The technique of implicit differentiation has been used in combination with linearized lifting-surface theory to derive analytical expressions for aerodynamic sensitivities (i.e., rates of change of lifting pressures with respect to general changes in aircraft geometry, including planform variations) for steady or oscillating planar or nonplanar lifting surfaces in subsonic, sonic, or supersonic flow. The geometric perturbation is defined in terms of a single variable, and the user need only provide simple expressions or similar means for defining the continuous or discontinuous global or local perturbation of interest. Example expressions are given for perturbations of the sweep, taper, and aspect ratio of a wing with trapezoidal semispan planform. In addition to direct computational use, the analytical method presented here should provide benchmark criteria for assessing the accuracy of aerodynamic sensitivities obtained by approximate methods such as finite geometry perturbation and differencing. The present process appears to be readily adaptable to more general surface-panel methods

Yates, E. Carson, Jr.

English

NASA Technical Reports Server

NASA Technical Memorandum 100502 
AERODYNAMIC SENSITIVITIES FROM SUBSONIC, SONIC, AND 
SUPERSONIC UNSTEADY, NONPLANAR LIFTING-SURFACE THEORY 
(UASA-Tfl-100502) A E R O D Y W A l f C  SBISITXVITIBS 1180-12459 
n c n  SUEIODIC, SCIIC A I D  S P P E E S O I X C  
D hST E A C  I ,  110 U t  LI E A B  L 1?TI YC-SP D FAC 1 FHZORI 
I K A S A )  1 8  p C S C L  O I A  Onclas 
G3/02 0107337 
E. Carson Yates, Jr. * 
September 1987 
National Aeronautics and 
Space Administration 
LmgIey Rasemch Center 
Hampton. Virginia 23665 
https://ntrs.nasa.gov/search.jsp?R=19880003077 2020-03-20T08:36:38+00:00Z
i 
Summary 
I 
' *  
The technique of implicit differentiation has been used in combination with 
linearized lifting-surface theory to derive analytical expressions for aerodynamic 
sensitivities (i.e., rates of change of lifting pressures with respect to general 
changes in aircraft geometry, including planform varictions) for steady or oscil- 
lating planar or nonplanar lifting surfaces in subsonic, sonic, or supersonic flow. 
The geometric perturbation is defined in terms of a single variable, and the user 
need only provide simple expressions or similar means for defining the continuous or 
discontinuous global or local perturbation of interest. Example expressions are 
given for perturbations of the sweep, taper, and aspect ratio of a wing with trape- 
zoidal semispan planform. 
method presented here should provide benchmark criteria for assessing the accuracy of 
aerodynamic sensitivities obtained by approximate methods such as finite geometry 
perturbation and differencing. 
to more general surface-panel methods. 
In addition to direct computational use, the analytical 
The present process appears to be readily adaptable 
t Nomenclature 
A 
coefficient of pressure-mode function fn(C)g (0) in pressure expansion m 
(Eq- (10)) 
'nm 
B = A T  
b(y) streamwise semichord at spanwise station y 
reference length (typlcally root semichord) 
complex amplitude of lifting-pressure coefficient 
b0 
Ai? 
P 
fn(i) nth chordwise pressure-mode funct€on 
gm(0) mth spanwise pressure-mode function 
Ij(kl),Kj(kl) 
K kernel function 
CI 
modified Ressel functCons of first and second kind, respectively 
kernel funct€ons for sidewash and upwash, respectively KV ,KW 
k = bow/U, reduced frequency 
= kro = kr/B, frequency parameter k l  
Lj(kl) modified Struve function 
a. arc length of lifting surface, root to tip, in cross-stream direction 
M free-stream Mach number 
tAil coordinates are normalized with respect to bo. 
1 
perturbation of lifting surface 
2 + 4 1 1 2  
= bo 
0 = 8r 
2 112 
= (Yt + z o )  
free-stream speed (in X direction) 
= (MR - xo)/Br 
complex amplitude of normalwash at point on lifting surface 
fixed streamwise, lateral, and vertical coordinates, respectively 
Cartesian coordinates 
x-E, y-n, z-t, respectively 
streamwise location of mean chord line 
streamwise location of leading edge 
coordinaty? 02 lifting surface transformed so that 
A A  
x = € = -1 at leading 
edge , x € +1 at trailing edge, and 6 = fl at tips 
=m 
lateral inclination (local dihedral) of lifting surface 
sweep angle of lifting-surface midchord line 
Cartesian coordinates 
* rI8 
circular frequency of oscillation 
Introduction 
Accurate and efficient computation of aerodynamic sensitivities (i.e., rates of 
change of surface pressures and weighted integrals of pressures with respect to 
changes in aircraft geometry) is of growing importance for aerodynamic shape optimi- 
zation as well as for multidisciplinary design synthesis (ref. 1). Such sensitivi- 
ties can, of course, be obtained approximately with any aerodynamic method by simply 
differencing the results of a solution for a base configuration and a solution for a 
geometrically perturbed configuration. 
