D-A36921 THEEFFECT OF TRUIIN NO TICES ON THE PRODUCTION OF I/J LFT ONAN AIRFOI U..U) AIR FORCE INST OF TECH WRGHT-PATTERSON AFB OH SCHOOL OF ENGI.. K W TUPPER UNCLA SFIED DEC 83 AF OT/GAE/AA/83D-24 F/G 20/4 NL l/immmmmmmmmmE EIIIIIEEEEIIIIE IIIIIIIIEIIIIIE mIIIIIIEEIIIIIE IIIIIIIIIIIII
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D-A36921 THEEFFECT OF TRUIIN NO TICES ON THE PRODUCTION OF I/JLFT ONAN AIRFOI U..U) A IR FORCE INST OF TECH
WRGHT-PATTERSON AFB OH SCHOOL OF ENGI.. K W TUPPERUNCLA SFIED DEC 83 AF OT/GAE/AA/83D-24 F/G 20/4 NL
II. Solution Development ....... ............... 3
Solution Overview ......................... 3Equations for Flow About a Cylinder . . . .. 5Joukowski Transformation ........ ...... 10Determination of Strength of Vortices 11Velocities Induced at Discrete Vorticesand on the Cylinder ............. .. 14Circulation About the Airfoil.... . ... 17Velocity in the Airfoil Frame .. . .. . 18Pressure, Lift, Vorticity Distributionon the Airfoil. ..... ............ ... 19Numerical Solution Process. ........... . 21
In the last section it was shown that the present method
compares favorably to the results of others for the con-
stant-a, impulsively-started airfoil. In this section the
results of applying the method to the previously-unstudied
problem of constant- flow is presented. The presentation
of these results is broken into four parts in order to more
systematically explore and understand the interplay of pos-
sible effects. These four parts deal with the effects and
selection of starting conditions, the general effect of
on the build-up of C, the effect of thickness, and the ef-
fect of camber, respectively.
Selection of Standard Starting Conditions. As was
shown in the previous section, it takes some finite time for
an airfoil at angle of attack, a, suddenly placed into motion
to build to a steady-state value of lift. It is not sur-
prising, then, to find that the onset of constant a demon-
strates a different result depending on the time delay from
onset of impulsive motion to onset of constant a. The dif-
ferences, however, were found to be predictable, and thus
separable, as the following will show.
To determine the effect the initial a and a start time
have on the C vs. a curve for an airfoil at constant a, aiI
15% thick symmetric Joukowski airfoil with a = 0.01 ,
t = 0.02 was started at various initial a's and allowed
to build lift at that a for varying lengths of time t . For0 *
initial a = 0 , one can see from Fig. 13 that the time t
41
I ,,
.07.
It.06
.05
C .04' -
.03.
Steady State(-
.02 t = 0.2
= 1.0 (-o--.a.-).01Ja
0.1 0.2 0.3 0.4 0.50
ia
Fig. 13. Effect of Start Time to Begin a15% Joukowski Airfoil, ao=0?a*--0.01
40
42
.... . . .. - - -I: .
at which the . begins has no effect on the C vs. a curve.
For initial a = 5 , the C vs. a curves were dependent
upon the value of t at which the a was begun. However,
as can be seen in Fig. 14, the slope of the C vs. a
curve for a 0.01 does not depend upon the value of t
at which the a was begun. Choosing At = 0.1 , the 15%
thick symmetric Joukowski airfoil was allowed to build lift
to within 90% of steady-state C at various initial a's
before starting an a = 0.01 . As seen in Fig. 15, the
slope of the C vs. a curves for initial a's of 20, 4 and
60 are all approximately equal. The dashed lines on Fig. 15
depict the C vs. a curves that would be obtained by starting
the constant -a motion at full steady-state lift values
rather than the 90% steady-state lift values depicted by
the solid lines. Note that the initial value of CY obtained
for each of the starting angles of attack of 2 , 4 and 6
is the same amount above the steady-state C2 curve, and is
therefore independent of initial angle of attack. This ini-
tial value of C, will be called the 'jump' condition. Thus,
by the foregoing analysis, C 2 vs. a curve slope effects due
to the vortex wake will be assumed independent of initial a
and t.
