An iterative method for wall interference assessment and/or correction is presented for transonic flow conditions in wind tunnels equipped with two component velocity measurements on a single interface. The iterative method does not require modeling of the test article and tunnel wall boundary conditions. Analytical proof for the convergence and stability of the iterative method is shown in the subsonic flow regime. The numerical solutions are given for both 2-D and axisymmetrical cases at transonic speeds with the application of global Mach number correction

Lo, C. F.

Sickles, W. L.

English

NASA Technical Reports Server

N89-20959 
TWO-MEASURED VARIABLE METHOD FOR WALL INTERFERENCE 
ASSESSMENT/CORRECTION 
C. F. Lo and W. L. Sickles 
Calspan Corporation/AEDC Division 
Arnold Air Force Station , Tennessee 
Summary 
An iterative method for wall interference assessment and/or 
correction is presented specifically for transonic flow conditions in 
wind tunnels equipped with two-component velocity measurements on a 
single interface. The iterative method does not require modeling of the 
test article and tunnel wall boundary conditions. Analytical proof for 
the convergence and stability of the iterative method is shown in the 
subsonic flow regime. The numerical solutions are given for both two- 
dimensional and axisymmetrical cases at transonic speeds with the 
application of global Mach number correction. 
Introduction 
The recognition of the shortcomings of the classical methods for 
determining wall interference has demanded that a variety of flow 
measurements near or on the tunnel wall be made to assess wall 
interference. For example, Arnold Engineering Development Center has 
designed an interface measurement system for the &foot Transonic Tunnel 
and the wall pressure measurement system is being planned for 
installation in the current 12-ft Wind Tunnel restoration project at 
NASA/Ames Research Center. It is well known that classical methods 
require a mathematical modeling of the wall boundary condition which is 
a difficult task, in particular, for a ventilated wall. The 
characteristics of a ventilated wall are not precisely known and 
generally vary with Mach number, model geometry, Reynolds number and 
wall configuration. Another shortcoming is the necessity to accurately 
represent the test article beyond the simplified representation used in 
the classical approach. 
Furthermore, the development of an adaptive wall tunnel requires 
that the boundary flow measurements be an integral part of the 
procedure. The boundary measurements are designed to determine 
adaptive-wall setting for free-air or noninterference condition. 
Techniques using only the measurements to assess the level of 
interference will complement the adaptive-wall process by determining i f  
a noninterference wall setting has been achieved or by correcting for 
residual interference. Therefore, several wall interference methods 
based solely on the measured flow boundary condition have been developed 
for different kinds of measurements, such as two velocity components at 
a single interface (Refs. 1-4) or one velocity component at two 
interfaces (Ref. 5). However, all methods are limited to subsonic flow. 
QIIE;iCEDUVG PAGE BLANK NOT FLLMED I 
853 
https://ntrs.nasa.gov/search.jsp?R=19890011588 2020-03-20T02:37:41+00:00Z
The purpose of this paper is to present a method for assessing wall 
interference for subsonic as well as transonic flow regimes based only 
on two velocity distributions measured on a single interface and no 
knowledge of the test article. This method as shown in Fig. 1 is 
divided into two tasks. The first task is the determination of the 
effective (equivalent) shape of the model from the two-measured 
variables by an iterative procedure. The second task is to predict the 
