AIAA 94-2200 Solution for Transonic Flow Over Wavy Wall Balbir S. Narang and Johnny Ho San Diego State University San Diego, CA

25th AIAA Fluid Dynamics Conference

June 20-23, 1994 / Colorado Springs, CO For perrnlsslon to copy or republlsh, contact the Amerlcan Institute of Aeronautlcs and Astronautlcs 370 L'Enfant Promenade, S.W., Washington, D.C. 20024

SOLUTION FOR TRANSONIC FLOW OVER WAVY WALL AIAA-94-2200-

Balbir S. Narang* and Johnny Ho+ San Diego State University San Diego, CA 92182-0183

Abstract been solved by a new method for solving non-linear partial differential equations. The solutions obtained for flow over a wavy wall show that local value of Mach number reaches a value of one for transonic parameter greater than 0.5. The experimental data obtained indicates that the flow becomes locally sonic for transonic parameter value of 0.52. The classical value of transonic Darameter is one for the flow to become

The transonic flow equation has

locally

An =

a, = ai,n =

L = v

M =

M, =

N =

T.P. =

u =

v =

u = x1 =

sonic.

Nomenclature arbitrary constants in equation ( 5 ) to be evaluated by the application of boundary conditions. acoustic velocity, 346.86 m/s arbitrary constants in equation (5) evaluated by setting coefficients of AiAn equal to zero wave length of the boundary

(N-1)/2 + 1

free stream Mach number

final value of n in the infinite series of equation (5) M: (y+l) E a / ( l - M , 2 ) 312

perturbation parameter induced axial velocity perturbation in the XI direction. induced velocity perturbation in the transverse flow direction. free stream velocity horizontal boundary coordinate

1 2 x l , x1 , etc. = values of x i along the horizontal direction

x2 = transverse boundary coordinate

*Professor. Aerosp. Engr. & Engr. Mech. Senior Member AIAA +Graduated with an M.S. Degree

W

1 Copyright 0 1999 American Institute of Aeronautics and

Astronautics, lnc. All rights reserved.

1 2 x2, x2, etc. = values of x2 along the

t$ = velocity potential a = 2 z / L an = 2 x n / L E = amplitude of the wavy wall

transverse boundary coordinate

c 2 112 = (l-M-)

Introdurtion It is intended here to solve the

perturbation flow equation1 over an infinitely long wavy wall. The object of this study is to utilize a new method of solving non-linear partial differential equations which Narang2 has developed. He used this method to solve the complete Navier Stokes equations for viscous incompressible flow to study the entrance flow between parallel plates. However, in order to obtain a solution of a non-linear differential equation, this method requires that first it be linearized so that the exact solution of the linearized equation can be obtained. The exact solution thus obtained is then modified to yield the solution of the non- linear equation. It appears that unlike the perturbation methods, the difference between the linear and non-linear solution of a given differential equation can be quite large. This can be proved to be true in the case of the solution of an ordinary non-linear differential equation2, viz.. yQ - xy' + y = o as well as many others. It is therefore not desirable to formulate our solution in the framework of some small parameters as is usually done in a typical perturbation method. It can also be shown that the solution obtained by the method discussed here can be reduced, as a special case, to that of the linearized partial differential equation. A particular flaw of this method of solution is that the boundary conditions used for obtaining the linear solution have also to be used without modification, for

obtaining the solution of the non-linear differential equation. In this case the boundary condition for vertical velocity is applied at x2 equal to 0.

A two-dimensional compressible, inviscid and irrotational perturbation flow is represented by the linear differential equation1 :

where $I is the perturbation velocity

potential and u = *and.= a x a x 2 . ~

For a wavy wall boundary: x2 - & sin a x 1 = 0 together with the linearized boundary condition:

U Eacos ( ax l ) (2 )

the solution of the above linear differential equation (1) is:

where A = -Ue/<.

M, is small enough so that local sonic velocity is not reached anywhere1. The non-linear perturbation equation of flow for high as well as low subsonic Mach numbers can be written as1:

+l = A e-acx2 cos (axl) ( 3 )

This solution is adequate as long as

This equation will be solved subject to the boundary condition given by equation (2).

