P~og. A~rospuee Sci. 1977, Vol. 18, pp. I - 57. Pergamon Press. Printed in Great Britain. AN OUTLINE OF THE TEChnIQUES AVAILABLE FOR THE MEASUREMENT OF SKIN FRICTION IN TURBULENT BOUNDARY LAYERS* K. G. Winter Royal Aircraft Establishment, Farnborough, Hants, U.K. Summary-The techniques covered include force-mdasur~ment balances, the use of the velocity profile, pressure measurements by surface pitot tubes or about obstacles, and the use of the analogies of heat transfer, mass transfer or surface oil-flow. Hot-wire or laser techniques for determin- ing the shear stress within the fluid are not included. The sources of error and ranges of application of the various techniaues are discussed. i. INTRODUCTION In most applications of fluid mechanics a knowledge of the drag created by fluid flowing over a solid surface is essential to the understanding of the performance of a system whether it be a ship or an aircraft or the flow through a pipe. Considerable effort has therefore been de- voted to the measurement of skin friction. This brief review concerns itself only with exter- nal flow and with measurements primarily related to the performance of aircraft. It was, however, the need to estimate the performance of ships which led to the first measure- ments at high Reynolds number. Probably the first systematic investigations were made over iO0 years ago by Froude (1872) who measured the drag of a series of planks towed at various speeds along a tank using the elegant apparatus shown in Fig. I. It is interesting to note that at that time even the qualitative effect of Reynolds number on skin friction was not gen- erally understood. Froude did apparently have a concept of a boundary layer and states: The investigation of skin friction may be separated into three primary divisions: (i) the law of the variation of resistance with the velocity; (2) the differences in resistance due to differences in the quality of surface; (3) the differences in the resistance per unit of surface due to dif- ferences in the length of surface. The necessity of investigating the latter of these conditions may not be at once apparent, it having been generally held that surface-friction varies directly with the area of the surface, and will be the sanm for a given area, whether the surface be long and narrow or short and broad. It has always seemed to me to be impossible that this should be the case, because the portion of the surface that goes first in the line of motion, in experiencing resistance from the water, must in turn co~mmnicate to the water motion in the direction in which it itself is travelling, and consequently the portion of the surface which succeeds the first will be rubbing not against stationary water, but against water partially moving in its own direction, and cannot therefore experience as much resistance from it. If this reasoning holds good, it is certain that doubling, for instance, the length of a surface, though it doubles the area, would not double the resistance for the resistance of the second half would not be as great as that of the first. *Notes prepared for von Karmon Institute for Fluid Dynamics Lecture Series on "Compressible Turbulent Boundary Layers", March 1 - 5, 1976~ Copyright ~ HMSO (London) 1976.
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P~og. A~rospuee Sci. 1977, Vol. 18, pp. I - 57. Pergamon Press. Printed in Great Britain.
AN OUTLINE OF THE TEChnIQUES AVAILABLE FOR THE MEASUREMENT OF SKIN FRICTION IN TURBULENT BOUNDARY LAYERS*
K. G. Winter
Royal Aircraft Establishment, Farnborough, Hants, U.K.
Summary-The techniques covered include force-mdasur~ment balances, the use of the velocity profile, pressure measurements by surface pitot tubes or about obstacles, and the use of the analogies of heat transfer, mass transfer or surface oil-flow. Hot-wire or laser techniques for determin- ing the shear stress within the fluid are not included. The sources of error and ranges of application of the various techniaues are discussed.
i. INTRODUCTION
In most applications of fluid mechanics a knowledge of the drag created by fluid flowing over
a solid surface is essential to the understanding of the performance of a system whether it be
a ship or an aircraft or the flow through a pipe. Considerable effort has therefore been de-
voted to the measurement of skin friction. This brief review concerns itself only with exter-
nal flow and with measurements primarily related to the performance of aircraft.
It was, however, the need to estimate the performance of ships which led to the first measure-
ments at high Reynolds number. Probably the first systematic investigations were made over
iO0 years ago by Froude (1872) who measured the drag of a series of planks towed at various
speeds along a tank using the elegant apparatus shown in Fig. I. It is interesting to note
that at that time even the qualitative effect of Reynolds number on skin friction was not gen-
erally understood. Froude did apparently have a concept of a boundary layer and states:
The investigation of skin friction may be separated into three primary divisions: (i) the law of the variation of resistance with the velocity; (2) the differences in resistance due to differences in the quality of
surface; (3) the differences in the resistance per unit of surface due to dif-
ferences in the length of surface.
The necessity of investigating the latter of these conditions may not be at once apparent, it having been generally held that surface-friction varies directly with the area of the surface, and will be the sanm for a given area, whether the surface be long and narrow or short and broad. It has always seemed to me to be impossible that this should be the case, because the portion of the surface that goes first in the line of motion, in experiencing resistance from the water, must in turn co~mmnicate to the water motion in the direction in which it itself is travelling, and consequently the portion of the surface which succeeds the first will be rubbing not against stationary water, but against water partially moving in its own direction, and cannot therefore experience as much resistance from it. If this reasoning holds good, it is certain that doubling, for instance, the length of a surface, though it doubles the area, would not double the resistance for the resistance of the second half would not be as great as that of the first.
*Notes prepared for von Karmon Institute for Fluid Dynamics Lecture Series on "Compressible Turbulent Boundary Layers", March 1 - 5, 1976~ Copyright ~ HMSO (London) 1976.
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Skin friction in turbulent boundary layers 3
Later Ke~f (1929) ~de measure~nts of local skin friction at several stations along the bot-
tom of a pontoon 77 m long using fairly large panels (309 x I010 m) ~unted on balances (Fig.
2). These measure~nts achieved Re~olds numbers of up to 5 x 108. ~e direct ~asurement
of skin friction by force balance was an essential step in setting ~ the basic skin friction
laws and these measurements of Ke~f together with those of others, notably Schoenherr (1932),
formed the basis for the generally-accepted skin friction esti~tion for incompressible flow
(see, for exa~le; Goldstein 1938). Because of our limited understanding of turbulent flows
-! Platte I-
o.~J'~30° ~ [ t ,- IO00mm ~ -I
.
