NASA Contractor Report 182051

ICASE Report No. 90-40

ICASE

ON THE INSTABILITY OF HYPERSONIC FLOW

PAST A FLAT PLATE

Nicholas Blackaby

Stephen Cowley

Philip Hall

Contract No. NAS1-18605

May 1990

Institute for Computer Applications in Science and Engineering

NASA Langley Research Center

Hampton, Virginia 23665-5225

Operated by the Universities Space Research Association

NASA

National Aeronaulics and

Space Adminislration

[-angley R_earch Center

Hampton, Virginia 23665-5225

(NA_A-CP-!0205I) UN THE _rNCTALSILITY

HYPFRSQNIC FLOW PAST A rL_] PLATE

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NgO-Z4ZS2

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ON THE INSTABILITY OF HYPERSONIC FLOW

PAST A FLAT PLATE 1

Nicholas Blackaby

Mathematics Department

Exeter University, Exeter EX4 4QE

United Kingdom

Stephen Cowley

Mathematics Department

Imperial College, London SW7 2BZ

United Kingdom

Philip Hall

Mathematics Department

Exeter University, Exeter EX4 4QE

United Kingdom

-" - i

: i

ABSTRACT

The instability of hypersonic boundary-layer flows over flat plates is considered. The

viscosity of the fluid is taken to be governed by Sutherland's law, which gives a much more

accurate representation of the temperature dependence of fluid viscosity at hypersonic speeds

than Chapman's approximate linear law; although at lower speeds the temperature variation

of the mean state is less pronounced so that the Chapman law can be used with some confidence. Attention is focussed on the so-called 'avorticity" mode of instability of the viscous

hypersonic boundary layer. This is thought to be the fastest growing inviscid disturbance

at hypersonic speeds; it is also believed to have an asymptotically larger growth rate than

any viscous or centrifugal instability. As a starting point we investigate the instability of

the hypersonic boundary layer which exists far downstream from the leading edge of the

plate, In this regime the shock that is attached to the leading edge of the plate plays no

role, so that the basic boundary layer is non-interactive. It is shown that the vorticity mode

of instability_his flow operates on a significantly different lengthscale than that obtained

if a Chapman viscosity law is assumed (see Smith and Brown, 1989). In particular, we find

that the growth rate predicted by a linear viscosity law overestimates the size of the growth

rate by 0(M2). Next, the development of the vorticity mode as the wavenumber decreases is

described, and it is shown that acoustic modes emerge when the wavenumber has decreased

from it's O(1) initial value to O(M-_). Finally, the inviscid instability of the boundary

layer near the leading edge in the interaction zone is discussed and particular attention is

focussed on the strong interaction region which occurs sufficiently close to the leading edge.

We find that the vorticity mode in this regime is again unstable, and that it is concentrated

in the transition layer at the edge of the boundary layer where the temperature adjusts

from its large, 0(M2), value in the viscous boundary layer, to its O(1) free stream value.

The existence of the shock indirectly, but significantly, influences the instability problem by

modifying the basic flow structure in this layer. , - _ .

IResearch was supported by the National Aeronautics and Space Administration under NASA Contract

No. NASI-18605 while the authors were in residenceat the InstituteforComputer Applicationsin Science

and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665.
±
1. INTRODUCTION

Our concern is with the inviscid mode of instability of hypersonic boundary-layer flows

over flat plates. In the first instance we will consider a regime where there are no shocks

present, and then we will show how the instability problem is significantly modified by

the presence of a shock in the flow field. The motivation for this and related work on

hypersonic boundary-layer instability theory is the renewed interest in hypersonic flight

which has been stimulated by plans to build a successor to the Space Shuttle. A primary

concern with such a vehicle is the question of where transition will occur over a wide

range of Mach numbers and whether it can be controlled. At the largest relevant Mach

numbers, say Mach 20-25, the extremely high temperatures associated with the flow would

destroy the vehicle unless it were cooled, so that it is of interest to know the effect of

the wall temperature on the instability properties of the flow. The purpose of this paper

is to determine the inviscid instability characteristics of physically realistic hypersonic

boundary-layer flows. We note here in passing that there is a simple generalization of

Rayleigh's (incompressible) inflection point theorem to compressible flows (Lees & Lin,

1946), and that many compressible boundary layers turn out to be inviscidly unstable even

though their incompressible counterparts ar'e stable. This is a significant result because

the growth rates of inviscid disturbances are generally much larger than those of viscous

or centrifugal instabilities so that they are the likely cause of transition to turbulence

in most situations. The modes which we discuss in this paper can be referred to as

generalized inflection point modes because, when neutral, their phase speed is equal to the

fluid velocity at the generalized inflection point. Furthermore the eigenfunctions of the

modes are localized around that point.

For convenience we will concentrate on high-Reynolds-number flow past a flat plate,

although many aspects of our analysis are applicable to other boundary-layer flows (e.g.

flow past a wedge). The underlying steady flows that we study depend upon the Mach

number, the Prandtl number and the choice of the viscosity law; the complications arising

from real gas effects are not investigated. Throughout we assume that the fluid viscosity

is Newtonian and is adequately described by Sutherland's formula. We will also take

the Prandtl number to be one, noting that it is relatively straightforward to relax this

restriction. Moreover the relaxation of the latter restriction does not significantly alter the

qualitative features of the results we present here.
Reshotko (1976) and Mack (1987) have reviewed earlier work on the linear instability

of high-Reynolds-number compressible flows. Many of these studies are based on the OrrSommerfeld equation; for a critique of the mathematical rigor of this approach see Smith

(1979, 1989). Here we examine the finear stability of high-Reynolds-number flows by means

of formal asymptotic expansions; for example Smith (1989) has applied triple-deck theory

to the lower-branch viscous Tollmien-Schlichting modes of compressible boundary layers.

Seddougui, Bowles & Smith (1989) have extended this theory to include the effects of

severe wall cooling, while Cowley & Hall (1988) have shown how such modes cart interact

with a shock at large Mach number. However, the viscous modes have relatively small

growth rates, and our main concern here will be with inviscid modes. The nature of the

asymptotic expansion procedure in these investigations clearly depends on the nature of

the mode of instability. In fact the third type of instability responsible for boundarylayer transition, the G6rtler vortex mode, develops an asymptotic structure at high Mach

numbers closely related to that found here; see Hall and Fu (1989).

When a quasi-parallel approximation is formally justifiable because the Reynolds number is large, inviscid modes satisfy the compressible Rayleigh equation. Numerical solutions

to this equation have been reported by, inter alia, Mack (1984, 1987) for boundary-layer

flows, Grosch & Jackson (1989) for shear flows, and Papageorgiou (1989) for wake flows.

For fluids satisfying a Chapman viscosity law, high-Mach-number asymptotic solutions to

this equation for the so-called "acoustic" boundary-layer modes have been obtained by

Cowley & Hall (1989), while Smith & Brown (1989) have identified the asymptotic form

of the "vorticity" mode. Goldstein & Balsa (1989) have given art asymptotic solution for

high-Much-number, shear-layer modes of instability for a Chapman fluid. Though the basic

states investigated by the two latter pairs of authors are different, they found essentailly

the same most unstable eigenvalue because it corresponds to a disturbance trapped in a

thin layer where the overall features of the basic state are unimportant. However Goldstein

and Balsa did not spot the exact solution of the vorticity mode equation found by Smil;h

and Brown.