wall interference, such as a global Mach number correction, based on the 
effective shape predicted from Task 1. 
In the presentation of the paper, a proposed iterative procedure to 
accomplish Task 1 is described and convergence analytically proved for 
subsonic flow. Numerical validation is demonstrated for two-dimensional 
and axisymmetrical transonic flows using a transonic small disturbance 
equation, and the global Mach number corrections are determined for Task 2. 
Iterative Procedure of Effective Shape 
To accomplish Task 1, an iterative procedure using two measured 
distributions of velocity near the wall has been developed and is shown 
in Fig. 2. The procedure consists of two basic steps in which each 
step involves obtaining a solution for the region between a measurement 
interface (y=h) and an effective shape line (y=y1) near the model. In 
the first step, the velocity component, uO(yi), i s  obtained from a 
solution o f  the region where the boundary condi ions are provided by an 
initial guess of the effective shape vo(yl), and the measured velocity 
component t(h). With the result, uo(yl), the second step obtains the 
vt(h). These two steps are repeated until convergence has been achieved 
and the effective shape, v(yl), determined. 
Defining a merit function for the difference, vn+l(yl) - vn(y1), 
one can relax the effective shape between iterations by the form of 
solution v Y (yl) in the same region using the measured boundary condition 
with o as a relaxation factor. An updated effective shape, v(yl), is 
yielded by repeating Steps 1 and 2 with this relaxed form o f  velocity as 
input until the given criterion of merit function is satisfied. 
Predicted Effective Shape 
The iterative procedure for determining the effective shape i s  
extremely flexible and can be applied to a variety of tunnel flow 
conditions. Depending on the flow condition, the region between the two 
surfaces could be analytically or numerically solved for the appropriate 
fluid equations, such as the Prandtl-Glauert equation, transonic small 
disturbance equation, full potential equation, or Euler equations. To 
illustrate the validity of the iterative procedure, it is applied to the 
two-dimensional, subsonic, Prandtl-Glauert equation. For this linear 
problem the solution for each iterative step can be determined 
analytically. The analytical expressions of the iterative solutions were 
derived by the Fourier transform technique as 
854 
1 f "  
The convergence and stability of these iterative procedures has been 
proved analytically in Appendix A and Appendix B, respectively. The 
results for a NACA0012 airfoil at M = 0.6 in an open-jet tunnel are 
shown in Fig. 3.  It can be seen that the predicted effective shape 
compares well with an independent reference calculation from the 
original airfoil profile. 
For transonic flow, the numerical solution of the transonic small 
disturbance equation was solved for the procedure as shown in Fig. 4.  
The results for a two-dimensional 12% parabolic airfoil at M = 0.8 and 
an axisyrnrnetrical body of revolution o f  a 10% parabolic arc profile at M 
= 0.975 are given in Figs. 5 and 6, respectively. For both two- 
dimensional and axisymrnetrical cases, the agreement between the 
predicted effective shape and the reference calculation are very good. 
Now that effective shape has been determined, the next task is to 
determine the correction. 
Global Mach Number Correction 
The concept of global Mach number correction assumes that there 
exists an equivalent Mach number in free air where the calculated pres- 
sure distribution at a corrected Mach number will match the tunnel 
pressure distribution on the model or in a surface near the model as on 
