A finite difference technique is used to solve the transonic small disturbance flow equation making use of shock capturing to treat wave discontinuities. Thus the nonlinear effects of thickness and angle of attack are considered. Such an approach is made feasible by the development of a new code called CAP-TSD (Computational Aeroelasticity Program - Transonic Small Disturbance), and is based on a fully implicit approximate factorization (AF) finite difference method to solve the time dependent transonic small disturbance equation. The application of the CAP-TSD code to the calculation of low to moderate supersonic steady and unsteady flows is presented. In particular, comparisons with exact linear theory solutions are made for steady and unsteady cases to evaluate shock capturing and other features of the current method. In addition, steady solutions obtained from an Euler code are used to evaluate the small disturbance aspects of the code. Steady and unsteady pressure comparisons are made with measurements for an F-15 wing model and for the RAE tailplane model

Batina, John T.

Bennett, Robert M.

Bland, Samuel R.

Gibbons, Michael D.

Mabey, Dennis G.

English

NASA Technical Reports Server

CALCULATION OF STEADY AND UNSTEADY PRESSURES AT 
SUPERSONIC SPEEDS WITH CAP-TSD 
Robert M. Bennett, Samuel R. Bland, 
and John T. Batina 
Unsteady Aerodynamics Branch 
NASA Langley Research Center 
Hampton, Virginia 
Michael D. Gibbons 
Planning Research Corp. 
Hampton, Virginia 
and 
Dennis G. Mabey 
Bedford, England 
Ro ya I Aircraft Est ab I ish m en t 
117 
https://ntrs.nasa.gov/search.jsp?R=19890009869 2020-03-20T04:16:18+00:00Z
INTRODUCTION 
Currently there is considerable interest in the development of methods for calculating 
supersonic unsteady aerodynamics for application to advanced configurations. Although flutter 
is oftentimes critical at transonic speeds, critical conditions can also be encountered at both low 
and high supersonic speeds depending upon configuration and operating envelopes. Linear theory 
has been the primary analytical tool for analyzing flutter in the low supersonic region with 
recent efforts directed towards refining the potential gradient methods. In linear theory the 
loading on a wing is determined by the relationship of the Mach lines to the planform which can 
lead to complex logic for configurations. In this study a finite-difference technique is used to 
solve the transonic small-disturbance flow equation making use of shock-capturing to treat 
wave discontinuities. Thus the nonlinear effects of thickness and angle of attack are considered. 
Such an approach is made feasible by the development of an efficient new code by the Unsteady 
Aerodynamics Branch of the NASA Langley Research Center. The new code is called CAP-TSD for 
Computational Aeroelastlcity Program - Transonic Small Disturbance and is based on a fully 
lmpllclt approximate-factorizatlon (AF) finite-dlfference method to solve the time-dependent 
transonlc small-dlsturbance equation and has been descrlbed by Batina' In an earlier 
presentation at this workshop. This paper presents the appllcatlon of the CAP-TSD code to the 
caiculatlon of low to moderate supersonlc steady and unsteady flows. in partlcular, comparisons 
wlth exact llnear theory solutlons are made for steady and unsteady cases to evaluate the shock 
capturing and other features of the current method. in addition, steady solutlons obtained from 
an Euler code are used to evaluate the small disturbance aspects of the code. Steady and unsteady 
pressure comparisons are made with measurements for an F-5 wing model and for the RAE 
tailplane model. (Fig. 1 .). 
+Batina et al., NASA CP- 3022,, 1989, Paper No. 4. 
INTRODUCTION 
0 Significant interest in unsteady supersonic aerodynamics 
Configurations can be flutter critical at supersonic speeds 
New configurations are under development 
Linear theory currently used 
Mach box, potential gradient, etc 
Treats mach lines and mach cones explicitly = complex logic 
for configurations 
0 Present study applies CAP-TSD to wings 