are dependent upon the size of the perturbation (step size) selected and are subject 
to cancellation errors if the step size is too small. Calculating sensitivtties by 
analytical methods avoids such problems. 
The resulting sensitivity values, however, 
Some capabilities to generate sensitivities and to solve the inverse 
(aerodynamic-design) problem have been developed from surface-panel methods (e.g., 
2 
I 
refs. 2 to 6 ) ,  but these have been limited to steady state, and applications appear 
to have been generally limited to geometric perturbations of lifting surfaces and 
only in the direction normal to the chord plane. j 
I This paper presents expressions for aerodynamic sensitivities derived from 
subsonic, sonic, and supersonic unsteady, nonplanar lifting-surface theory (refs. 7 
and 8 )  for general perturbations, including planform variations. The geometric 
perturbation is defined in terms of a single variable. Moreover, without loss of 
generality, perturbations are performed by varying only the normalwash distribution 
and/or the streamwise planform coordinates. 
accomplishes considerable simplification of the resulting expressions and avoids a 
third-order singularity along the line r = 0. Note that aspect ratio and even the 
spanwise location of a leading-edge or trailing-edge break point can be varied by 
the user provide simple analytical expressions or similar means for defining the 
bation of sweep angle). In contrast, the method of reference 6 requires that the 
perturbation be made up of individual perturbations of each of the panel nodes - a 
computationally expensive operation. 
I '  
I 
Performing the perturbations in this way 
I 
I perturbing only streamwise coordinates. The present formulation requires only that 
I continuous or discontinuous global or local perturbations of interest (e.g., pertur- 
I 
In addition to the steady-state limiting condition, the expressions derived here 
can be used, for example, to track flutter and other dynamic-response characteristics 
in multidisciplinary design processes. Such computations of aerodynamic sensitivi- 
ties, both steady and unsteady, should be useful in both conceptual and preliminary 
design. On the other hand, lifting-surface-theory calculations are themselves quite 
fast so that computation of sensitivities by differencing is not prohibitively expen- 
sive. Even so, the analytical expressions presented here provide convenient means 
for computation as well as accurate benchmark criteria for assessing the accuracy of 
differencing (e.g., as a function of perturbation size) so that, when needed, differ- 
encing can be used with better understanding and confidence. Hopefully, the present 
development will serve heuristic purposes as well. 
Theoretical Development 
Subsonic Flow 
The integral equation relatfng lifting-pressure and normalwash distribution on a 
nonplanar Lifting surface (Fig. 1) oscillating in subsonic flow (Eq. (1)  of ref. 7 )  
can be written 
where 
and KV and Kw are given, respectively, by 
3 
-ikx 
KV(kxo, ky,, kzOs M) = e a2E/%az 
and 
and 
.. -ik,  u 
(see Eqs. (4) t o  ( 6 )  of r e f .  7). Carrying ou t  t h e  d i f f e r e n t i a t i o n  ind ica t ed  in 
Eqs. (2 )  and ( 3 )  leads to 
-ikx y z U -iklu U - iklu 
du - kl  0' e ={l1(kl)  + Ql(k l )  - 12 j 1 ue 
O m  L-7 3 
KV = e 
r 
0 
and 
, 
4 
(see E q s .  (8) to (11) of ref. 7). The physical variables are related to the 
transformed variables in Eq. (1) by 
t n  
Y = -  Y 
bo 
n = -  
The normalwash cN is defined by the shape and/or motion of the lifting surface. 
It is customary to represent the unknown pressure distribution @ by a linear 
I P combination of chosen shape functions. Thus 
and the basic aerodynamic sensitivity is 
where Q represents the geometric perturbation of the lifting surface and should not 
be confused with the function Ql(kl) in equations ( 5 )  and (8).  Now 
purely geometrical; consequently, attention is focused on evaluating 
1 -  without loss of generality, th_e perturbation Q will be performed by varying only 
the normalwash distribution WN and/or the streamwise coordinates x and 5 in 
order to avoid having to deal with spanwise derivatives of Uv and % which in- 
volve a third-order singularity at r = 0. Variations of x and 5 may be defined 
in terms of changes in b and xL or x, (Eq. ( 9 ) ) .  Note that even a geometrical 
parameter such as the spanwise location of a leading-edge break point may he varied 
by changes in xL(y) only. 
