The widely transonic swept-wing code, FL022, was found to have an error in the transformed flow equation in the computational domain. The revised version of the code correctly accounted for the non-straight leading edge geometry and its effect on the pressure distribution

Chang, I. C.

Tauber, M.

English

NASA Technical Reports Server

r 
I NASA Technical Memorandum 84391 
ASA-TM-84391 19840001982 
Numerical Calculation of the 
Transonic Potential Flow Past a 
Cranked Wing 
1- Chung Chang and Michael Tauber 
September 1983 
NI\S/\ 
National Aeronau tics and 
Space Administration 
_ ___ 0_-
- - -- - --
, 1 198 
LANGl f.::Y ,?r"<; AR" N r 
lI8RA ~y A,)A 
HA~.'~TON. VI GINIA 
https://ntrs.nasa.gov/search.jsp?R=19840001982 2020-03-21T00:29:13+00:00Z
I 
I 
NASA Techn ica l Memorandum 84391 
Numerical Calculation of the 
Transonic Potential Flow Past a 
Cranked Wing 
I - Chung Chang and Michae l Tauber, Ames Research Center, Moffett Field, Californ ia 94035 
NI\S/\ 
National Aeronautics and 
Space Administration 
Ames Research Center 
Moffett Field . California 94035 
N fy - /0 0 <H #-. 
) 
" 
"t· ..... 
'. ,". 
, .' 
'~.;' , 
" Ii, !. 
, SUMMARY 
I " ~,': 
The widely used transonic swept-wing code, FL022, was found to have an error 
in the transformed flow equation in 'the computational domain. The revised version of 
the code correctly accounted for the non-straight-leading-edge geometry and its 
effect on the pressure distribution. 
INTRODUCTION 
Transonic flow calculations have been made for many years. Among the wide 
variety of numerical procedures that are now available, a series of computer pro-
grams known as the FLO codes have been developed by'Professor Jameson and his 
coworkers. One of them, the widely used swept wing code, FL022 (ref. 1), has been 
known to generate inaccurate resul~s for wings with non-straight-1eading-edges. 
Recently, the code was used to calculate the flow past an ONERA cranked wing model 
mounted on a wall (ref. 2), and the predictions were found to be very poor in the 
vicinity of the crank. The reason 'for the poor predictions was identified as an 
error associated with the scaling ~etween the physical space and the computational 
domain. In fact, the FL022 code is dimensional for wings having non-straight-
leading-edges. 
ANALYSIS 
The governing equations (ref. 1) for steady, inviscid, compressible isentropic 
flow in Cartesian coordinates (x,y,z) are the potential-flow equation 
(a2 - u2)~ + (a2 - v2)~ + (a2 - w2)~ - 2uv~ - 2uw~ - 2vw~ = 0 (1) 
xx yy zz xy xz yz 
the Bernoulli equation 
1 2 2' (u2 + v2 + w2) '+y a_, l'= constarit 
" 
and the isentropic relation 
..E... = constant 
pY 
,', 
Here, ~ is the velocity potential (defined, such that V~ = (u,v,w) is the velocity 
vector), a is the speed of' sound, y' is 1.4 for air, p is the pressure, and p is 
the density. Let the reduced potential, G, be defined through the relation 
. ':. G ;. 4>. -..:. ~ cos a' - y sin a 
1-
The term G satisfies equation (1), where a is the angle of attack of the wing. 
(Far-field flow speed and density are taken as unity.) 
The flow equation (1) can be written in terms of 