Both in the above mentioned boundary-layer analyses, and the hypersonic G5rtler

vortex instability analysis of Hall & Fu (1989), one of the key asymptotic regions for the

case of a Chapman viscosity law is a logarithmically thin layer which develops due to the

ezponential decay of the underlying steady temperature field away from the wall. However,

Chapman's viscosity law is not exact, and was introduced as a useful interpolation law

2
which greatly simplified steady boundary-layer calculations (e.g. see Stewartson 1964). At

the large temperatures typical in hypersonic flows, it differs significantly from the more

precise Sutherland's formula. In fact Chapman's law is simply a linear approximation to

the viscosity-temperature dependence of the fluid; it is therefore of questionable validity

in the hypersonic limit. At high Mach numbers the steady temperature field in a fluid

satisfying Sutherland's formula initially decays algebraically away from the wall, before

reverting to exponential decay in an asymptotic region "far" from the wall (e.g. Freeman

& Lam 1959). The effect of this algebraic decay is to change significantly the scalings in the

transition region; in particular the asymptotic expansions proceed in inverse powers of M

rather than _/_og(M). Moreover the wavelength of the most unstable disturbance varies

by a factor of x/_og(M) in the two cases. We note that a similar difference in scalings is

evident in the interaction region of steady hypersonic flow past a flat plate. In that case

Lee & Cheng (1969) have shown that the shock-heating transition layer is logarithmically

thin for Chapman's viscosity law, whereas for a power-law viscosity formula, and hence

for Sutherland's formula, the scaling for the transition layer is algebraic (Bush 1966).

The flows which we consider here are appropriate to different stages of hypersonic flow

past a semi-infinite flat plate. In the first instance we shall consider the instability of a

non-interactive flow which is appropriate to large distances downstream of the leading edge

of the plate. Here the attached shock at the leading edge has no effect on the flow field and

the basic state is the Sutherland law counterpart of that discussed by Smith and Brown

(1989). This basic state, and the Rayleigh equation which governs the inviscid instability,

are discussed in Section 2. The dispersion relationship associated with this equation is

then solved in Section 3. We shall discuss the growth rate of the mode over the whole

range of unstable wavenumbers and discuss how the disturbance is related to the acoustic

mode at small wavenumbers.

Then in Section 4 we go on to discuss the basic state in the so-called "strong interaction

region" further upstream. The description of the basic flow in this regime for a power law

fluid is due to Luniev (1959) and Bush (1966). We shall discuss how the interactive system

for the flow can be formulated when the Mach number is of order R_. However this

system can only be solved numerically and we are not aware of any published results on

that problem. Nevertheless we can of course still consider the instability of that state,

and we derive the appropriate (quasi-parallel) stability equations in this regime. The

strong hypersonic interaction limit then corresponds to taking the further limit of the
streamwisevariable tending to zero. In that casea similarity solution for the basic i_:ow

can be found, Bush (1966), and a re-scaled Rayleigh equation for the disturbance is found.

The solution of that equation is discussed in' Section 5. We could have instead considered

the weak hypersonic limit further downstream where Bush and Cross (1967) have given an

appropriate asymptotic description. We choose to concentrate on the strong interaction

regime because it is, to a certain extent, simpler. Further, if the flow is unstable in this

regime it is likely that growing disturbances will originate here. Finally in Section 6 we

shall draw some conclusions.

m_
2. NON-INTERACTIVE STEADY FLOWS

We begin by considering the stability of steady hypersonic flow far downstream from

any leading-edge interaction region; in particular, if L is the distance from the leading

edge, and Uoo, _Zoo, Poo and/?too, are the velocity, sound speed, density and shear viscosity

of the free stream flow, then we assume that the Reynolds number,

/5o0 (2.1a)

is larger than whatever power of the Mach number,

0_

M = --, (2.1b)

Zoo

is necessary for interactive and/or non-parallel effects to be negligible (see below for a more

precise restriction). We adopt a non-dimensionalisation based on coordinates Lx (where z

is in the direction of flow and y is normal to plate), velocities _'0o u, time Lt/Uo_, pressure

Poo _2 p, density _5oop, temperature Too T, and shear and bulk viscosities /500# and /500#'

respectively, where the subscript oo denotes the value of the quantity in the free-stream.

On the assumption that the fluid is a perfect gas with a constant ratio of specific heats %

the governing equations of the flow are

Op

0-7+ v. (p__)= o, (2.2a)

p-_t=-_+ [2v. (z__)+ v((J - 2_z)v .u)],

DT 2DP 1 (7 - 1)M2 _,

P-_-- = (7 - 1)M -_- + p------r-r_V.(#VT) + . R

7M2p. = pT,

(2.2b)

where

e,j=_ _+0x_/'

and Pr is the constant Prandtl number. For a shear viscosity obeying Sutherland's law

(2.3a)

#-- + '

(2.3c)

5
110.4

where C _ _ for air temperatures measured in degrees Kelvin. In the numerical calT_

culations discussed in the next section we took 2_, = 216.9. The boundary-layer equation

can be recovered by first substituting

_0 y

= R÷ pdy, v= R-_V, (2.4a,b)

where the Dorodnitsyn-Howarth variable, ¢, is introduced for convenience, and then taking

the limit R ---* c¢.

For steady two-dimensional flow over a flat plate, a similarity solution to these equations exists. With

C

= x/(1 + C)x' u = ¢¢, pV = -(¢, + C,¢¢), (2.5a)

¢ = vl (l + C)x f(r_), T = T(rl), p - p(n), (2.5b)

the governing similarity equations are found to be

pT = 1, (2.6a)

//" + T + Cf'' = o, (2.,3b)

1 T, , (_ - 1) 2M2T_ f,,2 0, (2.6c) f T' + -p--_r -_-_ T + T + C =

subject to the boundary conditions

f(0) = f'(0)= 0, "f'(c_)= T(oo)= 1, (2.6d)

_ :2

.... ?

and T(0) = T_ (fixed wall- temperature), or T'(0) = 0 (insulated wall). (2.6e)

For simplicity we will focus attention on Pr = 1, and denote by T, the wall temperature

when the boundary is insulated. Then, as is well known (e.g. Stewartson 1964), the energy

=

equation can be integrated to yield

.....

T = 1 + ((Tb - 1) + _(3 ,:_ 1)M2(Tb + f'))(1 - f..'),

where T_ = T_T, and T, = 1 + 1⁄2('y - 1)M 2.

(2.v)
The solution to (2.6) in the limit of large Mach number has been examined by Freeman

& Lain (1959). They showed that two asymptotic regions develop, defined as where the

coordinates r/and _ = M_, respectively, are order one.

_=o(1)

In this region we write ] = M-_]0(_) + ..., then using (2.7) it follows from (2.6b)

that

with

and

L !

/0/_' + _ (T_+ f')_(1- f');

f0(0) =/G(0) = 0, (2.sb)

24 72_

f0 _ _ + _ + (T, + 1)(-y - 1)_ 3 - (T, + 1)(_/- 1)_ 4 + "'" as _ _ oo. (2.8c)

We note that as _ --* oo, then f6 _ 1 algebraically. This is of course different than

the corresponding result for a Chapman fluid, in that case the correction term to the

free-stream speed is exponentially small. This difference is significant because it leads to

inviscid instabilities which are unstable at very different wavelengths.