,], I ~Jfh~ngun~ -.-"- ~" . . . . . . . . . . F - - - ~
variation of g with displacement of skin-friction balance
von Karman constant
viscosity
kinematic viscosity
mass concentration
density
T
Subscripts
e
m
~C Prandtl number -~
shearing stress
, also o =
edge of boundary layer
intermediate-enthalpy conditions
adiabatic wall conditions
½ (~ - I)M e e
1 + ~(y - I)M 2 e
w wall conditions
Superscript
i refers to incompressible flow.
Skin friction in turbulent boundary layers 7
3. DIRECT MEASUREMENT
Apart from the experiments of Schutz-Grunow (1940) interest in the direct measuren~ent of skin
friction* lapsed until the increasing speed of aircraft called for precise measurements in
compressible flows. As a result there have been numerous skin-friction balances designed over
the past few years, and much ingenuity has been exercised in design to overcome the essential
problem of obtaining accurate measurements of the shear forces which are very small, three
orders less than the inertial forces.
The problems which have to be considered are listed below.
(i) Provision of a transducer for measuring small forces or deflections, and the compromise
between the requirement to measure local properties and the necessity of having an ele-
ment of sufficient size that the force on it can be measured accurately.
(2) The effect of the necessary gaps around the floating element.
(3) The effects of misalignment of the floating element.
(4) Forces arising from pressure gradients.
(5) The effects of gravity or of acceleration if the balance is to be used in a moving
vehicle.
(6) Effects of temperature changes.
(7) Effects of heat transfer.
(8) Use with boundary-layer injection or suction.
(9) Effects of leaks.
(i0) Protection of the measuring system against transient normal forces during starting and
stopping if the balance is to be used in a supersonic tunnel.
The basic choice to be made is the size of the floating element which dictates the sensitivity
required of the measuring system which may he passive (displacement) or active (force-feed-
back). To illustrate the sensitivity required the table below shows over a range of Maeh
number the force in milligrams to be measured by a balance with a head of IOnnn diameter in a
zero-pressure gradient flow at one atmosphere stagnation pressure and a Reynolds number of
IO million.
M 0.I 0.5 i 2 3
Force mg 16 290 680 540 210
For adverse pressure-gradient flows the forces will be even less. A very sensitive transducer
is therefore needed and a variety of transducers and sizes of floating element has been used.
The table on the following page lists some of the designs developed in the past few years.
The balance used by Schultz-Grunow (1940) is included for historical interest. In this
balance the floating element was rather large and was mounted on offset torsional pivots and
restrained by a torsion bar. With the exception of the balance of Ozarapoglu (1973) (Fig. 5)
in which the floating element is supported on air bearings, the remaining balances have either
a parallel-linkage supporting arrangement (Fig. 3) or effectively a pivot below the floating
element (Fig. 4). A similar arrangement to that of Schultz-Grunow was used in a small balance
by Kovalenko and Nesterovich (1973). In their balance the floating element was pivoted about
an axis normal to its surface with the axis offset to one side of the element. The most popu-
lar device for detecting the position of the floating element is a Linear Variable Differential
*It might be noted that the classical work of Wieghard (1942) on the drag of surface excres- cences made use of direct force measurements.
Reference
Test conditions
Size of floating
Type of suspension/
Force
element (nmO
position/ force transducer
range
Schultz-Grunow
1940
Dhawan
1953
Coles
1953
Weiler & Hartwig 1952
Lyons
1957
U = 2Om/s
1.6 x 106 < Re
< 16 x 106
x
Low speed 6 x 104 < Re
< 60 x I0"
x
Subsonic 0.2 < M < 0.8, 0.3 x 106 < Re x
Supersonic 1.24 < M < 1.44
M = 1.97 0.4 x 106 < Re
< I0 x 106
x
M = 2.57 0.4 x 106 < Re
< 9 x 106
x
M = 3.70 0.5 x 106 < Re
< 8 x 106
x
M = 4.54 0.4 x i0
< Re
< 8 x 106
x
Supersonic wind tunnel
Supersonic flight
< 1.2
x 106
300
x 500*
11.5 × 63
2 x 20
6.2 x 37.9
25 dia
50 dia
Optical/manually-operated
offset torsion bar
Parallel interchangeable
LVDT
Parallel linkage
LVDT
Translation of element
support by micrometer
LVDT
Double parallel inter-
connected linkage to eliminate
sensitivity to linear and
rotational accelerations
LVDT
20 mg
to
800 mg
3 g
30 g
Smith &
Walker
1958
0,ii < M < 0.32
106 < Re
< 40 < 106
x
50 dia
Parallel linkage
LVDT
Kelvin current balance
14 g
MacArthur
1963
Moulic
1963
Shock tunnel
M = 6 Low density
6.4 dia
Parallel linkage
Piezoelectric beams
0.25 x 25
Side flexure pivot
LVDT
5 g
20 mg
Young
1965
Supersonic flow with heat transfer and surface
roughness
25 dia
Parallel linkage
LVDT
Dershin et al.
1966
*Estimated from sketch
Supersonic flow with mass transfer
"Pointed ellipse"
Parallel linkage
LVDT
Moore &
McVey
Brown &
Joubert
Fowke
Bruno,
Yanta &
Risher
Winter &
Gaudet
Hastings &
Sawyer
Paros
(Kistler)
Miller
Franklin
Morsy
van Kuren
Ozarapoglu
1966
1969
1969
1969
19
70
19
70
19
70
1971
1973
1974
1974
1975
High temperature hypersonic flows
Low-speed adverse pressure gradients
Supersonic speeds
Supersonic speeds including flows with heat transfer
0.2 <
M <
2.8
16 x
106 <
Re
< 200 ×
106
X
M=
4
iO x
106 <
Re
< 30 x
106
x
Used in a
wide range of conditions including
flight.
Cooling system available
Low-speed flow.
Favourable pressure gradient
Subsonic wind tunnel and water channel
Low speed flow past circular cylinder
High-temperature hypersonic flows with heat
transfer.
Floating element water cooled
Low-speed.