the current procedure and simulation. This equivalent or corrected Mach 
number (Mc) is computed using the effective shape determined from the 
iterative procedure. Setting the free-air Mach number equal t o  the tunnel  
855 
, 
Mach number (Mf = Mt), free-air calculations are performed exterior to 
the surface near the model. Comparisons of the pressure distribution for 
the axisymmetric case are shown in Fig. 7. Correcting the free-air Mach 
number from Mf = 0.975 to Mc = 0.925 resulted in a very good match 
between the free-air pressure distribution determined at Mc and the 
tunnel pressure distribution determined from the iterative procedure as 
shown in Fig. 7. Therefore, these tunnel da ta  a t  M = 0.975 correspond 
to those in free air at M = 0.925. 
Concluding Remarks 
A method of the wall interference assessment/correction for 
transonic flow conditions has been developed using a two-component 
measured flow boundary condition approach successfully. The method 
avoids the difficulty of modeling the test article and the tunnel wall 
boundary conditions. The prediction of interferences includes subsonic 
and transonic flow regimes without the limitation on the selection of 
equations. 
1. 
2. 
3.  
4. 
5. 
References 
Lo, C. F., "Tunnel Interference Assessment by Boundary 
Measurements," AIAA Journal, April 1978. 
Lo, C. F. and Sickles, W. L., "The Adaptive Wall Method for Two- 
Dimensional and Axisymmetrical Wind Tunnel," AEDC-TR-79-55, 
November 1979. 
Kraft, E. M. and Dahm, W.J.F., "Direct Assessment Wall Interference 
in a Two-Dimensional Subsonic Wind Tunnel," AIAA Paper No. 82-0187, 
January 1982. 
Mokry, M., Chan, Y. Y., and Jones, 0. J., "Two-Dimensional Wind 
Tunnel Wall Interference," AGARDograph No. 281, November 1983. 
Schairer, E. T., "Two-Dimensional Wind Tunnel Interference from 
Measurements on Two Contours," Journal of Aircraft, June 1984. 
856 
I Experiment I 
Flow Two Measured Flow Variables Flow 
at  Interface h 
ut(h),vt(h) 
I I 
Analytic Methods of  
Interference Prediction 
Task 1: 
Determination of 
Effective Model Shape 
V(Y 1) 
From Measured Variables 
I 1 
Task 2 :  
Use Effective Shape to  
Compute Wall Interference 
by Various Numerical 
Methods 
1 
Wal I Interference 
Assesment / Correction 
I 
Figure 1. Prediction methods of wall interference assessment/correction 
using two measured variables at interface. 
857 
Initial Selection 
Effective Shape, vO(yl) 
Solution of Region 
between Two Interfaces with B.C. 
vt(h). U"(Y1) 
No 
I ................ u"(yl) .... 
1 I bY1 0 
I Effective Shape I 
Figure  2 .  I t e r a t i v e  procedure for  o b t a i n i n g  e f f e c t i v e  f low shape a t  
y = y1 by using measured v a r i a b l e s ,  ut (h)  and v t ( h ) .  
858 
3 
e 
2 
1 
- 1  
Reference Result 
- - - - - - - Initial Guess 
_--- Iteration Result 
0 1 
X/C 
2 
Figure 3.  Comparison of results between i t e r a t ive  procedure w i t h  analytic 
expressions and reference calculation for NACA 0012, M = 0.6. 
859 
Computer 
Determine Effective 
Shape, v( f yl) 
Select Initial 
Effective Shapes 
v 
Step 1 computations 
n 
Effective Shape 
V"+'(Yl) 
Figure 4. Two-variable numerical procedure t o  determine effect ive shape. 
860 
ORIGINAL PAGE IS 
OF POOR QUALITY 
-6.0?.o -1.5 -1.0 -0.5 0.0 0 .5  1.0 1.5 2.0 2.5 3.0 
0 Tunnel Reference Calculation 
Figure 5.  
x/c 
Predicted e f f e c t i v e  shape f o r  two-dimensional case over 12% parabol ic  
a i r f o i l  a t  M t  = 0 .8 ,  IY, = 0 ,  y1 = 0.3,  h = 1, v o  = (x ,y1)  = 0 ,  open 
j e t  t unne l .  
XJC 
0 Tunnel Reference Calculation 
0 Converged Effective Shape 
5 2.0 2.5 : 0 
Figure 6 .  Predicted equiva len t  shape f o r  axisymmetric case over 10% parabol ic  a r c  
body a t  M t  = 0.975, r l  = 0 . 1 ,  h = 0.5 ,  v O ( x , r l )  = 0 ,  open j e t  tunnel .  