,7= o(1)

Here, we write

then gl satisfies the equation

gl (2.9) f = ,7+ M---¢+ M---,2+ ...,

( v/1- 1⁄2('r- 1)(T + 1)gl rig11' + 2

l+C-1⁄2('r-1)(T_+l)gl ) =0,

(2.10a)

subject to

gl "

24

I)"V3 as 77_0, and gl_0 as 7/-*co. (2.10b) (T, + 1)(_/

In Figure (2.1a) we show the function f0 for the adiabatic case T, = 1 with 3" = 1.4, we

note the algebraic approach of the function to the free-stream speed. In Figure (2.1b)

I

we show the transition layer function 91 corresponding to the same case. We note the
exponentialdecay 0fthis function for large 77. It is also worth pointing out that for a

Chapman fluid the transition-layer equation corresponding to (2.10a) is linear, and it's

solution can be expressed in terms of the exponential function. It was this simplification

that enabled Smith and Brown (1989) to spot the exact solution of the neutral vorticity

mode in their study of the instability problem in this layer. We shall see below that for

Sutherland's law we are not able to find a similar exact solution of the stability equation.

Sufficiently far downstream the quasi-parallel assumption is valid for inviscid instability modes. It is then appropriate to seek perturbations of the form

u = f'(_) +... + A_(_)exp

i (I+C) (ax+/sz-ct) +..., (2.11a)

p= 7M 2 +...+A;5(_?)exp i i (l+C)(ax+13z-ct) )+..., (2.11b)

with simiiar expressions for the other flow quantites. If the disturbance amplitude, A:_ is

sufficiently small the pressure perturbation ;5 satisfies the compressible Rayleigh equation

d2# 2f" d_ (c_2 +/52)T( T _ a2M2(f' - c) 2

dzl 2 f' - c dTI (_2 +/52) )t5 = 0, (2.12a)

where for convenience we have rescaled the quantities a,/5, c by dividing them by x{. The

conditions that there is no normal velocity at the wall, and that the disturbance is confined

to the boundary layer, can be expressed as

_' = 0 on 77= 0, j5 --_ 0 as 77---. oo. (2.12b)

Equation (2.12a) and boundary conditions (2.12b) specify a temporal-stability eigenrelation

c = c(a,/5); alternatively from a spatial Stability standpoint, the eigenrelation can be

regarded as a = a(c, t3).

Z

J

-=

tit
3. THE FAR DOWNSTREAM BEHAVIOUR OF THE INVISCID MODES

In this section we discuss the asymptotic form of unstable solutions to (2.12a) for the

region far downstream of the leading edge of the plate. In a previous investigation Cowley

and Hall (1988), hereafter referred to as CH, studied the so-called acoustic modes of (2.12a)

in this region on the assumption that the viscosity satisfies Chapman's law. Simultaneously

Smith and Brown (1989), hereafter referred to as SB, investigated the vorticity mode for

Chapman's law. The main difference between these two types of modes is that the acoustic

modes have a ,,_ M-2, whilst the vorticity mode, at least while it is close to neutral, has

a ,_ v/21ogM 2. Moreover the vorticity mode is centred at the adjustment layer at the

edge of the boundary layer whilst the acoustic one is concentrated in the m_n part of the

boundary layer.

However, as indicated above, at high Mach numbers the temperature variations in the

boundary layer are large, so that a linear temperature-viscosity law is a bad approximation;

Sutherland's law should be used to give a better representation of the viscosity. It is then

important to see how the asymptotic structures developed by CH and SB change. We shall

see that there are significant differences.

In the first instance we derive the asymptotic structure of the solution for a twodimensional vorticity mode of (2.12a). We determine the neutral values of a and c for

this mode, and find the limiting form of the mode when the further hmit a _ 0 is taken.

This limiting solution points to a sequence of distinguished asymptotic limits. Within the

sequence of asymptotic limits mentioned above the scaling, a _ M -3/2, appropriate to

an acoustic mode emerges; we therefore discuss the latter mode as a limiting case of the

vorticity mode.

Consider then the solution of (2.12a) which has the eigenfunction trapped in the

temperature adjustment layer at the edge of the boundary layer. We seek a solution which

has f_ = 0 and a = O(1), so that the wavelength of the vorticity mode is comparable with

the depth of the adjustment layer in terms of the Dorodnitsyn-Howarth variable. From

(2.7) and (2.9), the velocity field, u, and temperature field, T, of the underlying steady

flow expand as

2G(_)

= 1 + (_/_ 1)(T_ + 1)M 2 + "'" (3.1a, b)

'i'=I-G+...,

9
where 7/ = V_17 and the function a = 1⁄2(T, + 1)("/- 1)g_ satisfies a rescaled version of

(2.10a), and has the asymptotic behaviour

9 B

G = -L'Zl/_+ --173-_/-'-'''w+ (2C + 1) +... as _7---* 0, (3.2a)

G ---. 0 exponentially as ff ---, oo. (3.2b)

Here B is a constant to be calculated numerically.

Next we expand a, c and _5in the forms

1

a -- _--_6+ ...,

2

c=l+

(7- 1)(T_ + 1)M 2

15=p+ ...,

(3.3a, b, c)

where we have assumed that the disturbance moves downstream with the fluid speed in

the adjustment layer. On substituting for _ and T from (3.1), and using (3.3), we find

that the zeroth order approximation to (2.12a) in the adjustment layer is,

d2p 2G' dp 62(1 - G)25 = 0. (3.4)

d_ 2 G - _ d_]

We note that three-dimensional disturbances satisfy (3.4) but with 6 replaced by

62 + _2 so that in the present regime it is sufficient for us to consider only two-dimensional

modes. Equation (3.4) is to be solved subject to i_ vanishing in the limits _ ---* 0 and

r/---* oo, so that the disturbance is confined to the adjustment layer. For _ >> 1 it follows

from (3.4) that _ decays like exp(-6r_), whilst for _ << 1 a WKB solution of (3.4) can be

expressed in the form

/3 ,,_ exp[- ] O(17)]d_, (35)

where

96 2 B6

0 ,,_ --_-_ + i7 + [73_J7 + 2C6 +... as rj --, O. (3.6)

First, we restrict our attention to the neutral case. _ is then real and can be evaluated

by finding the fluid speed correct to order M-2 at the generalized inflection point which

is located where

_. T - O,

10

2

=

z

=
i.e. where

G" 2G'

b+ -0.

G' 1 - G

A numerical solution to (2.10a) using a Runge-Kutta method shows that this occurs when

_7-----1.971510, in which case

= -.993934 (3.7)

The corresponding real value of & is obtained by integrating (3.4) from _ = 0 to _7- oo with

an appropriate treatment at the generalised'inflection point. Such a calculation predicts

that the neutral value of & is

& _ 0.645065. (3.8)

However, of greater significance are the unstable eigenmodes. Figure (3.1) illustrates

the dependence of the growth rate, _t_, on the real wavenumber &. We see that the growth

rate attains it's maximum value of __ 0.256853 at & "_ 0.143619; further it turns out that

the acoustic modes have smaller growth rates (see below) so that this is the most unstable

inviscid mode for a hypersonic boundary layer. In Figure (3.2) we show the eigenfunction

of the vorticity mode equation at different values of the wavenumber. This figure indicates

that as the wavenumber decreases the eigenfunction starts to expand out of the transition

layer. We note here that ?_ as defined above is independent of T_ and that the growth rate

is obtained from (3.3b) by dividing _, by _(Tb + 1)(_, -- 1). It follows that wall cooling

has a destabilizing effect on the vorticity mode.

The structure of the growth-rate curves at small wavenumbers is of interest because

for sufficiently small values of the wavenumber we expect that the vorticity mode will

develop a structure similar to that of the acoustic mode. In the following discussion we

will isolate the different significant regimes which occur in the small wavenumber limit.

For simplicity we shall now concentrate on the adiabatic problem and take T_ = 1.

