A PARAMETRIC STUDY OF PLANFORM AND AEROELASTIC EFFECTS ON AERODYNAMIC CENTER,

a- AND q- STABILITY DERIVATIVES

APPENDIX C

METHOD FOR COMPUTING THE AERODYNAMIC

INFLUENCE COEFFICIENT MATRIX OF NONPLANAR WING-BODY-TAIL CONFIGURATIONS

by J. Roskam, C. Lan, and S. Mehrotra

CRINC-FRL 72-01 3

October 1972

Prepared under NASA Grant NGR 17-002-071 by

The FI ight Research Laboratory Depaftment of Aerospace Engineering

The University of Kansas

Lawrence, Kansas 66044 for

Langley Research Center National Aeronautics and Space Administration

https://ntrs.nasa.gov/search.jsp?R=19730013172 2020-06-03T04:09:15+00:00Z

TABLE OF CONTENTS

Chapter

1

2

3

4

5

Page

1 Introduction ,+ * e e a e ,+ e . . . e .. . Symbols . . . . . . . . . o . . . o . . . . . . ~ ~ . . . ~ . ~ e ~ , + ~ ~ e a 2

Method of Analysis .. . e . . . . .,+ . . *.. e . e e e . 4

Validation of the [A] -Matrix Routine. e . . . e e e 18

References....e.a..o.a~~~...~..o..ae~~.e~~~ 30

- Title -

.. . I l l

1. INTRODUCTION

The purpose of this appendix is to derive expressions for computing the Aerody- namic influence coefficient matrix (or [AI matrix) for nonplanar wing-body-tail configura- tions. An aerodynamic influence coefficient a i j (i.e. an element of the A matrix) is defined as the load in lbs. (Newtons) induced on panel i as a result of a unit angle of attack (rad) on panel j . Simulation of wing-body interference is not attempted. Fuselage, wing and tail thickness are assumed to be small with the result that the thickness effect on the flow-field is negligible. No dihedral effects are considered, even though these can be easily included by computing both downwash and sidewash. Camber effects may also be inclwded. Symbols used in this appendix are defined in Section 2.

The method for determining the aerodynamic influence coefficient matrix is based on the " I ifting" solution to the small perturbation, steady potential flow equation. Expressions for the various velocity functions, needed for computing the downwash are presented in Section 3 of this appendix. In section 4, a comparison of pressure distribu- tions and stability derivatives derived from this investigation, with already existing results i s made.

2. SYMBOLS

T h e units used for the physic.al quantities defined in this paper are given both in the International System of Units (SI) and the U .S. Customary Units.

Symbol

a

CAI &

b;k

CP

FA

F

K

L

U

V

w

Description

Downwash influence coefficient

Ae rody nam i c in f I ven ce coe ff ic ien t

Aspect ratio

jk / f l

Pressure coefficient

* Aerodynamrc Pressure force on a panel I

Velocity function

Constant

Slope of leading or trailing edges of a panel

Free stream Mach number

Dynamic pressure

Area of a panel

Perturbation velocity in x-direction

Perturbation velocity in y-direction

Perturbation velocity in z-direction, downwash

Dimension

Nondimensional

ft2 rad-’ (m2 rad-’)

Nondimensional

Nond imensiona I

Nondimensional

Lbs. (Newton)

Nondimensional

Nondimensional

N on d i m e ns i ona i

Lbs. ft’* (Newton m-*)

ft2 (m*)

Nondimensional

Nondimensional

Nondimensional

(x, Y I 4 Rectangular Cartesian coordinate system ft. (m)

ft. (m) Z Ordinates of Camber line C

2

Symbol

Greek

Description

cy Angle of attack

C = ( a - - d Z

dx

Dimensional

rad

rad

P = qlvl m 2 - 1 1 Nondimensional

4) Velocity potential Nondimension al

(5 , rt) Integration variables in Cartesian system ft. (m)

Spanwise station in fraction of the semi-span Y q = - b/2

Subscripts

I Aerodynamic panel number

j Aerodynamic panel number

k Panel corner point

k Aerodynamic panel number

( >' Referred to primed system of coordinates

Matrices

Square matrix (n x n)

Column matrix (n x 1) [ I

1

{ I { IT Row matrix (1 x n)

Diagonal matrix (n x n)

3

3. METHOD OF ANALYSIS

The aerodynamic inf luence coefficient method is based on the "lifting" solution to the small perturbation, steady potential flow equation:

2 (1 1 (1 - Moo)$xx -I- 4 -I- 4zz = O

YY

where $ is the perturbation velocity potential and M, the freestream Mach number. Note that the same equation is also satisfied by the perturbed velocity components. I t is well known that the lifting solution of the above equation can be obtained with a vortex distribution with strength to be determined by satisfying a boundary condition of flow tangency on the wing surface. In this appendix the Woodward's method of solution (Reference l ) , has been used.

