Recent experience in calculating unsteady transonic flow by means of viscous-inviscid interactions with the XTRAN2L computer code is examined. The boundary layer method for attached flows is based upon the work of Rizzetta. The nonisentropic corrections of Fuglsang and Williams are also incorporated along with the viscous interaction for some cases and initial results are presented. For unsteady flows, the inverse boundary layer equations developed by Vatsa and Carter are used in a quasi-steady manner and preliminary results are presented

Howlett, James T.

English

NASA Technical Reports Server

~89-19249 
VISCOUS FLOW CALCULATIONS FOR THE 
AGARD STANDARD CONFIGURATION AIRFOILS 
WITH EXPERIMENTAL COMPARISONS 
JAMES T. HOWLETT 
UNSTEADY AERODYNAMICS BRANCH 
NASA LANGLEY RESEARCH CENTER 
PRECEDiNG PAGE BLANK NOT FILMED 
313 
https://ntrs.nasa.gov/search.jsp?R=19890009878 2020-03-20T04:14:45+00:00Z
This paper reports on recent experience in calculating unsteady transonic flows by 
means of viscous-inviscid interactions with the XTRAN2L computer code (ref. 1). The 
boundary-layer method for attached flows is based upon the work of Rizzetta (ref. 2) as 
implemented in reference 3. The non-isentropic corrections of Fuglsang and Williams (ref. 
4) have also been incorporated along with the viscous interaction for some cases and initial 
results are presented. For unsteady separated flows, the inverse boundary-layer equations 
developed by Vatsa and Carter in reference 5 are used in a quasi-steady manner and 
preliminary results are presented. Currently, efforts are underway to include the viscous 
interactions in 3-D calculations in a stripwise fashion although no results for the 3-D 
work are presented herein. 
UNSTEADY TRANSONIC VISCOUS-INVISCID INTERACTIONS 
0 ATTACHED FLOW (ARC, RIZZETTA) 
- THEORY WELL-DEVELOPED 
- EXTENSIVE APPLICATIONS 
0 NONISENTROPIC CORRECTIONS WITH VISCOUS EFFECTS 
- INITIAL THEORY 
- LIMITED APPLICATIONS 
0 SEPARATED FLOW 
- PRELIMINARY THEORY 
- LIMITED APPLICATIONS 
0 3-D FLOW 
- XTRAN3 
- CAP-TSD 
314 
The inviscid code used in this study is the XTRAN2L computer code described in reference 
1. The viscous boundary layer analysis is based upon Green's lag-entrainment equations as 
described in reference 2. For attached flow, the equations are used in the direct form: 
pressure from the inviscid analysis is specified and the equations are integrated to obtain 
the boundary- layer displacement thickness 6'. For separated flows, the equations are 
inverted as described in reference 5 and the mass flow m is specified as input. In the 
inverse method, the boundary- layer displacement thickness 6' is updated using Carter's 
method (ref. 6). For both the direct and the inverse method, the effect of the viscous 
boundary layer is included in the inviscid analysis by means of the airfoil surface tangency 
boundary condition. 
BOUNDARY LAYER ANALYSIS 
DlRECT 
n n  = H  0 g'n+l 
lNVERSE 
X 
U *n *n v 
g*"+l = 6  + a 6  (- - 1) (CARTER) 
* 
$ = F  + F + 6  Y x t x  
315 
Non-isentropic modifications to the transonic small disturbance (TSD) equation were 
developed by Fuglsang and Williams in reference 4. These modifications include a 
streamwise flux that satisfies the Prandtl relations at shock jumps, convection of shock 
generated entropy in the wake, and an entropy correction in the pressure coefficient. The 
non-isentropic modifications have been incorporated into the computer code along with the 
viscous-inviscid interactions and some initial calculations are presented. 
NONISENTROPIC MODIFICATIONS TO TSD EQUATION 
(FUGLSANG AND WILLIAMS) 
0 MODIFIED STREAMWISE FLUX SATISWING PRANDTL RELATION AT SHOCK JUMPS 
0 CONVECTION OF SHOCK GENERATED ENTROPY IN WAKE 
0 ENTROPY CORRECTION IN PRESSURE COEFFICIENT 
316 
This figure illustrates one of the numerical difficulities encountered with the 
interacting boundary-layer calculation. The original version of the computer code included 
an upwind switch in the evaluation of the pressure gradient term $ X X  for input to the 
boundary-layer equations. This upwind switch introduced a discontinuity in the unsteady 
forces when the shock moved across a grid point. The dashed line in the figure shows this 
discontinuity in the moment coefficient for a typical case. The purpose of upwind switching 
in computational fluid dynamics is to account properly for the domain of dependence in the 
numerical solution of partial differential equations. However, the present application 
merely requires numerically computing the derivative of a known function. Hence, upwind 
switching is not required. The solid line in the figure shows that the moment coefficient 
varies smoothly in time when upwind switching is not used. For all results presented in 
this paper, upwind switching is not used in the boundary-layer calculation. However, the 
inviscid solution algorithm does use upwind switching in the standard manner. 