The choice of At also shows some effect on the C vs.
a curve and was investigated. To do this, a 15% thick sym-
metric Joukowski airfoil at a = 0.01 was run at At
values of 0.2, 0.1, 0.02 and 0.004. Figure 16 depicts a
comparison of C£ vs. a curves for these four values of At
43
I , .
0.7-
0.6
CZ0.5,
0.4.
ta = 0. 2 (%C~start = 53% (,----)
ta = 1. 0(%C~start = 58%)
0.3.*t-= 3. 0(%C start = 64%)(--a)
5.1 5.2 5.'3 51.4 5.5
a0
Fig. 14. Effect of Start Time to Begina
15% Joukowski Airfoil, a0=50, a*=0.01
44
1.0
.9 /
.8 "
.7.[7
.77
.6-C,
/ = 20 (-.---,o-)
0•0 4 {
a°= 4° (-o----)
.3 a0 = 60 (6-)
.2
2 3 4 5 6 7 8 9
a 0
Fig. 15. Effect of Initial Angle of Attack to Begin a(C9 at 90% Steady-State Value)
45
* ,a.
.15-
. 10.
CL
.05 At
0.2 KQ-Q.0.1 (.-)
0.02 (0-)
.5 1.00
a
Fig. 16. Effect of At on C~ vs.* a Curve Slope
15% Joukowski Airfoil, a0 =00, a =0.01
46
Note that the slope of the curves, C2 , reduces as Ata*
is reduced, but the reduction is negligible below At = 0.02.
As a result of the above analysis concerning initial a
and t at which a is begun, all constant-a computer runs0 o *
assumed an initial a = 00 and t = 0 for a start-up. A
standard At = 0.02 was chosen as a reasonable value based
upon the information presented in Fig. 5 for impulsive-start
motion and Fig.16 for constant-a motion. While a At less
than 0.02 would produce more accurate results, the increased
computer time required at the smaller At values was judged
excessive for the slight increase in accuracy that could be
obtained.
General Effect of a on C * To determine the effect
an a has on the production of lift on an airfoil, a 15%
thick symmetric Joukowski airfoil was chosen as a represen-
tative airfoil shape. Using the selected values of initial
a = 00 and At = 0.02 , the airfoil was subjected to
various values of a ranging from 0.005 to 0.035. Figure 17
depicts the C vs. a curves obtained for small angles of
attack. Comparing with the C£ vs. a curve for the steady-
state case, one can see that as the value of a is increased,
the slope of the C2 vs. a curve, C. , is reduced. As thea
motion progresses to larger values of a, the slopes of the
curves increase slightly (see Fig. 18).
Effect of Airfoil Thickness on a Effect. The general
effect a has on the production of lift on an airfoil has been
shown. This effect was shown for a specific airfoil only.
47
-- _ _ _ _ _ _ _ __ _ _ 4.* .
0.3
0.2a=0.035
0.02
0.1100
0.005
1.0 2.0
0at
Fig. 17. Effects of a on CVS. a
15% Joukowski Airfoil
48
0.8
0.6
C7
0.4 7. ,
Steady0.2
ci = 0.035 (.---)
ci = 0.02 ('--)
0i ci
O
Fig. 18. C vs. a Slope Change as a Increases
at Constant a
49
APO2
To determine how airfoil thickness may influence this effect,
several symmetric Joukowski airfoils of varying thickness
were subjected to the same a conditions. Once again, values* 0
of At = 0.02 , initial a = 0 were used. Symmetric
Joukowski airfoils of 7%, 15% and 25.5% thickness, as well
as a flat plate airfoil, were subjected to a = 0.02. A
C vs. a curve can be plotted for each of these airfoils.