ber of quadrilateral panels as shown in Figure (1). O n each panel, the incidence and camber effects a r e represented by a constant planar pressure vortex distribution with an unknown strength A u. The velocity potential a t any point in the flow field due to this vortex distribution on one panel is given by:

In Woodward's method, the entire wing or airplane planform is divided into a num-

z ' (x ' - S > dSdq 4 (XI, y ' , z ' ) = - KA' 27r 17-

Pane I

(1 - K)Au z'dcdq fl (y ' - d2 + -I-

2 T

where K = 0.5 for Ma< 1 and K = 1 .O for M, > 1 . The primed coordinate system is a local coordinate system as shown in Figure (2). There a re many ways of finding the in- tegrals in Equation (2). A simple way is to regard each panel as a n algebraic sum of four semi-infinite triungular regions as shown in Figure (3a).

Thus, the velocity potential a t any point P is given by: $ (P) = $l (semi-infinite triangular region from corner I )

- 4 (semi-infinite triangular region from corner 2)

- 4 (semi-ikfinite triangular region from corner 3)

I- Cb4 (semi-infinite triangular region from corner 4)

/ 2

3

4

NUMBER 0 F PANELS IN FUSELAGE, NUMBER OF PANELS I N WING

NUMBER OF PANELS I N TAIL

= 20 = 60

= 20

1 Figure 1 Example of Quadrilateral Panel Distribution

5

* P

Figure 2 Definition of Panel Coordinate System

6

a) Positive Leading Edge Slopes

b) Negative Leading Edge Slopes

Figure 3 Decomposition of Area of Integration

7

A similar approach can also be used when the leading and trailing edges of the panel have negative slopes and when points 2 and 4 are coincident, i .e., the panel i s triangular. In the latter case, the same scheme as for the positive leading edge shape can be used, However, when the leading edge slope i s negative, the scheme shown in Figure (3b) is convenient.

semi-infinite triangular region with a positive leading edge slope. If the equation of the leading edge is given by 4 = !qs , the velocity potential due to a constant pressure vor- tex disfribution in the semi-infinite triangular region from the corner k is as follows:

It is clear then that the fundamental region of integration for Equation (2) is the

Z' (x' - e ) d( dv q1 e1

K A u @k= rbqfLq

' - + (l-M,)[ 2 y' - q) 2 2 + z' 1 1

where x' = x - Xk, y' = yp - yk and z' = z . In subsonic flow, t1 = 00 and

In supersonic flow, the integration limits are as shown in Figure (4) and are given below: = 00 .

P P

2 1/2 2 1/2 Lx' - (M,- 1) y' - {{Ma- 1) [(x' - L Y ' ) ~ + (L2 + 1 - M3 z I 2 ] }

2 2 L +1-M,

8

Figure 4 Limits of Integration in Supersonic Flow

9

Equations (3) have been integrated in closed form by Woodward (Reference 1)

The result is:

The corresponding velocity components are:

The velocity functions F's are defined as follows:

ti - 1 z'd' F = Re { t a n

LrtL - x l y ' 3

for M, -= 1 - 1 y' F4 = tan 2'

F4 = 0 for M, L 1

F5 = fog - Lr' for M, I 1 r

F5 = 0 for >1

for M,<1

for M,>1

2 2 1/2 where: r' = (2' + y ' )

1/2 d' = [x12 + (1 - k,) I

z 2 x = Lx' -I- (1 - M,)y'

y = X I - Ly' I

L

k 2 2 2 ' 2 1 '1/2 r = [(XI - Ly') f (L + 1 - M,)z

Let p be the downwash contro. point of a pane , Then the downwash a t p of panel i due to a unit vortex distribution on the semi-infinite triangular region from the corner k of panel j is given by:

K P f 1)F2(ijk) - b! [Fl(ijk) - F5(ijk)l - ytjk F6(ijk)} Wijk (P) = (b'jk - Ik ( 8)

where t h e plus sign i s for the subsonic case and the minus sign for the supersonic case. The quantit ies, b! and yf . in Equation (8) are defined as follows:

Jk I Jk

1 1

I for k = 1 , 2

for k = 3, 4 I

X ' i j k and Zl i jk can also be defined in a s imi la r way as:

and also

(9)

Equation 8 indicates that only F F , F and F velocity functions a re needed to evaluate the downwash Wk . Simplified expressions of these velocity functions can be derived by using equations supersonic cases are presented in Table 1 .

due to a uni t vortex distribution on the jth panel, represented by a.. as :