Adverse pressure gradients
19 dia
127 dia
20.3 dia
368 dia
7.9 dia
9.4 dia
25 dia
16 dia
50.1 x
3.2
12 x
12
127 dia
Flexure pivot
3 g
Pneumatic position sensor
High temperature motor
Parallel linkage
200 mg
LVDT
Flexure pivot
200 g
LVDT
Permanent magnet plus coil
Flexure pivot
2 g
LVDT
Var iab i e
by changing
Motor-driven spring
loading
spring
18OO g
Parallel linkage
Resistance strain gauges
Parallel linkage
500 mg
LVDT
Pivoted about crossed-spring
IOO mg
flexure.
Differential capaclty.to
Permanent magnet plus coil
IO g
Parallel linkage
200 mg
LVDT
Pivoted.
Variable geometry
i g
electronic valve.
Jewelled pivots.
130 mg
Clock springs
LVDT
Parallel linkage
5 g
LVDT
Air bearings
I g
LVDT
o"
= rt
o"
0 m m
10 K.G. Winter
Atrflow Dashpot ~ /Floating element "~ _ _ 25rim dia ,./ - t ~ /
I I
x/////////// j[~L/7~ v o T Cover
Fig. 3. Parallel-linkage balance (D.R.L.).
Transformer (LVDT) which is capable of a resolution of displacement as small as 0.05 ~m. In
many balances the force is determined from displacement of the element against a spring as
indicated by the signal from the LVDT. There are disadvantages in the use of a displacement
balance since the necessary gaps round the element vary with the load and this variabion may
produce spurious effects. In balances of the nulling type the position transducer is used to
provide the signal to a force system which maintains the floating element at a given position
The force system is usually either a Kelvin current balance or a coil and permanent magnet.
As an alternative to separate position and force transducers the Kelvin current balance may
be arranged to serve both purposes as shown by Franklin (1960).
Airflov _ ~ ~Floating element
_ 9ram dla/~
::;:~t:#¢¢ ~ H ~ ; ~ H'clt Sh''ld
I I ~ t L ~ _ . ~ _ _ ~ V , ~ I I ,-.t.,-,, . i . j " " ' . . ' ~ T I ~ / A I l K / / ~ I " ~ / / / ~ F o ~ . b,,oo,, • p,",',': """ I zllv//y/ z, IN ]--,,,to"o,-,,,
(; tossed - sprinq pivot
Fig. 4. Pivoted balance (Kistler).
The difficulty of measuring small forces in a displacement-type balance was overcome by
MacArthur (1963) by supporting his floating element on a piezo-electric crystal and Winter
and Gaudet (1970) were able to use resistance strain gauges in a large balance. Franklin
(1973) has obtained high sensitivity by using a variable geomatry electronic valve. Moore
and McVey (1966) have investigated a wide variety of position and force transducers for ap-
plication at high skin temperatures.
Skin-friction balances have been used in flight on rockets (Fenter and Lyons, 1957) and on
high-speed aircraft (Garringer and Saltzman, 1967; Fisher and Saltzmann, 1973). To eliminate
their sensitivity to linear accelerations of the flight vehicle it is necessary to arrange a
mass balance for the measuring system as is done in the Kistler gauge (Paros, 1970). Weilet
(~954) and Lyons (1957) produced designs which were also insensitive to rotational accelera-
Skin friction in turbulent boundary layers I!
Airflov Position ~ Floating adjuster 127 mm d i a / e l e m c n t
These three ranges correspond roughly to the three regions in the velocity profile, the vis-
cous sub-layer yUT/V < 6, the transition region 6 < yu /~ < 60 and the logarithmic region
60 < yu /v < 500.
As was pointed out subsequently by Head and Vasanta Ram (1971) the expressions (5-2) do not
quite match at the changeover and also the expression for the outermost region is inconvenient
to use because of its implicit form. Furthermore Twd2/4pv2 varies by more than four orders
of magnitude over the full range of Patel's calibration. They therefore suggested the use of
two alternative forms of the calibration, the first
*The 4 in the denominator of the logarithms was originally adopted by Preston, since if the height of the centre-line of the pitot tube is taken as the relevant variable in the wall similarity parameter (y = d/2),
~ d2w ~ I ~ 2
4pv 2
26 K.G. Winter
Ap Apd 2 - - v s . - -
T w Pv 2
is tabulated in Table I, and the second is in effect a Clauser plot for a Preston tube.
the calibration is expressed as
If
j
then
w ~p ~p 1 Ud cf = = ÷ F
~ u 2 ½0u 2 ~ou 2 2 ~ •
u This expression is shown as a chart in Fig. 17 where U -~
variables with cf as a parameter.
I d__~__) ~ dU = and- are used as the ½PU2 "o
o8[
0.~ ~p
V
0.. ~
0.~
0.~
i ~ K i l l I l ~ n R i 3 ~ i s n ~ i K i l
0.2
0.,
2.0 2.5 3.0 3.5 4.0 4.5 5.0
Fig. 17. Preston tube calibration chart: Head and Vasanta Ram
Bradshaw and Unsworth (1973) give a further alternative, but implicit, expression,
du I duT~ AP=Tw 96 + 60 log s--~+ 23.7 log5-~-j;
du --< IOOO. valid over the range 50 < u
(5-3)
Patel also investigated the limiting pressure gradients, both favourable and adverse for which
his calibration might be expected to apply. His proposed limits are based on values of the
parameter
Skin friction in turbulent boundary layers 27
Table . ww as a function of Apd2/pu2 (Head and Vasanta Ram).
du dA Maximum e r r o r 3% 0 > ~ > - 0 . 0 0 5 ~ < 200 d-~x < O; p
du dA > -0.007 ~ ~ 200 d~x < O. Maximum error 6% O > Ap v
As Patel points out these limitations are a rough guide only.