It is now assumed that the modified potential function, e , satisfying equation (4) should consist of the linear solution as well as higher order harmonics along with a weighting function (factorial in the denominator) to accelerate convergence. It is expected

that this function, 4. should be able to express flow fields up to a maximum local Mach values of 1.3 as entropy increase for such flows could be neglected, see Kuen-Shih, et. al. 131. Accordingly, the function is assumed to have the form:

e = X A n &i n [(fn(x1))2i-2/ N M

n=l i=l

(2i-2)!1 ehaCx2 cos ( a x l ) ( 5 ) The requirement for convergence to a solution of the differential equation (4) dictated that

fn(xl) = sin (an xi) for T.P. S 0.5 = cos (an xi) for T.P. > 0.5 ( 6 )

This choice of fn (xi) makes physical sense because at lower values of T.P. the non-linear addition to the linear potential, 91, in equation ( 5 ) should provide contribution to ulinear which is in phase with the boundary (ulinear is in phase with the boundary) and vice versa for T.P. > 0.5 to provide for transition to supersonic flow behavior.

are arbitrary and these will be deter- mined from the boundary conditions, just as the arbitrary constant A of equation (3) was determined from the boundary condition.

In order to obtain convergent solution of equation (4). it was found necessary that its linear terms in the perturbation potential, e , be multiplied by Z (which is equal to one) in a functional form given by the equation below. This makes all the terms of equation (4) homogeneous in coefficients AiAn.

The constants An in equation (5) W

It can easily be seen from equation (2) that Z is equal to one. Z can now be written as:

~

n = l i=l

v (7)

2

The perturbation equation becomes: 2 - a

ax, a x , - Z[(l-M_) 2 f 2 1 - d

The substitution of the potential function of equation (5) in equation (8) yields an algebraic equation which is quadratic throughout in all unknown coefficients aisn.

(5) can be separated into two parts as indicated below in equation (9). The first part $1 represents the solution of the linearized differential equation (1) and the second part represents the contribution to @ due to the non- linearity, viz:

The potential function, 0, of equation

N . M .

(2i-2)!]e-"cx2 cos ( a x l ) (9) v

The decomposition of 9 into two parts above in equation (9) is obtained from equation (5). For a value of i = 1 and n = 1, the expansion of equation (5) yields Ala l , l e -ucx2 cos(ax1). Now by writing Ala1.1 = A + (Ala1.1 - A) = A + A1 al, l , the term becomes Ae-u5x2 cos(ax1) f A;

a;,le-acx2 cos(ax1). The first of these two terms is the @! of equation (3). values of A i and 4 are identically the same as those of An and ai,n respectively except when i = 1 and n = 1. The second part of @ of equation (9) decays with decreasing perturbations and grows with increasing perturbations.

I 1

The

ical S o l u m The differential equation (5) is

solved for T.P. values of 0.3281, 0.474, 0.5, 0.7225, 1.0858, 2.972, 4.596, 6.2241 and 18.427 subject to the linearized boundary

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condition at the wall given by equation (2).

n of The C- Substituting equations (5) and (7)

in equation (8) yields the following algebraic equation:

/(2i-2)! )

e2 [ C A n x a i + i , n [( f: ti-'/(2i-1)! + f n

N M

n = l i=l

3

It is clear from the above equation that 1

i.e. A:, A1A2, etc. The value of the

-2 a fl f l sin (ax 1)1 - al, a 2 for a value of N equal to 15 there are 120 distinct quadratic combinations of AkAi

(12) 1

cos (ax 1 summation index M is chosen to be equal to 8. There will therefore be 120 unknown constants ai,,,. These constants

2 aisn will also appear quadratically i.e. al. a1.1. a1.2, etc. in equation (10). For the purpose of numerical calculation of the coefficients ai,n the flow field is divided into fifteen equal intervals. Ax1 and A x 2 were assumed to be the equal. numerical evaluation of the constants

For

ai,* equation (IO) can now be written as:

Performing the matrix operation in equation (1 1) and rearrangement gives 120 distinctly different terms as shown below:

+ A 1 5 A 1 5 ( ~ i , 1 2 0 ] = 0 1= 1 ( 1 3 )

In equation (13) above all of the

Ji' 1 i g120.1 g120.2 . ' ' . ~120 ,120 A 1 5 A 1 5

. . . .

