One of the most important uses of method that calculate unsteady aerodynamic loads is to predict and analyze the aeroelastic responses of flight vehicles. Currently, methods based on transonic small disturbance potential aerodynamics are the primary tools for aeroelastic analysis. Flow solutions obtained using isentropic potential theory can be highly inaccurate and even multivalued, because they do not model the effects of entropy that is produced when shock waves are in the flow field. From the results that are presented, it is concluded that nonisentropic potential methods more accurately model Euler solutions than do isentropic methods. The primary effects of modeling shock generated entropy are: (1) to eliminate mulitple flow solutions when strong shock waves are in the flow field; and (2) to bring the strengths and locations of computed shock waves into better agreement with those calculated using Euler method and those measured during experiments

Whitlow, Woodrow, Jr.

English

NASA Technical Reports Server

APPLICATION OF A FULL POTENTIAL METHOD TO AGARD 
STANDARD AIRFOILS 
WOODROW WHITLOW, JR. 
UNSTEADY AERODYNAMICS BRANCH 
NASA LANGLEY RESEARCH CENTER 
PRECEDkNG PAGE BLANK NOT FLMED 
157 
https://ntrs.nasa.gov/search.jsp?R=19890009871 2020-03-20T04:16:14+00:00Z
OUTLINE 
An outline of the presentation is shown here. First, the motivation for performing this 
research is discussed. Next, the formulation of an isentropic full potential method is presented, 
followed by a nonisentropic method. Since the methods that are presented use body-fitted grids, 
methods for modeling the motions of dynamic grids are presented. Computed results for the 
NACA 0012 and NLR 7301 airfoils are shown. Summary statements about this effort are 
presented, and some conclusions are made. 
0 MOTIVATION 
0 ISENTROPIC FULL POTENTIAL 
0 NONEENTROPIC FULL POTENTIAL 
0 DYNAMIC GRIDS 
0 RESULTS (NACA 0012, NLR 7301) 
0 SUMMARY 
0 CONCLUSIONS 
158 
MOTIVATION 
One of the most important uses of methods that calculate unsteady aerodynamic loads is to 
predict and analyze the aeroelastic responses of flight vehicles. Currently, methods based on 
transonic small disturbance (TSD) potential aerodynamics are the the primary tools for 
aeroelastic analysis. Theoretically, TSD methods are limited to thin bodies undergoing small 
amplitude motions. Full potential (FP) methods do not have these limitations, but flow 
solutions obtained using isentropic potential theory--TSD or FP--can be highly inaccurate and 
even multivalued. This is because isentropic potential methods do not model the effects of 
entropy that is produced when shock waves are in the flow field. Thus, the goal of this effort is 
to develop an unsteady full potential method that models the effects of shock-generated entropy: 
0 TRANSONIC SMALL DISTURBANCE THEORY IS THE PRIMARY 
AERODYNAMIC TOOL FOR ANALYZING TRANSONIC AEROEIASTIC 
PHENOMENA 
0 TSD LIMITED TO THIN BODIES UNDERGOING SMALL AMPLITUDE 
MOTIONS 
0 SOLUTIONS FROM ISENTROPIC POTENTIAL THEORY CAN BE 
HIGHLY INACCURATE AND MULTIVALUED 
0 GOAL OF THE PRESENT EFFORT IS TO DEVELOP AN UNSTEADY 
FULL POTENTIAL METHOD THAT MODELS NONISENTROPIC 
EFFECTS 
159 
ISENTROPIC FULL POTENTIAL 
(GENERALIZED COORDINATES) 
Shown here is the formulation of the isentropic full potential method in generalized 
coordinates. The first equation is the continuity equation in strong conservation form, where 
z is computational time, and 6 and are the computational coordinate directions around and 
normal to the airfoil, respectively. Density p is given by the expression in the second equation, 
where y is  the ratio of specific heats, M is free stream Mach number, is the velocity 
potential, and the nondimensional physical time is denoted by t. The variables U and W are the 
contravariant velocities in the 6 and directions, respectively. The metrics of the body-fitted 
grid are A1 ,A2, and k. The Jacobian of the coordinate transformation is given by J, and is the 
density divided by the Jacobian. The density, biased to introduce artificial viscosity and capture 
shock waves, is denoted by7. 
