction force was multishy
plied by four
The spring scale had 12 ounce divisions but could be
read to the nearest sixth of an ounce Some of the measureshy
ments of force were under an ounce hence a considerable
spread of the measurements was noticed in the pre liminary
data and throughout the experiment However sufficient
points were obtained so that it was possible to draw a
reliable curve through the data in all casas An analysis
was made to determine the average deviation from Stokes
equation for the spheres It raa found that the average
deviation was 15 1 for the light oil 16 6 for the heavy
oil and 15 9 overall The maximum deviation was 89
50
Inspection of the other data shows that these deviations
are also representative of the cylinders and flat plates
The force measurement is the least accurate part of the
experiment Other insignificant errors are introduced by
a small variation in the temperature This variation was
held to about 10 from the temperature of the calibrated
correction curve The velocity measurements and the
dimensions of the cylinders spheres and pl~ tes are conshy
sidered go od enough so tha t no appreciable errors occur
In order to e l iminate the WL parameter for flat plates
in parallel f l ow an additional factor for the effect of
the edges was subtracted from the measured force Janour
(5 p 27) presented the foll owing equation for the edge
correction for one edge of a flat plate in parallel flow
F ~ lv~ bull (37 ) edge gc
In present work this equation as doubled because both
edges of the plates were submerged in fluid It is assumed
in appl ying this correction that the lowe r limit of a
Reynolds number of 10 proposed by Janour can be extended
close to 0 1
Analysis of Results
Forty of the points for the spheres were used to get
51
the correction factor for the wires The remaining thirty
points are well erouped about Stokes law
The data for cylinders for LD ratios of 16 24 and
32 did not seem to be se gregated therefore these data
were plotted together It would seem that in the low range
of Reyno l ds numbers an LD of 16 and greater can be con shy
sidered an ~nfini tely long cylinder The other LD ratios
of 2 4 6 a 12 provided fairly distinct and separate
lines The best straight lines were drawn through the data
for each of the LD ratios It was evident that in eaeh
case a slope of -1 on a lo g-log graph gave the best straight
line which would indicate that the force varies directly
as the velocity It was possible to develop an empirical
expression relating dra g coefficient Reynolds number and
LD The following equation was obtained from the straight
line plots of Re vs fd for the various LD ratios
(38 )
Equation (38) applies for Reyno l ds numbers from 01 to 10
and for LD ratios of 2 to 16 For LD ratios greater
than 16
10 re (39 )
The data for flat plates in parallel flow is plotted
in Figure 15 after the correction factor for tho edge
52
effect was subtracted When the edge correction is made
no effect of WL ratio is indicated This result would be
expected The data followed a straight line with a slope
of -1 up to a Reynolds number of 2 After that a curve was
dravm connecting the line to that obtained by Janour The
equation for the straight section of the curve is
f - 6 (40)- Re
which applies for Reynolds numbers of 0 1 to 2 0 Here
a gain the force is proportional to the velocity Vfuen
determining drag force for flat plates in parallel flow
the force is first calculated from Equations (40) and (15 )
then the edge correction is added
The effect of the geometric ratios is clearly shown in
the data for flat plates in perpendicul ar flow which are
plotted in Figures 16 and 17 As with the other data the
best straight line was drawn through the various points
for eaoh of the WL ratios Again the line had a slope of
-1 The equation relating fd Re and wL was found t o be
rd 37 (w) -o 3o (41)Irel
which applies for Reynolds numbers of about 05 to 2 0 and
WL ratios of 1 to 4 It is possible but it has not been
proved that Equation (41) is suitable for higher WL ratios
The exponent on WL in Equation 41) is very close to that
53
on L D i n Equation ( 38 )~ It i s possible t ha t these
exponents are t he same but this cannot be sho~~ depound1nitely
until more accura te da ta are available It would be exshy
pected that a s the Reynolds number approaches zero t he
effect of geometric ratios would be the same for cylinders
and fla t pla tes in perpendicula r flow
It is seen in the t a bles of data that occasionally a
ne gative force was obtained because the correction applie d
due to t he wire dra g was greater than the mea sured force
These points obviously are incorrect This occurred only
for the smallest plates in the heavy oil at t he highest
velocities However these knom bad points occur in less
tha n 5~ of the data
It is clearl y shown that for cylinders and plates the
fd increases as L D or W L decreases This is in direct
contrast to Wiesel aberger s investigation However his
work is for hi gher Reynolds numbers at which a turbulent
wake forms bull
Comparison of Results with Other Data and Theoretical So l utions
The data for sphere~ a grees of course with Stokes
l aw since that law was used to determine the correction
factor for the wire Liebster (9 Pbull 548 ) has
54
substantiated Stokes equation
There are no experimental data with which to compare
the results of the cylinders Wieselsbergers minimum
Reynolds number of 4 is above the ran ge covered in the preshy
sent investigation The da ta for the highest LD ratios
(16 24 and 32) does agree almost exactly wi t h the solution
of Allen and Southwell (1 P bull 141) (LD =00) in the range
of Reynolds numbers from 0 1 to 1 0 Allen and Southwells
solution a greed with the data of Wieselsberger (16 p 22)
However the present data is above the theoretical solutions
of Lamb (8 p 112-121) throughout the range of Reynolds
numbers from 0 01 to 1 0 and above the solutions of
Bairstow Cave and Lang (2 p 404) I mai (4 p 157) and
Tomotika and Aoi (15 p 302) for Reynolds numbers of 0 1
to 1 0 Allen and Southwells solution a grees dth both
Wieselsberger 1 s a nd the present data Their solution and
the present data represent the best means for predicting
drag coefficients for flow over long cylinders for Reynolds
numbers of 0 01 to 10 It should be remembered that the
o t her solutions should a gree with eac h other since they
were all essentially derived by linearizing the Na viershy
Stokes equation
The data for flat plates in parallel flow is
55
considerably above the theoretical solutions of Janssen
(6 p 183 ) and Tomotika and Aoi (15 Pbull 302) However
Fi f~re 15 shows that a smooth transition occurs bet een
the present work and the data of Janour (5 P bull 31) The
present data considerably extend the experimental inforshy
mation previously available for laminar flow paral lel to
flat plates In the re gion of Reynol ds numbers less than
2 the drag coefficient is shown to be inversely proportional
to the Reynolds number Janours data covers a range of
Reynolds numbers from 11 to 1000 The results of the
present investigation line up with Janours results which
in turn on extrapolation to higher Reyno l ds numbers
(greater than 1000) make a smooth transition into Blasius
curve represented by Equation (10) At Reyno l ds numbers
greater than 20 000 the drag coefficient is inversely proshy
portional to the square root of the Reynolds number
The data for flat plates in perpendicular flow is conshy
siderably above the solutions of Tomotika and Aoi
(15 p 302) and Imai (4 p 157 However their solutions
f or cylinders and plates in parallel flow are also below
the present data Also it should be remembered that their
solutions are for infinitely wide plates If a value of
WL of above 100 is used in Equation (41) then the present
data and the solutions of Tomotika and Aoi are fairly close
56
The present results indicate that Equation (41~ can be
used with an accuracy of 15 to 20 within the limitations
of the equation (WL 1 to 4 Re = 0 05 to 2)
57
SUM RY AND CONCLUSIONS
Only a small amount of work has been done in the past
on the study of laminar flow over immersed bodies There
are many areas in the chemical process industries and the
field of aeronautics where this information would be very
helpful The purpose of the present investi gation wa s to
study the almost totally unexplored range of Reynol ds
numbers from 0 01 to 10
Drag coefficients have been determined for spheres
cylinders and flat plates in paralle l and perpendicular
flow The drag coefficients have been plotted as a
function of the Reynolds number with dimension ratios as
a parameter on lo g-log graphs The best straight lines
have been drawn through the data In all cases these lines
had a slope of -1 hich shows that the dra g coefficient is
inversely proportional to the Reynolds number at very low
Reynolds numbers for all shapes and dimension ratios The
following equations have been determined from the data
For cylinders
fd - 27 L -0 36 (38 ) - Re ())
which applies for Reynolds numbers of 0 01 to 1 and LD of
2 to 16 For LD greater than 16 the equation is
58
(39)
For flat plates in parallel flow a correction factor has
been applied to account for the edge effect The equation
which applies for Reyno l ds numbers of 0 1 to 2 is
f 6Re
(40)
For flat plates in perpendicular flow
f d
- 37 - Re (w) t -
0 bull 30 (41)
wbieh applies for W L of 1 to 4 and Reynolds numbers of
0 05 to 2
It is concluded tha t Equations (38-41) give the best
values of drag coefficients within an accuracy of 20~ for
the range of Reynolds numbers that were considered Also
it is evident that the dimension ratios are a n important
factor in determining the drag coefficient for a given
Reynolds number Furthermore the drag coefficient inshy
creases with decreasing values of L D or W L for a constant
Reynolds number The da ta obtained in this investi gation
compare favorably with the other experimental data and with
some of the theoretical sol utions It should be remembered
that when comparing the experimental data with theoretical
solutions that practically all of the solutions are for an
infinitely long cylinder or an infinitely wide plate
It is recommended tha t the present apparatus be
59
modified so that a force of 001 pound can be measured
Also it would improve tho accuracy to set up a constant
temperature bath so that the temperature of the oil can not
vary over 02degF A few check points on the present data
is all that is necessary to confirm the validity of
Equations (38- 41) It is also r ecommended that only SAE 140
oil be used and that 2 inches should be the minimum plate
width and cylinder length to be studi3d These conditions
would help to maintain the accuracy of the correction force
for the wire
60
~WMENCIATURE
Symbol Dimensions
A area sq ft
D diameter ft
F force lb f
L length ft
M mas s lb m Re Reynolds number Dvf= -ltr w width ft
a area sq ft
b characteristic length ft
d diameter ft
f drag coefficientfd
gravitation constant l b mft gc 2= 32 17 l b _ rsec
1 length ft
m mass l b bullm
p pressure lbrsqft
r radius ft
t time see
u velocity ft sec
v velocity ft sec
w width ft
61
Symbol Dimensions
X xbullcoordinate ft
y y- coordinate ft
o( vorticity
time sec
viscosity lb m ft -sec
kinematic viscosity ft 2sec
circumference diameter = 3 1416
3density lb m ft
function
stream function
Laplacian operator
infinity
Subscripts
c corrected
f force
1 l iquid
m mass
p projected
s solid
w wetted
62
BI BLIOGRAPHY
1 Allan D N de G and R v Southwell Re laxation methods applied to determine the motion in two di shymensions of a viscous fluid past a fixed cylinder Quarterly Journal of Mechanics and Applied Mathe shymatics 8 129-145 1955
2 Bairstow L B M Cave and E D Lang The reshysistance of a cylinder moving in a viscous fluid Philosophical Transactions of the Royal Society of London ser A 223383- 432 1923
3 Goldstein Sidney The steady flow of viscous fluid past a fixed spherical obstacle at small Reyno l ds numbers Proceedings of the Royal Society of London ser A 123225-235 1929
4 Imai I A new method of solving Oseens equations and its application to the flow past an inclined elliptic cylinder Proceedings of the Royal Society of London ser A 224 141-160 1954
5 Janour Zbynek Resistance of a plate in paralle l flow at low Reyno lds numbers Washington Nov 1951 40 p National Advisory Committee for Aeronautics Te chnica l Memorandum 1316)
6 Janssen E An analog solution of the Navier-Stokes equation for the case of flow past a f l at plate at low Reynolds numbers In 1956 Heat Transfer and Fluid Mechanics Institute (Preprints of Papers) p 173-183
7 Knudsen James G and Donal d L Katz Fluid Dynamics a nd Heat Transfer Ann Arbor University of Michigan 1953 243 p (Michi gan University Engineering Research Bulletin no 37)
8 La~b Horace On the uniform motion of a spherethrough a viscous fluid Philosophical Magazine and Journal of Science s~r 6 21112-121 1911
9 Liebster H Uben den widerstrand von kugeln Annalen Der Physik ser 4 82 541- 562 1 927
63
10 McAdams William H Heat transmission 3d ed New York McGraw- Hill 1954 532 p
11 Pai Shih- I Viscous f l ow theory I Laminar flow Princeton D Van Nostrand 1956 384 p
12 Prandtlbull Ludwi g Es sentials of fluid dynamics London Blackie amp Son 1954 452 p
13 Relf i F Discussion of the results of measure shyments of the resistance of wires with some additionshyal tests of the resistance of wires of small diame shyters In Technical report of the Advisory Committee for Aeronautics London) March 1914 p 47 - 51 (Report and memoranda no 102 )
14 Stokes George Gabriel Mathematical and physical papers Vol 3 Cambridge University Press 1922 413 p
15 Tomotika s and T Aoi The steady flow of a viscous fluid past an elliptic cylinder and a flat plate at smal l Reynolds numbers Quarterly Journal of Me chanics and Applie d Ma thematics 6 290- 312 1953
16 Wieselsbergo r c Versuche Ube r der luftwiderstand gerundeter und kant iger korper Er gebnisse der Aeroshydynamischen Versucbsansta l t Vol 2 G~ttingen 1923 80 p
FIGURE e- PHOTOGRAPH OF SPHERES CYLINDERS AND PLATES
34
holes were drilled so that each plate could be used for
two geometric ratios by changing the wires (See for
example plates la and lb in Table I
35
EXPERI MENTA L PROCEDURE
Viscosity and Density Calibration
A calibrated hydrometer measuring to the nearest
0002 was used to measure the density Table VI shows that
the effect of temperature on density is practically negli shy
gible in the small temperature range used
A Brookfield Synchro-lectric viscometer was used to
measure the viscosity of both the light and heavy oil
Figures 18 and 19 show the effect of temperature on visshy
cosity In addition the viscosity of the light oil was
checke d using the falling ball method and the equation
D2--ltA (f s bull fl) g (34) l 8v
The viscometer was calibrated by the National Bureau of bull
Standards and was accurate to l tb
Velocity Measurements
The velocity of movement through the oil was measured
by determining the rate of rotation of the pulleys with a
stop watch Usually the time for 10 revolutions was
measured at the highe r ve locities and for 5 revolutions at
the low velocities From this information and the di
amaters of the pulleys the velocities ere calculated
36
The time was measured to the nearest tenth of a second
Since the measured time was usually between 20 and 40
aeconds 1 the error in ~easuring velocity was considered to
be less tha~ 0 5~
force Measurements
The object connected to the scale 1 was dropped to the
bottom of the oil drum The motor was started and the scale
was read as the object vms being pulled towards the top of
the drum Two or three readings were taken for each object
at each velocity In nearly all cases these readings were
the same
37
ti XPER I MENTAL RE STJLTS
The dra g coefficient and the Reynolds number were
calculated by the use of Equations (l or (15) for each of
the spheres cylinders and plates from the measured
quantities of force and velocity a~d the values of the vis shy
cosity and density corresponding to the temperature of the
oil It was necessary to ~ubtract from the measured force
the force on the wire The corrected force measurement was
then used to determine the drag coefficient The force on
the wire has been determined as being proportional to the
velocity A correction curve relating force on the wire
and ve l ocity is plo tted in Figure 9 for the li ght oil and
Fi gure 10 for the heavy oil
The calculated drag coefficients Reynolds numbers
and velocities along with the measured force for the spheres
cylinders flat plates - parallel flow and flat plates shy
perpendicular flow have been tabulated in Tables II III
I V and v respectively
The calculated drag coefficients have been plotted as
a function of the Reynolds number on logarithic graph paper
with geometric ratios as a parameter
Drag coefficients for the spheres are plo tted in
Figure 11 The data for the cylinders are plotted in
CD_ bull 0 G 0
03
Tshy02
01
10 20 30 410 50 60 70 80
VELOCITY- FTJSEC
DRAG FORCE ON THE WIRE-LIGHT OIL
FIGURE 9
I -shy I -middot -- -shy -1shy _i-i I --~ I I _ -middot- shy I i
_I_ - _ middot- LL I l l tmiddot - middot1middot ~- - - - -+i middotshy I - --+-cl - l
1 1 I I IV jc---- --r--middotmiddottmiddot r-middotmiddot--tmiddotmiddot---shy _____ _L __ --~- --1shy middotmiddotr-r-middott- 1 -f-f-T- _~ +-L--1---~- 1--l
~- - shy I-+---Rmiddot-- I I I l i ~~ i -~~ ~- -T f i rshy ~-- --shy i- ----~-- shy - middot1 shy
I i I i I I 1--- -middot - fshy middot i----1---+-shy - i-middot -~+-- --~- --~-- ---- -t+ I v-~~ -middot j
i I middot 1_ _ I tmiddot---+-+1-+--li~+middot -+--+-+-1-+-+-+-+--tc--1-+-t-11-shy - middot --t- 1---t- t----tmiddotshy --~-- -middot i-shy I 1i - ~ i I i v i middotmiddotmiddot
[~v +L~ + ~ - I~~j-+ r V I ~t--- -~-- I +---~-- I f-middot ---1-- ~ -- --- ) Li --+--+--+-+-+-+--1--+--+---t---4 -1--1--+-+--+-l-i
tl~ I I Q Y +l~~ii-+-++++-middotHH-++-+-+-+--H--++ -i t Imiddot i i 1 j _V I f1 r-t~-middot l--r-tshy -~ 7 middot 1 -shy middot middotmiddot I
DRAG FORCE ON THE WIRE- HEAVY OIL
FIGURE 10
40
+shy l i~ltgt ~ bull r-rshy I i t _l
1 lf-1-1 l+r+ fJ-Ct I+ t li 1~t rtH r+l rf-l It llil I I
l l~pound 11 1 ~middot ~~middott ~ It lqf L
t I+--= ~r 17 -Er I _ ~ _pound~- sect Imiddot I+
iU=ff=t 1 +~ t_ - ~ r 111= t h=
I middot
t= IE I 1 1
plusmn~ kplusmni - -STOKE S EQ
(~ l h+middot
ru HmiddotHti+H1 11
c lffii l t~ 4 ~ ~middot ~ff l ~ ~h i ltlri
1 yen~ middot I ~ I I T ~ gt l+t H+h l+ i j l tfl-l Imiddotmiddot ft+ ++ l f+ Imiddotmiddot I+ I+ middott bulli I 1middot1 I ftt-1shy middot I middot r 11 I IH Ij ~ ~ middotishy J F 1= 6= ~
=f l~iit rtti l lit~ I FS lf~ l=i-+
l-11ffi tt lr 1 ~1 -t =l=Rttl 1ft i- 1 ~ I+ I
~~ lflJ
t I lfl m ~~WFB Lt
41plusmn811 IF I Hir tt ft itttplusmn i I~
1-+++middot
I ~ I (~ ffitrHf1 Ittmiddot ~ l r i H-t-r r HHt m 11 H++ I
bull I I
1_ _ F bullmiddot Imiddotmiddot t-- 1-T h iT
f-t+ ftt I+ I lt + T Imiddot 1
1t _plusmn middot~~ ~- 11shy
=a~ 1~ - =itf lttti
H I
=
DATA FOR SPHERES
FIGURE II
41
I -1---1-1-+--+--Ti-+-------+----r--shy --r--- -shy + t----+shy ----4-~---+-f----f--+-f--l--1 I t--shy --t-- ---+-shy
J-+-~f--~~ -___l_ ~---
i 1 L~L~-~tr-l----H~4-----~-f------+------+-----+----+---+middot-t-middot-H5000
middot~ ~ m- ~ - ~t plusmn~ 3t i t~ -f--- bullbull - ~~ h middot-
01 0~ 10
Re
-
DATA FOR CYLINDERS - LD = 6 8 AND 12
FIGURE I 4
44
Figures 12 13 and 14 The data for LD values of 16 24
and 32 were nearly the same and have been plotted to gether
i n Figure 12 In addition the curves for the other LD
ratios determined fro m Fib~res 13 and 14 have been drawn
in Figure 12 so that the effect of the length-to-diameter
is clearly shown Figure 13 shows the data for LD values
of 2 and 4 and the curves determined from this data
Firure 14 shows the data for LD values of 6 8 and 12
and the curves determined from this data
The data for flat plates in parallel flow are plotted
in Fi gure 15 A correction factor for the edge effect has
beon used so that the width-to-length ratio is not a
parameter in this plot A portion of the data of Janour
(5 p 31) is also shown in the diagram
The data for fla t plates in perpendicular flow is
plotted in Figures 16 a nd 17 Figure 16 shows the data for
WL values of 2 Also the curves for the three WL ratios
1 2 and 4 have been drawn in the fi gure Figure 17 shows
the data for WL values of 1 and 4 The curves determined
from the data have also been dravm in the figure
45
10~ ~ ~--- -shy
t==Ff1TR=+ iJ+--_-_--r_-_---+-+---+--+-+--_---_-~r-=r~=~+--=---=---=---=--~=--=_~1=_--=_~_-middot~~--+-+-t~ 1 Ll~+--+-- ---jtshyl~t L--+ I
I
P------ _l -- --1---L i
20 ~-- I ~g I --- - ---+-- r t L_shy
~ ~B 1) I --o-o- JONES - () - - ~~ p f---j- -~-- e e JANOU R
c gt ~c ~ ------ JANSSEN I 0 0 ~ I
IO ~2=i~~~~~~a=~~f=j= ---- TOM OTIKA bulll= I
~~n ~~--~~~~~~o~~~~~--4- NDCIgttl o shy
-
~--~~~~~+--+~+--4-r-~1+-~-middot+1~ ~ --H--~-~~os I i i i-4 ---~T I I f-- t --- li-------~--+-_--+--t-----~~-~_+---_-_-_--+------+-+-__+-[- +_- ___ _______ __+---+-r-+--H----_+--r--------+shy
it I+ ~ bull t ~1 ri j t++t+t++tft bullm H--~+H-t+t-++H-f+t+~HtttH t bull~H-IrttI-H
iH-H u nH m
I
t H+t-~ 1-r f-tj
i it iT -t middotHt I I I I Ill
~middot __
r middotshy
i I r-
f H- jLj f r H rr t~
II
t f f-l -t+tt ~ ==_ =~middot irE
I I
I
I
f
I --
i
t
1 r bull - r
~- ltt++l=tUtt~S-t+t+++~-++U +HJJm~-fl~HHtt1 tttn ll+t-Tt-~- ~ r fH T --r -1 t ---t- -tshy w _+ _ I-shy middotI
-shy -r- + Hbull Hshy t-I --r++ -t iHr -1 H-e-- -t I 1IT 1
1 H-rf-I IJftJ Jf+i+ ~ L
=+shy - tjshy rtmiddotshy ~ -
+ H 1-Jt I tt o =tt ~-
~1 l +fill l plusmn~ fplusmn -shy + I t-
DATA FOR FLAT PLATES PERPENDICULAR FLOW- WL= I 4
FIGURE 17
48
DI SCUSS ION OF RESULTS
Correction and Accuracy of Measurements
After a few pre liminary force measurements with the
spheres and a check with Stokes law (Equation 2) it was
apparent that the drag force on the wire was appreciable
and needed to be considered It was decided to take a
series of measurements with the spheres and calculate the
difference between the measured force and the force calcushy
lated from Stokes law The difference in force could then
be attributed to the drag on the wire If Stokes law is
followed the force on the wire should be proportional to
the velocity
A series of twenty measurements of the force on the
spheres was taken for each oil and the difference between
the measured force and that calcula ted by Stokes 1 law was
determined For each oil this difference as plo tted vs
the velocity The points grouped fairly ell around a
strai ght line nearly passing through the origin The
method of least squares was used to determine the equation
of the line best fitting the da t a The equa tion of the
line for the li bht oil tas found to be
Fe bullbull05605v - oooa (35)
which was determined at about 62 7degF Since the intercept
49
of the line is very close to zero it is believed that the
line is a good indication of the drag on the wire The
equation of the line for the heavy oil was found to be
F - 19llv I oo2o1 (36 ) c shy
which was determined at about 64 2deg The intercept of this
line is also quite close to zero These lines plotted in
Fi poundures 9 and 10 were used throughout the investigation
for the correction factor of the drag on the wires For
the cylinders and flat plates in parallel flow which were
pulled by two wires the values determined from Equations
35) and (36) were doubled For the plates in perpendicular
flow pulled by four wires the correction force was multishy
plied by four
The spring scale had 12 ounce divisions but could be
read to the nearest sixth of an ounce Some of the measureshy
ments of force were under an ounce hence a considerable
spread of the measurements was noticed in the pre liminary
data and throughout the experiment However sufficient
points were obtained so that it was possible to draw a
reliable curve through the data in all casas An analysis
was made to determine the average deviation from Stokes
equation for the spheres It raa found that the average
deviation was 15 1 for the light oil 16 6 for the heavy
oil and 15 9 overall The maximum deviation was 89
50
Inspection of the other data shows that these deviations
are also representative of the cylinders and flat plates
The force measurement is the least accurate part of the
experiment Other insignificant errors are introduced by
a small variation in the temperature This variation was
held to about 10 from the temperature of the calibrated
correction curve The velocity measurements and the
dimensions of the cylinders spheres and pl~ tes are conshy
sidered go od enough so tha t no appreciable errors occur
In order to e l iminate the WL parameter for flat plates
in parallel f l ow an additional factor for the effect of
the edges was subtracted from the measured force Janour
(5 p 27) presented the foll owing equation for the edge
correction for one edge of a flat plate in parallel flow
F ~ lv~ bull (37 ) edge gc
In present work this equation as doubled because both
edges of the plates were submerged in fluid It is assumed
in appl ying this correction that the lowe r limit of a
Reynolds number of 10 proposed by Janour can be extended
close to 0 1
Analysis of Results
Forty of the points for the spheres were used to get
51
the correction factor for the wires The remaining thirty
points are well erouped about Stokes law
The data for cylinders for LD ratios of 16 24 and
32 did not seem to be se gregated therefore these data
were plotted together It would seem that in the low range
of Reyno l ds numbers an LD of 16 and greater can be con shy
sidered an ~nfini tely long cylinder The other LD ratios
of 2 4 6 a 12 provided fairly distinct and separate
lines The best straight lines were drawn through the data
for each of the LD ratios It was evident that in eaeh
case a slope of -1 on a lo g-log graph gave the best straight
line which would indicate that the force varies directly
as the velocity It was possible to develop an empirical
expression relating dra g coefficient Reynolds number and
LD The following equation was obtained from the straight
line plots of Re vs fd for the various LD ratios
(38 )
Equation (38) applies for Reyno l ds numbers from 01 to 10
and for LD ratios of 2 to 16 For LD ratios greater
than 16
10 re (39 )
The data for flat plates in parallel flow is plotted
in Figure 15 after the correction factor for tho edge
52
effect was subtracted When the edge correction is made
no effect of WL ratio is indicated This result would be
expected The data followed a straight line with a slope
of -1 up to a Reynolds number of 2 After that a curve was
dravm connecting the line to that obtained by Janour The
equation for the straight section of the curve is
f - 6 (40)- Re
which applies for Reynolds numbers of 0 1 to 2 0 Here
a gain the force is proportional to the velocity Vfuen
determining drag force for flat plates in parallel flow
the force is first calculated from Equations (40) and (15 )
then the edge correction is added
The effect of the geometric ratios is clearly shown in
the data for flat plates in perpendicul ar flow which are
plotted in Figures 16 and 17 As with the other data the
best straight line was drawn through the various points
for eaoh of the WL ratios Again the line had a slope of
-1 The equation relating fd Re and wL was found t o be
rd 37 (w) -o 3o (41)Irel
which applies for Reynolds numbers of about 05 to 2 0 and
WL ratios of 1 to 4 It is possible but it has not been
proved that Equation (41) is suitable for higher WL ratios
The exponent on WL in Equation 41) is very close to that
53
on L D i n Equation ( 38 )~ It i s possible t ha t these
exponents are t he same but this cannot be sho~~ depound1nitely
until more accura te da ta are available It would be exshy
pected that a s the Reynolds number approaches zero t he
effect of geometric ratios would be the same for cylinders
and fla t pla tes in perpendicula r flow
It is seen in the t a bles of data that occasionally a
ne gative force was obtained because the correction applie d
due to t he wire dra g was greater than the mea sured force
These points obviously are incorrect This occurred only
for the smallest plates in the heavy oil at t he highest
velocities However these knom bad points occur in less
tha n 5~ of the data
It is clearl y shown that for cylinders and plates the
fd increases as L D or W L decreases This is in direct
contrast to Wiesel aberger s investigation However his
work is for hi gher Reynolds numbers at which a turbulent
wake forms bull
Comparison of Results with Other Data and Theoretical So l utions
The data for sphere~ a grees of course with Stokes
l aw since that law was used to determine the correction
factor for the wire Liebster (9 Pbull 548 ) has
54
substantiated Stokes equation
There are no experimental data with which to compare
the results of the cylinders Wieselsbergers minimum
Reynolds number of 4 is above the ran ge covered in the preshy
sent investigation The da ta for the highest LD ratios
(16 24 and 32) does agree almost exactly wi t h the solution
of Allen and Southwell (1 P bull 141) (LD =00) in the range
of Reynolds numbers from 0 1 to 1 0 Allen and Southwells
solution a greed with the data of Wieselsberger (16 p 22)
However the present data is above the theoretical solutions
of Lamb (8 p 112-121) throughout the range of Reynolds
numbers from 0 01 to 1 0 and above the solutions of
Bairstow Cave and Lang (2 p 404) I mai (4 p 157) and
Tomotika and Aoi (15 p 302) for Reynolds numbers of 0 1
to 1 0 Allen and Southwells solution a grees dth both
Wieselsberger 1 s a nd the present data Their solution and
the present data represent the best means for predicting
drag coefficients for flow over long cylinders for Reynolds
numbers of 0 01 to 10 It should be remembered that the
o t her solutions should a gree with eac h other since they
were all essentially derived by linearizing the Na viershy
Stokes equation
The data for flat plates in parallel flow is
55
considerably above the theoretical solutions of Janssen
(6 p 183 ) and Tomotika and Aoi (15 Pbull 302) However
Fi f~re 15 shows that a smooth transition occurs bet een
the present work and the data of Janour (5 P bull 31) The
present data considerably extend the experimental inforshy
mation previously available for laminar flow paral lel to
flat plates In the re gion of Reynol ds numbers less than
2 the drag coefficient is shown to be inversely proportional
to the Reynolds number Janours data covers a range of
Reynolds numbers from 11 to 1000 The results of the
present investigation line up with Janours results which
in turn on extrapolation to higher Reyno l ds numbers
(greater than 1000) make a smooth transition into Blasius
curve represented by Equation (10) At Reyno l ds numbers
greater than 20 000 the drag coefficient is inversely proshy
portional to the square root of the Reynolds number
The data for flat plates in perpendicular flow is conshy
siderably above the solutions of Tomotika and Aoi
(15 p 302) and Imai (4 p 157 However their solutions
f or cylinders and plates in parallel flow are also below
the present data Also it should be remembered that their
solutions are for infinitely wide plates If a value of
WL of above 100 is used in Equation (41) then the present
data and the solutions of Tomotika and Aoi are fairly close
56
The present results indicate that Equation (41~ can be
used with an accuracy of 15 to 20 within the limitations
of the equation (WL 1 to 4 Re = 0 05 to 2)
57
SUM RY AND CONCLUSIONS
Only a small amount of work has been done in the past
on the study of laminar flow over immersed bodies There
are many areas in the chemical process industries and the
field of aeronautics where this information would be very
helpful The purpose of the present investi gation wa s to
study the almost totally unexplored range of Reynol ds
numbers from 0 01 to 10
Drag coefficients have been determined for spheres
cylinders and flat plates in paralle l and perpendicular
flow The drag coefficients have been plotted as a
function of the Reynolds number with dimension ratios as
a parameter on lo g-log graphs The best straight lines
have been drawn through the data In all cases these lines
had a slope of -1 hich shows that the dra g coefficient is
inversely proportional to the Reynolds number at very low
Reynolds numbers for all shapes and dimension ratios The
following equations have been determined from the data
For cylinders
fd - 27 L -0 36 (38 ) - Re ())
which applies for Reynolds numbers of 0 01 to 1 and LD of
2 to 16 For LD greater than 16 the equation is
58
(39)
For flat plates in parallel flow a correction factor has
been applied to account for the edge effect The equation
which applies for Reyno l ds numbers of 0 1 to 2 is
f 6Re
(40)
For flat plates in perpendicular flow
f d
- 37 - Re (w) t -
0 bull 30 (41)
wbieh applies for W L of 1 to 4 and Reynolds numbers of
0 05 to 2
It is concluded tha t Equations (38-41) give the best
values of drag coefficients within an accuracy of 20~ for
the range of Reynolds numbers that were considered Also
it is evident that the dimension ratios are a n important
factor in determining the drag coefficient for a given
Reynolds number Furthermore the drag coefficient inshy
creases with decreasing values of L D or W L for a constant
Reynolds number The da ta obtained in this investi gation
compare favorably with the other experimental data and with
some of the theoretical sol utions It should be remembered
that when comparing the experimental data with theoretical
solutions that practically all of the solutions are for an
infinitely long cylinder or an infinitely wide plate
It is recommended tha t the present apparatus be
59
modified so that a force of 001 pound can be measured
Also it would improve tho accuracy to set up a constant
temperature bath so that the temperature of the oil can not
vary over 02degF A few check points on the present data
is all that is necessary to confirm the validity of
Equations (38- 41) It is also r ecommended that only SAE 140
oil be used and that 2 inches should be the minimum plate
width and cylinder length to be studi3d These conditions
would help to maintain the accuracy of the correction force
for the wire
60
~WMENCIATURE
Symbol Dimensions
A area sq ft
D diameter ft
F force lb f
L length ft
M mas s lb m Re Reynolds number Dvf= -ltr w width ft
a area sq ft
b characteristic length ft
d diameter ft
f drag coefficientfd
gravitation constant l b mft gc 2= 32 17 l b _ rsec
1 length ft
m mass l b bullm
p pressure lbrsqft
r radius ft
t time see
u velocity ft sec
v velocity ft sec
w width ft
61
Symbol Dimensions
X xbullcoordinate ft
y y- coordinate ft
o( vorticity
time sec
viscosity lb m ft -sec
kinematic viscosity ft 2sec
circumference diameter = 3 1416
3density lb m ft
function
stream function
Laplacian operator
infinity
Subscripts
c corrected
f force
1 l iquid
m mass
p projected
s solid
w wetted
62
BI BLIOGRAPHY
1 Allan D N de G and R v Southwell Re laxation methods applied to determine the motion in two di shymensions of a viscous fluid past a fixed cylinder Quarterly Journal of Mechanics and Applied Mathe shymatics 8 129-145 1955
2 Bairstow L B M Cave and E D Lang The reshysistance of a cylinder moving in a viscous fluid Philosophical Transactions of the Royal Society of London ser A 223383- 432 1923
3 Goldstein Sidney The steady flow of viscous fluid past a fixed spherical obstacle at small Reyno l ds numbers Proceedings of the Royal Society of London ser A 123225-235 1929
4 Imai I A new method of solving Oseens equations and its application to the flow past an inclined elliptic cylinder Proceedings of the Royal Society of London ser A 224 141-160 1954
5 Janour Zbynek Resistance of a plate in paralle l flow at low Reyno lds numbers Washington Nov 1951 40 p National Advisory Committee for Aeronautics Te chnica l Memorandum 1316)
6 Janssen E An analog solution of the Navier-Stokes equation for the case of flow past a f l at plate at low Reynolds numbers In 1956 Heat Transfer and Fluid Mechanics Institute (Preprints of Papers) p 173-183
7 Knudsen James G and Donal d L Katz Fluid Dynamics a nd Heat Transfer Ann Arbor University of Michigan 1953 243 p (Michi gan University Engineering Research Bulletin no 37)
8 La~b Horace On the uniform motion of a spherethrough a viscous fluid Philosophical Magazine and Journal of Science s~r 6 21112-121 1911
9 Liebster H Uben den widerstrand von kugeln Annalen Der Physik ser 4 82 541- 562 1 927
63
10 McAdams William H Heat transmission 3d ed New York McGraw- Hill 1954 532 p
11 Pai Shih- I Viscous f l ow theory I Laminar flow Princeton D Van Nostrand 1956 384 p
12 Prandtlbull Ludwi g Es sentials of fluid dynamics London Blackie amp Son 1954 452 p
13 Relf i F Discussion of the results of measure shyments of the resistance of wires with some additionshyal tests of the resistance of wires of small diame shyters In Technical report of the Advisory Committee for Aeronautics London) March 1914 p 47 - 51 (Report and memoranda no 102 )
14 Stokes George Gabriel Mathematical and physical papers Vol 3 Cambridge University Press 1922 413 p
15 Tomotika s and T Aoi The steady flow of a viscous fluid past an elliptic cylinder and a flat plate at smal l Reynolds numbers Quarterly Journal of Me chanics and Applie d Ma thematics 6 290- 312 1953
16 Wieselsbergo r c Versuche Ube r der luftwiderstand gerundeter und kant iger korper Er gebnisse der Aeroshydynamischen Versucbsansta l t Vol 2 G~ttingen 1923 80 p
FIGURE e- PHOTOGRAPH OF SPHERES CYLINDERS AND PLATES
34
holes were drilled so that each plate could be used for
two geometric ratios by changing the wires (See for
example plates la and lb in Table I
35
EXPERI MENTA L PROCEDURE
Viscosity and Density Calibration
A calibrated hydrometer measuring to the nearest
0002 was used to measure the density Table VI shows that
the effect of temperature on density is practically negli shy
gible in the small temperature range used
A Brookfield Synchro-lectric viscometer was used to
measure the viscosity of both the light and heavy oil
Figures 18 and 19 show the effect of temperature on visshy
cosity In addition the viscosity of the light oil was
checke d using the falling ball method and the equation
D2--ltA (f s bull fl) g (34) l 8v
The viscometer was calibrated by the National Bureau of bull
Standards and was accurate to l tb
Velocity Measurements
The velocity of movement through the oil was measured
by determining the rate of rotation of the pulleys with a
stop watch Usually the time for 10 revolutions was
measured at the highe r ve locities and for 5 revolutions at
the low velocities From this information and the di
amaters of the pulleys the velocities ere calculated
36
The time was measured to the nearest tenth of a second
Since the measured time was usually between 20 and 40
aeconds 1 the error in ~easuring velocity was considered to
be less tha~ 0 5~
force Measurements
The object connected to the scale 1 was dropped to the
bottom of the oil drum The motor was started and the scale
was read as the object vms being pulled towards the top of
the drum Two or three readings were taken for each object
at each velocity In nearly all cases these readings were
the same
37
ti XPER I MENTAL RE STJLTS
The dra g coefficient and the Reynolds number were
calculated by the use of Equations (l or (15) for each of
the spheres cylinders and plates from the measured
quantities of force and velocity a~d the values of the vis shy
cosity and density corresponding to the temperature of the
oil It was necessary to ~ubtract from the measured force
the force on the wire The corrected force measurement was
then used to determine the drag coefficient The force on
the wire has been determined as being proportional to the
velocity A correction curve relating force on the wire
and ve l ocity is plo tted in Figure 9 for the li ght oil and
Fi gure 10 for the heavy oil
The calculated drag coefficients Reynolds numbers
and velocities along with the measured force for the spheres
cylinders flat plates - parallel flow and flat plates shy
perpendicular flow have been tabulated in Tables II III
I V and v respectively
The calculated drag coefficients have been plotted as
a function of the Reynolds number on logarithic graph paper
with geometric ratios as a parameter
Drag coefficients for the spheres are plo tted in
Figure 11 The data for the cylinders are plotted in
CD_ bull 0 G 0
03
Tshy02
01
10 20 30 410 50 60 70 80
VELOCITY- FTJSEC
DRAG FORCE ON THE WIRE-LIGHT OIL
FIGURE 9
I -shy I -middot -- -shy -1shy _i-i I --~ I I _ -middot- shy I i
_I_ - _ middot- LL I l l tmiddot - middot1middot ~- - - - -+i middotshy I - --+-cl - l
1 1 I I IV jc---- --r--middotmiddottmiddot r-middotmiddot--tmiddotmiddot---shy _____ _L __ --~- --1shy middotmiddotr-r-middott- 1 -f-f-T- _~ +-L--1---~- 1--l
~- - shy I-+---Rmiddot-- I I I l i ~~ i -~~ ~- -T f i rshy ~-- --shy i- ----~-- shy - middot1 shy
I i I i I I 1--- -middot - fshy middot i----1---+-shy - i-middot -~+-- --~- --~-- ---- -t+ I v-~~ -middot j
i I middot 1_ _ I tmiddot---+-+1-+--li~+middot -+--+-+-1-+-+-+-+--tc--1-+-t-11-shy - middot --t- 1---t- t----tmiddotshy --~-- -middot i-shy I 1i - ~ i I i v i middotmiddotmiddot
[~v +L~ + ~ - I~~j-+ r V I ~t--- -~-- I +---~-- I f-middot ---1-- ~ -- --- ) Li --+--+--+-+-+-+--1--+--+---t---4 -1--1--+-+--+-l-i
tl~ I I Q Y +l~~ii-+-++++-middotHH-++-+-+-+--H--++ -i t Imiddot i i 1 j _V I f1 r-t~-middot l--r-tshy -~ 7 middot 1 -shy middot middotmiddot I
DRAG FORCE ON THE WIRE- HEAVY OIL
FIGURE 10
40
+shy l i~ltgt ~ bull r-rshy I i t _l
1 lf-1-1 l+r+ fJ-Ct I+ t li 1~t rtH r+l rf-l It llil I I
l l~pound 11 1 ~middot ~~middott ~ It lqf L
t I+--= ~r 17 -Er I _ ~ _pound~- sect Imiddot I+
iU=ff=t 1 +~ t_ - ~ r 111= t h=
I middot
t= IE I 1 1
plusmn~ kplusmni - -STOKE S EQ
(~ l h+middot
ru HmiddotHti+H1 11
c lffii l t~ 4 ~ ~middot ~ff l ~ ~h i ltlri
1 yen~ middot I ~ I I T ~ gt l+t H+h l+ i j l tfl-l Imiddotmiddot ft+ ++ l f+ Imiddotmiddot I+ I+ middott bulli I 1middot1 I ftt-1shy middot I middot r 11 I IH Ij ~ ~ middotishy J F 1= 6= ~
=f l~iit rtti l lit~ I FS lf~ l=i-+
l-11ffi tt lr 1 ~1 -t =l=Rttl 1ft i- 1 ~ I+ I
~~ lflJ
t I lfl m ~~WFB Lt
41plusmn811 IF I Hir tt ft itttplusmn i I~
1-+++middot
I ~ I (~ ffitrHf1 Ittmiddot ~ l r i H-t-r r HHt m 11 H++ I
bull I I
1_ _ F bullmiddot Imiddotmiddot t-- 1-T h iT
f-t+ ftt I+ I lt + T Imiddot 1
1t _plusmn middot~~ ~- 11shy
=a~ 1~ - =itf lttti
H I
=
DATA FOR SPHERES
FIGURE II
41
I -1---1-1-+--+--Ti-+-------+----r--shy --r--- -shy + t----+shy ----4-~---+-f----f--+-f--l--1 I t--shy --t-- ---+-shy
J-+-~f--~~ -___l_ ~---
i 1 L~L~-~tr-l----H~4-----~-f------+------+-----+----+---+middot-t-middot-H5000
middot~ ~ m- ~ - ~t plusmn~ 3t i t~ -f--- bullbull - ~~ h middot-
01 0~ 10
Re
-
DATA FOR CYLINDERS - LD = 6 8 AND 12
FIGURE I 4
44
Figures 12 13 and 14 The data for LD values of 16 24
and 32 were nearly the same and have been plotted to gether
i n Figure 12 In addition the curves for the other LD
ratios determined fro m Fib~res 13 and 14 have been drawn
in Figure 12 so that the effect of the length-to-diameter
is clearly shown Figure 13 shows the data for LD values
of 2 and 4 and the curves determined from this data
Firure 14 shows the data for LD values of 6 8 and 12
and the curves determined from this data
The data for flat plates in parallel flow are plotted
in Fi gure 15 A correction factor for the edge effect has
beon used so that the width-to-length ratio is not a
parameter in this plot A portion of the data of Janour
(5 p 31) is also shown in the diagram
The data for fla t plates in perpendicular flow is
plotted in Figures 16 a nd 17 Figure 16 shows the data for
WL values of 2 Also the curves for the three WL ratios
1 2 and 4 have been drawn in the fi gure Figure 17 shows
the data for WL values of 1 and 4 The curves determined
from the data have also been dravm in the figure
45
10~ ~ ~--- -shy
t==Ff1TR=+ iJ+--_-_--r_-_---+-+---+--+-+--_---_-~r-=r~=~+--=---=---=---=--~=--=_~1=_--=_~_-middot~~--+-+-t~ 1 Ll~+--+-- ---jtshyl~t L--+ I
I
P------ _l -- --1---L i
20 ~-- I ~g I --- - ---+-- r t L_shy
~ ~B 1) I --o-o- JONES - () - - ~~ p f---j- -~-- e e JANOU R
c gt ~c ~ ------ JANSSEN I 0 0 ~ I
IO ~2=i~~~~~~a=~~f=j= ---- TOM OTIKA bulll= I
~~n ~~--~~~~~~o~~~~~--4- NDCIgttl o shy
-
~--~~~~~+--+~+--4-r-~1+-~-middot+1~ ~ --H--~-~~os I i i i-4 ---~T I I f-- t --- li-------~--+-_--+--t-----~~-~_+---_-_-_--+------+-+-__+-[- +_- ___ _______ __+---+-r-+--H----_+--r--------+shy
it I+ ~ bull t ~1 ri j t++t+t++tft bullm H--~+H-t+t-++H-f+t+~HtttH t bull~H-IrttI-H
iH-H u nH m
I
t H+t-~ 1-r f-tj
i it iT -t middotHt I I I I Ill
~middot __
r middotshy
i I r-
f H- jLj f r H rr t~
II
t f f-l -t+tt ~ ==_ =~middot irE
I I
I
I
f
I --
i
t
1 r bull - r
~- ltt++l=tUtt~S-t+t+++~-++U +HJJm~-fl~HHtt1 tttn ll+t-Tt-~- ~ r fH T --r -1 t ---t- -tshy w _+ _ I-shy middotI
-shy -r- + Hbull Hshy t-I --r++ -t iHr -1 H-e-- -t I 1IT 1
1 H-rf-I IJftJ Jf+i+ ~ L
=+shy - tjshy rtmiddotshy ~ -
+ H 1-Jt I tt o =tt ~-
~1 l +fill l plusmn~ fplusmn -shy + I t-
DATA FOR FLAT PLATES PERPENDICULAR FLOW- WL= I 4
FIGURE 17
48
DI SCUSS ION OF RESULTS
Correction and Accuracy of Measurements
After a few pre liminary force measurements with the
spheres and a check with Stokes law (Equation 2) it was
apparent that the drag force on the wire was appreciable
and needed to be considered It was decided to take a
series of measurements with the spheres and calculate the
difference between the measured force and the force calcushy
lated from Stokes law The difference in force could then
be attributed to the drag on the wire If Stokes law is
followed the force on the wire should be proportional to
the velocity
A series of twenty measurements of the force on the
spheres was taken for each oil and the difference between
the measured force and that calcula ted by Stokes 1 law was
determined For each oil this difference as plo tted vs
the velocity The points grouped fairly ell around a
strai ght line nearly passing through the origin The
method of least squares was used to determine the equation
of the line best fitting the da t a The equa tion of the
line for the li bht oil tas found to be
Fe bullbull05605v - oooa (35)
which was determined at about 62 7degF Since the intercept
49
of the line is very close to zero it is believed that the
line is a good indication of the drag on the wire The
equation of the line for the heavy oil was found to be
F - 19llv I oo2o1 (36 ) c shy
which was determined at about 64 2deg The intercept of this
line is also quite close to zero These lines plotted in
Fi poundures 9 and 10 were used throughout the investigation
for the correction factor of the drag on the wires For
the cylinders and flat plates in parallel flow which were
pulled by two wires the values determined from Equations
35) and (36) were doubled For the plates in perpendicular
flow pulled by four wires the correction force was multishy
plied by four
The spring scale had 12 ounce divisions but could be
read to the nearest sixth of an ounce Some of the measureshy
ments of force were under an ounce hence a considerable
spread of the measurements was noticed in the pre liminary
data and throughout the experiment However sufficient
points were obtained so that it was possible to draw a
reliable curve through the data in all casas An analysis
was made to determine the average deviation from Stokes
equation for the spheres It raa found that the average
deviation was 15 1 for the light oil 16 6 for the heavy
oil and 15 9 overall The maximum deviation was 89
50
Inspection of the other data shows that these deviations
are also representative of the cylinders and flat plates
The force measurement is the least accurate part of the
experiment Other insignificant errors are introduced by
a small variation in the temperature This variation was
held to about 10 from the temperature of the calibrated
correction curve The velocity measurements and the
dimensions of the cylinders spheres and pl~ tes are conshy
sidered go od enough so tha t no appreciable errors occur
In order to e l iminate the WL parameter for flat plates
in parallel f l ow an additional factor for the effect of
the edges was subtracted from the measured force Janour
(5 p 27) presented the foll owing equation for the edge
correction for one edge of a flat plate in parallel flow
F ~ lv~ bull (37 ) edge gc
In present work this equation as doubled because both
edges of the plates were submerged in fluid It is assumed
in appl ying this correction that the lowe r limit of a
Reynolds number of 10 proposed by Janour can be extended
close to 0 1
Analysis of Results
Forty of the points for the spheres were used to get
51
the correction factor for the wires The remaining thirty
points are well erouped about Stokes law
The data for cylinders for LD ratios of 16 24 and
32 did not seem to be se gregated therefore these data
were plotted together It would seem that in the low range
of Reyno l ds numbers an LD of 16 and greater can be con shy
sidered an ~nfini tely long cylinder The other LD ratios
of 2 4 6 a 12 provided fairly distinct and separate
lines The best straight lines were drawn through the data
for each of the LD ratios It was evident that in eaeh
case a slope of -1 on a lo g-log graph gave the best straight
line which would indicate that the force varies directly
as the velocity It was possible to develop an empirical
expression relating dra g coefficient Reynolds number and
LD The following equation was obtained from the straight
line plots of Re vs fd for the various LD ratios
(38 )
Equation (38) applies for Reyno l ds numbers from 01 to 10
and for LD ratios of 2 to 16 For LD ratios greater
than 16
10 re (39 )
The data for flat plates in parallel flow is plotted
in Figure 15 after the correction factor for tho edge
52
effect was subtracted When the edge correction is made
no effect of WL ratio is indicated This result would be
expected The data followed a straight line with a slope
of -1 up to a Reynolds number of 2 After that a curve was
dravm connecting the line to that obtained by Janour The
equation for the straight section of the curve is
f - 6 (40)- Re
which applies for Reynolds numbers of 0 1 to 2 0 Here
a gain the force is proportional to the velocity Vfuen
determining drag force for flat plates in parallel flow
the force is first calculated from Equations (40) and (15 )
then the edge correction is added
The effect of the geometric ratios is clearly shown in
the data for flat plates in perpendicul ar flow which are
plotted in Figures 16 and 17 As with the other data the
best straight line was drawn through the various points
for eaoh of the WL ratios Again the line had a slope of
-1 The equation relating fd Re and wL was found t o be
rd 37 (w) -o 3o (41)Irel
which applies for Reynolds numbers of about 05 to 2 0 and
WL ratios of 1 to 4 It is possible but it has not been
proved that Equation (41) is suitable for higher WL ratios
The exponent on WL in Equation 41) is very close to that
53
on L D i n Equation ( 38 )~ It i s possible t ha t these
exponents are t he same but this cannot be sho~~ depound1nitely
until more accura te da ta are available It would be exshy
pected that a s the Reynolds number approaches zero t he
effect of geometric ratios would be the same for cylinders
and fla t pla tes in perpendicula r flow
It is seen in the t a bles of data that occasionally a
ne gative force was obtained because the correction applie d
due to t he wire dra g was greater than the mea sured force
These points obviously are incorrect This occurred only
for the smallest plates in the heavy oil at t he highest
velocities However these knom bad points occur in less
tha n 5~ of the data
It is clearl y shown that for cylinders and plates the
fd increases as L D or W L decreases This is in direct
contrast to Wiesel aberger s investigation However his
work is for hi gher Reynolds numbers at which a turbulent
wake forms bull
Comparison of Results with Other Data and Theoretical So l utions
The data for sphere~ a grees of course with Stokes
l aw since that law was used to determine the correction
factor for the wire Liebster (9 Pbull 548 ) has
54
substantiated Stokes equation
There are no experimental data with which to compare
the results of the cylinders Wieselsbergers minimum
Reynolds number of 4 is above the ran ge covered in the preshy
sent investigation The da ta for the highest LD ratios
(16 24 and 32) does agree almost exactly wi t h the solution
of Allen and Southwell (1 P bull 141) (LD =00) in the range
of Reynolds numbers from 0 1 to 1 0 Allen and Southwells
solution a greed with the data of Wieselsberger (16 p 22)
However the present data is above the theoretical solutions
of Lamb (8 p 112-121) throughout the range of Reynolds
numbers from 0 01 to 1 0 and above the solutions of
Bairstow Cave and Lang (2 p 404) I mai (4 p 157) and
Tomotika and Aoi (15 p 302) for Reynolds numbers of 0 1
to 1 0 Allen and Southwells solution a grees dth both
Wieselsberger 1 s a nd the present data Their solution and
the present data represent the best means for predicting
drag coefficients for flow over long cylinders for Reynolds
numbers of 0 01 to 10 It should be remembered that the
o t her solutions should a gree with eac h other since they
were all essentially derived by linearizing the Na viershy
Stokes equation
The data for flat plates in parallel flow is
55
considerably above the theoretical solutions of Janssen
(6 p 183 ) and Tomotika and Aoi (15 Pbull 302) However
Fi f~re 15 shows that a smooth transition occurs bet een
the present work and the data of Janour (5 P bull 31) The
present data considerably extend the experimental inforshy
mation previously available for laminar flow paral lel to
flat plates In the re gion of Reynol ds numbers less than
2 the drag coefficient is shown to be inversely proportional
to the Reynolds number Janours data covers a range of
Reynolds numbers from 11 to 1000 The results of the
present investigation line up with Janours results which
in turn on extrapolation to higher Reyno l ds numbers
(greater than 1000) make a smooth transition into Blasius
curve represented by Equation (10) At Reyno l ds numbers
greater than 20 000 the drag coefficient is inversely proshy
portional to the square root of the Reynolds number
The data for flat plates in perpendicular flow is conshy
siderably above the solutions of Tomotika and Aoi
(15 p 302) and Imai (4 p 157 However their solutions
f or cylinders and plates in parallel flow are also below