1' 2 5 6

7 and 9 through 13. These expressions for both subsonic and

Referring back to equation 8, it follows that the downwash a t point p of panel i can be written Ij1

(1 8) aij, - - wijl - wij2 - wij3 $- wij4

I n this report o n i j symmetric flight configurations are considered. Le the mirror image of the jth panel with respect to the l ine of symmetry be denoted by jtt' panel. The down- wash a t point p due to the pan I j ' is ai jz. The total downwash a t p due to a unit vortex distribution on both jth and j i t E panels IS then:

a.. = aij l f aij2 'I

12

Table 1 Velocity functions for computing downwash

according to equation 8

Subsonic Case: - x' -t- d'

F' = Rein (p, r l )

I b'I &12 + z' 2

- b'y1)2 -t- (bt2 -t- l ) rJ2

I

Supersonic Case:

7

73

i f b' > 1

F5 = 0

Case 2. C' =GI2 -t- z'* and b' < 1

F1 = O

if y' < b ' r ' F2 = 0

F2 = if y' = b ' t ' 7r

i f y' > b'e' 7r F2 = JrFf

F5 = 0

F6 = 0

Note: I f 8' = and b' 3'1, all functions are zero, 14

F1 =O.

F2 = 0.

F2 = 0

i f y' 6 0

i f y' > 0 and 4 ' < (b'y'

i f y' ) 0, 6 ' > (b'y' +dl - b'2 1.' I ) anc y '> b'6 '

F2 = 0

F5 = 0.

F6= 0.

Note: I f f '< and b' >/ 1 , all functions are zeroo

15

The boundary conditions require that the flow be tangential to the camber line. Since the downwash at p of panel i due to the vortex distribution on a l l panels is:

W N aij (Au)~

where N, is the number of panels on one-half of the wing, fuselage and tail, the tan- gency condition can be written as:

W N

( - dzC - a ) i = a.. (Au)~ 'J j = l dx

or in matrix form:

C a} = Ca.. l(Au\ dz

{ - i!x - 'J

Solving this equation for {Au} , the unknown vortex strengths Au are obtained as:

+ - - - I - Note that Au = u -u . Since C = -2u, i t follows that AC = C -C = 2Au.

Hence:

P P P P

The total force on any pan61 i i s given by:

FA, = tjS. ( AC ) P i I

I

where S i i s the area of panel i andq i s the dynamic pressure. I n matrix form, the panel forces can be written as:

16

0 s2 . .

0 w 'N

dx

Where [A] i s the aerodynamic influence coefficient matrix and i s given by:

, - L41= - 2 p&Lj]-l and

The computer program for generating this (Appendix A of the Summary Report).

[ A ] matrix i s included in Reference 2

17

program, Reference 3 . Both programs employed a total of 80 panels with 50 panels on the halfwing, 10

on the half fuselage and 20 on one side of the tail. The pressure distributions are com- pared in Figure 6 . Note that the computed ACp is plotted at the panel centroid for the K.U. program, but a t the panel quarter chord for the Vortex Lattice program. It. is seen that the results from both programs agree reasonably well The computation of stability derivatives is compared in Table 2. The experimental data are obtained from Reference 4. To determine the effect of the number of panels on the numerical outcome, f 00 and 120 panel solutions were computed. Table 2. shows that there was little change by using a panel scheme of more than 100.

1

Table 2. Comparison of Computed

Stability Derivatives with Experiment for

d C d dCL -0.0395 -0.0541 -0.0429 -0.0438 -0.065

In comparison with experimental results, it is assumed that for a high wing con- figuration, the wing may be placed directly above the fuselage at a distance of one half fuselage diameter, so that two lifting surfaces (wing and fuselage) will overlap in some region. A similar aerodynamic representation has been assumed for a low wing configur- ation or for a high or low tail . Several other configurations presented in References 5 through 8 were investigated with the K.U. program. Their geometries a re shown in Figures 7 through 10 respectively. The total number of panels used on a half configuration was

on with experimental results. It is seen that the com- ement with experiments.

18

I ,

-

f 3

t Y

0

v Z

-CL w

Lu ai w u. w cr: I

. 3 - 3 .

+ s Y u a, s 0

E e u)

g b

IC

73 a,

. 3 Ls 0

v)

.- + ., 3 0) IC .- r5 V

U I x

-u 0 9

I u) S

.- +

.- 3 d

rc 0

2 c

d Q

p! 3 cn

19

1.5

1 .o

ACP

0.5

0

Figure 6a Comparison of wing pressure distribution a t r) = 0.1

1.5

1 .o

0

YC LOCATION Comparison of wing pressure distribution at q = 0.3 Figure 6b

20

2.5

2.0

1.5

1 .o

0.5

0 1 .2 1 .o .4 .6 .8

X / C LOCATION

0

Figure 6c Comparison of’wing pressure distribution at q = 0.5

21

ACp '

3 .O

2.5

2.0

1.5

1 .o

0.5

0

j . e . . : 8 . . I ,

TEX LATTICE' PROG

. . . . .