The key to understanding the subsequent regimes when & is related to inverse powers

of the Much number is to write down the small & asymptotic structure of (3.4). Figure

(3.3) is a schematic illustration of the different regions in _ space which emerge in the limit

& --, 0. Also shown in this figure is the wall layer in which the temperature becomes of

order M 2. For the moment & is not considered to be sufficiently small for & to be O(M -_)

for some positive ¢; it then turns out that the regions I and IV shown in figure (3.3) are

quite passive. However, at sufficiently small values of & the wall layer structure of the

basic state will enter the problem - see below.

11
After some careful numerical calculations at small values of & we deduced theft

expands in the form

Cl _2

= + .... (3.9)

Also, from (3.2b,4) it follows that for rl ,-, &-: the pressure decays like exp(-&f_). This

suggests that the pressure in III should be expanded in the form

^ I ^ ^ 3 ^111 ^

= 1 q- aVpl 'k ave2 -4- ... "-k a'p6 q- &(/37 -- f_)q-

^ 10 ^

^ IX ^ 1_ ^ ^ 13 ^

+ a'r P:: + &"r P12 + a'rP:3 + &2P14 + ...,

(3,10)

where we have anticipated the form of several terms in this expansion in that Pl, P2,...,/310

are taken as constants, whilst /511,/512, etc. are functions of f_. At order &_ in :;his

expansion we find that Pll satisfies

,, 2G' Ph m _ _ 0,

so that after use of the exponential mathching condition it follows that

2f _

/511= --_II - Gdf) -/%4 f)+/%11, (3.11)

where _II is another constant. /512and 1313satisfysimilarequations with forcing functions

coming from the higher order terms in the expansion of the wavespeed.

The order &2 term in (3.10) is then found to satisfy

A,,= _ (/%3:--p2- (- - - +

P{4 (1 G) 2 2G' c2^ c3 t_2 C4 2_2_3 _23_

=which may be integrated twice to give/5:4. However it is enough for our purposes to note

that when f_ --_ 0, P14 0-6 so that as fl ---* 0 the order ^ ,..r &2 ,'_ a, and terms in (3.10) become

comparable when r/= O(_); thus, as anticipated earlier, the sublayer III is of depth &_.

Within this layer we define

1

Z=&-vO

and expand the pressure as

:= =

^.k^

p= 1 + a,p: + ... + &P7 + &$Ps(Z) + ...,

i2

7

w
where we have, by matching with the solution in II, again anticipated several terms in this

expansion. At order &} we find that 158 satisfies

72 -t 81

15;' + Z(_lZ 4 + 9) ph = _"

We write the solution for 158 in the form

z 81dZ 1

/5; _ (Cl % 9Z4)2( (cl Z4 ÷ 9) 2 - -I_-_" )'

co

where the constant of integration has been chosen to satisfy/5_ .._, -1 as Z ---* oo in order

^&

to match with the solution in II. Finally we consider Region IV and define _] = a, z. The

zeroth order approximation to (3.4) here is

8 81

p,, -{- -p_ - _-_p -- 0, (3.12) z

which must be solved subject to p ---*0 as z ---, 0. The appropriate solution then has p

constant as z _ oo so a match with III can only be achieved if

/o `'_ (6lZ 81dZ4 + 9) 2 - _1 = 0. (3.13)

It is assumed that cl in (3.13) is complex so that after a little manipulation we obtain

( 8x/2),_ . 3i7r

_1 = - 3--_r" exp(---_-),

which corresponds to an unstable mode of (2.12a).

= O(MNext we consider the situation when a is so small that IV in Figure (3.3) merges with

the wall layer of the basic state. At this stage the wall layer is of thickness M- _ so that

a ,-_ M-_. The wavenumber a is then written in the form

o_0

O_-- s d-...,

Mr

where for the moment we shall not be precise about the size of the first correction term in

this expansion. The zeroth order approximation to (2.12a) in the wall layer then becomes

2(1- _0) )/5= O, (3.14)

_, - 1)2(1 + - - 1)(1+ I10 -- 1 p 4

13
where a dash denotes a derivative with respect to the wall layer variable _ = M_'r/, and

T0 is the first term in the expansion of _ in that layer. The above equation is again to be

solved subject to/Y(0) = 0. For large _ it has the asymptotic solutions

/5 --_ NO = constant and /3_7 "-_ N1 = constant.

For most values of a0 the constant No is nonzero, and the structure in layers I, II, III

survives intact. Thus for these values of a0 the wavespeed c expands as

dl

c= 1 M_ +'"' (3.15)

and dl will be complex so that the wave growth rate is of order M- _.

However, equation (3.14) has a countable infinite set of eigenvahes for which the

constant No = 0. In this case the eigenfunction takes on its greatest value in the wall

layer, i.e. where the steady velocity field is adjusting to it's free-stream value. These

eigenvalues correspond to the acoustic inviscid modes and are the counterpart of those

discussed by CH. A numerical solution of (3.14) yielded the sequence a0 = 2.47, 7.17,

12.19, 17.33, 22.54, 27.79, .... The first three eigenfunctions associated with this sequence

of eigenfunctions are shown in Figure (3.4). Thus at a countable discrete set of points the

acoustic modes emerge as almost neutral "vorticity" modes. In fact we could seek acoustic

modes with a ,_ O(1), i.e. for a wavenumber much larger than currently assumed, for

which the eigenfunction is concentrated in the _ = O(1) wall layer. These modes have a

fast variation in this layer which may be described by the WKB method. At certain values

of M these eigenvalues coalesce with the neutral vorticity mode discussed earlier, and an

analysis outlined in CH (see also SB) can be performed to describe the 'splitting' of the

eigenvalues in this region, we do not pursue that calculation here but simply determine

the Structure of the vortlcity m0dein=the vicinity of the acoustic mode. Thus the smMi a

structure outlined above remains valid for a = O(M- _) if the acoustic modes are avoided;

it follows that c_ continues to grow in the further limit aM} ---* O, where no neutral acom_tic

modes exist.

In order to examine the interaction of the vorticity mode with the acoustic modes in

a little more detail, we expand the wavenumber in the form

45 41

4 - M3/2 -b M25/14 -[- • "",

14

7

=
whilst c still expands as in (3.15). Here we are denoting by a_} any eigenvalue of (3.14)

corresponding to No = 0. In the wall layer 15expands as

15= M2/TQo(1 "+ ¢[M]) + Q1 +..., (3.16)

where we have anticipated the possible occurrence of further eigenfunctions between Q0, Q1

by inserting the factor (1 + @[M]). For our purposes it is not necessary for us to calculate

@ here. If we substitute the above expansion into Rayleigh's equation we see that Q0

satisfies (3.14) with a = a_}. The derivative of Qo vanishes at _ = 0 whilst for _ >> 1

Qo "-' qo_- 7

where q0 is a constant. At higher order in Rayleigh's equation we find that Q1 satisfies

,, 2a_) ,

Q1 aO --1 Q1

(a_)2(7 --1)2(1 _ _2)(1 _ 2(1 --riO)

4 (7 -- T o) )Qi

2(1 - _0)

_ i 5(7 - 1)2(1- - (7 - i) iT )Q°"

The solution of this equation which has Q_(0) = 0 is such that for large

QI "" aoqoa_.

The constant ao depends on a_} and 7, and can only be determined numerically. However it

can be shown that ao is alternatively positive and negative at successive values of a_). The

region III of Figure (3.3) becomes of depth O(M-3/14) when a is O(M-3/2). The solution

in this layer is calculated using the procedure outlined above; the only significant change

is that in matching regions III and IV we must now account for the fact that Qo _ _-7

for large _. After some manipulation we find that the eigenrelation obtained when this

matching has been carried out is

where

1 1 1

al = -_d 1 + ai;l/4e_/4' (3.17)

- = ----f9 a_ _4/7 12 (3.18a, b, c, d) a*l = 9A2a°a°a_7D 2 , di Ddi, D , _--_, , A=(7_l)-_.