0 Computational Aeroelasticity Program - Lransonic Small Disturbance 
Previous presentation'described application to configurations 
Comparisons presented: Linear theory, F-5 model, and RAE tailplane 
model +Pitt , this CP 
Figure 1 
CAP-TSD 
As previously indicated, CAP-TSD is based on a fully implicit approximate-factorization (AF) 
algorithm. The program solves the modified transonic small disturbance (TSD) equation 
including the +tt term. The program has been developed for configurations and is highly 
vectorized. A TSD code called XTRAN3S has been previously developed and used extensively: 
however, the partially-explicit alternating direction implicit algorithm used in XTRAN3S was 
unstable for swept and tapered wings at supersonic speeds. The AF algorithm used in CAP-TSD 
is stable for such cases. The finite-difference method treats discrete waves by shock capturing. 
For all the cases presented here, a finite-difference grid was used that consisted of 90 x 30 x 
60 points in the x-y-z directions giving a total of 162,000 grid points. The grid extended 10 
root chord lengths ahead and aft of the wing, nearly 13 chord lengths above and below, and one 
sernispan outboard of the tip. On the wihg, 50 points were used along each chord and 20 points 
along the span. This grid is one that would be used for a subsonic freestream and extends 
further than may be necessary for the supersonic cases. The outer boundary conditions for the 
cases considered herein are the "reflecting" outer boundary conditions, but care has been taken 
to ensure that the wing is located within the "Mach diamond" such that waves do not reflect from 
the outer boundaries back onto the wing surface. For the oscillating wings the calculations were 
made at 360 steps per cycle which corresponds to a time step of the order of 0.1. Only two 
cycles of motion were calculated because the flow field converges rapidly for supersonic flow. 
The second cycle was used for Fourier analysis. One Newton iteration per step was used for flow 
field convergence. (Fig. 2.) 
CAP - TSD 
0 Fu I I y i m p I i ci t ap p r ox i m ate- f acto r iz at i o n (A F) algor i t h m 
Solves general frequency modified transonic small disturbance (TSD) 
equation 
Developed for configurations 
0 Vectorized 
Stable for supersonic flow 
0 Treats discontinuous waves by shock-capturing 
0 Reflecting outer boundary conditions and large grid extent used 
for supersonic flow 
0 Fine grid: 90x30~60 (x-y-z) 162,000 points 
Figure 2 
119 
COMPARISON OF CAP-TSD AND EXACT LINEAR THEORY 
RECTANGULAR WING, AR = 4, M = 1.30 
6 -  
The first comparison with linear theory is for tip loading on a wing in steady flow. The loading 
within the tip Mach line for a wing with a supersonic leading edge has an exact conical flow 
solution. A typical result is shown in figure 3 and is compared with a corresponding CAP-TSD 
calculation for a rectangular wing. For this case the loading has a discontinuous slope at the 
Mach line which is smeared by the finite-difference scheme but the overall trend of the loading 
is reproduced. 
A c P  
a 
-
2 -  
-- 
- 1 = 0.77 
~ 
0 0.2 0.4 0.6 0.8 1.0 
xic 
Figure 3 
120 
COMPARISON OF CAP-TSD AND EXACT LINEAR THEORY 
L.E. SWEEP = 40°, TAPER RATIO = 0.5, M = 1.12 
The next comparison is for a more complex situation. For a wing with a subsonic leading edge, 
the loading at the tip Mach line has a jump. A loading of this type is shown in figure 4 for a 
hexagonal wing planform of 40" leading-edge sweep, taper ratio of 0.50, and at M = 1.1 2. The 
over-all trend of the loading again is given by CAP-TSD but there is considerable smearing of 
the jump discontinuity. For this severe test case further development of the finite differencing 
is desirable, but the agreement of the overall trend is encouraging. 
Exact 
- 1 = 0.77 
I I I 
0 .2 .4 .6 .8 1.0 
xlc 
Figure 4 
121 
COMPARISON OF CAP-TSD AND EXACT LINEAR THEORY 
2-D FLAT PLATE PITCHING ABOUT 41.3% C, M = 1.30 
For unsteady flow, exact solutions for a two-dimensional flat plate airfoil are given in NACA 
Report 846 by Garrick and Rubinow (1946). Solutions for a rectangular wing of aspect-ratio- 
