AIAA 93-2981

An Analytical Description of Hypersonic Boundary Layer Stability

D. Bower and C. S. Liu University at Buffalo, S Buffalo, New York

JNY

AIAA 24th Fluid Dynamics Conference July 6-9, 1993 / Orlando, FL

b For permission to copy or republish, contact the American lnstltute of Aeronautics and Astronautics 370 L'Enfant Promenade, S.W., Washington, D.C. 20024

AN ANALYTICAL DESCRIPTION OF HYPERSONIC BOUNDARY LAYER STABILITY

Daniel R. Bower'and Ching Shi Liut University at Buffalo, SUNY

Department of Mechanical and Aerospace Engineering Buffalo, New York 14260

i,

Abstract The neutral modes of a hypersonic boundary layer flow over an adiabatic flat plate are consid- ered. A formulation of the governing second or- der linear equation for the pressure disturbance is developed that lends itself to the application of the WKB method over the entire boundary layer. This formulation provides analytic eigen- values and eigenfunction relations for the pres- sure disturbances and is applicable to flows at moderate Mach numbers as well. Solutions are determined for the cases of the wave speed c = 0 and c = 1 and show good qualitative agreement with the numerical computations of Mack ', as well as the high Mach number results of Smith and Brown G.

I, 1 Introduction

The stability of hypersonic boundary layers has always been a complex and difficult subject. In recent years, mo- tivated by the potential application to the design of high speed vehicles, much effort and thus much progress has been acheived towards the understanding of the onset and growth of instabilities in hypersonic boundary layers. The basic analysis for compressible stability theory was first developed by Lees and Lin I ; however, most ofour current Understanding is due to the outstanding works of Mack 2,3,4, who provided numerical computations for the invis- cid and viscous stability problem and found that the invis- cid dist,urbances to be the dominant ones at higher Mach numbers. These inviscid modes are two-dimensional, thus in the limit Re - 00, the stability equations may be re- duced to the compressible Rayliegh equation. This equa- tion was analyzed by Mack 3, who obtained afull numerical solution In the limit Mm - 00, Cowley and Hall and Smith and Brown ' examined the equation, and developed a solution for the acoustic modes for a fluid satisfying the Chapman viscosity relation. In addition, Smith and Brown provide an asymptotic solution for the vorticity mode in the limit Mm - CD by means of the triple deck method.

'Graduate Assistant, Student Member AIAA [Associate Professor

'Copyright 0 1 9 9 3 by the American Institute of Aeronautics

'W

Aeronautics and Astronautics, Inc. AU rights reserved.

Using a similar technique, Balsa and Goldstein ' analyzed supersonic mixing layers, Grubin and Trigub examined the problem for fluids with a Prandtl number other than unity, and provided calculat.ions for the vorticity mode in boundary layer flows of Helium. More recently, Blackaby, Cowley and Hall studied the vorticity mode in the limit M, - co for fluids with the viscosity specified by Suther- land's law. They also examined this mode in the leading edge interaction zone.

Because of the complexity of the problem, even in lin- ear theory, many of the results rely heavily on numerical computations and/or asymptotic methods. It is therefore of interest to develop a simplified model for this problem that can give more analytical details for a wide range of free stream Mach numbers. In the present study, for sim- plicity, we concentrate on the two dimensional disturbance and consider high Reynoldsnumber flows past an adiabatic flat plate. The viscosity of the fluid is assumed to obey the Chapman's relation and the Prandt,l number is assumed to be unity. For the mean velocity, an analytic expression is developed t,o approximate the usual Blasius profile. With these simplifications, the problem can be treated analyti- cally by means of t,he well known WKB method.

2 Formulation 2.1 Governing Equations

We consider the flow of a parallel, zero pressure gradient hypersonic boundary layer over a flat plate. The surface is assumed to be adiabatic, and the fluid is assumed to be a model fluid, i.e. Praudtl number of unity, and the viscosity is assumed to be proportional to the absolute temperature, pJpw z TITm. The region of interest is considered to be far away from the leading edge, such that the flow is assumed to be locally parallel, with no shock present. The disturbance is assumed to be an inviscid, two-dimensional normal mode.