EFFECT OF UPWIND SWITCH IN 
BOUNDARY LAYER CALCULATION 
NACA 64A010A M 0.796 
0.004 ' Upwind switch in 
boundary layer -.- 
-Not used 
I I I 
1 .o 
Nondimensional time 
0 
0 
317 
This figure shows steady pressure distributions for the NACA 64A006 airfoil for a range 
of Mach numbers. For the subsonic cases, the viscous and inviscid results are nearly 
identical except for small differences near the trailing edge where the viscous results more 
closely match the experiment. At M = 0.850, a shock wave develops near midchord and the 
viscous calculation agrees much better with the experiment than the inviscid result. For M 
= 0.875, the shock strengthens and moves aft. In this case, both the viscous and inviscid 
calculations exhibit differences from the experiment although the viscous result is closer to 
the experiment in the vicinity of the shock. Downstream of the shock the viscous result is 
in good agreement with the experiment. 
STEADY PRESSURE DISTRIBUTION FOR 
NACA 64A006 AIRFOIL 
M = 0.800 
-.4y I I I I I 
0 .2 .4 .6 .a 1.0 
M = 0.850 
.I 
.6 
. 4  
-C, 
.2 
C 
- .2 
- . 4  
Viscous 
Inviscid 
Experimental upper 
0 Experimental lower 
--- 
M = 0.825 
I I I I I 
.2 .4 .6 .a 1.0 
M = 0.875 
.4 .4 
CP' 
c P. -C, - - - - - - - - - - p  b P  
.2 .2 
0 0 
-.2 - .2 
-.4 - .4 
0 .2 .4 .6 .B 1.0 0 .2 .* .6 .8 1.0 
x/c x/c 
318 
ORlGlNAL PAGE IS 
OF POOR QUALITY 
This figure shows plots of the boundary-layer displacement thickness for several mach 
numbers for steady calculations for the NACA 64A006 airfoil. For the two lowest values of 
Mach number, the results are subcritical and the displacement thickness increases 
smoothly in the downstream direction as Mach number is increased. At M = 0.850, the 
displacement thickness shows a slight increase due to the shock wave near midchord. For M 
= 0.875, the calculated result has a strong shock near 60% chord and the displacement 
thickness increases significantly across this shock. A further increase in Mach number to 
0.960 moves the shock off the trailing edge and the displacement thickness increases slowly 
as the trailing edge is approached. 
EFFECT OF MACH NUMBER ON DISPLACEMENT THICKNESS 
FOR NACA 64A006 AIRFOIL 
M = 0.800 
M = 0.825 
M = 0.850 
M = 0.875 
M = 0.960 
_ _ _ _  - _ - -  - -  
-- 
-__-- 
9 
;p-3 9 12- 
I 
i 
I 
I 
1D - 
I 
8- 
I 
3 i .il .6 0 .L 
xlc 
319 
Unsteady pressure distributions are plotted for the NACA 64A006 airfoil with an 
oscillating flap for M = 0.850and k = 0.242. The calculated mean pressures are similar to 
the steady pressure distributions for this Mach number. The effect of the viscous boundary 
layer can be seen in the results for the upper surface pressure distribution in the lower left 
hand side of the figure. The viscous unsteady pressure distributions agree very well with 
the experimental results, especially in the vicinity of the shock where the inviscid 
calculation shows the largest discrepancy. The interacting viscous boundary layer gives a 
substantial improvement over inviscid calculations in predicting the unsteady pressure 
distributions for this airfoil. 