4 oscillating in pitch at M = 1.30 about the 41.3% chord are presented in NACA TN 3076 by 
Nelson, et ai. (1954). Several cases have been run with CAP-TSD for the rectangular wing. 
Since the inboard portion of the wing has two dimensional flow, the inboard solutions can be 
compared to the 2-0 results of Garrick. and Rubinow. . The comparison for unsteady lift is shown 
in figure 5. The agreement with the exact solution up to k = 1.0 is excellent. 
0 
C 
'a 2 - 
-2 O 3  I mag inar y 
0 .2 .4 .6 .a 1.0 
k 
Figure 5 
122 
COMPARISON OF CAP-TSD AND EXACT LINEAR THEORY 
FLAT PLATE WING, PITCHING ABOUT 41.3% C, M = 1.30, K = 0.111 
For the oscillating rectangular wing of aspect ratio 4 at M = 1.30, the spanwise distribution of 
lift magnitude and phase over the wing for K = 0.116 is shown in figure 6. Good agreement is 
evident with some slight overprediction at the tip. Note that the phase angle is shown with a 
highly expanded scale. These cases indicate that the overall loading is well reproduced by CAP- 
TSD in supersonic unsteady flows. 
5 
4 
3 
P1.1 
1 
2 
0 
- 2  
-6 
-8 
I I I I 
.2 .4 .6 .a 1.0 0 .2 .4 .6 .8 1.0 
- 
i 
0 
rl 
Figure 6 
123 
COMPARISON OF CAP-TSD AND EXPERIMENT 
F-5 MODEL, STEADY FLOW, M = 1.10, a o  = 0" 
The first comparisons with experimental data are for an F-5 wing model tested by the Dutch 
NLR. The F-5 wing model is a typical supersonic wing with a panel aspect ratio of 1.58, a 
leading edge sweep angle of 31.9" and a taper ratio of 0.28. The airfoil section is a modified 
NACA 65A004.8 which has a slightly drooped nose and is symmetric aft of 40% chord. Subsonic 
and transonic calculations for this model have been made previously by several investigators 
using TSD codes and have been in generally good agreement with the experimental data. 
Calculations with CAP-TSD are compared with steady flow data for a0 = 0" and for Mach 
numbers of 1.1 0 in figure 7. Generally good agreement is demonstrated. There is a shock on the 
lower surface near the leading edge which is swept aft slightly more than the leading edge. Some 
deviation of the results obtained with CAP-TSD from the data in this region is noted. 
.6 7 - 
j 
.4 L t, 
r 
- CAP-TSD 
.. I I .. 
;d Experiment (NLR) -.4 4 -, U 
1 I I I I I 1 -.6 
1 [ = 0.34 - 1 = 0.17 
1 = 0.94 - I l l  I l l  q = 0.69 
1 n = 0.04 1 
I 
'\ 0 
0 3  
I I -.6 0 .2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0 0 .2 .4 .6 .0 1.0 
XIC xlc XlC 
Figure 7 
I 124 
ORIGINAL PAGE IS 
OF POOR QUALITY 
COMPARISON OF CAP-TSD AND EXPERIMENT 
F-5 MODEL, STEADY FLOW, M = 1.32, a o  = 0" 
Corresponding calculations for the F-5 wing at M = 1.32 are shown in figure 8. Generally good 
agreement is also evident at this higher Mach number. The shock on the lower leading-edge 
surface that was present for M = 1.1 0 is no longer evident. 
Figure 8 
125 
COMPARISON OF CAP-TSD FL057MG (EULER), AND EXPERIMENT 
F-5 MODEL, STEADY FLOW, M = 1.10, uo = 0" 
A further comparison for M 5 1.1 0 with a steady-flow Euler code (FL057MG) is given in figure 
9. The Euler calculation uses a C-type grid which wraps around the nose of the airfoil and is 
able to resolve the leading-edge shock in more detail. The pressures over the remainder of the 
wing are in reasonable agreement taking into account the coarseness of the grid for the Euler 
calculation (approximately 24,000 points). 
- 1 = 0.49 r ,- FL057MG 
.2 
-cp 0 
0 .2 .4 .6 .8 1.0 
x Jc 
- 
1 = 0.84 
r 
0 .2 .4 .6 .8 1.0 
x Jc x Jc 
Figure 9 
126 
COMPARISON OF CAP-TSD AND EXPERIMENT 
F-5 MODEL OSCILLATING IN PITCH ABOUT 50% Cr 
M = 1.10, k = 0.116, a o  = O", a1 = 0.267' 
UPPER SURFACE 
Unsteady pressures are calculated with CAP-TSD for one reduced frequency for each of these two 
Mach numbers for the F-5 model wing. Upper and lower surface pressures are compared with 
experiment separately. The upper surface pressures for M = 1.1 0 and k = 0.1 16 are shown in 
figure 10. There is generally good agreement with the data. A typical supersonic loading is 
evident with only modest variations along the chord. 
- 
1 = 0.17 
- 
'1 = 0.69 
- 1 = 0.49 