I 
86 1 
-0.20 
-0.18 
-0.16 
-0.14 
-0.12 
-0.10 
-0.08 
c p  -0.06 
-0. @I 
-0.02 
-0.00 
0.02 
0.011 
0.06 
0.08 
0.10 
-2.0 -1.5 -1.0 -I 
0 Free Air at M=Mt 
A Free Air at M = M, 
0 Procedure results 
5 
x/c 
Figure 7. Pressure distributions for tunnel flow at Mt = 0.975 and 
free flight f l o w  at M, = 0.925. 
862 
Appendix A. Convergence o f  the I te ra t ive  Procedure 
For the convenience of mathematical manipulation, the Fourier 
Transform technique is applied to prove the convergence of the iterative 
procedure. In the transform plane, the solution for u(y1) at y = y1 for 
Step 1 in Eq. (1) is of the form 
- 
(A-1) - u ( p h )  i - I f l l  u I n )  (p,y,) = - - L’ (ps ’ , )  tanh pp(h-yl) 
c o s h p p ( h - y l )  p 
The boundary conditions applied to obtain the solution u(n)(yl) are 
ut(h) measured and v(n)(yl) selected. For Step 2, the solution for 
v(y1) o f  Eq. (2) can be expressed in the transform plane as 
where u(n)(yl) is obtained from Eq. (A-1) of step 1 and vt(h1) is 
measured. 
Substituting u(n)(yl) from Eq. (A-1) into Eq. (A-2), one obtains a 
single expression which can be used to start the iterative procedure as 
described in the previous section 
This single expression, Eq. (A-3), can be utilized to prove the 
convergence o f  the iterative procedure. For simplicity the iterative 
expression, Eq. ( A - 3 )  i s  written in the following function form, 
(A-3a) 
(A-3b) 
2 K = tunh p p  (h -v , )  
Before carrying successive substitution of each iteration, we can obtain 
the converged solution directily from Eq. (A-3a) as v(n+l) = v(n) = v 
1 
u = G -  
1 - K  
- 
(A- 4 1 
863 
It should be noted that the solution o f  Eq. (A-4) in the physical plane 
is an integral equation of the first kind which i s  unstable and 
difficult to solve (see Appendix B) .  Thus, we choose Eq. (A-3)  to 
obtain the present solution in an iterative fashion. 
V ( 0 )  ( y l ) ,  the n-th iteration gives Let the initial guess be 
d n )  = G + GK + G K ~  + ... 
Since K = tanhz pp ( h - y l  
- 
1 - 
l im 
n + m  I - K  
U ( n '  = G - . 
which converges to Eq. (A-4) as we wish to prove. 
(A -5 )  
I 864 
Appendix B. Stability o f  the Iterative Procedure 
Before examining the stability of the iterative procedure, we 
investigate the explicit solution Eq. ( A - 4 )  in the physical plane. 
Equation ( A - 4 )  in the physical plane can be expressed as an integral 
equation o f  the first kind 
The effects of measurement errors of ut, vt on the effective shape wil 
be evaluated by the following assumed errors 
where a is a parameter. The measurement error will vanish as a 
approaches infinity. 
The result of effective shape due to vterr of Eq. (B-2) obtained 
from Eq. (B-1) is 
A s  the vterr decreases to the vanishing small as a + 00, the error of 
effective shape is magnified to an arbitrarily large value 
This demonstrates that the formulation o f  Eq. ( B - 1 )  is not stable, since 
a small measurement error in the boundary flow quantities will result in 
the effective shape of E q .  (B-4)  growing t o  an arbitrarily large amount. 
On the other hand, we will demonstrate that the iterative procedure is 
stable as follows. 
We assume that the measurement error of vt and ut of the forms 
1 
ut h , h )  = - cos an 
e r r  a 
1 
ut  (x ,h)  = - cos ax e r r  a 
The error of the effective shape obtained from Eqs. (1) and (2) is: 
cosax 1 sinax tanhaph 
err a coshaph a coshaph Ut (x)  = - - P -  
(B-5) 
865 
when the measured error vanishes as a * 00, the error of effective shape 
Eq. (8-6) will also vanish. T h i s  indicates that the iterative procedure 
i s  stable as the measured input i s  perturbed. 
866 


Two-measured variable method for wall interference assessment/correction

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server