(1962) law o f t h e w a l l i n p r e s s u r e g r a d i e n t s i s a c c e p t e d ,
u _ f YUT, &
u ~ T T
He notes that if Townsend's
v ~T where & - --
pu~ ~y
and that in general AT, which is equal to &p only at the wall, might therefore be expected to
be the controlling parameter, l~s use a prior~ in a boundary layer investigation is, of
course, not possible. The limit on du /~ is perhaps also rather sweeping since the limiting T
value might be expected to depend on the Reynolds number of the boundary layer, tending t@
increase as a boundary-layer Reynolds number increases. The limitation for favourable gradi-
ents of dAp/dX < 0 was introduced to ensure that a boundary layer, if it is in a condition
where it may be subject to relaminarization, should only he approaching that state. The phy-
sical features of the flow which lead to the limitations have been discussed by Patel and
Head (1968) and are illustrated in Figs. 18 and 19. Fig. 18 presents profiles from Newman's
(1951) experiments (from the Stanford tabulations) and shows how the velocity profiles depart
from the law of the wall for zero pressure gradient as the adverse pressure gradient increase~
This departure is predicted by Townsend's velocity profile for a linear shear-stress gradient
away from the wall. On the other hand the recent reassessment by Galbraith and Head (1975)
of eddy - viscosity profiles implies that the mixing length increases in strong adverse pres-
sure gradients and that this delays the departure of velocity profiles from the conventional
law of the wall.
The profiles as plotted in Fig. 18 differ from those in Fig. 4 of Patel and Head and add
weight to the conclusion of Galbraith and Head. The difference is presumably due to a differ-
ent derivation of the skin-friction coefficient in the Stanford tabulations from that used by
Patel and Head. It is interesting to note, on the basis of Fig. 18, that the logarithmic re-
gion is entirely absent from the curve for the largest pressure gradient but the profile ap-
pears to follow a blending curve of the form suggested by van Driest (1956) without allowance
for the normal shear-stress gradient; hence Patel's calibration, for a sufficiently small
Preston tube might be applicable.
Skin friction in turbulent boundary layers 29
I00 --
90--
80 -
70 -
u 6 0 -
Uz
5 0 - -
4 0 - -
3 0 - -
20
o
o o o
o
o
o
o
o
o :r O
+ o +
o +
A~ o 2 . 0.0075 o + x 0 . 0 2 1 4 + + 0 . 0 9 0 5 o . ~,
o 0.2980 : . : /
o / 0 +
0 4- ,
0 4- ÷
o ..r.. xo. x o.~.-'- .~X + "4~'x
i O ÷
0 2 3 4 ~ 6 s l O 2 3 ~ , 6 102 YUr
v
I IIII lJ I 2 3 4 56 8103 2
Fig. 18. Velocity profiles in adverse pressure gradient: Newman.
Figure 19 shows a series of profiles in a strongly favourable pressure gradient. As the flow
progresses into the favourable gradine (x 0 - x B increasing) the velocity profiles, whilst re-
maining turbulent, first fall below the line of the law of the wall and then lie above the
line as the flow starts to revert to a laminar state. The departure from the law of the wall
= is evident at x 0 - x B = -2 in where Ap -0.01, in accord with Patel's limiting value for 5p
of -0.005.
Brown and Joubert (1967) also investigated the limitations on the use of Preston tubes in ad-
verse pressure gradients, basing the analysis of their experimental data on the dimensional
analysis of Perry et a~. (1966). This analysis indicates that the first departure from the
law of the wall due to a pressure gradient will be in the form of a half-power region, that is
U T
where a = I dp pdx'
and that the half-power region will start at a constant value of ay/u 2 = 1.41. They therefore
analysed their results for a range of Preston tubes in terms of ud/u 2 and obtained the follow- T
ing table of errors:
Preston tube error % I 2 3 ~5 7
ad - - 1.35 1.74 2.06 2.55 2.98 u 2 T
30 K.O. Winter
30
20
I0
22in. en'try 124in.entry A 0
cf U cf - A (f~/s) (ft/s) 8G.18 0 .00492 87.32 0 . 0 0 4 6 7 0.0178
77.93 0 .00537 78 .65 0.00521 0.0212
68.91 0 .00578 68.32 0 .00578 0 .0244
0 U Ur
59.65 0.00690 59.72 0.00597 0.0288
53.98 0 .00527 54.24 0.00516 0.0275
50.02 0 . 0 0 4 4 9 52.97 0.00:387 0 .0204
0 4 8 . 6 5 0 . 0 0 3 8 6 52.41 0.00318 0.0103
0 47.74 0.00351 51.87 0 .003055 0 .0038
0 2 I0 I0 ~ 10 3 10 4
yur v
Fig. 19. Velocity profiles in accelerating flow: Patel and Head.
which are broadly in agreement with their proposed limit of the log law as ~y/u 2 = 1.41. They
also suggest that, since the log law region will vanish when the outer and inner limits become
the same, that is when y = 30v/u = 1.41 u~/u or av/u 3 = A 0.05, Preston tubes should not T T p
be used for stronger pressure gradients than this. Since, however, Patel's calibration extends
to the viscous sub-layer this restriction may not be necessary for very small tubes. Criteria
based on the results of Brown and Joubert are shown in Fig. 20 in terms of du /v vs. A In T
terms of this mapping Patel's criteria for 3% accuracy are in good agreement at their inter-
section but are conservative as general criteria, but his criteria for 6% accuracy overestimate
the permissible limits. Ozarapoglu measured velocity profiles in strong adverse pressure grad-
ients and determined the outer limit of the logarithmic region. His critical values are also
shown on Fig. 20 and are roughly in agreement with Patel's 6% criterion for low values of A P
but indicate that Preston tubes may be used up to larger values of d than accepted previously. P
This is confirmed by recent results of Chu and Young (1975).
Patel's calibration expressions have an upper limit of y* = 5.3 which corresponds to a value
of duT/9 of approximately iOOO. This should be regarded as an upper limit for which the cali-
bration was obtained in Patel's experiments and not an upper limit for the application of the
technique, which will be the outer limit of the log-law region. This limit derived from the
experiments of Winter and Gaudet in zero pressure gradient is shown as a function of the Rey-
nolds number based on momentum thickness in Fig. 21.
Skin friction in turbulent boundary layers 31
-0"12
r-- -O' lO l
~ . . . . ~ Oza r s poglu
I - 0 , 0 8 I
,B-,,
Ap I
I - 0 " 0 6 ~., •
Pig. 20.