6 
U = 5  +A,$  +A2@ 
t 5 
J = 5  x z  6 - 5  z x  6 
A 
P = ?  
W = <  + A $  +A,@ 
2 5  6 t 
(u 
p = biased density 
160 
ISENTROPIC FULL POTENTIAL 
(FACTORIZATION) 
The isentropic full potential formulation is linearized and factored as shown in this figure. 
Here, 4, and Ly represent differential operators in the €, and c directions. The right side of the 
factored equation is the discretized form of the continuity equation plus some other terms not 
shown. The subscripts i and j represent grid points in the €, and c directions, respectively, and 
the superscripts n and n-1 represent computational time levels. Solutions for t$ are advanced 
in time by adding the potential from the previous time step 0” to the correction A& 
L L A $ =  
5 5  
N 
] + ... i,j- 112 (74- 112,j + (pw>” i,j+ 112 - (PW) P U  
h = AT 
161 
FLUX BIASED DIFFERENCING 
The flow equations are discretized spatially using flux biased differencing. Artificial viscosity, 
necessary to capture shock waves, is introduced into the difference equations by defining the 
biased density p as shown in the figure. The (pq)- term is the difference between the actual 
flux pq and the sonic flux p'q' in supersonic regions and is zero in subsonic regions. 
Expressions for the total speed q, sonic speed q', and sonic density p' are shown in the figure. 
N 1 
0 
q 2 = A q 2 + 2 A @ @  + A q 2  
l 5  2 5 r  3 r  
1 2 -  
p* = (q* )Y-1 
162 
CHARACTERISTICS OF FLUX BIASED DIFFERENCING 
Flux biased differencing (a) accurately tracks sonic conditions and automatically specifies the 
correct amount of artificial viscosity, (b) produces no velocity overshoots at shock waves, 
allowing for larger time steps for unsteady calculations, (c) produces well defined, monotone 
shock profiles with a maximum two point transition between the upstream and downstream 
states, and (d) dissipates expansion shocks, ruling out solutions with such nonphysical 
characteristics. 
0 AUTOMATICALLY SPECIFIES CORRECT AMOUNT OF ARTIFICIAL 
VISCOSITY 
0 ALLOWS FOR LARGER TIME STEPS 
0 PRODUCES TWO POINT SHOCK WAVES 
0 DISSIPATES EXPANSION SHOCKS 
163 
NONISENTROPIC DENSITY 
To model the jump in entropy across shock waves, the density downstream of shock waves is 
modified to Pie-*S/R. The entropy jump is a function of the normal Mach number upstream of 
the shock wav'e Mn. The isentropic density is given by the expression in the figure and is the 
same as that shown in the formulation of the isentropic potential method. When the expression 
for the nonisentropic density is inserted into the continuity equation, the equation at the bottom 
of the figure is obtained. Except across shock waves and wakes, the last part of that equation 
vanishes. Thus, except at those locations, it is necessary only to solve the classical full 
potential equation. 
-AS 
R 
P = Pie 
2 
n AS = f(M ) 
164 
NONISENTROPIC FULL POTENTIAL 
(FACTORIZATION) 
When nonisentropic effects are modeled, the factored equation becomes as shown at the top of 
the figure. The nonisentropic biased density for U > 0 is given by the expression at the bottom 
of the figure. 
-AS -AS 
-AS 
165 
DYNAMIC GRIDS 
To apply the airfoil surface boundary conditions at the instantaneous surface position requires a 
new grid at each time step. For this work, the body-fitted grids were generated using an elliptic 
method. Using this method, the resources required to generate the grids can be more than those 
necessary to do the aerodynamic calculations. Thus, an efficient grid interpolation method is 
used to generate the required grids. To simulate harmonic motions, the elliptic method is used 
to calculate grids at the extreme airfoil positions, and the grids for all other airfoil positions 
are generated using linear interpolation. Interior grid points are redistributed at each time 
step, while points on the outer boundary remain fixed. 