Plotting the average slopes of these curves, C£ , versus air-
foil thickness ratio t/c (where t is the maximum airfoil
thickness), one can determine the effect of airfoil thickness
on the C vs. a curve slope reduction due to a. Figure 19
depicts C vs. t/c for a = 0.02 . One can see that a2a
has a greater effect on lift curve slope reduction for thin
airfoils than for thick airfoils. This effect is consistent
with results previously presented. Note that in Fig. 9,
where a = 0 , for any given value of UAt/ c, the slope
of the CI/C ss curve is slightly greater for the 25.5% thick
symmetric Joukowski airfoil than for the flat plate. Al-
though the value of C2 /C£ is less for the airfoil withss
thickness, the rate at which C£/C2 is increasing is greater.ss
This implies that, under similar a conditions, C will in-
crease at a faster rate for a thick airfoil than for a thin
airfoil. Figure 19 confirms that conclusion.
Effect of Airfoil Camber on a Effect. In much the same
way as airfoil thickness effects are calculated, airfoil
camber effects can also be explored. Joukowski airfoils of
15% thickness at various camber ratios were subjected to an
50
('40
44
I1I
44
u,.
_4
U
-
-4
0-.,
S4
L)2
51U
51I
= 0.02 As before, initial a was 0 , At = 0.02
Plotting average C versus camber ratio (maximum camber/
chord), camber effects can be shown. Figure 20 depicts C2
vs. camber ratio for 15% thick Joukowski airfoils of various
camber ratios. One can see that a has a greater effect on
lift curve slope reduction for less cambered airfoils than
for highly cambered airfoils.
52
4 N4
0
44E $4
rz
-H .
Li 44I
co %.0
532
IV. Conclusion
It has been shown that as an airfoil pitches at a con-
stant a, the airfoil trailing vortex wake causes the slope
of the C vs. a curve to be less than the slope of the CI vs. a
curve for steady-state a. The greater the value of a for a
given airfoil, the greater the slope reduction of the C. vs. a
curve caused by the vortex wake. This effect becomes less
pronounced as airfoil thickness increases. Similarly, the
effect is also less pronounced as airfoil camber increases.
Using the results from the previous section, the follow-
ing predictions of constant-a effect may be made.
For a flat plate, the reduction in C. may be approx-a
imately calculated by
C( .15 1 + 2 2 (49)
a( + 0.00008)0.15
(See Fig. 21 for a comparison of this prediction with numer-
ical data.) This prediction may be approximately corrected
for thickness by adding a correction term derived from Fig.
19. Thus
t 0.75Ca ,t/c) = ( ) + C(*) (50)aL a (0
where t/c is the airfoil thickness to chord ratio and
C£ (cI) is the C. vs. a curve slope for a flat plate pre-
dicted by Eq. (49). A further approximate correction may
be made for camber by adding another correction term, de-
rived from Fig. 20. Thus
54
4 -,,__ __ . I 1 I I
CN 4-4
+ 0
U, 0-- 4
ji,
o to-. 440
00
+ I
o n
5 ISI U"
II
.I1(
1 0• -4
55
C (a* t/cmc/c) e 3 0(mc/c-0.09) + C (a *,t/c) (51)
where mc/c is the airfoil camber ratio and C (a/) is
the C vs. a curve slope for an airfoil with thickness pre-
dicted by Eq. (50).
The amount that CI increases immediately after an air-
foil begins constant-a motion is referred to as the 'jump'
condition. This value for a flat plate can be predicted by
AC(a) = 3.47a* (52)
where AC (a) is the 'jump' condition change. Thickness
effects on AC (a ) can be approximated by the equation
AC (a*,t/c) = [l + 2(t/c)]Ac2 (d*) (53)3
where t/c is the airfoil thickness ratio and AC2 (a ) is the
'jump' condition for a flat plate defined by Eq. (52). A
final approximate correction to the 'jump' condition can be
made by
AC( t/c, mc/c) AC(a, t/c) - 1.3(-mc) (54)
where mc/c is the airfoil camber ratio and AC2 (cL, t/c) is
as defined by Eq. (53).