For the linear stability problem for a compressible boundary layer over an adiabatic flat plate, the governing equation is (see Mack 3,

where, -2 M = ~ ( U - C ) ~ M ~

,,. / /

Here n is the wavenumber in t!ie free stream direction and ,is taken to be real in this analysis; c the wave speed. In general, c is complex and determines the temporal insta- bility of the perturbation; p, M are the mean flow density an,' -locity profiles, and M, is the Mach number of the frt earn. All of the properties are non-dimensionalized w i w e s p e c t to the free stream values. - M is referred to as the relative Mach number. It is

the local mach number of the mean flow in the direction of the wavenumber a , relative to the phase velocity; or conversely, can be viewed as the Mach number of the disturbance relative t c t h e sound speed of the mean flow. If c is complex, then M is complex. In this investigation, we consider real values of c only, i.e. a neutral disturbance, hence ;i? is real only. The formultion of the governing dif- ferential equation in t.erms of the relative Mach number provides a convenient parameter to work with since it con- tains all of the physical parameters of the flow.

In Eq. ( l ) , the existence of the parameter c, will change the character of the equation as the value of c varies. The value of this wave speed, as suggested by Lees and Lin classifies the neutral wave as either subsonic, for which 1 - 1/MW < c < 1 + l /A4, , sonic, for which c = 1 f l/Mw, and supersonic, for which c < 1 - l/M,. Since both the density and velocity profiles vary throughout the boundary layer, and are always less than one (except a t the edge of the boundary layer), the supersonic case represents the situation where z2 > 1 over some portion of the boundary layer, and a multiplicity of solutions can occur, as first noted by Mack.

2.hdMethod of Analysis

Recently Eq. (1) has been analyzed in the high Mach num- ber limit by utilizing the Howarth-Dorodnitsyn change of variable (Smith and Brown, Blackaby et al. , Grubin and Trigub) and also by application of the triple deck method (Cowley and Hall). In the present study, we shall approach this equation directly by means of the WKB method. The results are then applicable over a wide range of free stream Mach numbers.

Eq. (1) can be rewritten as

1-Z2

Introducing the transformation,

Eq. (2) can then be reduced to the canonical form

d 2 p - - a2Q (7)p = 0 dV2

where,

-2 Clearly the relative sonic point (34 = 1) corresponds

to a turning point of Eq. (3) , i.e. where Q(7) = 0. An approximate solution for Eq. (4) can be obtained by means of the WKB method (see e.g. Bender and Orszag In). The WKB approximation yields solutions on each side of the turning point, as well as an approximate solution through the turning point. The solut.ion is given as

where C is a constant to be determined. PI corresponds to the solution in the relative sonic portion of the bound- ary layer, Prr that near the turning point, and PIII that in the relative supersonic region. With the stability prob- lem formulated in this fashion, a single differential equa- tion governs the behavior of the pressure disturbance over the en t i re boundary layer. Once a relative Mach num- ber distribution is established for the problem at hand, the solution is reduced to an evaluation of the integrals J m d t and J a d t over the relative subsonic and supersonic regions, respectively. It should be noted that these equations can be reduced to an uniformly valid ap- proximation, by means of the Langer transformation (see Bender and Orszag). This single equation is valid over the entire boundary layer and is given by

where,

S n = / m d v

Note that this solution retains the multimode character first noted by Mack. In the supersonic region, the solution has a sinusiodal form, and hence there are mutiple values of the argument of the sine term which will satisfy the boundary conditions (e.g to satisfy the boundary condition ? = 0, the sine term can only have t,he argument YT) hence the eigenvalue relation can be shown to be

- n = ( 1 , 2 , 3 , . . .) ( 10) 1 2%-1

Z T

J,". m d t + ; Q " = [

/' ,

3 Solutions for Model Flows

The mean velocity is usually assumed to be the well known 9lasius profile. In order to develop analytical results for : problem, this profile is approximated by

L U = tanh (0) (11)

where 0 = 3y

Fig. 1 shows the difference in relative Mach number profile based on the Blasius profile with that based on the approx- imate profile for c = 1. Clearly they have the same quali- tat.ive features and thus Eq. (11) represents a reasonablc approximation for the Blasius profile in establishing the relative Mach number distribution. For a Prandtl number of unity, and viscosity that is proportional to the absolute temperature, the density is given by

These simplified profiles are used to obtain solutions for various values of the wave speed c.