the present data Also it should be remembered that their
solutions are for infinitely wide plates If a value of
WL of above 100 is used in Equation (41) then the present
data and the solutions of Tomotika and Aoi are fairly close
56
The present results indicate that Equation (41~ can be
used with an accuracy of 15 to 20 within the limitations
of the equation (WL 1 to 4 Re = 0 05 to 2)
57
SUM RY AND CONCLUSIONS
Only a small amount of work has been done in the past
on the study of laminar flow over immersed bodies There
are many areas in the chemical process industries and the
field of aeronautics where this information would be very
helpful The purpose of the present investi gation wa s to
study the almost totally unexplored range of Reynol ds
numbers from 0 01 to 10
Drag coefficients have been determined for spheres
cylinders and flat plates in paralle l and perpendicular
flow The drag coefficients have been plotted as a
function of the Reynolds number with dimension ratios as
a parameter on lo g-log graphs The best straight lines
have been drawn through the data In all cases these lines
had a slope of -1 hich shows that the dra g coefficient is
inversely proportional to the Reynolds number at very low
Reynolds numbers for all shapes and dimension ratios The
following equations have been determined from the data
For cylinders
fd - 27 L -0 36 (38 ) - Re ())
which applies for Reynolds numbers of 0 01 to 1 and LD of
2 to 16 For LD greater than 16 the equation is
58
(39)
For flat plates in parallel flow a correction factor has
been applied to account for the edge effect The equation
which applies for Reyno l ds numbers of 0 1 to 2 is
f 6Re
(40)
For flat plates in perpendicular flow
f d
- 37 - Re (w) t -
0 bull 30 (41)
wbieh applies for W L of 1 to 4 and Reynolds numbers of
0 05 to 2
It is concluded tha t Equations (38-41) give the best
values of drag coefficients within an accuracy of 20~ for
the range of Reynolds numbers that were considered Also
it is evident that the dimension ratios are a n important
factor in determining the drag coefficient for a given
Reynolds number Furthermore the drag coefficient inshy
creases with decreasing values of L D or W L for a constant
Reynolds number The da ta obtained in this investi gation
compare favorably with the other experimental data and with
some of the theoretical sol utions It should be remembered
that when comparing the experimental data with theoretical
solutions that practically all of the solutions are for an
infinitely long cylinder or an infinitely wide plate
It is recommended tha t the present apparatus be
59
modified so that a force of 001 pound can be measured
Also it would improve tho accuracy to set up a constant
temperature bath so that the temperature of the oil can not
vary over 02degF A few check points on the present data
is all that is necessary to confirm the validity of
Equations (38- 41) It is also r ecommended that only SAE 140
oil be used and that 2 inches should be the minimum plate
width and cylinder length to be studi3d These conditions
would help to maintain the accuracy of the correction force
for the wire
60
~WMENCIATURE
Symbol Dimensions
A area sq ft
D diameter ft
F force lb f
L length ft
M mas s lb m Re Reynolds number Dvf= -ltr w width ft
a area sq ft
b characteristic length ft
d diameter ft
f drag coefficientfd
gravitation constant l b mft gc 2= 32 17 l b _ rsec
1 length ft
m mass l b bullm
p pressure lbrsqft
r radius ft
t time see
u velocity ft sec
v velocity ft sec
w width ft
61
Symbol Dimensions
X xbullcoordinate ft
y y- coordinate ft
o( vorticity
time sec
viscosity lb m ft -sec
kinematic viscosity ft 2sec
circumference diameter = 3 1416
3density lb m ft
function
stream function
Laplacian operator
infinity
Subscripts
c corrected
f force
1 l iquid
m mass
p projected
s solid
w wetted
62
BI BLIOGRAPHY
1 Allan D N de G and R v Southwell Re laxation methods applied to determine the motion in two di shymensions of a viscous fluid past a fixed cylinder Quarterly Journal of Mechanics and Applied Mathe shymatics 8 129-145 1955
2 Bairstow L B M Cave and E D Lang The reshysistance of a cylinder moving in a viscous fluid Philosophical Transactions of the Royal Society of London ser A 223383- 432 1923
3 Goldstein Sidney The steady flow of viscous fluid past a fixed spherical obstacle at small Reyno l ds numbers Proceedings of the Royal Society of London ser A 123225-235 1929
4 Imai I A new method of solving Oseens equations and its application to the flow past an inclined elliptic cylinder Proceedings of the Royal Society of London ser A 224 141-160 1954
5 Janour Zbynek Resistance of a plate in paralle l flow at low Reyno lds numbers Washington Nov 1951 40 p National Advisory Committee for Aeronautics Te chnica l Memorandum 1316)
6 Janssen E An analog solution of the Navier-Stokes equation for the case of flow past a f l at plate at low Reynolds numbers In 1956 Heat Transfer and Fluid Mechanics Institute (Preprints of Papers) p 173-183
7 Knudsen James G and Donal d L Katz Fluid Dynamics a nd Heat Transfer Ann Arbor University of Michigan 1953 243 p (Michi gan University Engineering Research Bulletin no 37)
8 La~b Horace On the uniform motion of a spherethrough a viscous fluid Philosophical Magazine and Journal of Science s~r 6 21112-121 1911
9 Liebster H Uben den widerstrand von kugeln Annalen Der Physik ser 4 82 541- 562 1 927
63
10 McAdams William H Heat transmission 3d ed New York McGraw- Hill 1954 532 p
11 Pai Shih- I Viscous f l ow theory I Laminar flow Princeton D Van Nostrand 1956 384 p
12 Prandtlbull Ludwi g Es sentials of fluid dynamics London Blackie amp Son 1954 452 p
13 Relf i F Discussion of the results of measure shyments of the resistance of wires with some additionshyal tests of the resistance of wires of small diame shyters In Technical report of the Advisory Committee for Aeronautics London) March 1914 p 47 - 51 (Report and memoranda no 102 )
14 Stokes George Gabriel Mathematical and physical papers Vol 3 Cambridge University Press 1922 413 p
15 Tomotika s and T Aoi The steady flow of a viscous fluid past an elliptic cylinder and a flat plate at smal l Reynolds numbers Quarterly Journal of Me chanics and Applie d Ma thematics 6 290- 312 1953
16 Wieselsbergo r c Versuche Ube r der luftwiderstand gerundeter und kant iger korper Er gebnisse der Aeroshydynamischen Versucbsansta l t Vol 2 G~ttingen 1923 80 p
FIGURE e- PHOTOGRAPH OF SPHERES CYLINDERS AND PLATES
34
holes were drilled so that each plate could be used for
two geometric ratios by changing the wires (See for
example plates la and lb in Table I
35
EXPERI MENTA L PROCEDURE
Viscosity and Density Calibration
A calibrated hydrometer measuring to the nearest
0002 was used to measure the density Table VI shows that
the effect of temperature on density is practically negli shy
gible in the small temperature range used
A Brookfield Synchro-lectric viscometer was used to
measure the viscosity of both the light and heavy oil
Figures 18 and 19 show the effect of temperature on visshy
cosity In addition the viscosity of the light oil was
checke d using the falling ball method and the equation
D2--ltA (f s bull fl) g (34) l 8v
The viscometer was calibrated by the National Bureau of bull
Standards and was accurate to l tb
Velocity Measurements
The velocity of movement through the oil was measured
by determining the rate of rotation of the pulleys with a
stop watch Usually the time for 10 revolutions was
measured at the highe r ve locities and for 5 revolutions at
the low velocities From this information and the di
amaters of the pulleys the velocities ere calculated
36
The time was measured to the nearest tenth of a second
Since the measured time was usually between 20 and 40
aeconds 1 the error in ~easuring velocity was considered to
be less tha~ 0 5~
force Measurements
The object connected to the scale 1 was dropped to the
bottom of the oil drum The motor was started and the scale
was read as the object vms being pulled towards the top of
the drum Two or three readings were taken for each object
at each velocity In nearly all cases these readings were
the same
37
ti XPER I MENTAL RE STJLTS
The dra g coefficient and the Reynolds number were
calculated by the use of Equations (l or (15) for each of
the spheres cylinders and plates from the measured
quantities of force and velocity a~d the values of the vis shy
cosity and density corresponding to the temperature of the
oil It was necessary to ~ubtract from the measured force
the force on the wire The corrected force measurement was
then used to determine the drag coefficient The force on
the wire has been determined as being proportional to the
velocity A correction curve relating force on the wire
and ve l ocity is plo tted in Figure 9 for the li ght oil and
Fi gure 10 for the heavy oil
The calculated drag coefficients Reynolds numbers
and velocities along with the measured force for the spheres
cylinders flat plates - parallel flow and flat plates shy
perpendicular flow have been tabulated in Tables II III
I V and v respectively
The calculated drag coefficients have been plotted as
a function of the Reynolds number on logarithic graph paper
with geometric ratios as a parameter
Drag coefficients for the spheres are plo tted in
Figure 11 The data for the cylinders are plotted in
CD_ bull 0 G 0
03
Tshy02
01
10 20 30 410 50 60 70 80
VELOCITY- FTJSEC
DRAG FORCE ON THE WIRE-LIGHT OIL
FIGURE 9
I -shy I -middot -- -shy -1shy _i-i I --~ I I _ -middot- shy I i
_I_ - _ middot- LL I l l tmiddot - middot1middot ~- - - - -+i middotshy I - --+-cl - l
1 1 I I IV jc---- --r--middotmiddottmiddot r-middotmiddot--tmiddotmiddot---shy _____ _L __ --~- --1shy middotmiddotr-r-middott- 1 -f-f-T- _~ +-L--1---~- 1--l
~- - shy I-+---Rmiddot-- I I I l i ~~ i -~~ ~- -T f i rshy ~-- --shy i- ----~-- shy - middot1 shy
I i I i I I 1--- -middot - fshy middot i----1---+-shy - i-middot -~+-- --~- --~-- ---- -t+ I v-~~ -middot j
i I middot 1_ _ I tmiddot---+-+1-+--li~+middot -+--+-+-1-+-+-+-+--tc--1-+-t-11-shy - middot --t- 1---t- t----tmiddotshy --~-- -middot i-shy I 1i - ~ i I i v i middotmiddotmiddot
[~v +L~ + ~ - I~~j-+ r V I ~t--- -~-- I +---~-- I f-middot ---1-- ~ -- --- ) Li --+--+--+-+-+-+--1--+--+---t---4 -1--1--+-+--+-l-i
tl~ I I Q Y +l~~ii-+-++++-middotHH-++-+-+-+--H--++ -i t Imiddot i i 1 j _V I f1 r-t~-middot l--r-tshy -~ 7 middot 1 -shy middot middotmiddot I
DRAG FORCE ON THE WIRE- HEAVY OIL
FIGURE 10
40
+shy l i~ltgt ~ bull r-rshy I i t _l
1 lf-1-1 l+r+ fJ-Ct I+ t li 1~t rtH r+l rf-l It llil I I
l l~pound 11 1 ~middot ~~middott ~ It lqf L
t I+--= ~r 17 -Er I _ ~ _pound~- sect Imiddot I+
iU=ff=t 1 +~ t_ - ~ r 111= t h=
I middot
t= IE I 1 1
plusmn~ kplusmni - -STOKE S EQ
(~ l h+middot
ru HmiddotHti+H1 11
c lffii l t~ 4 ~ ~middot ~ff l ~ ~h i ltlri
1 yen~ middot I ~ I I T ~ gt l+t H+h l+ i j l tfl-l Imiddotmiddot ft+ ++ l f+ Imiddotmiddot I+ I+ middott bulli I 1middot1 I ftt-1shy middot I middot r 11 I IH Ij ~ ~ middotishy J F 1= 6= ~
=f l~iit rtti l lit~ I FS lf~ l=i-+
l-11ffi tt lr 1 ~1 -t =l=Rttl 1ft i- 1 ~ I+ I
~~ lflJ
t I lfl m ~~WFB Lt
41plusmn811 IF I Hir tt ft itttplusmn i I~
1-+++middot
I ~ I (~ ffitrHf1 Ittmiddot ~ l r i H-t-r r HHt m 11 H++ I
bull I I
1_ _ F bullmiddot Imiddotmiddot t-- 1-T h iT
f-t+ ftt I+ I lt + T Imiddot 1
1t _plusmn middot~~ ~- 11shy
=a~ 1~ - =itf lttti
H I
=
DATA FOR SPHERES
FIGURE II
41
I -1---1-1-+--+--Ti-+-------+----r--shy --r--- -shy + t----+shy ----4-~---+-f----f--+-f--l--1 I t--shy --t-- ---+-shy
J-+-~f--~~ -___l_ ~---
i 1 L~L~-~tr-l----H~4-----~-f------+------+-----+----+---+middot-t-middot-H5000
middot~ ~ m- ~ - ~t plusmn~ 3t i t~ -f--- bullbull - ~~ h middot-
01 0~ 10
Re
-
DATA FOR CYLINDERS - LD = 6 8 AND 12
FIGURE I 4
44
Figures 12 13 and 14 The data for LD values of 16 24
and 32 were nearly the same and have been plotted to gether
i n Figure 12 In addition the curves for the other LD
ratios determined fro m Fib~res 13 and 14 have been drawn
in Figure 12 so that the effect of the length-to-diameter
is clearly shown Figure 13 shows the data for LD values
of 2 and 4 and the curves determined from this data
Firure 14 shows the data for LD values of 6 8 and 12
and the curves determined from this data
The data for flat plates in parallel flow are plotted
in Fi gure 15 A correction factor for the edge effect has
beon used so that the width-to-length ratio is not a
parameter in this plot A portion of the data of Janour
(5 p 31) is also shown in the diagram
The data for fla t plates in perpendicular flow is
plotted in Figures 16 a nd 17 Figure 16 shows the data for
WL values of 2 Also the curves for the three WL ratios
1 2 and 4 have been drawn in the fi gure Figure 17 shows
the data for WL values of 1 and 4 The curves determined
from the data have also been dravm in the figure
45
10~ ~ ~--- -shy
t==Ff1TR=+ iJ+--_-_--r_-_---+-+---+--+-+--_---_-~r-=r~=~+--=---=---=---=--~=--=_~1=_--=_~_-middot~~--+-+-t~ 1 Ll~+--+-- ---jtshyl~t L--+ I
I
P------ _l -- --1---L i
20 ~-- I ~g I --- - ---+-- r t L_shy
~ ~B 1) I --o-o- JONES - () - - ~~ p f---j- -~-- e e JANOU R
c gt ~c ~ ------ JANSSEN I 0 0 ~ I
IO ~2=i~~~~~~a=~~f=j= ---- TOM OTIKA bulll= I
~~n ~~--~~~~~~o~~~~~--4- NDCIgttl o shy
-
~--~~~~~+--+~+--4-r-~1+-~-middot+1~ ~ --H--~-~~os I i i i-4 ---~T I I f-- t --- li-------~--+-_--+--t-----~~-~_+---_-_-_--+------+-+-__+-[- +_- ___ _______ __+---+-r-+--H----_+--r--------+shy
it I+ ~ bull t ~1 ri j t++t+t++tft bullm H--~+H-t+t-++H-f+t+~HtttH t bull~H-IrttI-H
iH-H u nH m
I
t H+t-~ 1-r f-tj
i it iT -t middotHt I I I I Ill
~middot __
r middotshy
i I r-
f H- jLj f r H rr t~
II
t f f-l -t+tt ~ ==_ =~middot irE
I I
I
I
f
I --
i
t
1 r bull - r
~- ltt++l=tUtt~S-t+t+++~-++U +HJJm~-fl~HHtt1 tttn ll+t-Tt-~- ~ r fH T --r -1 t ---t- -tshy w _+ _ I-shy middotI
-shy -r- + Hbull Hshy t-I --r++ -t iHr -1 H-e-- -t I 1IT 1
1 H-rf-I IJftJ Jf+i+ ~ L
=+shy - tjshy rtmiddotshy ~ -
+ H 1-Jt I tt o =tt ~-
~1 l +fill l plusmn~ fplusmn -shy + I t-
DATA FOR FLAT PLATES PERPENDICULAR FLOW- WL= I 4
FIGURE 17
48
DI SCUSS ION OF RESULTS
Correction and Accuracy of Measurements
After a few pre liminary force measurements with the
spheres and a check with Stokes law (Equation 2) it was
apparent that the drag force on the wire was appreciable
and needed to be considered It was decided to take a
series of measurements with the spheres and calculate the
difference between the measured force and the force calcushy
lated from Stokes law The difference in force could then
be attributed to the drag on the wire If Stokes law is
followed the force on the wire should be proportional to
the velocity
A series of twenty measurements of the force on the
spheres was taken for each oil and the difference between
the measured force and that calcula ted by Stokes 1 law was
determined For each oil this difference as plo tted vs
the velocity The points grouped fairly ell around a
strai ght line nearly passing through the origin The
method of least squares was used to determine the equation
of the line best fitting the da t a The equa tion of the
line for the li bht oil tas found to be
Fe bullbull05605v - oooa (35)
which was determined at about 62 7degF Since the intercept
49
of the line is very close to zero it is believed that the
line is a good indication of the drag on the wire The
equation of the line for the heavy oil was found to be
F - 19llv I oo2o1 (36 ) c shy
which was determined at about 64 2deg The intercept of this
line is also quite close to zero These lines plotted in
Fi poundures 9 and 10 were used throughout the investigation
for the correction factor of the drag on the wires For
the cylinders and flat plates in parallel flow which were
pulled by two wires the values determined from Equations
35) and (36) were doubled For the plates in perpendicular
flow pulled by four wires the correction force was multishy
plied by four
The spring scale had 12 ounce divisions but could be
read to the nearest sixth of an ounce Some of the measureshy
ments of force were under an ounce hence a considerable
spread of the measurements was noticed in the pre liminary
data and throughout the experiment However sufficient
points were obtained so that it was possible to draw a
reliable curve through the data in all casas An analysis
was made to determine the average deviation from Stokes
equation for the spheres It raa found that the average
deviation was 15 1 for the light oil 16 6 for the heavy
oil and 15 9 overall The maximum deviation was 89
50
Inspection of the other data shows that these deviations
are also representative of the cylinders and flat plates
The force measurement is the least accurate part of the
experiment Other insignificant errors are introduced by
a small variation in the temperature This variation was
held to about 10 from the temperature of the calibrated
correction curve The velocity measurements and the
dimensions of the cylinders spheres and pl~ tes are conshy
sidered go od enough so tha t no appreciable errors occur
In order to e l iminate the WL parameter for flat plates
in parallel f l ow an additional factor for the effect of
the edges was subtracted from the measured force Janour
(5 p 27) presented the foll owing equation for the edge
correction for one edge of a flat plate in parallel flow
F ~ lv~ bull (37 ) edge gc
In present work this equation as doubled because both
edges of the plates were submerged in fluid It is assumed
in appl ying this correction that the lowe r limit of a
Reynolds number of 10 proposed by Janour can be extended
close to 0 1
Analysis of Results
Forty of the points for the spheres were used to get
51
the correction factor for the wires The remaining thirty
points are well erouped about Stokes law
The data for cylinders for LD ratios of 16 24 and
32 did not seem to be se gregated therefore these data
were plotted together It would seem that in the low range
of Reyno l ds numbers an LD of 16 and greater can be con shy
sidered an ~nfini tely long cylinder The other LD ratios
of 2 4 6 a 12 provided fairly distinct and separate
lines The best straight lines were drawn through the data
for each of the LD ratios It was evident that in eaeh
case a slope of -1 on a lo g-log graph gave the best straight
line which would indicate that the force varies directly
as the velocity It was possible to develop an empirical
expression relating dra g coefficient Reynolds number and
LD The following equation was obtained from the straight
line plots of Re vs fd for the various LD ratios
(38 )
Equation (38) applies for Reyno l ds numbers from 01 to 10
and for LD ratios of 2 to 16 For LD ratios greater
than 16
10 re (39 )
The data for flat plates in parallel flow is plotted
in Figure 15 after the correction factor for tho edge
52
effect was subtracted When the edge correction is made
no effect of WL ratio is indicated This result would be
expected The data followed a straight line with a slope
of -1 up to a Reynolds number of 2 After that a curve was
dravm connecting the line to that obtained by Janour The
equation for the straight section of the curve is
f - 6 (40)- Re
which applies for Reynolds numbers of 0 1 to 2 0 Here
a gain the force is proportional to the velocity Vfuen
determining drag force for flat plates in parallel flow
the force is first calculated from Equations (40) and (15 )
then the edge correction is added
The effect of the geometric ratios is clearly shown in
the data for flat plates in perpendicul ar flow which are
plotted in Figures 16 and 17 As with the other data the
best straight line was drawn through the various points
for eaoh of the WL ratios Again the line had a slope of
-1 The equation relating fd Re and wL was found t o be
rd 37 (w) -o 3o (41)Irel
which applies for Reynolds numbers of about 05 to 2 0 and
WL ratios of 1 to 4 It is possible but it has not been
proved that Equation (41) is suitable for higher WL ratios
The exponent on WL in Equation 41) is very close to that
53
on L D i n Equation ( 38 )~ It i s possible t ha t these
exponents are t he same but this cannot be sho~~ depound1nitely
until more accura te da ta are available It would be exshy
pected that a s the Reynolds number approaches zero t he
effect of geometric ratios would be the same for cylinders
and fla t pla tes in perpendicula r flow
It is seen in the t a bles of data that occasionally a
ne gative force was obtained because the correction applie d
due to t he wire dra g was greater than the mea sured force
These points obviously are incorrect This occurred only
for the smallest plates in the heavy oil at t he highest
velocities However these knom bad points occur in less
tha n 5~ of the data
It is clearl y shown that for cylinders and plates the
fd increases as L D or W L decreases This is in direct
contrast to Wiesel aberger s investigation However his
work is for hi gher Reynolds numbers at which a turbulent
wake forms bull
Comparison of Results with Other Data and Theoretical So l utions
The data for sphere~ a grees of course with Stokes
l aw since that law was used to determine the correction
factor for the wire Liebster (9 Pbull 548 ) has
54
substantiated Stokes equation
There are no experimental data with which to compare
the results of the cylinders Wieselsbergers minimum
Reynolds number of 4 is above the ran ge covered in the preshy
sent investigation The da ta for the highest LD ratios
(16 24 and 32) does agree almost exactly wi t h the solution
of Allen and Southwell (1 P bull 141) (LD =00) in the range
of Reynolds numbers from 0 1 to 1 0 Allen and Southwells
solution a greed with the data of Wieselsberger (16 p 22)
However the present data is above the theoretical solutions
of Lamb (8 p 112-121) throughout the range of Reynolds
numbers from 0 01 to 1 0 and above the solutions of
Bairstow Cave and Lang (2 p 404) I mai (4 p 157) and
Tomotika and Aoi (15 p 302) for Reynolds numbers of 0 1
to 1 0 Allen and Southwells solution a grees dth both
Wieselsberger 1 s a nd the present data Their solution and
the present data represent the best means for predicting
drag coefficients for flow over long cylinders for Reynolds
numbers of 0 01 to 10 It should be remembered that the
o t her solutions should a gree with eac h other since they
were all essentially derived by linearizing the Na viershy
Stokes equation
The data for flat plates in parallel flow is
55
considerably above the theoretical solutions of Janssen
(6 p 183 ) and Tomotika and Aoi (15 Pbull 302) However
Fi f~re 15 shows that a smooth transition occurs bet een
the present work and the data of Janour (5 P bull 31) The
present data considerably extend the experimental inforshy
mation previously available for laminar flow paral lel to
flat plates In the re gion of Reynol ds numbers less than
2 the drag coefficient is shown to be inversely proportional
to the Reynolds number Janours data covers a range of
Reynolds numbers from 11 to 1000 The results of the
present investigation line up with Janours results which
in turn on extrapolation to higher Reyno l ds numbers
(greater than 1000) make a smooth transition into Blasius
curve represented by Equation (10) At Reyno l ds numbers
greater than 20 000 the drag coefficient is inversely proshy
portional to the square root of the Reynolds number
The data for flat plates in perpendicular flow is conshy
siderably above the solutions of Tomotika and Aoi
(15 p 302) and Imai (4 p 157 However their solutions
f or cylinders and plates in parallel flow are also below
the present data Also it should be remembered that their
solutions are for infinitely wide plates If a value of
WL of above 100 is used in Equation (41) then the present
data and the solutions of Tomotika and Aoi are fairly close
56
The present results indicate that Equation (41~ can be
used with an accuracy of 15 to 20 within the limitations
of the equation (WL 1 to 4 Re = 0 05 to 2)
57
SUM RY AND CONCLUSIONS
Only a small amount of work has been done in the past
on the study of laminar flow over immersed bodies There
are many areas in the chemical process industries and the
field of aeronautics where this information would be very
helpful The purpose of the present investi gation wa s to
study the almost totally unexplored range of Reynol ds
numbers from 0 01 to 10
Drag coefficients have been determined for spheres
cylinders and flat plates in paralle l and perpendicular
flow The drag coefficients have been plotted as a
function of the Reynolds number with dimension ratios as
a parameter on lo g-log graphs The best straight lines
have been drawn through the data In all cases these lines
had a slope of -1 hich shows that the dra g coefficient is
inversely proportional to the Reynolds number at very low
Reynolds numbers for all shapes and dimension ratios The
following equations have been determined from the data
For cylinders
fd - 27 L -0 36 (38 ) - Re ())
which applies for Reynolds numbers of 0 01 to 1 and LD of
2 to 16 For LD greater than 16 the equation is
58
(39)
For flat plates in parallel flow a correction factor has
been applied to account for the edge effect The equation
which applies for Reyno l ds numbers of 0 1 to 2 is
f 6Re
(40)
For flat plates in perpendicular flow
f d
- 37 - Re (w) t -
0 bull 30 (41)
wbieh applies for W L of 1 to 4 and Reynolds numbers of
0 05 to 2
It is concluded tha t Equations (38-41) give the best
values of drag coefficients within an accuracy of 20~ for
the range of Reynolds numbers that were considered Also
it is evident that the dimension ratios are a n important
factor in determining the drag coefficient for a given
Reynolds number Furthermore the drag coefficient inshy
creases with decreasing values of L D or W L for a constant
Reynolds number The da ta obtained in this investi gation
compare favorably with the other experimental data and with
some of the theoretical sol utions It should be remembered
that when comparing the experimental data with theoretical
solutions that practically all of the solutions are for an
infinitely long cylinder or an infinitely wide plate
It is recommended tha t the present apparatus be
59
modified so that a force of 001 pound can be measured
Also it would improve tho accuracy to set up a constant
temperature bath so that the temperature of the oil can not
vary over 02degF A few check points on the present data
is all that is necessary to confirm the validity of
Equations (38- 41) It is also r ecommended that only SAE 140
oil be used and that 2 inches should be the minimum plate
width and cylinder length to be studi3d These conditions
would help to maintain the accuracy of the correction force
for the wire
60
~WMENCIATURE
Symbol Dimensions
A area sq ft
D diameter ft
F force lb f
L length ft
M mas s lb m Re Reynolds number Dvf= -ltr w width ft
a area sq ft
b characteristic length ft
d diameter ft
f drag coefficientfd
gravitation constant l b mft gc 2= 32 17 l b _ rsec
1 length ft
m mass l b bullm
p pressure lbrsqft
r radius ft
t time see
u velocity ft sec
v velocity ft sec
w width ft
61
Symbol Dimensions
X xbullcoordinate ft
y y- coordinate ft
o( vorticity
time sec
viscosity lb m ft -sec
kinematic viscosity ft 2sec
circumference diameter = 3 1416
3density lb m ft
function
stream function
Laplacian operator
infinity
Subscripts
c corrected
f force
1 l iquid
m mass
p projected
s solid
w wetted
62
BI BLIOGRAPHY
1 Allan D N de G and R v Southwell Re laxation methods applied to determine the motion in two di shymensions of a viscous fluid past a fixed cylinder Quarterly Journal of Mechanics and Applied Mathe shymatics 8 129-145 1955
2 Bairstow L B M Cave and E D Lang The reshysistance of a cylinder moving in a viscous fluid Philosophical Transactions of the Royal Society of London ser A 223383- 432 1923
3 Goldstein Sidney The steady flow of viscous fluid past a fixed spherical obstacle at small Reyno l ds numbers Proceedings of the Royal Society of London ser A 123225-235 1929
4 Imai I A new method of solving Oseens equations and its application to the flow past an inclined elliptic cylinder Proceedings of the Royal Society of London ser A 224 141-160 1954
5 Janour Zbynek Resistance of a plate in paralle l flow at low Reyno lds numbers Washington Nov 1951 40 p National Advisory Committee for Aeronautics Te chnica l Memorandum 1316)
6 Janssen E An analog solution of the Navier-Stokes equation for the case of flow past a f l at plate at low Reynolds numbers In 1956 Heat Transfer and Fluid Mechanics Institute (Preprints of Papers) p 173-183
7 Knudsen James G and Donal d L Katz Fluid Dynamics a nd Heat Transfer Ann Arbor University of Michigan 1953 243 p (Michi gan University Engineering Research Bulletin no 37)
8 La~b Horace On the uniform motion of a spherethrough a viscous fluid Philosophical Magazine and Journal of Science s~r 6 21112-121 1911
9 Liebster H Uben den widerstrand von kugeln Annalen Der Physik ser 4 82 541- 562 1 927
63
10 McAdams William H Heat transmission 3d ed New York McGraw- Hill 1954 532 p
11 Pai Shih- I Viscous f l ow theory I Laminar flow Princeton D Van Nostrand 1956 384 p
12 Prandtlbull Ludwi g Es sentials of fluid dynamics London Blackie amp Son 1954 452 p
13 Relf i F Discussion of the results of measure shyments of the resistance of wires with some additionshyal tests of the resistance of wires of small diame shyters In Technical report of the Advisory Committee for Aeronautics London) March 1914 p 47 - 51 (Report and memoranda no 102 )
14 Stokes George Gabriel Mathematical and physical papers Vol 3 Cambridge University Press 1922 413 p
15 Tomotika s and T Aoi The steady flow of a viscous fluid past an elliptic cylinder and a flat plate at smal l Reynolds numbers Quarterly Journal of Me chanics and Applie d Ma thematics 6 290- 312 1953
16 Wieselsbergo r c Versuche Ube r der luftwiderstand gerundeter und kant iger korper Er gebnisse der Aeroshydynamischen Versucbsansta l t Vol 2 G~ttingen 1923 80 p
FIGURE e- PHOTOGRAPH OF SPHERES CYLINDERS AND PLATES
34
holes were drilled so that each plate could be used for
two geometric ratios by changing the wires (See for
example plates la and lb in Table I
35
EXPERI MENTA L PROCEDURE
Viscosity and Density Calibration
A calibrated hydrometer measuring to the nearest
0002 was used to measure the density Table VI shows that
the effect of temperature on density is practically negli shy
gible in the small temperature range used
A Brookfield Synchro-lectric viscometer was used to
measure the viscosity of both the light and heavy oil
Figures 18 and 19 show the effect of temperature on visshy
cosity In addition the viscosity of the light oil was
checke d using the falling ball method and the equation
D2--ltA (f s bull fl) g (34) l 8v
The viscometer was calibrated by the National Bureau of bull
Standards and was accurate to l tb
Velocity Measurements
The velocity of movement through the oil was measured
by determining the rate of rotation of the pulleys with a
stop watch Usually the time for 10 revolutions was
measured at the highe r ve locities and for 5 revolutions at
the low velocities From this information and the di
amaters of the pulleys the velocities ere calculated
36
The time was measured to the nearest tenth of a second
Since the measured time was usually between 20 and 40
aeconds 1 the error in ~easuring velocity was considered to
be less tha~ 0 5~
force Measurements
The object connected to the scale 1 was dropped to the
bottom of the oil drum The motor was started and the scale
was read as the object vms being pulled towards the top of
the drum Two or three readings were taken for each object
at each velocity In nearly all cases these readings were
the same
37
ti XPER I MENTAL RE STJLTS
The dra g coefficient and the Reynolds number were
calculated by the use of Equations (l or (15) for each of
the spheres cylinders and plates from the measured
quantities of force and velocity a~d the values of the vis shy
cosity and density corresponding to the temperature of the
oil It was necessary to ~ubtract from the measured force
the force on the wire The corrected force measurement was
then used to determine the drag coefficient The force on
the wire has been determined as being proportional to the
velocity A correction curve relating force on the wire
and ve l ocity is plo tted in Figure 9 for the li ght oil and
Fi gure 10 for the heavy oil
The calculated drag coefficients Reynolds numbers
and velocities along with the measured force for the spheres
cylinders flat plates - parallel flow and flat plates shy
perpendicular flow have been tabulated in Tables II III
I V and v respectively
The calculated drag coefficients have been plotted as
a function of the Reynolds number on logarithic graph paper
with geometric ratios as a parameter
Drag coefficients for the spheres are plo tted in
Figure 11 The data for the cylinders are plotted in
CD_ bull 0 G 0
03
Tshy02
01
10 20 30 410 50 60 70 80
VELOCITY- FTJSEC
DRAG FORCE ON THE WIRE-LIGHT OIL
FIGURE 9
I -shy I -middot -- -shy -1shy _i-i I --~ I I _ -middot- shy I i
_I_ - _ middot- LL I l l tmiddot - middot1middot ~- - - - -+i middotshy I - --+-cl - l
1 1 I I IV jc---- --r--middotmiddottmiddot r-middotmiddot--tmiddotmiddot---shy _____ _L __ --~- --1shy middotmiddotr-r-middott- 1 -f-f-T- _~ +-L--1---~- 1--l
~- - shy I-+---Rmiddot-- I I I l i ~~ i -~~ ~- -T f i rshy ~-- --shy i- ----~-- shy - middot1 shy
I i I i I I 1--- -middot - fshy middot i----1---+-shy - i-middot -~+-- --~- --~-- ---- -t+ I v-~~ -middot j
i I middot 1_ _ I tmiddot---+-+1-+--li~+middot -+--+-+-1-+-+-+-+--tc--1-+-t-11-shy - middot --t- 1---t- t----tmiddotshy --~-- -middot i-shy I 1i - ~ i I i v i middotmiddotmiddot
[~v +L~ + ~ - I~~j-+ r V I ~t--- -~-- I +---~-- I f-middot ---1-- ~ -- --- ) Li --+--+--+-+-+-+--1--+--+---t---4 -1--1--+-+--+-l-i
tl~ I I Q Y +l~~ii-+-++++-middotHH-++-+-+-+--H--++ -i t Imiddot i i 1 j _V I f1 r-t~-middot l--r-tshy -~ 7 middot 1 -shy middot middotmiddot I
DRAG FORCE ON THE WIRE- HEAVY OIL
FIGURE 10
40
+shy l i~ltgt ~ bull r-rshy I i t _l
1 lf-1-1 l+r+ fJ-Ct I+ t li 1~t rtH r+l rf-l It llil I I
l l~pound 11 1 ~middot ~~middott ~ It lqf L
t I+--= ~r 17 -Er I _ ~ _pound~- sect Imiddot I+
iU=ff=t 1 +~ t_ - ~ r 111= t h=
I middot
t= IE I 1 1
plusmn~ kplusmni - -STOKE S EQ
(~ l h+middot
ru HmiddotHti+H1 11
c lffii l t~ 4 ~ ~middot ~ff l ~ ~h i ltlri
1 yen~ middot I ~ I I T ~ gt l+t H+h l+ i j l tfl-l Imiddotmiddot ft+ ++ l f+ Imiddotmiddot I+ I+ middott bulli I 1middot1 I ftt-1shy middot I middot r 11 I IH Ij ~ ~ middotishy J F 1= 6= ~
=f l~iit rtti l lit~ I FS lf~ l=i-+
l-11ffi tt lr 1 ~1 -t =l=Rttl 1ft i- 1 ~ I+ I
~~ lflJ
t I lfl m ~~WFB Lt
41plusmn811 IF I Hir tt ft itttplusmn i I~
1-+++middot
I ~ I (~ ffitrHf1 Ittmiddot ~ l r i H-t-r r HHt m 11 H++ I
bull I I
1_ _ F bullmiddot Imiddotmiddot t-- 1-T h iT
f-t+ ftt I+ I lt + T Imiddot 1
1t _plusmn middot~~ ~- 11shy
=a~ 1~ - =itf lttti
H I
=
DATA FOR SPHERES
FIGURE II
41
I -1---1-1-+--+--Ti-+-------+----r--shy --r--- -shy + t----+shy ----4-~---+-f----f--+-f--l--1 I t--shy --t-- ---+-shy
J-+-~f--~~ -___l_ ~---
i 1 L~L~-~tr-l----H~4-----~-f------+------+-----+----+---+middot-t-middot-H5000
middot~ ~ m- ~ - ~t plusmn~ 3t i t~ -f--- bullbull - ~~ h middot-
01 0~ 10
Re
-
DATA FOR CYLINDERS - LD = 6 8 AND 12
FIGURE I 4
44
Figures 12 13 and 14 The data for LD values of 16 24
and 32 were nearly the same and have been plotted to gether
i n Figure 12 In addition the curves for the other LD
ratios determined fro m Fib~res 13 and 14 have been drawn
in Figure 12 so that the effect of the length-to-diameter
is clearly shown Figure 13 shows the data for LD values
of 2 and 4 and the curves determined from this data
Firure 14 shows the data for LD values of 6 8 and 12
and the curves determined from this data
The data for flat plates in parallel flow are plotted
in Fi gure 15 A correction factor for the edge effect has
beon used so that the width-to-length ratio is not a
parameter in this plot A portion of the data of Janour
(5 p 31) is also shown in the diagram
The data for fla t plates in perpendicular flow is
plotted in Figures 16 a nd 17 Figure 16 shows the data for
WL values of 2 Also the curves for the three WL ratios
1 2 and 4 have been drawn in the fi gure Figure 17 shows
the data for WL values of 1 and 4 The curves determined
from the data have also been dravm in the figure
45
10~ ~ ~--- -shy
t==Ff1TR=+ iJ+--_-_--r_-_---+-+---+--+-+--_---_-~r-=r~=~+--=---=---=---=--~=--=_~1=_--=_~_-middot~~--+-+-t~ 1 Ll~+--+-- ---jtshyl~t L--+ I
I
P------ _l -- --1---L i
20 ~-- I ~g I --- - ---+-- r t L_shy
~ ~B 1) I --o-o- JONES - () - - ~~ p f---j- -~-- e e JANOU R
c gt ~c ~ ------ JANSSEN I 0 0 ~ I
IO ~2=i~~~~~~a=~~f=j= ---- TOM OTIKA bulll= I
~~n ~~--~~~~~~o~~~~~--4- NDCIgttl o shy
-
~--~~~~~+--+~+--4-r-~1+-~-middot+1~ ~ --H--~-~~os I i i i-4 ---~T I I f-- t --- li-------~--+-_--+--t-----~~-~_+---_-_-_--+------+-+-__+-[- +_- ___ _______ __+---+-r-+--H----_+--r--------+shy
it I+ ~ bull t ~1 ri j t++t+t++tft bullm H--~+H-t+t-++H-f+t+~HtttH t bull~H-IrttI-H
iH-H u nH m
I
t H+t-~ 1-r f-tj
i it iT -t middotHt I I I I Ill
~middot __
r middotshy
i I r-
f H- jLj f r H rr t~
II
t f f-l -t+tt ~ ==_ =~middot irE
I I
I
I
f
I --
i
t
1 r bull - r
~- ltt++l=tUtt~S-t+t+++~-++U +HJJm~-fl~HHtt1 tttn ll+t-Tt-~- ~ r fH T --r -1 t ---t- -tshy w _+ _ I-shy middotI
-shy -r- + Hbull Hshy t-I --r++ -t iHr -1 H-e-- -t I 1IT 1
1 H-rf-I IJftJ Jf+i+ ~ L
=+shy - tjshy rtmiddotshy ~ -
+ H 1-Jt I tt o =tt ~-
~1 l +fill l plusmn~ fplusmn -shy + I t-
DATA FOR FLAT PLATES PERPENDICULAR FLOW- WL= I 4
FIGURE 17
48
DI SCUSS ION OF RESULTS
Correction and Accuracy of Measurements
After a few pre liminary force measurements with the
spheres and a check with Stokes law (Equation 2) it was
apparent that the drag force on the wire was appreciable
and needed to be considered It was decided to take a
series of measurements with the spheres and calculate the
difference between the measured force and the force calcushy
lated from Stokes law The difference in force could then
be attributed to the drag on the wire If Stokes law is
followed the force on the wire should be proportional to
the velocity
A series of twenty measurements of the force on the
spheres was taken for each oil and the difference between
the measured force and that calcula ted by Stokes 1 law was
determined For each oil this difference as plo tted vs
the velocity The points grouped fairly ell around a
strai ght line nearly passing through the origin The
method of least squares was used to determine the equation
of the line best fitting the da t a The equa tion of the
line for the li bht oil tas found to be
Fe bullbull05605v - oooa (35)
which was determined at about 62 7degF Since the intercept
49
of the line is very close to zero it is believed that the
line is a good indication of the drag on the wire The
equation of the line for the heavy oil was found to be
F - 19llv I oo2o1 (36 ) c shy
which was determined at about 64 2deg The intercept of this
line is also quite close to zero These lines plotted in
Fi poundures 9 and 10 were used throughout the investigation
for the correction factor of the drag on the wires For
the cylinders and flat plates in parallel flow which were
pulled by two wires the values determined from Equations
35) and (36) were doubled For the plates in perpendicular
flow pulled by four wires the correction force was multishy
plied by four
The spring scale had 12 ounce divisions but could be
read to the nearest sixth of an ounce Some of the measureshy
ments of force were under an ounce hence a considerable
spread of the measurements was noticed in the pre liminary
data and throughout the experiment However sufficient
points were obtained so that it was possible to draw a
reliable curve through the data in all casas An analysis
was made to determine the average deviation from Stokes
equation for the spheres It raa found that the average
deviation was 15 1 for the light oil 16 6 for the heavy
oil and 15 9 overall The maximum deviation was 89
50
Inspection of the other data shows that these deviations
are also representative of the cylinders and flat plates
The force measurement is the least accurate part of the
experiment Other insignificant errors are introduced by
a small variation in the temperature This variation was
held to about 10 from the temperature of the calibrated
correction curve The velocity measurements and the
dimensions of the cylinders spheres and pl~ tes are conshy
sidered go od enough so tha t no appreciable errors occur
In order to e l iminate the WL parameter for flat plates
in parallel f l ow an additional factor for the effect of
the edges was subtracted from the measured force Janour
(5 p 27) presented the foll owing equation for the edge
correction for one edge of a flat plate in parallel flow
F ~ lv~ bull (37 ) edge gc
In present work this equation as doubled because both
edges of the plates were submerged in fluid It is assumed
in appl ying this correction that the lowe r limit of a
Reynolds number of 10 proposed by Janour can be extended
close to 0 1
Analysis of Results
Forty of the points for the spheres were used to get
51
the correction factor for the wires The remaining thirty
points are well erouped about Stokes law
The data for cylinders for LD ratios of 16 24 and
32 did not seem to be se gregated therefore these data
were plotted together It would seem that in the low range
of Reyno l ds numbers an LD of 16 and greater can be con shy
sidered an ~nfini tely long cylinder The other LD ratios
of 2 4 6 a 12 provided fairly distinct and separate
lines The best straight lines were drawn through the data
for each of the LD ratios It was evident that in eaeh
case a slope of -1 on a lo g-log graph gave the best straight
line which would indicate that the force varies directly
as the velocity It was possible to develop an empirical
expression relating dra g coefficient Reynolds number and
LD The following equation was obtained from the straight
line plots of Re vs fd for the various LD ratios
(38 )
Equation (38) applies for Reyno l ds numbers from 01 to 10
and for LD ratios of 2 to 16 For LD ratios greater
than 16
10 re (39 )
The data for flat plates in parallel flow is plotted
in Figure 15 after the correction factor for tho edge
52
effect was subtracted When the edge correction is made
no effect of WL ratio is indicated This result would be
expected The data followed a straight line with a slope
of -1 up to a Reynolds number of 2 After that a curve was
dravm connecting the line to that obtained by Janour The
equation for the straight section of the curve is
f - 6 (40)- Re
which applies for Reynolds numbers of 0 1 to 2 0 Here
a gain the force is proportional to the velocity Vfuen
determining drag force for flat plates in parallel flow
the force is first calculated from Equations (40) and (15 )
then the edge correction is added
The effect of the geometric ratios is clearly shown in
the data for flat plates in perpendicul ar flow which are
plotted in Figures 16 and 17 As with the other data the
best straight line was drawn through the various points
for eaoh of the WL ratios Again the line had a slope of
-1 The equation relating fd Re and wL was found t o be
rd 37 (w) -o 3o (41)Irel
which applies for Reynolds numbers of about 05 to 2 0 and
WL ratios of 1 to 4 It is possible but it has not been
proved that Equation (41) is suitable for higher WL ratios
The exponent on WL in Equation 41) is very close to that
53
on L D i n Equation ( 38 )~ It i s possible t ha t these
exponents are t he same but this cannot be sho~~ depound1nitely
until more accura te da ta are available It would be exshy
pected that a s the Reynolds number approaches zero t he
effect of geometric ratios would be the same for cylinders
and fla t pla tes in perpendicula r flow
It is seen in the t a bles of data that occasionally a
ne gative force was obtained because the correction applie d
due to t he wire dra g was greater than the mea sured force
These points obviously are incorrect This occurred only
for the smallest plates in the heavy oil at t he highest
velocities However these knom bad points occur in less
tha n 5~ of the data
It is clearl y shown that for cylinders and plates the
fd increases as L D or W L decreases This is in direct
contrast to Wiesel aberger s investigation However his
work is for hi gher Reynolds numbers at which a turbulent
wake forms bull
Comparison of Results with Other Data and Theoretical So l utions
The data for sphere~ a grees of course with Stokes
l aw since that law was used to determine the correction
factor for the wire Liebster (9 Pbull 548 ) has
54
substantiated Stokes equation
There are no experimental data with which to compare
the results of the cylinders Wieselsbergers minimum
Reynolds number of 4 is above the ran ge covered in the preshy
sent investigation The da ta for the highest LD ratios
(16 24 and 32) does agree almost exactly wi t h the solution
of Allen and Southwell (1 P bull 141) (LD =00) in the range
of Reynolds numbers from 0 1 to 1 0 Allen and Southwells
solution a greed with the data of Wieselsberger (16 p 22)
However the present data is above the theoretical solutions
of Lamb (8 p 112-121) throughout the range of Reynolds
numbers from 0 01 to 1 0 and above the solutions of
Bairstow Cave and Lang (2 p 404) I mai (4 p 157) and
Tomotika and Aoi (15 p 302) for Reynolds numbers of 0 1
to 1 0 Allen and Southwells solution a grees dth both
Wieselsberger 1 s a nd the present data Their solution and
the present data represent the best means for predicting
drag coefficients for flow over long cylinders for Reynolds
numbers of 0 01 to 10 It should be remembered that the
o t her solutions should a gree with eac h other since they
were all essentially derived by linearizing the Na viershy
Stokes equation
The data for flat plates in parallel flow is
55
considerably above the theoretical solutions of Janssen
(6 p 183 ) and Tomotika and Aoi (15 Pbull 302) However
Fi f~re 15 shows that a smooth transition occurs bet een
the present work and the data of Janour (5 P bull 31) The
present data considerably extend the experimental inforshy
mation previously available for laminar flow paral lel to
flat plates In the re gion of Reynol ds numbers less than
2 the drag coefficient is shown to be inversely proportional
to the Reynolds number Janours data covers a range of
Reynolds numbers from 11 to 1000 The results of the
present investigation line up with Janours results which
in turn on extrapolation to higher Reyno l ds numbers
(greater than 1000) make a smooth transition into Blasius
curve represented by Equation (10) At Reyno l ds numbers
greater than 20 000 the drag coefficient is inversely proshy
portional to the square root of the Reynolds number
The data for flat plates in perpendicular flow is conshy
siderably above the solutions of Tomotika and Aoi
(15 p 302) and Imai (4 p 157 However their solutions
f or cylinders and plates in parallel flow are also below
the present data Also it should be remembered that their
solutions are for infinitely wide plates If a value of
WL of above 100 is used in Equation (41) then the present
data and the solutions of Tomotika and Aoi are fairly close
56
The present results indicate that Equation (41~ can be
used with an accuracy of 15 to 20 within the limitations
of the equation (WL 1 to 4 Re = 0 05 to 2)
57
SUM RY AND CONCLUSIONS
Only a small amount of work has been done in the past
on the study of laminar flow over immersed bodies There
are many areas in the chemical process industries and the
field of aeronautics where this information would be very
helpful The purpose of the present investi gation wa s to
study the almost totally unexplored range of Reynol ds
numbers from 0 01 to 10
Drag coefficients have been determined for spheres
cylinders and flat plates in paralle l and perpendicular
flow The drag coefficients have been plotted as a
function of the Reynolds number with dimension ratios as
a parameter on lo g-log graphs The best straight lines
have been drawn through the data In all cases these lines
had a slope of -1 hich shows that the dra g coefficient is
inversely proportional to the Reynolds number at very low
Reynolds numbers for all shapes and dimension ratios The
following equations have been determined from the data
For cylinders
fd - 27 L -0 36 (38 ) - Re ())
which applies for Reynolds numbers of 0 01 to 1 and LD of
2 to 16 For LD greater than 16 the equation is
58
(39)
For flat plates in parallel flow a correction factor has
been applied to account for the edge effect The equation
which applies for Reyno l ds numbers of 0 1 to 2 is
f 6Re
(40)
For flat plates in perpendicular flow
f d
- 37 - Re (w) t -
0 bull 30 (41)
wbieh applies for W L of 1 to 4 and Reynolds numbers of
0 05 to 2
It is concluded tha t Equations (38-41) give the best
values of drag coefficients within an accuracy of 20~ for
the range of Reynolds numbers that were considered Also
it is evident that the dimension ratios are a n important
factor in determining the drag coefficient for a given
Reynolds number Furthermore the drag coefficient inshy
creases with decreasing values of L D or W L for a constant
Reynolds number The da ta obtained in this investi gation
compare favorably with the other experimental data and with
some of the theoretical sol utions It should be remembered
that when comparing the experimental data with theoretical
solutions that practically all of the solutions are for an
infinitely long cylinder or an infinitely wide plate
It is recommended tha t the present apparatus be
59
modified so that a force of 001 pound can be measured
Also it would improve tho accuracy to set up a constant
temperature bath so that the temperature of the oil can not
vary over 02degF A few check points on the present data
is all that is necessary to confirm the validity of
Equations (38- 41) It is also r ecommended that only SAE 140
oil be used and that 2 inches should be the minimum plate
width and cylinder length to be studi3d These conditions
would help to maintain the accuracy of the correction force
for the wire
60
~WMENCIATURE
Symbol Dimensions
A area sq ft
D diameter ft
F force lb f
L length ft
M mas s lb m Re Reynolds number Dvf= -ltr w width ft
a area sq ft
b characteristic length ft
d diameter ft
f drag coefficientfd
gravitation constant l b mft gc 2= 32 17 l b _ rsec
1 length ft
m mass l b bullm
p pressure lbrsqft
r radius ft
t time see
u velocity ft sec
v velocity ft sec
w width ft
61
Symbol Dimensions
X xbullcoordinate ft
y y- coordinate ft
o( vorticity
time sec
viscosity lb m ft -sec
kinematic viscosity ft 2sec
circumference diameter = 3 1416
3density lb m ft
function
stream function
Laplacian operator
infinity
Subscripts
c corrected
f force
1 l iquid
m mass
p projected
s solid
w wetted
62
BI BLIOGRAPHY
1 Allan D N de G and R v Southwell Re laxation methods applied to determine the motion in two di shymensions of a viscous fluid past a fixed cylinder Quarterly Journal of Mechanics and Applied Mathe shymatics 8 129-145 1955
2 Bairstow L B M Cave and E D Lang The reshysistance of a cylinder moving in a viscous fluid Philosophical Transactions of the Royal Society of London ser A 223383- 432 1923
3 Goldstein Sidney The steady flow of viscous fluid past a fixed spherical obstacle at small Reyno l ds numbers Proceedings of the Royal Society of London ser A 123225-235 1929
4 Imai I A new method of solving Oseens equations and its application to the flow past an inclined elliptic cylinder Proceedings of the Royal Society of London ser A 224 141-160 1954
5 Janour Zbynek Resistance of a plate in paralle l flow at low Reyno lds numbers Washington Nov 1951 40 p National Advisory Committee for Aeronautics Te chnica l Memorandum 1316)
6 Janssen E An analog solution of the Navier-Stokes equation for the case of flow past a f l at plate at low Reynolds numbers In 1956 Heat Transfer and Fluid Mechanics Institute (Preprints of Papers) p 173-183
7 Knudsen James G and Donal d L Katz Fluid Dynamics a nd Heat Transfer Ann Arbor University of Michigan 1953 243 p (Michi gan University Engineering Research Bulletin no 37)
8 La~b Horace On the uniform motion of a spherethrough a viscous fluid Philosophical Magazine and Journal of Science s~r 6 21112-121 1911
9 Liebster H Uben den widerstrand von kugeln Annalen Der Physik ser 4 82 541- 562 1 927
63
10 McAdams William H Heat transmission 3d ed New York McGraw- Hill 1954 532 p
11 Pai Shih- I Viscous f l ow theory I Laminar flow Princeton D Van Nostrand 1956 384 p
12 Prandtlbull Ludwi g Es sentials of fluid dynamics London Blackie amp Son 1954 452 p
13 Relf i F Discussion of the results of measure shyments of the resistance of wires with some additionshyal tests of the resistance of wires of small diame shyters In Technical report of the Advisory Committee for Aeronautics London) March 1914 p 47 - 51 (Report and memoranda no 102 )
14 Stokes George Gabriel Mathematical and physical papers Vol 3 Cambridge University Press 1922 413 p
15 Tomotika s and T Aoi The steady flow of a viscous fluid past an elliptic cylinder and a flat plate at smal l Reynolds numbers Quarterly Journal of Me chanics and Applie d Ma thematics 6 290- 312 1953
16 Wieselsbergo r c Versuche Ube r der luftwiderstand gerundeter und kant iger korper Er gebnisse der Aeroshydynamischen Versucbsansta l t Vol 2 G~ttingen 1923 80 p
FIGURE e- PHOTOGRAPH OF SPHERES CYLINDERS AND PLATES
34
holes were drilled so that each plate could be used for
two geometric ratios by changing the wires (See for
example plates la and lb in Table I
35
EXPERI MENTA L PROCEDURE
Viscosity and Density Calibration
A calibrated hydrometer measuring to the nearest
0002 was used to measure the density Table VI shows that
the effect of temperature on density is practically negli shy
gible in the small temperature range used
A Brookfield Synchro-lectric viscometer was used to
measure the viscosity of both the light and heavy oil
Figures 18 and 19 show the effect of temperature on visshy
cosity In addition the viscosity of the light oil was
checke d using the falling ball method and the equation
D2--ltA (f s bull fl) g (34) l 8v
The viscometer was calibrated by the National Bureau of bull
Standards and was accurate to l tb
Velocity Measurements
The velocity of movement through the oil was measured
by determining the rate of rotation of the pulleys with a
stop watch Usually the time for 10 revolutions was
measured at the highe r ve locities and for 5 revolutions at
the low velocities From this information and the di