UNSTEADY PRESSURE DISTRIBUTION FOR NACA 64A006 AIRFOIL 
M = 0.850 K = 0.242 
Viscous 
Inviscid ---- Viscous 50 r 
0 Experimental real 
a Experimental imag 
- - - -inviscid I 
0 Experimental upper 
0 Experimental lower 
ME AN '\ \ 
-.4 0 1 .2 .4 .6 .a 1.0 
Viscous 
Inviscid 
0 Experiment01 real 
0 Experimental imag ::t 10 ---- /r 
0 
-lot 
10 
AEP 
0 
-1  0 
- 2 0 1  , , ' J ,  LIFT,ING , 
- 30 
.a 1.0 0 .2 .4 .6 
Viscous 
Inviscid ---- 
0 Exper im en tat reo1 
0 Experiman tal imog 
10 
0u 
CP 
0 
-1 0 
LOWER 
0 .- .4 I A I .a I 1.0 1 -20Ll  30 0 -2 .I .6 .& 1.0 
UPPER 
- 5 O L  I 
x/c x/c  
-20 t 
320 
a 
ORIGINAL PAGE 1s 
’ OF POOR QUALITY 
This figure shows plots of the unsteady lift as a function of mach number for the NACA 
64A006 airfoil for a reduced frequency of 0.060. The results indicate that the ViSCOUS 
boundary layer corrects up to 25% of the differences between the inviscid results and the 
experiments for the lower values of Mach number. The linear theory results, also shown O n  
the figure, are competitive with the CFD calculations for predicting the unsteady lift for 
most of the cases investigated for this airfoil. 
COMPARISON OF UNSTEADY LIFT FOR 
NACA 64A006 AIRFOIL 
K = 0.060 
0 EXPERIMENT 
0 XTRAN2L 
0. PRESENT METHOD 
-- - 0 LINEAR - -_ - 
01 
I 
REAL PART 
-3 
-4 
-5 
‘\ \, 
\ -  - 
\ 
- . T- - IMAGINARY PART - 
\ 
\ 
‘0 
-6 I I I I I I 
0.78 0.80 0.82 0.84 0.86 0.88 
MACH NUMBER 
0 
0.78 0.80 0.82 0.84 0.86 0.88 
MACH NUMBER 
321 
322 
The steady pressure distributions for the MBB-A3 supercritical airfoil at the 
supercritical design point (M = 0.765, a0 = 1.50) are plotted in this figure. The calculated 
cases are for the actual experimental values of Mach number and angle of attack rather than 
values adjusted to match flow conditions in the wind tunnel as is frequently done in 
comparison with this particular data. The experiment shows supercritical flow without a 
discernable shock wave typical of flow at the design point. The region of supercritical flow 
terminates at about x/c = 0.53. The viscous calculation indicates a moderate strength shock 
wave at nearly the same location. Away from the shock, agreement between the viscous 
calculation and the experiment is very good, although some discrepancies are noted near the 
leading edge on the lower surface. For this case, the inclusion of viscous effects yields a 
significant improvement in the calculation of the steady pressure distribution. 
STEADY PRESSURE DISTRIBUTION 
FOR MBB-A3 AIRFOIL 
M = 0.765 QO * 1.5' 
Viscous 
-Inviscid --- 
0 Experimental  upper 
0 Experimental  lower 
-1.01 I 1 I 1 I 
0 .2 . 4  .6 .B 1 .o 
x/c 
ORIGINAL PAGE IS 
OF POOR QUALlrV 
Steady pressure distributions for the NACA 64A010A (Ames) airfoil at M = 0.796 and 
a0 = 00 have been calculated with the inviscid code, the viscous interaction theory, and non- 
isentropic corrections to the viscous interaction results. As the figure shows, the viscous 
calculation agrees better with the experimental results in so far as shock location and 
strength is concerned. The non-isentropic corrections move the shock position downstream 
about 1% chord and increase the shock strength slightly. In general however, differences 
between the experiment and all three calculated results are small. 
STEADY PRESSURE DISTRIBUTION FOR NACA 64A010 AIRFOIL 
M 0.796 
WITH NON-ISENTROPIC CORRECTIONS 
1.6 
1.2 
.e 
a c: . 4  
C 
- . 4  
- . E  
En t ropy+l / i scous  
Viscous 
inviscid 
C, s t a r  
_ _ - - _ _ _ _ - _  
-- 
0 Experimental 
I 
.2 .4 .6 .8 I .o 
w c  
323 
This figure shows plots of the unsteady pressure distributions for the NACA 64A010A 
(Ames) airfoil oscillating in pitch at M = 0.796 and k = 0.025. In contrast to the small 
effect the non-isentropic corrections have on the steady pressure distributions, these 
corrections to the viscous interaction theory give substantially better agreement with the 
unsteady experimental results. In the vicinity of the shock wave, the modified theory 
matches the experimental points very well, whereas the inviscid calculation is quite 
different. The significant improvement of calculated unsteady pressure distributions for 
this case due to the inclusion of non-isentropic and viscous effects is particularly 
interesting because the shock wave has only moderate strength. 