56
'
V. Recommendation
The assumption that the trailing vortex wake of an air-
foil undergoing a constant rate of change of angle of attack
has a negligible effect on the production of lift on the air-
foil is not, in general, valid. Although the effect is not
jlarge (see Eq. (49)), it should be accounted for in the in-
vestigation of dynamic stall of airfoils. The methods de-
veloped by Docken (4) and Lawrence (5) could be modified to
include the techniques presented in this thesis to more ac-
curately predict the potential flow field about a pitching
airfoil at any instant in time. Incorporating the calcula-
tion of wake vortex effects outlined in this thesis into
Lawrence's work would significantly contribute to the solu-
tion of the dynamic stall problem for an airfoil undergoing
a constant rate of change of angle of attack.
57
I,
APPEND I X: Computer Prog ram
CC This program computes circulation, pressure differenceC distribution, vorticity distribution, coefficient ofC lift, and trailing vortex wake shape for a 2-D JouKowsKiC airfoil in an incompressible, inviscid free stream atC tangle of attack. The angle of attack may be a constantC value, or it may be changed at a constant rate for theC number of time steps desired. All output values areC computed ,assuming a trailing vortex wake made up of dis-C crete point vortices of constant strength, each of whichC influences the motion of all the other vortices and theC flow about the airfoil. For the constant rate-of-chanae"I of ingle-of-attack case, coefficient of lift can beC ;ound ,as a function of the rate of change of angle ofC attack. Variables in the program are defined as follows:C
C alf a - angle of attackC iallf-2 - initial angle of attackC alfdot - time rate of change of angle of attackC beta - the beta parameter of a JouKowsKi airfoilC cal fa - COSINE of alfaC chord - airfoil chord length
cl - coefficient of liftC ciss - steady-state coefficient of liftC countt - 'an integer counter used to determine whichC time steps will record output in certainC fiesC ct.heta - COSINE of theta1C d'a.. fa - increrental change of alfaC ~oeizo - array of incremental values of coefficientC of pressure along the airfoil chordC deld - distance on the x-axis in the cylinderC plane behind the cylinder where the firstC shed vortex is placedC delgam - array of values of strengths of gamma forC each individual vortex pairC dgoams - a sum of vortex strengthsC dsl - incremental distance along airfoil lowerC surfaceC dsu - incremental distance along airfoil upperC surfaceC dt - incremental unit of timeC Z - complex number; derivative of theC JoukowsKi transformajtionC dz&-c - rn,,agnitude of c ZD
'S e..t,.a - x .' u~e of trailing vortex position
58
-W 0
C eta2 - x-value of trailing vortex image positiong a mmnI - airfoil circulation
C i,.jK,1 - integer values used in iterationsC intgrl - value of the integration of velocityC differences between upper and lowerC airfoil surfacesC lastdt - last time increment at which alfa changesC mag2 - distance from a vortex to the center of theC cylinder in the cylinder-centered planeC malfa - maximum value of alfaC maxdt - first time increment at which alfa changesC from alfaOC max.t - last time stepC MU - complex number; distance between the originC in the displaced-cylinder plane and the