In Figure (3.5) we have shown solutions of (3.17) for -5 < &_ < 5. At both ends of this

Figure dl approaches e- _" which gives the required matching with the unstable vorticity

15
modesaway from the acoustic modes. We further note that for small a_ the growth rate

tends to zero. More precisely from (3.17) we find that for small a_

_1/8e-i_/4

dl ~ (_i) 1/2 + +...

2

so that, dependent on on the sign of al, the growth rate goes to zero like M-}(_)112 or

M-}(&i) 11/8. : - :_ :

Strictly we should now rescale the above expansions when a_ becomes sufficiently

small. We do not pursue that rescaled problem here since our main concern is with completing a discussion of the structure of the vorticity mode at all lengthscales, in particular

we wish to see if the vorticity mode connects with any other neutral states. We have

already seen that ci continues increasing as the wavenumber decreases through O(M-3/2)

values, apart from small neighborhoods of the acoustic modes where it decrease towards

zero. Thus c_ tends to infinity on the present scaling when ao tends to zero. Thus the

main significance of the a ,,_ M -3/2 regime is that it is at this stage where the acoustic

modes emerge; however the small a structure, developed initially for the a = O(1) scaling,

survives this regime intact in the limit of ao tending to zero.

= OIM-÷)

The next significant stage in the development of the vorticity mode occurs when the

temperature in the upper layer, i.e. where r_ ~ a-1, becomes such that _ ~ (_ - c)2M 2.

The pressure eigenfunction in the upper layer then has its decay to zero modified. This

situation occurs when

with

c=1-_+..., (3.1.8)

l

a - M7/4 -J- .... (3.19)

In this case the pressure in the upper layer decays like exp(-/(1- d_l)_M-7/4_): - This

means that in the rescaled region corresponding to III of Figure (3.2), where we write

Z = M1/4rl, the pressure perturbation expands as

P2(z)

15=1+...+ M2 + ....

and P2 Satisiles the equation

-,, 24,4 -, 9A2(_/- 1)2/2

P2 - Z[dlZ4 - 3A]P2 = Z8

16
The solution of this equation which enablesus to match with the behaviour discussed

above is

]5_ 3A f = 9A2('7 - 1)2l 2

-- (aT1 -- Z-'4)(- z (all Z4 3A) 2

In the wall layer there are now no acoustic mode eigenfunctions which lead to decay at

infinity so that matching leads to the eigenrelation

/0 _ 9A2(7---1)2 dZ lv/1---_dl (dl z4- 3A) 2 + _1 .-- 0.

A more convenient form for the eigenrelation is found by writing

3(3A)1/4(_/- 1)27r

so that

As expected in the limit [-_ oo we obtain dl _ e-3_/7[- _ which is the limiting form of

the solution obtained for a ,,, M- }. It is a simple matter to show from the above equation

that dl_ is a monotonic decreasing function of [ and in particular for [ _ 0 we find that

dl _-_e -'_/3 (3.20)

1 ?

The main implication of this limit is that the M- • layer for the a ,,_ M- T modes decreases

_Z

in size like _1 ' when [1 --_ 0. The next stage is when the M-_ and M-_ layers merge

which occurs when _1 = O(M) and a = O(M- _).

O(M-t)

Here we expand c and a in the forms

L

c=co+..., a-- + ....

M}

Now there are just three regions, of depth M- _, M 0, and M}, for us to consider. Interestingly the extent of the perturbation away from the wall has been reduced by a factor

of M_ from the order M1⁄4 scale of the case a ,,_ M- _. This reduction in depth is caused

by (_ - c) remaining O(1) in this layer, and if we write 77= M÷Y, the dominant terms in

the equation for the pressure in that layer give

/Sty + L2(1 - c0)2i0 ----0 (3.21)

17
If co has a positive imaginary part, it follows that

15= exp(-aY),

with a = iL(1 - co). Thus where Y -+ 0 we have that _ --, -M-_a. This means that

d_

the required solution for/5 in the 77= O(1) region is

(T

= 1 +...-- M---TIp÷ + _?] + .... (3.22)

Here the constant terms are small compared to M 0 but large compared to M- _; in principle they may be found by matching requirements. It remains to consider the small layer

when 77= M-_, and 15expands as

1p

_5= 1 +... + _--_ 2(()+ ....

]52 is found to satisfy

uo_co.P_- 2u_ _ i L2(_/-1)(l+u0)(1-u0)(l(_/-1)(l+u0)(1-u0)-(u0-c) 2)

so that

= /j Z2(_/- 1)(1 + u0)(1 - u0) (_/-1)(1 + u0)(1 - u0) (u0- c0)2 (

2(u0 C0) 2 2

A match with the core solution is achieved if

-- (uo -- co)2)d,_ ¢.

(3.22)

(1- co) 2/0 °° L2(')' - 1)(1 + uo)(1- uo) 4(uo - co) 2

= -o" = -iL(1 - co).

((_/-- 1)(1 + u0)(1 -- u0) -- 2(u0 -- c0)2)d(

¢) (3.,.3)

This is the required eigenrelation to determine the complex wavespeed co as a function

of the scaled wavenumber L. A large L analysis of this equation shows that the limiting

small t solution of the a = O(M- _) case is retrieved. :::_

In Figure (3.6) we show co as a function of L for 100 < L <: 400 and the case of an

insulated wall, We see that the maximum value of c_ for a two-dimensional mode occurs

for L ,,_ 225 and a neutral mode exists for L _ Ii0. We postpone until the final section a

discussion of the implications of the results we have found above.

18

w

l

w

_--=

I

f_
4 The Inviscid Instability Problem in Interactive Boundary Layers.

We consider the hypersonic flow of a Sutherland law fluid past the semi infinite wall

defined by the positive x axis. The oncoming flow has constant velocity at infinity and

the leading edge of the wall is sharp. This results in the formation of a shock wave which

acts as an upper bound for the flow disturbances in the sub-layers below, the uppermost

of which is inviscid. The reader is referred to Figure (4.1) for a description of the different

parts of the flow field.

This region has been studied by for example, Bush (1966), and since viscosity is

negligible here the use of a power-viscosity-temperature law does not alter the well-known

governing equations for this region. Below the expansions and equations governing the

flow are given. Apart from some minor differences, our formulation is essentially the same

as that of, amongst others, Stewartson (1955, 1964) and Bush (1966) and so the reader is

referred to those papers for more details.

The flow quantities and coordinates have been non-dimensionalised using upstream

flow quantities and L a typical streamwise lengthscale. We then scale the non-dimensionalised normal coordinate y appropriately

Y

y = -- (4.1) M'

where M is the Mach Number and Y is taken to be O(1). Note that as yet we do not

scale the downstream variable x. As in previous studies we consider the two-dimensional

problem and this enables the introduction of a streamfunction ¢ defined by

pu = Cy , pv = -¢_ (4.2a, b)

and

Further we expand the velocity, temperature, density, pressure and streamfunction as

T=TI+... ,

Ul

u= l +...

P--Pl-F'" ,

Vl

V: m+...,

M

I)1

1

.

IV1

(4.3a, b,c,d,e,f)

19
If the above expansions are substituted into the inviscid Navier-Stokes equations wriiten

in Von-Mises coordinates we obtain, after some manipulation, the governing equations for

this layer

Vl, =-Pie, , Vl,_, = 0"_ , Pl = E(¢I)pl. (4a, b,c)

The function E(¢1) is evaluated from the initial conditions which for these hyperbolic

equations are given at the shock. Conventionally, we define the shock by Y = f(x) where

f is unknown at present; we note that ¢1 = f on the shock.