amaters of the pulleys the velocities ere calculated
36
The time was measured to the nearest tenth of a second
Since the measured time was usually between 20 and 40
aeconds 1 the error in ~easuring velocity was considered to
be less tha~ 0 5~
force Measurements
The object connected to the scale 1 was dropped to the
bottom of the oil drum The motor was started and the scale
was read as the object vms being pulled towards the top of
the drum Two or three readings were taken for each object
at each velocity In nearly all cases these readings were
the same
37
ti XPER I MENTAL RE STJLTS
The dra g coefficient and the Reynolds number were
calculated by the use of Equations (l or (15) for each of
the spheres cylinders and plates from the measured
quantities of force and velocity a~d the values of the vis shy
cosity and density corresponding to the temperature of the
oil It was necessary to ~ubtract from the measured force
the force on the wire The corrected force measurement was
then used to determine the drag coefficient The force on
the wire has been determined as being proportional to the
velocity A correction curve relating force on the wire
and ve l ocity is plo tted in Figure 9 for the li ght oil and
Fi gure 10 for the heavy oil
The calculated drag coefficients Reynolds numbers
and velocities along with the measured force for the spheres
cylinders flat plates - parallel flow and flat plates shy
perpendicular flow have been tabulated in Tables II III
I V and v respectively
The calculated drag coefficients have been plotted as
a function of the Reynolds number on logarithic graph paper
with geometric ratios as a parameter
Drag coefficients for the spheres are plo tted in
Figure 11 The data for the cylinders are plotted in
CD_ bull 0 G 0
03
Tshy02
01
10 20 30 410 50 60 70 80
VELOCITY- FTJSEC
DRAG FORCE ON THE WIRE-LIGHT OIL
FIGURE 9
I -shy I -middot -- -shy -1shy _i-i I --~ I I _ -middot- shy I i
_I_ - _ middot- LL I l l tmiddot - middot1middot ~- - - - -+i middotshy I - --+-cl - l
1 1 I I IV jc---- --r--middotmiddottmiddot r-middotmiddot--tmiddotmiddot---shy _____ _L __ --~- --1shy middotmiddotr-r-middott- 1 -f-f-T- _~ +-L--1---~- 1--l
~- - shy I-+---Rmiddot-- I I I l i ~~ i -~~ ~- -T f i rshy ~-- --shy i- ----~-- shy - middot1 shy
I i I i I I 1--- -middot - fshy middot i----1---+-shy - i-middot -~+-- --~- --~-- ---- -t+ I v-~~ -middot j
i I middot 1_ _ I tmiddot---+-+1-+--li~+middot -+--+-+-1-+-+-+-+--tc--1-+-t-11-shy - middot --t- 1---t- t----tmiddotshy --~-- -middot i-shy I 1i - ~ i I i v i middotmiddotmiddot
[~v +L~ + ~ - I~~j-+ r V I ~t--- -~-- I +---~-- I f-middot ---1-- ~ -- --- ) Li --+--+--+-+-+-+--1--+--+---t---4 -1--1--+-+--+-l-i
tl~ I I Q Y +l~~ii-+-++++-middotHH-++-+-+-+--H--++ -i t Imiddot i i 1 j _V I f1 r-t~-middot l--r-tshy -~ 7 middot 1 -shy middot middotmiddot I
DRAG FORCE ON THE WIRE- HEAVY OIL
FIGURE 10
40
+shy l i~ltgt ~ bull r-rshy I i t _l
1 lf-1-1 l+r+ fJ-Ct I+ t li 1~t rtH r+l rf-l It llil I I
l l~pound 11 1 ~middot ~~middott ~ It lqf L
t I+--= ~r 17 -Er I _ ~ _pound~- sect Imiddot I+
iU=ff=t 1 +~ t_ - ~ r 111= t h=
I middot
t= IE I 1 1
plusmn~ kplusmni - -STOKE S EQ
(~ l h+middot
ru HmiddotHti+H1 11
c lffii l t~ 4 ~ ~middot ~ff l ~ ~h i ltlri
1 yen~ middot I ~ I I T ~ gt l+t H+h l+ i j l tfl-l Imiddotmiddot ft+ ++ l f+ Imiddotmiddot I+ I+ middott bulli I 1middot1 I ftt-1shy middot I middot r 11 I IH Ij ~ ~ middotishy J F 1= 6= ~
=f l~iit rtti l lit~ I FS lf~ l=i-+
l-11ffi tt lr 1 ~1 -t =l=Rttl 1ft i- 1 ~ I+ I
~~ lflJ
t I lfl m ~~WFB Lt
41plusmn811 IF I Hir tt ft itttplusmn i I~
1-+++middot
I ~ I (~ ffitrHf1 Ittmiddot ~ l r i H-t-r r HHt m 11 H++ I
bull I I
1_ _ F bullmiddot Imiddotmiddot t-- 1-T h iT
f-t+ ftt I+ I lt + T Imiddot 1
1t _plusmn middot~~ ~- 11shy
=a~ 1~ - =itf lttti
H I
=
DATA FOR SPHERES
FIGURE II
41
I -1---1-1-+--+--Ti-+-------+----r--shy --r--- -shy + t----+shy ----4-~---+-f----f--+-f--l--1 I t--shy --t-- ---+-shy
J-+-~f--~~ -___l_ ~---
i 1 L~L~-~tr-l----H~4-----~-f------+------+-----+----+---+middot-t-middot-H5000
middot~ ~ m- ~ - ~t plusmn~ 3t i t~ -f--- bullbull - ~~ h middot-
01 0~ 10
Re
-
DATA FOR CYLINDERS - LD = 6 8 AND 12
FIGURE I 4
44
Figures 12 13 and 14 The data for LD values of 16 24
and 32 were nearly the same and have been plotted to gether
i n Figure 12 In addition the curves for the other LD
ratios determined fro m Fib~res 13 and 14 have been drawn
in Figure 12 so that the effect of the length-to-diameter
is clearly shown Figure 13 shows the data for LD values
of 2 and 4 and the curves determined from this data
Firure 14 shows the data for LD values of 6 8 and 12
and the curves determined from this data
The data for flat plates in parallel flow are plotted
in Fi gure 15 A correction factor for the edge effect has
beon used so that the width-to-length ratio is not a
parameter in this plot A portion of the data of Janour
(5 p 31) is also shown in the diagram
The data for fla t plates in perpendicular flow is
plotted in Figures 16 a nd 17 Figure 16 shows the data for
WL values of 2 Also the curves for the three WL ratios
1 2 and 4 have been drawn in the fi gure Figure 17 shows
the data for WL values of 1 and 4 The curves determined
from the data have also been dravm in the figure
45
10~ ~ ~--- -shy
t==Ff1TR=+ iJ+--_-_--r_-_---+-+---+--+-+--_---_-~r-=r~=~+--=---=---=---=--~=--=_~1=_--=_~_-middot~~--+-+-t~ 1 Ll~+--+-- ---jtshyl~t L--+ I
I
P------ _l -- --1---L i
20 ~-- I ~g I --- - ---+-- r t L_shy
~ ~B 1) I --o-o- JONES - () - - ~~ p f---j- -~-- e e JANOU R
c gt ~c ~ ------ JANSSEN I 0 0 ~ I
IO ~2=i~~~~~~a=~~f=j= ---- TOM OTIKA bulll= I
~~n ~~--~~~~~~o~~~~~--4- NDCIgttl o shy
-
~--~~~~~+--+~+--4-r-~1+-~-middot+1~ ~ --H--~-~~os I i i i-4 ---~T I I f-- t --- li-------~--+-_--+--t-----~~-~_+---_-_-_--+------+-+-__+-[- +_- ___ _______ __+---+-r-+--H----_+--r--------+shy
it I+ ~ bull t ~1 ri j t++t+t++tft bullm H--~+H-t+t-++H-f+t+~HtttH t bull~H-IrttI-H
iH-H u nH m
I
t H+t-~ 1-r f-tj
i it iT -t middotHt I I I I Ill
~middot __
r middotshy
i I r-
f H- jLj f r H rr t~
II
t f f-l -t+tt ~ ==_ =~middot irE
I I
I
I
f
I --
i
t
1 r bull - r
~- ltt++l=tUtt~S-t+t+++~-++U +HJJm~-fl~HHtt1 tttn ll+t-Tt-~- ~ r fH T --r -1 t ---t- -tshy w _+ _ I-shy middotI
-shy -r- + Hbull Hshy t-I --r++ -t iHr -1 H-e-- -t I 1IT 1
1 H-rf-I IJftJ Jf+i+ ~ L
=+shy - tjshy rtmiddotshy ~ -
+ H 1-Jt I tt o =tt ~-
~1 l +fill l plusmn~ fplusmn -shy + I t-
DATA FOR FLAT PLATES PERPENDICULAR FLOW- WL= I 4
FIGURE 17
48
DI SCUSS ION OF RESULTS
Correction and Accuracy of Measurements
After a few pre liminary force measurements with the
spheres and a check with Stokes law (Equation 2) it was
apparent that the drag force on the wire was appreciable
and needed to be considered It was decided to take a
series of measurements with the spheres and calculate the
difference between the measured force and the force calcushy
lated from Stokes law The difference in force could then
be attributed to the drag on the wire If Stokes law is
followed the force on the wire should be proportional to
the velocity
A series of twenty measurements of the force on the
spheres was taken for each oil and the difference between
the measured force and that calcula ted by Stokes 1 law was
determined For each oil this difference as plo tted vs
the velocity The points grouped fairly ell around a
strai ght line nearly passing through the origin The
method of least squares was used to determine the equation
of the line best fitting the da t a The equa tion of the
line for the li bht oil tas found to be
Fe bullbull05605v - oooa (35)
which was determined at about 62 7degF Since the intercept
49
of the line is very close to zero it is believed that the
line is a good indication of the drag on the wire The
equation of the line for the heavy oil was found to be
F - 19llv I oo2o1 (36 ) c shy
which was determined at about 64 2deg The intercept of this
line is also quite close to zero These lines plotted in
Fi poundures 9 and 10 were used throughout the investigation
for the correction factor of the drag on the wires For
the cylinders and flat plates in parallel flow which were
pulled by two wires the values determined from Equations
35) and (36) were doubled For the plates in perpendicular
flow pulled by four wires the correction force was multishy
plied by four
The spring scale had 12 ounce divisions but could be
read to the nearest sixth of an ounce Some of the measureshy
ments of force were under an ounce hence a considerable
spread of the measurements was noticed in the pre liminary
data and throughout the experiment However sufficient
points were obtained so that it was possible to draw a
reliable curve through the data in all casas An analysis
was made to determine the average deviation from Stokes
equation for the spheres It raa found that the average
deviation was 15 1 for the light oil 16 6 for the heavy
oil and 15 9 overall The maximum deviation was 89
50
Inspection of the other data shows that these deviations
are also representative of the cylinders and flat plates
The force measurement is the least accurate part of the
experiment Other insignificant errors are introduced by
a small variation in the temperature This variation was
held to about 10 from the temperature of the calibrated
correction curve The velocity measurements and the
dimensions of the cylinders spheres and pl~ tes are conshy
sidered go od enough so tha t no appreciable errors occur
In order to e l iminate the WL parameter for flat plates
in parallel f l ow an additional factor for the effect of
the edges was subtracted from the measured force Janour
(5 p 27) presented the foll owing equation for the edge
correction for one edge of a flat plate in parallel flow
F ~ lv~ bull (37 ) edge gc
In present work this equation as doubled because both
edges of the plates were submerged in fluid It is assumed
in appl ying this correction that the lowe r limit of a
Reynolds number of 10 proposed by Janour can be extended
close to 0 1
Analysis of Results
Forty of the points for the spheres were used to get
51
the correction factor for the wires The remaining thirty
points are well erouped about Stokes law
The data for cylinders for LD ratios of 16 24 and
32 did not seem to be se gregated therefore these data
were plotted together It would seem that in the low range
of Reyno l ds numbers an LD of 16 and greater can be con shy
sidered an ~nfini tely long cylinder The other LD ratios
of 2 4 6 a 12 provided fairly distinct and separate
lines The best straight lines were drawn through the data
for each of the LD ratios It was evident that in eaeh
case a slope of -1 on a lo g-log graph gave the best straight
line which would indicate that the force varies directly
as the velocity It was possible to develop an empirical
expression relating dra g coefficient Reynolds number and
LD The following equation was obtained from the straight
line plots of Re vs fd for the various LD ratios
(38 )
Equation (38) applies for Reyno l ds numbers from 01 to 10
and for LD ratios of 2 to 16 For LD ratios greater
than 16
10 re (39 )
The data for flat plates in parallel flow is plotted
in Figure 15 after the correction factor for tho edge
52
effect was subtracted When the edge correction is made
no effect of WL ratio is indicated This result would be
expected The data followed a straight line with a slope
of -1 up to a Reynolds number of 2 After that a curve was
dravm connecting the line to that obtained by Janour The
equation for the straight section of the curve is
f - 6 (40)- Re
which applies for Reynolds numbers of 0 1 to 2 0 Here
a gain the force is proportional to the velocity Vfuen
determining drag force for flat plates in parallel flow
the force is first calculated from Equations (40) and (15 )
then the edge correction is added
The effect of the geometric ratios is clearly shown in
the data for flat plates in perpendicul ar flow which are
plotted in Figures 16 and 17 As with the other data the
best straight line was drawn through the various points
for eaoh of the WL ratios Again the line had a slope of
-1 The equation relating fd Re and wL was found t o be
rd 37 (w) -o 3o (41)Irel
which applies for Reynolds numbers of about 05 to 2 0 and
WL ratios of 1 to 4 It is possible but it has not been
proved that Equation (41) is suitable for higher WL ratios
The exponent on WL in Equation 41) is very close to that
53
on L D i n Equation ( 38 )~ It i s possible t ha t these
exponents are t he same but this cannot be sho~~ depound1nitely
until more accura te da ta are available It would be exshy
pected that a s the Reynolds number approaches zero t he
effect of geometric ratios would be the same for cylinders
and fla t pla tes in perpendicula r flow
It is seen in the t a bles of data that occasionally a
ne gative force was obtained because the correction applie d
due to t he wire dra g was greater than the mea sured force
These points obviously are incorrect This occurred only
for the smallest plates in the heavy oil at t he highest
velocities However these knom bad points occur in less
tha n 5~ of the data
It is clearl y shown that for cylinders and plates the
fd increases as L D or W L decreases This is in direct
contrast to Wiesel aberger s investigation However his
work is for hi gher Reynolds numbers at which a turbulent
wake forms bull
Comparison of Results with Other Data and Theoretical So l utions
The data for sphere~ a grees of course with Stokes
l aw since that law was used to determine the correction
factor for the wire Liebster (9 Pbull 548 ) has
54
substantiated Stokes equation
There are no experimental data with which to compare
the results of the cylinders Wieselsbergers minimum
Reynolds number of 4 is above the ran ge covered in the preshy
sent investigation The da ta for the highest LD ratios
(16 24 and 32) does agree almost exactly wi t h the solution
of Allen and Southwell (1 P bull 141) (LD =00) in the range
of Reynolds numbers from 0 1 to 1 0 Allen and Southwells
solution a greed with the data of Wieselsberger (16 p 22)
However the present data is above the theoretical solutions
of Lamb (8 p 112-121) throughout the range of Reynolds
numbers from 0 01 to 1 0 and above the solutions of
Bairstow Cave and Lang (2 p 404) I mai (4 p 157) and
Tomotika and Aoi (15 p 302) for Reynolds numbers of 0 1
to 1 0 Allen and Southwells solution a grees dth both
Wieselsberger 1 s a nd the present data Their solution and
the present data represent the best means for predicting
drag coefficients for flow over long cylinders for Reynolds
numbers of 0 01 to 10 It should be remembered that the
o t her solutions should a gree with eac h other since they
were all essentially derived by linearizing the Na viershy
Stokes equation
The data for flat plates in parallel flow is
55
considerably above the theoretical solutions of Janssen
(6 p 183 ) and Tomotika and Aoi (15 Pbull 302) However
Fi f~re 15 shows that a smooth transition occurs bet een
the present work and the data of Janour (5 P bull 31) The
present data considerably extend the experimental inforshy
mation previously available for laminar flow paral lel to
flat plates In the re gion of Reynol ds numbers less than
2 the drag coefficient is shown to be inversely proportional
to the Reynolds number Janours data covers a range of
Reynolds numbers from 11 to 1000 The results of the
present investigation line up with Janours results which
in turn on extrapolation to higher Reyno l ds numbers
(greater than 1000) make a smooth transition into Blasius
curve represented by Equation (10) At Reyno l ds numbers
greater than 20 000 the drag coefficient is inversely proshy
portional to the square root of the Reynolds number
The data for flat plates in perpendicular flow is conshy
siderably above the solutions of Tomotika and Aoi
(15 p 302) and Imai (4 p 157 However their solutions
f or cylinders and plates in parallel flow are also below
the present data Also it should be remembered that their
solutions are for infinitely wide plates If a value of
WL of above 100 is used in Equation (41) then the present
data and the solutions of Tomotika and Aoi are fairly close
56
The present results indicate that Equation (41~ can be
used with an accuracy of 15 to 20 within the limitations
of the equation (WL 1 to 4 Re = 0 05 to 2)
57
SUM RY AND CONCLUSIONS
Only a small amount of work has been done in the past
on the study of laminar flow over immersed bodies There
are many areas in the chemical process industries and the
field of aeronautics where this information would be very
helpful The purpose of the present investi gation wa s to
study the almost totally unexplored range of Reynol ds
numbers from 0 01 to 10
Drag coefficients have been determined for spheres
cylinders and flat plates in paralle l and perpendicular
flow The drag coefficients have been plotted as a
function of the Reynolds number with dimension ratios as
a parameter on lo g-log graphs The best straight lines
have been drawn through the data In all cases these lines
had a slope of -1 hich shows that the dra g coefficient is
inversely proportional to the Reynolds number at very low
Reynolds numbers for all shapes and dimension ratios The
following equations have been determined from the data
For cylinders
fd - 27 L -0 36 (38 ) - Re ())
which applies for Reynolds numbers of 0 01 to 1 and LD of
2 to 16 For LD greater than 16 the equation is
58
(39)
For flat plates in parallel flow a correction factor has
been applied to account for the edge effect The equation
which applies for Reyno l ds numbers of 0 1 to 2 is
f 6Re
(40)
For flat plates in perpendicular flow
f d
- 37 - Re (w) t -
0 bull 30 (41)
wbieh applies for W L of 1 to 4 and Reynolds numbers of
0 05 to 2
It is concluded tha t Equations (38-41) give the best
values of drag coefficients within an accuracy of 20~ for
the range of Reynolds numbers that were considered Also
it is evident that the dimension ratios are a n important
factor in determining the drag coefficient for a given
Reynolds number Furthermore the drag coefficient inshy
creases with decreasing values of L D or W L for a constant
Reynolds number The da ta obtained in this investi gation
compare favorably with the other experimental data and with
some of the theoretical sol utions It should be remembered
that when comparing the experimental data with theoretical
solutions that practically all of the solutions are for an
infinitely long cylinder or an infinitely wide plate
It is recommended tha t the present apparatus be
59
modified so that a force of 001 pound can be measured
Also it would improve tho accuracy to set up a constant
temperature bath so that the temperature of the oil can not
vary over 02degF A few check points on the present data
is all that is necessary to confirm the validity of
Equations (38- 41) It is also r ecommended that only SAE 140
oil be used and that 2 inches should be the minimum plate
width and cylinder length to be studi3d These conditions
would help to maintain the accuracy of the correction force
for the wire
60
~WMENCIATURE
Symbol Dimensions
A area sq ft
D diameter ft
F force lb f
L length ft
M mas s lb m Re Reynolds number Dvf= -ltr w width ft
a area sq ft
b characteristic length ft
d diameter ft
f drag coefficientfd
gravitation constant l b mft gc 2= 32 17 l b _ rsec
1 length ft
m mass l b bullm
p pressure lbrsqft
r radius ft
t time see
u velocity ft sec
v velocity ft sec
w width ft
61
Symbol Dimensions
X xbullcoordinate ft
y y- coordinate ft
o( vorticity
time sec
viscosity lb m ft -sec
kinematic viscosity ft 2sec
circumference diameter = 3 1416
3density lb m ft
function
stream function
Laplacian operator
infinity
Subscripts
c corrected
f force
1 l iquid
m mass
p projected
s solid
w wetted
62
BI BLIOGRAPHY
1 Allan D N de G and R v Southwell Re laxation methods applied to determine the motion in two di shymensions of a viscous fluid past a fixed cylinder Quarterly Journal of Mechanics and Applied Mathe shymatics 8 129-145 1955
2 Bairstow L B M Cave and E D Lang The reshysistance of a cylinder moving in a viscous fluid Philosophical Transactions of the Royal Society of London ser A 223383- 432 1923
3 Goldstein Sidney The steady flow of viscous fluid past a fixed spherical obstacle at small Reyno l ds numbers Proceedings of the Royal Society of London ser A 123225-235 1929
4 Imai I A new method of solving Oseens equations and its application to the flow past an inclined elliptic cylinder Proceedings of the Royal Society of London ser A 224 141-160 1954
5 Janour Zbynek Resistance of a plate in paralle l flow at low Reyno lds numbers Washington Nov 1951 40 p National Advisory Committee for Aeronautics Te chnica l Memorandum 1316)
6 Janssen E An analog solution of the Navier-Stokes equation for the case of flow past a f l at plate at low Reynolds numbers In 1956 Heat Transfer and Fluid Mechanics Institute (Preprints of Papers) p 173-183
7 Knudsen James G and Donal d L Katz Fluid Dynamics a nd Heat Transfer Ann Arbor University of Michigan 1953 243 p (Michi gan University Engineering Research Bulletin no 37)
8 La~b Horace On the uniform motion of a spherethrough a viscous fluid Philosophical Magazine and Journal of Science s~r 6 21112-121 1911
9 Liebster H Uben den widerstrand von kugeln Annalen Der Physik ser 4 82 541- 562 1 927
63
10 McAdams William H Heat transmission 3d ed New York McGraw- Hill 1954 532 p
11 Pai Shih- I Viscous f l ow theory I Laminar flow Princeton D Van Nostrand 1956 384 p
12 Prandtlbull Ludwi g Es sentials of fluid dynamics London Blackie amp Son 1954 452 p
13 Relf i F Discussion of the results of measure shyments of the resistance of wires with some additionshyal tests of the resistance of wires of small diame shyters In Technical report of the Advisory Committee for Aeronautics London) March 1914 p 47 - 51 (Report and memoranda no 102 )
14 Stokes George Gabriel Mathematical and physical papers Vol 3 Cambridge University Press 1922 413 p
15 Tomotika s and T Aoi The steady flow of a viscous fluid past an elliptic cylinder and a flat plate at smal l Reynolds numbers Quarterly Journal of Me chanics and Applie d Ma thematics 6 290- 312 1953
16 Wieselsbergo r c Versuche Ube r der luftwiderstand gerundeter und kant iger korper Er gebnisse der Aeroshydynamischen Versucbsansta l t Vol 2 G~ttingen 1923 80 p
FIGURE e- PHOTOGRAPH OF SPHERES CYLINDERS AND PLATES
34
holes were drilled so that each plate could be used for
two geometric ratios by changing the wires (See for
example plates la and lb in Table I
35
EXPERI MENTA L PROCEDURE
Viscosity and Density Calibration
A calibrated hydrometer measuring to the nearest
0002 was used to measure the density Table VI shows that
the effect of temperature on density is practically negli shy
gible in the small temperature range used
A Brookfield Synchro-lectric viscometer was used to
measure the viscosity of both the light and heavy oil
Figures 18 and 19 show the effect of temperature on visshy
cosity In addition the viscosity of the light oil was
checke d using the falling ball method and the equation
D2--ltA (f s bull fl) g (34) l 8v
The viscometer was calibrated by the National Bureau of bull
Standards and was accurate to l tb
Velocity Measurements
The velocity of movement through the oil was measured
by determining the rate of rotation of the pulleys with a
stop watch Usually the time for 10 revolutions was
measured at the highe r ve locities and for 5 revolutions at
the low velocities From this information and the di
amaters of the pulleys the velocities ere calculated
36
The time was measured to the nearest tenth of a second
Since the measured time was usually between 20 and 40
aeconds 1 the error in ~easuring velocity was considered to
be less tha~ 0 5~
force Measurements
The object connected to the scale 1 was dropped to the
bottom of the oil drum The motor was started and the scale
was read as the object vms being pulled towards the top of
the drum Two or three readings were taken for each object
at each velocity In nearly all cases these readings were
the same
37
ti XPER I MENTAL RE STJLTS
The dra g coefficient and the Reynolds number were
calculated by the use of Equations (l or (15) for each of
the spheres cylinders and plates from the measured
quantities of force and velocity a~d the values of the vis shy
cosity and density corresponding to the temperature of the
oil It was necessary to ~ubtract from the measured force
the force on the wire The corrected force measurement was
then used to determine the drag coefficient The force on
the wire has been determined as being proportional to the
velocity A correction curve relating force on the wire
and ve l ocity is plo tted in Figure 9 for the li ght oil and
Fi gure 10 for the heavy oil
The calculated drag coefficients Reynolds numbers
and velocities along with the measured force for the spheres
cylinders flat plates - parallel flow and flat plates shy
perpendicular flow have been tabulated in Tables II III
I V and v respectively
The calculated drag coefficients have been plotted as
a function of the Reynolds number on logarithic graph paper
with geometric ratios as a parameter
Drag coefficients for the spheres are plo tted in
Figure 11 The data for the cylinders are plotted in
CD_ bull 0 G 0
03
Tshy02
01
10 20 30 410 50 60 70 80
VELOCITY- FTJSEC
DRAG FORCE ON THE WIRE-LIGHT OIL
FIGURE 9
I -shy I -middot -- -shy -1shy _i-i I --~ I I _ -middot- shy I i
_I_ - _ middot- LL I l l tmiddot - middot1middot ~- - - - -+i middotshy I - --+-cl - l
1 1 I I IV jc---- --r--middotmiddottmiddot r-middotmiddot--tmiddotmiddot---shy _____ _L __ --~- --1shy middotmiddotr-r-middott- 1 -f-f-T- _~ +-L--1---~- 1--l
~- - shy I-+---Rmiddot-- I I I l i ~~ i -~~ ~- -T f i rshy ~-- --shy i- ----~-- shy - middot1 shy
I i I i I I 1--- -middot - fshy middot i----1---+-shy - i-middot -~+-- --~- --~-- ---- -t+ I v-~~ -middot j
i I middot 1_ _ I tmiddot---+-+1-+--li~+middot -+--+-+-1-+-+-+-+--tc--1-+-t-11-shy - middot --t- 1---t- t----tmiddotshy --~-- -middot i-shy I 1i - ~ i I i v i middotmiddotmiddot
[~v +L~ + ~ - I~~j-+ r V I ~t--- -~-- I +---~-- I f-middot ---1-- ~ -- --- ) Li --+--+--+-+-+-+--1--+--+---t---4 -1--1--+-+--+-l-i
tl~ I I Q Y +l~~ii-+-++++-middotHH-++-+-+-+--H--++ -i t Imiddot i i 1 j _V I f1 r-t~-middot l--r-tshy -~ 7 middot 1 -shy middot middotmiddot I
DRAG FORCE ON THE WIRE- HEAVY OIL
FIGURE 10
40
+shy l i~ltgt ~ bull r-rshy I i t _l
1 lf-1-1 l+r+ fJ-Ct I+ t li 1~t rtH r+l rf-l It llil I I
l l~pound 11 1 ~middot ~~middott ~ It lqf L
t I+--= ~r 17 -Er I _ ~ _pound~- sect Imiddot I+
iU=ff=t 1 +~ t_ - ~ r 111= t h=
I middot
t= IE I 1 1
plusmn~ kplusmni - -STOKE S EQ
(~ l h+middot
ru HmiddotHti+H1 11
c lffii l t~ 4 ~ ~middot ~ff l ~ ~h i ltlri
1 yen~ middot I ~ I I T ~ gt l+t H+h l+ i j l tfl-l Imiddotmiddot ft+ ++ l f+ Imiddotmiddot I+ I+ middott bulli I 1middot1 I ftt-1shy middot I middot r 11 I IH Ij ~ ~ middotishy J F 1= 6= ~
=f l~iit rtti l lit~ I FS lf~ l=i-+
l-11ffi tt lr 1 ~1 -t =l=Rttl 1ft i- 1 ~ I+ I
~~ lflJ
t I lfl m ~~WFB Lt
41plusmn811 IF I Hir tt ft itttplusmn i I~
1-+++middot
I ~ I (~ ffitrHf1 Ittmiddot ~ l r i H-t-r r HHt m 11 H++ I
bull I I
1_ _ F bullmiddot Imiddotmiddot t-- 1-T h iT
f-t+ ftt I+ I lt + T Imiddot 1
1t _plusmn middot~~ ~- 11shy
=a~ 1~ - =itf lttti
H I
=
DATA FOR SPHERES
FIGURE II
41
I -1---1-1-+--+--Ti-+-------+----r--shy --r--- -shy + t----+shy ----4-~---+-f----f--+-f--l--1 I t--shy --t-- ---+-shy
J-+-~f--~~ -___l_ ~---
i 1 L~L~-~tr-l----H~4-----~-f------+------+-----+----+---+middot-t-middot-H5000
middot~ ~ m- ~ - ~t plusmn~ 3t i t~ -f--- bullbull - ~~ h middot-
01 0~ 10
Re
-
DATA FOR CYLINDERS - LD = 6 8 AND 12
FIGURE I 4
44
Figures 12 13 and 14 The data for LD values of 16 24
and 32 were nearly the same and have been plotted to gether
i n Figure 12 In addition the curves for the other LD
ratios determined fro m Fib~res 13 and 14 have been drawn
in Figure 12 so that the effect of the length-to-diameter
is clearly shown Figure 13 shows the data for LD values
of 2 and 4 and the curves determined from this data
Firure 14 shows the data for LD values of 6 8 and 12
and the curves determined from this data
The data for flat plates in parallel flow are plotted
in Fi gure 15 A correction factor for the edge effect has
beon used so that the width-to-length ratio is not a
parameter in this plot A portion of the data of Janour
(5 p 31) is also shown in the diagram
The data for fla t plates in perpendicular flow is
plotted in Figures 16 a nd 17 Figure 16 shows the data for
WL values of 2 Also the curves for the three WL ratios
1 2 and 4 have been drawn in the fi gure Figure 17 shows
the data for WL values of 1 and 4 The curves determined
from the data have also been dravm in the figure
45
10~ ~ ~--- -shy
t==Ff1TR=+ iJ+--_-_--r_-_---+-+---+--+-+--_---_-~r-=r~=~+--=---=---=---=--~=--=_~1=_--=_~_-middot~~--+-+-t~ 1 Ll~+--+-- ---jtshyl~t L--+ I
I
P------ _l -- --1---L i
20 ~-- I ~g I --- - ---+-- r t L_shy
~ ~B 1) I --o-o- JONES - () - - ~~ p f---j- -~-- e e JANOU R
c gt ~c ~ ------ JANSSEN I 0 0 ~ I
IO ~2=i~~~~~~a=~~f=j= ---- TOM OTIKA bulll= I
~~n ~~--~~~~~~o~~~~~--4- NDCIgttl o shy
-
~--~~~~~+--+~+--4-r-~1+-~-middot+1~ ~ --H--~-~~os I i i i-4 ---~T I I f-- t --- li-------~--+-_--+--t-----~~-~_+---_-_-_--+------+-+-__+-[- +_- ___ _______ __+---+-r-+--H----_+--r--------+shy
it I+ ~ bull t ~1 ri j t++t+t++tft bullm H--~+H-t+t-++H-f+t+~HtttH t bull~H-IrttI-H
iH-H u nH m
I
t H+t-~ 1-r f-tj
i it iT -t middotHt I I I I Ill
~middot __
r middotshy
i I r-
f H- jLj f r H rr t~
II
t f f-l -t+tt ~ ==_ =~middot irE
I I
I
I
f
I --
i
t
1 r bull - r
~- ltt++l=tUtt~S-t+t+++~-++U +HJJm~-fl~HHtt1 tttn ll+t-Tt-~- ~ r fH T --r -1 t ---t- -tshy w _+ _ I-shy middotI
-shy -r- + Hbull Hshy t-I --r++ -t iHr -1 H-e-- -t I 1IT 1
1 H-rf-I IJftJ Jf+i+ ~ L
=+shy - tjshy rtmiddotshy ~ -
+ H 1-Jt I tt o =tt ~-
~1 l +fill l plusmn~ fplusmn -shy + I t-
DATA FOR FLAT PLATES PERPENDICULAR FLOW- WL= I 4
FIGURE 17
48
DI SCUSS ION OF RESULTS
Correction and Accuracy of Measurements
After a few pre liminary force measurements with the
spheres and a check with Stokes law (Equation 2) it was
apparent that the drag force on the wire was appreciable
and needed to be considered It was decided to take a
series of measurements with the spheres and calculate the
difference between the measured force and the force calcushy
lated from Stokes law The difference in force could then
be attributed to the drag on the wire If Stokes law is
followed the force on the wire should be proportional to
the velocity
A series of twenty measurements of the force on the
spheres was taken for each oil and the difference between
the measured force and that calcula ted by Stokes 1 law was
determined For each oil this difference as plo tted vs
the velocity The points grouped fairly ell around a
strai ght line nearly passing through the origin The
method of least squares was used to determine the equation
of the line best fitting the da t a The equa tion of the
line for the li bht oil tas found to be
Fe bullbull05605v - oooa (35)
which was determined at about 62 7degF Since the intercept
49
of the line is very close to zero it is believed that the
line is a good indication of the drag on the wire The
equation of the line for the heavy oil was found to be
F - 19llv I oo2o1 (36 ) c shy
which was determined at about 64 2deg The intercept of this
line is also quite close to zero These lines plotted in
Fi poundures 9 and 10 were used throughout the investigation
for the correction factor of the drag on the wires For
the cylinders and flat plates in parallel flow which were
pulled by two wires the values determined from Equations
35) and (36) were doubled For the plates in perpendicular
flow pulled by four wires the correction force was multishy
plied by four
The spring scale had 12 ounce divisions but could be
read to the nearest sixth of an ounce Some of the measureshy
ments of force were under an ounce hence a considerable
spread of the measurements was noticed in the pre liminary
data and throughout the experiment However sufficient
points were obtained so that it was possible to draw a
reliable curve through the data in all casas An analysis
was made to determine the average deviation from Stokes
equation for the spheres It raa found that the average
deviation was 15 1 for the light oil 16 6 for the heavy
oil and 15 9 overall The maximum deviation was 89
50
Inspection of the other data shows that these deviations
are also representative of the cylinders and flat plates
The force measurement is the least accurate part of the
experiment Other insignificant errors are introduced by
a small variation in the temperature This variation was
held to about 10 from the temperature of the calibrated
correction curve The velocity measurements and the
dimensions of the cylinders spheres and pl~ tes are conshy
sidered go od enough so tha t no appreciable errors occur
In order to e l iminate the WL parameter for flat plates
in parallel f l ow an additional factor for the effect of
the edges was subtracted from the measured force Janour
(5 p 27) presented the foll owing equation for the edge
correction for one edge of a flat plate in parallel flow
F ~ lv~ bull (37 ) edge gc
In present work this equation as doubled because both
edges of the plates were submerged in fluid It is assumed
in appl ying this correction that the lowe r limit of a
Reynolds number of 10 proposed by Janour can be extended
close to 0 1
Analysis of Results
Forty of the points for the spheres were used to get
51
the correction factor for the wires The remaining thirty
points are well erouped about Stokes law
The data for cylinders for LD ratios of 16 24 and
32 did not seem to be se gregated therefore these data
were plotted together It would seem that in the low range
of Reyno l ds numbers an LD of 16 and greater can be con shy
sidered an ~nfini tely long cylinder The other LD ratios
of 2 4 6 a 12 provided fairly distinct and separate
lines The best straight lines were drawn through the data
for each of the LD ratios It was evident that in eaeh
case a slope of -1 on a lo g-log graph gave the best straight
line which would indicate that the force varies directly
as the velocity It was possible to develop an empirical
expression relating dra g coefficient Reynolds number and
LD The following equation was obtained from the straight
line plots of Re vs fd for the various LD ratios
(38 )
Equation (38) applies for Reyno l ds numbers from 01 to 10
and for LD ratios of 2 to 16 For LD ratios greater
than 16
10 re (39 )
The data for flat plates in parallel flow is plotted
in Figure 15 after the correction factor for tho edge
52
effect was subtracted When the edge correction is made
no effect of WL ratio is indicated This result would be
expected The data followed a straight line with a slope
of -1 up to a Reynolds number of 2 After that a curve was
dravm connecting the line to that obtained by Janour The
equation for the straight section of the curve is
f - 6 (40)- Re
which applies for Reynolds numbers of 0 1 to 2 0 Here
a gain the force is proportional to the velocity Vfuen
determining drag force for flat plates in parallel flow
the force is first calculated from Equations (40) and (15 )
then the edge correction is added
The effect of the geometric ratios is clearly shown in
the data for flat plates in perpendicul ar flow which are
plotted in Figures 16 and 17 As with the other data the
best straight line was drawn through the various points
for eaoh of the WL ratios Again the line had a slope of
-1 The equation relating fd Re and wL was found t o be
rd 37 (w) -o 3o (41)Irel
which applies for Reynolds numbers of about 05 to 2 0 and
WL ratios of 1 to 4 It is possible but it has not been
proved that Equation (41) is suitable for higher WL ratios
The exponent on WL in Equation 41) is very close to that
53
on L D i n Equation ( 38 )~ It i s possible t ha t these
exponents are t he same but this cannot be sho~~ depound1nitely
until more accura te da ta are available It would be exshy
pected that a s the Reynolds number approaches zero t he
effect of geometric ratios would be the same for cylinders
and fla t pla tes in perpendicula r flow
It is seen in the t a bles of data that occasionally a
ne gative force was obtained because the correction applie d
due to t he wire dra g was greater than the mea sured force
These points obviously are incorrect This occurred only
for the smallest plates in the heavy oil at t he highest
velocities However these knom bad points occur in less
tha n 5~ of the data
It is clearl y shown that for cylinders and plates the
fd increases as L D or W L decreases This is in direct
contrast to Wiesel aberger s investigation However his
work is for hi gher Reynolds numbers at which a turbulent
wake forms bull
Comparison of Results with Other Data and Theoretical So l utions
The data for sphere~ a grees of course with Stokes
l aw since that law was used to determine the correction
factor for the wire Liebster (9 Pbull 548 ) has
54
substantiated Stokes equation
There are no experimental data with which to compare
the results of the cylinders Wieselsbergers minimum
Reynolds number of 4 is above the ran ge covered in the preshy
sent investigation The da ta for the highest LD ratios
(16 24 and 32) does agree almost exactly wi t h the solution
of Allen and Southwell (1 P bull 141) (LD =00) in the range
of Reynolds numbers from 0 1 to 1 0 Allen and Southwells
solution a greed with the data of Wieselsberger (16 p 22)
However the present data is above the theoretical solutions
of Lamb (8 p 112-121) throughout the range of Reynolds
numbers from 0 01 to 1 0 and above the solutions of
Bairstow Cave and Lang (2 p 404) I mai (4 p 157) and
Tomotika and Aoi (15 p 302) for Reynolds numbers of 0 1
to 1 0 Allen and Southwells solution a grees dth both
Wieselsberger 1 s a nd the present data Their solution and
the present data represent the best means for predicting
drag coefficients for flow over long cylinders for Reynolds
numbers of 0 01 to 10 It should be remembered that the
o t her solutions should a gree with eac h other since they
were all essentially derived by linearizing the Na viershy
Stokes equation
The data for flat plates in parallel flow is
55
considerably above the theoretical solutions of Janssen
(6 p 183 ) and Tomotika and Aoi (15 Pbull 302) However
Fi f~re 15 shows that a smooth transition occurs bet een
the present work and the data of Janour (5 P bull 31) The
present data considerably extend the experimental inforshy
mation previously available for laminar flow paral lel to
flat plates In the re gion of Reynol ds numbers less than
2 the drag coefficient is shown to be inversely proportional
to the Reynolds number Janours data covers a range of
Reynolds numbers from 11 to 1000 The results of the
present investigation line up with Janours results which
in turn on extrapolation to higher Reyno l ds numbers
(greater than 1000) make a smooth transition into Blasius
curve represented by Equation (10) At Reyno l ds numbers
greater than 20 000 the drag coefficient is inversely proshy
portional to the square root of the Reynolds number
The data for flat plates in perpendicular flow is conshy
siderably above the solutions of Tomotika and Aoi
(15 p 302) and Imai (4 p 157 However their solutions
f or cylinders and plates in parallel flow are also below
the present data Also it should be remembered that their
solutions are for infinitely wide plates If a value of
WL of above 100 is used in Equation (41) then the present
data and the solutions of Tomotika and Aoi are fairly close
56
The present results indicate that Equation (41~ can be
used with an accuracy of 15 to 20 within the limitations
of the equation (WL 1 to 4 Re = 0 05 to 2)
57
SUM RY AND CONCLUSIONS
Only a small amount of work has been done in the past
on the study of laminar flow over immersed bodies There
are many areas in the chemical process industries and the
field of aeronautics where this information would be very
helpful The purpose of the present investi gation wa s to
study the almost totally unexplored range of Reynol ds
numbers from 0 01 to 10
Drag coefficients have been determined for spheres
cylinders and flat plates in paralle l and perpendicular
flow The drag coefficients have been plotted as a
function of the Reynolds number with dimension ratios as
a parameter on lo g-log graphs The best straight lines
have been drawn through the data In all cases these lines
had a slope of -1 hich shows that the dra g coefficient is
inversely proportional to the Reynolds number at very low
Reynolds numbers for all shapes and dimension ratios The
following equations have been determined from the data
For cylinders
fd - 27 L -0 36 (38 ) - Re ())
which applies for Reynolds numbers of 0 01 to 1 and LD of
2 to 16 For LD greater than 16 the equation is
58
(39)
For flat plates in parallel flow a correction factor has
been applied to account for the edge effect The equation
which applies for Reyno l ds numbers of 0 1 to 2 is
f 6Re
(40)
For flat plates in perpendicular flow
f d
- 37 - Re (w) t -
0 bull 30 (41)
wbieh applies for W L of 1 to 4 and Reynolds numbers of
0 05 to 2
It is concluded tha t Equations (38-41) give the best
values of drag coefficients within an accuracy of 20~ for
the range of Reynolds numbers that were considered Also
it is evident that the dimension ratios are a n important
factor in determining the drag coefficient for a given
Reynolds number Furthermore the drag coefficient inshy
creases with decreasing values of L D or W L for a constant
Reynolds number The da ta obtained in this investi gation
compare favorably with the other experimental data and with
some of the theoretical sol utions It should be remembered
that when comparing the experimental data with theoretical
solutions that practically all of the solutions are for an
infinitely long cylinder or an infinitely wide plate
It is recommended tha t the present apparatus be
59
modified so that a force of 001 pound can be measured
Also it would improve tho accuracy to set up a constant
temperature bath so that the temperature of the oil can not
vary over 02degF A few check points on the present data
is all that is necessary to confirm the validity of
Equations (38- 41) It is also r ecommended that only SAE 140
oil be used and that 2 inches should be the minimum plate
width and cylinder length to be studi3d These conditions
would help to maintain the accuracy of the correction force
for the wire
60
~WMENCIATURE
Symbol Dimensions
A area sq ft
D diameter ft
F force lb f
L length ft
M mas s lb m Re Reynolds number Dvf= -ltr w width ft
a area sq ft
b characteristic length ft
d diameter ft
f drag coefficientfd
gravitation constant l b mft gc 2= 32 17 l b _ rsec
1 length ft
m mass l b bullm
p pressure lbrsqft
r radius ft
t time see
u velocity ft sec
v velocity ft sec
w width ft
61
Symbol Dimensions
X xbullcoordinate ft
y y- coordinate ft
o( vorticity
time sec
viscosity lb m ft -sec
kinematic viscosity ft 2sec
circumference diameter = 3 1416
3density lb m ft
function
stream function
Laplacian operator
infinity
Subscripts
c corrected
f force
1 l iquid
m mass
p projected
s solid
w wetted
62
BI BLIOGRAPHY
1 Allan D N de G and R v Southwell Re laxation methods applied to determine the motion in two di shymensions of a viscous fluid past a fixed cylinder Quarterly Journal of Mechanics and Applied Mathe shymatics 8 129-145 1955
2 Bairstow L B M Cave and E D Lang The reshysistance of a cylinder moving in a viscous fluid Philosophical Transactions of the Royal Society of London ser A 223383- 432 1923
3 Goldstein Sidney The steady flow of viscous fluid past a fixed spherical obstacle at small Reyno l ds numbers Proceedings of the Royal Society of London ser A 123225-235 1929
4 Imai I A new method of solving Oseens equations and its application to the flow past an inclined elliptic cylinder Proceedings of the Royal Society of London ser A 224 141-160 1954
5 Janour Zbynek Resistance of a plate in paralle l flow at low Reyno lds numbers Washington Nov 1951 40 p National Advisory Committee for Aeronautics Te chnica l Memorandum 1316)
6 Janssen E An analog solution of the Navier-Stokes equation for the case of flow past a f l at plate at low Reynolds numbers In 1956 Heat Transfer and Fluid Mechanics Institute (Preprints of Papers) p 173-183
7 Knudsen James G and Donal d L Katz Fluid Dynamics a nd Heat Transfer Ann Arbor University of Michigan 1953 243 p (Michi gan University Engineering Research Bulletin no 37)
8 La~b Horace On the uniform motion of a spherethrough a viscous fluid Philosophical Magazine and Journal of Science s~r 6 21112-121 1911
9 Liebster H Uben den widerstrand von kugeln Annalen Der Physik ser 4 82 541- 562 1 927
63
10 McAdams William H Heat transmission 3d ed New York McGraw- Hill 1954 532 p
11 Pai Shih- I Viscous f l ow theory I Laminar flow Princeton D Van Nostrand 1956 384 p
12 Prandtlbull Ludwi g Es sentials of fluid dynamics London Blackie amp Son 1954 452 p
13 Relf i F Discussion of the results of measure shyments of the resistance of wires with some additionshyal tests of the resistance of wires of small diame shyters In Technical report of the Advisory Committee for Aeronautics London) March 1914 p 47 - 51 (Report and memoranda no 102 )
14 Stokes George Gabriel Mathematical and physical papers Vol 3 Cambridge University Press 1922 413 p
15 Tomotika s and T Aoi The steady flow of a viscous fluid past an elliptic cylinder and a flat plate at smal l Reynolds numbers Quarterly Journal of Me chanics and Applie d Ma thematics 6 290- 312 1953
16 Wieselsbergo r c Versuche Ube r der luftwiderstand gerundeter und kant iger korper Er gebnisse der Aeroshydynamischen Versucbsansta l t Vol 2 G~ttingen 1923 80 p
FIGURE e- PHOTOGRAPH OF SPHERES CYLINDERS AND PLATES
34
holes were drilled so that each plate could be used for
two geometric ratios by changing the wires (See for
example plates la and lb in Table I
35
EXPERI MENTA L PROCEDURE
Viscosity and Density Calibration
A calibrated hydrometer measuring to the nearest
0002 was used to measure the density Table VI shows that
the effect of temperature on density is practically negli shy
gible in the small temperature range used
A Brookfield Synchro-lectric viscometer was used to
measure the viscosity of both the light and heavy oil
Figures 18 and 19 show the effect of temperature on visshy
cosity In addition the viscosity of the light oil was
checke d using the falling ball method and the equation
D2--ltA (f s bull fl) g (34) l 8v
The viscometer was calibrated by the National Bureau of bull
Standards and was accurate to l tb
Velocity Measurements
The velocity of movement through the oil was measured
by determining the rate of rotation of the pulleys with a
stop watch Usually the time for 10 revolutions was
measured at the highe r ve locities and for 5 revolutions at
the low velocities From this information and the di
amaters of the pulleys the velocities ere calculated
36
The time was measured to the nearest tenth of a second
Since the measured time was usually between 20 and 40
aeconds 1 the error in ~easuring velocity was considered to
be less tha~ 0 5~
force Measurements
The object connected to the scale 1 was dropped to the
bottom of the oil drum The motor was started and the scale
was read as the object vms being pulled towards the top of
the drum Two or three readings were taken for each object
at each velocity In nearly all cases these readings were
the same
37
ti XPER I MENTAL RE STJLTS
The dra g coefficient and the Reynolds number were
calculated by the use of Equations (l or (15) for each of
the spheres cylinders and plates from the measured
quantities of force and velocity a~d the values of the vis shy
cosity and density corresponding to the temperature of the
oil It was necessary to ~ubtract from the measured force
the force on the wire The corrected force measurement was
then used to determine the drag coefficient The force on
the wire has been determined as being proportional to the
velocity A correction curve relating force on the wire
and ve l ocity is plo tted in Figure 9 for the li ght oil and
Fi gure 10 for the heavy oil
The calculated drag coefficients Reynolds numbers
and velocities along with the measured force for the spheres
cylinders flat plates - parallel flow and flat plates shy
perpendicular flow have been tabulated in Tables II III
I V and v respectively
The calculated drag coefficients have been plotted as
a function of the Reynolds number on logarithic graph paper
with geometric ratios as a parameter
Drag coefficients for the spheres are plo tted in
Figure 11 The data for the cylinders are plotted in
CD_ bull 0 G 0
03
Tshy02
01
10 20 30 410 50 60 70 80
VELOCITY- FTJSEC
DRAG FORCE ON THE WIRE-LIGHT OIL
FIGURE 9
I -shy I -middot -- -shy -1shy _i-i I --~ I I _ -middot- shy I i
_I_ - _ middot- LL I l l tmiddot - middot1middot ~- - - - -+i middotshy I - --+-cl - l
1 1 I I IV jc---- --r--middotmiddottmiddot r-middotmiddot--tmiddotmiddot---shy _____ _L __ --~- --1shy middotmiddotr-r-middott- 1 -f-f-T- _~ +-L--1---~- 1--l
~- - shy I-+---Rmiddot-- I I I l i ~~ i -~~ ~- -T f i rshy ~-- --shy i- ----~-- shy - middot1 shy
I i I i I I 1--- -middot - fshy middot i----1---+-shy - i-middot -~+-- --~- --~-- ---- -t+ I v-~~ -middot j
i I middot 1_ _ I tmiddot---+-+1-+--li~+middot -+--+-+-1-+-+-+-+--tc--1-+-t-11-shy - middot --t- 1---t- t----tmiddotshy --~-- -middot i-shy I 1i - ~ i I i v i middotmiddotmiddot
[~v +L~ + ~ - I~~j-+ r V I ~t--- -~-- I +---~-- I f-middot ---1-- ~ -- --- ) Li --+--+--+-+-+-+--1--+--+---t---4 -1--1--+-+--+-l-i
tl~ I I Q Y +l~~ii-+-++++-middotHH-++-+-+-+--H--++ -i t Imiddot i i 1 j _V I f1 r-t~-middot l--r-tshy -~ 7 middot 1 -shy middot middotmiddot I
DRAG FORCE ON THE WIRE- HEAVY OIL
FIGURE 10
40
+shy l i~ltgt ~ bull r-rshy I i t _l
1 lf-1-1 l+r+ fJ-Ct I+ t li 1~t rtH r+l rf-l It llil I I
l l~pound 11 1 ~middot ~~middott ~ It lqf L
t I+--= ~r 17 -Er I _ ~ _pound~- sect Imiddot I+
iU=ff=t 1 +~ t_ - ~ r 111= t h=
I middot
t= IE I 1 1
plusmn~ kplusmni - -STOKE S EQ
(~ l h+middot
ru HmiddotHti+H1 11
c lffii l t~ 4 ~ ~middot ~ff l ~ ~h i ltlri
1 yen~ middot I ~ I I T ~ gt l+t H+h l+ i j l tfl-l Imiddotmiddot ft+ ++ l f+ Imiddotmiddot I+ I+ middott bulli I 1middot1 I ftt-1shy middot I middot r 11 I IH Ij ~ ~ middotishy J F 1= 6= ~
=f l~iit rtti l lit~ I FS lf~ l=i-+
l-11ffi tt lr 1 ~1 -t =l=Rttl 1ft i- 1 ~ I+ I
~~ lflJ
t I lfl m ~~WFB Lt
41plusmn811 IF I Hir tt ft itttplusmn i I~
1-+++middot
I ~ I (~ ffitrHf1 Ittmiddot ~ l r i H-t-r r HHt m 11 H++ I
bull I I
1_ _ F bullmiddot Imiddotmiddot t-- 1-T h iT
f-t+ ftt I+ I lt + T Imiddot 1
1t _plusmn middot~~ ~- 11shy
=a~ 1~ - =itf lttti
H I
=
DATA FOR SPHERES
FIGURE II
41
I -1---1-1-+--+--Ti-+-------+----r--shy --r--- -shy + t----+shy ----4-~---+-f----f--+-f--l--1 I t--shy --t-- ---+-shy
J-+-~f--~~ -___l_ ~---
i 1 L~L~-~tr-l----H~4-----~-f------+------+-----+----+---+middot-t-middot-H5000
middot~ ~ m- ~ - ~t plusmn~ 3t i t~ -f--- bullbull - ~~ h middot-
01 0~ 10
Re
-
DATA FOR CYLINDERS - LD = 6 8 AND 12
FIGURE I 4
44
Figures 12 13 and 14 The data for LD values of 16 24
and 32 were nearly the same and have been plotted to gether
i n Figure 12 In addition the curves for the other LD
ratios determined fro m Fib~res 13 and 14 have been drawn
in Figure 12 so that the effect of the length-to-diameter
is clearly shown Figure 13 shows the data for LD values
of 2 and 4 and the curves determined from this data
Firure 14 shows the data for LD values of 6 8 and 12
and the curves determined from this data
The data for flat plates in parallel flow are plotted
in Fi gure 15 A correction factor for the edge effect has
beon used so that the width-to-length ratio is not a
parameter in this plot A portion of the data of Janour
(5 p 31) is also shown in the diagram
The data for fla t plates in perpendicular flow is
plotted in Figures 16 a nd 17 Figure 16 shows the data for
WL values of 2 Also the curves for the three WL ratios
1 2 and 4 have been drawn in the fi gure Figure 17 shows
the data for WL values of 1 and 4 The curves determined
from the data have also been dravm in the figure
45
10~ ~ ~--- -shy
t==Ff1TR=+ iJ+--_-_--r_-_---+-+---+--+-+--_---_-~r-=r~=~+--=---=---=---=--~=--=_~1=_--=_~_-middot~~--+-+-t~ 1 Ll~+--+-- ---jtshyl~t L--+ I
I
P------ _l -- --1---L i
20 ~-- I ~g I --- - ---+-- r t L_shy
~ ~B 1) I --o-o- JONES - () - - ~~ p f---j- -~-- e e JANOU R
c gt ~c ~ ------ JANSSEN I 0 0 ~ I
IO ~2=i~~~~~~a=~~f=j= ---- TOM OTIKA bulll= I
~~n ~~--~~~~~~o~~~~~--4- NDCIgttl o shy
-
~--~~~~~+--+~+--4-r-~1+-~-middot+1~ ~ --H--~-~~os I i i i-4 ---~T I I f-- t --- li-------~--+-_--+--t-----~~-~_+---_-_-_--+------+-+-__+-[- +_- ___ _______ __+---+-r-+--H----_+--r--------+shy
it I+ ~ bull t ~1 ri j t++t+t++tft bullm H--~+H-t+t-++H-f+t+~HtttH t bull~H-IrttI-H
iH-H u nH m
I
t H+t-~ 1-r f-tj
i it iT -t middotHt I I I I Ill
~middot __
r middotshy
i I r-
f H- jLj f r H rr t~
II
t f f-l -t+tt ~ ==_ =~middot irE
I I
I
I
f
I --
i
t
1 r bull - r
~- ltt++l=tUtt~S-t+t+++~-++U +HJJm~-fl~HHtt1 tttn ll+t-Tt-~- ~ r fH T --r -1 t ---t- -tshy w _+ _ I-shy middotI
-shy -r- + Hbull Hshy t-I --r++ -t iHr -1 H-e-- -t I 1IT 1
1 H-rf-I IJftJ Jf+i+ ~ L
=+shy - tjshy rtmiddotshy ~ -
+ H 1-Jt I tt o =tt ~-
~1 l +fill l plusmn~ fplusmn -shy + I t-
DATA FOR FLAT PLATES PERPENDICULAR FLOW- WL= I 4
FIGURE 17
48
DI SCUSS ION OF RESULTS
Correction and Accuracy of Measurements
After a few pre liminary force measurements with the
spheres and a check with Stokes law (Equation 2) it was
apparent that the drag force on the wire was appreciable
and needed to be considered It was decided to take a
series of measurements with the spheres and calculate the
difference between the measured force and the force calcushy
lated from Stokes law The difference in force could then
be attributed to the drag on the wire If Stokes law is
followed the force on the wire should be proportional to
the velocity
A series of twenty measurements of the force on the
spheres was taken for each oil and the difference between
the measured force and that calcula ted by Stokes 1 law was
determined For each oil this difference as plo tted vs
the velocity The points grouped fairly ell around a
strai ght line nearly passing through the origin The
method of least squares was used to determine the equation
of the line best fitting the da t a The equa tion of the
line for the li bht oil tas found to be
Fe bullbull05605v - oooa (35)
which was determined at about 62 7degF Since the intercept
49
of the line is very close to zero it is believed that the