0 SO I00 150 l u u uT._.dd
I
Criteria for limits of Preston tube.
~v~
IO 40 i
6
+I
2
i0 3 6 6
4
2
SO 2 #
6
4
1 s
• , ,,p**| , * , i,,,i| * * , ,,,,,|
1(3 Z 4 6 8103 2 4 6 I1104 2 4 6 8 102 Rel) 105
Pig. 21.
' ~ ' ~'~'lO 6
Outer limit of logarithmic region of flat plate.
Several ways have been proposed for transforming the pressure and friction parameters used in
the callbration of a Preston tube to enable the calibration to be applied to compressible flow,
The first of these was probably that due to Fenter and Stalmaeh (1957); this was derived from
their compressible form of the law of the wall already referred to, that is
U ~ sin_ 1 ~ = A log , UT w w
+ B. (5-4)
32 K.G. Winter
Fenter and Stalmach assume that (5-4) may be applied in place of the incompressible form at
y = d/2* where d is the Preston tube diameter, and recast (5-4) as
or
dU i ( i ) r w 2 w o T sin-i o~ u ..... A log + B
~w
Ow ~e I Red sin-I <°~ U I p e ~w o ~
- - Re d log ~ _~e Re d + B ~w ~ ~w
~ee --~w Red(cf) ~ A log --~w Red(of) ~ +7-7B A log 23/2 I . (5-5)
The Preston tube calibration is thus expressed as
~w Redc~ vs. P e ~w o T
u where ~ is obtained from the usual expression for pitot tubes in compressible flow, together
with the assumption of constant total temperature through the boundary layer.
Sinalla (1965) obtained a simpler expression by evaluating the Preston tube parameters at the
"intermediate temperature" using Monoghan's (1955) expression for adiabatic flow
T = I + 0.35r(y - I)M 2 o
m e
From the data of Fenter and Stalmach he obtained the empirical equation
w /p2d2u2\O 873 Omd2"r 0.0290 m (5-6)
IJm m
in which the p r e s s u r e r i s e Ap used in i n c o m p r e s s i b l e f low i s r ep i aced by tOmU2 where u i s the
v e l o c i t y i n d i c a t e d by the P res ton tube.
Hopkins and Keener (1966) also used an intermediate temperature hypothesis but instead of re-
placing Ap by ~OmU2 they used Ap = ~ou 2 = (y/2)PeM2. Their calibration expression is
O d2T m w ~2 m
- 0.0228 OOmd2U2~ 0.883
(5-7)
Allen (1973) investigated the accuracy of these various calibrations for adiabatic flow using
a wide range of sizes of Preston tubes over the Math number range 2 to 4.6, comparing the re-
suits from the Preston tubes with skin friction measured by a balance.** He also made use of
*They also investigated the transformation to the calibration for incompressible flow that would be obtained if the average p~tot pressure over the opening of the Preston tube were taken as the relevant quantity. They conclude in accordance with Hsu's (1955) result for incompres- sible flow that the difference from simply taking values at d/2 could be neglected.
**Alien has recently discovered that the balance, against which he calibrates his Preston tubes, read erroneously. His revised equation (5-9) is log F 2 = 0.01239 (log Pl) 2 + O.7814 log F I - 0.4723.
Skin friction in turbulent boundary layers 33
the data of Fenter and Stalmach (M = 1.7 to 3.7), Hopkins and Keener (M = 2.4 to 3.4) for
adiabatic flows, and of Keener and Hopkins 41969) (M = 7) for non-adiabatlc flows. In addi-
tion to the calibration laws listed above he also devised a calibration for compressible flow
based on Patel's expression for incompressible flow for his highest Reynolds number range, by
using the intermediate temperature hypothesis with Ap = ½PmU2 to obtain
m - 2 1 . 9 5 l o g ÷ 4 . 1 ( 5 - 8 ) 2 2
The general conclusion of this experiment was that of these various calibrations those of
Fenter and Stalmach, and of Pateli which are based on logarithmic profiles gave the best re-
suits over the widest range of parameters. Since both of these calibrations are implicit,
Allen proposed an interpolation formula which gave a slightly improved rms deviation from the
d a t a . This formula is
log F 2 = 0.01659 (log FI) 2 + 0.7665 log F I - 0.4681 (5-9)
Pm d u Pm }ae u - Re d where F 1 ~m Pe Vm
F2 = 42Pm~ w)~ d = 0(~ / m , e ,
II e
The Sommer and Short (1955) value for the intermediate temperature is used, that is
= i + 0.035M 2 + 0.45 T e
and Sutherland's formula is used for the dependence of Viscosity on temperature.
The last word written so far on Preston tube calibrations in compressible flow appears to be
that of Bradshaw and Unsworth (1973, 1974). They express justifiable doubts about placing re-
liance either on the assumption that the pressure read by a Preston tube may be taken as that
registered by a small pitot tube placed at the position of its centre, or on the concept of
calculating fluid properties at an intermediate temperature. The propose an empirical cali-
bration law based entirely on dimensional grounds as
for compressible adiabatic flow. In 45-10) fi represents a calibration for incompressible
flow and fc a compressibility correction 4~ is the speed of sound at wall conditions). Equa-
tion 45-10) is, of course, an implicit form of the calibration but is used because the func-
tional relationships are clear in this form so that subsequent ~endment on the basis of new
e~eri~ntal data can easily be ~de. As noted earlier Bradshaw and Unsworth devised a fit to
Patel's results over the range 50 < du /~ < I~ as T
2
= 96 + 60 log ~ + 23.7 log ~ (5-3) T w
and expressed the compressibility correction based on Allen's results as
34 K.G. Winter
f C
u T W
where M = Y a
w
= IO~M~ - 2 (5-11)
They tentatively claim an accuracy of about 2% at low speed and small du ~v decreasing to TW/ W
about 10% for du w/~ w up to i000 and M up to 0.I. It is the author's view that the table of
Head and Vasanta Ram provides the most satisfactory calibration for incompressible flow; in
particular it extends to lower Reynolds numbers than the formula of Bradshaw and Unsworth.