U N ST E A D Y P R E SS U R E D I ST R I B UT IO N WITH N 0 N - I S E NT R 0 PIC 
CORRECTIONS FOR NACA 64A010 AIRFOIL 
M 0.796 K = 0.025 
Viscous + ENTROPY 
- - - -Inviscid 
Viscous + ENTROPY 
inviscid 
0 Experimental  r e a l  
0 Experimental  h a g  
---- 
20 
0 
-10 
- 2 0  
40 
Viscous + ENTROPY 
Inviscid 
Experimental  r e a l  
Experimental  im ag 
20 
A?, 
10 
0 
- 1  0 
-201 I I I I 1 
Viscoils + ENTROPY 
Inviscid 
0 Experimental  r e a l  40r C Experimentai  imag ---- 
324 
ORIGINAL PAGE IS 
OF POOR QUALITY 
This figure presents comparisons of unsteady lift for calculated and experimental 
results for the NACA 64A010A (Ames) airfoil oscillating in pitch. The non-isentropic and 
viscous corrections give substantial improvements in the calculated values for low values of 
reduced frequency. This is especially evident in the imaginary part of the unsteady lift. 
Neither the inviscid nor viscous calculations predict the upward trend of the experimental 
results for low values of reduced frequency. The non-isentropic and viscous calculations 
show this low frequency upward trend very well, although some discrepancies are evident 
in the mid-frequency range. This significant improvement in lift predictions for low 
frequency cases re-emphasizes the importance of non-isentropic corrections even when the 
flow field does not exhibit strong shock waves. 
-5  
C 
'a 
I I I I I 
COMPARISON OF UNSTEADY LIFT WITH NON-ISENTROPIC 
CORRECTIONS FOR NACA 64A010 AIRFOIL 
M = 0.796 
Experiment 
Inviscid 
Viscous 
Ent ropytviscous 
- - - - - - - - - - - 
- 
16- 
1v - 
I I I I I 
I 2 3 Y 
- Experiment 
- - - - - - - - - - - Inviscid 
Viscous 
Entropytviscous - 'r 
" 1  'pa 0 I 
I , /' Imaginary part 
-'r#' 
325 
The inverse boundary-layer code has been used to calculate several test cases in which 
the flow is separated, or close to separation, in order to demonstrate the capabilities of the 
method. This figure shows plots of the steady pressure distributions for the NLR 7301 
airfoil at M = 0.70 and a0 = 30. Note the mean angle of attack is the actual experimental 
angle of 30 and not the corrected value of 20 which is specified in the AGARD conditions for 
this case. As the figure shows, the inverse boundary-layer code predicts a pressure 
distribution which agrees reasonably well with the experiment for this very difficult case. 
The calculated shock wave is about 5% chord upstream of the experiment and slightly 
weaker. The calculation also indicates a small region of separation at the trailing edge 
whereas the experimental pressures show no evidence of trailing edge separation. 
STEADY PRESSURE DISTRIBUTION CALCULATED WITH INVERSE BL CODE 
FOR NLR 7301 AIRFOIL 
M = 0.70 a0 30 
Viscous upper 
Viscous lower 
Experimental upper 
0. Experimental lower 
- -  
-cP 
326 
-0.4 
0 1 .o 
x/c 
This figure shows unsteady pressure distributions for the NACA 001 2 airfoil oscillating 
in pitch at M = 0.599. The mean angle of attack is 4.860 and the reduced frequency is k = 
0.081. For the viscous calculation with the inverse boundary -layer method, transition is 
specified to be at 20% chord. The mean pressure distributions show the calculated results 
underestimating the suction peak near the leading edge with the inviscid calculation being 
slightly closer to the experiment in this region. Over the rest of the airfoil, both viscous 
and inviscid calculations agree well with the experiment. The unsteady pressure 
distributions on the airfoil upper surface are well predicted by both viscous and inviscid 
calculations with the viscous shock location slightly upstream of the inviscid result. The 
viscous calculation indicates that the flow is very close to separation near the maximum 
angle of attack of 7.30. In fact, as shown in the next figure, a small change in the specified 
position of transition for the viscous calculation can result in flow separation during part of 
the oscillation cycle. 