tional domain generated through a sequence of mappings 
The first three mappings consist of the first shearing 
-
x x 
y 
-
z = z 
the square-root transformation 
2(x + iy) 
and t he second shearing transformation 
x 
Xl } 
Y1 - S(XI,Z1) 
Z1 
Y 
Z 
G and solved in a computa-
from the physical space . 
transformation 
(2) 
(3) 
where the singular line (the Jacobian vanishes) of the square-root transformation is 
represented as x = xs(z), y = ys(z) and the entire wing surface is described as 
Y1 = S(X1 ,Z1)' 
It is useful to introduce a scaling mapping between the first shearing transfor-
mation and the square-root mapping 
x 
x/SCAL } 
:::::2 
(4) y 
z = 
that will account for the exact scaling relation between the physical space and the 
computa tional domain. The parameters SCAL and SCALZ are real variables that are 
defined in the subroutine COORD of FL022, and set up the stretched coordinates 
(see ref. 1). The square-root transformation becomes 
The flow equation, in terms of G and using the mappings (2), (3), (4), and (5), 
becomes an equation of the form 
o 
2 
(6) 
r 
I 
which is equivalent to equation (9) of the original analysis (ref. 1) if the follow-
ing notations are introduced: 
o ::: XI-x 
)1 YI -Y 
~ ::: -(x'o + Y~)1) s 
n ::: X'fl - y~o s 
a ::: 
-(oSx + )1) 
B ::: 0 - fl Sx 
Q SCAL/SCALZ 
Y ::: n - ~Sx - QSz 
X ::: X'Xl- - + y'XI- -s xx s xy 
'¥ =: 
-y'XI -- + X'XI--s xx s xy 
It SxXI - - + Xl --xx xy 
L SxXI -- + Xl- -xy xx 
U uo + VlJ + w~ 
and V ua + vB +wy 
Then the coefficients of equation (6) can be written as 
A 
B 
C Q2 (a 2 - w2) 
D (oa + llS + ~y )a2 - uv 
E Uw) 
F Vw) 
+ 2vw'¥ 
and R2 =: -SCAL(a 2 - w2) (x "a + y"B) + U2S + w2Q2S + 2UwQS + (u 2 - v 2)1t + 2UV L 
s s xx zz xz 
- 2uw(S X + '¥ ) - 2vw(S '¥ - X) - a 2 [(o2 + )12 + ~2)S + Q2S + 2~QS ] 
x x xx zz xz 
+ (a2 - w2) [(y'X - x''¥) - S (x' X + y''¥)] 
s s x s s 
3 
In the original version of the FL022 code, the real variable SCAL was omit ted 
in the expression for Rl and R2 , which is equivalent to setting this variable 
to unit y . This did not have any effect for straight leading-edge geometries, but 
caused err ors for non-straight-leading-edges, or those with curvature. 
RESULTS 
The f l ow over the ONERA cranked wing (fig. 1) was calculated at a Mach number 
of 0 . 85 . The ref lection-plate model had an NACA 0012 airfoil and was set a t a n 
angle of at t ack of zero degrees. Comparisons w'ith the ONERA experimental data and the 
or i ginal FL022 results are shown in figures 2-6. The correction improves th e corre-
la t ion significantly near the leading-edge crank (station #3). 
CONCLUSIONS 
The widely used transonic swept-wing computer code, FL022, wa s found t o have an 
error in the transformed-flow equation in the computational domain. The revi sed 
version of the code correctly ac counted for the discontinuous leading-edge geometry 
and its effec t on the pressure distribution. 
REFERENCES 
1 . Jameson, A.; and Caughey, D.: Numerical Calculation of the Transonic Flow Past 
a Swept Wing . ERDA Research and Development Report COO-3077-l40, Courant 
Institute of Mathematical Sciences, New York University, New York, Jun e 1 977. 