For compressible flow the use of the Fenter-Stalmach functions appear to result in somewhat
less scatter than the Bradshaw-Unsworth formula, though it is admitted that only a small sam-
ple of data has been examined.
One aspect of the calibration of Preston tubes which has not been adequately explained is its
sensitivity to the Reynolds number of the static pressure hole to which the Preston-tube pres-
sure is referred.
Equation (5-10) is for adiabatic flow. For flow with heat transfer a further parameter
8q = q/0wCpTwU w where q is the heat flow per unit area, will enter into the equation. There
is a need for further systematic experiments to determine the effect of heat transfer. David-
son (1961) attempted this for M = 5 but found inconsistent results; Holmes and Luxton (1967)
in an experiment at low speed and Yanta et a~. (1969) at M = 4.8 found that their results were
best correlated by use of the intermediate temperature concept.
The measurements at supersonic speeds of Yanta et al. for favourable pressure gradients and of
Naleid (1958) and Hill (1963) for adverse pressure gradients give satisfactory results but
there do not appear to be any general criteria for the use of Preston tubes in pressure
gradients in compressible flow.
The Preston tube has also been used in flows with transpiration. Stevenson (1964) developed
a calibration equation using a power law approximation to the velocity profile (4-12) to find
a mean dynamic pressure over the opening of the tube, as had been done by Hsu for impermeable
surfaces. If the law of the wall is taken locally as
flu) I T + - I = C , v w
then an approximate expression for a Preston tube calibration can be derived as
(Apd2~ ~ = C 2 v (du~l l + 2 n ~ + cIdUTl l+n (5-12)
where C and n will depend upon the Reynolds number of the boundary layer but should have the
same values as for an impermeable wall. Typical values are C = 8.4, n = 1/7. Stevenson found
fair agreement between the skin friction deduced from Preston tube readings by means of (5-12)
and that from the law of the wall and from momentum traverses for the boundary layer growing
on a porous cylinder.
Skin friction in turbulent boundary layers 35
Simpson and Whitten (1968) suggest an alternative form of a Preston tube calibration for a
transpired boundary layer in which in addition to the usual shearing-stress and pressure-dif-
ference parameters they introduce a third parameter Vw(0w/Ap)l to take account of the tran-
spiration, but they do not give an explicit result.
For three-dimensional flows the Preston tube has been used, for example by East and Hoxey,
see Fig. 16. They set the tubes in the flow direction obtained from surveys of the boundary
layer by a yawmeter - in fact in the flow direction at a height ~ times the diameter of the
Preston tube. Their results, evaluated with Patel's calibration, appear to be satisfactory.
In two experiments on the flow past obstacles on a flat plate, one a circular cylinder and
the other an inclined fence, Prahlad (1968) showed that in the powerful favourable pressure
gradients which may develop in a three-dimensional flow the Preston tube should only be used
with caution. He used a vectorial pressure grad p and resolved this in the wall shear-stress
direction to show that at the points where the Preston tube failed, negative values of A as P
high as 0.04 were 0tbained, considerably in excess of Patel's criterion.
Prahlad (1972) also studied the yaw characteristics of Preston tubes, of surface tubes with
openings cut off obliquely at 4~ and of double tubes in the form of a 90 ° Conrad probe and
gives detailed charts of their characteristics in low-speed flow. Rajaratnamand Muralidhar
(1968) present calibration curves for a three-tube yawmeter used as a Preston tube.
Other variants of the Preston tube have been suggested, notably that of Rao et al. (1970) who
used two tubes of different diameter, though the tubes were not mounted in contact with the
wall. They argued that if it is assumed, as has been done so often, that the pitot tubes read
a mean dynamic pressure over their faces of area A], A 2 the difference between the reading of
two tubes in a two-dimensional flow is
Ap = 2 ~i (log y)2dA] - ~2 (log y)2dA2
AI A2
+ 0u~BA + 0u~A 2 log i log y dA 1 I At - ~2 log y dA 2 .
Thus if two tubes are chosen so that
(5-13)
i f i f A~I log y dA I ffi A-~ log y dA 2 ,
A1 A 2
a calibration may be obtained which is dependent of B and also of a static pressure measure-
ment. Rao et aZ. give an example for the flow on a flat plate in which the skin friction ob-
tained from the dual-pitot-tube calibration was within 5% of that obtained from either of the
tubes separately. The technique might be applied to rough wall flows in which B will be un-
known a priori.
An improvement to the dual pitot tube of Rao et al. has been explored by Gupta (1975) in which
two probes of different diameters rest on the wall slde-by-side and themouth of the smaller
one is chamfered at 45 ° away from the larger one, as shown in Fig. 22. In a limited investi-
gation a t low speeds Oupta showed t h a t dev i ce ( a ) , w l t k a d i ame t e r r a t i o o f 0 .78 , gave a p r e s -
su re d i f f e r e n c e of 71% of t h a t f o r t he l a r g e r tube u s e d a s a P r e s t o n t ube , and d ev i ce (b ) ,
36 K.G. Winter
with a diameter ratio of 0.64, gave 80%. Device (c) is suggested for use in three-dimensional
flows in which the pressure difference between the outer tubes is first used to align the de-
vice with the flow and, when the probe is aligned with the flow, the pressure difference be-
tween the centre tube and the outer tubes may be used to determine the skin friction. Bertelrud
(1976) has explored the use of a combined pitot-statie tube as a Preston tube.
0.7 mm
A PPreston ,, "
O.9mm Ca)
0 . 4 5 r a m 45"
-- ~ 0.80 - /, A PPrest°n I' "~
0.7mm
0.7mm
r
0.9mm
(b)
Fig. 22. Modified Preston tubes: Gupta.
(c)
6. OBSTACLES IN TWO-DIMENSIONAL FLOW
Because of the wall-similarity of the flow in a turbulent boundary layer the pressures around
almost any obstacle can be used to derive skin friction. Some of'the devices which have been
used are illustrated in Fig. 23 which shows their sensitivity Ap/~ where £ is a representative
height or diameter. The sensitivity of a Preston tube is also shown for comparison, and it
can be seen that the sensitivities are all of the same order as that of the Preston tube and
cover a range of about 2:1, at a given Reynolds number. The fence or the square ridge have
higher sensitivities than a Preston tube since the pressure difference they create is enhanced
by utilizing the suction at the downstream face as well as the pressure rise at the upstream
face. The devices are considered below in more detail.