line is a good indication of the drag on the wire The
equation of the line for the heavy oil was found to be
F - 19llv I oo2o1 (36 ) c shy
which was determined at about 64 2deg The intercept of this
line is also quite close to zero These lines plotted in
Fi poundures 9 and 10 were used throughout the investigation
for the correction factor of the drag on the wires For
the cylinders and flat plates in parallel flow which were
pulled by two wires the values determined from Equations
35) and (36) were doubled For the plates in perpendicular
flow pulled by four wires the correction force was multishy
plied by four
The spring scale had 12 ounce divisions but could be
read to the nearest sixth of an ounce Some of the measureshy
ments of force were under an ounce hence a considerable
spread of the measurements was noticed in the pre liminary
data and throughout the experiment However sufficient
points were obtained so that it was possible to draw a
reliable curve through the data in all casas An analysis
was made to determine the average deviation from Stokes
equation for the spheres It raa found that the average
deviation was 15 1 for the light oil 16 6 for the heavy
oil and 15 9 overall The maximum deviation was 89
50
Inspection of the other data shows that these deviations
are also representative of the cylinders and flat plates
The force measurement is the least accurate part of the
experiment Other insignificant errors are introduced by
a small variation in the temperature This variation was
held to about 10 from the temperature of the calibrated
correction curve The velocity measurements and the
dimensions of the cylinders spheres and pl~ tes are conshy
sidered go od enough so tha t no appreciable errors occur
In order to e l iminate the WL parameter for flat plates
in parallel f l ow an additional factor for the effect of
the edges was subtracted from the measured force Janour
(5 p 27) presented the foll owing equation for the edge
correction for one edge of a flat plate in parallel flow
F ~ lv~ bull (37 ) edge gc
In present work this equation as doubled because both
edges of the plates were submerged in fluid It is assumed
in appl ying this correction that the lowe r limit of a
Reynolds number of 10 proposed by Janour can be extended
close to 0 1
Analysis of Results
Forty of the points for the spheres were used to get
51
the correction factor for the wires The remaining thirty
points are well erouped about Stokes law
The data for cylinders for LD ratios of 16 24 and
32 did not seem to be se gregated therefore these data
were plotted together It would seem that in the low range
of Reyno l ds numbers an LD of 16 and greater can be con shy
sidered an ~nfini tely long cylinder The other LD ratios
of 2 4 6 a 12 provided fairly distinct and separate
lines The best straight lines were drawn through the data
for each of the LD ratios It was evident that in eaeh
case a slope of -1 on a lo g-log graph gave the best straight
line which would indicate that the force varies directly
as the velocity It was possible to develop an empirical
expression relating dra g coefficient Reynolds number and
LD The following equation was obtained from the straight
line plots of Re vs fd for the various LD ratios
(38 )
Equation (38) applies for Reyno l ds numbers from 01 to 10
and for LD ratios of 2 to 16 For LD ratios greater
than 16
10 re (39 )
The data for flat plates in parallel flow is plotted
in Figure 15 after the correction factor for tho edge
52
effect was subtracted When the edge correction is made
no effect of WL ratio is indicated This result would be
expected The data followed a straight line with a slope
of -1 up to a Reynolds number of 2 After that a curve was
dravm connecting the line to that obtained by Janour The
equation for the straight section of the curve is
f - 6 (40)- Re
which applies for Reynolds numbers of 0 1 to 2 0 Here
a gain the force is proportional to the velocity Vfuen
determining drag force for flat plates in parallel flow
the force is first calculated from Equations (40) and (15 )
then the edge correction is added
The effect of the geometric ratios is clearly shown in
the data for flat plates in perpendicul ar flow which are
plotted in Figures 16 and 17 As with the other data the
best straight line was drawn through the various points
for eaoh of the WL ratios Again the line had a slope of
-1 The equation relating fd Re and wL was found t o be
rd 37 (w) -o 3o (41)Irel
which applies for Reynolds numbers of about 05 to 2 0 and
WL ratios of 1 to 4 It is possible but it has not been
proved that Equation (41) is suitable for higher WL ratios
The exponent on WL in Equation 41) is very close to that
53
on L D i n Equation ( 38 )~ It i s possible t ha t these
exponents are t he same but this cannot be sho~~ depound1nitely
until more accura te da ta are available It would be exshy
pected that a s the Reynolds number approaches zero t he
effect of geometric ratios would be the same for cylinders
and fla t pla tes in perpendicula r flow
It is seen in the t a bles of data that occasionally a
ne gative force was obtained because the correction applie d
due to t he wire dra g was greater than the mea sured force
These points obviously are incorrect This occurred only
for the smallest plates in the heavy oil at t he highest
velocities However these knom bad points occur in less
tha n 5~ of the data
It is clearl y shown that for cylinders and plates the
fd increases as L D or W L decreases This is in direct
contrast to Wiesel aberger s investigation However his
work is for hi gher Reynolds numbers at which a turbulent
wake forms bull
Comparison of Results with Other Data and Theoretical So l utions
The data for sphere~ a grees of course with Stokes
l aw since that law was used to determine the correction
factor for the wire Liebster (9 Pbull 548 ) has
54
substantiated Stokes equation
There are no experimental data with which to compare
the results of the cylinders Wieselsbergers minimum
Reynolds number of 4 is above the ran ge covered in the preshy
sent investigation The da ta for the highest LD ratios
(16 24 and 32) does agree almost exactly wi t h the solution
of Allen and Southwell (1 P bull 141) (LD =00) in the range
of Reynolds numbers from 0 1 to 1 0 Allen and Southwells
solution a greed with the data of Wieselsberger (16 p 22)
However the present data is above the theoretical solutions
of Lamb (8 p 112-121) throughout the range of Reynolds
numbers from 0 01 to 1 0 and above the solutions of
Bairstow Cave and Lang (2 p 404) I mai (4 p 157) and
Tomotika and Aoi (15 p 302) for Reynolds numbers of 0 1
to 1 0 Allen and Southwells solution a grees dth both
Wieselsberger 1 s a nd the present data Their solution and
the present data represent the best means for predicting
drag coefficients for flow over long cylinders for Reynolds
numbers of 0 01 to 10 It should be remembered that the
o t her solutions should a gree with eac h other since they
were all essentially derived by linearizing the Na viershy
Stokes equation
The data for flat plates in parallel flow is
55
considerably above the theoretical solutions of Janssen
(6 p 183 ) and Tomotika and Aoi (15 Pbull 302) However
Fi f~re 15 shows that a smooth transition occurs bet een
the present work and the data of Janour (5 P bull 31) The
present data considerably extend the experimental inforshy
mation previously available for laminar flow paral lel to
flat plates In the re gion of Reynol ds numbers less than
2 the drag coefficient is shown to be inversely proportional
to the Reynolds number Janours data covers a range of
Reynolds numbers from 11 to 1000 The results of the
present investigation line up with Janours results which
in turn on extrapolation to higher Reyno l ds numbers
(greater than 1000) make a smooth transition into Blasius
curve represented by Equation (10) At Reyno l ds numbers
greater than 20 000 the drag coefficient is inversely proshy
portional to the square root of the Reynolds number
The data for flat plates in perpendicular flow is conshy
siderably above the solutions of Tomotika and Aoi
(15 p 302) and Imai (4 p 157 However their solutions
f or cylinders and plates in parallel flow are also below
the present data Also it should be remembered that their
solutions are for infinitely wide plates If a value of
WL of above 100 is used in Equation (41) then the present
data and the solutions of Tomotika and Aoi are fairly close
56
The present results indicate that Equation (41~ can be
used with an accuracy of 15 to 20 within the limitations
of the equation (WL 1 to 4 Re = 0 05 to 2)
57
SUM RY AND CONCLUSIONS
Only a small amount of work has been done in the past
on the study of laminar flow over immersed bodies There
are many areas in the chemical process industries and the
field of aeronautics where this information would be very
helpful The purpose of the present investi gation wa s to
study the almost totally unexplored range of Reynol ds
numbers from 0 01 to 10
Drag coefficients have been determined for spheres
cylinders and flat plates in paralle l and perpendicular
flow The drag coefficients have been plotted as a
function of the Reynolds number with dimension ratios as
a parameter on lo g-log graphs The best straight lines
have been drawn through the data In all cases these lines
had a slope of -1 hich shows that the dra g coefficient is
inversely proportional to the Reynolds number at very low
Reynolds numbers for all shapes and dimension ratios The
following equations have been determined from the data
For cylinders
fd - 27 L -0 36 (38 ) - Re ())
which applies for Reynolds numbers of 0 01 to 1 and LD of
2 to 16 For LD greater than 16 the equation is
58
(39)
For flat plates in parallel flow a correction factor has
been applied to account for the edge effect The equation
which applies for Reyno l ds numbers of 0 1 to 2 is
f 6Re
(40)
For flat plates in perpendicular flow
f d
- 37 - Re (w) t -
0 bull 30 (41)
wbieh applies for W L of 1 to 4 and Reynolds numbers of
0 05 to 2
It is concluded tha t Equations (38-41) give the best
values of drag coefficients within an accuracy of 20~ for
the range of Reynolds numbers that were considered Also
it is evident that the dimension ratios are a n important
factor in determining the drag coefficient for a given
Reynolds number Furthermore the drag coefficient inshy
creases with decreasing values of L D or W L for a constant
Reynolds number The da ta obtained in this investi gation
compare favorably with the other experimental data and with
some of the theoretical sol utions It should be remembered
that when comparing the experimental data with theoretical
solutions that practically all of the solutions are for an
infinitely long cylinder or an infinitely wide plate
It is recommended tha t the present apparatus be
59
modified so that a force of 001 pound can be measured
Also it would improve tho accuracy to set up a constant
temperature bath so that the temperature of the oil can not
vary over 02degF A few check points on the present data
is all that is necessary to confirm the validity of
Equations (38- 41) It is also r ecommended that only SAE 140
oil be used and that 2 inches should be the minimum plate
width and cylinder length to be studi3d These conditions
would help to maintain the accuracy of the correction force
for the wire
60
~WMENCIATURE
Symbol Dimensions
A area sq ft
D diameter ft
F force lb f
L length ft
M mas s lb m Re Reynolds number Dvf= -ltr w width ft
a area sq ft
b characteristic length ft
d diameter ft
f drag coefficientfd
gravitation constant l b mft gc 2= 32 17 l b _ rsec
1 length ft
m mass l b bullm
p pressure lbrsqft
r radius ft
t time see
u velocity ft sec
v velocity ft sec
w width ft
61
Symbol Dimensions
X xbullcoordinate ft
y y- coordinate ft
o( vorticity
time sec
viscosity lb m ft -sec
kinematic viscosity ft 2sec
circumference diameter = 3 1416
3density lb m ft
function
stream function
Laplacian operator
infinity
Subscripts
c corrected
f force
1 l iquid
m mass
p projected
s solid
w wetted
62
BI BLIOGRAPHY
1 Allan D N de G and R v Southwell Re laxation methods applied to determine the motion in two di shymensions of a viscous fluid past a fixed cylinder Quarterly Journal of Mechanics and Applied Mathe shymatics 8 129-145 1955
2 Bairstow L B M Cave and E D Lang The reshysistance of a cylinder moving in a viscous fluid Philosophical Transactions of the Royal Society of London ser A 223383- 432 1923
3 Goldstein Sidney The steady flow of viscous fluid past a fixed spherical obstacle at small Reyno l ds numbers Proceedings of the Royal Society of London ser A 123225-235 1929
4 Imai I A new method of solving Oseens equations and its application to the flow past an inclined elliptic cylinder Proceedings of the Royal Society of London ser A 224 141-160 1954
5 Janour Zbynek Resistance of a plate in paralle l flow at low Reyno lds numbers Washington Nov 1951 40 p National Advisory Committee for Aeronautics Te chnica l Memorandum 1316)
6 Janssen E An analog solution of the Navier-Stokes equation for the case of flow past a f l at plate at low Reynolds numbers In 1956 Heat Transfer and Fluid Mechanics Institute (Preprints of Papers) p 173-183
7 Knudsen James G and Donal d L Katz Fluid Dynamics a nd Heat Transfer Ann Arbor University of Michigan 1953 243 p (Michi gan University Engineering Research Bulletin no 37)
8 La~b Horace On the uniform motion of a spherethrough a viscous fluid Philosophical Magazine and Journal of Science s~r 6 21112-121 1911
9 Liebster H Uben den widerstrand von kugeln Annalen Der Physik ser 4 82 541- 562 1 927
63
10 McAdams William H Heat transmission 3d ed New York McGraw- Hill 1954 532 p
11 Pai Shih- I Viscous f l ow theory I Laminar flow Princeton D Van Nostrand 1956 384 p
12 Prandtlbull Ludwi g Es sentials of fluid dynamics London Blackie amp Son 1954 452 p
13 Relf i F Discussion of the results of measure shyments of the resistance of wires with some additionshyal tests of the resistance of wires of small diame shyters In Technical report of the Advisory Committee for Aeronautics London) March 1914 p 47 - 51 (Report and memoranda no 102 )
14 Stokes George Gabriel Mathematical and physical papers Vol 3 Cambridge University Press 1922 413 p
15 Tomotika s and T Aoi The steady flow of a viscous fluid past an elliptic cylinder and a flat plate at smal l Reynolds numbers Quarterly Journal of Me chanics and Applie d Ma thematics 6 290- 312 1953
16 Wieselsbergo r c Versuche Ube r der luftwiderstand gerundeter und kant iger korper Er gebnisse der Aeroshydynamischen Versucbsansta l t Vol 2 G~ttingen 1923 80 p
FIGURE e- PHOTOGRAPH OF SPHERES CYLINDERS AND PLATES
34
holes were drilled so that each plate could be used for
two geometric ratios by changing the wires (See for
example plates la and lb in Table I
35
EXPERI MENTA L PROCEDURE
Viscosity and Density Calibration
A calibrated hydrometer measuring to the nearest
0002 was used to measure the density Table VI shows that
the effect of temperature on density is practically negli shy
gible in the small temperature range used
A Brookfield Synchro-lectric viscometer was used to
measure the viscosity of both the light and heavy oil
Figures 18 and 19 show the effect of temperature on visshy
cosity In addition the viscosity of the light oil was
checke d using the falling ball method and the equation
D2--ltA (f s bull fl) g (34) l 8v
The viscometer was calibrated by the National Bureau of bull
Standards and was accurate to l tb
Velocity Measurements
The velocity of movement through the oil was measured
by determining the rate of rotation of the pulleys with a
stop watch Usually the time for 10 revolutions was
measured at the highe r ve locities and for 5 revolutions at
the low velocities From this information and the di
amaters of the pulleys the velocities ere calculated
36
The time was measured to the nearest tenth of a second
Since the measured time was usually between 20 and 40
aeconds 1 the error in ~easuring velocity was considered to
be less tha~ 0 5~
force Measurements
The object connected to the scale 1 was dropped to the
bottom of the oil drum The motor was started and the scale
was read as the object vms being pulled towards the top of
the drum Two or three readings were taken for each object
at each velocity In nearly all cases these readings were
the same
37
ti XPER I MENTAL RE STJLTS
The dra g coefficient and the Reynolds number were
calculated by the use of Equations (l or (15) for each of
the spheres cylinders and plates from the measured
quantities of force and velocity a~d the values of the vis shy
cosity and density corresponding to the temperature of the
oil It was necessary to ~ubtract from the measured force
the force on the wire The corrected force measurement was
then used to determine the drag coefficient The force on
the wire has been determined as being proportional to the
velocity A correction curve relating force on the wire
and ve l ocity is plo tted in Figure 9 for the li ght oil and
Fi gure 10 for the heavy oil
The calculated drag coefficients Reynolds numbers
and velocities along with the measured force for the spheres
cylinders flat plates - parallel flow and flat plates shy
perpendicular flow have been tabulated in Tables II III
I V and v respectively
The calculated drag coefficients have been plotted as
a function of the Reynolds number on logarithic graph paper
with geometric ratios as a parameter
Drag coefficients for the spheres are plo tted in
Figure 11 The data for the cylinders are plotted in
CD_ bull 0 G 0
03
Tshy02
01
10 20 30 410 50 60 70 80
VELOCITY- FTJSEC
DRAG FORCE ON THE WIRE-LIGHT OIL
FIGURE 9
I -shy I -middot -- -shy -1shy _i-i I --~ I I _ -middot- shy I i
_I_ - _ middot- LL I l l tmiddot - middot1middot ~- - - - -+i middotshy I - --+-cl - l
1 1 I I IV jc---- --r--middotmiddottmiddot r-middotmiddot--tmiddotmiddot---shy _____ _L __ --~- --1shy middotmiddotr-r-middott- 1 -f-f-T- _~ +-L--1---~- 1--l
~- - shy I-+---Rmiddot-- I I I l i ~~ i -~~ ~- -T f i rshy ~-- --shy i- ----~-- shy - middot1 shy
I i I i I I 1--- -middot - fshy middot i----1---+-shy - i-middot -~+-- --~- --~-- ---- -t+ I v-~~ -middot j
i I middot 1_ _ I tmiddot---+-+1-+--li~+middot -+--+-+-1-+-+-+-+--tc--1-+-t-11-shy - middot --t- 1---t- t----tmiddotshy --~-- -middot i-shy I 1i - ~ i I i v i middotmiddotmiddot
[~v +L~ + ~ - I~~j-+ r V I ~t--- -~-- I +---~-- I f-middot ---1-- ~ -- --- ) Li --+--+--+-+-+-+--1--+--+---t---4 -1--1--+-+--+-l-i
tl~ I I Q Y +l~~ii-+-++++-middotHH-++-+-+-+--H--++ -i t Imiddot i i 1 j _V I f1 r-t~-middot l--r-tshy -~ 7 middot 1 -shy middot middotmiddot I
DRAG FORCE ON THE WIRE- HEAVY OIL
FIGURE 10
40
+shy l i~ltgt ~ bull r-rshy I i t _l
1 lf-1-1 l+r+ fJ-Ct I+ t li 1~t rtH r+l rf-l It llil I I
l l~pound 11 1 ~middot ~~middott ~ It lqf L
t I+--= ~r 17 -Er I _ ~ _pound~- sect Imiddot I+
iU=ff=t 1 +~ t_ - ~ r 111= t h=
I middot
t= IE I 1 1
plusmn~ kplusmni - -STOKE S EQ
(~ l h+middot
ru HmiddotHti+H1 11
c lffii l t~ 4 ~ ~middot ~ff l ~ ~h i ltlri
1 yen~ middot I ~ I I T ~ gt l+t H+h l+ i j l tfl-l Imiddotmiddot ft+ ++ l f+ Imiddotmiddot I+ I+ middott bulli I 1middot1 I ftt-1shy middot I middot r 11 I IH Ij ~ ~ middotishy J F 1= 6= ~
=f l~iit rtti l lit~ I FS lf~ l=i-+
l-11ffi tt lr 1 ~1 -t =l=Rttl 1ft i- 1 ~ I+ I
~~ lflJ
t I lfl m ~~WFB Lt
41plusmn811 IF I Hir tt ft itttplusmn i I~
1-+++middot
I ~ I (~ ffitrHf1 Ittmiddot ~ l r i H-t-r r HHt m 11 H++ I
bull I I
1_ _ F bullmiddot Imiddotmiddot t-- 1-T h iT
f-t+ ftt I+ I lt + T Imiddot 1
1t _plusmn middot~~ ~- 11shy
=a~ 1~ - =itf lttti
H I
=
DATA FOR SPHERES
FIGURE II
41
I -1---1-1-+--+--Ti-+-------+----r--shy --r--- -shy + t----+shy ----4-~---+-f----f--+-f--l--1 I t--shy --t-- ---+-shy
J-+-~f--~~ -___l_ ~---
i 1 L~L~-~tr-l----H~4-----~-f------+------+-----+----+---+middot-t-middot-H5000
middot~ ~ m- ~ - ~t plusmn~ 3t i t~ -f--- bullbull - ~~ h middot-
01 0~ 10
Re
-
DATA FOR CYLINDERS - LD = 6 8 AND 12
FIGURE I 4
44
Figures 12 13 and 14 The data for LD values of 16 24
and 32 were nearly the same and have been plotted to gether
i n Figure 12 In addition the curves for the other LD
ratios determined fro m Fib~res 13 and 14 have been drawn
in Figure 12 so that the effect of the length-to-diameter
is clearly shown Figure 13 shows the data for LD values
of 2 and 4 and the curves determined from this data
Firure 14 shows the data for LD values of 6 8 and 12
and the curves determined from this data
The data for flat plates in parallel flow are plotted
in Fi gure 15 A correction factor for the edge effect has
beon used so that the width-to-length ratio is not a
parameter in this plot A portion of the data of Janour
(5 p 31) is also shown in the diagram
The data for fla t plates in perpendicular flow is
plotted in Figures 16 a nd 17 Figure 16 shows the data for
WL values of 2 Also the curves for the three WL ratios
1 2 and 4 have been drawn in the fi gure Figure 17 shows
the data for WL values of 1 and 4 The curves determined
from the data have also been dravm in the figure
45
10~ ~ ~--- -shy
t==Ff1TR=+ iJ+--_-_--r_-_---+-+---+--+-+--_---_-~r-=r~=~+--=---=---=---=--~=--=_~1=_--=_~_-middot~~--+-+-t~ 1 Ll~+--+-- ---jtshyl~t L--+ I
I
P------ _l -- --1---L i
20 ~-- I ~g I --- - ---+-- r t L_shy
~ ~B 1) I --o-o- JONES - () - - ~~ p f---j- -~-- e e JANOU R
c gt ~c ~ ------ JANSSEN I 0 0 ~ I
IO ~2=i~~~~~~a=~~f=j= ---- TOM OTIKA bulll= I
~~n ~~--~~~~~~o~~~~~--4- NDCIgttl o shy
-
~--~~~~~+--+~+--4-r-~1+-~-middot+1~ ~ --H--~-~~os I i i i-4 ---~T I I f-- t --- li-------~--+-_--+--t-----~~-~_+---_-_-_--+------+-+-__+-[- +_- ___ _______ __+---+-r-+--H----_+--r--------+shy
it I+ ~ bull t ~1 ri j t++t+t++tft bullm H--~+H-t+t-++H-f+t+~HtttH t bull~H-IrttI-H
iH-H u nH m
I
t H+t-~ 1-r f-tj
i it iT -t middotHt I I I I Ill
~middot __
r middotshy
i I r-
f H- jLj f r H rr t~
II
t f f-l -t+tt ~ ==_ =~middot irE
I I
I
I
f
I --
i
t
1 r bull - r
~- ltt++l=tUtt~S-t+t+++~-++U +HJJm~-fl~HHtt1 tttn ll+t-Tt-~- ~ r fH T --r -1 t ---t- -tshy w _+ _ I-shy middotI
-shy -r- + Hbull Hshy t-I --r++ -t iHr -1 H-e-- -t I 1IT 1
1 H-rf-I IJftJ Jf+i+ ~ L
=+shy - tjshy rtmiddotshy ~ -
+ H 1-Jt I tt o =tt ~-
~1 l +fill l plusmn~ fplusmn -shy + I t-
DATA FOR FLAT PLATES PERPENDICULAR FLOW- WL= I 4
FIGURE 17
48
DI SCUSS ION OF RESULTS
Correction and Accuracy of Measurements
After a few pre liminary force measurements with the
spheres and a check with Stokes law (Equation 2) it was
apparent that the drag force on the wire was appreciable
and needed to be considered It was decided to take a
series of measurements with the spheres and calculate the
difference between the measured force and the force calcushy
lated from Stokes law The difference in force could then
be attributed to the drag on the wire If Stokes law is
followed the force on the wire should be proportional to
the velocity
A series of twenty measurements of the force on the
spheres was taken for each oil and the difference between
the measured force and that calcula ted by Stokes 1 law was
determined For each oil this difference as plo tted vs
the velocity The points grouped fairly ell around a
strai ght line nearly passing through the origin The
method of least squares was used to determine the equation
of the line best fitting the da t a The equa tion of the
line for the li bht oil tas found to be
Fe bullbull05605v - oooa (35)
which was determined at about 62 7degF Since the intercept
49
of the line is very close to zero it is believed that the
line is a good indication of the drag on the wire The
equation of the line for the heavy oil was found to be
F - 19llv I oo2o1 (36 ) c shy
which was determined at about 64 2deg The intercept of this
line is also quite close to zero These lines plotted in
Fi poundures 9 and 10 were used throughout the investigation
for the correction factor of the drag on the wires For
the cylinders and flat plates in parallel flow which were
pulled by two wires the values determined from Equations
35) and (36) were doubled For the plates in perpendicular
flow pulled by four wires the correction force was multishy
plied by four
The spring scale had 12 ounce divisions but could be
read to the nearest sixth of an ounce Some of the measureshy
ments of force were under an ounce hence a considerable
spread of the measurements was noticed in the pre liminary
data and throughout the experiment However sufficient
points were obtained so that it was possible to draw a
reliable curve through the data in all casas An analysis
was made to determine the average deviation from Stokes
equation for the spheres It raa found that the average
deviation was 15 1 for the light oil 16 6 for the heavy
oil and 15 9 overall The maximum deviation was 89
50
Inspection of the other data shows that these deviations
are also representative of the cylinders and flat plates
The force measurement is the least accurate part of the
experiment Other insignificant errors are introduced by
a small variation in the temperature This variation was
held to about 10 from the temperature of the calibrated
correction curve The velocity measurements and the
dimensions of the cylinders spheres and pl~ tes are conshy
sidered go od enough so tha t no appreciable errors occur
In order to e l iminate the WL parameter for flat plates
in parallel f l ow an additional factor for the effect of
the edges was subtracted from the measured force Janour
(5 p 27) presented the foll owing equation for the edge
correction for one edge of a flat plate in parallel flow
F ~ lv~ bull (37 ) edge gc
In present work this equation as doubled because both
edges of the plates were submerged in fluid It is assumed
in appl ying this correction that the lowe r limit of a
Reynolds number of 10 proposed by Janour can be extended
close to 0 1
Analysis of Results
Forty of the points for the spheres were used to get
51
the correction factor for the wires The remaining thirty
points are well erouped about Stokes law
The data for cylinders for LD ratios of 16 24 and
32 did not seem to be se gregated therefore these data
were plotted together It would seem that in the low range
of Reyno l ds numbers an LD of 16 and greater can be con shy
sidered an ~nfini tely long cylinder The other LD ratios
of 2 4 6 a 12 provided fairly distinct and separate
lines The best straight lines were drawn through the data
for each of the LD ratios It was evident that in eaeh
case a slope of -1 on a lo g-log graph gave the best straight
line which would indicate that the force varies directly
as the velocity It was possible to develop an empirical
expression relating dra g coefficient Reynolds number and
LD The following equation was obtained from the straight
line plots of Re vs fd for the various LD ratios
(38 )
Equation (38) applies for Reyno l ds numbers from 01 to 10
and for LD ratios of 2 to 16 For LD ratios greater
than 16
10 re (39 )
The data for flat plates in parallel flow is plotted
in Figure 15 after the correction factor for tho edge
52
effect was subtracted When the edge correction is made
no effect of WL ratio is indicated This result would be
expected The data followed a straight line with a slope
of -1 up to a Reynolds number of 2 After that a curve was
dravm connecting the line to that obtained by Janour The
equation for the straight section of the curve is
f - 6 (40)- Re
which applies for Reynolds numbers of 0 1 to 2 0 Here
a gain the force is proportional to the velocity Vfuen
determining drag force for flat plates in parallel flow
the force is first calculated from Equations (40) and (15 )
then the edge correction is added
The effect of the geometric ratios is clearly shown in
the data for flat plates in perpendicul ar flow which are
plotted in Figures 16 and 17 As with the other data the
best straight line was drawn through the various points
for eaoh of the WL ratios Again the line had a slope of
-1 The equation relating fd Re and wL was found t o be
rd 37 (w) -o 3o (41)Irel
which applies for Reynolds numbers of about 05 to 2 0 and
WL ratios of 1 to 4 It is possible but it has not been
proved that Equation (41) is suitable for higher WL ratios
The exponent on WL in Equation 41) is very close to that
53
on L D i n Equation ( 38 )~ It i s possible t ha t these
exponents are t he same but this cannot be sho~~ depound1nitely
until more accura te da ta are available It would be exshy
pected that a s the Reynolds number approaches zero t he
effect of geometric ratios would be the same for cylinders
and fla t pla tes in perpendicula r flow
It is seen in the t a bles of data that occasionally a
ne gative force was obtained because the correction applie d
due to t he wire dra g was greater than the mea sured force
These points obviously are incorrect This occurred only
for the smallest plates in the heavy oil at t he highest
velocities However these knom bad points occur in less
tha n 5~ of the data
It is clearl y shown that for cylinders and plates the
fd increases as L D or W L decreases This is in direct
contrast to Wiesel aberger s investigation However his
work is for hi gher Reynolds numbers at which a turbulent
wake forms bull
Comparison of Results with Other Data and Theoretical So l utions
The data for sphere~ a grees of course with Stokes
l aw since that law was used to determine the correction
factor for the wire Liebster (9 Pbull 548 ) has
54
substantiated Stokes equation
There are no experimental data with which to compare
the results of the cylinders Wieselsbergers minimum
Reynolds number of 4 is above the ran ge covered in the preshy
sent investigation The da ta for the highest LD ratios
(16 24 and 32) does agree almost exactly wi t h the solution
of Allen and Southwell (1 P bull 141) (LD =00) in the range
of Reynolds numbers from 0 1 to 1 0 Allen and Southwells
solution a greed with the data of Wieselsberger (16 p 22)
However the present data is above the theoretical solutions
of Lamb (8 p 112-121) throughout the range of Reynolds
numbers from 0 01 to 1 0 and above the solutions of
Bairstow Cave and Lang (2 p 404) I mai (4 p 157) and
Tomotika and Aoi (15 p 302) for Reynolds numbers of 0 1
to 1 0 Allen and Southwells solution a grees dth both
Wieselsberger 1 s a nd the present data Their solution and
the present data represent the best means for predicting
drag coefficients for flow over long cylinders for Reynolds
numbers of 0 01 to 10 It should be remembered that the
o t her solutions should a gree with eac h other since they
were all essentially derived by linearizing the Na viershy
Stokes equation
The data for flat plates in parallel flow is
55
considerably above the theoretical solutions of Janssen
(6 p 183 ) and Tomotika and Aoi (15 Pbull 302) However
Fi f~re 15 shows that a smooth transition occurs bet een
the present work and the data of Janour (5 P bull 31) The
present data considerably extend the experimental inforshy
mation previously available for laminar flow paral lel to
flat plates In the re gion of Reynol ds numbers less than
2 the drag coefficient is shown to be inversely proportional
to the Reynolds number Janours data covers a range of
Reynolds numbers from 11 to 1000 The results of the
present investigation line up with Janours results which
in turn on extrapolation to higher Reyno l ds numbers
(greater than 1000) make a smooth transition into Blasius
curve represented by Equation (10) At Reyno l ds numbers
greater than 20 000 the drag coefficient is inversely proshy
portional to the square root of the Reynolds number
The data for flat plates in perpendicular flow is conshy
siderably above the solutions of Tomotika and Aoi
(15 p 302) and Imai (4 p 157 However their solutions
f or cylinders and plates in parallel flow are also below
the present data Also it should be remembered that their
solutions are for infinitely wide plates If a value of
WL of above 100 is used in Equation (41) then the present
data and the solutions of Tomotika and Aoi are fairly close
56
The present results indicate that Equation (41~ can be
used with an accuracy of 15 to 20 within the limitations
of the equation (WL 1 to 4 Re = 0 05 to 2)
57
SUM RY AND CONCLUSIONS
Only a small amount of work has been done in the past
on the study of laminar flow over immersed bodies There
are many areas in the chemical process industries and the
field of aeronautics where this information would be very
helpful The purpose of the present investi gation wa s to
study the almost totally unexplored range of Reynol ds
numbers from 0 01 to 10
Drag coefficients have been determined for spheres
cylinders and flat plates in paralle l and perpendicular
flow The drag coefficients have been plotted as a
function of the Reynolds number with dimension ratios as
a parameter on lo g-log graphs The best straight lines
have been drawn through the data In all cases these lines
had a slope of -1 hich shows that the dra g coefficient is
inversely proportional to the Reynolds number at very low
Reynolds numbers for all shapes and dimension ratios The
following equations have been determined from the data
For cylinders
fd - 27 L -0 36 (38 ) - Re ())
which applies for Reynolds numbers of 0 01 to 1 and LD of
2 to 16 For LD greater than 16 the equation is
58
(39)
For flat plates in parallel flow a correction factor has
been applied to account for the edge effect The equation
which applies for Reyno l ds numbers of 0 1 to 2 is
f 6Re
(40)
For flat plates in perpendicular flow
f d
- 37 - Re (w) t -
0 bull 30 (41)
wbieh applies for W L of 1 to 4 and Reynolds numbers of
0 05 to 2
It is concluded tha t Equations (38-41) give the best
values of drag coefficients within an accuracy of 20~ for
the range of Reynolds numbers that were considered Also
it is evident that the dimension ratios are a n important
factor in determining the drag coefficient for a given
Reynolds number Furthermore the drag coefficient inshy
creases with decreasing values of L D or W L for a constant
Reynolds number The da ta obtained in this investi gation
compare favorably with the other experimental data and with
some of the theoretical sol utions It should be remembered
that when comparing the experimental data with theoretical
solutions that practically all of the solutions are for an
infinitely long cylinder or an infinitely wide plate
It is recommended tha t the present apparatus be
59
modified so that a force of 001 pound can be measured
Also it would improve tho accuracy to set up a constant
temperature bath so that the temperature of the oil can not
vary over 02degF A few check points on the present data
is all that is necessary to confirm the validity of
Equations (38- 41) It is also r ecommended that only SAE 140
oil be used and that 2 inches should be the minimum plate
width and cylinder length to be studi3d These conditions
would help to maintain the accuracy of the correction force
for the wire
60
~WMENCIATURE
Symbol Dimensions
A area sq ft
D diameter ft
F force lb f
L length ft
M mas s lb m Re Reynolds number Dvf= -ltr w width ft
a area sq ft
b characteristic length ft
d diameter ft
f drag coefficientfd
gravitation constant l b mft gc 2= 32 17 l b _ rsec
1 length ft
m mass l b bullm
p pressure lbrsqft
r radius ft
t time see
u velocity ft sec
v velocity ft sec
w width ft
61
Symbol Dimensions
X xbullcoordinate ft
y y- coordinate ft
o( vorticity
time sec
viscosity lb m ft -sec
kinematic viscosity ft 2sec
circumference diameter = 3 1416
3density lb m ft
function
stream function
Laplacian operator
infinity
Subscripts
c corrected
f force
1 l iquid
m mass
p projected
s solid
w wetted
62
BI BLIOGRAPHY
1 Allan D N de G and R v Southwell Re laxation methods applied to determine the motion in two di shymensions of a viscous fluid past a fixed cylinder Quarterly Journal of Mechanics and Applied Mathe shymatics 8 129-145 1955
2 Bairstow L B M Cave and E D Lang The reshysistance of a cylinder moving in a viscous fluid Philosophical Transactions of the Royal Society of London ser A 223383- 432 1923
3 Goldstein Sidney The steady flow of viscous fluid past a fixed spherical obstacle at small Reyno l ds numbers Proceedings of the Royal Society of London ser A 123225-235 1929
4 Imai I A new method of solving Oseens equations and its application to the flow past an inclined elliptic cylinder Proceedings of the Royal Society of London ser A 224 141-160 1954
5 Janour Zbynek Resistance of a plate in paralle l flow at low Reyno lds numbers Washington Nov 1951 40 p National Advisory Committee for Aeronautics Te chnica l Memorandum 1316)
6 Janssen E An analog solution of the Navier-Stokes equation for the case of flow past a f l at plate at low Reynolds numbers In 1956 Heat Transfer and Fluid Mechanics Institute (Preprints of Papers) p 173-183
7 Knudsen James G and Donal d L Katz Fluid Dynamics a nd Heat Transfer Ann Arbor University of Michigan 1953 243 p (Michi gan University Engineering Research Bulletin no 37)
8 La~b Horace On the uniform motion of a spherethrough a viscous fluid Philosophical Magazine and Journal of Science s~r 6 21112-121 1911
9 Liebster H Uben den widerstrand von kugeln Annalen Der Physik ser 4 82 541- 562 1 927
63
10 McAdams William H Heat transmission 3d ed New York McGraw- Hill 1954 532 p
11 Pai Shih- I Viscous f l ow theory I Laminar flow Princeton D Van Nostrand 1956 384 p
12 Prandtlbull Ludwi g Es sentials of fluid dynamics London Blackie amp Son 1954 452 p
13 Relf i F Discussion of the results of measure shyments of the resistance of wires with some additionshyal tests of the resistance of wires of small diame shyters In Technical report of the Advisory Committee for Aeronautics London) March 1914 p 47 - 51 (Report and memoranda no 102 )
14 Stokes George Gabriel Mathematical and physical papers Vol 3 Cambridge University Press 1922 413 p
15 Tomotika s and T Aoi The steady flow of a viscous fluid past an elliptic cylinder and a flat plate at smal l Reynolds numbers Quarterly Journal of Me chanics and Applie d Ma thematics 6 290- 312 1953
16 Wieselsbergo r c Versuche Ube r der luftwiderstand gerundeter und kant iger korper Er gebnisse der Aeroshydynamischen Versucbsansta l t Vol 2 G~ttingen 1923 80 p
FIGURE e- PHOTOGRAPH OF SPHERES CYLINDERS AND PLATES
34
holes were drilled so that each plate could be used for
two geometric ratios by changing the wires (See for
example plates la and lb in Table I
35
EXPERI MENTA L PROCEDURE
Viscosity and Density Calibration
A calibrated hydrometer measuring to the nearest
0002 was used to measure the density Table VI shows that
the effect of temperature on density is practically negli shy
gible in the small temperature range used
A Brookfield Synchro-lectric viscometer was used to
measure the viscosity of both the light and heavy oil
Figures 18 and 19 show the effect of temperature on visshy
cosity In addition the viscosity of the light oil was
checke d using the falling ball method and the equation
D2--ltA (f s bull fl) g (34) l 8v
The viscometer was calibrated by the National Bureau of bull
Standards and was accurate to l tb
Velocity Measurements
The velocity of movement through the oil was measured
by determining the rate of rotation of the pulleys with a
stop watch Usually the time for 10 revolutions was
measured at the highe r ve locities and for 5 revolutions at
the low velocities From this information and the di
amaters of the pulleys the velocities ere calculated
36
The time was measured to the nearest tenth of a second
Since the measured time was usually between 20 and 40
aeconds 1 the error in ~easuring velocity was considered to
be less tha~ 0 5~
force Measurements
The object connected to the scale 1 was dropped to the
bottom of the oil drum The motor was started and the scale
was read as the object vms being pulled towards the top of
the drum Two or three readings were taken for each object
at each velocity In nearly all cases these readings were
the same
37
ti XPER I MENTAL RE STJLTS
The dra g coefficient and the Reynolds number were
calculated by the use of Equations (l or (15) for each of
the spheres cylinders and plates from the measured
quantities of force and velocity a~d the values of the vis shy
cosity and density corresponding to the temperature of the
oil It was necessary to ~ubtract from the measured force
the force on the wire The corrected force measurement was
then used to determine the drag coefficient The force on
the wire has been determined as being proportional to the
velocity A correction curve relating force on the wire
and ve l ocity is plo tted in Figure 9 for the li ght oil and
Fi gure 10 for the heavy oil
The calculated drag coefficients Reynolds numbers
and velocities along with the measured force for the spheres
cylinders flat plates - parallel flow and flat plates shy
perpendicular flow have been tabulated in Tables II III
I V and v respectively
The calculated drag coefficients have been plotted as
a function of the Reynolds number on logarithic graph paper
with geometric ratios as a parameter
Drag coefficients for the spheres are plo tted in
Figure 11 The data for the cylinders are plotted in
CD_ bull 0 G 0
03
Tshy02
01
10 20 30 410 50 60 70 80
VELOCITY- FTJSEC
DRAG FORCE ON THE WIRE-LIGHT OIL
FIGURE 9
I -shy I -middot -- -shy -1shy _i-i I --~ I I _ -middot- shy I i
_I_ - _ middot- LL I l l tmiddot - middot1middot ~- - - - -+i middotshy I - --+-cl - l
1 1 I I IV jc---- --r--middotmiddottmiddot r-middotmiddot--tmiddotmiddot---shy _____ _L __ --~- --1shy middotmiddotr-r-middott- 1 -f-f-T- _~ +-L--1---~- 1--l
~- - shy I-+---Rmiddot-- I I I l i ~~ i -~~ ~- -T f i rshy ~-- --shy i- ----~-- shy - middot1 shy
I i I i I I 1--- -middot - fshy middot i----1---+-shy - i-middot -~+-- --~- --~-- ---- -t+ I v-~~ -middot j
i I middot 1_ _ I tmiddot---+-+1-+--li~+middot -+--+-+-1-+-+-+-+--tc--1-+-t-11-shy - middot --t- 1---t- t----tmiddotshy --~-- -middot i-shy I 1i - ~ i I i v i middotmiddotmiddot
[~v +L~ + ~ - I~~j-+ r V I ~t--- -~-- I +---~-- I f-middot ---1-- ~ -- --- ) Li --+--+--+-+-+-+--1--+--+---t---4 -1--1--+-+--+-l-i
tl~ I I Q Y +l~~ii-+-++++-middotHH-++-+-+-+--H--++ -i t Imiddot i i 1 j _V I f1 r-t~-middot l--r-tshy -~ 7 middot 1 -shy middot middotmiddot I
DRAG FORCE ON THE WIRE- HEAVY OIL
FIGURE 10
40
+shy l i~ltgt ~ bull r-rshy I i t _l
1 lf-1-1 l+r+ fJ-Ct I+ t li 1~t rtH r+l rf-l It llil I I
l l~pound 11 1 ~middot ~~middott ~ It lqf L
t I+--= ~r 17 -Er I _ ~ _pound~- sect Imiddot I+
iU=ff=t 1 +~ t_ - ~ r 111= t h=
I middot
t= IE I 1 1
plusmn~ kplusmni - -STOKE S EQ
(~ l h+middot
ru HmiddotHti+H1 11
c lffii l t~ 4 ~ ~middot ~ff l ~ ~h i ltlri
1 yen~ middot I ~ I I T ~ gt l+t H+h l+ i j l tfl-l Imiddotmiddot ft+ ++ l f+ Imiddotmiddot I+ I+ middott bulli I 1middot1 I ftt-1shy middot I middot r 11 I IH Ij ~ ~ middotishy J F 1= 6= ~
=f l~iit rtti l lit~ I FS lf~ l=i-+
l-11ffi tt lr 1 ~1 -t =l=Rttl 1ft i- 1 ~ I+ I
~~ lflJ
t I lfl m ~~WFB Lt
41plusmn811 IF I Hir tt ft itttplusmn i I~
1-+++middot
I ~ I (~ ffitrHf1 Ittmiddot ~ l r i H-t-r r HHt m 11 H++ I
bull I I
1_ _ F bullmiddot Imiddotmiddot t-- 1-T h iT
f-t+ ftt I+ I lt + T Imiddot 1
1t _plusmn middot~~ ~- 11shy
=a~ 1~ - =itf lttti
H I
=
DATA FOR SPHERES
FIGURE II
41
I -1---1-1-+--+--Ti-+-------+----r--shy --r--- -shy + t----+shy ----4-~---+-f----f--+-f--l--1 I t--shy --t-- ---+-shy
J-+-~f--~~ -___l_ ~---
i 1 L~L~-~tr-l----H~4-----~-f------+------+-----+----+---+middot-t-middot-H5000
middot~ ~ m- ~ - ~t plusmn~ 3t i t~ -f--- bullbull - ~~ h middot-
01 0~ 10
Re
-
DATA FOR CYLINDERS - LD = 6 8 AND 12
FIGURE I 4
44
Figures 12 13 and 14 The data for LD values of 16 24
and 32 were nearly the same and have been plotted to gether
i n Figure 12 In addition the curves for the other LD
ratios determined fro m Fib~res 13 and 14 have been drawn
in Figure 12 so that the effect of the length-to-diameter
is clearly shown Figure 13 shows the data for LD values
of 2 and 4 and the curves determined from this data
Firure 14 shows the data for LD values of 6 8 and 12
and the curves determined from this data
The data for flat plates in parallel flow are plotted
in Fi gure 15 A correction factor for the edge effect has
beon used so that the width-to-length ratio is not a
parameter in this plot A portion of the data of Janour
(5 p 31) is also shown in the diagram
The data for fla t plates in perpendicular flow is
plotted in Figures 16 a nd 17 Figure 16 shows the data for
WL values of 2 Also the curves for the three WL ratios
1 2 and 4 have been drawn in the fi gure Figure 17 shows
the data for WL values of 1 and 4 The curves determined
from the data have also been dravm in the figure
45
10~ ~ ~--- -shy
t==Ff1TR=+ iJ+--_-_--r_-_---+-+---+--+-+--_---_-~r-=r~=~+--=---=---=---=--~=--=_~1=_--=_~_-middot~~--+-+-t~ 1 Ll~+--+-- ---jtshyl~t L--+ I
I
P------ _l -- --1---L i
20 ~-- I ~g I --- - ---+-- r t L_shy
~ ~B 1) I --o-o- JONES - () - - ~~ p f---j- -~-- e e JANOU R
c gt ~c ~ ------ JANSSEN I 0 0 ~ I
IO ~2=i~~~~~~a=~~f=j= ---- TOM OTIKA bulll= I
~~n ~~--~~~~~~o~~~~~--4- NDCIgttl o shy
-
~--~~~~~+--+~+--4-r-~1+-~-middot+1~ ~ --H--~-~~os I i i i-4 ---~T I I f-- t --- li-------~--+-_--+--t-----~~-~_+---_-_-_--+------+-+-__+-[- +_- ___ _______ __+---+-r-+--H----_+--r--------+shy
it I+ ~ bull t ~1 ri j t++t+t++tft bullm H--~+H-t+t-++H-f+t+~HtttH t bull~H-IrttI-H
iH-H u nH m
I
t H+t-~ 1-r f-tj
i it iT -t middotHt I I I I Ill
~middot __
r middotshy
i I r-
f H- jLj f r H rr t~
II
t f f-l -t+tt ~ ==_ =~middot irE
I I
I
I
f
I --
i
t
1 r bull - r
~- ltt++l=tUtt~S-t+t+++~-++U +HJJm~-fl~HHtt1 tttn ll+t-Tt-~- ~ r fH T --r -1 t ---t- -tshy w _+ _ I-shy middotI
-shy -r- + Hbull Hshy t-I --r++ -t iHr -1 H-e-- -t I 1IT 1
1 H-rf-I IJftJ Jf+i+ ~ L
=+shy - tjshy rtmiddotshy ~ -
+ H 1-Jt I tt o =tt ~-
~1 l +fill l plusmn~ fplusmn -shy + I t-
DATA FOR FLAT PLATES PERPENDICULAR FLOW- WL= I 4
FIGURE 17
48
DI SCUSS ION OF RESULTS
Correction and Accuracy of Measurements
After a few pre liminary force measurements with the
spheres and a check with Stokes law (Equation 2) it was
apparent that the drag force on the wire was appreciable
and needed to be considered It was decided to take a
series of measurements with the spheres and calculate the
difference between the measured force and the force calcushy
lated from Stokes law The difference in force could then
be attributed to the drag on the wire If Stokes law is
followed the force on the wire should be proportional to
the velocity
A series of twenty measurements of the force on the
spheres was taken for each oil and the difference between
the measured force and that calcula ted by Stokes 1 law was
determined For each oil this difference as plo tted vs
the velocity The points grouped fairly ell around a
strai ght line nearly passing through the origin The
method of least squares was used to determine the equation
of the line best fitting the da t a The equa tion of the
line for the li bht oil tas found to be
Fe bullbull05605v - oooa (35)
which was determined at about 62 7degF Since the intercept
49
of the line is very close to zero it is believed that the
line is a good indication of the drag on the wire The
equation of the line for the heavy oil was found to be
F - 19llv I oo2o1 (36 ) c shy
which was determined at about 64 2deg The intercept of this
line is also quite close to zero These lines plotted in
Fi poundures 9 and 10 were used throughout the investigation
for the correction factor of the drag on the wires For
the cylinders and flat plates in parallel flow which were
pulled by two wires the values determined from Equations
35) and (36) were doubled For the plates in perpendicular
flow pulled by four wires the correction force was multishy
plied by four
The spring scale had 12 ounce divisions but could be
read to the nearest sixth of an ounce Some of the measureshy
ments of force were under an ounce hence a considerable
spread of the measurements was noticed in the pre liminary
data and throughout the experiment However sufficient
points were obtained so that it was possible to draw a
reliable curve through the data in all casas An analysis
was made to determine the average deviation from Stokes
equation for the spheres It raa found that the average
deviation was 15 1 for the light oil 16 6 for the heavy
oil and 15 9 overall The maximum deviation was 89
50
Inspection of the other data shows that these deviations
are also representative of the cylinders and flat plates
The force measurement is the least accurate part of the
experiment Other insignificant errors are introduced by
a small variation in the temperature This variation was
held to about 10 from the temperature of the calibrated
correction curve The velocity measurements and the
dimensions of the cylinders spheres and pl~ tes are conshy
sidered go od enough so tha t no appreciable errors occur
In order to e l iminate the WL parameter for flat plates
in parallel f l ow an additional factor for the effect of
the edges was subtracted from the measured force Janour
(5 p 27) presented the foll owing equation for the edge
correction for one edge of a flat plate in parallel flow
F ~ lv~ bull (37 ) edge gc
In present work this equation as doubled because both
edges of the plates were submerged in fluid It is assumed
in appl ying this correction that the lowe r limit of a
Reynolds number of 10 proposed by Janour can be extended
close to 0 1
Analysis of Results
Forty of the points for the spheres were used to get
51
the correction factor for the wires The remaining thirty
points are well erouped about Stokes law
The data for cylinders for LD ratios of 16 24 and
32 did not seem to be se gregated therefore these data
were plotted together It would seem that in the low range
of Reyno l ds numbers an LD of 16 and greater can be con shy
sidered an ~nfini tely long cylinder The other LD ratios
of 2 4 6 a 12 provided fairly distinct and separate
lines The best straight lines were drawn through the data
for each of the LD ratios It was evident that in eaeh
case a slope of -1 on a lo g-log graph gave the best straight
line which would indicate that the force varies directly
as the velocity It was possible to develop an empirical
expression relating dra g coefficient Reynolds number and
LD The following equation was obtained from the straight
line plots of Re vs fd for the various LD ratios
(38 )
Equation (38) applies for Reyno l ds numbers from 01 to 10
and for LD ratios of 2 to 16 For LD ratios greater
than 16
10 re (39 )
The data for flat plates in parallel flow is plotted
in Figure 15 after the correction factor for tho edge
52
effect was subtracted When the edge correction is made
no effect of WL ratio is indicated This result would be
expected The data followed a straight line with a slope
of -1 up to a Reynolds number of 2 After that a curve was
dravm connecting the line to that obtained by Janour The
equation for the straight section of the curve is
f - 6 (40)- Re
which applies for Reynolds numbers of 0 1 to 2 0 Here
a gain the force is proportional to the velocity Vfuen
determining drag force for flat plates in parallel flow
the force is first calculated from Equations (40) and (15 )
then the edge correction is added
The effect of the geometric ratios is clearly shown in
the data for flat plates in perpendicul ar flow which are
plotted in Figures 16 and 17 As with the other data the
best straight line was drawn through the various points
for eaoh of the WL ratios Again the line had a slope of
-1 The equation relating fd Re and wL was found t o be
rd 37 (w) -o 3o (41)Irel
which applies for Reynolds numbers of about 05 to 2 0 and
WL ratios of 1 to 4 It is possible but it has not been
proved that Equation (41) is suitable for higher WL ratios
The exponent on WL in Equation 41) is very close to that
53
on L D i n Equation ( 38 )~ It i s possible t ha t these
exponents are t he same but this cannot be sho~~ depound1nitely
until more accura te da ta are available It would be exshy
pected that a s the Reynolds number approaches zero t he
effect of geometric ratios would be the same for cylinders
and fla t pla tes in perpendicula r flow
It is seen in the t a bles of data that occasionally a
ne gative force was obtained because the correction applie d
due to t he wire dra g was greater than the mea sured force
These points obviously are incorrect This occurred only
for the smallest plates in the heavy oil at t he highest
velocities However these knom bad points occur in less
tha n 5~ of the data
It is clearl y shown that for cylinders and plates the
fd increases as L D or W L decreases This is in direct
contrast to Wiesel aberger s investigation However his
work is for hi gher Reynolds numbers at which a turbulent
wake forms bull
Comparison of Results with Other Data and Theoretical So l utions
The data for sphere~ a grees of course with Stokes
l aw since that law was used to determine the correction
factor for the wire Liebster (9 Pbull 548 ) has
54
substantiated Stokes equation
There are no experimental data with which to compare
the results of the cylinders Wieselsbergers minimum
Reynolds number of 4 is above the ran ge covered in the preshy
sent investigation The da ta for the highest LD ratios
(16 24 and 32) does agree almost exactly wi t h the solution
of Allen and Southwell (1 P bull 141) (LD =00) in the range
of Reynolds numbers from 0 1 to 1 0 Allen and Southwells
solution a greed with the data of Wieselsberger (16 p 22)
However the present data is above the theoretical solutions
of Lamb (8 p 112-121) throughout the range of Reynolds
numbers from 0 01 to 1 0 and above the solutions of
Bairstow Cave and Lang (2 p 404) I mai (4 p 157) and
Tomotika and Aoi (15 p 302) for Reynolds numbers of 0 1
to 1 0 Allen and Southwells solution a grees dth both
Wieselsberger 1 s a nd the present data Their solution and
the present data represent the best means for predicting
drag coefficients for flow over long cylinders for Reynolds
numbers of 0 01 to 10 It should be remembered that the
o t her solutions should a gree with eac h other since they
were all essentially derived by linearizing the Na viershy
Stokes equation
The data for flat plates in parallel flow is
55
considerably above the theoretical solutions of Janssen
(6 p 183 ) and Tomotika and Aoi (15 Pbull 302) However
Fi f~re 15 shows that a smooth transition occurs bet een
the present work and the data of Janour (5 P bull 31) The
present data considerably extend the experimental inforshy
mation previously available for laminar flow paral lel to
flat plates In the re gion of Reynol ds numbers less than
2 the drag coefficient is shown to be inversely proportional
to the Reynolds number Janours data covers a range of