(11) where gi,j are the expressions containing the unknown quadratic combinations of ai,j, for example, for a value of xi = xi and x i = x2 the element 81.1 is (g1.1 is one of 120 elements of the coefficient A1A 1):

1 81.1 = (-L3 e-"Cxz /UE)

1 2 afl fl sin (ax1)]] -

2 1 (M_ (y+l)e-2acx2/U)

[(a2,1 fl fl cos ( a x l ) - a 1 a s in

1.1

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120 120

i= 1 i= 1 120 solution functions, Cgi,l , Cgi12 ,

etc., whose coefficients are the 120 distinct combinations of arbitrary constants AiAj, are independent of each other, and therefore, are deemed to

individually and are therefore set equal to zero. This situation is analogous to the case of a linear differential equation whose general solution is the summation of independent solution functions with each function multiplied by an arbitrary constant and whose each solution func- tion satisfies the differential equation.

equations is solved by using the subroutine developed by M. J. D. Powell[4]. Convergence to the final values of ai,n is obtained when the sum of the squares of the residuals of the equations has been minimized. In this analysis these values ranged between I O - l O and solution required initial guesses for the values of ai+. These values were always assumed to be equal to one.

satisfy the differential equation (8) v

This system of 120 algebraic

The starting of the

Determination o f Coe fficients AB

been solved for and the only unknowns

1 ' 2 1 (ax1)) X a2,1 [( f l f l f fi ) cos ( a x l )

At this stage all the ai,n's have d'

4

in equation ( 5 ) for potential function

coefficients An are determined by making the perturbation potential, 0 , in equation (5) satisfy the boundary condition for transverse velocity at x2 = 0, given by equation (2). The 15 different values of x i used were. the same as those used for determination of the constants ai,,.

v are the arbitrary constants An. The

I Facllrtp . . The experiments were carried out

in the supersonic wind tunnel facility at San Diego State University. The super- sonic wind tunnel is a conventional blow down facility which has nozzle blocks for test section Mach numbers of 1, 2, 2.5, 3 and 4.5. The nominal test section of the nozzles is 15.24 cm x 15.24 cm x 25.40 cm long. The wind tunnel could be operated continuously from 10 seconds to 30 seconds depending upon the Mach number of the nozzle block.

Au!zd.d

from an aluminum block of 25.40 cm x 15.24 cm which was of 1.27 cm nominal thickness. The wavelength of the boundary was 9.8167 cm with an amplitude of 0.160 cm. sine boundary consisted of 3 half wave lengths. Thirty-one, equally spaced holes were drilled along the center-line of the boundary for pressure taps. In addition, one hole was drilled 4.9149 cm forward of the sine boundary for a pressure tab to determine M, and another hole was drilled trailing the boundary by 1.6866 cm to determine the exit Mach number.

An aluminum box of 15.24 cm x 15.24 cm cross section and 25.40 cm in length was constructed to attach to the end of the wind tunnel test section. The bottom surface of the box was the sine boundary with a flat leading surface of 6.350 cm length. The inside surfaces of the box were flush with those of the test section.

The sine boundary was milled --

The length of the

. . and Condltlons Clear glass tube mercury mano-

meters were used for obtaining pressure readings. In order to obtain lower subsonic Mach numbers, the test section was narrowed by attaching a 1.27 cm thick, 15.24 cm wide and 45.72 cm long plate to the top of the inner surface. leading edge of this plate extended 20.32 cm into the original test section of the wind tunnel. The plate's leading edge was tapered to smooth out the flow. higher subsonic Mach numbers a two inch thick plate was used instead.