The Strong Interaction Zone

The solution of these equations (and the corresponding ones for the lower layers) can

be investigated analytically in the large x- and small x-limits using expansion procedures,

following Stewartson (1955, 1964), Brown and Stewartson (1975). The formulation below

follows the latter paper closely.

Let us consider the solution of (4.4a,b,c) for small x, i.e. close to the leading edge of

the plate, where the shock is attached. Here we write

f : al x3/4+... ,

vl = al_l(C)x -1/4 + "'"

pl = a_l(C)x -1/2 +"" ,

and Pl = Pl(_) +"" •

The similarity variable _ introduced in the above equations is defined by

(4.5a, b, c, d)

¢ _ ¢1 (4.6)

alx3/4"

The scalings (4.5b,c,d) are implied by the Rankine-Hugoniot relations, which relate

flow quantities either side of the shock Y = fix), and by (4.5a) which follows from matching

normal Velocities and pressure across the sub-layers beneath the shock; see Stewartson

(1964) and references therein.

The resulting equations for the leading order terms are

el _--7

U1 --

3_ _ 9("/- 1)" (4.7a, b,c,d)

vl = 7_-_pl and el = S('/+ 1),+1'

3 9 "/+1

2('/+1) ' Pl= S('/+l) ' Pl='/_I on #=1 ,

with

2O

L

=

l
which are the conditions at the shock. We require their solution for _ < 1. In particular,

for given 7 we can solve numerically for VI(0) and _1(0).

Substitution of (4.5b,d) into (4c) and recalling the definition of _ gives E(¢I) ,-_

pla 813 I _b-213 1 as _bl -+ 0. This implies that, as _ _ 0, _1 _,,_213v and this produces a

higher order correction in V1 if 7 > _. For a realistic gas 7 is larger than this value,

with the result that there must be a viscous sublayer beneath the present layer. Bush

(1966) found that this layer enables one to match the inviscid-layer solution on to the

boundary layer solution if one used a power-viscosity/temperature law (# ,,_ T ° ,ca < 1),

rather than the linear law used by Stewartson and most other research studies in the field.

Stewartson (1955) hoped that by using idealized assumed physical properties it would

help to "understand the behaviour of more realistic fluids". His assumption resulted in

discontinuous derivatives, or "kinks", in properties between the Inviscid Zone and the

Boundary Layer, see Fig. 7.4, page 167 of Stewartson (1964). Despite the results of Bush

(1966) (and a few subsequent authors) the linear viscosity law is still extensively used

in hypersonic shock/boundary-layer research; it is argued that these 'kinks' are only a

slight nuisance that can probably be explained away by deeper analysis and appeal to the

argument of Stewartson concerning the assumption of ideal fluid properties. The analysis

of Bush was extended to the linear case by Lee and Cheng (1969). They found that, by

consideration of a second order boundary-layer correction, coupled to Bush's transitional

layer analysis, the two layer structure of Stewartson gives the correct basis for the flow

structure.

Not surprisingly the extensive use of Chapman's law in hypersonic calculations has

encouraged researchers in stability theory to make the same rather severe approximation to

the fluid viscosity; we believe that the calculations given in this paper are the first to take

account of a realistic viscosity-temperature dependence in a first mode stability analysis.

We note that our flow solutions in the layers beneath the shock are in full agreement with

Bush, but he considered a general power law whereas we concentrate on # ,_ T 1/2, the

leading order form for Sutherland's law at high temperatures. For convenience we will

use the notation of Stewartson and Brown (1975). We wiU next consider the viscousadjustment-layer after discussing the third sub-layer, the viscous boundary layer.

Viscous Boundary Layer

This is adjacent to the surface of the plate and enables velocities to be reduced to

zero on the surface. First, the full asymptotic formulation is described briefly, then the

21
asymptotic behaviour for small x is discussed in the next subsection. We assume that the

boundary layer exists where ¢ ,-_ M-a, %bbeing the stream-function defined earlier. We

define the hypersonic parameter r by

Re = rM 5 (4.8)

and take r to be order one. Note that the power of the Mach number is six in the definitions

of the hypersonic parameters of those using a linear viscosity law. We write

V1 1

o,,

u=Ul+... , V=M+ , P=_--_P1,

1

p = _--_R1 + .-- , T = M281 and # = Mpt. (4.9a, b,c,d,e,f)

These scalings are standard apart from that for the viscosity # which follows immediately

from the high temperature form of Sutherland's law:

#= (I+C)T t/2. (4.10)

We make the assumptions that the Prandtl number is unity and that the wall is

insulating, this enables us to carry out a relatively simple analysis for the viscous solutions.

The solutions capture the new structure of the inviscid modes due to the effect of using

a power-viscosity law in the viscous sublayers. The relaxation of these assumptions does

not substantially alter the conclusions obtained below.

We again transfer to yon Mises coordinates; using the streamfunction _ as an independent variable instead of Y. The energy and momentum equations are combined to

yield the boundary-layer equations in these coordinates

and

uwi,= (_- i)(i- u_)p_.+ __

2 "YPi

together with #1 = 811/2

/

r _1 U1Ulk9

1

el - _ - (1- u_),

2

from (4.10), and with usual boundary conditions

(4.11a, b)

UI=0 on _I, =0_

UI-_I as _I'---*oo.

22

(4.12a, b)

c

T
Boundary-Layer Solution for the Strong Interaction Zone

Now let x be small so that the hypersonic interaction parameter becomes large. We

define the similarity variable ¢ by

1/4alxl/4

_ [4_/(1+ C)_1(0)]1/2( 2 )

rl/2 ¢, (4.13)

where al and _1(0) were introduced when considering the inviscid region. If we now write

= g(¢)+ ..., (4.14)

we obtain the similarity equation for the boundary-layer solution

-egg'= (1-g2)+g (1_g2)1/2/

and

g(O) = 0 , g(oo) = 1 . (4.15a, b,c)

This is a modified Faulkner-Skan equation, but the decay of g at large ¢ is now

algebraic rather than the usual exponential behaviour,

a

g,-_l--_---$+... , as

and a > 0 can be determined analytically.

oo, (4.16)

Matching normal velocities across the sub-layers in the usual manner yields the leading

order coefficient al in the small x expansion for the as yet unknown shock location

a2 3(1 + C) (2(0'- 1) 1/2 1/4/oaO _g2 (4..)

The Viscous Transition Layer

Here the velocities and temperatures in the outer inviscid flow match with those in

the inner boundary-layer flow. We give a brief account of the arguments used to deduce

the position and properties of this region and again consider the strong interaction zone.

Earlier it was noted that the inviscid density function Pl "_ (2/a_ as ( _ 0. It can

easily be deduced that ul and T1 grow like inverse powers of _ as ( --, 0. In fact

(x314_ 213"r T ,._ T1 "_ X-1/2¢ -2/3_ _'_ X -1/2 71/ , (4.18)

23
as ¢1 ---+ 0.

Now let us consider the boundary-layer temperature as the top of the layer is _pproached. We have written T = M20z and found that 81 = _-_(1 - U2). We know the

large ¢ behaviour of U1 and it follows immediately that

so that

1 X X

X

T = m201 ~ _ as ¢---, oo. (4.19)

Matching the limiting temperatures (4.18) (4.19) gives the position of the viscous transition

layer,

97

¢1 ~ M-Xx (3-_')/'4 , A = 6_/---__1 (4.20a, b)

This is in full agreement with Bush's result. Note that the powers involved in the asymptotics are functions of _,, the ratio of specific heat capacities, in the new layer. Further

investigation of the small ¢1 limit of the inviscid solutions implies the following expansions

for flow properties in the transition layer

U = 1+1Vl 6-,-_ Ul+''"

T= M,,--k'r-,TI +... ,

together with .........