Reynolds numbers from 11 to 1000 The results of the
present investigation line up with Janours results which
in turn on extrapolation to higher Reyno l ds numbers
(greater than 1000) make a smooth transition into Blasius
curve represented by Equation (10) At Reyno l ds numbers
greater than 20 000 the drag coefficient is inversely proshy
portional to the square root of the Reynolds number
The data for flat plates in perpendicular flow is conshy
siderably above the solutions of Tomotika and Aoi
(15 p 302) and Imai (4 p 157 However their solutions
f or cylinders and plates in parallel flow are also below
the present data Also it should be remembered that their
solutions are for infinitely wide plates If a value of
WL of above 100 is used in Equation (41) then the present
data and the solutions of Tomotika and Aoi are fairly close
56
The present results indicate that Equation (41~ can be
used with an accuracy of 15 to 20 within the limitations
of the equation (WL 1 to 4 Re = 0 05 to 2)
57
SUM RY AND CONCLUSIONS
Only a small amount of work has been done in the past
on the study of laminar flow over immersed bodies There
are many areas in the chemical process industries and the
field of aeronautics where this information would be very
helpful The purpose of the present investi gation wa s to
study the almost totally unexplored range of Reynol ds
numbers from 0 01 to 10
Drag coefficients have been determined for spheres
cylinders and flat plates in paralle l and perpendicular
flow The drag coefficients have been plotted as a
function of the Reynolds number with dimension ratios as
a parameter on lo g-log graphs The best straight lines
have been drawn through the data In all cases these lines
had a slope of -1 hich shows that the dra g coefficient is
inversely proportional to the Reynolds number at very low
Reynolds numbers for all shapes and dimension ratios The
following equations have been determined from the data
For cylinders
fd - 27 L -0 36 (38 ) - Re ())
which applies for Reynolds numbers of 0 01 to 1 and LD of
2 to 16 For LD greater than 16 the equation is
58
(39)
For flat plates in parallel flow a correction factor has
been applied to account for the edge effect The equation
which applies for Reyno l ds numbers of 0 1 to 2 is
f 6Re
(40)
For flat plates in perpendicular flow
f d
- 37 - Re (w) t -
0 bull 30 (41)
wbieh applies for W L of 1 to 4 and Reynolds numbers of
0 05 to 2
It is concluded tha t Equations (38-41) give the best
values of drag coefficients within an accuracy of 20~ for
the range of Reynolds numbers that were considered Also
it is evident that the dimension ratios are a n important
factor in determining the drag coefficient for a given
Reynolds number Furthermore the drag coefficient inshy
creases with decreasing values of L D or W L for a constant
Reynolds number The da ta obtained in this investi gation
compare favorably with the other experimental data and with
some of the theoretical sol utions It should be remembered
that when comparing the experimental data with theoretical
solutions that practically all of the solutions are for an
infinitely long cylinder or an infinitely wide plate
It is recommended tha t the present apparatus be
59
modified so that a force of 001 pound can be measured
Also it would improve tho accuracy to set up a constant
temperature bath so that the temperature of the oil can not
vary over 02degF A few check points on the present data
is all that is necessary to confirm the validity of
Equations (38- 41) It is also r ecommended that only SAE 140
oil be used and that 2 inches should be the minimum plate
width and cylinder length to be studi3d These conditions
would help to maintain the accuracy of the correction force
for the wire
60
~WMENCIATURE
Symbol Dimensions
A area sq ft
D diameter ft
F force lb f
L length ft
M mas s lb m Re Reynolds number Dvf= -ltr w width ft
a area sq ft
b characteristic length ft
d diameter ft
f drag coefficientfd
gravitation constant l b mft gc 2= 32 17 l b _ rsec
1 length ft
m mass l b bullm
p pressure lbrsqft
r radius ft
t time see
u velocity ft sec
v velocity ft sec
w width ft
61
Symbol Dimensions
X xbullcoordinate ft
y y- coordinate ft
o( vorticity
time sec
viscosity lb m ft -sec
kinematic viscosity ft 2sec
circumference diameter = 3 1416
3density lb m ft
function
stream function
Laplacian operator
infinity
Subscripts
c corrected
f force
1 l iquid
m mass
p projected
s solid
w wetted
62
BI BLIOGRAPHY
1 Allan D N de G and R v Southwell Re laxation methods applied to determine the motion in two di shymensions of a viscous fluid past a fixed cylinder Quarterly Journal of Mechanics and Applied Mathe shymatics 8 129-145 1955
2 Bairstow L B M Cave and E D Lang The reshysistance of a cylinder moving in a viscous fluid Philosophical Transactions of the Royal Society of London ser A 223383- 432 1923
3 Goldstein Sidney The steady flow of viscous fluid past a fixed spherical obstacle at small Reyno l ds numbers Proceedings of the Royal Society of London ser A 123225-235 1929
4 Imai I A new method of solving Oseens equations and its application to the flow past an inclined elliptic cylinder Proceedings of the Royal Society of London ser A 224 141-160 1954
5 Janour Zbynek Resistance of a plate in paralle l flow at low Reyno lds numbers Washington Nov 1951 40 p National Advisory Committee for Aeronautics Te chnica l Memorandum 1316)
6 Janssen E An analog solution of the Navier-Stokes equation for the case of flow past a f l at plate at low Reynolds numbers In 1956 Heat Transfer and Fluid Mechanics Institute (Preprints of Papers) p 173-183
7 Knudsen James G and Donal d L Katz Fluid Dynamics a nd Heat Transfer Ann Arbor University of Michigan 1953 243 p (Michi gan University Engineering Research Bulletin no 37)
8 La~b Horace On the uniform motion of a spherethrough a viscous fluid Philosophical Magazine and Journal of Science s~r 6 21112-121 1911
9 Liebster H Uben den widerstrand von kugeln Annalen Der Physik ser 4 82 541- 562 1 927
63
10 McAdams William H Heat transmission 3d ed New York McGraw- Hill 1954 532 p
11 Pai Shih- I Viscous f l ow theory I Laminar flow Princeton D Van Nostrand 1956 384 p
12 Prandtlbull Ludwi g Es sentials of fluid dynamics London Blackie amp Son 1954 452 p
13 Relf i F Discussion of the results of measure shyments of the resistance of wires with some additionshyal tests of the resistance of wires of small diame shyters In Technical report of the Advisory Committee for Aeronautics London) March 1914 p 47 - 51 (Report and memoranda no 102 )
14 Stokes George Gabriel Mathematical and physical papers Vol 3 Cambridge University Press 1922 413 p
15 Tomotika s and T Aoi The steady flow of a viscous fluid past an elliptic cylinder and a flat plate at smal l Reynolds numbers Quarterly Journal of Me chanics and Applie d Ma thematics 6 290- 312 1953
16 Wieselsbergo r c Versuche Ube r der luftwiderstand gerundeter und kant iger korper Er gebnisse der Aeroshydynamischen Versucbsansta l t Vol 2 G~ttingen 1923 80 p
FIGURE e- PHOTOGRAPH OF SPHERES CYLINDERS AND PLATES
34
holes were drilled so that each plate could be used for
two geometric ratios by changing the wires (See for
example plates la and lb in Table I
35
EXPERI MENTA L PROCEDURE
Viscosity and Density Calibration
A calibrated hydrometer measuring to the nearest
0002 was used to measure the density Table VI shows that
the effect of temperature on density is practically negli shy
gible in the small temperature range used
A Brookfield Synchro-lectric viscometer was used to
measure the viscosity of both the light and heavy oil
Figures 18 and 19 show the effect of temperature on visshy
cosity In addition the viscosity of the light oil was
checke d using the falling ball method and the equation
D2--ltA (f s bull fl) g (34) l 8v
The viscometer was calibrated by the National Bureau of bull
Standards and was accurate to l tb
Velocity Measurements
The velocity of movement through the oil was measured
by determining the rate of rotation of the pulleys with a
stop watch Usually the time for 10 revolutions was
measured at the highe r ve locities and for 5 revolutions at
the low velocities From this information and the di
amaters of the pulleys the velocities ere calculated
36
The time was measured to the nearest tenth of a second
Since the measured time was usually between 20 and 40
aeconds 1 the error in ~easuring velocity was considered to
be less tha~ 0 5~
force Measurements
The object connected to the scale 1 was dropped to the
bottom of the oil drum The motor was started and the scale
was read as the object vms being pulled towards the top of
the drum Two or three readings were taken for each object
at each velocity In nearly all cases these readings were
the same
37
ti XPER I MENTAL RE STJLTS
The dra g coefficient and the Reynolds number were
calculated by the use of Equations (l or (15) for each of
the spheres cylinders and plates from the measured
quantities of force and velocity a~d the values of the vis shy
cosity and density corresponding to the temperature of the
oil It was necessary to ~ubtract from the measured force
the force on the wire The corrected force measurement was
then used to determine the drag coefficient The force on
the wire has been determined as being proportional to the
velocity A correction curve relating force on the wire
and ve l ocity is plo tted in Figure 9 for the li ght oil and
Fi gure 10 for the heavy oil
The calculated drag coefficients Reynolds numbers
and velocities along with the measured force for the spheres
cylinders flat plates - parallel flow and flat plates shy
perpendicular flow have been tabulated in Tables II III
I V and v respectively
The calculated drag coefficients have been plotted as
a function of the Reynolds number on logarithic graph paper
with geometric ratios as a parameter
Drag coefficients for the spheres are plo tted in
Figure 11 The data for the cylinders are plotted in
CD_ bull 0 G 0
03
Tshy02
01
10 20 30 410 50 60 70 80
VELOCITY- FTJSEC
DRAG FORCE ON THE WIRE-LIGHT OIL
FIGURE 9
I -shy I -middot -- -shy -1shy _i-i I --~ I I _ -middot- shy I i
_I_ - _ middot- LL I l l tmiddot - middot1middot ~- - - - -+i middotshy I - --+-cl - l
1 1 I I IV jc---- --r--middotmiddottmiddot r-middotmiddot--tmiddotmiddot---shy _____ _L __ --~- --1shy middotmiddotr-r-middott- 1 -f-f-T- _~ +-L--1---~- 1--l
~- - shy I-+---Rmiddot-- I I I l i ~~ i -~~ ~- -T f i rshy ~-- --shy i- ----~-- shy - middot1 shy
I i I i I I 1--- -middot - fshy middot i----1---+-shy - i-middot -~+-- --~- --~-- ---- -t+ I v-~~ -middot j
i I middot 1_ _ I tmiddot---+-+1-+--li~+middot -+--+-+-1-+-+-+-+--tc--1-+-t-11-shy - middot --t- 1---t- t----tmiddotshy --~-- -middot i-shy I 1i - ~ i I i v i middotmiddotmiddot
[~v +L~ + ~ - I~~j-+ r V I ~t--- -~-- I +---~-- I f-middot ---1-- ~ -- --- ) Li --+--+--+-+-+-+--1--+--+---t---4 -1--1--+-+--+-l-i
tl~ I I Q Y +l~~ii-+-++++-middotHH-++-+-+-+--H--++ -i t Imiddot i i 1 j _V I f1 r-t~-middot l--r-tshy -~ 7 middot 1 -shy middot middotmiddot I
DRAG FORCE ON THE WIRE- HEAVY OIL
FIGURE 10
40
+shy l i~ltgt ~ bull r-rshy I i t _l
1 lf-1-1 l+r+ fJ-Ct I+ t li 1~t rtH r+l rf-l It llil I I
l l~pound 11 1 ~middot ~~middott ~ It lqf L
t I+--= ~r 17 -Er I _ ~ _pound~- sect Imiddot I+
iU=ff=t 1 +~ t_ - ~ r 111= t h=
I middot
t= IE I 1 1
plusmn~ kplusmni - -STOKE S EQ
(~ l h+middot
ru HmiddotHti+H1 11
c lffii l t~ 4 ~ ~middot ~ff l ~ ~h i ltlri
1 yen~ middot I ~ I I T ~ gt l+t H+h l+ i j l tfl-l Imiddotmiddot ft+ ++ l f+ Imiddotmiddot I+ I+ middott bulli I 1middot1 I ftt-1shy middot I middot r 11 I IH Ij ~ ~ middotishy J F 1= 6= ~
=f l~iit rtti l lit~ I FS lf~ l=i-+
l-11ffi tt lr 1 ~1 -t =l=Rttl 1ft i- 1 ~ I+ I
~~ lflJ
t I lfl m ~~WFB Lt
41plusmn811 IF I Hir tt ft itttplusmn i I~
1-+++middot
I ~ I (~ ffitrHf1 Ittmiddot ~ l r i H-t-r r HHt m 11 H++ I
bull I I
1_ _ F bullmiddot Imiddotmiddot t-- 1-T h iT
f-t+ ftt I+ I lt + T Imiddot 1
1t _plusmn middot~~ ~- 11shy
=a~ 1~ - =itf lttti
H I
=
DATA FOR SPHERES
FIGURE II
41
I -1---1-1-+--+--Ti-+-------+----r--shy --r--- -shy + t----+shy ----4-~---+-f----f--+-f--l--1 I t--shy --t-- ---+-shy
J-+-~f--~~ -___l_ ~---
i 1 L~L~-~tr-l----H~4-----~-f------+------+-----+----+---+middot-t-middot-H5000
middot~ ~ m- ~ - ~t plusmn~ 3t i t~ -f--- bullbull - ~~ h middot-
01 0~ 10
Re
-
DATA FOR CYLINDERS - LD = 6 8 AND 12
FIGURE I 4
44
Figures 12 13 and 14 The data for LD values of 16 24
and 32 were nearly the same and have been plotted to gether
i n Figure 12 In addition the curves for the other LD
ratios determined fro m Fib~res 13 and 14 have been drawn
in Figure 12 so that the effect of the length-to-diameter
is clearly shown Figure 13 shows the data for LD values
of 2 and 4 and the curves determined from this data
Firure 14 shows the data for LD values of 6 8 and 12
and the curves determined from this data
The data for flat plates in parallel flow are plotted
in Fi gure 15 A correction factor for the edge effect has
beon used so that the width-to-length ratio is not a
parameter in this plot A portion of the data of Janour
(5 p 31) is also shown in the diagram
The data for fla t plates in perpendicular flow is
plotted in Figures 16 a nd 17 Figure 16 shows the data for
WL values of 2 Also the curves for the three WL ratios
1 2 and 4 have been drawn in the fi gure Figure 17 shows
the data for WL values of 1 and 4 The curves determined
from the data have also been dravm in the figure
45
10~ ~ ~--- -shy
t==Ff1TR=+ iJ+--_-_--r_-_---+-+---+--+-+--_---_-~r-=r~=~+--=---=---=---=--~=--=_~1=_--=_~_-middot~~--+-+-t~ 1 Ll~+--+-- ---jtshyl~t L--+ I
I
P------ _l -- --1---L i
20 ~-- I ~g I --- - ---+-- r t L_shy
~ ~B 1) I --o-o- JONES - () - - ~~ p f---j- -~-- e e JANOU R
c gt ~c ~ ------ JANSSEN I 0 0 ~ I
IO ~2=i~~~~~~a=~~f=j= ---- TOM OTIKA bulll= I
~~n ~~--~~~~~~o~~~~~--4- NDCIgttl o shy
-
~--~~~~~+--+~+--4-r-~1+-~-middot+1~ ~ --H--~-~~os I i i i-4 ---~T I I f-- t --- li-------~--+-_--+--t-----~~-~_+---_-_-_--+------+-+-__+-[- +_- ___ _______ __+---+-r-+--H----_+--r--------+shy
it I+ ~ bull t ~1 ri j t++t+t++tft bullm H--~+H-t+t-++H-f+t+~HtttH t bull~H-IrttI-H
iH-H u nH m
I
t H+t-~ 1-r f-tj
i it iT -t middotHt I I I I Ill
~middot __
r middotshy
i I r-
f H- jLj f r H rr t~
II
t f f-l -t+tt ~ ==_ =~middot irE
I I
I
I
f
I --
i
t
1 r bull - r
~- ltt++l=tUtt~S-t+t+++~-++U +HJJm~-fl~HHtt1 tttn ll+t-Tt-~- ~ r fH T --r -1 t ---t- -tshy w _+ _ I-shy middotI
-shy -r- + Hbull Hshy t-I --r++ -t iHr -1 H-e-- -t I 1IT 1
1 H-rf-I IJftJ Jf+i+ ~ L
=+shy - tjshy rtmiddotshy ~ -
+ H 1-Jt I tt o =tt ~-
~1 l +fill l plusmn~ fplusmn -shy + I t-
DATA FOR FLAT PLATES PERPENDICULAR FLOW- WL= I 4
FIGURE 17
48
DI SCUSS ION OF RESULTS
Correction and Accuracy of Measurements
After a few pre liminary force measurements with the
spheres and a check with Stokes law (Equation 2) it was
apparent that the drag force on the wire was appreciable
and needed to be considered It was decided to take a
series of measurements with the spheres and calculate the
difference between the measured force and the force calcushy
lated from Stokes law The difference in force could then
be attributed to the drag on the wire If Stokes law is
followed the force on the wire should be proportional to
the velocity
A series of twenty measurements of the force on the
spheres was taken for each oil and the difference between
the measured force and that calcula ted by Stokes 1 law was
determined For each oil this difference as plo tted vs
the velocity The points grouped fairly ell around a
strai ght line nearly passing through the origin The
method of least squares was used to determine the equation
of the line best fitting the da t a The equa tion of the
line for the li bht oil tas found to be
Fe bullbull05605v - oooa (35)
which was determined at about 62 7degF Since the intercept
49
of the line is very close to zero it is believed that the
line is a good indication of the drag on the wire The
equation of the line for the heavy oil was found to be
F - 19llv I oo2o1 (36 ) c shy
which was determined at about 64 2deg The intercept of this
line is also quite close to zero These lines plotted in
Fi poundures 9 and 10 were used throughout the investigation
for the correction factor of the drag on the wires For
the cylinders and flat plates in parallel flow which were
pulled by two wires the values determined from Equations
35) and (36) were doubled For the plates in perpendicular
flow pulled by four wires the correction force was multishy
plied by four
The spring scale had 12 ounce divisions but could be
read to the nearest sixth of an ounce Some of the measureshy
ments of force were under an ounce hence a considerable
spread of the measurements was noticed in the pre liminary
data and throughout the experiment However sufficient
points were obtained so that it was possible to draw a
reliable curve through the data in all casas An analysis
was made to determine the average deviation from Stokes
equation for the spheres It raa found that the average
deviation was 15 1 for the light oil 16 6 for the heavy
oil and 15 9 overall The maximum deviation was 89
50
Inspection of the other data shows that these deviations
are also representative of the cylinders and flat plates
The force measurement is the least accurate part of the
experiment Other insignificant errors are introduced by
a small variation in the temperature This variation was
held to about 10 from the temperature of the calibrated
correction curve The velocity measurements and the
dimensions of the cylinders spheres and pl~ tes are conshy
sidered go od enough so tha t no appreciable errors occur
In order to e l iminate the WL parameter for flat plates
in parallel f l ow an additional factor for the effect of
the edges was subtracted from the measured force Janour
(5 p 27) presented the foll owing equation for the edge
correction for one edge of a flat plate in parallel flow
F ~ lv~ bull (37 ) edge gc
In present work this equation as doubled because both
edges of the plates were submerged in fluid It is assumed
in appl ying this correction that the lowe r limit of a
Reynolds number of 10 proposed by Janour can be extended
close to 0 1
Analysis of Results
Forty of the points for the spheres were used to get
51
the correction factor for the wires The remaining thirty
points are well erouped about Stokes law
The data for cylinders for LD ratios of 16 24 and
32 did not seem to be se gregated therefore these data
were plotted together It would seem that in the low range
of Reyno l ds numbers an LD of 16 and greater can be con shy
sidered an ~nfini tely long cylinder The other LD ratios
of 2 4 6 a 12 provided fairly distinct and separate
lines The best straight lines were drawn through the data
for each of the LD ratios It was evident that in eaeh
case a slope of -1 on a lo g-log graph gave the best straight
line which would indicate that the force varies directly
as the velocity It was possible to develop an empirical
expression relating dra g coefficient Reynolds number and
LD The following equation was obtained from the straight
line plots of Re vs fd for the various LD ratios
(38 )
Equation (38) applies for Reyno l ds numbers from 01 to 10
and for LD ratios of 2 to 16 For LD ratios greater
than 16
10 re (39 )
The data for flat plates in parallel flow is plotted
in Figure 15 after the correction factor for tho edge
52
effect was subtracted When the edge correction is made
no effect of WL ratio is indicated This result would be
expected The data followed a straight line with a slope
of -1 up to a Reynolds number of 2 After that a curve was
dravm connecting the line to that obtained by Janour The
equation for the straight section of the curve is
f - 6 (40)- Re
which applies for Reynolds numbers of 0 1 to 2 0 Here
a gain the force is proportional to the velocity Vfuen
determining drag force for flat plates in parallel flow
the force is first calculated from Equations (40) and (15 )
then the edge correction is added
The effect of the geometric ratios is clearly shown in
the data for flat plates in perpendicul ar flow which are
plotted in Figures 16 and 17 As with the other data the
best straight line was drawn through the various points
for eaoh of the WL ratios Again the line had a slope of
-1 The equation relating fd Re and wL was found t o be
rd 37 (w) -o 3o (41)Irel
which applies for Reynolds numbers of about 05 to 2 0 and
WL ratios of 1 to 4 It is possible but it has not been
proved that Equation (41) is suitable for higher WL ratios
The exponent on WL in Equation 41) is very close to that
53
on L D i n Equation ( 38 )~ It i s possible t ha t these
exponents are t he same but this cannot be sho~~ depound1nitely
until more accura te da ta are available It would be exshy
pected that a s the Reynolds number approaches zero t he
effect of geometric ratios would be the same for cylinders
and fla t pla tes in perpendicula r flow
It is seen in the t a bles of data that occasionally a
ne gative force was obtained because the correction applie d
due to t he wire dra g was greater than the mea sured force
These points obviously are incorrect This occurred only
for the smallest plates in the heavy oil at t he highest
velocities However these knom bad points occur in less
tha n 5~ of the data
It is clearl y shown that for cylinders and plates the
fd increases as L D or W L decreases This is in direct
contrast to Wiesel aberger s investigation However his
work is for hi gher Reynolds numbers at which a turbulent
wake forms bull
Comparison of Results with Other Data and Theoretical So l utions
The data for sphere~ a grees of course with Stokes
l aw since that law was used to determine the correction
factor for the wire Liebster (9 Pbull 548 ) has
54
substantiated Stokes equation
There are no experimental data with which to compare
the results of the cylinders Wieselsbergers minimum
Reynolds number of 4 is above the ran ge covered in the preshy
sent investigation The da ta for the highest LD ratios
(16 24 and 32) does agree almost exactly wi t h the solution
of Allen and Southwell (1 P bull 141) (LD =00) in the range
of Reynolds numbers from 0 1 to 1 0 Allen and Southwells
solution a greed with the data of Wieselsberger (16 p 22)
However the present data is above the theoretical solutions
of Lamb (8 p 112-121) throughout the range of Reynolds
numbers from 0 01 to 1 0 and above the solutions of
Bairstow Cave and Lang (2 p 404) I mai (4 p 157) and
Tomotika and Aoi (15 p 302) for Reynolds numbers of 0 1
to 1 0 Allen and Southwells solution a grees dth both
Wieselsberger 1 s a nd the present data Their solution and
the present data represent the best means for predicting
drag coefficients for flow over long cylinders for Reynolds
numbers of 0 01 to 10 It should be remembered that the
o t her solutions should a gree with eac h other since they
were all essentially derived by linearizing the Na viershy
Stokes equation
The data for flat plates in parallel flow is
55
considerably above the theoretical solutions of Janssen
(6 p 183 ) and Tomotika and Aoi (15 Pbull 302) However
Fi f~re 15 shows that a smooth transition occurs bet een
the present work and the data of Janour (5 P bull 31) The
present data considerably extend the experimental inforshy
mation previously available for laminar flow paral lel to
flat plates In the re gion of Reynol ds numbers less than
2 the drag coefficient is shown to be inversely proportional
to the Reynolds number Janours data covers a range of
Reynolds numbers from 11 to 1000 The results of the
present investigation line up with Janours results which
in turn on extrapolation to higher Reyno l ds numbers
(greater than 1000) make a smooth transition into Blasius
curve represented by Equation (10) At Reyno l ds numbers
greater than 20 000 the drag coefficient is inversely proshy
portional to the square root of the Reynolds number
The data for flat plates in perpendicular flow is conshy
siderably above the solutions of Tomotika and Aoi
(15 p 302) and Imai (4 p 157 However their solutions
f or cylinders and plates in parallel flow are also below
the present data Also it should be remembered that their
solutions are for infinitely wide plates If a value of
WL of above 100 is used in Equation (41) then the present
data and the solutions of Tomotika and Aoi are fairly close
56
The present results indicate that Equation (41~ can be
used with an accuracy of 15 to 20 within the limitations
of the equation (WL 1 to 4 Re = 0 05 to 2)
57
SUM RY AND CONCLUSIONS
Only a small amount of work has been done in the past
on the study of laminar flow over immersed bodies There
are many areas in the chemical process industries and the
field of aeronautics where this information would be very
helpful The purpose of the present investi gation wa s to
study the almost totally unexplored range of Reynol ds
numbers from 0 01 to 10
Drag coefficients have been determined for spheres
cylinders and flat plates in paralle l and perpendicular
flow The drag coefficients have been plotted as a
function of the Reynolds number with dimension ratios as
a parameter on lo g-log graphs The best straight lines
have been drawn through the data In all cases these lines
had a slope of -1 hich shows that the dra g coefficient is
inversely proportional to the Reynolds number at very low
Reynolds numbers for all shapes and dimension ratios The
following equations have been determined from the data
For cylinders
fd - 27 L -0 36 (38 ) - Re ())
which applies for Reynolds numbers of 0 01 to 1 and LD of
2 to 16 For LD greater than 16 the equation is
58
(39)
For flat plates in parallel flow a correction factor has
been applied to account for the edge effect The equation
which applies for Reyno l ds numbers of 0 1 to 2 is
f 6Re
(40)
For flat plates in perpendicular flow
f d
- 37 - Re (w) t -
0 bull 30 (41)
wbieh applies for W L of 1 to 4 and Reynolds numbers of
0 05 to 2
It is concluded tha t Equations (38-41) give the best
values of drag coefficients within an accuracy of 20~ for
the range of Reynolds numbers that were considered Also
it is evident that the dimension ratios are a n important
factor in determining the drag coefficient for a given
Reynolds number Furthermore the drag coefficient inshy
creases with decreasing values of L D or W L for a constant
Reynolds number The da ta obtained in this investi gation
compare favorably with the other experimental data and with
some of the theoretical sol utions It should be remembered
that when comparing the experimental data with theoretical
solutions that practically all of the solutions are for an
infinitely long cylinder or an infinitely wide plate
It is recommended tha t the present apparatus be
59
modified so that a force of 001 pound can be measured
Also it would improve tho accuracy to set up a constant
temperature bath so that the temperature of the oil can not
vary over 02degF A few check points on the present data
is all that is necessary to confirm the validity of
Equations (38- 41) It is also r ecommended that only SAE 140
oil be used and that 2 inches should be the minimum plate
width and cylinder length to be studi3d These conditions
would help to maintain the accuracy of the correction force
for the wire
60
~WMENCIATURE
Symbol Dimensions
A area sq ft
D diameter ft
F force lb f
L length ft
M mas s lb m Re Reynolds number Dvf= -ltr w width ft
a area sq ft
b characteristic length ft
d diameter ft
f drag coefficientfd
gravitation constant l b mft gc 2= 32 17 l b _ rsec
1 length ft
m mass l b bullm
p pressure lbrsqft
r radius ft
t time see
u velocity ft sec
v velocity ft sec
w width ft
61
Symbol Dimensions
X xbullcoordinate ft
y y- coordinate ft
o( vorticity
time sec
viscosity lb m ft -sec
kinematic viscosity ft 2sec
circumference diameter = 3 1416
3density lb m ft
function
stream function
Laplacian operator
infinity
Subscripts
c corrected
f force
1 l iquid
m mass
p projected
s solid
w wetted
62
BI BLIOGRAPHY
1 Allan D N de G and R v Southwell Re laxation methods applied to determine the motion in two di shymensions of a viscous fluid past a fixed cylinder Quarterly Journal of Mechanics and Applied Mathe shymatics 8 129-145 1955
2 Bairstow L B M Cave and E D Lang The reshysistance of a cylinder moving in a viscous fluid Philosophical Transactions of the Royal Society of London ser A 223383- 432 1923
3 Goldstein Sidney The steady flow of viscous fluid past a fixed spherical obstacle at small Reyno l ds numbers Proceedings of the Royal Society of London ser A 123225-235 1929
4 Imai I A new method of solving Oseens equations and its application to the flow past an inclined elliptic cylinder Proceedings of the Royal Society of London ser A 224 141-160 1954
5 Janour Zbynek Resistance of a plate in paralle l flow at low Reyno lds numbers Washington Nov 1951 40 p National Advisory Committee for Aeronautics Te chnica l Memorandum 1316)
6 Janssen E An analog solution of the Navier-Stokes equation for the case of flow past a f l at plate at low Reynolds numbers In 1956 Heat Transfer and Fluid Mechanics Institute (Preprints of Papers) p 173-183
7 Knudsen James G and Donal d L Katz Fluid Dynamics a nd Heat Transfer Ann Arbor University of Michigan 1953 243 p (Michi gan University Engineering Research Bulletin no 37)
8 La~b Horace On the uniform motion of a spherethrough a viscous fluid Philosophical Magazine and Journal of Science s~r 6 21112-121 1911
9 Liebster H Uben den widerstrand von kugeln Annalen Der Physik ser 4 82 541- 562 1 927
63
10 McAdams William H Heat transmission 3d ed New York McGraw- Hill 1954 532 p
11 Pai Shih- I Viscous f l ow theory I Laminar flow Princeton D Van Nostrand 1956 384 p
12 Prandtlbull Ludwi g Es sentials of fluid dynamics London Blackie amp Son 1954 452 p
13 Relf i F Discussion of the results of measure shyments of the resistance of wires with some additionshyal tests of the resistance of wires of small diame shyters In Technical report of the Advisory Committee for Aeronautics London) March 1914 p 47 - 51 (Report and memoranda no 102 )
14 Stokes George Gabriel Mathematical and physical papers Vol 3 Cambridge University Press 1922 413 p
15 Tomotika s and T Aoi The steady flow of a viscous fluid past an elliptic cylinder and a flat plate at smal l Reynolds numbers Quarterly Journal of Me chanics and Applie d Ma thematics 6 290- 312 1953
16 Wieselsbergo r c Versuche Ube r der luftwiderstand gerundeter und kant iger korper Er gebnisse der Aeroshydynamischen Versucbsansta l t Vol 2 G~ttingen 1923 80 p
FIGURE e- PHOTOGRAPH OF SPHERES CYLINDERS AND PLATES
34
holes were drilled so that each plate could be used for
two geometric ratios by changing the wires (See for
example plates la and lb in Table I
35
EXPERI MENTA L PROCEDURE
Viscosity and Density Calibration
A calibrated hydrometer measuring to the nearest
0002 was used to measure the density Table VI shows that
the effect of temperature on density is practically negli shy
gible in the small temperature range used
A Brookfield Synchro-lectric viscometer was used to
measure the viscosity of both the light and heavy oil
Figures 18 and 19 show the effect of temperature on visshy
cosity In addition the viscosity of the light oil was
checke d using the falling ball method and the equation
D2--ltA (f s bull fl) g (34) l 8v
The viscometer was calibrated by the National Bureau of bull
Standards and was accurate to l tb
Velocity Measurements
The velocity of movement through the oil was measured
by determining the rate of rotation of the pulleys with a
stop watch Usually the time for 10 revolutions was
measured at the highe r ve locities and for 5 revolutions at
the low velocities From this information and the di
amaters of the pulleys the velocities ere calculated
36
The time was measured to the nearest tenth of a second
Since the measured time was usually between 20 and 40
aeconds 1 the error in ~easuring velocity was considered to
be less tha~ 0 5~
force Measurements
The object connected to the scale 1 was dropped to the
bottom of the oil drum The motor was started and the scale
was read as the object vms being pulled towards the top of
the drum Two or three readings were taken for each object
at each velocity In nearly all cases these readings were
the same
37
ti XPER I MENTAL RE STJLTS
The dra g coefficient and the Reynolds number were
calculated by the use of Equations (l or (15) for each of
the spheres cylinders and plates from the measured
quantities of force and velocity a~d the values of the vis shy
cosity and density corresponding to the temperature of the
oil It was necessary to ~ubtract from the measured force
the force on the wire The corrected force measurement was
then used to determine the drag coefficient The force on
the wire has been determined as being proportional to the
velocity A correction curve relating force on the wire
and ve l ocity is plo tted in Figure 9 for the li ght oil and
Fi gure 10 for the heavy oil
The calculated drag coefficients Reynolds numbers
and velocities along with the measured force for the spheres
cylinders flat plates - parallel flow and flat plates shy
perpendicular flow have been tabulated in Tables II III
I V and v respectively
The calculated drag coefficients have been plotted as
a function of the Reynolds number on logarithic graph paper
with geometric ratios as a parameter
Drag coefficients for the spheres are plo tted in
Figure 11 The data for the cylinders are plotted in
CD_ bull 0 G 0
03
Tshy02
01
10 20 30 410 50 60 70 80
VELOCITY- FTJSEC
DRAG FORCE ON THE WIRE-LIGHT OIL
FIGURE 9
I -shy I -middot -- -shy -1shy _i-i I --~ I I _ -middot- shy I i
_I_ - _ middot- LL I l l tmiddot - middot1middot ~- - - - -+i middotshy I - --+-cl - l
1 1 I I IV jc---- --r--middotmiddottmiddot r-middotmiddot--tmiddotmiddot---shy _____ _L __ --~- --1shy middotmiddotr-r-middott- 1 -f-f-T- _~ +-L--1---~- 1--l
~- - shy I-+---Rmiddot-- I I I l i ~~ i -~~ ~- -T f i rshy ~-- --shy i- ----~-- shy - middot1 shy
I i I i I I 1--- -middot - fshy middot i----1---+-shy - i-middot -~+-- --~- --~-- ---- -t+ I v-~~ -middot j
i I middot 1_ _ I tmiddot---+-+1-+--li~+middot -+--+-+-1-+-+-+-+--tc--1-+-t-11-shy - middot --t- 1---t- t----tmiddotshy --~-- -middot i-shy I 1i - ~ i I i v i middotmiddotmiddot
[~v +L~ + ~ - I~~j-+ r V I ~t--- -~-- I +---~-- I f-middot ---1-- ~ -- --- ) Li --+--+--+-+-+-+--1--+--+---t---4 -1--1--+-+--+-l-i
tl~ I I Q Y +l~~ii-+-++++-middotHH-++-+-+-+--H--++ -i t Imiddot i i 1 j _V I f1 r-t~-middot l--r-tshy -~ 7 middot 1 -shy middot middotmiddot I
DRAG FORCE ON THE WIRE- HEAVY OIL
FIGURE 10
40
+shy l i~ltgt ~ bull r-rshy I i t _l
1 lf-1-1 l+r+ fJ-Ct I+ t li 1~t rtH r+l rf-l It llil I I
l l~pound 11 1 ~middot ~~middott ~ It lqf L
t I+--= ~r 17 -Er I _ ~ _pound~- sect Imiddot I+
iU=ff=t 1 +~ t_ - ~ r 111= t h=
I middot
t= IE I 1 1
plusmn~ kplusmni - -STOKE S EQ
(~ l h+middot
ru HmiddotHti+H1 11
c lffii l t~ 4 ~ ~middot ~ff l ~ ~h i ltlri
1 yen~ middot I ~ I I T ~ gt l+t H+h l+ i j l tfl-l Imiddotmiddot ft+ ++ l f+ Imiddotmiddot I+ I+ middott bulli I 1middot1 I ftt-1shy middot I middot r 11 I IH Ij ~ ~ middotishy J F 1= 6= ~
=f l~iit rtti l lit~ I FS lf~ l=i-+
l-11ffi tt lr 1 ~1 -t =l=Rttl 1ft i- 1 ~ I+ I
~~ lflJ
t I lfl m ~~WFB Lt
41plusmn811 IF I Hir tt ft itttplusmn i I~
1-+++middot
I ~ I (~ ffitrHf1 Ittmiddot ~ l r i H-t-r r HHt m 11 H++ I
bull I I
1_ _ F bullmiddot Imiddotmiddot t-- 1-T h iT
f-t+ ftt I+ I lt + T Imiddot 1
1t _plusmn middot~~ ~- 11shy
=a~ 1~ - =itf lttti
H I
=
DATA FOR SPHERES
FIGURE II
41
I -1---1-1-+--+--Ti-+-------+----r--shy --r--- -shy + t----+shy ----4-~---+-f----f--+-f--l--1 I t--shy --t-- ---+-shy
J-+-~f--~~ -___l_ ~---
i 1 L~L~-~tr-l----H~4-----~-f------+------+-----+----+---+middot-t-middot-H5000
middot~ ~ m- ~ - ~t plusmn~ 3t i t~ -f--- bullbull - ~~ h middot-
01 0~ 10
Re
-
DATA FOR CYLINDERS - LD = 6 8 AND 12
FIGURE I 4
44
Figures 12 13 and 14 The data for LD values of 16 24
and 32 were nearly the same and have been plotted to gether
i n Figure 12 In addition the curves for the other LD
ratios determined fro m Fib~res 13 and 14 have been drawn
in Figure 12 so that the effect of the length-to-diameter
is clearly shown Figure 13 shows the data for LD values
of 2 and 4 and the curves determined from this data
Firure 14 shows the data for LD values of 6 8 and 12
and the curves determined from this data
The data for flat plates in parallel flow are plotted
in Fi gure 15 A correction factor for the edge effect has
beon used so that the width-to-length ratio is not a
parameter in this plot A portion of the data of Janour
(5 p 31) is also shown in the diagram
The data for fla t plates in perpendicular flow is
plotted in Figures 16 a nd 17 Figure 16 shows the data for
WL values of 2 Also the curves for the three WL ratios
1 2 and 4 have been drawn in the fi gure Figure 17 shows
the data for WL values of 1 and 4 The curves determined
from the data have also been dravm in the figure
45
10~ ~ ~--- -shy
t==Ff1TR=+ iJ+--_-_--r_-_---+-+---+--+-+--_---_-~r-=r~=~+--=---=---=---=--~=--=_~1=_--=_~_-middot~~--+-+-t~ 1 Ll~+--+-- ---jtshyl~t L--+ I
I
P------ _l -- --1---L i
20 ~-- I ~g I --- - ---+-- r t L_shy
~ ~B 1) I --o-o- JONES - () - - ~~ p f---j- -~-- e e JANOU R
c gt ~c ~ ------ JANSSEN I 0 0 ~ I
IO ~2=i~~~~~~a=~~f=j= ---- TOM OTIKA bulll= I
~~n ~~--~~~~~~o~~~~~--4- NDCIgttl o shy
-
~--~~~~~+--+~+--4-r-~1+-~-middot+1~ ~ --H--~-~~os I i i i-4 ---~T I I f-- t --- li-------~--+-_--+--t-----~~-~_+---_-_-_--+------+-+-__+-[- +_- ___ _______ __+---+-r-+--H----_+--r--------+shy
it I+ ~ bull t ~1 ri j t++t+t++tft bullm H--~+H-t+t-++H-f+t+~HtttH t bull~H-IrttI-H
iH-H u nH m
I
t H+t-~ 1-r f-tj
i it iT -t middotHt I I I I Ill
~middot __
r middotshy
i I r-
f H- jLj f r H rr t~
II
t f f-l -t+tt ~ ==_ =~middot irE
I I
I
I
f
I --
i
t
1 r bull - r
~- ltt++l=tUtt~S-t+t+++~-++U +HJJm~-fl~HHtt1 tttn ll+t-Tt-~- ~ r fH T --r -1 t ---t- -tshy w _+ _ I-shy middotI
-shy -r- + Hbull Hshy t-I --r++ -t iHr -1 H-e-- -t I 1IT 1
1 H-rf-I IJftJ Jf+i+ ~ L
=+shy - tjshy rtmiddotshy ~ -
+ H 1-Jt I tt o =tt ~-
~1 l +fill l plusmn~ fplusmn -shy + I t-
DATA FOR FLAT PLATES PERPENDICULAR FLOW- WL= I 4
FIGURE 17
48
DI SCUSS ION OF RESULTS
Correction and Accuracy of Measurements
After a few pre liminary force measurements with the
spheres and a check with Stokes law (Equation 2) it was
apparent that the drag force on the wire was appreciable
and needed to be considered It was decided to take a
series of measurements with the spheres and calculate the
difference between the measured force and the force calcushy
lated from Stokes law The difference in force could then
be attributed to the drag on the wire If Stokes law is
followed the force on the wire should be proportional to
the velocity
A series of twenty measurements of the force on the
spheres was taken for each oil and the difference between
the measured force and that calcula ted by Stokes 1 law was
determined For each oil this difference as plo tted vs
the velocity The points grouped fairly ell around a
strai ght line nearly passing through the origin The
method of least squares was used to determine the equation
of the line best fitting the da t a The equa tion of the
line for the li bht oil tas found to be
Fe bullbull05605v - oooa (35)
which was determined at about 62 7degF Since the intercept
49
of the line is very close to zero it is believed that the
line is a good indication of the drag on the wire The
equation of the line for the heavy oil was found to be
F - 19llv I oo2o1 (36 ) c shy
which was determined at about 64 2deg The intercept of this
line is also quite close to zero These lines plotted in
Fi poundures 9 and 10 were used throughout the investigation
for the correction factor of the drag on the wires For
the cylinders and flat plates in parallel flow which were
pulled by two wires the values determined from Equations
35) and (36) were doubled For the plates in perpendicular
flow pulled by four wires the correction force was multishy
plied by four
The spring scale had 12 ounce divisions but could be
read to the nearest sixth of an ounce Some of the measureshy
ments of force were under an ounce hence a considerable
spread of the measurements was noticed in the pre liminary
data and throughout the experiment However sufficient
points were obtained so that it was possible to draw a
reliable curve through the data in all casas An analysis
was made to determine the average deviation from Stokes
equation for the spheres It raa found that the average
deviation was 15 1 for the light oil 16 6 for the heavy
oil and 15 9 overall The maximum deviation was 89
50
Inspection of the other data shows that these deviations
are also representative of the cylinders and flat plates
The force measurement is the least accurate part of the
experiment Other insignificant errors are introduced by
a small variation in the temperature This variation was
held to about 10 from the temperature of the calibrated
correction curve The velocity measurements and the
dimensions of the cylinders spheres and pl~ tes are conshy
sidered go od enough so tha t no appreciable errors occur
In order to e l iminate the WL parameter for flat plates
in parallel f l ow an additional factor for the effect of
the edges was subtracted from the measured force Janour
(5 p 27) presented the foll owing equation for the edge
correction for one edge of a flat plate in parallel flow
F ~ lv~ bull (37 ) edge gc
In present work this equation as doubled because both
edges of the plates were submerged in fluid It is assumed
in appl ying this correction that the lowe r limit of a
Reynolds number of 10 proposed by Janour can be extended
close to 0 1
Analysis of Results
Forty of the points for the spheres were used to get
51
the correction factor for the wires The remaining thirty
points are well erouped about Stokes law
The data for cylinders for LD ratios of 16 24 and
32 did not seem to be se gregated therefore these data
were plotted together It would seem that in the low range
of Reyno l ds numbers an LD of 16 and greater can be con shy
sidered an ~nfini tely long cylinder The other LD ratios
of 2 4 6 a 12 provided fairly distinct and separate
lines The best straight lines were drawn through the data
for each of the LD ratios It was evident that in eaeh
case a slope of -1 on a lo g-log graph gave the best straight
line which would indicate that the force varies directly
as the velocity It was possible to develop an empirical
expression relating dra g coefficient Reynolds number and
LD The following equation was obtained from the straight
line plots of Re vs fd for the various LD ratios
(38 )
Equation (38) applies for Reyno l ds numbers from 01 to 10
and for LD ratios of 2 to 16 For LD ratios greater
than 16
10 re (39 )
The data for flat plates in parallel flow is plotted
in Figure 15 after the correction factor for tho edge
52
effect was subtracted When the edge correction is made
no effect of WL ratio is indicated This result would be
expected The data followed a straight line with a slope
of -1 up to a Reynolds number of 2 After that a curve was
dravm connecting the line to that obtained by Janour The
equation for the straight section of the curve is
f - 6 (40)- Re
which applies for Reynolds numbers of 0 1 to 2 0 Here
a gain the force is proportional to the velocity Vfuen
determining drag force for flat plates in parallel flow
the force is first calculated from Equations (40) and (15 )
then the edge correction is added
The effect of the geometric ratios is clearly shown in
the data for flat plates in perpendicul ar flow which are
plotted in Figures 16 and 17 As with the other data the
best straight line was drawn through the various points
for eaoh of the WL ratios Again the line had a slope of
-1 The equation relating fd Re and wL was found t o be
rd 37 (w) -o 3o (41)Irel
which applies for Reynolds numbers of about 05 to 2 0 and
WL ratios of 1 to 4 It is possible but it has not been
proved that Equation (41) is suitable for higher WL ratios
The exponent on WL in Equation 41) is very close to that
53
on L D i n Equation ( 38 )~ It i s possible t ha t these
exponents are t he same but this cannot be sho~~ depound1nitely
until more accura te da ta are available It would be exshy
pected that a s the Reynolds number approaches zero t he
effect of geometric ratios would be the same for cylinders
and fla t pla tes in perpendicula r flow
It is seen in the t a bles of data that occasionally a
ne gative force was obtained because the correction applie d
due to t he wire dra g was greater than the mea sured force
These points obviously are incorrect This occurred only
for the smallest plates in the heavy oil at t he highest
velocities However these knom bad points occur in less
tha n 5~ of the data
It is clearl y shown that for cylinders and plates the
fd increases as L D or W L decreases This is in direct
contrast to Wiesel aberger s investigation However his
work is for hi gher Reynolds numbers at which a turbulent
wake forms bull
Comparison of Results with Other Data and Theoretical So l utions
The data for sphere~ a grees of course with Stokes
l aw since that law was used to determine the correction
factor for the wire Liebster (9 Pbull 548 ) has
54
substantiated Stokes equation
There are no experimental data with which to compare
the results of the cylinders Wieselsbergers minimum
Reynolds number of 4 is above the ran ge covered in the preshy
sent investigation The da ta for the highest LD ratios
(16 24 and 32) does agree almost exactly wi t h the solution
of Allen and Southwell (1 P bull 141) (LD =00) in the range
of Reynolds numbers from 0 1 to 1 0 Allen and Southwells
solution a greed with the data of Wieselsberger (16 p 22)
However the present data is above the theoretical solutions
of Lamb (8 p 112-121) throughout the range of Reynolds
numbers from 0 01 to 1 0 and above the solutions of
Bairstow Cave and Lang (2 p 404) I mai (4 p 157) and
Tomotika and Aoi (15 p 302) for Reynolds numbers of 0 1
to 1 0 Allen and Southwells solution a grees dth both
Wieselsberger 1 s a nd the present data Their solution and
the present data represent the best means for predicting
drag coefficients for flow over long cylinders for Reynolds
numbers of 0 01 to 10 It should be remembered that the
o t her solutions should a gree with eac h other since they
were all essentially derived by linearizing the Na viershy
Stokes equation
The data for flat plates in parallel flow is
55
considerably above the theoretical solutions of Janssen
(6 p 183 ) and Tomotika and Aoi (15 Pbull 302) However
Fi f~re 15 shows that a smooth transition occurs bet een
the present work and the data of Janour (5 P bull 31) The
present data considerably extend the experimental inforshy
mation previously available for laminar flow paral lel to
flat plates In the re gion of Reynol ds numbers less than
2 the drag coefficient is shown to be inversely proportional
to the Reynolds number Janours data covers a range of
Reynolds numbers from 11 to 1000 The results of the
present investigation line up with Janours results which
in turn on extrapolation to higher Reyno l ds numbers
(greater than 1000) make a smooth transition into Blasius
curve represented by Equation (10) At Reyno l ds numbers
greater than 20 000 the drag coefficient is inversely proshy
portional to the square root of the Reynolds number
The data for flat plates in perpendicular flow is conshy
siderably above the solutions of Tomotika and Aoi
(15 p 302) and Imai (4 p 157 However their solutions
f or cylinders and plates in parallel flow are also below
the present data Also it should be remembered that their
solutions are for infinitely wide plates If a value of
WL of above 100 is used in Equation (41) then the present
data and the solutions of Tomotika and Aoi are fairly close
56
The present results indicate that Equation (41~ can be
used with an accuracy of 15 to 20 within the limitations
of the equation (WL 1 to 4 Re = 0 05 to 2)
57
SUM RY AND CONCLUSIONS
Only a small amount of work has been done in the past
on the study of laminar flow over immersed bodies There
are many areas in the chemical process industries and the
field of aeronautics where this information would be very
helpful The purpose of the present investi gation wa s to
study the almost totally unexplored range of Reynol ds
numbers from 0 01 to 10
Drag coefficients have been determined for spheres
cylinders and flat plates in paralle l and perpendicular
flow The drag coefficients have been plotted as a
function of the Reynolds number with dimension ratios as
a parameter on lo g-log graphs The best straight lines
have been drawn through the data In all cases these lines
had a slope of -1 hich shows that the dra g coefficient is
inversely proportional to the Reynolds number at very low
Reynolds numbers for all shapes and dimension ratios The
following equations have been determined from the data
For cylinders
fd - 27 L -0 36 (38 ) - Re ())
which applies for Reynolds numbers of 0 01 to 1 and LD of
2 to 16 For LD greater than 16 the equation is
58
(39)
For flat plates in parallel flow a correction factor has
been applied to account for the edge effect The equation
which applies for Reyno l ds numbers of 0 1 to 2 is
f 6Re
(40)
For flat plates in perpendicular flow
f d
- 37 - Re (w) t -
0 bull 30 (41)
wbieh applies for W L of 1 to 4 and Reynolds numbers of
0 05 to 2
It is concluded tha t Equations (38-41) give the best
values of drag coefficients within an accuracy of 20~ for
the range of Reynolds numbers that were considered Also
it is evident that the dimension ratios are a n important
factor in determining the drag coefficient for a given
Reynolds number Furthermore the drag coefficient inshy
creases with decreasing values of L D or W L for a constant
Reynolds number The da ta obtained in this investi gation
compare favorably with the other experimental data and with
some of the theoretical sol utions It should be remembered
that when comparing the experimental data with theoretical
solutions that practically all of the solutions are for an
infinitely long cylinder or an infinitely wide plate
It is recommended tha t the present apparatus be
59
modified so that a force of 001 pound can be measured
Also it would improve tho accuracy to set up a constant
temperature bath so that the temperature of the oil can not
vary over 02degF A few check points on the present data
is all that is necessary to confirm the validity of
Equations (38- 41) It is also r ecommended that only SAE 140
oil be used and that 2 inches should be the minimum plate
width and cylinder length to be studi3d These conditions
would help to maintain the accuracy of the correction force
for the wire
60
~WMENCIATURE
Symbol Dimensions
A area sq ft
D diameter ft
F force lb f
L length ft
M mas s lb m Re Reynolds number Dvf= -ltr w width ft
a area sq ft
b characteristic length ft
d diameter ft
f drag coefficientfd
gravitation constant l b mft gc 2= 32 17 l b _ rsec
1 length ft
m mass l b bullm
p pressure lbrsqft
r radius ft
t time see
u velocity ft sec
v velocity ft sec
w width ft
61
Symbol Dimensions
X xbullcoordinate ft
y y- coordinate ft
o( vorticity
time sec
viscosity lb m ft -sec
kinematic viscosity ft 2sec
circumference diameter = 3 1416
3density lb m ft
function
stream function
Laplacian operator
infinity
Subscripts
c corrected
f force
1 l iquid
m mass
p projected
s solid
w wetted
62
BI BLIOGRAPHY
1 Allan D N de G and R v Southwell Re laxation methods applied to determine the motion in two di shymensions of a viscous fluid past a fixed cylinder Quarterly Journal of Mechanics and Applied Mathe shymatics 8 129-145 1955
2 Bairstow L B M Cave and E D Lang The reshysistance of a cylinder moving in a viscous fluid Philosophical Transactions of the Royal Society of London ser A 223383- 432 1923
3 Goldstein Sidney The steady flow of viscous fluid past a fixed spherical obstacle at small Reyno l ds numbers Proceedings of the Royal Society of London ser A 123225-235 1929
4 Imai I A new method of solving Oseens equations and its application to the flow past an inclined elliptic cylinder Proceedings of the Royal Society of London ser A 224 141-160 1954
5 Janour Zbynek Resistance of a plate in paralle l flow at low Reyno lds numbers Washington Nov 1951 40 p National Advisory Committee for Aeronautics Te chnica l Memorandum 1316)
6 Janssen E An analog solution of the Navier-Stokes equation for the case of flow past a f l at plate at low Reynolds numbers In 1956 Heat Transfer and Fluid Mechanics Institute (Preprints of Papers) p 173-183
7 Knudsen James G and Donal d L Katz Fluid Dynamics a nd Heat Transfer Ann Arbor University of Michigan 1953 243 p (Michi gan University Engineering Research Bulletin no 37)
8 La~b Horace On the uniform motion of a spherethrough a viscous fluid Philosophical Magazine and Journal of Science s~r 6 21112-121 1911
9 Liebster H Uben den widerstrand von kugeln Annalen Der Physik ser 4 82 541- 562 1 927
63
10 McAdams William H Heat transmission 3d ed New York McGraw- Hill 1954 532 p
11 Pai Shih- I Viscous f l ow theory I Laminar flow Princeton D Van Nostrand 1956 384 p
12 Prandtlbull Ludwi g Es sentials of fluid dynamics London Blackie amp Son 1954 452 p
13 Relf i F Discussion of the results of measure shyments of the resistance of wires with some additionshyal tests of the resistance of wires of small diame shyters In Technical report of the Advisory Committee for Aeronautics London) March 1914 p 47 - 51 (Report and memoranda no 102 )
14 Stokes George Gabriel Mathematical and physical papers Vol 3 Cambridge University Press 1922 413 p
15 Tomotika s and T Aoi The steady flow of a viscous fluid past an elliptic cylinder and a flat plate at smal l Reynolds numbers Quarterly Journal of Me chanics and Applie d Ma thematics 6 290- 312 1953
16 Wieselsbergo r c Versuche Ube r der luftwiderstand gerundeter und kant iger korper Er gebnisse der Aeroshydynamischen Versucbsansta l t Vol 2 G~ttingen 1923 80 p
FIGURE e- PHOTOGRAPH OF SPHERES CYLINDERS AND PLATES
34
holes were drilled so that each plate could be used for
two geometric ratios by changing the wires (See for
example plates la and lb in Table I
35
EXPERI MENTA L PROCEDURE
Viscosity and Density Calibration
A calibrated hydrometer measuring to the nearest
0002 was used to measure the density Table VI shows that
the effect of temperature on density is practically negli shy
gible in the small temperature range used
A Brookfield Synchro-lectric viscometer was used to
measure the viscosity of both the light and heavy oil