The

For

Figures one through five show the values of local Mach numbers for various transonic parameters. It can be seen that the (nonlinear) numerical solutions of equation (8) are fairly close to the Linear solution. Figure 1 also provides a linear check of the solution of equation (8) for the limiting case of small value of transonic parameter. However, convergent solutions of equation, (8), for transonic parameter values greater than 0.5 could not be obtained unless fn(X) was set equal to cos (anx). This suggests that somewhere over the boundary local Mach number equal to one or greater should have been obtained for T.P. values greater than 0.5. This, however, does not occur until T.P. values equal to or greater than one are achieved. The reason for this apparent anomaly is that the calculation of the values, of ai,,, from equation (13) satisfy the exact differential equation (8) but the evaluation of the coefficients A,. for perturbation velocities, satisfy only the approximate boundary condition at x2 equal to zero.

supersonic locally if it were possible to satisfy the boundary conditions at the actual boundary rather than at x2 = 0. The experimental data of Figure 4 does lend credibility to this statement as a local maximum Mach value of 1.097 is obtained on the first crest of the boundary at T.P. equal to 0.7225. Experimental data of Figures 2 and 4

The flow would have become

5

indicate that a local Mach value of one is achieved between T.P. values of 0.474 and 0.7225. approximate value of T.P. for the first local Occurrence of sonic velocity on the first crest of the boundary, it was assumed that:

(Mlocd)- = a + b(T.P.) + c(T.P.)~ where a, b and c are constants. T.P. values of 0.3281, 0.4740 and 0.7225 and corresponding experimental (Mlocal),a, values of 0.827, 0.920 and 1.097 respectively, it was determined that (Mlocal),, equal to one occurs when T.P. equals 0.5904. The linear interpolation between T.P. values of 0.4740 and 0.7225 and their respective (Mlocal),,, yield a T.P. value of 0.5863 which is within 0.7644% of 0.5904 for local occurrence of sonic velocity.

(8) unlike the experimental results, are for an infinite boundary which has no leading edge effects. It is therefore more desirable to use the experimental results of the second crest of the boundary rather than the first crest to more accurately determine the experimental value of T.P. for (Mlocal),a, equal to one for comparison with numerical results. Quadratic interpolation from Figures 1. 2 and 4 for the second crest yields a T.P. value of 0.5197 for the local occurrence of sonic velocity (while linear interpolation from figures 2 and 4 yields a TP. value of 0.5217). This value is within 3.94% of the numerically obtained value of 0.5. At least part of this difference between the numerical and experimental values can be attributed to the fact that the theoretical model of equation (8) representing fluid flow is very approximate while the experimental results are closer to the representation of real flow. It is actually surprising that the two values are this close. Beauchamp's5 value of transonic parameter for sonic flow is 0.897 while Chang's3 value is 0.745. to point out that all of these values are

In order to obtain an

Based on

The numerical results of equation

It is interesting

v' less than one which is the classical value.

The peak of the experimental results for local Mach numbers in Figure 4 shows a shift relative to the peak of linear flow. This shift would be typical of a supersonic flow. The shift of the peak of the numerical results is delayed until a T.P. value of 4.596 due to the satisfaction of only the approximate boundary conditions This shift can be seen in Figures 6, 7, 8 and 9. Peak shift does not take place in Chang3 until after a T.P. value of 1.086.

trend towards zig zag behavior of the local values of Mach number with increasing values of transonic parameter. Both the numerical and experimental results in Figure 7 show a remarkable similarity in their curious zig zag behavior for the local values of Mach number although the values for these two cases are quite different. This behavior is not seen in Beauchamp5 and Chang3 probably due to the necessity of having to use smoothing schemes to get their numerical solutions.

Figure 10 shows solution of the linear equation, the equation number (8) and the complete potential flow equation in perturbation form1 without dropping any of the terms. In the complete equation, the linear terms were multiplied by Z 2 and the terms containing first powers of perturbation velocities were multiplied by 2.

Figure 11 shows the convergence behavior of the numerical solutions for various values of transonic parameter. The graphs show the sum of the series for a particular value of n, for example, at n equal to 10 the graphs show the sum of the first ten terms of the series for a particular value of transonic parameter. Estimate of the truncation errors is also shown in the Figure. Truncation error is equal to I(va1ue of the last term of the series - average value of the last two terms)/average value of the last two terms1 x 100.