6.1. Fence
The sub-layer fence was probably first suggested by Konstantinov (1955) and has been used by
Head and Rechenhert (1962), and by Vagt and Fernholz (1973). As well as giving a relatively
large pressure difference it has the advantage that it may be made sufficiently small to re-
main within the viscous sub-layer and hence to be used with confidence in flows with strong
pressure gradients. Because of its fore-and-aft symmetry it can also be used quantitatively
Skin friction in turbulent boundary layers 37
in flows which are separating. Its small size has the disadvantage that its geometry is dif-
ficult to define, and the recommended technique is to calibrate each particular fence against
a Preston tube in a well-behaved flow, thus using a Preston-tube calibration as the primary
standard giving the surface shearing stress. However, the calibration curve for the fence for
hu /~ up to ii shown in Fig. 23, and taken from Rechenberg (1962), can be joined by a specula- T
tlve fairing between hu /9 = ii and I00 to the line representing the results of Good and T
Joubert (1968) who investigated the characteristics of large fences. The calibration for the
square ridge in Fig. 23 is taken from the excrescence drag measurements of Gaudet and Winter
(1973). The fact that this is quite close to the results of Good and Joubert suggests that
the geometry of a large fence is not too critical, but the sensitivity of the drag of ridges
shown by Gaudet and Winter to the rounding of the upper corners indicates that the radius of
the corners should be made as small as possible. The sub-layer fence is also attractive as
a device for compressible flows since its small height will minimize the effect of variations
of density away from the wall.
4 x I0 z
2
8
6
4
¥ 2
tO 8 6
4
Z
I
Fence , i ~ '~
/~-Submerged . - - - step
i i | i i i | i
2 4 6 810 2 u~t 4 6 8102
Y __
$ M 'e
5...i"" " ~ l x k with ~ / cut-cut
Static -hole / pair / /
| d J i |
2 ,4 6 8 tO 3
Fig. 23. Sensitivities of various obstacles as skin-friction meters.
6.2 Razor blade
In order to explore the flow in a viscous sub-layer of very small thickness Stanton et al.
(1920) used pitot tubes with a rectangular opening of width much greater than height with the
test wall forming the inner surface of the tube. They determined the effective height of the
tubes which would make their readings consistent with a linear velocity profile at the wall
and thus produced a skin-friction meter which became known as a Stanton tube. Other workers
subsequently used larger tubes and showed that it was not necessary that the tube shouldbe
sufficiently small to be within the sub-layer for it to act as a skin-friction meter. A re-
view of the investigations up to 1954 covering both experimental and theoretical work was
given by Trilling and H~kkinen (1955). Hool (1955) suggested that a Stanton-type tube could
readily be formed by attaching a portion of a razor blade over a static-pressure hole and this
simple device has since become widely used.
A detailed saudy of the use of razor-blades at low speeds was made by East (1966) with a view
to their use in three-dimensional flows. He calibrated against a Preston tube with skin fric-
tion determined from Patel's calibration. Th~ ,,~e nf a pnTe~nn of a razor blade has obvious
38 K.G. Winter
attractions since the skin friction can readily be measured in any experiment in which static
pressure holes are provided to measure the pressure distribution.* East simplified the tech-
nique by forming his pressure holes in magnets so that the portion of the razor blade could
simply be placed over the hole, obviating the need for adhesive, and enabling the height of
the opening formed by the blade to be determined from previous measurements of its thickness.
The calibration obtained by East is shown in Fig. 24 and has the equation
y* = -0.23 + O.61x* + O.O165~ .2 (6-1)
ph2T where x* = log ph2Ap and y* = log w
u2 u2
4.0
3 . 0
f "" b l a d e
/ O.O0~in
* 2 .0 - f z ~ °
*-"0.23*0.~18~ % +0.0165x*
~, 1 I I 2.0 3.0 4)~phZ ~ 5.0 6.0
x•= L°q'° k - ~ - )
-Fig. 24. Calibration of Standard razor-blade surface-pitot tubes: East.
Equation (6-1) is for the "standard" position of the razor blades that is with the edge of the
razor blade over the leading edge of the static hole. The change in the pressure rise for
various fore-and-aft positions is shown in Fig. 25. An additional error can arise from the
dependence of the static pressure on the size of hole used, and East suggested as a standard
that d/h = 6, and that the breadth of the blade should exceed 30 times the height to avoid end
effects. Smith et al. (1962) had assumed in their calibrations at supersonic speeds that re-
sults would be unaffected by grinding away the upper portion of the blade to minimize the in-
teractions which might occur between blades in an array. East showed that the removal of the
1.2
, o " % 4 x .. x
0.8 - Pos i t ion ~ _ x + ,9 h A B ' T ~ k . z ~ ° x
, ~_ 0 .60 .O020 inx +
O . O 0 ~ O i n o • = ~ " ~ 0.4 -0"0153in ~' •
I I I I ~r -3 -2 -t 0 7
0.2
0
- d ( a p p r o x )
I I I I I I I 2 3 4 5 6
Ax/h
Fig. 25. The effect on Ap of varying the razor-blade longitudinal posi- tion relative to the static hole; East.
*The use of "plug-in" Preston tubes by Peake, et al. (1971) should however be noted.
Skin friction in turbulent boundary layers 39
upper portion of the blade in fact altered the calibration. Inspection of a typical blade of
thickness 0.2 uln indicates that this is to be expected since the edge is composed of two cham-
fers, the first being of total angle about 20 ° and length of 0.25 mm, and the second of angle
about 12 ° and length 0.85 mm, so that the flow over a blade is not likely to attach before the
first shoulder and perhaps not before the second shoulder. Gaudet (unpublished) reanalysed
East's results and other unpublished results and found an empirical expression for the effec-
tive height of a blade
hef f = h + Ah,
h - h Ah t
where h h ' (6-2) t
h being the height to the edge of the blade, and h t the total height. His expression for
Fig. 32. Distribution of oil-film thickness under flow approaching separation: Tanner and Blows.