-1 
w, 
ab (n) is 
e . Moreover, aa nm 
5 
By use of Eq. ( l o ) ,  Eq. (1) may be expressed as 
bo GN 1 1 anmFnm - 8n - - t u  n m  0 
where 
Solution of Eq. (12) requires, of course, that the equatfon be written for at least 
as many points x,y,z on the lifting surface as there are values of an, to be 
calculated. Differentiating Eq. (12) with respect to 0 gives 
~ 
in which the values of 
base configuration, and 
Equation (14) shows that if the perturbation 
then 
solution of Eq. (12) with the perturbation normalwash 
Fnm- and anm are obtained from solution of Eq. (12) for the 
a(W,/U)/aQ is geometrically determfned and known a priori. 
9 does not involve a planform change, 
a ( W , / l J ) / @  input in place of 
aFnm/q = 0 ,  and the required values of aanm/W msy be obtained from a direct 
ij”. 
Differentiating Eq. (13) with respect to r) gives 
1 Note that 
(1 5 )  
d 
6 
is the only quantity in Eq. (15) that depends on the particular perturbation 
Q. 
Geometrical Example 
The example of a planar wing ! Lth trapezoidal seniispan planform serves to 
illustrate the nature of ax/aQ for perturbations in aspect ratio A, taper ratio 
A, and midchord sweep angle A. The semispan is invariant. The first of 
Eqs. (9) gives 
Then with A and A invariant, 
With A and A invariant, 
ax = ax 
aq a x  Y + -  2-- - =  A ( A  A tan A )  
With A and A invariant, 
For use in Eq. (15) these expressions l e a d  to 
7 
~ 
and these may be combined into a single expression for use in Eq. (15). 
where for Q = A, 
co = 0, c1 = - + A tan A, c2 = o 2 
A co = 1, c1 = T tan A, c2 = -1 
for 0 = A, 
A 2 
co = 0, c1 = 4 (1 + A)(1 + tan A), c2 = 0 
Derivative of the Kernel Function 
The required expressions for aKV/axo and a%/axo in Eq. (15) are obtained by 
differentiating Eqs. (5) and (61, respectively. Thus 
2 2  
- =  as ikx 0 M 18 82{3(1 
e 
axO 
a 
kMP -i- 
- ik\ 
The singularities of KV and KW ( E q s .  ( 5 )  and (a ) ) ,  and in particular the 
r singularity, remain in aKV/axo and aKv/h0 (Eqs. (17) and (18)) only for 
unsteady flow. For both steady and unsteady flow, however, a$/axo contains a 
third-order singularity (R'3) at the point As indicated earlier, 
a third-order singularity along the line to '2 fl has been avoided hv not DerturhCng 
-2 
0 
= yo = z0 = 0 .  
I 8 
in y direction. The RW3 singularity amy be dealt with in Eq. (15) by the 
classtcal analytical extraction technique. 
however, converges slowly. 
also included in the extraction. 
The remaining numerical quadrature, 
Results can be improved if second-derivative terms are 
I 
Eqs. (17) and (18) contain, respectively, the factors yozo/R5 and zo/R 2 5  
which also appear to indicate third-order singularity. However, if the normalwash 
control points (x,y,z) are not located at points of normalwash discontinuity (as, in- 
deed, they should not be!), then as yo goes to zero, zo behaves as yo (Pig. 1). 
Consequently, y0zo/R5 behaves as R-*, and zz/R5 behaves as R'l. Thus these 
terms require no special attention in Eq. (15)  beyond the aforementioned extraction 
of the third-order singularity. 
ues of aanm/aO which, in turn, permit the evaluation of a@p/W from Eq. (11). 
If this procedure is employed or a number of different perturbations, only the sim- 
reevaluated for each. 