2 . Phil ippe , J .; a nd Cha ttot , J .: Experimental and Theoretical Studies on Heli -
copter Blade Tips a t ONERA. Presented at the 6th European Rotorcraft a nd 
Powered Lift Air craf t Fo rum, 1980. 
4 
--------_. - -_. 
r---· 
ONERA WING ON WALL 
FMACH = 0.85 ALPHA = 0 
-, 
C 1 
1 /. 2.1 C )0 1 
I· 3.0 C 
Figure 1 .- The ONERA cranked wing on wall . 
5 
-1.2 
- .8 
-.4 
.4 
.8 
1.2 
o ORIGINAL 
- REVISED 
o ONERA DATA 
c __ ----=-=====--
STATION = 1 
MACH = 0.8500 
CL = 0 
AL=O 
CD = 0.0188 CM = 0 
Figure 2. - Pressure distribution at span station 1 . 
6 
----- _ . - ----- --
--- ---._--- -
-1.2 
-.8 
- .4 
o 
.4 
.8 
1.2 
o ORIGINAL 
REVISED 
o ONERA DATA 
c ___ ---=========-
STATION = 2 
MACH = 0.8500 
CL = 0 
AL = O 
CD = 0.0118 CM = 0 
Figure 3.- Pressure distribution at span station 2. 
7 
-1.2 
-.8 
-.4 
o 
.4 
.8 
1.2 
o ORIGINAL 
REVISED 
o ONERA DATA 
o 
o 
c __ -----========-
STATION = 3 
MACH = 0.8500 AL = 0 
CL = 0 CD = 0.0047 CM = 0 
Figure 4 .- Pressure distribution a t span sta tion 3. 
8 
'. 
-1.2 
-.8 
-.4 
o 
.4 
.8 
1.2 
o ORIGINAL 
REVISED 
o ONERA DATA 
c ___ ---~ 
STATION = 4 
MACH = 0.8500 AL = 0 
CL=O CD =-0.0045 CM=O 
Figure 5. - Pressure distribution at span station 4. 
9 
1 
I 
-1.2 
-.8 
-.4 
o 
.4 
.8 
1.2 
o ORIGINAL 
REVISED 
o ONERA DATA 
C ========------
STATION = 5 
MACH = 0.8500 
CL = 0 
AL=O 
CD = -0.0222 CM = 0 
Fi gure 6 . - Pressure distribution at span sta tion 5 . 
10 
'. 
' . 
1. Report No. 2. Government Accession No. 3 . 
. 
Recipient's Catalog No. 
NASA TM-84391 
4. Title and Subtitle 5. Report Date 
NUMERICAL CALCULATION OF THE TRANSONIC POTENTIAL September 1983 
FLOW PAST A CRANKED WING 6. Performing Organization Code 
7. Author(s) 8. Performing Organization Report No. 
: I-Chung Chang and Michael Tauber A- 9452 
10. Wor k Unit No. 
9. PeriorfTling Organi zation Name and Addre~s T- 3334Y 
Ames Research Center 11 . Contract or Grant No . 
Moffett Field, California 94035 
13. Type of Report a'1d Per iod Covered 
1 ~. Sponsoring t-gerlcy Name and Addres~ 
Technical Memorandum 
National Aeronautics and Space Administration 14. Sponsoring Agency Code 
Washington, D.C. 20546 505-31- 21 
1:i. Su pp!ernentary ~OICS 
Point of Contact : I-Chung Chang, Ames Research Center, MiS 227- 8, Moffett 
Field, California 94035 (415) 965- 6396 or FTS 448- 6396 
Hi AllsI'act 
The widely used transonic swept-wing code, FL022 , was found to have 
a n error in the transformed flow equation in the computational doma in. The 
revised version of the code correctly accounted for the non-straight-leading-
edge geometry and its effect on the pressure distribution. 
17. Key Words (Su~gested by Author(s)) 103. Distribution Statement 
Transonic flow Unl imited 
Supercritical wing 
Aerodynamics 
Subject Category : 05 
In. Security Classif . (of this report) 20. Secu,ity Classif . (of this page) 21 . No . of Pages 22. Price . 
Unclassified Unclassified 13 A02 
"For sale by the National Technical Information Service, Springfield , Virginia 22161 
f\) 
w 
0') 
w 
J 



Numerical calculation of the transonic potential flow past a cranked wing

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server