Figures 18 and 19 show the effect of temperature on visshy
cosity In addition the viscosity of the light oil was
checke d using the falling ball method and the equation
D2--ltA (f s bull fl) g (34) l 8v
The viscometer was calibrated by the National Bureau of bull
Standards and was accurate to l tb
Velocity Measurements
The velocity of movement through the oil was measured
by determining the rate of rotation of the pulleys with a
stop watch Usually the time for 10 revolutions was
measured at the highe r ve locities and for 5 revolutions at
the low velocities From this information and the di
amaters of the pulleys the velocities ere calculated
36
The time was measured to the nearest tenth of a second
Since the measured time was usually between 20 and 40
aeconds 1 the error in ~easuring velocity was considered to
be less tha~ 0 5~
force Measurements
The object connected to the scale 1 was dropped to the
bottom of the oil drum The motor was started and the scale
was read as the object vms being pulled towards the top of
the drum Two or three readings were taken for each object
at each velocity In nearly all cases these readings were
the same
37
ti XPER I MENTAL RE STJLTS
The dra g coefficient and the Reynolds number were
calculated by the use of Equations (l or (15) for each of
the spheres cylinders and plates from the measured
quantities of force and velocity a~d the values of the vis shy
cosity and density corresponding to the temperature of the
oil It was necessary to ~ubtract from the measured force
the force on the wire The corrected force measurement was
then used to determine the drag coefficient The force on
the wire has been determined as being proportional to the
velocity A correction curve relating force on the wire
and ve l ocity is plo tted in Figure 9 for the li ght oil and
Fi gure 10 for the heavy oil
The calculated drag coefficients Reynolds numbers
and velocities along with the measured force for the spheres
cylinders flat plates - parallel flow and flat plates shy
perpendicular flow have been tabulated in Tables II III
I V and v respectively
The calculated drag coefficients have been plotted as
a function of the Reynolds number on logarithic graph paper
with geometric ratios as a parameter
Drag coefficients for the spheres are plo tted in
Figure 11 The data for the cylinders are plotted in
CD_ bull 0 G 0
03
Tshy02
01
10 20 30 410 50 60 70 80
VELOCITY- FTJSEC
DRAG FORCE ON THE WIRE-LIGHT OIL
FIGURE 9
I -shy I -middot -- -shy -1shy _i-i I --~ I I _ -middot- shy I i
_I_ - _ middot- LL I l l tmiddot - middot1middot ~- - - - -+i middotshy I - --+-cl - l
1 1 I I IV jc---- --r--middotmiddottmiddot r-middotmiddot--tmiddotmiddot---shy _____ _L __ --~- --1shy middotmiddotr-r-middott- 1 -f-f-T- _~ +-L--1---~- 1--l
~- - shy I-+---Rmiddot-- I I I l i ~~ i -~~ ~- -T f i rshy ~-- --shy i- ----~-- shy - middot1 shy
I i I i I I 1--- -middot - fshy middot i----1---+-shy - i-middot -~+-- --~- --~-- ---- -t+ I v-~~ -middot j
i I middot 1_ _ I tmiddot---+-+1-+--li~+middot -+--+-+-1-+-+-+-+--tc--1-+-t-11-shy - middot --t- 1---t- t----tmiddotshy --~-- -middot i-shy I 1i - ~ i I i v i middotmiddotmiddot
[~v +L~ + ~ - I~~j-+ r V I ~t--- -~-- I +---~-- I f-middot ---1-- ~ -- --- ) Li --+--+--+-+-+-+--1--+--+---t---4 -1--1--+-+--+-l-i
tl~ I I Q Y +l~~ii-+-++++-middotHH-++-+-+-+--H--++ -i t Imiddot i i 1 j _V I f1 r-t~-middot l--r-tshy -~ 7 middot 1 -shy middot middotmiddot I
DRAG FORCE ON THE WIRE- HEAVY OIL
FIGURE 10
40
+shy l i~ltgt ~ bull r-rshy I i t _l
1 lf-1-1 l+r+ fJ-Ct I+ t li 1~t rtH r+l rf-l It llil I I
l l~pound 11 1 ~middot ~~middott ~ It lqf L
t I+--= ~r 17 -Er I _ ~ _pound~- sect Imiddot I+
iU=ff=t 1 +~ t_ - ~ r 111= t h=
I middot
t= IE I 1 1
plusmn~ kplusmni - -STOKE S EQ
(~ l h+middot
ru HmiddotHti+H1 11
c lffii l t~ 4 ~ ~middot ~ff l ~ ~h i ltlri
1 yen~ middot I ~ I I T ~ gt l+t H+h l+ i j l tfl-l Imiddotmiddot ft+ ++ l f+ Imiddotmiddot I+ I+ middott bulli I 1middot1 I ftt-1shy middot I middot r 11 I IH Ij ~ ~ middotishy J F 1= 6= ~
=f l~iit rtti l lit~ I FS lf~ l=i-+
l-11ffi tt lr 1 ~1 -t =l=Rttl 1ft i- 1 ~ I+ I
~~ lflJ
t I lfl m ~~WFB Lt
41plusmn811 IF I Hir tt ft itttplusmn i I~
1-+++middot
I ~ I (~ ffitrHf1 Ittmiddot ~ l r i H-t-r r HHt m 11 H++ I
bull I I
1_ _ F bullmiddot Imiddotmiddot t-- 1-T h iT
f-t+ ftt I+ I lt + T Imiddot 1
1t _plusmn middot~~ ~- 11shy
=a~ 1~ - =itf lttti
H I
=
DATA FOR SPHERES
FIGURE II
41
I -1---1-1-+--+--Ti-+-------+----r--shy --r--- -shy + t----+shy ----4-~---+-f----f--+-f--l--1 I t--shy --t-- ---+-shy
J-+-~f--~~ -___l_ ~---
i 1 L~L~-~tr-l----H~4-----~-f------+------+-----+----+---+middot-t-middot-H5000
middot~ ~ m- ~ - ~t plusmn~ 3t i t~ -f--- bullbull - ~~ h middot-
01 0~ 10
Re
-
DATA FOR CYLINDERS - LD = 6 8 AND 12
FIGURE I 4
44
Figures 12 13 and 14 The data for LD values of 16 24
and 32 were nearly the same and have been plotted to gether
i n Figure 12 In addition the curves for the other LD
ratios determined fro m Fib~res 13 and 14 have been drawn
in Figure 12 so that the effect of the length-to-diameter
is clearly shown Figure 13 shows the data for LD values
of 2 and 4 and the curves determined from this data
Firure 14 shows the data for LD values of 6 8 and 12
and the curves determined from this data
The data for flat plates in parallel flow are plotted
in Fi gure 15 A correction factor for the edge effect has
beon used so that the width-to-length ratio is not a
parameter in this plot A portion of the data of Janour
(5 p 31) is also shown in the diagram
The data for fla t plates in perpendicular flow is
plotted in Figures 16 a nd 17 Figure 16 shows the data for
WL values of 2 Also the curves for the three WL ratios
1 2 and 4 have been drawn in the fi gure Figure 17 shows
the data for WL values of 1 and 4 The curves determined
from the data have also been dravm in the figure
45
10~ ~ ~--- -shy
t==Ff1TR=+ iJ+--_-_--r_-_---+-+---+--+-+--_---_-~r-=r~=~+--=---=---=---=--~=--=_~1=_--=_~_-middot~~--+-+-t~ 1 Ll~+--+-- ---jtshyl~t L--+ I
I
P------ _l -- --1---L i
20 ~-- I ~g I --- - ---+-- r t L_shy
~ ~B 1) I --o-o- JONES - () - - ~~ p f---j- -~-- e e JANOU R
c gt ~c ~ ------ JANSSEN I 0 0 ~ I
IO ~2=i~~~~~~a=~~f=j= ---- TOM OTIKA bulll= I
~~n ~~--~~~~~~o~~~~~--4- NDCIgttl o shy
-
~--~~~~~+--+~+--4-r-~1+-~-middot+1~ ~ --H--~-~~os I i i i-4 ---~T I I f-- t --- li-------~--+-_--+--t-----~~-~_+---_-_-_--+------+-+-__+-[- +_- ___ _______ __+---+-r-+--H----_+--r--------+shy
it I+ ~ bull t ~1 ri j t++t+t++tft bullm H--~+H-t+t-++H-f+t+~HtttH t bull~H-IrttI-H
iH-H u nH m
I
t H+t-~ 1-r f-tj
i it iT -t middotHt I I I I Ill
~middot __
r middotshy
i I r-
f H- jLj f r H rr t~
II
t f f-l -t+tt ~ ==_ =~middot irE
I I
I
I
f
I --
i
t
1 r bull - r
~- ltt++l=tUtt~S-t+t+++~-++U +HJJm~-fl~HHtt1 tttn ll+t-Tt-~- ~ r fH T --r -1 t ---t- -tshy w _+ _ I-shy middotI
-shy -r- + Hbull Hshy t-I --r++ -t iHr -1 H-e-- -t I 1IT 1
1 H-rf-I IJftJ Jf+i+ ~ L
=+shy - tjshy rtmiddotshy ~ -
+ H 1-Jt I tt o =tt ~-
~1 l +fill l plusmn~ fplusmn -shy + I t-
DATA FOR FLAT PLATES PERPENDICULAR FLOW- WL= I 4
FIGURE 17
48
DI SCUSS ION OF RESULTS
Correction and Accuracy of Measurements
After a few pre liminary force measurements with the
spheres and a check with Stokes law (Equation 2) it was
apparent that the drag force on the wire was appreciable
and needed to be considered It was decided to take a
series of measurements with the spheres and calculate the
difference between the measured force and the force calcushy
lated from Stokes law The difference in force could then
be attributed to the drag on the wire If Stokes law is
followed the force on the wire should be proportional to
the velocity
A series of twenty measurements of the force on the
spheres was taken for each oil and the difference between
the measured force and that calcula ted by Stokes 1 law was
determined For each oil this difference as plo tted vs
the velocity The points grouped fairly ell around a
strai ght line nearly passing through the origin The
method of least squares was used to determine the equation
of the line best fitting the da t a The equa tion of the
line for the li bht oil tas found to be
Fe bullbull05605v - oooa (35)
which was determined at about 62 7degF Since the intercept
49
of the line is very close to zero it is believed that the
line is a good indication of the drag on the wire The
equation of the line for the heavy oil was found to be
F - 19llv I oo2o1 (36 ) c shy
which was determined at about 64 2deg The intercept of this
line is also quite close to zero These lines plotted in
Fi poundures 9 and 10 were used throughout the investigation
for the correction factor of the drag on the wires For
the cylinders and flat plates in parallel flow which were
pulled by two wires the values determined from Equations
35) and (36) were doubled For the plates in perpendicular
flow pulled by four wires the correction force was multishy
plied by four
The spring scale had 12 ounce divisions but could be
read to the nearest sixth of an ounce Some of the measureshy
ments of force were under an ounce hence a considerable
spread of the measurements was noticed in the pre liminary
data and throughout the experiment However sufficient
points were obtained so that it was possible to draw a
reliable curve through the data in all casas An analysis
was made to determine the average deviation from Stokes
equation for the spheres It raa found that the average
deviation was 15 1 for the light oil 16 6 for the heavy
oil and 15 9 overall The maximum deviation was 89
50
Inspection of the other data shows that these deviations
are also representative of the cylinders and flat plates
The force measurement is the least accurate part of the
experiment Other insignificant errors are introduced by
a small variation in the temperature This variation was
held to about 10 from the temperature of the calibrated
correction curve The velocity measurements and the
dimensions of the cylinders spheres and pl~ tes are conshy
sidered go od enough so tha t no appreciable errors occur
In order to e l iminate the WL parameter for flat plates
in parallel f l ow an additional factor for the effect of
the edges was subtracted from the measured force Janour
(5 p 27) presented the foll owing equation for the edge
correction for one edge of a flat plate in parallel flow
F ~ lv~ bull (37 ) edge gc
In present work this equation as doubled because both
edges of the plates were submerged in fluid It is assumed
in appl ying this correction that the lowe r limit of a
Reynolds number of 10 proposed by Janour can be extended
close to 0 1
Analysis of Results
Forty of the points for the spheres were used to get
51
the correction factor for the wires The remaining thirty
points are well erouped about Stokes law
The data for cylinders for LD ratios of 16 24 and
32 did not seem to be se gregated therefore these data
were plotted together It would seem that in the low range
of Reyno l ds numbers an LD of 16 and greater can be con shy
sidered an ~nfini tely long cylinder The other LD ratios
of 2 4 6 a 12 provided fairly distinct and separate
lines The best straight lines were drawn through the data
for each of the LD ratios It was evident that in eaeh
case a slope of -1 on a lo g-log graph gave the best straight
line which would indicate that the force varies directly
as the velocity It was possible to develop an empirical
expression relating dra g coefficient Reynolds number and
LD The following equation was obtained from the straight
line plots of Re vs fd for the various LD ratios
(38 )
Equation (38) applies for Reyno l ds numbers from 01 to 10
and for LD ratios of 2 to 16 For LD ratios greater
than 16
10 re (39 )
The data for flat plates in parallel flow is plotted
in Figure 15 after the correction factor for tho edge
52
effect was subtracted When the edge correction is made
no effect of WL ratio is indicated This result would be
expected The data followed a straight line with a slope
of -1 up to a Reynolds number of 2 After that a curve was
dravm connecting the line to that obtained by Janour The
equation for the straight section of the curve is
f - 6 (40)- Re
which applies for Reynolds numbers of 0 1 to 2 0 Here
a gain the force is proportional to the velocity Vfuen
determining drag force for flat plates in parallel flow
the force is first calculated from Equations (40) and (15 )
then the edge correction is added
The effect of the geometric ratios is clearly shown in
the data for flat plates in perpendicul ar flow which are
plotted in Figures 16 and 17 As with the other data the
best straight line was drawn through the various points
for eaoh of the WL ratios Again the line had a slope of
-1 The equation relating fd Re and wL was found t o be
rd 37 (w) -o 3o (41)Irel
which applies for Reynolds numbers of about 05 to 2 0 and
WL ratios of 1 to 4 It is possible but it has not been
proved that Equation (41) is suitable for higher WL ratios
The exponent on WL in Equation 41) is very close to that
53
on L D i n Equation ( 38 )~ It i s possible t ha t these
exponents are t he same but this cannot be sho~~ depound1nitely
until more accura te da ta are available It would be exshy
pected that a s the Reynolds number approaches zero t he
effect of geometric ratios would be the same for cylinders
and fla t pla tes in perpendicula r flow
It is seen in the t a bles of data that occasionally a
ne gative force was obtained because the correction applie d
due to t he wire dra g was greater than the mea sured force
These points obviously are incorrect This occurred only
for the smallest plates in the heavy oil at t he highest
velocities However these knom bad points occur in less
tha n 5~ of the data
It is clearl y shown that for cylinders and plates the
fd increases as L D or W L decreases This is in direct
contrast to Wiesel aberger s investigation However his
work is for hi gher Reynolds numbers at which a turbulent
wake forms bull
Comparison of Results with Other Data and Theoretical So l utions
The data for sphere~ a grees of course with Stokes
l aw since that law was used to determine the correction
factor for the wire Liebster (9 Pbull 548 ) has
54
substantiated Stokes equation
There are no experimental data with which to compare
the results of the cylinders Wieselsbergers minimum
Reynolds number of 4 is above the ran ge covered in the preshy
sent investigation The da ta for the highest LD ratios
(16 24 and 32) does agree almost exactly wi t h the solution
of Allen and Southwell (1 P bull 141) (LD =00) in the range
of Reynolds numbers from 0 1 to 1 0 Allen and Southwells
solution a greed with the data of Wieselsberger (16 p 22)
However the present data is above the theoretical solutions
of Lamb (8 p 112-121) throughout the range of Reynolds
numbers from 0 01 to 1 0 and above the solutions of
Bairstow Cave and Lang (2 p 404) I mai (4 p 157) and
Tomotika and Aoi (15 p 302) for Reynolds numbers of 0 1
to 1 0 Allen and Southwells solution a grees dth both
Wieselsberger 1 s a nd the present data Their solution and
the present data represent the best means for predicting
drag coefficients for flow over long cylinders for Reynolds
numbers of 0 01 to 10 It should be remembered that the
o t her solutions should a gree with eac h other since they
were all essentially derived by linearizing the Na viershy
Stokes equation
The data for flat plates in parallel flow is
55
considerably above the theoretical solutions of Janssen
(6 p 183 ) and Tomotika and Aoi (15 Pbull 302) However
Fi f~re 15 shows that a smooth transition occurs bet een
the present work and the data of Janour (5 P bull 31) The
present data considerably extend the experimental inforshy
mation previously available for laminar flow paral lel to
flat plates In the re gion of Reynol ds numbers less than
2 the drag coefficient is shown to be inversely proportional
to the Reynolds number Janours data covers a range of
Reynolds numbers from 11 to 1000 The results of the
present investigation line up with Janours results which
in turn on extrapolation to higher Reyno l ds numbers
(greater than 1000) make a smooth transition into Blasius
curve represented by Equation (10) At Reyno l ds numbers
greater than 20 000 the drag coefficient is inversely proshy
portional to the square root of the Reynolds number
The data for flat plates in perpendicular flow is conshy
siderably above the solutions of Tomotika and Aoi
(15 p 302) and Imai (4 p 157 However their solutions
f or cylinders and plates in parallel flow are also below
the present data Also it should be remembered that their
solutions are for infinitely wide plates If a value of
WL of above 100 is used in Equation (41) then the present
data and the solutions of Tomotika and Aoi are fairly close
56
The present results indicate that Equation (41~ can be
used with an accuracy of 15 to 20 within the limitations
of the equation (WL 1 to 4 Re = 0 05 to 2)
57
SUM RY AND CONCLUSIONS
Only a small amount of work has been done in the past
on the study of laminar flow over immersed bodies There
are many areas in the chemical process industries and the
field of aeronautics where this information would be very
helpful The purpose of the present investi gation wa s to
study the almost totally unexplored range of Reynol ds
numbers from 0 01 to 10
Drag coefficients have been determined for spheres
cylinders and flat plates in paralle l and perpendicular
flow The drag coefficients have been plotted as a
function of the Reynolds number with dimension ratios as
a parameter on lo g-log graphs The best straight lines
have been drawn through the data In all cases these lines
had a slope of -1 hich shows that the dra g coefficient is
inversely proportional to the Reynolds number at very low
Reynolds numbers for all shapes and dimension ratios The
following equations have been determined from the data
For cylinders
fd - 27 L -0 36 (38 ) - Re ())
which applies for Reynolds numbers of 0 01 to 1 and LD of
2 to 16 For LD greater than 16 the equation is
58
(39)
For flat plates in parallel flow a correction factor has
been applied to account for the edge effect The equation
which applies for Reyno l ds numbers of 0 1 to 2 is
f 6Re
(40)
For flat plates in perpendicular flow
f d
- 37 - Re (w) t -
0 bull 30 (41)
wbieh applies for W L of 1 to 4 and Reynolds numbers of
0 05 to 2
It is concluded tha t Equations (38-41) give the best
values of drag coefficients within an accuracy of 20~ for
the range of Reynolds numbers that were considered Also
it is evident that the dimension ratios are a n important
factor in determining the drag coefficient for a given
Reynolds number Furthermore the drag coefficient inshy
creases with decreasing values of L D or W L for a constant
Reynolds number The da ta obtained in this investi gation
compare favorably with the other experimental data and with
some of the theoretical sol utions It should be remembered
that when comparing the experimental data with theoretical
solutions that practically all of the solutions are for an
infinitely long cylinder or an infinitely wide plate
It is recommended tha t the present apparatus be
59
modified so that a force of 001 pound can be measured
Also it would improve tho accuracy to set up a constant
temperature bath so that the temperature of the oil can not
vary over 02degF A few check points on the present data
is all that is necessary to confirm the validity of
Equations (38- 41) It is also r ecommended that only SAE 140
oil be used and that 2 inches should be the minimum plate
width and cylinder length to be studi3d These conditions
would help to maintain the accuracy of the correction force
for the wire
60
~WMENCIATURE
Symbol Dimensions
A area sq ft
D diameter ft
F force lb f
L length ft
M mas s lb m Re Reynolds number Dvf= -ltr w width ft
a area sq ft
b characteristic length ft
d diameter ft
f drag coefficientfd
gravitation constant l b mft gc 2= 32 17 l b _ rsec
1 length ft
m mass l b bullm
p pressure lbrsqft
r radius ft
t time see
u velocity ft sec
v velocity ft sec
w width ft
61
Symbol Dimensions
X xbullcoordinate ft
y y- coordinate ft
o( vorticity
time sec
viscosity lb m ft -sec
kinematic viscosity ft 2sec
circumference diameter = 3 1416
3density lb m ft
function
stream function
Laplacian operator
infinity
Subscripts
c corrected
f force
1 l iquid
m mass
p projected
s solid
w wetted
62
BI BLIOGRAPHY
1 Allan D N de G and R v Southwell Re laxation methods applied to determine the motion in two di shymensions of a viscous fluid past a fixed cylinder Quarterly Journal of Mechanics and Applied Mathe shymatics 8 129-145 1955
2 Bairstow L B M Cave and E D Lang The reshysistance of a cylinder moving in a viscous fluid Philosophical Transactions of the Royal Society of London ser A 223383- 432 1923
3 Goldstein Sidney The steady flow of viscous fluid past a fixed spherical obstacle at small Reyno l ds numbers Proceedings of the Royal Society of London ser A 123225-235 1929
4 Imai I A new method of solving Oseens equations and its application to the flow past an inclined elliptic cylinder Proceedings of the Royal Society of London ser A 224 141-160 1954
5 Janour Zbynek Resistance of a plate in paralle l flow at low Reyno lds numbers Washington Nov 1951 40 p National Advisory Committee for Aeronautics Te chnica l Memorandum 1316)
6 Janssen E An analog solution of the Navier-Stokes equation for the case of flow past a f l at plate at low Reynolds numbers In 1956 Heat Transfer and Fluid Mechanics Institute (Preprints of Papers) p 173-183
7 Knudsen James G and Donal d L Katz Fluid Dynamics a nd Heat Transfer Ann Arbor University of Michigan 1953 243 p (Michi gan University Engineering Research Bulletin no 37)
8 La~b Horace On the uniform motion of a spherethrough a viscous fluid Philosophical Magazine and Journal of Science s~r 6 21112-121 1911
9 Liebster H Uben den widerstrand von kugeln Annalen Der Physik ser 4 82 541- 562 1 927
63
10 McAdams William H Heat transmission 3d ed New York McGraw- Hill 1954 532 p
11 Pai Shih- I Viscous f l ow theory I Laminar flow Princeton D Van Nostrand 1956 384 p
12 Prandtlbull Ludwi g Es sentials of fluid dynamics London Blackie amp Son 1954 452 p
13 Relf i F Discussion of the results of measure shyments of the resistance of wires with some additionshyal tests of the resistance of wires of small diame shyters In Technical report of the Advisory Committee for Aeronautics London) March 1914 p 47 - 51 (Report and memoranda no 102 )
14 Stokes George Gabriel Mathematical and physical papers Vol 3 Cambridge University Press 1922 413 p
15 Tomotika s and T Aoi The steady flow of a viscous fluid past an elliptic cylinder and a flat plate at smal l Reynolds numbers Quarterly Journal of Me chanics and Applie d Ma thematics 6 290- 312 1953
16 Wieselsbergo r c Versuche Ube r der luftwiderstand gerundeter und kant iger korper Er gebnisse der Aeroshydynamischen Versucbsansta l t Vol 2 G~ttingen 1923 80 p
FIGURE e- PHOTOGRAPH OF SPHERES CYLINDERS AND PLATES
34
holes were drilled so that each plate could be used for
two geometric ratios by changing the wires (See for
example plates la and lb in Table I
35
EXPERI MENTA L PROCEDURE
Viscosity and Density Calibration
A calibrated hydrometer measuring to the nearest
0002 was used to measure the density Table VI shows that
the effect of temperature on density is practically negli shy
gible in the small temperature range used
A Brookfield Synchro-lectric viscometer was used to
measure the viscosity of both the light and heavy oil
Figures 18 and 19 show the effect of temperature on visshy
cosity In addition the viscosity of the light oil was
checke d using the falling ball method and the equation
D2--ltA (f s bull fl) g (34) l 8v
The viscometer was calibrated by the National Bureau of bull
Standards and was accurate to l tb
Velocity Measurements
The velocity of movement through the oil was measured
by determining the rate of rotation of the pulleys with a
stop watch Usually the time for 10 revolutions was
measured at the highe r ve locities and for 5 revolutions at
the low velocities From this information and the di
amaters of the pulleys the velocities ere calculated
36
The time was measured to the nearest tenth of a second
Since the measured time was usually between 20 and 40
aeconds 1 the error in ~easuring velocity was considered to
be less tha~ 0 5~
force Measurements
The object connected to the scale 1 was dropped to the
bottom of the oil drum The motor was started and the scale
was read as the object vms being pulled towards the top of
the drum Two or three readings were taken for each object
at each velocity In nearly all cases these readings were
the same
37
ti XPER I MENTAL RE STJLTS
The dra g coefficient and the Reynolds number were
calculated by the use of Equations (l or (15) for each of
the spheres cylinders and plates from the measured
quantities of force and velocity a~d the values of the vis shy
cosity and density corresponding to the temperature of the
oil It was necessary to ~ubtract from the measured force
the force on the wire The corrected force measurement was
then used to determine the drag coefficient The force on
the wire has been determined as being proportional to the
velocity A correction curve relating force on the wire
and ve l ocity is plo tted in Figure 9 for the li ght oil and
Fi gure 10 for the heavy oil
The calculated drag coefficients Reynolds numbers
and velocities along with the measured force for the spheres
cylinders flat plates - parallel flow and flat plates shy
perpendicular flow have been tabulated in Tables II III
I V and v respectively
The calculated drag coefficients have been plotted as
a function of the Reynolds number on logarithic graph paper
with geometric ratios as a parameter
Drag coefficients for the spheres are plo tted in
Figure 11 The data for the cylinders are plotted in
CD_ bull 0 G 0
03
Tshy02
01
10 20 30 410 50 60 70 80
VELOCITY- FTJSEC
DRAG FORCE ON THE WIRE-LIGHT OIL
FIGURE 9
I -shy I -middot -- -shy -1shy _i-i I --~ I I _ -middot- shy I i
_I_ - _ middot- LL I l l tmiddot - middot1middot ~- - - - -+i middotshy I - --+-cl - l
1 1 I I IV jc---- --r--middotmiddottmiddot r-middotmiddot--tmiddotmiddot---shy _____ _L __ --~- --1shy middotmiddotr-r-middott- 1 -f-f-T- _~ +-L--1---~- 1--l
~- - shy I-+---Rmiddot-- I I I l i ~~ i -~~ ~- -T f i rshy ~-- --shy i- ----~-- shy - middot1 shy
I i I i I I 1--- -middot - fshy middot i----1---+-shy - i-middot -~+-- --~- --~-- ---- -t+ I v-~~ -middot j
i I middot 1_ _ I tmiddot---+-+1-+--li~+middot -+--+-+-1-+-+-+-+--tc--1-+-t-11-shy - middot --t- 1---t- t----tmiddotshy --~-- -middot i-shy I 1i - ~ i I i v i middotmiddotmiddot
[~v +L~ + ~ - I~~j-+ r V I ~t--- -~-- I +---~-- I f-middot ---1-- ~ -- --- ) Li --+--+--+-+-+-+--1--+--+---t---4 -1--1--+-+--+-l-i
tl~ I I Q Y +l~~ii-+-++++-middotHH-++-+-+-+--H--++ -i t Imiddot i i 1 j _V I f1 r-t~-middot l--r-tshy -~ 7 middot 1 -shy middot middotmiddot I
DRAG FORCE ON THE WIRE- HEAVY OIL
FIGURE 10
40
+shy l i~ltgt ~ bull r-rshy I i t _l
1 lf-1-1 l+r+ fJ-Ct I+ t li 1~t rtH r+l rf-l It llil I I
l l~pound 11 1 ~middot ~~middott ~ It lqf L
t I+--= ~r 17 -Er I _ ~ _pound~- sect Imiddot I+
iU=ff=t 1 +~ t_ - ~ r 111= t h=
I middot
t= IE I 1 1
plusmn~ kplusmni - -STOKE S EQ
(~ l h+middot
ru HmiddotHti+H1 11
c lffii l t~ 4 ~ ~middot ~ff l ~ ~h i ltlri
1 yen~ middot I ~ I I T ~ gt l+t H+h l+ i j l tfl-l Imiddotmiddot ft+ ++ l f+ Imiddotmiddot I+ I+ middott bulli I 1middot1 I ftt-1shy middot I middot r 11 I IH Ij ~ ~ middotishy J F 1= 6= ~
=f l~iit rtti l lit~ I FS lf~ l=i-+
l-11ffi tt lr 1 ~1 -t =l=Rttl 1ft i- 1 ~ I+ I
~~ lflJ
t I lfl m ~~WFB Lt
41plusmn811 IF I Hir tt ft itttplusmn i I~
1-+++middot
I ~ I (~ ffitrHf1 Ittmiddot ~ l r i H-t-r r HHt m 11 H++ I
bull I I
1_ _ F bullmiddot Imiddotmiddot t-- 1-T h iT
f-t+ ftt I+ I lt + T Imiddot 1
1t _plusmn middot~~ ~- 11shy
=a~ 1~ - =itf lttti
H I
=
DATA FOR SPHERES
FIGURE II
41
I -1---1-1-+--+--Ti-+-------+----r--shy --r--- -shy + t----+shy ----4-~---+-f----f--+-f--l--1 I t--shy --t-- ---+-shy
J-+-~f--~~ -___l_ ~---
i 1 L~L~-~tr-l----H~4-----~-f------+------+-----+----+---+middot-t-middot-H5000
middot~ ~ m- ~ - ~t plusmn~ 3t i t~ -f--- bullbull - ~~ h middot-
01 0~ 10
Re
-
DATA FOR CYLINDERS - LD = 6 8 AND 12
FIGURE I 4
44
Figures 12 13 and 14 The data for LD values of 16 24
and 32 were nearly the same and have been plotted to gether
i n Figure 12 In addition the curves for the other LD
ratios determined fro m Fib~res 13 and 14 have been drawn
in Figure 12 so that the effect of the length-to-diameter
is clearly shown Figure 13 shows the data for LD values
of 2 and 4 and the curves determined from this data
Firure 14 shows the data for LD values of 6 8 and 12
and the curves determined from this data
The data for flat plates in parallel flow are plotted
in Fi gure 15 A correction factor for the edge effect has
beon used so that the width-to-length ratio is not a
parameter in this plot A portion of the data of Janour
(5 p 31) is also shown in the diagram
The data for fla t plates in perpendicular flow is
plotted in Figures 16 a nd 17 Figure 16 shows the data for
WL values of 2 Also the curves for the three WL ratios
1 2 and 4 have been drawn in the fi gure Figure 17 shows
the data for WL values of 1 and 4 The curves determined
from the data have also been dravm in the figure
45
10~ ~ ~--- -shy
t==Ff1TR=+ iJ+--_-_--r_-_---+-+---+--+-+--_---_-~r-=r~=~+--=---=---=---=--~=--=_~1=_--=_~_-middot~~--+-+-t~ 1 Ll~+--+-- ---jtshyl~t L--+ I
I
P------ _l -- --1---L i
20 ~-- I ~g I --- - ---+-- r t L_shy
~ ~B 1) I --o-o- JONES - () - - ~~ p f---j- -~-- e e JANOU R
c gt ~c ~ ------ JANSSEN I 0 0 ~ I
IO ~2=i~~~~~~a=~~f=j= ---- TOM OTIKA bulll= I
~~n ~~--~~~~~~o~~~~~--4- NDCIgttl o shy
-
~--~~~~~+--+~+--4-r-~1+-~-middot+1~ ~ --H--~-~~os I i i i-4 ---~T I I f-- t --- li-------~--+-_--+--t-----~~-~_+---_-_-_--+------+-+-__+-[- +_- ___ _______ __+---+-r-+--H----_+--r--------+shy
it I+ ~ bull t ~1 ri j t++t+t++tft bullm H--~+H-t+t-++H-f+t+~HtttH t bull~H-IrttI-H
iH-H u nH m
I
t H+t-~ 1-r f-tj
i it iT -t middotHt I I I I Ill
~middot __
r middotshy
i I r-
f H- jLj f r H rr t~
II
t f f-l -t+tt ~ ==_ =~middot irE
I I
I
I
f
I --
i
t
1 r bull - r
~- ltt++l=tUtt~S-t+t+++~-++U +HJJm~-fl~HHtt1 tttn ll+t-Tt-~- ~ r fH T --r -1 t ---t- -tshy w _+ _ I-shy middotI
-shy -r- + Hbull Hshy t-I --r++ -t iHr -1 H-e-- -t I 1IT 1
1 H-rf-I IJftJ Jf+i+ ~ L
=+shy - tjshy rtmiddotshy ~ -
+ H 1-Jt I tt o =tt ~-
~1 l +fill l plusmn~ fplusmn -shy + I t-
DATA FOR FLAT PLATES PERPENDICULAR FLOW- WL= I 4
FIGURE 17
48
DI SCUSS ION OF RESULTS
Correction and Accuracy of Measurements
After a few pre liminary force measurements with the
spheres and a check with Stokes law (Equation 2) it was
apparent that the drag force on the wire was appreciable
and needed to be considered It was decided to take a
series of measurements with the spheres and calculate the
difference between the measured force and the force calcushy
lated from Stokes law The difference in force could then
be attributed to the drag on the wire If Stokes law is
followed the force on the wire should be proportional to
the velocity
A series of twenty measurements of the force on the
spheres was taken for each oil and the difference between
the measured force and that calcula ted by Stokes 1 law was
determined For each oil this difference as plo tted vs
the velocity The points grouped fairly ell around a
strai ght line nearly passing through the origin The
method of least squares was used to determine the equation
of the line best fitting the da t a The equa tion of the
line for the li bht oil tas found to be
Fe bullbull05605v - oooa (35)
which was determined at about 62 7degF Since the intercept
49
of the line is very close to zero it is believed that the
line is a good indication of the drag on the wire The
equation of the line for the heavy oil was found to be
F - 19llv I oo2o1 (36 ) c shy
which was determined at about 64 2deg The intercept of this
line is also quite close to zero These lines plotted in
Fi poundures 9 and 10 were used throughout the investigation
for the correction factor of the drag on the wires For
the cylinders and flat plates in parallel flow which were
pulled by two wires the values determined from Equations
35) and (36) were doubled For the plates in perpendicular
flow pulled by four wires the correction force was multishy
plied by four
The spring scale had 12 ounce divisions but could be
read to the nearest sixth of an ounce Some of the measureshy
ments of force were under an ounce hence a considerable
spread of the measurements was noticed in the pre liminary
data and throughout the experiment However sufficient
points were obtained so that it was possible to draw a
reliable curve through the data in all casas An analysis
was made to determine the average deviation from Stokes
equation for the spheres It raa found that the average
deviation was 15 1 for the light oil 16 6 for the heavy
oil and 15 9 overall The maximum deviation was 89
50
Inspection of the other data shows that these deviations
are also representative of the cylinders and flat plates
The force measurement is the least accurate part of the
experiment Other insignificant errors are introduced by
a small variation in the temperature This variation was
held to about 10 from the temperature of the calibrated
correction curve The velocity measurements and the
dimensions of the cylinders spheres and pl~ tes are conshy
sidered go od enough so tha t no appreciable errors occur
In order to e l iminate the WL parameter for flat plates
in parallel f l ow an additional factor for the effect of
the edges was subtracted from the measured force Janour
(5 p 27) presented the foll owing equation for the edge
correction for one edge of a flat plate in parallel flow
F ~ lv~ bull (37 ) edge gc
In present work this equation as doubled because both
edges of the plates were submerged in fluid It is assumed
in appl ying this correction that the lowe r limit of a
Reynolds number of 10 proposed by Janour can be extended
close to 0 1
Analysis of Results
Forty of the points for the spheres were used to get
51
the correction factor for the wires The remaining thirty
points are well erouped about Stokes law
The data for cylinders for LD ratios of 16 24 and
32 did not seem to be se gregated therefore these data
were plotted together It would seem that in the low range
of Reyno l ds numbers an LD of 16 and greater can be con shy
sidered an ~nfini tely long cylinder The other LD ratios
of 2 4 6 a 12 provided fairly distinct and separate
lines The best straight lines were drawn through the data
for each of the LD ratios It was evident that in eaeh
case a slope of -1 on a lo g-log graph gave the best straight
line which would indicate that the force varies directly
as the velocity It was possible to develop an empirical
expression relating dra g coefficient Reynolds number and
LD The following equation was obtained from the straight
line plots of Re vs fd for the various LD ratios
(38 )
Equation (38) applies for Reyno l ds numbers from 01 to 10
and for LD ratios of 2 to 16 For LD ratios greater
than 16
10 re (39 )
The data for flat plates in parallel flow is plotted
in Figure 15 after the correction factor for tho edge
52
effect was subtracted When the edge correction is made
no effect of WL ratio is indicated This result would be
expected The data followed a straight line with a slope
of -1 up to a Reynolds number of 2 After that a curve was
dravm connecting the line to that obtained by Janour The
equation for the straight section of the curve is
f - 6 (40)- Re
which applies for Reynolds numbers of 0 1 to 2 0 Here
a gain the force is proportional to the velocity Vfuen
determining drag force for flat plates in parallel flow
the force is first calculated from Equations (40) and (15 )
then the edge correction is added
The effect of the geometric ratios is clearly shown in
the data for flat plates in perpendicul ar flow which are
plotted in Figures 16 and 17 As with the other data the
best straight line was drawn through the various points
for eaoh of the WL ratios Again the line had a slope of
-1 The equation relating fd Re and wL was found t o be
rd 37 (w) -o 3o (41)Irel
which applies for Reynolds numbers of about 05 to 2 0 and
WL ratios of 1 to 4 It is possible but it has not been
proved that Equation (41) is suitable for higher WL ratios
The exponent on WL in Equation 41) is very close to that
53
on L D i n Equation ( 38 )~ It i s possible t ha t these
exponents are t he same but this cannot be sho~~ depound1nitely
until more accura te da ta are available It would be exshy
pected that a s the Reynolds number approaches zero t he
effect of geometric ratios would be the same for cylinders
and fla t pla tes in perpendicula r flow
It is seen in the t a bles of data that occasionally a
ne gative force was obtained because the correction applie d
due to t he wire dra g was greater than the mea sured force
These points obviously are incorrect This occurred only
for the smallest plates in the heavy oil at t he highest
velocities However these knom bad points occur in less
tha n 5~ of the data
It is clearl y shown that for cylinders and plates the
fd increases as L D or W L decreases This is in direct
contrast to Wiesel aberger s investigation However his
work is for hi gher Reynolds numbers at which a turbulent
wake forms bull
Comparison of Results with Other Data and Theoretical So l utions
The data for sphere~ a grees of course with Stokes
l aw since that law was used to determine the correction
factor for the wire Liebster (9 Pbull 548 ) has
54
substantiated Stokes equation
There are no experimental data with which to compare
the results of the cylinders Wieselsbergers minimum
Reynolds number of 4 is above the ran ge covered in the preshy
sent investigation The da ta for the highest LD ratios
(16 24 and 32) does agree almost exactly wi t h the solution
of Allen and Southwell (1 P bull 141) (LD =00) in the range
of Reynolds numbers from 0 1 to 1 0 Allen and Southwells
solution a greed with the data of Wieselsberger (16 p 22)
However the present data is above the theoretical solutions
of Lamb (8 p 112-121) throughout the range of Reynolds
numbers from 0 01 to 1 0 and above the solutions of
Bairstow Cave and Lang (2 p 404) I mai (4 p 157) and
Tomotika and Aoi (15 p 302) for Reynolds numbers of 0 1
to 1 0 Allen and Southwells solution a grees dth both
Wieselsberger 1 s a nd the present data Their solution and
the present data represent the best means for predicting
drag coefficients for flow over long cylinders for Reynolds
numbers of 0 01 to 10 It should be remembered that the
o t her solutions should a gree with eac h other since they
were all essentially derived by linearizing the Na viershy
Stokes equation
The data for flat plates in parallel flow is
55
considerably above the theoretical solutions of Janssen
(6 p 183 ) and Tomotika and Aoi (15 Pbull 302) However
Fi f~re 15 shows that a smooth transition occurs bet een
the present work and the data of Janour (5 P bull 31) The
present data considerably extend the experimental inforshy
mation previously available for laminar flow paral lel to
flat plates In the re gion of Reynol ds numbers less than
2 the drag coefficient is shown to be inversely proportional
to the Reynolds number Janours data covers a range of
Reynolds numbers from 11 to 1000 The results of the
present investigation line up with Janours results which
in turn on extrapolation to higher Reyno l ds numbers
(greater than 1000) make a smooth transition into Blasius
curve represented by Equation (10) At Reyno l ds numbers
greater than 20 000 the drag coefficient is inversely proshy
portional to the square root of the Reynolds number
The data for flat plates in perpendicular flow is conshy
siderably above the solutions of Tomotika and Aoi
(15 p 302) and Imai (4 p 157 However their solutions
f or cylinders and plates in parallel flow are also below
the present data Also it should be remembered that their
solutions are for infinitely wide plates If a value of
WL of above 100 is used in Equation (41) then the present
data and the solutions of Tomotika and Aoi are fairly close
56
The present results indicate that Equation (41~ can be
used with an accuracy of 15 to 20 within the limitations
of the equation (WL 1 to 4 Re = 0 05 to 2)
57
SUM RY AND CONCLUSIONS
Only a small amount of work has been done in the past
on the study of laminar flow over immersed bodies There
are many areas in the chemical process industries and the
field of aeronautics where this information would be very
helpful The purpose of the present investi gation wa s to
study the almost totally unexplored range of Reynol ds
numbers from 0 01 to 10
Drag coefficients have been determined for spheres
cylinders and flat plates in paralle l and perpendicular
flow The drag coefficients have been plotted as a
function of the Reynolds number with dimension ratios as
a parameter on lo g-log graphs The best straight lines
have been drawn through the data In all cases these lines
had a slope of -1 hich shows that the dra g coefficient is
inversely proportional to the Reynolds number at very low
Reynolds numbers for all shapes and dimension ratios The
following equations have been determined from the data
For cylinders
fd - 27 L -0 36 (38 ) - Re ())
which applies for Reynolds numbers of 0 01 to 1 and LD of
2 to 16 For LD greater than 16 the equation is
58
(39)
For flat plates in parallel flow a correction factor has
been applied to account for the edge effect The equation
which applies for Reyno l ds numbers of 0 1 to 2 is
f 6Re
(40)
For flat plates in perpendicular flow
f d
- 37 - Re (w) t -
0 bull 30 (41)
wbieh applies for W L of 1 to 4 and Reynolds numbers of
0 05 to 2
It is concluded tha t Equations (38-41) give the best
values of drag coefficients within an accuracy of 20~ for
the range of Reynolds numbers that were considered Also
it is evident that the dimension ratios are a n important
factor in determining the drag coefficient for a given
Reynolds number Furthermore the drag coefficient inshy
creases with decreasing values of L D or W L for a constant
Reynolds number The da ta obtained in this investi gation
compare favorably with the other experimental data and with
some of the theoretical sol utions It should be remembered
that when comparing the experimental data with theoretical
solutions that practically all of the solutions are for an
infinitely long cylinder or an infinitely wide plate
It is recommended tha t the present apparatus be
59
modified so that a force of 001 pound can be measured
Also it would improve tho accuracy to set up a constant
temperature bath so that the temperature of the oil can not
vary over 02degF A few check points on the present data
is all that is necessary to confirm the validity of
Equations (38- 41) It is also r ecommended that only SAE 140
oil be used and that 2 inches should be the minimum plate
width and cylinder length to be studi3d These conditions
would help to maintain the accuracy of the correction force
for the wire
60
~WMENCIATURE
Symbol Dimensions
A area sq ft
D diameter ft
F force lb f
L length ft
M mas s lb m Re Reynolds number Dvf= -ltr w width ft
a area sq ft
b characteristic length ft
d diameter ft
f drag coefficientfd
gravitation constant l b mft gc 2= 32 17 l b _ rsec
1 length ft
m mass l b bullm
p pressure lbrsqft
r radius ft
t time see
u velocity ft sec
v velocity ft sec
w width ft
61
Symbol Dimensions
X xbullcoordinate ft
y y- coordinate ft
o( vorticity
time sec
viscosity lb m ft -sec
kinematic viscosity ft 2sec
circumference diameter = 3 1416
3density lb m ft
function
stream function
Laplacian operator
infinity
Subscripts
c corrected
f force
1 l iquid
m mass
p projected
s solid
w wetted
62
BI BLIOGRAPHY
1 Allan D N de G and R v Southwell Re laxation methods applied to determine the motion in two di shymensions of a viscous fluid past a fixed cylinder Quarterly Journal of Mechanics and Applied Mathe shymatics 8 129-145 1955
2 Bairstow L B M Cave and E D Lang The reshysistance of a cylinder moving in a viscous fluid Philosophical Transactions of the Royal Society of London ser A 223383- 432 1923
3 Goldstein Sidney The steady flow of viscous fluid past a fixed spherical obstacle at small Reyno l ds numbers Proceedings of the Royal Society of London ser A 123225-235 1929
4 Imai I A new method of solving Oseens equations and its application to the flow past an inclined elliptic cylinder Proceedings of the Royal Society of London ser A 224 141-160 1954
5 Janour Zbynek Resistance of a plate in paralle l flow at low Reyno lds numbers Washington Nov 1951 40 p National Advisory Committee for Aeronautics Te chnica l Memorandum 1316)
6 Janssen E An analog solution of the Navier-Stokes equation for the case of flow past a f l at plate at low Reynolds numbers In 1956 Heat Transfer and Fluid Mechanics Institute (Preprints of Papers) p 173-183
7 Knudsen James G and Donal d L Katz Fluid Dynamics a nd Heat Transfer Ann Arbor University of Michigan 1953 243 p (Michi gan University Engineering Research Bulletin no 37)
8 La~b Horace On the uniform motion of a spherethrough a viscous fluid Philosophical Magazine and Journal of Science s~r 6 21112-121 1911
9 Liebster H Uben den widerstrand von kugeln Annalen Der Physik ser 4 82 541- 562 1 927
63
10 McAdams William H Heat transmission 3d ed New York McGraw- Hill 1954 532 p
11 Pai Shih- I Viscous f l ow theory I Laminar flow Princeton D Van Nostrand 1956 384 p
12 Prandtlbull Ludwi g Es sentials of fluid dynamics London Blackie amp Son 1954 452 p
13 Relf i F Discussion of the results of measure shyments of the resistance of wires with some additionshyal tests of the resistance of wires of small diame shyters In Technical report of the Advisory Committee for Aeronautics London) March 1914 p 47 - 51 (Report and memoranda no 102 )
14 Stokes George Gabriel Mathematical and physical papers Vol 3 Cambridge University Press 1922 413 p
15 Tomotika s and T Aoi The steady flow of a viscous fluid past an elliptic cylinder and a flat plate at smal l Reynolds numbers Quarterly Journal of Me chanics and Applie d Ma thematics 6 290- 312 1953
16 Wieselsbergo r c Versuche Ube r der luftwiderstand gerundeter und kant iger korper Er gebnisse der Aeroshydynamischen Versucbsansta l t Vol 2 G~ttingen 1923 80 p
FIGURE e- PHOTOGRAPH OF SPHERES CYLINDERS AND PLATES
34
holes were drilled so that each plate could be used for
two geometric ratios by changing the wires (See for
example plates la and lb in Table I
35
EXPERI MENTA L PROCEDURE
Viscosity and Density Calibration
A calibrated hydrometer measuring to the nearest
0002 was used to measure the density Table VI shows that
the effect of temperature on density is practically negli shy
gible in the small temperature range used
A Brookfield Synchro-lectric viscometer was used to
measure the viscosity of both the light and heavy oil
Figures 18 and 19 show the effect of temperature on visshy
cosity In addition the viscosity of the light oil was
checke d using the falling ball method and the equation
D2--ltA (f s bull fl) g (34) l 8v
The viscometer was calibrated by the National Bureau of bull
Standards and was accurate to l tb
Velocity Measurements
The velocity of movement through the oil was measured
by determining the rate of rotation of the pulleys with a
stop watch Usually the time for 10 revolutions was
measured at the highe r ve locities and for 5 revolutions at
the low velocities From this information and the di
amaters of the pulleys the velocities ere calculated
36
The time was measured to the nearest tenth of a second
Since the measured time was usually between 20 and 40
aeconds 1 the error in ~easuring velocity was considered to
be less tha~ 0 5~
force Measurements
The object connected to the scale 1 was dropped to the
bottom of the oil drum The motor was started and the scale
was read as the object vms being pulled towards the top of
the drum Two or three readings were taken for each object
at each velocity In nearly all cases these readings were
the same
37
ti XPER I MENTAL RE STJLTS
The dra g coefficient and the Reynolds number were
calculated by the use of Equations (l or (15) for each of
the spheres cylinders and plates from the measured
quantities of force and velocity a~d the values of the vis shy
cosity and density corresponding to the temperature of the
oil It was necessary to ~ubtract from the measured force
the force on the wire The corrected force measurement was
then used to determine the drag coefficient The force on
the wire has been determined as being proportional to the
velocity A correction curve relating force on the wire
and ve l ocity is plo tted in Figure 9 for the li ght oil and
Fi gure 10 for the heavy oil
The calculated drag coefficients Reynolds numbers
and velocities along with the measured force for the spheres
cylinders flat plates - parallel flow and flat plates shy
perpendicular flow have been tabulated in Tables II III
I V and v respectively
The calculated drag coefficients have been plotted as
a function of the Reynolds number on logarithic graph paper
with geometric ratios as a parameter
Drag coefficients for the spheres are plo tted in
Figure 11 The data for the cylinders are plotted in
CD_ bull 0 G 0
03
Tshy02
01
10 20 30 410 50 60 70 80
VELOCITY- FTJSEC
DRAG FORCE ON THE WIRE-LIGHT OIL
FIGURE 9
I -shy I -middot -- -shy -1shy _i-i I --~ I I _ -middot- shy I i
_I_ - _ middot- LL I l l tmiddot - middot1middot ~- - - - -+i middotshy I - --+-cl - l
1 1 I I IV jc---- --r--middotmiddottmiddot r-middotmiddot--tmiddotmiddot---shy _____ _L __ --~- --1shy middotmiddotr-r-middott- 1 -f-f-T- _~ +-L--1---~- 1--l
~- - shy I-+---Rmiddot-- I I I l i ~~ i -~~ ~- -T f i rshy ~-- --shy i- ----~-- shy - middot1 shy
I i I i I I 1--- -middot - fshy middot i----1---+-shy - i-middot -~+-- --~- --~-- ---- -t+ I v-~~ -middot j
i I middot 1_ _ I tmiddot---+-+1-+--li~+middot -+--+-+-1-+-+-+-+--tc--1-+-t-11-shy - middot --t- 1---t- t----tmiddotshy --~-- -middot i-shy I 1i - ~ i I i v i middotmiddotmiddot
[~v +L~ + ~ - I~~j-+ r V I ~t--- -~-- I +---~-- I f-middot ---1-- ~ -- --- ) Li --+--+--+-+-+-+--1--+--+---t---4 -1--1--+-+--+-l-i
tl~ I I Q Y +l~~ii-+-++++-middotHH-++-+-+-+--H--++ -i t Imiddot i i 1 j _V I f1 r-t~-middot l--r-tshy -~ 7 middot 1 -shy middot middotmiddot I
DRAG FORCE ON THE WIRE- HEAVY OIL
FIGURE 10
40
+shy l i~ltgt ~ bull r-rshy I i t _l
1 lf-1-1 l+r+ fJ-Ct I+ t li 1~t rtH r+l rf-l It llil I I
l l~pound 11 1 ~middot ~~middott ~ It lqf L
t I+--= ~r 17 -Er I _ ~ _pound~- sect Imiddot I+
iU=ff=t 1 +~ t_ - ~ r 111= t h=
I middot
t= IE I 1 1
plusmn~ kplusmni - -STOKE S EQ
(~ l h+middot
ru HmiddotHti+H1 11
c lffii l t~ 4 ~ ~middot ~ff l ~ ~h i ltlri
1 yen~ middot I ~ I I T ~ gt l+t H+h l+ i j l tfl-l Imiddotmiddot ft+ ++ l f+ Imiddotmiddot I+ I+ middott bulli I 1middot1 I ftt-1shy middot I middot r 11 I IH Ij ~ ~ middotishy J F 1= 6= ~
=f l~iit rtti l lit~ I FS lf~ l=i-+
l-11ffi tt lr 1 ~1 -t =l=Rttl 1ft i- 1 ~ I+ I
~~ lflJ
t I lfl m ~~WFB Lt
41plusmn811 IF I Hir tt ft itttplusmn i I~
1-+++middot
I ~ I (~ ffitrHf1 Ittmiddot ~ l r i H-t-r r HHt m 11 H++ I
bull I I
1_ _ F bullmiddot Imiddotmiddot t-- 1-T h iT
f-t+ ftt I+ I lt + T Imiddot 1
1t _plusmn middot~~ ~- 11shy
=a~ 1~ - =itf lttti
H I
=
DATA FOR SPHERES
FIGURE II
41
I -1---1-1-+--+--Ti-+-------+----r--shy --r--- -shy + t----+shy ----4-~---+-f----f--+-f--l--1 I t--shy --t-- ---+-shy
J-+-~f--~~ -___l_ ~---
i 1 L~L~-~tr-l----H~4-----~-f------+------+-----+----+---+middot-t-middot-H5000
middot~ ~ m- ~ - ~t plusmn~ 3t i t~ -f--- bullbull - ~~ h middot-
01 0~ 10
Re
-
DATA FOR CYLINDERS - LD = 6 8 AND 12
FIGURE I 4
44
Figures 12 13 and 14 The data for LD values of 16 24
and 32 were nearly the same and have been plotted to gether
i n Figure 12 In addition the curves for the other LD
ratios determined fro m Fib~res 13 and 14 have been drawn
in Figure 12 so that the effect of the length-to-diameter
is clearly shown Figure 13 shows the data for LD values
of 2 and 4 and the curves determined from this data
Firure 14 shows the data for LD values of 6 8 and 12
and the curves determined from this data
The data for flat plates in parallel flow are plotted
in Fi gure 15 A correction factor for the edge effect has
beon used so that the width-to-length ratio is not a
parameter in this plot A portion of the data of Janour
(5 p 31) is also shown in the diagram
The data for fla t plates in perpendicular flow is
plotted in Figures 16 a nd 17 Figure 16 shows the data for
WL values of 2 Also the curves for the three WL ratios
1 2 and 4 have been drawn in the fi gure Figure 17 shows
the data for WL values of 1 and 4 The curves determined
from the data have also been dravm in the figure
45
10~ ~ ~--- -shy
t==Ff1TR=+ iJ+--_-_--r_-_---+-+---+--+-+--_---_-~r-=r~=~+--=---=---=---=--~=--=_~1=_--=_~_-middot~~--+-+-t~ 1 Ll~+--+-- ---jtshyl~t L--+ I
I
P------ _l -- --1---L i
20 ~-- I ~g I --- - ---+-- r t L_shy
~ ~B 1) I --o-o- JONES - () - - ~~ p f---j- -~-- e e JANOU R
c gt ~c ~ ------ JANSSEN I 0 0 ~ I
IO ~2=i~~~~~~a=~~f=j= ---- TOM OTIKA bulll= I
~~n ~~--~~~~~~o~~~~~--4- NDCIgttl o shy
-
~--~~~~~+--+~+--4-r-~1+-~-middot+1~ ~ --H--~-~~os I i i i-4 ---~T I I f-- t --- li-------~--+-_--+--t-----~~-~_+---_-_-_--+------+-+-__+-[- +_- ___ _______ __+---+-r-+--H----_+--r--------+shy
it I+ ~ bull t ~1 ri j t++t+t++tft bullm H--~+H-t+t-++H-f+t+~HtttH t bull~H-IrttI-H
iH-H u nH m
I
t H+t-~ 1-r f-tj
i it iT -t middotHt I I I I Ill
~middot __
r middotshy
i I r-
f H- jLj f r H rr t~
II
t f f-l -t+tt ~ ==_ =~middot irE
I I
I
I
f
I --
i
t
1 r bull - r
~- ltt++l=tUtt~S-t+t+++~-++U +HJJm~-fl~HHtt1 tttn ll+t-Tt-~- ~ r fH T --r -1 t ---t- -tshy w _+ _ I-shy middotI
-shy -r- + Hbull Hshy t-I --r++ -t iHr -1 H-e-- -t I 1IT 1
1 H-rf-I IJftJ Jf+i+ ~ L
=+shy - tjshy rtmiddotshy ~ -
+ H 1-Jt I tt o =tt ~-
~1 l +fill l plusmn~ fplusmn -shy + I t-
DATA FOR FLAT PLATES PERPENDICULAR FLOW- WL= I 4
FIGURE 17
48
DI SCUSS ION OF RESULTS
Correction and Accuracy of Measurements
After a few pre liminary force measurements with the
spheres and a check with Stokes law (Equation 2) it was
apparent that the drag force on the wire was appreciable
and needed to be considered It was decided to take a
series of measurements with the spheres and calculate the
difference between the measured force and the force calcushy
lated from Stokes law The difference in force could then
be attributed to the drag on the wire If Stokes law is
followed the force on the wire should be proportional to
the velocity