2 
The use of Eqs. (15)  to (18) permits Eq. (14) to be solved for the required val- 
ple geometric expressions ab -f /aQ in Eq. (11) and axo/W in Eq. (15)  need be 
The expressions for Kv and as well as for aKv/axo and a%/axo simpli- 
fy considerably for steady flow (k = 0) or for planar geometry (zo = 0). 
steady flow, k = 0, and 
Thus for 
0 
and 
so that 
4 5 
= 38 y0zo/R 
k=O 
and c 
2 2  
aKW 
axO 
-
k=O 
(29) 
9 
For planar geometry, zo = 0, and 
and 
so that 
Kv 
Kw 
= o  aKV I- 
2 =o 
0 
z =o 
0 
'iklu d j }  ( 2 4 )  
U 1 ue -ik 1~ 
z 10 + f i [ l ( k l )  YO - O m  $ 
0 
kMR 2 2  -iT 
- 
z =o 
0 
2 =o = -e - l + i -  R3 82( 7"). 8 ikxoM a% 
0 
Finally, for planar steady flow, k = zo = 0, and 
X 
= +(l + $) 
YO k=z =O 0 
Kw 
and 
Alternative Formulation for Subsonic, Sonic, 
and Supersonic Speeds 
( 2 5 )  
I An alternative form of the kernel function has been used to define the kernel 
€or sonic and supersonic speeds as well as €or subsonic (ref. 
present notation 
Thus in the 
~ 
where 
-ikxo 2 4 
K = e  (KITl/P + K2T2*/P ) 
I j .  
I 
and 
K1 in Eq. (31) should not be confused with the modified Ressel function Kl(kl) in 
Eqs. ( 7 )  and ( 8 ) .  The limits of integration s1 and 5 2  are determined by the time 
required for transmission of small disturbances from sending point (€,n,<) to receiv- 
ing point (x,y,z) with consideration of the limits imposed by Mach lines, if any. 
The derivative aK/axo required for the generation of aerodvnamic sensitivities 
may be obtained from Eq. (28) .  
- ikK - =  
0 
aK e 
axO P 
and 
s=s 2 -iks 
0 
1 S'S 
0 
(34) 
(35)  
(36) 
11 
The derivatives of sl and 92 indicated in Eqs.  (35) and (36) may be obtained from 
the expressions for sl and 82 that are pertinent to subsonic, sonic, or super- 
sonic flow (ref. 8). Thus for subsonic flow ' 
2 2 
0 a P  
- ax a i" - ;I) = >(3f32 f - 1) J 
For sonic flow, 
s2 = m, and a l l  t h e  detivatCves are zero as  i n  Eq. (38 ) .  
For supersonfc Flow, 
12 
as 1 2 2  
2 
( - r = M p / R ,  2 2  2 - = G h + y ) ,  a r = ( l - M ? ) / B  
0 
R a P2 
2 L($) = -Mpxo/R , a($) = -2M 2 2  p xo/R 4 , 
0 
ax 
0 
ax 
Substltutlon of the appropriate expressions from E q s .  ( 3 7 )  to (41) into F q s .  (35) 
and ( 3 6 )  and the latter two equations into Fq. (34) produces the kernel-functlon 
derivative needed €or subsonic, sonic, or supersonic speeds. The kernel derivative 
expression is used, in turn, to evaluate aFnm/aO from 
which corresponds to E q .  ( 1 5 ) .  
Note that the weight factors (two radicals) in Eq. ( 4 2 1 ,  as well as in 
E q s .  (lo), (111, (131, and (15), are appropriate for subsonic .speeds. 
factors and different pressure-mode functions €,(SI and g,(rl) are used for sonic 
or supersonic speeds (see, e.g., ref. 9 ) ,  and the domain of integration is llmlted to 
the portions of lifting surface lying within the Mach forecone with vertex at the 
receiving point (x,y,z). 
Different 
13 
Concluding Remarks 
The technique of implicit differentiation has been used in combination with 
linearized lifting-surface theory to derive analytical expressions for aerodynamic 
sensitivities (i.e., rates of change of lifting pressures with respect to general 
changes in aircraft geometry, including planform variations) for steady or oscil- 
lating planar or nonplanar lifting surfaces in subsonic, sonic, or supersonic flow. 
The geometric perturbation is defined in terms of a single variable, and the user 
need only provide simple expressions or similar means for defining the continuous or 
discontinuous global or local perturbations of interest. Example expressions are 
given for perturbations of the sweep, taper, and aspect ratio of a wing with trape- 
zoidal semispan planform. In addition to direct computational use, the analytical 
method presented here should provide benchmark criteria for assessing the accuracy of 
aerodynamic sensitivities obtained by approximate methods such as finite geometry 
perturbation and differencing. 
to more general surface-panel methods. 