There appears to be a definite

W

W

6

It has been shown that locally sonic flows begin at transonic parameter values greater than 0.5 for infinitely long boundaries while for finite bodies the sonic flows begin with T.P. values of 0.5904 or greater.

The new method developed to solve differential equations cannot be utilized unless linearized solution is available. Whenever practical, solutions generated by this method would provide a theoretical bench mark.

v

Acknowledeement The author is grateful to the SDSU

Computer Center and the San Diego Super Computer Center for providing computing time to carry out the numerical work.

References Iliepmann, H. W. and Roshko, A.,

Elements of Gas Dynamics, John Wiley & Sons., Inc., New York, 1960.

2Narang, B. S . , "Exact Solution for Entrance Region Flow Between Parallel Plates", International Journal of Heat and Fluid Flow. Vol. 4. No. 3, pp. 177-181 (1983).

3Chang. Keun-Shik and Kwon, 0. J.. "Mixed Transonic Flow Over a Wavy Wall with Embedded Shock Waves", Transaction of The Japan Society for Aeronautical and Space Sciences, Vol. 25, No. 70, Feb., 1983.

4Rabinowitz. Phillip, Editor, Numerical Methads for Non-lhex Aleebraic E m , Gordon and Breach Science Publishers, new York, 1970.

5Beauchamp. Philip and Murman, Earl1 M., "Wavy Wall Solutions of the Euler Equations", AlAA Journal. Vol. 24, No. 12, 1986.

v

0.9 .' ML(nontines0 - ML(exPer1ment) - ML (linear)

Dimensionless Horizontal Coordinate, X1 / L

Fig 2 Local Mach vs. horizontal distance at T.P. = 0.474.

0.6 0 0.2 0.4 0.6 0.8 I 1.2 1.4

Dimensioniess Horizontal Coordinate. X1 I L

Fig. 3 Local Mach vs. horlionlal dklance at T.P = 0.500.

7

+ ~~(non i ineer ] - MLlexperlmanl) '--ML (linear) 1 3

1.2- - i1.1- I

z 0 1 - z

3 0.9 - .c

30.0- J -10.7

0.6 0.7 - 0.6 ! L

0 0.2 0.4 0.8 0.8 1 7.2 7.4

Dimensionless Horizontal Coordinate, XI I L

Flg. 4 Local Mach VI. horizonlal distance a1 T.P. - 0.723. 0 0 2 0.4 0.6 0.0 1 1.2 1 . 4

Dimensionless Horizontal Coordinate. X i L

fig. 7 Local Mach number "9. horizonid dlstencs a1 T.P = 6.224

0.6 ! I

0 0.2 0.4 0.6 0.8 I 1.2 1.4 Dimensionless Horizontal Coordinate. Xl / L

Fig. 5 Local Mach number vs. horlzonlai distance a1 T.R = 1.086.

60 - u (linear) *" u (nonilnear) .. E

Fig. 6 Local horlzonlal PerlurbaliorI velocity w. horlzonlal dislance at T.P. = 4.596.

- 60 2 E -u llinear) + u (nonlinear1 I

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Dlmsnsioniaal Horiwntsi Cooldlnats, X i I L

Fig. 8 Local horizontal penurbalion velocity v8. horizonlal dislance al T.P = 6.581.

- u (linear) + u (nonllnesr)

P

I.80 7 0 0.1 0.2 0.3 0.4 0.5 0.8 0.7 0.8 0.9 7

Dimansbnlsrs Horlwntsi CoordinfM X i I L

Fig. 9 Local horlzontal perturbation velocity vs. horlzonlai distance a1 TP I 18.427.

8

- v e u (linear) * u (nonlinear) u(fUl1 equation)

5 2-

: . 1 -

I 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Dlmsnrionlesa tlorizomd Coordlme. X i I L

Fig. 10 Local horizontal penurbation velocity v6. horizontal dlstsnce st TP = 2.972.

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9