C center of the cylinderC pi - the constant 3.14159265C RHO - complex. r number; a position in the cylinder-C centered planeC RHOP - complex number; a position in the displaced-C cylinder planeC sl.fa -.SINE of alfaC ssgam - steady state value of circulationC stheta - SINE of thetaC sumsqr - the square of the distance of a trailingC vortex from the origin in the cylinder planeC sumu,suMv - sum of the velocities on the cylinder inC the x and y directions, respectively, in-C duced by trailing vortices and their imagesC t - integer counter for number of time stepsC theta - angle measured counterclockwise from theC '.-axis in the cylinder planeC totutotv - sum of velocities ,at 'a vortex location *.nC the x and y directions, respectively, in-r - duced by trailing vortices and their imagesC UIv - velocities at a vorte.' location in theC x and y directions, respectivelyC ua,va - velocities at a vortex location in theC x and y directions, respectively, in theC airfoil planeC usurf,vsurf - velocities on the cylinder in the x and yC directions, respectivelyC utheta - velocity tangent to the cylinderC vordis - vorticityC vsqare - velocity on upper surface of airfoilC squared minus velocity on lower surfaceC of airfoil squaredC x'y - position on airfoilC .,vort.yvort - position of a trailing vortex
Z- compl.ex number' a position on the airfoilC zet: .- y v'lue of trailing vorte. positionC :qt,.2 - y v' ,-e of trailing vortex image position
59
1 (t'mmL. ".. .. -] . , k. i )- -l rf :-
C zt - distance 'alona x-axis fror, origin toC cylinder in displaced-cylinder plone
C Z - complex number; 'a position in -the waKe inC the airfoil planeCC FILES:*
-~ C
C INPUT - unformatted list of input variablesC OUTPUT - list of C1 vs. C1 steady-state for angle ofC attack and time stepC PRES' - pressure distribution at specified time
C VORT - vorticity distribution at specified time.C WAKE - position of trailing vortices at specified time
WRITE( 18,72) zt,betard'af', deid ,SCrg'IrIT,afdot,dtWRTTE(19,72) ztpbetavd'aliardeld,ssgaiT~,afdotrdtbet a be tia* p i/180d al1fia =dal a *p i/i80QifQO=alfia*p i/iSOMU=CMPLX(zt-COS(beta),SIN(beta))
CC Cialculiate coordinates of points on the airfoil* (x,y)I C DO 15 i=-180,180
thetea=i*pi/180RHOP=CMPLX(COS(theta-beta) ,SIN(theta-beta) )+MUZ=RHOP+t**2/RHO'x( i )REAL(Zy( i)=AIMAG(Z)
CC Calcu'a te pressure dist-ibution 'and unsteady 'aerodynam.c
0 force on the 'airfoil.C
DO 32 i=176,0,-4theta =i*oi/18Octheta=COS(theta)sthetaSIN(theta)dsu=S0RT((x(i)-x(i+4))**2+(y(i)-y(i+4))**2)*2/chorddsl=SQRT( (x(-l*i)-x(-1*(i+4)))**2+(Y(-l*j)-Y(-1*
+ (i+4)))**2)*2/chord
+ +intgrl(tpi+4)vsq'areutheta( i )**2-utheta(-l*i )**2IF(ti.EG.1) THEN.
1. Kramer, von M. "Die Zunahme des Maximalauftriebes vonTragflugeln bei plotzlicher AnstellwinkelvergroBerung(Boeneffekt)," Zeitschrift fur Flugtechnik und Motor-luftschiffahrt, 7: 185-189 (April 1932).
2. Deekens, A.C. and W.R. Kuebler, Jr. "A Smoke TunnelInvestigation of Dynamic Separation," Aeronautics Digest,Fall 1978. USAFA-TR-79-1, Air Force Academy, CO.Feb 1979.
3. Daley, D.C. The Experimental Investigation of Dynamic
Stall. Thesis, AFIT/GAE/AA/82D-6. Air Force Instituteof Technology, Wright-Patterson AFB, OH, 1983.
4. Docken, R.G., Jr. Gust Response Prediction of an Air-foil using a Modified vonKarman-Pohlhausen Technique.Thesis, AFIT/GAE/AA/82D-9, Air Force Institute ofTechnology, Wright-Patterson AFB, OH, 1982.
5. Lawrence, Maj J.S. Investigation of Effects Contributingto Dynamic Stall Usa a Momentum-Integral Method.Thesis, AFIT/GAE/AA/83D-12. Air Force Institute ofTechnology, Wright-Patterson AFB, OH, 1983.
6. Shung, Yeou-Kuang. Numerical Solution for Some ProblemsConcerning Unsteady Motion of Airfoils. Ph.D. disserta-tion. University of Colorado, Boulder, CO, 1977.