The present process appears to be readily adaptahle 
Acknowledgement 
The author extends his sincere appreciation to Robert N. Desmarais for checking 
the derivation of equations and for valuable and stimulating discussions relative to 
numerical implementation. 
14 
Rof oralices 
1 . Sobieszczanski-Sobieski , Jaroslaw: The Case for Aerodynamic Sensitivity 
Analysis. In "Sensitivity Analysis in Engineering." NASA CP-2457, 1987. 
2. Bristow, De Re; and Hawk, J. D.: Subsonic Panel Method for the Efficient 
Analysis of Multiple Geometry Perturbations. NASA CP-3528, 1982. 
3. Rristow, D. R.; Hawk, J. De; and Thomas, J. L.: Subsonic 3-D Surface Panel 
Method for Rapid Analysis of Multiple Geometry Perturbations. 
Paper 52-0993, 1982. 
AIAA 
4. Hawk, J. D.; and Bristow, De R.: Subsonic Surface Panel Method for Airframe 
Analysis and Wing Design. AIAA Paper 83-0341, 1983. 
5. Rristow, D. Re; and Hawk, J. D.: Subsonic Panel Method for Designing Wing 
Surfaces From Pressure Distribution. NASA CR-3713, 1983. 
6. Hawk, J. Dennis; and Rristow, Dean Re: Development of MCAERO Wing Design Panel 
Method With Interactive Graphics Module. NASA CR-3775, 1984. 
7. Yates, E. Carson, Jr.: A Kernel-Function Formulation for Nonplanar Lifting 
Surfaces Oscillating in Subsonic Flow. AIAA Journal, Vole 4, No. 8, Aug. 1966, 
pp. 1486-1488. 
i 
8. Harder, Robert L.; and Rodden, Willtan P.: Kernel Function for Nonplanar 
Oscillating Surfaces in Supersonic Flow. Journal of Aircraft, Val. 8 ,  No. 8 ,  
Aug. 1971, pp. 677-679. 
9. Richardson, J. R.: A Method for Calculating the Liftinp; Forces on Wings 
(Unsteady Subsonic and Supersonic Lifting-Surface Theory). R. & M. No. 3157, 
British A.R.C., 1955. 
1 5  
3 
16 
Report Documentation Page 
1. Report No. 
NASA TM-100502 
2. Government Accession No. 
Aerodynamic Sensitivities From Subsonic, Sonic, and 
Supersonic Unsteady, Nonplanar Lifting-Surface Theory 
17. Key Words (Suggested by Authorls)) 
7. Authorb) 
18. Distribution Statement 
E. Carson Yates, Jr. 
19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No. of pages 
Uncl ass i f i ed Unclassified 17 
9. Performing Organization Name and Address 
22. Price 
A02 
NASA Langley Research Center 
Hampton, VA 23665-5225 
12. Sponsoring Agency Name and Address 
National Aeronautics and Space Administration 
Washington, DC 20546-0001 
15. Supplementary Notes 
3. Recipient's Catalog No. 
5. Report Date 
September 1987 
6. Performing Organization Code 
8. Performing Organization Report No. 
10. Work Unit No. 
505-63-1 1 
11. Contract or Grant No. 
13. Type of Report and Period Covered 
Technical Memorandum 
14. Sponsoring Agency Code 
16. Abstract 
The technique of implicit differentiation has been used in combination with 
linearized lifting-surface theory to derive analytical expressions for aerodynamic 
sensitivities (i.e., rates of change of lifting pressures with respect to general 
changes in aircraft geometry, including planform variations) for steady or oscil- 
lating planar or nonplanar lifting surfaces in subsonic. sonic, or supersonic flow. 
The geometric perturbation is defined in terms of a single variable, and the user 
need only provide simple expressions or similar means for defining the continuous 0 1  
discontinuous global or local perturbation of interest. Example expressions are 
given for perturbations of the sweep, taper, and aspect ratio of a wing with trape- 
zoidal semispan planform. In addition to direct computational use, the analytical 
method presented here should provide benchmark criteria for assessing the accuracy 
of aerodynamic sensitivities obtained by approximate methods such as finite 
geometry perturbation and differencing. The present process appears to be readily 
adaptable to more general surface-panel methods. 
Aerodynamic sensitivities 
Unsteady aerodynamics 
Lifting-surface theory 
Unclassified - Unlimited 
Subject Category - 02 


Aerodynamic sensitivities from subsonic, sonic and supersonic unsteady, nonplanar lifting-surface theory

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