_/pl

p=

T

We define the scaled streamfunction

¢1 = T/M -_ (= Mg;).

(4.21a, b, c)

(4.22)

These are then substituted into the Parabolised Navier-Stokes equations in Von- Mises

co-ordinates to obtain the equations for ul

ul= TIPI= +'7Pl 0 ( 1_ OUl_, .,

(1+ C) - 7Pl r 077 T{ 0r} / (4"2°/

where, as we are taking the Prandtl number to be unity and the wail to be insulating, the

corresponding energy equation can be integrated to yield

T1 = (1 - )ul. (4.24)

24

i

=

=

=

|

@

=
The pressure Pl is independent of the normal co-ordinates, as usual in viscous regions

governed by the boundary-layer equations, so the pressure is expanded

al2Pl(0)

Pl- _ +""

X 2

in the two viscous layers, this being the limiting form as the inviscid layer is descended.

We see that (from (4.23)) in this adjustment layer we still have the high temperature

form of Sutherland's law, in contrast with the shock-free adjustment layer of section 2

where the full form is needed. Again, a similarity solution for the flow exists in this

viscous sub-layer with similarity variable z defined by

rl = _xAz, (4.25a)

where

(c + 1) a pl(0)

and

3-)_

A - (4.25b, c)

4

Writing Ul = x_-2G(z) leads to the governing equation for the flow solution in this

crucial region

- 2)a (a - )za, - + (4.26)

4 - J(1- )a

Note that from the definition of Ul, (4.21a), we expect that G < 0. We require that this

solution for ul matches to the corresponding solutions above and below, i.e. in the inviscid

sub-layer and viscous boundary layer respectively, so we must investigate asymptotically

the small and large z limiting forms of G.

As z _ 0 we find that G has the behaviour

576-_ 2

G = -goz -4 -k glz -v -b ..-, go : (_ _ 1)(3"/- 1) 2.

(4.27a, b)

Thus the transition layer solution matches onto the large ¢ form of g. The exponent v

of z in the first correction term can be calculated analytically whilst the coefficient gl

must be determined from a numerical solution. In fact v satisfies a quadratic equation

with coefficients functions of % With _, = 1.4 we find that v = 0.6267... so that we have

relatively small correction terms. The value of go for this choice of gamma is 275.6 ... and

so G grows quickly as x --* 0.

25
We now investigate the behaviour of G for large z we find that

G _ -Aoz-_ + Alz-2-_ + "", (4.2s)

and so again we have a relatively small correction term. Note that G decays algebraically, in

contrast to the rapid exponential decay of the Blasius and the "Modified Blasius" functions

which arise in the shock-free, far-downstream cases (the former after employing the linear

Chapman viscosity law and the latter from the use of the more realistic Sutherland's law

- see earlier). The second coefficient A1 is a function of gamma and can be determined in

terms of A0; in fact

1_

8A_ (3"7 -{- 1)

A1 =

9_(_- 1)_(3_- 1)"

The constant A0 is not fixed by the asymptotics. Instead we must choose it so that t:he

velocity solution matches with the appropriate limiting form of the inviscid solution. We

obtain, after an elementary matching argument, and a little manipulation, the result

311- _-- 2 _ I-_-_- JA0 = (1% C)- _ "" alr _ Pl0 e_ (4.29)

For the value "_ = 1.4 we find AO = 0.1751 .... In our numerical calculations we took

without loss of generality the hypersonic interaction parameter to be unity. Now that the

base flow for this region has been found (at leading order) we can consider its stability

characteristics; in particular we are interested here in the linear stability of inviscid modes

concentrated (trapped) within this adjustment layer. These so-called vorticity-modes have

been discussed earlier for other flows.

The Vortlclty Mode in the Strong Interaction Zone

The scalings for these modes appear complicated but follow in a straight-forward

manner after applying the usual vorticity mode arguments to the flow field discussed

above. The modes of wavenumber k have wavelength comparable with the thickness of

the transition region and a Rayleigh analysis suggests we require 0¢_ ,-_ T2k 2. This can

be seen, for example, from (2.12a) and (3.4) studied earlier. From (4.215), and recaUing

the definition of A in (4.20b), we have T ,-_ M 4_-6, and also from (4.22), ¢ ,-_ M -x-1

Note that with these scaling we are primarily concerned with obtaining the correct Much

number dependence; the small x dependence will be incorporated later. We deduce that

2!

t

=

w

26

I
k _ M 7-3_ :>> 1, and thus this represents a short wavelength mode and so we introduce

a fast x-scale

X = M "r- 3_ x. (4.30)

The time-scale can also be similarly deduced from Rayleigh analysis, after noting that the

vorticity modes propagate in a frame moving with uniform velocity and recalling the ul

scaling (4.21a) . We are lead to the introduction of a short time scale.

T = M _-lt. (4.31)

We note that these scales are compatible with our earlier analysis of the non-interactive

case. At leading order we have the multiple-scales

Or _ Or -I- M7-3XOx , Ot '+ M)_-Io_, - M7-3XOx •

Note that non-parallel effects are O(0=) ,-, O(1) and are negligible in comparison with the

direct growth effects of order k(u - 1) = O(M )'-1) >> 1; for our choice of "7 = 1.4 we

find that A = 1.7027. The wavespeed _ of the vorticity mode propagating in the frame

moving with uniform (nondimensionalised) velocity u = 1 is related to its stationary-frame

wavespeed c by

c--1 ÷ M4(2__ ) -F..- (4.32)

which follows immediately from (4.21a). Note that it represents a small perturbation about

the velocity of the propagating frame. The remainder of the analysis follows the classical inviscid mode approach for formulating the pressure equation for the linear wave-like

disturbance, now that the scales have been deduced. We assume that the flow previously

discussed for this region is perturbed by an infinitesimal disturbance proportional to

e_ (x - _) (4.33)

containing all the X and _- dependence of the disturbance. We note that the basic flow

properties are independent of r and X allowing such an assumption. After a little manipulation we will obtain the so-called vorticity-made-Rayleigh equation for the amplitude of

the disturbance pressure p,

/_,, - -------=p, k2p - 0, (4.34)

ul - c ,,/2p2

27
where we require p to decay to zero for large and small _. We now return to the Strong

Interaction Zone and investigate the :asymptotic behaviour of (4.34) for small x.

We recall the leading order small x forms

a p (O) Pl -- j. , r/ ----

Xa

Ul = x_-2G.

(4.35a, b, c)

We need the small-x dependence of 0 and k but need not worry about p as (4.34) is linear.

We find that

_- x _- 2C,

and

r_

k - (7 - 1)x 3(_- 1)/4K:' (4.36a, b)

are the necessary behaviours. The former follows immediately from (4.35c) and the latter

results from considering the coefficient of 15in (4.34). We finally arrive at the renormalised

problem for p; the vorticity-mode pressure-amplitude equation for the strong-interactic_n

zone

2G'

i5,_ G - C p" - 1C2G2i5 = 0, (4.37)

to be solved subject to 15 vanishing in the limits z _ 0 and z ---* oo, so that, as in previous

analyses, the disturbance is confined to the adjustment layer. We discuss our numerical

solution of this eigenvalue problem and its limiting small wavenumber form in the next

section.

28
5 The Solution of the Strong Interaction Vorticity Mode Equation.