The figure shows grid interpolation for an airfoil pitching about a point XP. A polar coordinate 
system centered at xp is used. Using the subscripts 1 and 2 to denote the minimum and 
maximum pitch angles, the position of each grid point at any time z is given by the expressions 
for r(z) and e(z). The interpolated grid points are then obtained from the expressions for x(z) 
and z(z) given in the figure. 
0 
0 OUTER BOUNDARY FIXED 
ASSUME LINEAR VARIATION BETWEEN EXTREME POSITIONS 
a(z) - a 
1 
2 1 
r(z) = rl + (r2 - rl> 
a - a  
a(z) - a 
2 1 
P ~ ( z )  = r(z)cose(z) + x 
z(z) = r(z)sine(z) 
166 
NACA 0012 
STEADY FLOW SOLUTIONS 
M = 0.84, a = 0" 
This figure shows steady pressures that are calculated on the NACA 0012 airfoil using the 
isentropic and nonisentropic full potential (FP) methods and an Euler method for M = 0.84, 
a = 0". For this airfoil, the flow conditions are in' the region where multiple solutions are 
known to occur. The isentropic FP result is asymmetric with negative lift. The Euler solution 
is symmetric with zero lift. The nonisentropic FP calculation yields a symmetric, nonlifting 
pressure distribution that shows good agreement with the Euler pressures. Thus, the 
nonisentropic FP method eliminates the phenomenon of multiple flow solutions. 
-1.2 
-.8 
-.4 
cP O 
.4 
.8 
1.2 
lsentroplc FP upper ---- Isentroplc FP lower -- Nonisentroplc FP 
Euler --- 
I 
I I I I I 
I .2  .4 .6 .8 1.0 
x f c  
167 
NACA 0012 
ISENTROPIC COMPUTATIONS 
M = 0.755, a = 0.016' + 2.5l0SIN(Kz), K = 0.814 
This figure shows comparisons of calculated isentropic transonic small disturbance and full 
potential (FP) unsteady pressures with experimental data. Generally, the FP pressures are in 
good agreement with experiment, but the shock wave is too strong and located too far aft on the 
ai rfo il . 
-1.2 
- . 8  
- . 4  
CP 0 
.4 
.0  
1.2- 
0 . 2  .4 .6 .8 1.0 
. I C  
-1.2 
- . 8  
-.4 
CP O 
.4 
. B  
1.2 
0 . 2  . 4  .8 . 8  1.0 
r l c  
0 Experlmsnt upper 
0 Experiment lower  - FP upper ---- FP lower  
TSD upper 
TSD lower 
-- 
-1 .2 
- .8  
- .4  
CP O 
.4 
0 .2 . 4  .6 .0 1.0 
1.2 
r l c  
-1 .2 
- .8  
-.4 
CP O L . 4  
' 1 .2  8L.o 
X I C  
I 168 
ORIGf?JAL PAGE 1s 
OF POOR Q U A L m  NACA 0012 
NONEENTROPIC COMPUTATIONS 
M = 0.755, 01 = 0.016' + 2.5IoSIN(Kz), K = 0.0814 
This figure shows comparisons of isentropic and nonisentropic full potential (FP) pressures 
with experimental data. Generally, the effects of modeling the shock-generated entropy are to 
weaken the shock wave and move it forward on the airfoil. As a result, the nonisentropic FP 
calculations show improved agreement with the experimental data. 