0 .2 .4 .6 .8 1 .o X/C LOCATION

Figure 6d Comparison of Wing Pressure Distribution at q = 0.7

22

3 e 5

3 .O

2 e o

ACP

1.5

1 .o

0.5

0 . .+

. . ..

0 - .2 .4 .6

X/C LOCATION

.% 1 .o

Figure 6e Comparison of wing pressure distribution a t T]= 0.9

23

1.5

1.0

0.5

0 0 .2 04 .6 * 8 1 .o

X/C LOCATION Figure 6f Comparison of horizontal tail pressure distribution at q = 0.448

0.5

-0.5

Figure 69 Comparison of fuselage pressure distribution at q= 0.5

24

B- 4.0 in. (0.1 01 7)

MOMENT REFERENCE POINT

f

-

(0.0420)

I

(0.0420)

ALL BRACKETED DIMENSIONS IN METERS

NOTE: CONFIGURATION WBH 2 OF REFERENCE 5

Figure 7 Aerodynamic Representation for Geometry of WBH2

25

26

rMOMENT REFERENCE F 'OINT

ALL BRACKETED DIMENSIONS IN METERS

NOTE: CONFIGURLTION WBH 4-OF REFERENCE 7

in. 524)

Figure 9 Aerodynamic Representation for Geometry of WBH 4

27

S 0

0 S a, a, Q a, & 0

.-

.I-

.I-

v)

L

.- E

2

S x U e

28

co m N 0

I

ln 4 I

3 at e U S

ln

- 4

E" .- L al Q X

W

-f 3 G

2 e

.- *- + ln

L .- V

U 1

u v .- 6

2

Eo

c x -0 2

-a a, 3 Q

.I-

u rc 0 S 0

U a

v)

L .- Eo v

c3

al

U I-

z5

0. 7

o', c3

03 Q 'hl

0 I

- h O -a0

1 0 9 m b-

0 9 0

- .- U

I u)

I I u) c

+

.-

.- 3

z!

- .- U .I-

I= 0 .- 4- e 3 0)

rc S

.- u"

8 A I rn c .- 3

I I

L t L sq 33 z

c3

29

5. REFERENCES

1 .

2.

3.

4.

5.

6.

7.

,8.

. . odward, F.A., "Analysis and Design of Wing-Body Combinations a t Subsonic

and Supersonic Speeds, I' Journal of Aircraft, Vole 5., No. 6, Nov. - Dec., 1968. Roskam, J., Lan, C., and Mehrotra, S.; "A Computer Program for Calculating cy-

and q- Stability Derivatives and Induced Drag for Thin Elastic Aeroplanes at Subsonic and Supersonic Speeds," NASA CR-172229; Prepared by the Flight Research Laboratory of t h e University of Kansas under NASA Grant NGR 17-002-071, October, 1972. Appendix A of the Summary Report, NASA CR- 21 17, Margason, R.J. and Lamar, J .E., "Vortex-Lattice Fortran Program for Estimating Subsonic Aerodynamic Characteristics of Complex Planforms .I1 NASA TN D-6142, 1971.

Tinling, E.T., and Lopez, A. E,, "Subsonic Static Aerodynamic Characteristics of an Airplane Model Having a Triangular Wing of Aspect Ratio 3. I--Effects of Horizontal-tail location and size on the longitudinal characteristics .'I

NACA TN 4041 , 1957.

Fuller,' E.D., "Static Stability Characteristics a t Mach Numbers from 1 .90 to 4.63 of a 76O Swept Arrow Wing Model With Variations in Horizontal-Tail Height, Wing Height and Dihedral, Spearman, M.Ld, e t al . , "Investigation of Aerodynamic Characteristics in Pitch and Sides1 ip of a 45 Swept-back-wing Airplane Model with Various Vertical Locations of Wing and Horizontal Tail," NACA RM L54l.06, 1955. Sleernan, W.C., J r . , "An Experimental St$y a t High Subsonic Speeds of Several Tail Configurations on a Model Having a 45 Sweptback Wing." NACA RM L57C08, 1957. Stivers, L.S., Jr., and Lippmann, G.W., "Effects of Vertical Locations of Wing and Horizontal Tail o n the Aerodynamic Characteristics in Pi tch a t Mach Numbers from 0.60 to 'i .40 of an Airpfane Configuration with a n Unswept wing." NACA RM A571 1 0, 1 957.

NASA TN D-4076, '1 967.

30