A series of twenty measurements of the force on the
spheres was taken for each oil and the difference between
the measured force and that calcula ted by Stokes 1 law was
determined For each oil this difference as plo tted vs
the velocity The points grouped fairly ell around a
strai ght line nearly passing through the origin The
method of least squares was used to determine the equation
of the line best fitting the da t a The equa tion of the
line for the li bht oil tas found to be
Fe bullbull05605v - oooa (35)
which was determined at about 62 7degF Since the intercept
49
of the line is very close to zero it is believed that the
line is a good indication of the drag on the wire The
equation of the line for the heavy oil was found to be
F - 19llv I oo2o1 (36 ) c shy
which was determined at about 64 2deg The intercept of this
line is also quite close to zero These lines plotted in
Fi poundures 9 and 10 were used throughout the investigation
for the correction factor of the drag on the wires For
the cylinders and flat plates in parallel flow which were
pulled by two wires the values determined from Equations
35) and (36) were doubled For the plates in perpendicular
flow pulled by four wires the correction force was multishy
plied by four
The spring scale had 12 ounce divisions but could be
read to the nearest sixth of an ounce Some of the measureshy
ments of force were under an ounce hence a considerable
spread of the measurements was noticed in the pre liminary
data and throughout the experiment However sufficient
points were obtained so that it was possible to draw a
reliable curve through the data in all casas An analysis
was made to determine the average deviation from Stokes
equation for the spheres It raa found that the average
deviation was 15 1 for the light oil 16 6 for the heavy
oil and 15 9 overall The maximum deviation was 89
50
Inspection of the other data shows that these deviations
are also representative of the cylinders and flat plates
The force measurement is the least accurate part of the
experiment Other insignificant errors are introduced by
a small variation in the temperature This variation was
held to about 10 from the temperature of the calibrated
correction curve The velocity measurements and the
dimensions of the cylinders spheres and pl~ tes are conshy
sidered go od enough so tha t no appreciable errors occur
In order to e l iminate the WL parameter for flat plates
in parallel f l ow an additional factor for the effect of
the edges was subtracted from the measured force Janour
(5 p 27) presented the foll owing equation for the edge
correction for one edge of a flat plate in parallel flow
F ~ lv~ bull (37 ) edge gc
In present work this equation as doubled because both
edges of the plates were submerged in fluid It is assumed
in appl ying this correction that the lowe r limit of a
Reynolds number of 10 proposed by Janour can be extended
close to 0 1
Analysis of Results
Forty of the points for the spheres were used to get
51
the correction factor for the wires The remaining thirty
points are well erouped about Stokes law
The data for cylinders for LD ratios of 16 24 and
32 did not seem to be se gregated therefore these data
were plotted together It would seem that in the low range
of Reyno l ds numbers an LD of 16 and greater can be con shy
sidered an ~nfini tely long cylinder The other LD ratios
of 2 4 6 a 12 provided fairly distinct and separate
lines The best straight lines were drawn through the data
for each of the LD ratios It was evident that in eaeh
case a slope of -1 on a lo g-log graph gave the best straight
line which would indicate that the force varies directly
as the velocity It was possible to develop an empirical
expression relating dra g coefficient Reynolds number and
LD The following equation was obtained from the straight
line plots of Re vs fd for the various LD ratios
(38 )
Equation (38) applies for Reyno l ds numbers from 01 to 10
and for LD ratios of 2 to 16 For LD ratios greater
than 16
10 re (39 )
The data for flat plates in parallel flow is plotted
in Figure 15 after the correction factor for tho edge
52
effect was subtracted When the edge correction is made
no effect of WL ratio is indicated This result would be
expected The data followed a straight line with a slope
of -1 up to a Reynolds number of 2 After that a curve was
dravm connecting the line to that obtained by Janour The
equation for the straight section of the curve is
f - 6 (40)- Re
which applies for Reynolds numbers of 0 1 to 2 0 Here
a gain the force is proportional to the velocity Vfuen
determining drag force for flat plates in parallel flow
the force is first calculated from Equations (40) and (15 )
then the edge correction is added
The effect of the geometric ratios is clearly shown in
the data for flat plates in perpendicul ar flow which are
plotted in Figures 16 and 17 As with the other data the
best straight line was drawn through the various points
for eaoh of the WL ratios Again the line had a slope of
-1 The equation relating fd Re and wL was found t o be
rd 37 (w) -o 3o (41)Irel
which applies for Reynolds numbers of about 05 to 2 0 and
WL ratios of 1 to 4 It is possible but it has not been
proved that Equation (41) is suitable for higher WL ratios
The exponent on WL in Equation 41) is very close to that
53
on L D i n Equation ( 38 )~ It i s possible t ha t these
exponents are t he same but this cannot be sho~~ depound1nitely
until more accura te da ta are available It would be exshy
pected that a s the Reynolds number approaches zero t he
effect of geometric ratios would be the same for cylinders
and fla t pla tes in perpendicula r flow
It is seen in the t a bles of data that occasionally a
ne gative force was obtained because the correction applie d
due to t he wire dra g was greater than the mea sured force
These points obviously are incorrect This occurred only
for the smallest plates in the heavy oil at t he highest
velocities However these knom bad points occur in less
tha n 5~ of the data
It is clearl y shown that for cylinders and plates the
fd increases as L D or W L decreases This is in direct
contrast to Wiesel aberger s investigation However his
work is for hi gher Reynolds numbers at which a turbulent
wake forms bull
Comparison of Results with Other Data and Theoretical So l utions
The data for sphere~ a grees of course with Stokes
l aw since that law was used to determine the correction
factor for the wire Liebster (9 Pbull 548 ) has
54
substantiated Stokes equation
There are no experimental data with which to compare
the results of the cylinders Wieselsbergers minimum
Reynolds number of 4 is above the ran ge covered in the preshy
sent investigation The da ta for the highest LD ratios
(16 24 and 32) does agree almost exactly wi t h the solution
of Allen and Southwell (1 P bull 141) (LD =00) in the range
of Reynolds numbers from 0 1 to 1 0 Allen and Southwells
solution a greed with the data of Wieselsberger (16 p 22)
However the present data is above the theoretical solutions
of Lamb (8 p 112-121) throughout the range of Reynolds
numbers from 0 01 to 1 0 and above the solutions of
Bairstow Cave and Lang (2 p 404) I mai (4 p 157) and
Tomotika and Aoi (15 p 302) for Reynolds numbers of 0 1
to 1 0 Allen and Southwells solution a grees dth both
Wieselsberger 1 s a nd the present data Their solution and
the present data represent the best means for predicting
drag coefficients for flow over long cylinders for Reynolds
numbers of 0 01 to 10 It should be remembered that the
o t her solutions should a gree with eac h other since they
were all essentially derived by linearizing the Na viershy
Stokes equation
The data for flat plates in parallel flow is
55
considerably above the theoretical solutions of Janssen
(6 p 183 ) and Tomotika and Aoi (15 Pbull 302) However
Fi f~re 15 shows that a smooth transition occurs bet een
the present work and the data of Janour (5 P bull 31) The
present data considerably extend the experimental inforshy
mation previously available for laminar flow paral lel to
flat plates In the re gion of Reynol ds numbers less than
2 the drag coefficient is shown to be inversely proportional
to the Reynolds number Janours data covers a range of
Reynolds numbers from 11 to 1000 The results of the
present investigation line up with Janours results which
in turn on extrapolation to higher Reyno l ds numbers
(greater than 1000) make a smooth transition into Blasius
curve represented by Equation (10) At Reyno l ds numbers
greater than 20 000 the drag coefficient is inversely proshy
portional to the square root of the Reynolds number
The data for flat plates in perpendicular flow is conshy
siderably above the solutions of Tomotika and Aoi
(15 p 302) and Imai (4 p 157 However their solutions
f or cylinders and plates in parallel flow are also below
the present data Also it should be remembered that their
solutions are for infinitely wide plates If a value of
WL of above 100 is used in Equation (41) then the present
data and the solutions of Tomotika and Aoi are fairly close
56
The present results indicate that Equation (41~ can be
used with an accuracy of 15 to 20 within the limitations
of the equation (WL 1 to 4 Re = 0 05 to 2)
57
SUM RY AND CONCLUSIONS
Only a small amount of work has been done in the past
on the study of laminar flow over immersed bodies There
are many areas in the chemical process industries and the
field of aeronautics where this information would be very
helpful The purpose of the present investi gation wa s to
study the almost totally unexplored range of Reynol ds
numbers from 0 01 to 10
Drag coefficients have been determined for spheres
cylinders and flat plates in paralle l and perpendicular
flow The drag coefficients have been plotted as a
function of the Reynolds number with dimension ratios as
a parameter on lo g-log graphs The best straight lines
have been drawn through the data In all cases these lines
had a slope of -1 hich shows that the dra g coefficient is
inversely proportional to the Reynolds number at very low
Reynolds numbers for all shapes and dimension ratios The
following equations have been determined from the data
For cylinders
fd - 27 L -0 36 (38 ) - Re ())
which applies for Reynolds numbers of 0 01 to 1 and LD of
2 to 16 For LD greater than 16 the equation is
58
(39)
For flat plates in parallel flow a correction factor has
been applied to account for the edge effect The equation
which applies for Reyno l ds numbers of 0 1 to 2 is
f 6Re
(40)
For flat plates in perpendicular flow
f d
- 37 - Re (w) t -
0 bull 30 (41)
wbieh applies for W L of 1 to 4 and Reynolds numbers of
0 05 to 2
It is concluded tha t Equations (38-41) give the best
values of drag coefficients within an accuracy of 20~ for
the range of Reynolds numbers that were considered Also
it is evident that the dimension ratios are a n important
factor in determining the drag coefficient for a given
Reynolds number Furthermore the drag coefficient inshy
creases with decreasing values of L D or W L for a constant
Reynolds number The da ta obtained in this investi gation
compare favorably with the other experimental data and with
some of the theoretical sol utions It should be remembered
that when comparing the experimental data with theoretical
solutions that practically all of the solutions are for an
infinitely long cylinder or an infinitely wide plate
It is recommended tha t the present apparatus be
59
modified so that a force of 001 pound can be measured
Also it would improve tho accuracy to set up a constant
temperature bath so that the temperature of the oil can not
vary over 02degF A few check points on the present data
is all that is necessary to confirm the validity of
Equations (38- 41) It is also r ecommended that only SAE 140
oil be used and that 2 inches should be the minimum plate
width and cylinder length to be studi3d These conditions
would help to maintain the accuracy of the correction force
for the wire
60
~WMENCIATURE
Symbol Dimensions
A area sq ft
D diameter ft
F force lb f
L length ft
M mas s lb m Re Reynolds number Dvf= -ltr w width ft
a area sq ft
b characteristic length ft
d diameter ft
f drag coefficientfd
gravitation constant l b mft gc 2= 32 17 l b _ rsec
1 length ft
m mass l b bullm
p pressure lbrsqft
r radius ft
t time see
u velocity ft sec
v velocity ft sec
w width ft
61
Symbol Dimensions
X xbullcoordinate ft
y y- coordinate ft
o( vorticity
time sec
viscosity lb m ft -sec
kinematic viscosity ft 2sec
circumference diameter = 3 1416
3density lb m ft
function
stream function
Laplacian operator
infinity
Subscripts
c corrected
f force
1 l iquid
m mass
p projected
s solid
w wetted
62
BI BLIOGRAPHY
1 Allan D N de G and R v Southwell Re laxation methods applied to determine the motion in two di shymensions of a viscous fluid past a fixed cylinder Quarterly Journal of Mechanics and Applied Mathe shymatics 8 129-145 1955
2 Bairstow L B M Cave and E D Lang The reshysistance of a cylinder moving in a viscous fluid Philosophical Transactions of the Royal Society of London ser A 223383- 432 1923
3 Goldstein Sidney The steady flow of viscous fluid past a fixed spherical obstacle at small Reyno l ds numbers Proceedings of the Royal Society of London ser A 123225-235 1929
4 Imai I A new method of solving Oseens equations and its application to the flow past an inclined elliptic cylinder Proceedings of the Royal Society of London ser A 224 141-160 1954
5 Janour Zbynek Resistance of a plate in paralle l flow at low Reyno lds numbers Washington Nov 1951 40 p National Advisory Committee for Aeronautics Te chnica l Memorandum 1316)
6 Janssen E An analog solution of the Navier-Stokes equation for the case of flow past a f l at plate at low Reynolds numbers In 1956 Heat Transfer and Fluid Mechanics Institute (Preprints of Papers) p 173-183
7 Knudsen James G and Donal d L Katz Fluid Dynamics a nd Heat Transfer Ann Arbor University of Michigan 1953 243 p (Michi gan University Engineering Research Bulletin no 37)
8 La~b Horace On the uniform motion of a spherethrough a viscous fluid Philosophical Magazine and Journal of Science s~r 6 21112-121 1911
9 Liebster H Uben den widerstrand von kugeln Annalen Der Physik ser 4 82 541- 562 1 927
63
10 McAdams William H Heat transmission 3d ed New York McGraw- Hill 1954 532 p
11 Pai Shih- I Viscous f l ow theory I Laminar flow Princeton D Van Nostrand 1956 384 p
12 Prandtlbull Ludwi g Es sentials of fluid dynamics London Blackie amp Son 1954 452 p
13 Relf i F Discussion of the results of measure shyments of the resistance of wires with some additionshyal tests of the resistance of wires of small diame shyters In Technical report of the Advisory Committee for Aeronautics London) March 1914 p 47 - 51 (Report and memoranda no 102 )
14 Stokes George Gabriel Mathematical and physical papers Vol 3 Cambridge University Press 1922 413 p
15 Tomotika s and T Aoi The steady flow of a viscous fluid past an elliptic cylinder and a flat plate at smal l Reynolds numbers Quarterly Journal of Me chanics and Applie d Ma thematics 6 290- 312 1953
16 Wieselsbergo r c Versuche Ube r der luftwiderstand gerundeter und kant iger korper Er gebnisse der Aeroshydynamischen Versucbsansta l t Vol 2 G~ttingen 1923 80 p
FIGURE e- PHOTOGRAPH OF SPHERES CYLINDERS AND PLATES
34
holes were drilled so that each plate could be used for
two geometric ratios by changing the wires (See for
example plates la and lb in Table I
35
EXPERI MENTA L PROCEDURE
Viscosity and Density Calibration
A calibrated hydrometer measuring to the nearest
0002 was used to measure the density Table VI shows that
the effect of temperature on density is practically negli shy
gible in the small temperature range used
A Brookfield Synchro-lectric viscometer was used to
measure the viscosity of both the light and heavy oil
Figures 18 and 19 show the effect of temperature on visshy
cosity In addition the viscosity of the light oil was
checke d using the falling ball method and the equation
D2--ltA (f s bull fl) g (34) l 8v
The viscometer was calibrated by the National Bureau of bull
Standards and was accurate to l tb
Velocity Measurements
The velocity of movement through the oil was measured
by determining the rate of rotation of the pulleys with a
stop watch Usually the time for 10 revolutions was
measured at the highe r ve locities and for 5 revolutions at
the low velocities From this information and the di
amaters of the pulleys the velocities ere calculated
36
The time was measured to the nearest tenth of a second
Since the measured time was usually between 20 and 40
aeconds 1 the error in ~easuring velocity was considered to
be less tha~ 0 5~
force Measurements
The object connected to the scale 1 was dropped to the
bottom of the oil drum The motor was started and the scale
was read as the object vms being pulled towards the top of
the drum Two or three readings were taken for each object
at each velocity In nearly all cases these readings were
the same
37
ti XPER I MENTAL RE STJLTS
The dra g coefficient and the Reynolds number were
calculated by the use of Equations (l or (15) for each of
the spheres cylinders and plates from the measured
quantities of force and velocity a~d the values of the vis shy
cosity and density corresponding to the temperature of the
oil It was necessary to ~ubtract from the measured force
the force on the wire The corrected force measurement was
then used to determine the drag coefficient The force on
the wire has been determined as being proportional to the
velocity A correction curve relating force on the wire
and ve l ocity is plo tted in Figure 9 for the li ght oil and
Fi gure 10 for the heavy oil
The calculated drag coefficients Reynolds numbers
and velocities along with the measured force for the spheres
cylinders flat plates - parallel flow and flat plates shy
perpendicular flow have been tabulated in Tables II III
I V and v respectively
The calculated drag coefficients have been plotted as
a function of the Reynolds number on logarithic graph paper
with geometric ratios as a parameter
Drag coefficients for the spheres are plo tted in
Figure 11 The data for the cylinders are plotted in
CD_ bull 0 G 0
03
Tshy02
01
10 20 30 410 50 60 70 80
VELOCITY- FTJSEC
DRAG FORCE ON THE WIRE-LIGHT OIL
FIGURE 9
I -shy I -middot -- -shy -1shy _i-i I --~ I I _ -middot- shy I i
_I_ - _ middot- LL I l l tmiddot - middot1middot ~- - - - -+i middotshy I - --+-cl - l
1 1 I I IV jc---- --r--middotmiddottmiddot r-middotmiddot--tmiddotmiddot---shy _____ _L __ --~- --1shy middotmiddotr-r-middott- 1 -f-f-T- _~ +-L--1---~- 1--l
~- - shy I-+---Rmiddot-- I I I l i ~~ i -~~ ~- -T f i rshy ~-- --shy i- ----~-- shy - middot1 shy
I i I i I I 1--- -middot - fshy middot i----1---+-shy - i-middot -~+-- --~- --~-- ---- -t+ I v-~~ -middot j
i I middot 1_ _ I tmiddot---+-+1-+--li~+middot -+--+-+-1-+-+-+-+--tc--1-+-t-11-shy - middot --t- 1---t- t----tmiddotshy --~-- -middot i-shy I 1i - ~ i I i v i middotmiddotmiddot
[~v +L~ + ~ - I~~j-+ r V I ~t--- -~-- I +---~-- I f-middot ---1-- ~ -- --- ) Li --+--+--+-+-+-+--1--+--+---t---4 -1--1--+-+--+-l-i
tl~ I I Q Y +l~~ii-+-++++-middotHH-++-+-+-+--H--++ -i t Imiddot i i 1 j _V I f1 r-t~-middot l--r-tshy -~ 7 middot 1 -shy middot middotmiddot I
DRAG FORCE ON THE WIRE- HEAVY OIL
FIGURE 10
40
+shy l i~ltgt ~ bull r-rshy I i t _l
1 lf-1-1 l+r+ fJ-Ct I+ t li 1~t rtH r+l rf-l It llil I I
l l~pound 11 1 ~middot ~~middott ~ It lqf L
t I+--= ~r 17 -Er I _ ~ _pound~- sect Imiddot I+
iU=ff=t 1 +~ t_ - ~ r 111= t h=
I middot
t= IE I 1 1
plusmn~ kplusmni - -STOKE S EQ
(~ l h+middot
ru HmiddotHti+H1 11
c lffii l t~ 4 ~ ~middot ~ff l ~ ~h i ltlri
1 yen~ middot I ~ I I T ~ gt l+t H+h l+ i j l tfl-l Imiddotmiddot ft+ ++ l f+ Imiddotmiddot I+ I+ middott bulli I 1middot1 I ftt-1shy middot I middot r 11 I IH Ij ~ ~ middotishy J F 1= 6= ~
=f l~iit rtti l lit~ I FS lf~ l=i-+
l-11ffi tt lr 1 ~1 -t =l=Rttl 1ft i- 1 ~ I+ I
~~ lflJ
t I lfl m ~~WFB Lt
41plusmn811 IF I Hir tt ft itttplusmn i I~
1-+++middot
I ~ I (~ ffitrHf1 Ittmiddot ~ l r i H-t-r r HHt m 11 H++ I
bull I I
1_ _ F bullmiddot Imiddotmiddot t-- 1-T h iT
f-t+ ftt I+ I lt + T Imiddot 1
1t _plusmn middot~~ ~- 11shy
=a~ 1~ - =itf lttti
H I
=
DATA FOR SPHERES
FIGURE II
41
I -1---1-1-+--+--Ti-+-------+----r--shy --r--- -shy + t----+shy ----4-~---+-f----f--+-f--l--1 I t--shy --t-- ---+-shy
J-+-~f--~~ -___l_ ~---
i 1 L~L~-~tr-l----H~4-----~-f------+------+-----+----+---+middot-t-middot-H5000
middot~ ~ m- ~ - ~t plusmn~ 3t i t~ -f--- bullbull - ~~ h middot-
01 0~ 10
Re
-
DATA FOR CYLINDERS - LD = 6 8 AND 12
FIGURE I 4
44
Figures 12 13 and 14 The data for LD values of 16 24
and 32 were nearly the same and have been plotted to gether
i n Figure 12 In addition the curves for the other LD
ratios determined fro m Fib~res 13 and 14 have been drawn
in Figure 12 so that the effect of the length-to-diameter
is clearly shown Figure 13 shows the data for LD values
of 2 and 4 and the curves determined from this data
Firure 14 shows the data for LD values of 6 8 and 12
and the curves determined from this data
The data for flat plates in parallel flow are plotted
in Fi gure 15 A correction factor for the edge effect has
beon used so that the width-to-length ratio is not a
parameter in this plot A portion of the data of Janour
(5 p 31) is also shown in the diagram
The data for fla t plates in perpendicular flow is
plotted in Figures 16 a nd 17 Figure 16 shows the data for
WL values of 2 Also the curves for the three WL ratios
1 2 and 4 have been drawn in the fi gure Figure 17 shows
the data for WL values of 1 and 4 The curves determined
from the data have also been dravm in the figure
45
10~ ~ ~--- -shy
t==Ff1TR=+ iJ+--_-_--r_-_---+-+---+--+-+--_---_-~r-=r~=~+--=---=---=---=--~=--=_~1=_--=_~_-middot~~--+-+-t~ 1 Ll~+--+-- ---jtshyl~t L--+ I
I
P------ _l -- --1---L i
20 ~-- I ~g I --- - ---+-- r t L_shy
~ ~B 1) I --o-o- JONES - () - - ~~ p f---j- -~-- e e JANOU R
c gt ~c ~ ------ JANSSEN I 0 0 ~ I
IO ~2=i~~~~~~a=~~f=j= ---- TOM OTIKA bulll= I
~~n ~~--~~~~~~o~~~~~--4- NDCIgttl o shy
-
~--~~~~~+--+~+--4-r-~1+-~-middot+1~ ~ --H--~-~~os I i i i-4 ---~T I I f-- t --- li-------~--+-_--+--t-----~~-~_+---_-_-_--+------+-+-__+-[- +_- ___ _______ __+---+-r-+--H----_+--r--------+shy
it I+ ~ bull t ~1 ri j t++t+t++tft bullm H--~+H-t+t-++H-f+t+~HtttH t bull~H-IrttI-H
iH-H u nH m
I
t H+t-~ 1-r f-tj
i it iT -t middotHt I I I I Ill
~middot __
r middotshy
i I r-
f H- jLj f r H rr t~
II
t f f-l -t+tt ~ ==_ =~middot irE
I I
I
I
f
I --
i
t
1 r bull - r
~- ltt++l=tUtt~S-t+t+++~-++U +HJJm~-fl~HHtt1 tttn ll+t-Tt-~- ~ r fH T --r -1 t ---t- -tshy w _+ _ I-shy middotI
-shy -r- + Hbull Hshy t-I --r++ -t iHr -1 H-e-- -t I 1IT 1
1 H-rf-I IJftJ Jf+i+ ~ L
=+shy - tjshy rtmiddotshy ~ -
+ H 1-Jt I tt o =tt ~-
~1 l +fill l plusmn~ fplusmn -shy + I t-
DATA FOR FLAT PLATES PERPENDICULAR FLOW- WL= I 4
FIGURE 17
48
DI SCUSS ION OF RESULTS
Correction and Accuracy of Measurements
After a few pre liminary force measurements with the
spheres and a check with Stokes law (Equation 2) it was
apparent that the drag force on the wire was appreciable
and needed to be considered It was decided to take a
series of measurements with the spheres and calculate the
difference between the measured force and the force calcushy
lated from Stokes law The difference in force could then
be attributed to the drag on the wire If Stokes law is
followed the force on the wire should be proportional to
the velocity
A series of twenty measurements of the force on the
spheres was taken for each oil and the difference between
the measured force and that calcula ted by Stokes 1 law was
determined For each oil this difference as plo tted vs
the velocity The points grouped fairly ell around a
strai ght line nearly passing through the origin The
method of least squares was used to determine the equation
of the line best fitting the da t a The equa tion of the
line for the li bht oil tas found to be
Fe bullbull05605v - oooa (35)
which was determined at about 62 7degF Since the intercept
49
of the line is very close to zero it is believed that the
line is a good indication of the drag on the wire The
equation of the line for the heavy oil was found to be
F - 19llv I oo2o1 (36 ) c shy
which was determined at about 64 2deg The intercept of this
line is also quite close to zero These lines plotted in
Fi poundures 9 and 10 were used throughout the investigation
for the correction factor of the drag on the wires For
the cylinders and flat plates in parallel flow which were
pulled by two wires the values determined from Equations
35) and (36) were doubled For the plates in perpendicular
flow pulled by four wires the correction force was multishy
plied by four
The spring scale had 12 ounce divisions but could be
read to the nearest sixth of an ounce Some of the measureshy
ments of force were under an ounce hence a considerable
spread of the measurements was noticed in the pre liminary
data and throughout the experiment However sufficient
points were obtained so that it was possible to draw a
reliable curve through the data in all casas An analysis
was made to determine the average deviation from Stokes
equation for the spheres It raa found that the average
deviation was 15 1 for the light oil 16 6 for the heavy
oil and 15 9 overall The maximum deviation was 89
50
Inspection of the other data shows that these deviations
are also representative of the cylinders and flat plates
The force measurement is the least accurate part of the
experiment Other insignificant errors are introduced by
a small variation in the temperature This variation was
held to about 10 from the temperature of the calibrated
correction curve The velocity measurements and the
dimensions of the cylinders spheres and pl~ tes are conshy
sidered go od enough so tha t no appreciable errors occur
In order to e l iminate the WL parameter for flat plates
in parallel f l ow an additional factor for the effect of
the edges was subtracted from the measured force Janour
(5 p 27) presented the foll owing equation for the edge
correction for one edge of a flat plate in parallel flow
F ~ lv~ bull (37 ) edge gc
In present work this equation as doubled because both
edges of the plates were submerged in fluid It is assumed
in appl ying this correction that the lowe r limit of a
Reynolds number of 10 proposed by Janour can be extended
close to 0 1
Analysis of Results
Forty of the points for the spheres were used to get
51
the correction factor for the wires The remaining thirty
points are well erouped about Stokes law
The data for cylinders for LD ratios of 16 24 and
32 did not seem to be se gregated therefore these data
were plotted together It would seem that in the low range
of Reyno l ds numbers an LD of 16 and greater can be con shy
sidered an ~nfini tely long cylinder The other LD ratios
of 2 4 6 a 12 provided fairly distinct and separate
lines The best straight lines were drawn through the data
for each of the LD ratios It was evident that in eaeh
case a slope of -1 on a lo g-log graph gave the best straight
line which would indicate that the force varies directly
as the velocity It was possible to develop an empirical
expression relating dra g coefficient Reynolds number and
LD The following equation was obtained from the straight
line plots of Re vs fd for the various LD ratios
(38 )
Equation (38) applies for Reyno l ds numbers from 01 to 10
and for LD ratios of 2 to 16 For LD ratios greater
than 16
10 re (39 )
The data for flat plates in parallel flow is plotted
in Figure 15 after the correction factor for tho edge
52
effect was subtracted When the edge correction is made
no effect of WL ratio is indicated This result would be
expected The data followed a straight line with a slope
of -1 up to a Reynolds number of 2 After that a curve was
dravm connecting the line to that obtained by Janour The
equation for the straight section of the curve is
f - 6 (40)- Re
which applies for Reynolds numbers of 0 1 to 2 0 Here
a gain the force is proportional to the velocity Vfuen
determining drag force for flat plates in parallel flow
the force is first calculated from Equations (40) and (15 )
then the edge correction is added
The effect of the geometric ratios is clearly shown in
the data for flat plates in perpendicul ar flow which are
plotted in Figures 16 and 17 As with the other data the
best straight line was drawn through the various points
for eaoh of the WL ratios Again the line had a slope of
-1 The equation relating fd Re and wL was found t o be
rd 37 (w) -o 3o (41)Irel
which applies for Reynolds numbers of about 05 to 2 0 and
WL ratios of 1 to 4 It is possible but it has not been
proved that Equation (41) is suitable for higher WL ratios
The exponent on WL in Equation 41) is very close to that
53
on L D i n Equation ( 38 )~ It i s possible t ha t these
exponents are t he same but this cannot be sho~~ depound1nitely
until more accura te da ta are available It would be exshy
pected that a s the Reynolds number approaches zero t he
effect of geometric ratios would be the same for cylinders
and fla t pla tes in perpendicula r flow
It is seen in the t a bles of data that occasionally a
ne gative force was obtained because the correction applie d
due to t he wire dra g was greater than the mea sured force
These points obviously are incorrect This occurred only
for the smallest plates in the heavy oil at t he highest
velocities However these knom bad points occur in less
tha n 5~ of the data
It is clearl y shown that for cylinders and plates the
fd increases as L D or W L decreases This is in direct
contrast to Wiesel aberger s investigation However his
work is for hi gher Reynolds numbers at which a turbulent
wake forms bull
Comparison of Results with Other Data and Theoretical So l utions
The data for sphere~ a grees of course with Stokes
l aw since that law was used to determine the correction
factor for the wire Liebster (9 Pbull 548 ) has
54
substantiated Stokes equation
There are no experimental data with which to compare
the results of the cylinders Wieselsbergers minimum
Reynolds number of 4 is above the ran ge covered in the preshy
sent investigation The da ta for the highest LD ratios
(16 24 and 32) does agree almost exactly wi t h the solution
of Allen and Southwell (1 P bull 141) (LD =00) in the range
of Reynolds numbers from 0 1 to 1 0 Allen and Southwells
solution a greed with the data of Wieselsberger (16 p 22)
However the present data is above the theoretical solutions
of Lamb (8 p 112-121) throughout the range of Reynolds
numbers from 0 01 to 1 0 and above the solutions of
Bairstow Cave and Lang (2 p 404) I mai (4 p 157) and
Tomotika and Aoi (15 p 302) for Reynolds numbers of 0 1
to 1 0 Allen and Southwells solution a grees dth both
Wieselsberger 1 s a nd the present data Their solution and
the present data represent the best means for predicting
drag coefficients for flow over long cylinders for Reynolds
numbers of 0 01 to 10 It should be remembered that the
o t her solutions should a gree with eac h other since they
were all essentially derived by linearizing the Na viershy
Stokes equation
The data for flat plates in parallel flow is
55
considerably above the theoretical solutions of Janssen
(6 p 183 ) and Tomotika and Aoi (15 Pbull 302) However
Fi f~re 15 shows that a smooth transition occurs bet een
the present work and the data of Janour (5 P bull 31) The
present data considerably extend the experimental inforshy
mation previously available for laminar flow paral lel to
flat plates In the re gion of Reynol ds numbers less than
2 the drag coefficient is shown to be inversely proportional
to the Reynolds number Janours data covers a range of
Reynolds numbers from 11 to 1000 The results of the
present investigation line up with Janours results which
in turn on extrapolation to higher Reyno l ds numbers
(greater than 1000) make a smooth transition into Blasius
curve represented by Equation (10) At Reyno l ds numbers
greater than 20 000 the drag coefficient is inversely proshy
portional to the square root of the Reynolds number
The data for flat plates in perpendicular flow is conshy
siderably above the solutions of Tomotika and Aoi
(15 p 302) and Imai (4 p 157 However their solutions
f or cylinders and plates in parallel flow are also below
the present data Also it should be remembered that their
solutions are for infinitely wide plates If a value of
WL of above 100 is used in Equation (41) then the present
data and the solutions of Tomotika and Aoi are fairly close
56
The present results indicate that Equation (41~ can be
used with an accuracy of 15 to 20 within the limitations
of the equation (WL 1 to 4 Re = 0 05 to 2)
57
SUM RY AND CONCLUSIONS
Only a small amount of work has been done in the past
on the study of laminar flow over immersed bodies There
are many areas in the chemical process industries and the
field of aeronautics where this information would be very
helpful The purpose of the present investi gation wa s to
study the almost totally unexplored range of Reynol ds
numbers from 0 01 to 10
Drag coefficients have been determined for spheres
cylinders and flat plates in paralle l and perpendicular
flow The drag coefficients have been plotted as a
function of the Reynolds number with dimension ratios as
a parameter on lo g-log graphs The best straight lines
have been drawn through the data In all cases these lines
had a slope of -1 hich shows that the dra g coefficient is
inversely proportional to the Reynolds number at very low
Reynolds numbers for all shapes and dimension ratios The
following equations have been determined from the data
For cylinders
fd - 27 L -0 36 (38 ) - Re ())
which applies for Reynolds numbers of 0 01 to 1 and LD of
2 to 16 For LD greater than 16 the equation is
58
(39)
For flat plates in parallel flow a correction factor has
been applied to account for the edge effect The equation
which applies for Reyno l ds numbers of 0 1 to 2 is
f 6Re
(40)
For flat plates in perpendicular flow
f d
- 37 - Re (w) t -
0 bull 30 (41)
wbieh applies for W L of 1 to 4 and Reynolds numbers of
0 05 to 2
It is concluded tha t Equations (38-41) give the best
values of drag coefficients within an accuracy of 20~ for
the range of Reynolds numbers that were considered Also
it is evident that the dimension ratios are a n important
factor in determining the drag coefficient for a given
Reynolds number Furthermore the drag coefficient inshy
creases with decreasing values of L D or W L for a constant
Reynolds number The da ta obtained in this investi gation
compare favorably with the other experimental data and with
some of the theoretical sol utions It should be remembered
that when comparing the experimental data with theoretical
solutions that practically all of the solutions are for an
infinitely long cylinder or an infinitely wide plate
It is recommended tha t the present apparatus be
59
modified so that a force of 001 pound can be measured
Also it would improve tho accuracy to set up a constant
temperature bath so that the temperature of the oil can not
vary over 02degF A few check points on the present data
is all that is necessary to confirm the validity of
Equations (38- 41) It is also r ecommended that only SAE 140
oil be used and that 2 inches should be the minimum plate
width and cylinder length to be studi3d These conditions
would help to maintain the accuracy of the correction force
for the wire
60
~WMENCIATURE
Symbol Dimensions
A area sq ft
D diameter ft
F force lb f
L length ft
M mas s lb m Re Reynolds number Dvf= -ltr w width ft
a area sq ft
b characteristic length ft
d diameter ft
f drag coefficientfd
gravitation constant l b mft gc 2= 32 17 l b _ rsec
1 length ft
m mass l b bullm
p pressure lbrsqft
r radius ft
t time see
u velocity ft sec
v velocity ft sec
w width ft
61
Symbol Dimensions
X xbullcoordinate ft
y y- coordinate ft
o( vorticity
time sec
viscosity lb m ft -sec
kinematic viscosity ft 2sec
circumference diameter = 3 1416
3density lb m ft
function
stream function
Laplacian operator
infinity
Subscripts
c corrected
f force
1 l iquid
m mass
p projected
s solid
w wetted
62
BI BLIOGRAPHY
1 Allan D N de G and R v Southwell Re laxation methods applied to determine the motion in two di shymensions of a viscous fluid past a fixed cylinder Quarterly Journal of Mechanics and Applied Mathe shymatics 8 129-145 1955
2 Bairstow L B M Cave and E D Lang The reshysistance of a cylinder moving in a viscous fluid Philosophical Transactions of the Royal Society of London ser A 223383- 432 1923
3 Goldstein Sidney The steady flow of viscous fluid past a fixed spherical obstacle at small Reyno l ds numbers Proceedings of the Royal Society of London ser A 123225-235 1929
4 Imai I A new method of solving Oseens equations and its application to the flow past an inclined elliptic cylinder Proceedings of the Royal Society of London ser A 224 141-160 1954
5 Janour Zbynek Resistance of a plate in paralle l flow at low Reyno lds numbers Washington Nov 1951 40 p National Advisory Committee for Aeronautics Te chnica l Memorandum 1316)
6 Janssen E An analog solution of the Navier-Stokes equation for the case of flow past a f l at plate at low Reynolds numbers In 1956 Heat Transfer and Fluid Mechanics Institute (Preprints of Papers) p 173-183
7 Knudsen James G and Donal d L Katz Fluid Dynamics a nd Heat Transfer Ann Arbor University of Michigan 1953 243 p (Michi gan University Engineering Research Bulletin no 37)
8 La~b Horace On the uniform motion of a spherethrough a viscous fluid Philosophical Magazine and Journal of Science s~r 6 21112-121 1911
9 Liebster H Uben den widerstrand von kugeln Annalen Der Physik ser 4 82 541- 562 1 927
63
10 McAdams William H Heat transmission 3d ed New York McGraw- Hill 1954 532 p
11 Pai Shih- I Viscous f l ow theory I Laminar flow Princeton D Van Nostrand 1956 384 p
12 Prandtlbull Ludwi g Es sentials of fluid dynamics London Blackie amp Son 1954 452 p
13 Relf i F Discussion of the results of measure shyments of the resistance of wires with some additionshyal tests of the resistance of wires of small diame shyters In Technical report of the Advisory Committee for Aeronautics London) March 1914 p 47 - 51 (Report and memoranda no 102 )
14 Stokes George Gabriel Mathematical and physical papers Vol 3 Cambridge University Press 1922 413 p
15 Tomotika s and T Aoi The steady flow of a viscous fluid past an elliptic cylinder and a flat plate at smal l Reynolds numbers Quarterly Journal of Me chanics and Applie d Ma thematics 6 290- 312 1953
16 Wieselsbergo r c Versuche Ube r der luftwiderstand gerundeter und kant iger korper Er gebnisse der Aeroshydynamischen Versucbsansta l t Vol 2 G~ttingen 1923 80 p
FIGURE e- PHOTOGRAPH OF SPHERES CYLINDERS AND PLATES
34
holes were drilled so that each plate could be used for
two geometric ratios by changing the wires (See for
example plates la and lb in Table I
35
EXPERI MENTA L PROCEDURE
Viscosity and Density Calibration
A calibrated hydrometer measuring to the nearest
0002 was used to measure the density Table VI shows that
the effect of temperature on density is practically negli shy
gible in the small temperature range used
A Brookfield Synchro-lectric viscometer was used to
measure the viscosity of both the light and heavy oil
Figures 18 and 19 show the effect of temperature on visshy
cosity In addition the viscosity of the light oil was
checke d using the falling ball method and the equation
D2--ltA (f s bull fl) g (34) l 8v
The viscometer was calibrated by the National Bureau of bull
Standards and was accurate to l tb
Velocity Measurements
The velocity of movement through the oil was measured
by determining the rate of rotation of the pulleys with a
stop watch Usually the time for 10 revolutions was
measured at the highe r ve locities and for 5 revolutions at
the low velocities From this information and the di
amaters of the pulleys the velocities ere calculated
36
The time was measured to the nearest tenth of a second
Since the measured time was usually between 20 and 40
aeconds 1 the error in ~easuring velocity was considered to
be less tha~ 0 5~
force Measurements
The object connected to the scale 1 was dropped to the
bottom of the oil drum The motor was started and the scale
was read as the object vms being pulled towards the top of
the drum Two or three readings were taken for each object
at each velocity In nearly all cases these readings were
the same
37
ti XPER I MENTAL RE STJLTS
The dra g coefficient and the Reynolds number were
calculated by the use of Equations (l or (15) for each of
the spheres cylinders and plates from the measured
quantities of force and velocity a~d the values of the vis shy
cosity and density corresponding to the temperature of the
oil It was necessary to ~ubtract from the measured force
the force on the wire The corrected force measurement was
then used to determine the drag coefficient The force on
the wire has been determined as being proportional to the
velocity A correction curve relating force on the wire
and ve l ocity is plo tted in Figure 9 for the li ght oil and
Fi gure 10 for the heavy oil
The calculated drag coefficients Reynolds numbers
and velocities along with the measured force for the spheres
cylinders flat plates - parallel flow and flat plates shy
perpendicular flow have been tabulated in Tables II III
I V and v respectively
The calculated drag coefficients have been plotted as
a function of the Reynolds number on logarithic graph paper
with geometric ratios as a parameter
Drag coefficients for the spheres are plo tted in
Figure 11 The data for the cylinders are plotted in
CD_ bull 0 G 0
03
Tshy02
01
10 20 30 410 50 60 70 80
VELOCITY- FTJSEC
DRAG FORCE ON THE WIRE-LIGHT OIL
FIGURE 9
I -shy I -middot -- -shy -1shy _i-i I --~ I I _ -middot- shy I i
_I_ - _ middot- LL I l l tmiddot - middot1middot ~- - - - -+i middotshy I - --+-cl - l
1 1 I I IV jc---- --r--middotmiddottmiddot r-middotmiddot--tmiddotmiddot---shy _____ _L __ --~- --1shy middotmiddotr-r-middott- 1 -f-f-T- _~ +-L--1---~- 1--l
~- - shy I-+---Rmiddot-- I I I l i ~~ i -~~ ~- -T f i rshy ~-- --shy i- ----~-- shy - middot1 shy
I i I i I I 1--- -middot - fshy middot i----1---+-shy - i-middot -~+-- --~- --~-- ---- -t+ I v-~~ -middot j
i I middot 1_ _ I tmiddot---+-+1-+--li~+middot -+--+-+-1-+-+-+-+--tc--1-+-t-11-shy - middot --t- 1---t- t----tmiddotshy --~-- -middot i-shy I 1i - ~ i I i v i middotmiddotmiddot
[~v +L~ + ~ - I~~j-+ r V I ~t--- -~-- I +---~-- I f-middot ---1-- ~ -- --- ) Li --+--+--+-+-+-+--1--+--+---t---4 -1--1--+-+--+-l-i
tl~ I I Q Y +l~~ii-+-++++-middotHH-++-+-+-+--H--++ -i t Imiddot i i 1 j _V I f1 r-t~-middot l--r-tshy -~ 7 middot 1 -shy middot middotmiddot I
DRAG FORCE ON THE WIRE- HEAVY OIL
FIGURE 10
40
+shy l i~ltgt ~ bull r-rshy I i t _l
1 lf-1-1 l+r+ fJ-Ct I+ t li 1~t rtH r+l rf-l It llil I I
l l~pound 11 1 ~middot ~~middott ~ It lqf L
t I+--= ~r 17 -Er I _ ~ _pound~- sect Imiddot I+
iU=ff=t 1 +~ t_ - ~ r 111= t h=
I middot
t= IE I 1 1
plusmn~ kplusmni - -STOKE S EQ
(~ l h+middot
ru HmiddotHti+H1 11
c lffii l t~ 4 ~ ~middot ~ff l ~ ~h i ltlri
1 yen~ middot I ~ I I T ~ gt l+t H+h l+ i j l tfl-l Imiddotmiddot ft+ ++ l f+ Imiddotmiddot I+ I+ middott bulli I 1middot1 I ftt-1shy middot I middot r 11 I IH Ij ~ ~ middotishy J F 1= 6= ~
=f l~iit rtti l lit~ I FS lf~ l=i-+
l-11ffi tt lr 1 ~1 -t =l=Rttl 1ft i- 1 ~ I+ I
~~ lflJ
t I lfl m ~~WFB Lt
41plusmn811 IF I Hir tt ft itttplusmn i I~
1-+++middot
I ~ I (~ ffitrHf1 Ittmiddot ~ l r i H-t-r r HHt m 11 H++ I
bull I I
1_ _ F bullmiddot Imiddotmiddot t-- 1-T h iT
f-t+ ftt I+ I lt + T Imiddot 1
1t _plusmn middot~~ ~- 11shy
=a~ 1~ - =itf lttti
H I
=
DATA FOR SPHERES
FIGURE II
41
I -1---1-1-+--+--Ti-+-------+----r--shy --r--- -shy + t----+shy ----4-~---+-f----f--+-f--l--1 I t--shy --t-- ---+-shy
J-+-~f--~~ -___l_ ~---
i 1 L~L~-~tr-l----H~4-----~-f------+------+-----+----+---+middot-t-middot-H5000
middot~ ~ m- ~ - ~t plusmn~ 3t i t~ -f--- bullbull - ~~ h middot-
01 0~ 10
Re
-
DATA FOR CYLINDERS - LD = 6 8 AND 12
FIGURE I 4
44
Figures 12 13 and 14 The data for LD values of 16 24
and 32 were nearly the same and have been plotted to gether
i n Figure 12 In addition the curves for the other LD
ratios determined fro m Fib~res 13 and 14 have been drawn
in Figure 12 so that the effect of the length-to-diameter
is clearly shown Figure 13 shows the data for LD values
of 2 and 4 and the curves determined from this data
Firure 14 shows the data for LD values of 6 8 and 12
and the curves determined from this data
The data for flat plates in parallel flow are plotted
in Fi gure 15 A correction factor for the edge effect has
beon used so that the width-to-length ratio is not a
parameter in this plot A portion of the data of Janour
(5 p 31) is also shown in the diagram
The data for fla t plates in perpendicular flow is
plotted in Figures 16 a nd 17 Figure 16 shows the data for
WL values of 2 Also the curves for the three WL ratios
1 2 and 4 have been drawn in the fi gure Figure 17 shows
the data for WL values of 1 and 4 The curves determined
from the data have also been dravm in the figure
45
10~ ~ ~--- -shy
t==Ff1TR=+ iJ+--_-_--r_-_---+-+---+--+-+--_---_-~r-=r~=~+--=---=---=---=--~=--=_~1=_--=_~_-middot~~--+-+-t~ 1 Ll~+--+-- ---jtshyl~t L--+ I
I
P------ _l -- --1---L i
20 ~-- I ~g I --- - ---+-- r t L_shy
~ ~B 1) I --o-o- JONES - () - - ~~ p f---j- -~-- e e JANOU R
c gt ~c ~ ------ JANSSEN I 0 0 ~ I
IO ~2=i~~~~~~a=~~f=j= ---- TOM OTIKA bulll= I
~~n ~~--~~~~~~o~~~~~--4- NDCIgttl o shy
-
~--~~~~~+--+~+--4-r-~1+-~-middot+1~ ~ --H--~-~~os I i i i-4 ---~T I I f-- t --- li-------~--+-_--+--t-----~~-~_+---_-_-_--+------+-+-__+-[- +_- ___ _______ __+---+-r-+--H----_+--r--------+shy
it I+ ~ bull t ~1 ri j t++t+t++tft bullm H--~+H-t+t-++H-f+t+~HtttH t bull~H-IrttI-H
iH-H u nH m
I
t H+t-~ 1-r f-tj
i it iT -t middotHt I I I I Ill
~middot __
r middotshy
i I r-
f H- jLj f r H rr t~
II
t f f-l -t+tt ~ ==_ =~middot irE
I I
I
I
f
I --
i
t
1 r bull - r
~- ltt++l=tUtt~S-t+t+++~-++U +HJJm~-fl~HHtt1 tttn ll+t-Tt-~- ~ r fH T --r -1 t ---t- -tshy w _+ _ I-shy middotI
-shy -r- + Hbull Hshy t-I --r++ -t iHr -1 H-e-- -t I 1IT 1
1 H-rf-I IJftJ Jf+i+ ~ L
=+shy - tjshy rtmiddotshy ~ -
+ H 1-Jt I tt o =tt ~-
~1 l +fill l plusmn~ fplusmn -shy + I t-
DATA FOR FLAT PLATES PERPENDICULAR FLOW- WL= I 4
FIGURE 17
48
DI SCUSS ION OF RESULTS
Correction and Accuracy of Measurements
After a few pre liminary force measurements with the
spheres and a check with Stokes law (Equation 2) it was
apparent that the drag force on the wire was appreciable
and needed to be considered It was decided to take a
series of measurements with the spheres and calculate the
difference between the measured force and the force calcushy
lated from Stokes law The difference in force could then
be attributed to the drag on the wire If Stokes law is
followed the force on the wire should be proportional to
the velocity
A series of twenty measurements of the force on the
spheres was taken for each oil and the difference between
the measured force and that calcula ted by Stokes 1 law was
determined For each oil this difference as plo tted vs
the velocity The points grouped fairly ell around a
strai ght line nearly passing through the origin The
method of least squares was used to determine the equation
of the line best fitting the da t a The equa tion of the
line for the li bht oil tas found to be
Fe bullbull05605v - oooa (35)
which was determined at about 62 7degF Since the intercept
49
of the line is very close to zero it is believed that the
line is a good indication of the drag on the wire The
equation of the line for the heavy oil was found to be
F - 19llv I oo2o1 (36 ) c shy
which was determined at about 64 2deg The intercept of this
line is also quite close to zero These lines plotted in
Fi poundures 9 and 10 were used throughout the investigation
for the correction factor of the drag on the wires For
the cylinders and flat plates in parallel flow which were
pulled by two wires the values determined from Equations
35) and (36) were doubled For the plates in perpendicular
flow pulled by four wires the correction force was multishy
plied by four
The spring scale had 12 ounce divisions but could be
read to the nearest sixth of an ounce Some of the measureshy
ments of force were under an ounce hence a considerable
spread of the measurements was noticed in the pre liminary
data and throughout the experiment However sufficient
points were obtained so that it was possible to draw a
reliable curve through the data in all casas An analysis
was made to determine the average deviation from Stokes
equation for the spheres It raa found that the average
deviation was 15 1 for the light oil 16 6 for the heavy
oil and 15 9 overall The maximum deviation was 89
50
Inspection of the other data shows that these deviations
are also representative of the cylinders and flat plates
The force measurement is the least accurate part of the
experiment Other insignificant errors are introduced by
a small variation in the temperature This variation was
held to about 10 from the temperature of the calibrated
correction curve The velocity measurements and the
dimensions of the cylinders spheres and pl~ tes are conshy
sidered go od enough so tha t no appreciable errors occur
In order to e l iminate the WL parameter for flat plates
in parallel f l ow an additional factor for the effect of
the edges was subtracted from the measured force Janour
(5 p 27) presented the foll owing equation for the edge
correction for one edge of a flat plate in parallel flow
F ~ lv~ bull (37 ) edge gc
In present work this equation as doubled because both
edges of the plates were submerged in fluid It is assumed
in appl ying this correction that the lowe r limit of a
Reynolds number of 10 proposed by Janour can be extended
close to 0 1
Analysis of Results
Forty of the points for the spheres were used to get
51
the correction factor for the wires The remaining thirty
points are well erouped about Stokes law
The data for cylinders for LD ratios of 16 24 and
32 did not seem to be se gregated therefore these data
were plotted together It would seem that in the low range
of Reyno l ds numbers an LD of 16 and greater can be con shy
sidered an ~nfini tely long cylinder The other LD ratios
of 2 4 6 a 12 provided fairly distinct and separate
lines The best straight lines were drawn through the data
for each of the LD ratios It was evident that in eaeh
case a slope of -1 on a lo g-log graph gave the best straight
line which would indicate that the force varies directly
as the velocity It was possible to develop an empirical
expression relating dra g coefficient Reynolds number and
LD The following equation was obtained from the straight
line plots of Re vs fd for the various LD ratios
(38 )
Equation (38) applies for Reyno l ds numbers from 01 to 10
and for LD ratios of 2 to 16 For LD ratios greater
than 16
10 re (39 )
The data for flat plates in parallel flow is plotted
in Figure 15 after the correction factor for tho edge
52
effect was subtracted When the edge correction is made
no effect of WL ratio is indicated This result would be
expected The data followed a straight line with a slope
of -1 up to a Reynolds number of 2 After that a curve was
dravm connecting the line to that obtained by Janour The
equation for the straight section of the curve is
f - 6 (40)- Re
which applies for Reynolds numbers of 0 1 to 2 0 Here
a gain the force is proportional to the velocity Vfuen
determining drag force for flat plates in parallel flow
the force is first calculated from Equations (40) and (15 )
then the edge correction is added
The effect of the geometric ratios is clearly shown in
the data for flat plates in perpendicul ar flow which are
plotted in Figures 16 and 17 As with the other data the
best straight line was drawn through the various points
for eaoh of the WL ratios Again the line had a slope of
-1 The equation relating fd Re and wL was found t o be
rd 37 (w) -o 3o (41)Irel
which applies for Reynolds numbers of about 05 to 2 0 and
WL ratios of 1 to 4 It is possible but it has not been
proved that Equation (41) is suitable for higher WL ratios
The exponent on WL in Equation 41) is very close to that
53
on L D i n Equation ( 38 )~ It i s possible t ha t these
exponents are t he same but this cannot be sho~~ depound1nitely
until more accura te da ta are available It would be exshy
pected that a s the Reynolds number approaches zero t he
effect of geometric ratios would be the same for cylinders
and fla t pla tes in perpendicula r flow
It is seen in the t a bles of data that occasionally a
ne gative force was obtained because the correction applie d
due to t he wire dra g was greater than the mea sured force
These points obviously are incorrect This occurred only
for the smallest plates in the heavy oil at t he highest
velocities However these knom bad points occur in less
tha n 5~ of the data
It is clearl y shown that for cylinders and plates the
fd increases as L D or W L decreases This is in direct
contrast to Wiesel aberger s investigation However his
work is for hi gher Reynolds numbers at which a turbulent
wake forms bull
Comparison of Results with Other Data and Theoretical So l utions
The data for sphere~ a grees of course with Stokes
l aw since that law was used to determine the correction
factor for the wire Liebster (9 Pbull 548 ) has
54
substantiated Stokes equation
There are no experimental data with which to compare
the results of the cylinders Wieselsbergers minimum
Reynolds number of 4 is above the ran ge covered in the preshy
sent investigation The da ta for the highest LD ratios
(16 24 and 32) does agree almost exactly wi t h the solution
of Allen and Southwell (1 P bull 141) (LD =00) in the range
of Reynolds numbers from 0 1 to 1 0 Allen and Southwells
solution a greed with the data of Wieselsberger (16 p 22)
However the present data is above the theoretical solutions
of Lamb (8 p 112-121) throughout the range of Reynolds
numbers from 0 01 to 1 0 and above the solutions of
Bairstow Cave and Lang (2 p 404) I mai (4 p 157) and
Tomotika and Aoi (15 p 302) for Reynolds numbers of 0 1
to 1 0 Allen and Southwells solution a grees dth both
Wieselsberger 1 s a nd the present data Their solution and
the present data represent the best means for predicting
drag coefficients for flow over long cylinders for Reynolds
numbers of 0 01 to 10 It should be remembered that the
o t her solutions should a gree with eac h other since they
were all essentially derived by linearizing the Na viershy