3.1 Waves with c = 0

For the case c = 0, Eqs. (11) and (12) yeild

(13) - M, (tanh(0)) M =

/ I + q M & (1 - tanh(j)*)

W and thus

where

(15)

For this case, the appropriate boundary conditions for Eq. (1) are

p = 0 at y = 0, and p is bounded as y - 00

Using Eq. (13) and (14), the transformation given by Eq. (3) although somewhat tedious, can he evaluated. The result of integration is shown in Fig. 2.

To apply the WKB approximations given by Eqs (6-#), 'be integrals s m d t and s m d t must be evalu-

ed. For the relative supersonic region, the integral is ' U r i t t e i as

Using Eqs. (3), (13) and (14). this integral reduces to

I

This integral, also tedious, may he evaluated in a closed form. A similar method is employed to obtain the integral in the relative subsonic region, below the turning point. Once these integrals are evaluated, the eigenvalue relation and the eigenfunction can be determined by substitution into Eqs. (9), and (6-#), respectively.

Fig. 3 show the fourth mode of the eigenfunction for a free stream Mach number = 6. This solution has the ex- pected behavior, with the disturbance decaying to zero in the relative subsonic region near the wall, attaining an os- cillatory nature in the supersonic region. The amplitude of the disturbance eventually levels to a constant value in the free stream outside the boundary layer. The corresponding eigenvalue relation for the first five neutral modes is shown in Fig. 4, and clearly demonstrates the multiplicity of the solutions. The solution for the eigenvalue and the eigen- function exhibit all of the features expected for this wave speed, as inferred from the results obtained numerically by previous researchers. However, in this formulation these approximate solutions were obtained analytically, and are valid for a wide range of free stream Mach numbers.

3.2 Waves with c=l

We now apply the same procedure to the case when the disturbance has a wave speed equal to the mean velocity a t the edge of the boundary layer. As discussed by Mack, this mode is termed the non-inflectional neutral mode and was examined in the limit M - cc by Smith and Brown, and Cowley and Hall. In this situation, the turning point is at the edge of the boundary layer, and the relative su- personic region appears near the wall. Here, in contrast to the previous case there is the possibilty that waves may be trapped between the wall and the turning point. This case is also of importance since this neutral wave is ac- companied hy a family of unstble waves with c C 1 , and these subsonic waves (by the Lees and Lin criteria) are considered to be the most unstable ones.

Using Eqs. (11) and (12), the expression for the relative Mach number for the case c = 1 is found to be

(16) - M, (tanh(Q) - 1) M =

1 + q M & (1 - tanh(y)')

This profile is shown in Fig. 5 for several free stream Mach numbers. For this case, the appropriate boundary condi- tions for Eq. (1) are

p’(0) = 0, p ( 6 ) = 0

With this expression, Eq. (3) yeilds

where

z = tanh(y)

The evaluation of this integral is straightforward, and the expression for q = ~ ( r ) is shown in Fig. 6. To develop the eigenvalue relation and the associated eigenfunctuiou for the pressure perturbation, the same procedure as in the previous case is followed, i.e. an evaluation of the integral

in the supersonic region near the wall below the turning point, and

for the region above the turning point. The integrals which can be evaluated in closed form are rather tedious, and details are omitted here.

rring back to Fig. 5, it is interesting to note that

critical point (z2 = 0) at a constant position, namely at the edge of the boundary layer, the turning point of this problem moves out towards the edge of the bound- ary layer. In order for the solution to satisfy the bound- ary conditions both at the wall and the turning point, the wavelength of the disturbance must increase. The eigen- value relation obtained, shown in Fig 7, reflects this trend, with the wavenumber monotonically decreasing with in- creasing Mach number. Also shown in Fig. 7 are the asymptotic results of Cowley and Hall for the case c = 1 , which show good qualitative comparison with the results obtained here, especially a t large values of the free stream Mach numbers. Shown in Fig. 8 is the eigenfunctions cor- responding to the second mode of the eigenvalue relation, exhibiting the expected behavior.

as r;/ t e free stream Mach number increases, even with the

4 Summary and Conclusions

We have shown that the compressible form of the Rayleigh equation for two dimensional inviscid disturbances can be formluated in such a fashion that the WKB method may be utilized throughout the entire boundary layer. This method of analyzing the linear two dimensional stability prc.l-’nni provides an uniformly valid approximate solution f C disturbance and for the associated eigenvalue re- l a w The solutions obtained by using this method have

shown consistent qualitative agreement with the numerical computations of Mack, as well as the asymptotic results of Smith and Brown. This was shown for the case of c = 0 and c = 1, the latter case representing the situation when the relative sonic point is near t,he edge of the boundary layer, and the pertuhations are considered to be all acous- tic modes.