- 1 = 0.94 
-4 I I I I 
0 .2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0 
xlc xlc 
Figure 10 
xlc 
127 
COMPARISON OF CAP-TSD AND EXPERIMENT 
F-5 MODEL OSCILLATING IN PITCH ABOUT 50Ym Cr 
LOWER SURFACE 
M = 1.10, k = 0.116, a o  = 0', a1 = 0.267' 
The lower surface pressures corresponding to the upper surface results of the previous figure 
are shown in figure 11. In the region of the lower surface shock, a large peak in unsteady 
loading is apparent which appears to be due to an embedded transonic flow region. CAP-TSD 
gives reasonable trends for this'case even for the very large unsteady loading at the tip station. 
I 
4 1  c Imaginary 
-cP - -4 . O W  
Real lo 
-12 t 
- 16 
CAP-TSD k 
t 
1 = 0.34 t 
I 
- I  '- 
-16 
0 .2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0 
XIC xlc x I C  
Figure 11 
1 128 
COMPARISON OF CAP-TSD AND EXPERIMENT 
F-5 MODEL OSCILLATING IN PITCH ABOUT 50% Cr 
UPPER SURFACE 
M = 1.32, k = 0.198, a o  = Oo, a1 = 0.222O 
Unsteady pressures for the upper surface of the F-5 model oscillating in pitch at M = 1.32 are 
shown in figure 12. Again there is generally good agreement with the data in a similar manner 
to the agreement for M = 1.1 0 for the upper surface. 
- 
8 
4 
0 
-CP 
-4 
8 
4 - 
-CP 
0 
-4 
0 .2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0 
xlc xlc XIC 
Figure 12 
129 
- 4 8[ 
COMPARISON OF CAP-TSD AND EXPERIMENT 
F-5 MODEL OSCILLATING IN PITCH ABOUT 50% Cr 
LOWER SURFACE 
M = 1.32, k = 0.198, cco = Oo, CCI = 0.222O 
The lower surface pressures for the F-5 model at M = 1.32 are presented in figure 13. The 
large peak near the leading edge that was evident at M = l.10 does not appear in these results. 
Generally good agreement with the data is again obtained. 
fl = 0.17 1 = 0.49 - 
- 
q = 0.94 I l l  I l l  q = 0.69 
t 'I = 0.84 I 
t t 
-rj-*-n--- 
0 0 0  
-CP 
-4  
0 .2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0 
xlc xlc XIC 
Figure 13 
COMPARISON OF CAP-TSD AND EXPERIMENT 
RAE TAILPLANE MODEL, STEADY FLOW, M = 1.20, a o  = 0.0" 
The RAE tailplane model was built and tested by the British RAE. The tailplane is a planform 
that is typical of a tailplane for a supersonic fighter. It has a panel aspect ratio of 1.20, a taper 
ratio of 0.27, and a leading-edge sweep angle of 50.2". The airfoil is approximately a NACA 
64A010.2 which is thicker (10.2% thick) than is typical for supersonic wings. Only upper 
surface pressures were measured for this model. 
Calculated steady results from CAP-TSD for ao = 0' and M = 1.20 are compared with measured 
pressures in figure 14. The measured pressures shown are the mean values measured at 3 Hz 
(k < 0.015) as steady pressures were not measured for these Mach numbers. The agreement is 
quite good at all five span stations. 
r .8 r I [ Experiment (RAE) t 
-cp  . 2 L  
Lower 
CAP-TSD 
-.2 t 
I -- I---- 
-.4 1 I 1 I 
?I = 0.84 A 
j T ri = fi = 0.42 
1 
-c, 
.8 r I 1  .8 
.6 
.4 
.2 
0 
-.2 
-.4 I 
__ .. 
I 
0 .2 .4 .6 .8 1.0 
I I -11 0 .2 .4 .6 .8 1.0 
x IC  
Figure 14 
xlc 
131 
COMPARISON OF CAP-TSD AND EXPERIMENT 
RAE TAILPLANE MODEL, STEADY FLOW, M = 1.71, (XO = 0.14' 
i 
.6 
Experiment (RAE) 
i I 
The calculated steady results for M = 1.32 are compared with the experimental results in 
figure 15. Here the agreement is good inboard, but some deviation is shown near the tip. 
- 
-cP 
.e 
.6 
.4 
.2 
0 
-.2 
-.4 
c 
= 0.14 
o .2 .4 .6 .a 1.0 
xlc 
i' ' I 1  1 = 0.96 -l<yyl - 
o .2 .4 .6 .8 1.0 
xlc 
Figure 15 
I 132 
COMPARISON OF CAP-TSD, FL057MG (EULER) AND EXPERIMENT 
RAE TAILPLANE MODEL, .STEADY FLOW, M I 1.71, a o  = 0.14' 
Corresponding calculations for M = 1.71 with FL057MG using the Euler equations are presented 
in figure 16. The results from CAP-TSD and the Euler results are in close agreement. The 
results from FL057MG also show the deviation from the data near the tip. 
- - - 1 = 0.14 q = 0.66 q = 0.96 
.6 
.4 
.2 
-cp 0 
-.2 
-.4 
-.6 
0 .2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0 
xlc x I C  x I C  
Figure 16 
133 
COMPARISON OF CAP-TSD AND EXPERIMENT 
RAE TAILPLANE MODEL OSCILLATING IN PITCH ABOUT 68% Cr 