r 
- 1 . 2  
-.o 
- .4  
CP 
.4 
1 .2  . O L  0 . 2  .4  .e .o  1.0 
X I C  
r 
-1 .2  
-.o 
-.4 
CD 
0 
.4  
1.2 
0 .2  .4 .o  .8 1.0 
X l C  
0 Erp*rlment upper 
0 Experiment lower - Isentropic upper 
Iaentroplc lower -- Nonlsentroplc upper 
Nonlsentroplc lower  
---- 
--- 
-1.6 
-1 .2  
-.a 
-.4 
CP 
. 4  
0 . 2  .4 .e  .o  1.0 
a I c  
- 1 . 2  
- . o  
- 4  
CP 
. 4  
.o  
1.2: .4  . o  . 8  1.0 0 . 2  
X I C  
169 
NLR 7301 
STEADY STATE PRESSURES 
M = 0.721, cx = -0.19' 
Shown In this figure are comparisons of the full potential (FP) pressures with Euler 
calculations and experimental data for ,the NLR 7301 airfoil at M = 0.721, a = -0.19". For 
this case, the effects of shock-generated entropy are small, and the two FP solutions are nearly 
identical. Thus, the isentropic and nonisentropic FP calculations are shown as one line, 
designated "FP". Results obtained using the potential methods show very good agreement with 
experimental data and with thd Euler calculations. The shock lobation is slightly upstream of 
the experimental location and slightly downstream of the location predicted by the Euler method. 
-T 0 Experiment upper 0 Experiment lower 
CP 
-1.2 
-.8 
-.4 
0 
.4 
1 . 2 A  
0 .2 .4 .0 .0 1.0 
x/c 
170 
NLR 7301 
STEADY STATE PRESSURES 
M = 0.7, a = 2" 
This figure shows a comparison of isentropic potential and Euler pressures on the NLR 7301 
airfoil for M = 0.7, a = 2". The shock wave calculated with the isentropic method is too strong 
and located too far aft on the airfoil, suggesting that this case is outside the range of validity of 
the method. 
CP 
-2.0 
-1.6 
-1.2 
-.8 
-.4 
0 
.4 
- Isentropic FP upper ---_ Isentropic FP lower 
0 Eukr upper 
o Euler lower 
1 . 2 1  
0 .2 .4 .6 .8 1.0 
x/c 
171 
NLR 7301 
STEADY STATE PRESSURES 
M = 0.7, a = 2" 
Modeling the nonisentropic effects brings the potential flow solution into good agreement with 
the Euler results. The shock waves differ in location by only 3 percent chord and have nearly 
the same strength. In addition, the agreement of pressures on the lower surface is excellent. 
cP 
- Nonisentropk 
Nonisentropic 
0 Euler upper 
0 Euler lower 
-1.6 
- 1.2 
-.8 
-.4 
0 
.4 
FP 
FP 
upper 
lower 
1 . 2 w  
0 .2 .4 .6 .8 1.0 
XfC 
172 
SUMMARY 
An unsteady full potential method for calculating flows with strong shock waves has been 
presented. The method uses approximate factorization to advance the solutions in time and a 
linear interpolation method to model dynamic grid motion. Calculated results were presented 
for the NACA 0012 and NLR 7301 airfoils. 
0 PRESENTED AN UNSTEADY FULL POTENTIAL METHOD FOR FLOWS 
WITH STRONG SHOCKS 
0 USED APPROXIMATE FACTORIZATION TO ADVANCE SOLUTIONS 
IN TIME 
0 USED LINEAR INTERPOLATION TO MODEL DYNAMIC GRIDS 
0 PRESENTED ISENTROPIC AND NONISENTROPIC CALCULATIONS 
FOR NACA 0012 AND NLR 7301 AIRFOILS 
173 
CONCLUSIONS 
From the results that were presented, it can be concluded that nonisentropic potential methods 
more accurately model Euler solutions than do isentropic methods. The primary effects of 
modeling shock-generated entropy are (1) to eliminate multiple flow solutions when strong 
shock waves are in the flow field and (2) to bring the strengths and locations of computed shock 
waves into better agreement with those calculated using Euler methods and those measured 
during experiments. 
o MODELING NONISENTROPIC EFFECTS RESULTS IN A POTENTIAL 
METHOD THAT MORE ACCURATELY MODELS EULER SOLUTIONS 
0 PRIMARY EFFECTS OF MODELING SHOCK-GENERATED ENTROPY 
- MULTIPLE SOLUTIONS ELIMINATED 
- COMPUTED SHOCK WAVES IN BETTER AGREEMENT WITH 
EULER SOLUTIONS AND EXPERIMENT 
174 


Application of a full potential method to AGARD standard airfoils

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