The technique needs a special environment to make the interferometric measurements possible
but clearly could be valuable in specialized cases, and has the great advantage over many
methods that no knowledge of the properties of the test medium is required. It can be extended
to three-dimensional flows as shown in a subsequent paper by Tanner and Kulkarni (1976). As
Tanner and Blows point out care must be taken in flows near separation that the accumulation
of oil does not modify the flow. There may, however, be limitations on the combinations of
shear stress and film thicknesses for which the film will remain stable. Figure 34 shows a
Skin friction in turbulent boundary layers 5]
photograph taken by the present author of a very thick oil film on a flat plate at a speed of
35 m/s. The photograph shows the flow in the region of the transition front near the centre
of the plate where the wavelength of the pattern in the oil is clearly decreasing through the
transition region. In the lower right of the photograph the turbulent wedge spreading from
the intersection of the plate and the sidewall is indicated by the shorter wavelength of the
disturbances compared with those in laminar flow.
7 0 - -
8 0 - -
50 - -
4 0 - -
E
g -~ 3 0 - -
2O
I0
o
Fig. 33.
• 5 min + I O m i n o 2 0 min z~ 4 0 m i n {~ 6 0 m i n
_ T - 5 m i n - - ~ . ~ ' '
20 40 60 x mrn
IT-oo i
J
6u--J/
/
80
Limiting form of oil-film thickness distribution: Blows.
I ~oo
Tanner and
Surface of + polished wood /
Surface 1 painted black
!
3S role
Camera Flat plate
Tunne window
Laminar
TurNlent wedge from ~ ~ -
junction of plate leading edge end tunnel sidewall
I I I I m from leading edge 1.7 1.6 I.S 1.4
Trop-~ition region us indicated by surface pitot tube
F i g . 34 . S u r f a c e w a v e s i n t h i c k f i l m o f o i l .
52 K.G. Winter
These difficulties can be avoided by the use of a simplified version of the technique as de-
scribed by Tanner (1976), which may be applied to polished metal surfaces. In this version
only the leading edge of an oil film is used where the thickness varies linearly with down-
stream distance so that
~x = -- (9-4)
w ut
Hence by finding the gradient with distance of the film thickness the shearing stress can be
obtained. This is done by using two laser beams with a known separation, x, one to define
the leading edge of the film and the other to record the evolution of the thickness by count-
ing fringes as indicated by zero crossings of the output of a photocell on which the reflected
light is focused.
9.2. Liquid crystals
Another suggestion for a surface-coating technique is that of Klein and Margozzi (1969, 1970)
who have made an exploratory investigation of the use of liquid crystals. Liquid crystals ap-
pear to be viscous liquids and yet show many of the properties of solid crystals. One of
these properties is selective light scattering so that when illuminated with unpolarized white
light incident at a given angle liquid crystals reflect strongly only one light wave length
at each viewing angle. Small changes in conditions can cause a shift in the wavelength of the
reflected light in a reversible way. Klein and Margozzi showed that a mixture of liquid cry-
stals could be produced, the properties of which were primarily dependent upon the shearing
stress imposed on the mixture, although the mixture also exhibited sensitivity to temperature
and to the angle between the specimen and the direction of illumination and scattering.
Figure 35 shows a calibration obtained in the shear flow between a fixed annulus and a rotat-
ing annulus. The calibration shows a reversal of colour change for ~w greater than about 300
N/m 2 (3.06 x 10 -2 g/mm2). In practice this reversal would not often be of significance since
shearing stresses would rarely exceed the critical value. (For example in a wind tunnel at a
Mach number of unity and a stagnation pressure of one atmosphere a skin-friction coefficient
of 0.002 gives a shearing stress of about 75 N/m2.) However, it was found on making experi-
ments in a pipe that a film of sufficient thickness to exhibit the light-scattering properties
Fig. 35.
E
% 4
2 I 550
? I
l
o
IHIII IIIIII
560 570 580 590 600
Woveleng ' l 'h , nm
Wavelength of light scattered from liquid crystals under shear stress: Klein and Margozzi.
O \ \
o o~ \
\ \ \
\
Skin friction in turbulent boundary layers 53
produced marked ridges normal to the flow, and that the scattered light signal fluctuated con-
siderably making it extremely difficult to measure the wavelength. A great deal of further
development is therefore required before the technique can be considered for routine applica-
tion.
IO. CONCLUDING REMARKS
Of all the techniques reviewed it is apparent that none can be considered an absolute and re-
liable standard. The obvious technique of directly measuring the surface shearing stress by
a force balance is beset by many pitfalls which may be overcome in particular cases but the
possibility of specifying a priori the requirements to be met in general seems remote. An
analysis is given of the errors arising from various causes and this may provide some guidance
in design. The most reliable device at present seems to be the Preston tube because of its
simple geometry and because it has been investigated the most thoroughly. However, there is
still room for further work on the effects of pressure gradient, of flow unsteadiness and of
heat transfer and in three-dimensional flow. Potentially, sub-layer fences hold most promise
for devices of the pressure-measuring type if a design can be found with a geometry easy to
manufacture repeatably, which is very difficult because of their smell size. There is an
opportunity for the exercise of some ingenuity in devising instruments for use in three-dimen-
sional flows. In the long run the heated-element instrument is likely to prove the most re-
liable. For general application the discovery of a simple shear-sensitive surface-coating
agent would be most welcome.
REFERENCES
Allen, J.M. (1968) Use of Baronti-Libby transformations and Preston tube calibrations to determine skin friction from turbulent velocity profiles. NASA TN D-4853.
Allen, J.M. (1973) Evaluations of compressible-flow Preston tube calibrations. NASA TN D-7190.
Allen, J.M. (1976) Systematic study of error sources in skin-friction balance measurements. NASA TN D-8291.
Baronti, P. O. and Libby, P. A. (1966) Velocity profiles in turbulent compressible boundary layers. AIAA J. ~, 193-202.
Bellhouse, B. T. and Schultz, D. L. (1965) The measurement of skin friction in supersonic flow by means of heated thin film gauges. ARC R & M 3490.
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