M = 1.20, K = 0.346, 010 = O", 011 = .378" 
Unsteady pressures for pitching oscillations of the RAE tailplane model at M = 1.20 and 70 Hz 
are presented in figure 17. Good overall agreement for the in phase (or real) data is obtained 
but the out-of-phase components (imaginary) are under predicted somewhat. As previously 
indicated, pressures were measured only on one surface. 
.4 Experiment (RAE) 
- .2 
0 
-.2 
-% 
- . 4 1  I , , , , , 
1. 6 = 0.66 f 6 = 0.42 ?I = 0.84 A 
r r 
.4 1 
-.4  
0 .2 .4 .6 .8 1.0 
x lc  
0 .2 .4 .6 .8 1.0 
x l c  
Figure 17 
COMPARISON OF CAP-TSD AND EXPERIMENT 
RAE TAILPLANE MODEL OSCILLATING IN PITCH ABOUT 68% Cr 
M = 1.71, K = .270, a o  = .14O, = ,570' 
The unsteady results for the tailplane oscillating at 70 Hz at M = 1.71 are shown in figure 18. 
The agreement is comparable to that of M = 1.20. The unsteady loading has less of a peak near 
the leading edge for M = 1.71 than for M = 1.20. 
.4 
Experiment (RAE) 
- .2 
0 
-cP 
-.2 
-.4 ' I 1 I I I 1 1 1 1 1 1 
I fi = 0.66 fi = 0.84 1 
I I  
t ij = 0.42 
.6 
.4 
- .2 
0 
-cP 
-.2 
-.4 1-1 
o .2 .4 .6 .a 1.0 I,,,,, 0 .2 .4 .6 .8 1.0 
x l c  X I C  
Figure 18 
135 
COMPARISON OF CAP-TSD AND EXPERIMENT 
RAE TAILPLANE MODEL M = 1.71, a0 = 5.14, STEADY FLOW 
I 
Some measurements were made at 5" mean angle of attack for this model. Pressures on the 
lower surface were obtained by testing the upper surface at the corresponding negative angle of 
attack. The steady pressures for ao = 5" and M = 1.71 are compared with results from CAP- 
TSD in figure 19. Both the thickness loading and the pressure difference are reasonably well 
predicted even at this large angle of attack and Mach number. Again some deviation of the 
results from the data is evident on the upper surface ne'ar the tip. 
L ! , ,  
0 .2 .4 .6 .8 1.0 
x l c  x l c  
Figure 19 
136 
7 
CONCLUDING REMARKS 
A transonic unsteady aerodynamic and aeroelastic analysis code called CAP-TSD has been 
developed for application to realistic aircraft configurations. The new code now permits the 
calculation of unsteady flows about complete aircraft configurations for aeroelastic analysis in 
the flutter critical transonic speed range. It uses an AF algorithm that has been shown to be very 
efficient for steady or unsteady transonic flow problems including supersonic freestream flows. 
Results were presented for several wings that demonstrate the applicability of CAP-TSD for 
supersonic flows including embedded transonic flows. Comparisons with known exact analytical 
solutions from linear theory demonstrated that CAP-TSD gives reasonably good fidelity to 
approximating weak shock waves, but with smearing of strong discontinuities. Unsteady lift 
was well predicted for a two-dimensional flat plate airfoil and for an oscillating rectangular 
wing. CAP-TSD results for the thin F-5 wing were in good overall agreement with 
experimental steady and unsteady pressures and with a steady flow Euler code. One case with an 
embedded swept shock near the lower surface leading edge indicated that embedded transonic 
flows can be treated. Results for the RAE tailplane model were in good agreement with the 
measured data for both steady and unsteady cases, and with a steady calculation with an Euler 
code. Good agreement was also found for a steady flow case at a Mach number of 1.71 and five 
degrees mean angle of attack. The present study has demonstrated the applicability of the CAP- 
TSD code to flows with supersonic freestream with favorable comparisons of selected cases. 
Improvements to the shock capturing characteristics and the supersonic outer boundary 
conditions for supersonic flows are desirable. (Fig. 20.) 
CONCLUDING REMARKS 
a 
a 
0 
0 
CAP-TSD applied to wings in supersonic flow 
Results compare favorably with exact linear theory solutions 
Generally good agreement with experiment for F-5 and tailplane models 
Improvements needed in shock capturing and outer boundary conditions 
Figure 20 
137 


Calculation of steady and unsteady pressures at supersonic speeds with CAP-TSD

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