7. Spiegel, Murray R. Theory and Problems of Complex Vari-ables. New York: Schaum's Outline Series, McGraw-HillBook Company, 1964.
8. Kuethe, Arnold M. and Chuen-Yen Chow. Foundations ofAerodynamics: Bases of Aerodynamic Design (Third Edition).New Yorks John Wiley and Sons, Inc., 1976.
9. Milne-Thomson, L.M. Theoretical Aerodynamics (ThirdEdition). London: Maaillan and Company Limited, 1958.
10. Karamcheti, Krishnamurty. Principles of Ideal-FluidAerodynamics. New Yorks John Wiley and Sons, Inc., 1966.
11. Von Wagner, H. "Dynamischer Auftrieb von Tragflugeln,"
Zeitschrift fuer Angewandte Mathematik und Mechanik, 5:17 (February 1925).
12. Giesing, Joseph P. "Nonlinear Two-Dimensional UnsteadyPotential Flow with Lift," Journal of Aircraft, :135(March-April 1968).
4... 66 .6
13. Chow, Chuen-Yen and Ming-Ke Huang. "The Initial Liftand Drag of an Impulsively Started Airfoil of FiniteThickness," Journal of Fluid Mechanics, 118:393-409(May 1982).
V
67/
- *1
VITA
Captain Kenneth W. Tupper was born on 22 April 1952 in
Johnson City, New York. He graduated from high school in
Meadville, Pennsylvania, in 1970 and attended the United
States Air Force Academy from which he received the degree
of Bachelor of Science in Astronautical Engineering in June
1974. Upon graduation, he was commissioned a Second Lieuten-ant in the USAF. He completed pilot training and received
his wings in July 1975. He served as a KC-135 pilot and
flight instructor in the 924th Aerial Refueling Squadron,
Castle AFB, California, until entering the School of Engi-
neering, Air Force Institute of Technology, in June 1982.
Permanent address: 978 Northwood DriveMerced, California 95340
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1S. ABSTRACT lContinue on a d if syearn, and identfy by block numbar)
Title: THE EFFECT OF TRAILING VORTICES ON THE PRODUCTIONOF LIFT ON AN AIRFOIL UNDERGOING A CONSTANTRATE OF CHANGE OF ANGLE OF ATTACK
Thesis Advisor: Eric J. Jumper, Major, USAF
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This study explored the effect of a trailing vortexwake on the production of lift on an airfoil.undergoing aconstant rate of change of angle of attack, a. Thq studyshowed that when an airfoil encounters a constant-a flow,the trailing vortex wake acts to suppress the slope of theairfoil's C, vs. a curve. The change in magnitude of thiseffect as a function of airfoil thickness and camber wasalso investigated.
Potential flow theory was used to model the flow abouta two-dimensional circular cylinder, and that flow was trans-formed to flow about an airfoil by the Joukowski transforma-tion. The trailing vortex wake was modeled by a sequence ofdiscrete point vortices, and the pitching motion of the air-foil was modeled by a series of small incremental changesin angle of attack, Am, over a short period of time, At.The rate of change of angle of attack, i, was then definedas Am/At. After each time change At, a was changed by anamount a. A discrete vortex was introduced into the wake ata distance UAt behind the airfoil trailing edge, and abound vortex of equal strength but opposite sense was intro-duced to satisfy the Kutta condition and keep the total cir-culation in the flow field equal to zero. As each new vor-tex pair was introduced, all other trailing vortices wereassumed to move in the wake by a distance UAt, where U isthe velocity induced at a vortex position by all other trail-ing vortices, the bound vortices, and the free stream flow.The unsteady Bernoulli equation was solved using numericalintegration and differentiation techniques to determine pres-sure difference distribution, vorticity distribution, andcoefficient of lift on the airfoil for that instant in time.This information was then used to investigate the overalleffect of constant-6 flow a$ well as the effect of thicknessand camber on the constant-a problem, and simple rules forpredicting the effects were developed.