In the present, study, the governing equation is derived such that the perturbation is expressed in terms of the relative Mach number, a parameter which contains all of the flow field parameters. As a result, all that need to be calculated are the integrals s m d t and J m d t which are functions of the relative Mach number distri- bution only. Hence, this method could be applied to a variety of different problems, so long as the relative Mach number distribution could be determined for the problem, e.g., a flat plate with heat transfer. Of course, for cer- tain distrihntions, these integrals may not be evaluated in closed form ; then either an approximate integral may be obhined or the integration may be performed numerically. Thus, this approach provides a much simpler method than a full numerical integration of the governing equation.

5 Acknowledgements

The authors would like to acknowledge the contribution of Jim Sonnenmeier of the Department of Mechanical and Aerospace Engineering, whose help was invaluable in the preparation of this paper, and is greatly appreciated.

References [l] Lees, L . and Lin, C. C., (‘ Investigation of the stabil-

ity of the Laminar Boundary Layer in a Compressible Boundary Fluid,’’ NACA TN-1115, 1946.

[2] Mack, L. M., “Linear Stability Theory and the Prob- lem of Supersonic Boundary Layer Transition ,” AIAA Journal, Vol. 13, No.3, 1975, pp. 278-289.

[3] Mack, L. M., “ Boundary Layer Linear Stability The- ory ,’’ in Special Course on Stability and Transit ion o f Laminar Flow, AGARD Report No. 709, March 1984.

[4] Mack, L. M., ’‘ Review of Compressible Linear Stability Theory ,” in Stability of T i m e Dependent and Spatially Varying Flows, Springer, 1987.

[5] Cowley, S. and Hall, P., “On the Instability of Hyper- sonic Flow Past a Wedge,” Journal of Fluid Mechanics, Val. 214, 1990, pp. 17-42.

[6] Smith, F . T . and Brown, S. N. , “The Inviscid Instabil- ity of a Blasius Boundary Layer at Large Values of the Mach Number,’’ Journal of Fluid Mechanics, Val. 219, 1990, pp. 499-518.

“On the Instahili- ties of Supersonic Mixing Layers: A High Mach Number Asymptotic Theory,” Journal of Fluid Mechanics, Val. 216, 1990, pp. 585-611.

[a] Grubin, S. E. and Trigub V. N., “The AsymptoticThe- ory of Hypersonic Boundary Layer Stability,” Journal of Fluid Mechanics, Val. 246, 1993, pp. 361-380.

[7] Balsa, T. F. and Goldstein, M.,

4

3

ii?? 2

1

i a r

R C

60

7

4 0

20

M m = 7 : I

- M Y

re 1. Relative Mach number profiles for Bl&ius and Figure 2. 11 vs. with c = 0 for several free stream M(Q) velocity profiles with c = 1. Mach numbers.

1

P a

-1

0.3

0 . 6

0 . 4

a

c . 2

0

Y L/

Figure 3 . Pressure perturbation for M, = 6, n = 3 , Figure 4. Eigenvalue relation for c = 0 c = 0.

1

7zz 2

1

0 0 . 5 1

Y

Figure 5 . Relative Mach number profiles for several free stream Mach numbers with c = 1.

O 1

0.5

0 . :

0 9 3

0 . 2

0 . 1

0

M , Figure 7 . Eigenvalue relation for c = 1 compared with

the results of Smith and Brown. -------

6

5 . 5

5

17

4 . 5

3.5

,,' AI, = 12

,//-

,/" .. ..

x - /,I,,,,,/" I' - 6 .-.-

./"

/ ,/ firr, = 4

M , = 2.5

,/ - 2 . 5 -2 -1.5 - 1 - 0 . 5 0 - M

Figure 6. 17 vs. with c = 1 for several free stream Mach numbers.

0 0.5 1

Y

Figure 8. Pressure perturbation for M , = 8, n = 2, c = l .