UNSTEADY PRESSURE DISTRIBUTION CALCULATED WITH INVERSE BL CODE 
FOR NACA 0012 AIRFOIL 
M 0.599 a0 = 4.860 a1 2.440 K 0.081 
-CP 
--- Inviscid 
Experimental upper 
Experimental lower 
CP' - _ - - - - _ _ _ _ _ - _ - _ _  
-1.oM I I I I 1 
* o r  
Viscous 
Inviscid 
0 Experimental real  
20 'O[ ---- Experimental imag 
4 
Viscous 
Inviscid ---- 
A tP 
10 
c"P 
Lifting 
-20 lOL 
Viscous 
Inviscid ---- 
0 Experimental real  
20 0 Experimental im ag 
10 
Lower 
-20  ' O L  0 .2 .4 .(I .a 1.0 
x/c 
327 
This figure shows plots of the unsteady lift and moment coefficients for the NACA 0012 
airfoil as calculated by the inverse boundary-layer method with transition specified at 10% 
chord. With transition at this location, which is upstream of the shock wave, the boundary- 
layer displacement thickness increases significantly across the shock wave. This increased 
displacement thickness causes the flow to separate just after maximum lift. The lift 
coefficient plotted in the figure clearly shows the sudden decrease in lift associated with 
flow separation and the corresponding increase in lift upon reattachment. After the flow 
reattaches, the lift coefficient continues to vary smoothly throughout the rest of the cycle. 
The inverse boundary - layer method successfully captures this flow separation and 
reattachment without difficulty. 
UNSTEADY LIFT CALCULATED WITH INVERSE BL CODE 
FOR NACA 0012 AIRFOIL 
M = 0.599 a0 = 4.860 ai = 2.440 K = 0.81 
This paper has presented comparisons of calculated and experimental results for 
unsteady transonic flows over airfoils. The calculations include results obtained from a 
viscous-inviscid interaction method based upon the 2-D XTRAN2L inviscid transonic 
computer code. Non-isentropic corrections have been included in some cases. The viscous 
boundary-layer equations have been solved in the direct mode for attached flow and an 
inverse mode for separated flow. The results have demonstrated that pressures and forces 
calculated by the viscous-inviscid interaction method compare well with experimental 
results for steady and unsteady attached flows. The non-isentropic corrections with the 
viscous interaction method provided improved comparisons with experiments for unsteady 
low frequency oscillations in cases involving moderate strength shock waves. Initial 
applications of the inverse boundary- layer method have demonstrated that this method can 
calculate unsteady flow fields involving flow separation and reattachment. The results 
indicate that the viscous-inviscid interaction method can provide accurate predictions of 
viscous effects in unsteady transonic flow fields. 
CONCLUSIONS 
0 PRESSURES AND FORCES CALCULATED BY VISCOUS-INVISCID INTERACTION 
COMPARE WELL WITH EXPERIMENTS FOR STEADY AND UNSTEADY ATACHED FLOW 
0 NON-ISENTROPIC CORRECTIONS WITH VISCOUS INTERACTION YIELD IMPROVED 
COMPARISONS WITH EXPERIMENTS FOR UNSTEADY LOW FREQUENCY OSCILLATIONS 
0 INVERSE BOUNDARY LAYER METHOD HAS POTENTIAL FOR CALCULATING UNSTEADY 
FLOW SEPARATION AND REATTACHMENT 
329 
REFERENCES 
1. Whitlow, Woodrow, Jr.: XTRAN2L: A Program for Solving the General-Frequency 
Unsteady Transonic Small Disturbance Equation. NASA TM 85723, November 1983. 
2. Rizzetta, Donald P.: Procedures for the Computation of Unsteady Transonic Flows 
Including Viscous Effects. NASA CR 166249, January 1982. 
3. Howlett, James T.: Efficient Self-consistent Viscous-Inviscid Solutions for Unsteady 
Transonic Flow. AIAA Paper No. 85-0482, 23rd Aerospace Sciences Meeting, Reno, Nevada, 
January 14-1 7, 1985. 
4. Fuglsang, Dennis F; and Williams, Marc H.: Non-Isentropic Unsteady Transonic Small 
AlAA Paper No. 85-0600, AIAA/ASME/ASCE/AHS 26th Structures, Disturbance Theory. 
Structural Dynamics and Materials Conference, Orlando, Florida, April 1985. 
5. Vatsa, V. N.; and Carter, J. E.: Development of an Integral Boundary-Layer Technique 
for Separated Turbulent Flow. United Technologies Research Center, Report No. UTRC 81 -28, 
April 1981. 
6. Carter, James E.: A New Boundary-Layer Inviscid Iteration Technique for Separated 
Flow. AlAA Computational Fluid Dynamics Conference, Williamsburg, VA, July 23-25, 1979. 
330 


Viscous flow calculations for the AGARD standard configuration airfoils with experimental comparisons

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