Stokes equation
The data for flat plates in parallel flow is
55
considerably above the theoretical solutions of Janssen
(6 p 183 ) and Tomotika and Aoi (15 Pbull 302) However
Fi f~re 15 shows that a smooth transition occurs bet een
the present work and the data of Janour (5 P bull 31) The
present data considerably extend the experimental inforshy
mation previously available for laminar flow paral lel to
flat plates In the re gion of Reynol ds numbers less than
2 the drag coefficient is shown to be inversely proportional
to the Reynolds number Janours data covers a range of
Reynolds numbers from 11 to 1000 The results of the
present investigation line up with Janours results which
in turn on extrapolation to higher Reyno l ds numbers
(greater than 1000) make a smooth transition into Blasius
curve represented by Equation (10) At Reyno l ds numbers
greater than 20 000 the drag coefficient is inversely proshy
portional to the square root of the Reynolds number
The data for flat plates in perpendicular flow is conshy
siderably above the solutions of Tomotika and Aoi
(15 p 302) and Imai (4 p 157 However their solutions
f or cylinders and plates in parallel flow are also below
the present data Also it should be remembered that their
solutions are for infinitely wide plates If a value of
WL of above 100 is used in Equation (41) then the present
data and the solutions of Tomotika and Aoi are fairly close
56
The present results indicate that Equation (41~ can be
used with an accuracy of 15 to 20 within the limitations
of the equation (WL 1 to 4 Re = 0 05 to 2)
57
SUM RY AND CONCLUSIONS
Only a small amount of work has been done in the past
on the study of laminar flow over immersed bodies There
are many areas in the chemical process industries and the
field of aeronautics where this information would be very
helpful The purpose of the present investi gation wa s to
study the almost totally unexplored range of Reynol ds
numbers from 0 01 to 10
Drag coefficients have been determined for spheres
cylinders and flat plates in paralle l and perpendicular
flow The drag coefficients have been plotted as a
function of the Reynolds number with dimension ratios as
a parameter on lo g-log graphs The best straight lines
have been drawn through the data In all cases these lines
had a slope of -1 hich shows that the dra g coefficient is
inversely proportional to the Reynolds number at very low
Reynolds numbers for all shapes and dimension ratios The
following equations have been determined from the data
For cylinders
fd - 27 L -0 36 (38 ) - Re ())
which applies for Reynolds numbers of 0 01 to 1 and LD of
2 to 16 For LD greater than 16 the equation is
58
(39)
For flat plates in parallel flow a correction factor has
been applied to account for the edge effect The equation
which applies for Reyno l ds numbers of 0 1 to 2 is
f 6Re
(40)
For flat plates in perpendicular flow
f d
- 37 - Re (w) t -
0 bull 30 (41)
wbieh applies for W L of 1 to 4 and Reynolds numbers of
0 05 to 2
It is concluded tha t Equations (38-41) give the best
values of drag coefficients within an accuracy of 20~ for
the range of Reynolds numbers that were considered Also
it is evident that the dimension ratios are a n important
factor in determining the drag coefficient for a given
Reynolds number Furthermore the drag coefficient inshy
creases with decreasing values of L D or W L for a constant
Reynolds number The da ta obtained in this investi gation
compare favorably with the other experimental data and with
some of the theoretical sol utions It should be remembered
that when comparing the experimental data with theoretical
solutions that practically all of the solutions are for an
infinitely long cylinder or an infinitely wide plate
It is recommended tha t the present apparatus be
59
modified so that a force of 001 pound can be measured
Also it would improve tho accuracy to set up a constant
temperature bath so that the temperature of the oil can not
vary over 02degF A few check points on the present data
is all that is necessary to confirm the validity of
Equations (38- 41) It is also r ecommended that only SAE 140
oil be used and that 2 inches should be the minimum plate
width and cylinder length to be studi3d These conditions
would help to maintain the accuracy of the correction force
for the wire
60
~WMENCIATURE
Symbol Dimensions
A area sq ft
D diameter ft
F force lb f
L length ft
M mas s lb m Re Reynolds number Dvf= -ltr w width ft
a area sq ft
b characteristic length ft
d diameter ft
f drag coefficientfd
gravitation constant l b mft gc 2= 32 17 l b _ rsec
1 length ft
m mass l b bullm
p pressure lbrsqft
r radius ft
t time see
u velocity ft sec
v velocity ft sec
w width ft
61
Symbol Dimensions
X xbullcoordinate ft
y y- coordinate ft
o( vorticity
time sec
viscosity lb m ft -sec
kinematic viscosity ft 2sec
circumference diameter = 3 1416
3density lb m ft
function
stream function
Laplacian operator
infinity
Subscripts
c corrected
f force
1 l iquid
m mass
p projected
s solid
w wetted
62
BI BLIOGRAPHY
1 Allan D N de G and R v Southwell Re laxation methods applied to determine the motion in two di shymensions of a viscous fluid past a fixed cylinder Quarterly Journal of Mechanics and Applied Mathe shymatics 8 129-145 1955
2 Bairstow L B M Cave and E D Lang The reshysistance of a cylinder moving in a viscous fluid Philosophical Transactions of the Royal Society of London ser A 223383- 432 1923
3 Goldstein Sidney The steady flow of viscous fluid past a fixed spherical obstacle at small Reyno l ds numbers Proceedings of the Royal Society of London ser A 123225-235 1929
4 Imai I A new method of solving Oseens equations and its application to the flow past an inclined elliptic cylinder Proceedings of the Royal Society of London ser A 224 141-160 1954
5 Janour Zbynek Resistance of a plate in paralle l flow at low Reyno lds numbers Washington Nov 1951 40 p National Advisory Committee for Aeronautics Te chnica l Memorandum 1316)
6 Janssen E An analog solution of the Navier-Stokes equation for the case of flow past a f l at plate at low Reynolds numbers In 1956 Heat Transfer and Fluid Mechanics Institute (Preprints of Papers) p 173-183
7 Knudsen James G and Donal d L Katz Fluid Dynamics a nd Heat Transfer Ann Arbor University of Michigan 1953 243 p (Michi gan University Engineering Research Bulletin no 37)
8 La~b Horace On the uniform motion of a spherethrough a viscous fluid Philosophical Magazine and Journal of Science s~r 6 21112-121 1911
9 Liebster H Uben den widerstrand von kugeln Annalen Der Physik ser 4 82 541- 562 1 927
63
10 McAdams William H Heat transmission 3d ed New York McGraw- Hill 1954 532 p
11 Pai Shih- I Viscous f l ow theory I Laminar flow Princeton D Van Nostrand 1956 384 p
12 Prandtlbull Ludwi g Es sentials of fluid dynamics London Blackie amp Son 1954 452 p
13 Relf i F Discussion of the results of measure shyments of the resistance of wires with some additionshyal tests of the resistance of wires of small diame shyters In Technical report of the Advisory Committee for Aeronautics London) March 1914 p 47 - 51 (Report and memoranda no 102 )
14 Stokes George Gabriel Mathematical and physical papers Vol 3 Cambridge University Press 1922 413 p
15 Tomotika s and T Aoi The steady flow of a viscous fluid past an elliptic cylinder and a flat plate at smal l Reynolds numbers Quarterly Journal of Me chanics and Applie d Ma thematics 6 290- 312 1953
16 Wieselsbergo r c Versuche Ube r der luftwiderstand gerundeter und kant iger korper Er gebnisse der Aeroshydynamischen Versucbsansta l t Vol 2 G~ttingen 1923 80 p
FIGURE e- PHOTOGRAPH OF SPHERES CYLINDERS AND PLATES
34
holes were drilled so that each plate could be used for
two geometric ratios by changing the wires (See for
example plates la and lb in Table I
35
EXPERI MENTA L PROCEDURE
Viscosity and Density Calibration
A calibrated hydrometer measuring to the nearest
0002 was used to measure the density Table VI shows that
the effect of temperature on density is practically negli shy
gible in the small temperature range used
A Brookfield Synchro-lectric viscometer was used to
measure the viscosity of both the light and heavy oil
Figures 18 and 19 show the effect of temperature on visshy
cosity In addition the viscosity of the light oil was
checke d using the falling ball method and the equation
D2--ltA (f s bull fl) g (34) l 8v
The viscometer was calibrated by the National Bureau of bull
Standards and was accurate to l tb
Velocity Measurements
The velocity of movement through the oil was measured
by determining the rate of rotation of the pulleys with a
stop watch Usually the time for 10 revolutions was
measured at the highe r ve locities and for 5 revolutions at
the low velocities From this information and the di
amaters of the pulleys the velocities ere calculated
36
The time was measured to the nearest tenth of a second
Since the measured time was usually between 20 and 40
aeconds 1 the error in ~easuring velocity was considered to
be less tha~ 0 5~
force Measurements
The object connected to the scale 1 was dropped to the
bottom of the oil drum The motor was started and the scale
was read as the object vms being pulled towards the top of
the drum Two or three readings were taken for each object
at each velocity In nearly all cases these readings were
the same
37
ti XPER I MENTAL RE STJLTS
The dra g coefficient and the Reynolds number were
calculated by the use of Equations (l or (15) for each of
the spheres cylinders and plates from the measured
quantities of force and velocity a~d the values of the vis shy
cosity and density corresponding to the temperature of the
oil It was necessary to ~ubtract from the measured force
the force on the wire The corrected force measurement was
then used to determine the drag coefficient The force on
the wire has been determined as being proportional to the
velocity A correction curve relating force on the wire
and ve l ocity is plo tted in Figure 9 for the li ght oil and
Fi gure 10 for the heavy oil
The calculated drag coefficients Reynolds numbers
and velocities along with the measured force for the spheres
cylinders flat plates - parallel flow and flat plates shy
perpendicular flow have been tabulated in Tables II III
I V and v respectively
The calculated drag coefficients have been plotted as
a function of the Reynolds number on logarithic graph paper
with geometric ratios as a parameter
Drag coefficients for the spheres are plo tted in
Figure 11 The data for the cylinders are plotted in
CD_ bull 0 G 0
03
Tshy02
01
10 20 30 410 50 60 70 80
VELOCITY- FTJSEC
DRAG FORCE ON THE WIRE-LIGHT OIL
FIGURE 9
I -shy I -middot -- -shy -1shy _i-i I --~ I I _ -middot- shy I i
_I_ - _ middot- LL I l l tmiddot - middot1middot ~- - - - -+i middotshy I - --+-cl - l
1 1 I I IV jc---- --r--middotmiddottmiddot r-middotmiddot--tmiddotmiddot---shy _____ _L __ --~- --1shy middotmiddotr-r-middott- 1 -f-f-T- _~ +-L--1---~- 1--l
~- - shy I-+---Rmiddot-- I I I l i ~~ i -~~ ~- -T f i rshy ~-- --shy i- ----~-- shy - middot1 shy
I i I i I I 1--- -middot - fshy middot i----1---+-shy - i-middot -~+-- --~- --~-- ---- -t+ I v-~~ -middot j
i I middot 1_ _ I tmiddot---+-+1-+--li~+middot -+--+-+-1-+-+-+-+--tc--1-+-t-11-shy - middot --t- 1---t- t----tmiddotshy --~-- -middot i-shy I 1i - ~ i I i v i middotmiddotmiddot
[~v +L~ + ~ - I~~j-+ r V I ~t--- -~-- I +---~-- I f-middot ---1-- ~ -- --- ) Li --+--+--+-+-+-+--1--+--+---t---4 -1--1--+-+--+-l-i
tl~ I I Q Y +l~~ii-+-++++-middotHH-++-+-+-+--H--++ -i t Imiddot i i 1 j _V I f1 r-t~-middot l--r-tshy -~ 7 middot 1 -shy middot middotmiddot I
DRAG FORCE ON THE WIRE- HEAVY OIL
FIGURE 10
40
+shy l i~ltgt ~ bull r-rshy I i t _l
1 lf-1-1 l+r+ fJ-Ct I+ t li 1~t rtH r+l rf-l It llil I I
l l~pound 11 1 ~middot ~~middott ~ It lqf L
t I+--= ~r 17 -Er I _ ~ _pound~- sect Imiddot I+
iU=ff=t 1 +~ t_ - ~ r 111= t h=
I middot
t= IE I 1 1
plusmn~ kplusmni - -STOKE S EQ
(~ l h+middot
ru HmiddotHti+H1 11
c lffii l t~ 4 ~ ~middot ~ff l ~ ~h i ltlri
1 yen~ middot I ~ I I T ~ gt l+t H+h l+ i j l tfl-l Imiddotmiddot ft+ ++ l f+ Imiddotmiddot I+ I+ middott bulli I 1middot1 I ftt-1shy middot I middot r 11 I IH Ij ~ ~ middotishy J F 1= 6= ~
=f l~iit rtti l lit~ I FS lf~ l=i-+
l-11ffi tt lr 1 ~1 -t =l=Rttl 1ft i- 1 ~ I+ I
~~ lflJ
t I lfl m ~~WFB Lt
41plusmn811 IF I Hir tt ft itttplusmn i I~
1-+++middot
I ~ I (~ ffitrHf1 Ittmiddot ~ l r i H-t-r r HHt m 11 H++ I
bull I I
1_ _ F bullmiddot Imiddotmiddot t-- 1-T h iT
f-t+ ftt I+ I lt + T Imiddot 1
1t _plusmn middot~~ ~- 11shy
=a~ 1~ - =itf lttti
H I
=
DATA FOR SPHERES
FIGURE II
41
I -1---1-1-+--+--Ti-+-------+----r--shy --r--- -shy + t----+shy ----4-~---+-f----f--+-f--l--1 I t--shy --t-- ---+-shy
J-+-~f--~~ -___l_ ~---
i 1 L~L~-~tr-l----H~4-----~-f------+------+-----+----+---+middot-t-middot-H5000
middot~ ~ m- ~ - ~t plusmn~ 3t i t~ -f--- bullbull - ~~ h middot-
01 0~ 10
Re
-
DATA FOR CYLINDERS - LD = 6 8 AND 12
FIGURE I 4
44
Figures 12 13 and 14 The data for LD values of 16 24
and 32 were nearly the same and have been plotted to gether
i n Figure 12 In addition the curves for the other LD
ratios determined fro m Fib~res 13 and 14 have been drawn
in Figure 12 so that the effect of the length-to-diameter
is clearly shown Figure 13 shows the data for LD values
of 2 and 4 and the curves determined from this data
Firure 14 shows the data for LD values of 6 8 and 12
and the curves determined from this data
The data for flat plates in parallel flow are plotted
in Fi gure 15 A correction factor for the edge effect has
beon used so that the width-to-length ratio is not a
parameter in this plot A portion of the data of Janour
(5 p 31) is also shown in the diagram
The data for fla t plates in perpendicular flow is
plotted in Figures 16 a nd 17 Figure 16 shows the data for
WL values of 2 Also the curves for the three WL ratios
1 2 and 4 have been drawn in the fi gure Figure 17 shows
the data for WL values of 1 and 4 The curves determined
from the data have also been dravm in the figure
45
10~ ~ ~--- -shy
t==Ff1TR=+ iJ+--_-_--r_-_---+-+---+--+-+--_---_-~r-=r~=~+--=---=---=---=--~=--=_~1=_--=_~_-middot~~--+-+-t~ 1 Ll~+--+-- ---jtshyl~t L--+ I
I
P------ _l -- --1---L i
20 ~-- I ~g I --- - ---+-- r t L_shy
~ ~B 1) I --o-o- JONES - () - - ~~ p f---j- -~-- e e JANOU R
c gt ~c ~ ------ JANSSEN I 0 0 ~ I
IO ~2=i~~~~~~a=~~f=j= ---- TOM OTIKA bulll= I
~~n ~~--~~~~~~o~~~~~--4- NDCIgttl o shy
-
~--~~~~~+--+~+--4-r-~1+-~-middot+1~ ~ --H--~-~~os I i i i-4 ---~T I I f-- t --- li-------~--+-_--+--t-----~~-~_+---_-_-_--+------+-+-__+-[- +_- ___ _______ __+---+-r-+--H----_+--r--------+shy
it I+ ~ bull t ~1 ri j t++t+t++tft bullm H--~+H-t+t-++H-f+t+~HtttH t bull~H-IrttI-H
iH-H u nH m
I
t H+t-~ 1-r f-tj
i it iT -t middotHt I I I I Ill
~middot __
r middotshy
i I r-
f H- jLj f r H rr t~
II
t f f-l -t+tt ~ ==_ =~middot irE
I I
I
I
f
I --
i
t
1 r bull - r
~- ltt++l=tUtt~S-t+t+++~-++U +HJJm~-fl~HHtt1 tttn ll+t-Tt-~- ~ r fH T --r -1 t ---t- -tshy w _+ _ I-shy middotI
-shy -r- + Hbull Hshy t-I --r++ -t iHr -1 H-e-- -t I 1IT 1
1 H-rf-I IJftJ Jf+i+ ~ L
=+shy - tjshy rtmiddotshy ~ -
+ H 1-Jt I tt o =tt ~-
~1 l +fill l plusmn~ fplusmn -shy + I t-
DATA FOR FLAT PLATES PERPENDICULAR FLOW- WL= I 4
FIGURE 17
48
DI SCUSS ION OF RESULTS
Correction and Accuracy of Measurements
After a few pre liminary force measurements with the
spheres and a check with Stokes law (Equation 2) it was
apparent that the drag force on the wire was appreciable
and needed to be considered It was decided to take a
series of measurements with the spheres and calculate the
difference between the measured force and the force calcushy
lated from Stokes law The difference in force could then
be attributed to the drag on the wire If Stokes law is
followed the force on the wire should be proportional to
the velocity
A series of twenty measurements of the force on the
spheres was taken for each oil and the difference between
the measured force and that calcula ted by Stokes 1 law was
determined For each oil this difference as plo tted vs
the velocity The points grouped fairly ell around a
strai ght line nearly passing through the origin The
method of least squares was used to determine the equation
of the line best fitting the da t a The equa tion of the
line for the li bht oil tas found to be
Fe bullbull05605v - oooa (35)
which was determined at about 62 7degF Since the intercept
49
of the line is very close to zero it is believed that the
line is a good indication of the drag on the wire The
equation of the line for the heavy oil was found to be
F - 19llv I oo2o1 (36 ) c shy
which was determined at about 64 2deg The intercept of this
line is also quite close to zero These lines plotted in
Fi poundures 9 and 10 were used throughout the investigation
for the correction factor of the drag on the wires For
the cylinders and flat plates in parallel flow which were
pulled by two wires the values determined from Equations
35) and (36) were doubled For the plates in perpendicular
flow pulled by four wires the correction force was multishy
plied by four
The spring scale had 12 ounce divisions but could be
read to the nearest sixth of an ounce Some of the measureshy
ments of force were under an ounce hence a considerable
spread of the measurements was noticed in the pre liminary
data and throughout the experiment However sufficient
points were obtained so that it was possible to draw a
reliable curve through the data in all casas An analysis
was made to determine the average deviation from Stokes
equation for the spheres It raa found that the average
deviation was 15 1 for the light oil 16 6 for the heavy
oil and 15 9 overall The maximum deviation was 89
50
Inspection of the other data shows that these deviations
are also representative of the cylinders and flat plates
The force measurement is the least accurate part of the
experiment Other insignificant errors are introduced by
a small variation in the temperature This variation was
held to about 10 from the temperature of the calibrated
correction curve The velocity measurements and the
dimensions of the cylinders spheres and pl~ tes are conshy
sidered go od enough so tha t no appreciable errors occur
In order to e l iminate the WL parameter for flat plates
in parallel f l ow an additional factor for the effect of
the edges was subtracted from the measured force Janour
(5 p 27) presented the foll owing equation for the edge
correction for one edge of a flat plate in parallel flow
F ~ lv~ bull (37 ) edge gc
In present work this equation as doubled because both
edges of the plates were submerged in fluid It is assumed
in appl ying this correction that the lowe r limit of a
Reynolds number of 10 proposed by Janour can be extended
close to 0 1
Analysis of Results
Forty of the points for the spheres were used to get
51
the correction factor for the wires The remaining thirty
points are well erouped about Stokes law
The data for cylinders for LD ratios of 16 24 and
32 did not seem to be se gregated therefore these data
were plotted together It would seem that in the low range
of Reyno l ds numbers an LD of 16 and greater can be con shy
sidered an ~nfini tely long cylinder The other LD ratios
of 2 4 6 a 12 provided fairly distinct and separate
lines The best straight lines were drawn through the data
for each of the LD ratios It was evident that in eaeh
case a slope of -1 on a lo g-log graph gave the best straight
line which would indicate that the force varies directly
as the velocity It was possible to develop an empirical
expression relating dra g coefficient Reynolds number and
LD The following equation was obtained from the straight
line plots of Re vs fd for the various LD ratios
(38 )
Equation (38) applies for Reyno l ds numbers from 01 to 10
and for LD ratios of 2 to 16 For LD ratios greater
than 16
10 re (39 )
The data for flat plates in parallel flow is plotted
in Figure 15 after the correction factor for tho edge
52
effect was subtracted When the edge correction is made
no effect of WL ratio is indicated This result would be
expected The data followed a straight line with a slope
of -1 up to a Reynolds number of 2 After that a curve was
dravm connecting the line to that obtained by Janour The
equation for the straight section of the curve is
f - 6 (40)- Re
which applies for Reynolds numbers of 0 1 to 2 0 Here
a gain the force is proportional to the velocity Vfuen
determining drag force for flat plates in parallel flow
the force is first calculated from Equations (40) and (15 )
then the edge correction is added
The effect of the geometric ratios is clearly shown in
the data for flat plates in perpendicul ar flow which are
plotted in Figures 16 and 17 As with the other data the
best straight line was drawn through the various points
for eaoh of the WL ratios Again the line had a slope of
-1 The equation relating fd Re and wL was found t o be
rd 37 (w) -o 3o (41)Irel
which applies for Reynolds numbers of about 05 to 2 0 and
WL ratios of 1 to 4 It is possible but it has not been
proved that Equation (41) is suitable for higher WL ratios
The exponent on WL in Equation 41) is very close to that
53
on L D i n Equation ( 38 )~ It i s possible t ha t these
exponents are t he same but this cannot be sho~~ depound1nitely
until more accura te da ta are available It would be exshy
pected that a s the Reynolds number approaches zero t he
effect of geometric ratios would be the same for cylinders
and fla t pla tes in perpendicula r flow
It is seen in the t a bles of data that occasionally a
ne gative force was obtained because the correction applie d
due to t he wire dra g was greater than the mea sured force
These points obviously are incorrect This occurred only
for the smallest plates in the heavy oil at t he highest
velocities However these knom bad points occur in less
tha n 5~ of the data
It is clearl y shown that for cylinders and plates the
fd increases as L D or W L decreases This is in direct
contrast to Wiesel aberger s investigation However his
work is for hi gher Reynolds numbers at which a turbulent
wake forms bull
Comparison of Results with Other Data and Theoretical So l utions
The data for sphere~ a grees of course with Stokes
l aw since that law was used to determine the correction
factor for the wire Liebster (9 Pbull 548 ) has
54
substantiated Stokes equation
There are no experimental data with which to compare
the results of the cylinders Wieselsbergers minimum
Reynolds number of 4 is above the ran ge covered in the preshy
sent investigation The da ta for the highest LD ratios
(16 24 and 32) does agree almost exactly wi t h the solution
of Allen and Southwell (1 P bull 141) (LD =00) in the range
of Reynolds numbers from 0 1 to 1 0 Allen and Southwells
solution a greed with the data of Wieselsberger (16 p 22)
However the present data is above the theoretical solutions
of Lamb (8 p 112-121) throughout the range of Reynolds
numbers from 0 01 to 1 0 and above the solutions of
Bairstow Cave and Lang (2 p 404) I mai (4 p 157) and
Tomotika and Aoi (15 p 302) for Reynolds numbers of 0 1
to 1 0 Allen and Southwells solution a grees dth both
Wieselsberger 1 s a nd the present data Their solution and
the present data represent the best means for predicting
drag coefficients for flow over long cylinders for Reynolds
numbers of 0 01 to 10 It should be remembered that the
o t her solutions should a gree with eac h other since they
were all essentially derived by linearizing the Na viershy
Stokes equation
The data for flat plates in parallel flow is
55
considerably above the theoretical solutions of Janssen
(6 p 183 ) and Tomotika and Aoi (15 Pbull 302) However
Fi f~re 15 shows that a smooth transition occurs bet een
the present work and the data of Janour (5 P bull 31) The
present data considerably extend the experimental inforshy
mation previously available for laminar flow paral lel to
flat plates In the re gion of Reynol ds numbers less than
2 the drag coefficient is shown to be inversely proportional
to the Reynolds number Janours data covers a range of
Reynolds numbers from 11 to 1000 The results of the
present investigation line up with Janours results which
in turn on extrapolation to higher Reyno l ds numbers
(greater than 1000) make a smooth transition into Blasius
curve represented by Equation (10) At Reyno l ds numbers
greater than 20 000 the drag coefficient is inversely proshy
portional to the square root of the Reynolds number
The data for flat plates in perpendicular flow is conshy
siderably above the solutions of Tomotika and Aoi
(15 p 302) and Imai (4 p 157 However their solutions
f or cylinders and plates in parallel flow are also below
the present data Also it should be remembered that their
solutions are for infinitely wide plates If a value of
WL of above 100 is used in Equation (41) then the present
data and the solutions of Tomotika and Aoi are fairly close
56
The present results indicate that Equation (41~ can be
used with an accuracy of 15 to 20 within the limitations
of the equation (WL 1 to 4 Re = 0 05 to 2)
57
SUM RY AND CONCLUSIONS
Only a small amount of work has been done in the past
on the study of laminar flow over immersed bodies There
are many areas in the chemical process industries and the
field of aeronautics where this information would be very
helpful The purpose of the present investi gation wa s to
study the almost totally unexplored range of Reynol ds
numbers from 0 01 to 10
Drag coefficients have been determined for spheres
cylinders and flat plates in paralle l and perpendicular
flow The drag coefficients have been plotted as a
function of the Reynolds number with dimension ratios as
a parameter on lo g-log graphs The best straight lines
have been drawn through the data In all cases these lines
had a slope of -1 hich shows that the dra g coefficient is
inversely proportional to the Reynolds number at very low
Reynolds numbers for all shapes and dimension ratios The
following equations have been determined from the data
For cylinders
fd - 27 L -0 36 (38 ) - Re ())
which applies for Reynolds numbers of 0 01 to 1 and LD of
2 to 16 For LD greater than 16 the equation is
58
(39)
For flat plates in parallel flow a correction factor has
been applied to account for the edge effect The equation
which applies for Reyno l ds numbers of 0 1 to 2 is
f 6Re
(40)
For flat plates in perpendicular flow
f d
- 37 - Re (w) t -
0 bull 30 (41)
wbieh applies for W L of 1 to 4 and Reynolds numbers of
0 05 to 2
It is concluded tha t Equations (38-41) give the best
values of drag coefficients within an accuracy of 20~ for
the range of Reynolds numbers that were considered Also
it is evident that the dimension ratios are a n important
factor in determining the drag coefficient for a given
Reynolds number Furthermore the drag coefficient inshy
creases with decreasing values of L D or W L for a constant
Reynolds number The da ta obtained in this investi gation
compare favorably with the other experimental data and with
some of the theoretical sol utions It should be remembered
that when comparing the experimental data with theoretical
solutions that practically all of the solutions are for an
infinitely long cylinder or an infinitely wide plate
It is recommended tha t the present apparatus be
59
modified so that a force of 001 pound can be measured
Also it would improve tho accuracy to set up a constant
temperature bath so that the temperature of the oil can not
vary over 02degF A few check points on the present data
is all that is necessary to confirm the validity of
Equations (38- 41) It is also r ecommended that only SAE 140
oil be used and that 2 inches should be the minimum plate
width and cylinder length to be studi3d These conditions
would help to maintain the accuracy of the correction force
for the wire
60
~WMENCIATURE
Symbol Dimensions
A area sq ft
D diameter ft
F force lb f
L length ft
M mas s lb m Re Reynolds number Dvf= -ltr w width ft
a area sq ft
b characteristic length ft
d diameter ft
f drag coefficientfd
gravitation constant l b mft gc 2= 32 17 l b _ rsec
1 length ft
m mass l b bullm
p pressure lbrsqft
r radius ft
t time see
u velocity ft sec
v velocity ft sec
w width ft
61
Symbol Dimensions
X xbullcoordinate ft
y y- coordinate ft
o( vorticity
time sec
viscosity lb m ft -sec
kinematic viscosity ft 2sec
circumference diameter = 3 1416
3density lb m ft
function
stream function
Laplacian operator
infinity
Subscripts
c corrected
f force
1 l iquid
m mass
p projected
s solid
w wetted
62
BI BLIOGRAPHY
1 Allan D N de G and R v Southwell Re laxation methods applied to determine the motion in two di shymensions of a viscous fluid past a fixed cylinder Quarterly Journal of Mechanics and Applied Mathe shymatics 8 129-145 1955
2 Bairstow L B M Cave and E D Lang The reshysistance of a cylinder moving in a viscous fluid Philosophical Transactions of the Royal Society of London ser A 223383- 432 1923
3 Goldstein Sidney The steady flow of viscous fluid past a fixed spherical obstacle at small Reyno l ds numbers Proceedings of the Royal Society of London ser A 123225-235 1929
4 Imai I A new method of solving Oseens equations and its application to the flow past an inclined elliptic cylinder Proceedings of the Royal Society of London ser A 224 141-160 1954
5 Janour Zbynek Resistance of a plate in paralle l flow at low Reyno lds numbers Washington Nov 1951 40 p National Advisory Committee for Aeronautics Te chnica l Memorandum 1316)
6 Janssen E An analog solution of the Navier-Stokes equation for the case of flow past a f l at plate at low Reynolds numbers In 1956 Heat Transfer and Fluid Mechanics Institute (Preprints of Papers) p 173-183
7 Knudsen James G and Donal d L Katz Fluid Dynamics a nd Heat Transfer Ann Arbor University of Michigan 1953 243 p (Michi gan University Engineering Research Bulletin no 37)
8 La~b Horace On the uniform motion of a spherethrough a viscous fluid Philosophical Magazine and Journal of Science s~r 6 21112-121 1911
9 Liebster H Uben den widerstrand von kugeln Annalen Der Physik ser 4 82 541- 562 1 927
63
10 McAdams William H Heat transmission 3d ed New York McGraw- Hill 1954 532 p
11 Pai Shih- I Viscous f l ow theory I Laminar flow Princeton D Van Nostrand 1956 384 p
12 Prandtlbull Ludwi g Es sentials of fluid dynamics London Blackie amp Son 1954 452 p
13 Relf i F Discussion of the results of measure shyments of the resistance of wires with some additionshyal tests of the resistance of wires of small diame shyters In Technical report of the Advisory Committee for Aeronautics London) March 1914 p 47 - 51 (Report and memoranda no 102 )
14 Stokes George Gabriel Mathematical and physical papers Vol 3 Cambridge University Press 1922 413 p
15 Tomotika s and T Aoi The steady flow of a viscous fluid past an elliptic cylinder and a flat plate at smal l Reynolds numbers Quarterly Journal of Me chanics and Applie d Ma thematics 6 290- 312 1953
16 Wieselsbergo r c Versuche Ube r der luftwiderstand gerundeter und kant iger korper Er gebnisse der Aeroshydynamischen Versucbsansta l t Vol 2 G~ttingen 1923 80 p
FIGURE e- PHOTOGRAPH OF SPHERES CYLINDERS AND PLATES
34
holes were drilled so that each plate could be used for
two geometric ratios by changing the wires (See for
example plates la and lb in Table I
35
EXPERI MENTA L PROCEDURE
Viscosity and Density Calibration
A calibrated hydrometer measuring to the nearest
0002 was used to measure the density Table VI shows that
the effect of temperature on density is practically negli shy
gible in the small temperature range used
A Brookfield Synchro-lectric viscometer was used to
measure the viscosity of both the light and heavy oil
Figures 18 and 19 show the effect of temperature on visshy
cosity In addition the viscosity of the light oil was
checke d using the falling ball method and the equation
D2--ltA (f s bull fl) g (34) l 8v
The viscometer was calibrated by the National Bureau of bull
Standards and was accurate to l tb
Velocity Measurements
The velocity of movement through the oil was measured
by determining the rate of rotation of the pulleys with a
stop watch Usually the time for 10 revolutions was
measured at the highe r ve locities and for 5 revolutions at
the low velocities From this information and the di
amaters of the pulleys the velocities ere calculated
36
The time was measured to the nearest tenth of a second
Since the measured time was usually between 20 and 40
aeconds 1 the error in ~easuring velocity was considered to
be less tha~ 0 5~
force Measurements
The object connected to the scale 1 was dropped to the
bottom of the oil drum The motor was started and the scale
was read as the object vms being pulled towards the top of
the drum Two or three readings were taken for each object
at each velocity In nearly all cases these readings were
the same
37
ti XPER I MENTAL RE STJLTS
The dra g coefficient and the Reynolds number were
calculated by the use of Equations (l or (15) for each of
the spheres cylinders and plates from the measured
quantities of force and velocity a~d the values of the vis shy
cosity and density corresponding to the temperature of the
oil It was necessary to ~ubtract from the measured force
the force on the wire The corrected force measurement was
then used to determine the drag coefficient The force on
the wire has been determined as being proportional to the
velocity A correction curve relating force on the wire
and ve l ocity is plo tted in Figure 9 for the li ght oil and
Fi gure 10 for the heavy oil
The calculated drag coefficients Reynolds numbers
and velocities along with the measured force for the spheres
cylinders flat plates - parallel flow and flat plates shy
perpendicular flow have been tabulated in Tables II III
I V and v respectively
The calculated drag coefficients have been plotted as
a function of the Reynolds number on logarithic graph paper
with geometric ratios as a parameter
Drag coefficients for the spheres are plo tted in
Figure 11 The data for the cylinders are plotted in
CD_ bull 0 G 0
03
Tshy02
01
10 20 30 410 50 60 70 80
VELOCITY- FTJSEC
DRAG FORCE ON THE WIRE-LIGHT OIL
FIGURE 9
I -shy I -middot -- -shy -1shy _i-i I --~ I I _ -middot- shy I i
_I_ - _ middot- LL I l l tmiddot - middot1middot ~- - - - -+i middotshy I - --+-cl - l
1 1 I I IV jc---- --r--middotmiddottmiddot r-middotmiddot--tmiddotmiddot---shy _____ _L __ --~- --1shy middotmiddotr-r-middott- 1 -f-f-T- _~ +-L--1---~- 1--l
~- - shy I-+---Rmiddot-- I I I l i ~~ i -~~ ~- -T f i rshy ~-- --shy i- ----~-- shy - middot1 shy
I i I i I I 1--- -middot - fshy middot i----1---+-shy - i-middot -~+-- --~- --~-- ---- -t+ I v-~~ -middot j
i I middot 1_ _ I tmiddot---+-+1-+--li~+middot -+--+-+-1-+-+-+-+--tc--1-+-t-11-shy - middot --t- 1---t- t----tmiddotshy --~-- -middot i-shy I 1i - ~ i I i v i middotmiddotmiddot
[~v +L~ + ~ - I~~j-+ r V I ~t--- -~-- I +---~-- I f-middot ---1-- ~ -- --- ) Li --+--+--+-+-+-+--1--+--+---t---4 -1--1--+-+--+-l-i
tl~ I I Q Y +l~~ii-+-++++-middotHH-++-+-+-+--H--++ -i t Imiddot i i 1 j _V I f1 r-t~-middot l--r-tshy -~ 7 middot 1 -shy middot middotmiddot I
DRAG FORCE ON THE WIRE- HEAVY OIL
FIGURE 10
40
+shy l i~ltgt ~ bull r-rshy I i t _l
1 lf-1-1 l+r+ fJ-Ct I+ t li 1~t rtH r+l rf-l It llil I I
l l~pound 11 1 ~middot ~~middott ~ It lqf L
t I+--= ~r 17 -Er I _ ~ _pound~- sect Imiddot I+
iU=ff=t 1 +~ t_ - ~ r 111= t h=
I middot
t= IE I 1 1
plusmn~ kplusmni - -STOKE S EQ
(~ l h+middot
ru HmiddotHti+H1 11
c lffii l t~ 4 ~ ~middot ~ff l ~ ~h i ltlri
1 yen~ middot I ~ I I T ~ gt l+t H+h l+ i j l tfl-l Imiddotmiddot ft+ ++ l f+ Imiddotmiddot I+ I+ middott bulli I 1middot1 I ftt-1shy middot I middot r 11 I IH Ij ~ ~ middotishy J F 1= 6= ~
=f l~iit rtti l lit~ I FS lf~ l=i-+
l-11ffi tt lr 1 ~1 -t =l=Rttl 1ft i- 1 ~ I+ I
~~ lflJ
t I lfl m ~~WFB Lt
41plusmn811 IF I Hir tt ft itttplusmn i I~
1-+++middot
I ~ I (~ ffitrHf1 Ittmiddot ~ l r i H-t-r r HHt m 11 H++ I
bull I I
1_ _ F bullmiddot Imiddotmiddot t-- 1-T h iT
f-t+ ftt I+ I lt + T Imiddot 1
1t _plusmn middot~~ ~- 11shy
=a~ 1~ - =itf lttti
H I
=
DATA FOR SPHERES
FIGURE II
41
I -1---1-1-+--+--Ti-+-------+----r--shy --r--- -shy + t----+shy ----4-~---+-f----f--+-f--l--1 I t--shy --t-- ---+-shy
J-+-~f--~~ -___l_ ~---
i 1 L~L~-~tr-l----H~4-----~-f------+------+-----+----+---+middot-t-middot-H5000
middot~ ~ m- ~ - ~t plusmn~ 3t i t~ -f--- bullbull - ~~ h middot-
01 0~ 10
Re
-
DATA FOR CYLINDERS - LD = 6 8 AND 12
FIGURE I 4
44
Figures 12 13 and 14 The data for LD values of 16 24
and 32 were nearly the same and have been plotted to gether
i n Figure 12 In addition the curves for the other LD
ratios determined fro m Fib~res 13 and 14 have been drawn
in Figure 12 so that the effect of the length-to-diameter
is clearly shown Figure 13 shows the data for LD values
of 2 and 4 and the curves determined from this data
Firure 14 shows the data for LD values of 6 8 and 12
and the curves determined from this data
The data for flat plates in parallel flow are plotted
in Fi gure 15 A correction factor for the edge effect has
beon used so that the width-to-length ratio is not a
parameter in this plot A portion of the data of Janour
(5 p 31) is also shown in the diagram
The data for fla t plates in perpendicular flow is
plotted in Figures 16 a nd 17 Figure 16 shows the data for
WL values of 2 Also the curves for the three WL ratios
1 2 and 4 have been drawn in the fi gure Figure 17 shows
the data for WL values of 1 and 4 The curves determined
from the data have also been dravm in the figure
45
10~ ~ ~--- -shy
t==Ff1TR=+ iJ+--_-_--r_-_---+-+---+--+-+--_---_-~r-=r~=~+--=---=---=---=--~=--=_~1=_--=_~_-middot~~--+-+-t~ 1 Ll~+--+-- ---jtshyl~t L--+ I
I
P------ _l -- --1---L i
20 ~-- I ~g I --- - ---+-- r t L_shy
~ ~B 1) I --o-o- JONES - () - - ~~ p f---j- -~-- e e JANOU R
c gt ~c ~ ------ JANSSEN I 0 0 ~ I
IO ~2=i~~~~~~a=~~f=j= ---- TOM OTIKA bulll= I
~~n ~~--~~~~~~o~~~~~--4- NDCIgttl o shy
-
~--~~~~~+--+~+--4-r-~1+-~-middot+1~ ~ --H--~-~~os I i i i-4 ---~T I I f-- t --- li-------~--+-_--+--t-----~~-~_+---_-_-_--+------+-+-__+-[- +_- ___ _______ __+---+-r-+--H----_+--r--------+shy
it I+ ~ bull t ~1 ri j t++t+t++tft bullm H--~+H-t+t-++H-f+t+~HtttH t bull~H-IrttI-H
iH-H u nH m
I
t H+t-~ 1-r f-tj
i it iT -t middotHt I I I I Ill
~middot __
r middotshy
i I r-
f H- jLj f r H rr t~
II
t f f-l -t+tt ~ ==_ =~middot irE
I I
I
I
f
I --
i
t
1 r bull - r
~- ltt++l=tUtt~S-t+t+++~-++U +HJJm~-fl~HHtt1 tttn ll+t-Tt-~- ~ r fH T --r -1 t ---t- -tshy w _+ _ I-shy middotI
-shy -r- + Hbull Hshy t-I --r++ -t iHr -1 H-e-- -t I 1IT 1
1 H-rf-I IJftJ Jf+i+ ~ L
=+shy - tjshy rtmiddotshy ~ -
+ H 1-Jt I tt o =tt ~-
~1 l +fill l plusmn~ fplusmn -shy + I t-
DATA FOR FLAT PLATES PERPENDICULAR FLOW- WL= I 4
FIGURE 17
48
DI SCUSS ION OF RESULTS
Correction and Accuracy of Measurements
After a few pre liminary force measurements with the
spheres and a check with Stokes law (Equation 2) it was
apparent that the drag force on the wire was appreciable
and needed to be considered It was decided to take a
series of measurements with the spheres and calculate the
difference between the measured force and the force calcushy
lated from Stokes law The difference in force could then
be attributed to the drag on the wire If Stokes law is
followed the force on the wire should be proportional to
the velocity
A series of twenty measurements of the force on the
spheres was taken for each oil and the difference between
the measured force and that calcula ted by Stokes 1 law was
determined For each oil this difference as plo tted vs
the velocity The points grouped fairly ell around a
strai ght line nearly passing through the origin The
method of least squares was used to determine the equation
of the line best fitting the da t a The equa tion of the
line for the li bht oil tas found to be
Fe bullbull05605v - oooa (35)
which was determined at about 62 7degF Since the intercept
49
of the line is very close to zero it is believed that the
line is a good indication of the drag on the wire The
equation of the line for the heavy oil was found to be
F - 19llv I oo2o1 (36 ) c shy
which was determined at about 64 2deg The intercept of this
line is also quite close to zero These lines plotted in
Fi poundures 9 and 10 were used throughout the investigation
for the correction factor of the drag on the wires For
the cylinders and flat plates in parallel flow which were
pulled by two wires the values determined from Equations
35) and (36) were doubled For the plates in perpendicular
flow pulled by four wires the correction force was multishy
plied by four
The spring scale had 12 ounce divisions but could be
read to the nearest sixth of an ounce Some of the measureshy
ments of force were under an ounce hence a considerable
spread of the measurements was noticed in the pre liminary
data and throughout the experiment However sufficient
points were obtained so that it was possible to draw a
reliable curve through the data in all casas An analysis
was made to determine the average deviation from Stokes
equation for the spheres It raa found that the average
deviation was 15 1 for the light oil 16 6 for the heavy
oil and 15 9 overall The maximum deviation was 89
50
Inspection of the other data shows that these deviations
are also representative of the cylinders and flat plates
The force measurement is the least accurate part of the
experiment Other insignificant errors are introduced by
a small variation in the temperature This variation was
held to about 10 from the temperature of the calibrated
correction curve The velocity measurements and the
dimensions of the cylinders spheres and pl~ tes are conshy
sidered go od enough so tha t no appreciable errors occur
In order to e l iminate the WL parameter for flat plates
in parallel f l ow an additional factor for the effect of
the edges was subtracted from the measured force Janour
(5 p 27) presented the foll owing equation for the edge
correction for one edge of a flat plate in parallel flow
F ~ lv~ bull (37 ) edge gc
In present work this equation as doubled because both
edges of the plates were submerged in fluid It is assumed
in appl ying this correction that the lowe r limit of a
Reynolds number of 10 proposed by Janour can be extended
close to 0 1
Analysis of Results
Forty of the points for the spheres were used to get
51
the correction factor for the wires The remaining thirty
points are well erouped about Stokes law
The data for cylinders for LD ratios of 16 24 and
32 did not seem to be se gregated therefore these data
were plotted together It would seem that in the low range
of Reyno l ds numbers an LD of 16 and greater can be con shy
sidered an ~nfini tely long cylinder The other LD ratios
of 2 4 6 a 12 provided fairly distinct and separate
lines The best straight lines were drawn through the data
for each of the LD ratios It was evident that in eaeh
case a slope of -1 on a lo g-log graph gave the best straight
line which would indicate that the force varies directly
as the velocity It was possible to develop an empirical
expression relating dra g coefficient Reynolds number and
LD The following equation was obtained from the straight
line plots of Re vs fd for the various LD ratios
(38 )
Equation (38) applies for Reyno l ds numbers from 01 to 10
and for LD ratios of 2 to 16 For LD ratios greater
than 16
10 re (39 )
The data for flat plates in parallel flow is plotted
in Figure 15 after the correction factor for tho edge
52
effect was subtracted When the edge correction is made
no effect of WL ratio is indicated This result would be
expected The data followed a straight line with a slope
of -1 up to a Reynolds number of 2 After that a curve was
dravm connecting the line to that obtained by Janour The
equation for the straight section of the curve is
f - 6 (40)- Re
which applies for Reynolds numbers of 0 1 to 2 0 Here
a gain the force is proportional to the velocity Vfuen
determining drag force for flat plates in parallel flow
the force is first calculated from Equations (40) and (15 )
then the edge correction is added
The effect of the geometric ratios is clearly shown in
the data for flat plates in perpendicul ar flow which are
plotted in Figures 16 and 17 As with the other data the
best straight line was drawn through the various points
for eaoh of the WL ratios Again the line had a slope of
-1 The equation relating fd Re and wL was found t o be
rd 37 (w) -o 3o (41)Irel
which applies for Reynolds numbers of about 05 to 2 0 and
WL ratios of 1 to 4 It is possible but it has not been
proved that Equation (41) is suitable for higher WL ratios
The exponent on WL in Equation 41) is very close to that
53
on L D i n Equation ( 38 )~ It i s possible t ha t these
exponents are t he same but this cannot be sho~~ depound1nitely
until more accura te da ta are available It would be exshy
pected that a s the Reynolds number approaches zero t he
effect of geometric ratios would be the same for cylinders
and fla t pla tes in perpendicula r flow
It is seen in the t a bles of data that occasionally a
ne gative force was obtained because the correction applie d
due to t he wire dra g was greater than the mea sured force
These points obviously are incorrect This occurred only
for the smallest plates in the heavy oil at t he highest
velocities However these knom bad points occur in less
tha n 5~ of the data
It is clearl y shown that for cylinders and plates the
fd increases as L D or W L decreases This is in direct
contrast to Wiesel aberger s investigation However his
work is for hi gher Reynolds numbers at which a turbulent
wake forms bull
Comparison of Results with Other Data and Theoretical So l utions
The data for sphere~ a grees of course with Stokes
l aw since that law was used to determine the correction
factor for the wire Liebster (9 Pbull 548 ) has
54
substantiated Stokes equation
There are no experimental data with which to compare
the results of the cylinders Wieselsbergers minimum
Reynolds number of 4 is above the ran ge covered in the preshy
sent investigation The da ta for the highest LD ratios
(16 24 and 32) does agree almost exactly wi t h the solution
of Allen and Southwell (1 P bull 141) (LD =00) in the range
of Reynolds numbers from 0 1 to 1 0 Allen and Southwells
solution a greed with the data of Wieselsberger (16 p 22)
However the present data is above the theoretical solutions
of Lamb (8 p 112-121) throughout the range of Reynolds
numbers from 0 01 to 1 0 and above the solutions of
Bairstow Cave and Lang (2 p 404) I mai (4 p 157) and
Tomotika and Aoi (15 p 302) for Reynolds numbers of 0 1
to 1 0 Allen and Southwells solution a grees dth both
Wieselsberger 1 s a nd the present data Their solution and
the present data represent the best means for predicting
drag coefficients for flow over long cylinders for Reynolds
numbers of 0 01 to 10 It should be remembered that the
o t her solutions should a gree with eac h other since they
were all essentially derived by linearizing the Na viershy
Stokes equation
The data for flat plates in parallel flow is
55
considerably above the theoretical solutions of Janssen
(6 p 183 ) and Tomotika and Aoi (15 Pbull 302) However
Fi f~re 15 shows that a smooth transition occurs bet een
the present work and the data of Janour (5 P bull 31) The
present data considerably extend the experimental inforshy
mation previously available for laminar flow paral lel to
flat plates In the re gion of Reynol ds numbers less than
2 the drag coefficient is shown to be inversely proportional
to the Reynolds number Janours data covers a range of
Reynolds numbers from 11 to 1000 The results of the
present investigation line up with Janours results which
in turn on extrapolation to higher Reyno l ds numbers
(greater than 1000) make a smooth transition into Blasius
curve represented by Equation (10) At Reyno l ds numbers
greater than 20 000 the drag coefficient is inversely proshy
portional to the square root of the Reynolds number
The data for flat plates in perpendicular flow is conshy
siderably above the solutions of Tomotika and Aoi
(15 p 302) and Imai (4 p 157 However their solutions
f or cylinders and plates in parallel flow are also below
the present data Also it should be remembered that their
solutions are for infinitely wide plates If a value of
WL of above 100 is used in Equation (41) then the present
data and the solutions of Tomotika and Aoi are fairly close
56
The present results indicate that Equation (41~ can be
used with an accuracy of 15 to 20 within the limitations
of the equation (WL 1 to 4 Re = 0 05 to 2)
57
SUM RY AND CONCLUSIONS
Only a small amount of work has been done in the past
on the study of laminar flow over immersed bodies There
are many areas in the chemical process industries and the
field of aeronautics where this information would be very
helpful The purpose of the present investi gation wa s to
study the almost totally unexplored range of Reynol ds
numbers from 0 01 to 10
Drag coefficients have been determined for spheres
cylinders and flat plates in paralle l and perpendicular
flow The drag coefficients have been plotted as a
function of the Reynolds number with dimension ratios as
a parameter on lo g-log graphs The best straight lines
have been drawn through the data In all cases these lines
had a slope of -1 hich shows that the dra g coefficient is
inversely proportional to the Reynolds number at very low
Reynolds numbers for all shapes and dimension ratios The
following equations have been determined from the data
For cylinders
fd - 27 L -0 36 (38 ) - Re ())
which applies for Reynolds numbers of 0 01 to 1 and LD of
2 to 16 For LD greater than 16 the equation is
58
(39)
For flat plates in parallel flow a correction factor has
been applied to account for the edge effect The equation
which applies for Reyno l ds numbers of 0 1 to 2 is
f 6Re
(40)
For flat plates in perpendicular flow
f d
- 37 - Re (w) t -
0 bull 30 (41)
wbieh applies for W L of 1 to 4 and Reynolds numbers of
0 05 to 2
It is concluded tha t Equations (38-41) give the best
values of drag coefficients within an accuracy of 20~ for
the range of Reynolds numbers that were considered Also
it is evident that the dimension ratios are a n important
factor in determining the drag coefficient for a given
Reynolds number Furthermore the drag coefficient inshy
creases with decreasing values of L D or W L for a constant
Reynolds number The da ta obtained in this investi gation
compare favorably with the other experimental data and with
some of the theoretical sol utions It should be remembered
that when comparing the experimental data with theoretical
solutions that practically all of the solutions are for an
infinitely long cylinder or an infinitely wide plate
It is recommended tha t the present apparatus be
59
modified so that a force of 001 pound can be measured
Also it would improve tho accuracy to set up a constant
temperature bath so that the temperature of the oil can not
vary over 02degF A few check points on the present data
is all that is necessary to confirm the validity of
Equations (38- 41) It is also r ecommended that only SAE 140
oil be used and that 2 inches should be the minimum plate
width and cylinder length to be studi3d These conditions
would help to maintain the accuracy of the correction force
for the wire
60
~WMENCIATURE
Symbol Dimensions
A area sq ft
D diameter ft
F force lb f
L length ft
M mas s lb m Re Reynolds number Dvf= -ltr w width ft
a area sq ft
b characteristic length ft
d diameter ft
f drag coefficientfd
gravitation constant l b mft gc 2= 32 17 l b _ rsec
1 length ft
m mass l b bullm
p pressure lbrsqft
r radius ft
t time see
u velocity ft sec
v velocity ft sec
w width ft
61
Symbol Dimensions
X xbullcoordinate ft
y y- coordinate ft
o( vorticity
time sec
viscosity lb m ft -sec
kinematic viscosity ft 2sec
circumference diameter = 3 1416
3density lb m ft
function
stream function
Laplacian operator
infinity
Subscripts
c corrected
f force
1 l iquid
m mass
p projected
s solid
w wetted
62
BI BLIOGRAPHY
1 Allan D N de G and R v Southwell Re laxation methods applied to determine the motion in two di shymensions of a viscous fluid past a fixed cylinder Quarterly Journal of Mechanics and Applied Mathe shymatics 8 129-145 1955
2 Bairstow L B M Cave and E D Lang The reshysistance of a cylinder moving in a viscous fluid Philosophical Transactions of the Royal Society of London ser A 223383- 432 1923
3 Goldstein Sidney The steady flow of viscous fluid past a fixed spherical obstacle at small Reyno l ds numbers Proceedings of the Royal Society of London ser A 123225-235 1929
4 Imai I A new method of solving Oseens equations and its application to the flow past an inclined elliptic cylinder Proceedings of the Royal Society of London ser A 224 141-160 1954
5 Janour Zbynek Resistance of a plate in paralle l flow at low Reyno lds numbers Washington Nov 1951 40 p National Advisory Committee for Aeronautics Te chnica l Memorandum 1316)
6 Janssen E An analog solution of the Navier-Stokes equation for the case of flow past a f l at plate at low Reynolds numbers In 1956 Heat Transfer and Fluid Mechanics Institute (Preprints of Papers) p 173-183
7 Knudsen James G and Donal d L Katz Fluid Dynamics a nd Heat Transfer Ann Arbor University of Michigan 1953 243 p (Michi gan University Engineering Research Bulletin no 37)
8 La~b Horace On the uniform motion of a spherethrough a viscous fluid Philosophical Magazine and Journal of Science s~r 6 21112-121 1911
9 Liebster H Uben den widerstrand von kugeln Annalen Der Physik ser 4 82 541- 562 1 927
63
10 McAdams William H Heat transmission 3d ed New York McGraw- Hill 1954 532 p
11 Pai Shih- I Viscous f l ow theory I Laminar flow Princeton D Van Nostrand 1956 384 p
12 Prandtlbull Ludwi g Es sentials of fluid dynamics London Blackie amp Son 1954 452 p
13 Relf i F Discussion of the results of measure shyments of the resistance of wires with some additionshyal tests of the resistance of wires of small diame shyters In Technical report of the Advisory Committee for Aeronautics London) March 1914 p 47 - 51 (Report and memoranda no 102 )
14 Stokes George Gabriel Mathematical and physical papers Vol 3 Cambridge University Press 1922 413 p
15 Tomotika s and T Aoi The steady flow of a viscous fluid past an elliptic cylinder and a flat plate at smal l Reynolds numbers Quarterly Journal of Me chanics and Applie d Ma thematics 6 290- 312 1953
16 Wieselsbergo r c Versuche Ube r der luftwiderstand gerundeter und kant iger korper Er gebnisse der Aeroshydynamischen Versucbsansta l t Vol 2 G~ttingen 1923 80 p