The leading order asymptotes of equation (4.37) are found to be

15--, 15oz 2 exp 3z 3 as z --* 0

1 (3_/AOK:z (3_ - 2)) and 15--* 15ooz_ exp 3_/- 2 3_/ asz --* vo

(5.1a,b)

where 150 and poo are constants and the numbers go and A0 arise from the corresponding

asymptotes for G given earlier. Higher order terms in these expressions can be found

analytically, and, in fact, are needed for accurate numerical solutions to this eigenvalue

problem for C and K:, the scaled relative wavespeed and the scaled wavenumber of the

inviscid vorticity mode, discussed after briefly paying attention to the neutral case. There

C is real and takes the value of G evaluated at the generalized inflection point which of

course occurs where

GGz_ = 2G 2.

A numerical solution to (4.26) using a Runge-Kutta method shows that this occurs

when _ = 1.661432, where the new variable _ = £nz was introduced to 'stretch' the coordinate-scale for small z where G, and hence 15, vary rapidly. The resulting neutral value

of the wavespeed is then given by

C = -0.633318.

The corresponding real value of K: is obtained from a numerical solution of (4.37) for all z

with an appropriate treatment of the path of integration at the generalised inflection point

(g.i.p.). The particular numerical procedure chosen to calculate this neutral wavenumber,

and the complex eigenvalues C was to extend the line of integration into the complexplane, taking a triangular contour around and below the g.i.p. The neutral value of K:

was calculated to be

K: = .477957

Figure (5.1) shows the growth rate K:C_ plotted against K: we find that the maximum

growth rate _ .060918 occurs at/C "_ .156100 and that the growth rate goes to zero when

the wavenumber goes to zero. In Figure (5.2) we show the eigenfunction of the most

unstable mode, we note the exponential decay of the eigenfunction at both ends of the

range of integration.

29
We shall now show how the small wavenumber structure of the strong interaction

inviscid mode develops When the wavenumber tends to zero. In fact we simply give the

structure corresponding to the first small wavenumber regime discussed in Section 3. Some

discussion of the regimes encountered at even smaller wavenumbers will be given in the

next section.

The small-K7 behaviour

Numerical solutions of the pressure-amplitude equation, (4.37), indicate that C grows

as K7 -+ 0; exactly the same behaviour was found for the no shock problem studied in

Section 3. However, in the latter, the leading order dependence on the wavenumber was

indicated from the numerical solutions, whereas, in the present case, such guidance was

not possible due to a complicated dependence of C on KT. Instead, the equation was studied

analytically for small KT, and initial investigations suggested that

c ~ { (k2)-_4'_ 1}

as k ---, 0. Here the parameter e is a function of 7 and is defined by

4

4e= 1---- (5.2)

37

and we have replaced KT, C by k, c respectively here and henceforth in this sub-section.

4

It follows from (5.2) that special attention is required for values of 7 close to 7 = -

3

and, as this coincides with the range of physically relevant values of the ratio of specific

heat capacities, we solely consider the case of ]el << 1 in this section. We note that the

4

suggested form of c is defined and continuous at 7 = _ by appealing to "L'H6pital's Rule".

Below we outline the asymptotic regions and corresponding solutions required to resolve

the leading order behaviour of c as k ---+0.

When z is very large, we have the WKB asymptote

, ( 37A0. ,_a.az'_ p ~ z.-Vexp a-4--_ _z., )

and we note that for smaller z this form breaks down in a 'turning-point' region corresponding to

3"y

z ~ k - _-7=z-,>> 1.

In this region we define

_,= k,,-,z, P= Po(_)+... (5.3a, b)

3O
and, using the large-z asymptotes for G, we find that P0 satisfies

1

Po;; - A29- 4_Po= o. _ = --.

33'

(5.4a, b)

This equation for P0 has an analytic solution involving the modified Bessel Function

K, the required solution being

where

Po= Do, _K_ (1⁄2,)

1

77=4A0#9 _, u=l-2a and p=--. (5.5b, c,d)

2u

Here Do is an arbitrary constant. We can investigate the small 9 behaviour of P0 by

considering the limiting forms, for small arguments, of the modified Bessel functions. We

find that

D

(1-.)

r(2 - g)(A20_2)"

r(1- ,)(1 -,) 9+ ...] (5.6)

as 9 _ 0, the ordering of the second and third terms being dependent on the value of 3';

we note that the corresponding powers of 9 (the indicia1 roots of the associated Frobenius

4

solution) become equal when 7 = 3" Moreover, the coefficients of these terms are also

4

- _/_ = 1). Here P is the Gamma function. singular when 3' has this value (note that 7 = 3

That special attention is needed for vMues of 3' near four-thirds is now apparent and,

as mentioned earlier, it is exactly this range of 3, that we are interested in, physically.

1

Considering ¢ = - - a to be small, we see that 4

[ 94t- 1)q-lq-O(e)] Po; " 2DoA 2 ( 4e (5.7)

as 9 _ 0. We note that P0_; (and, in fact, P0) is defined, and moreover, continuous as

e - 0. It is convenient to recast the last result in terms of z to ease matching to the next

asymptotic region to be discussed. The result is that

Po, "_ 2DoA2k _ [k _ ( z4"- 1) k,'--_;.- 1) ]

4e + ( 4e + 1 (45)

as z--, 0.

The solution for 15 = P0(Y) +'", will continue to be the leading order behaviour of P

until contributions from the 'middle'-term (proportional to/5 ) become leading order. The

31
location of this next region of interest can be easily located, by analogy with the respective

analysis of Section 3, as we expect c >> 1 as k --_ 0. Here z _ c -1/4, using the small-z

asymptote of G, and we expand

P = 2Do[1 + k2(_23/2p(9) -{-...], c = _(52 -4-...

where (52, = _f2(k; e), is a small parameter. The/5 form is implied from the PO solution for

small 9. The new, scaled normal variable, _ is taken to be O(1) and is defined by

z = _21/4_ (<< 1). (5.10)

Substituting into the pressure-amplitude equation, (4.37), yields the equation satisfied by

P

8go g2

P_ + _(go + @4) P_ = _-_ (5.11)

where go arises from the small-z asymptote of G. This can easily be integrated to give

P_(o) = (e + _)2 o. (@4+ go)2 + 91 , (5.1_)

with D1 an arbitrary constant to be determined by matching with the previous region. As

---, oo we see that Pg ---, DI_ 2, thus P, -. D1_2821/4 and matching with (5.8) gives

k2Dl _2 _f72/4 = Aok _-'° + 1

4e

at leading order. We choose D1 = -A2O - 2 and this determines the size of c for small k

--4C (5.13)

4

Taking the limiting value as e ---*0, we have that when 3' = -

3

67/4= -Ink 2- 1 >> 1 (5.13)

recalling that In x _ -co as x --, 0; the choice of D1 < 0 ensuring that (52 is real and we

allow _ to be complex. When e is small we see that _f2 is large and has the form suggested

at the start of this section. Thus if2, as given by (5.13), leads us to conclude that, foJ:

the physically realistic values of 3" around four-thirds, the scaled, relative wavespeed c

i

1

32
increases as the scaled wavenumber k decreases, this growth having logarithmic magnitude

in contrast to the much faster algebraic growth found in the no-shock analysis presented

in Section 3. We note that the corresponding frequency, proportional to ck, tends to zero

as k decreases. The k- dependence of c, for k --* 0, can be deduced in a similar fashion

for other values of % and we note that, mathematically, the no-shock result of Section 3

is the limiting case 3' ---* oo.

All that remains now is to determine _ and this is easily deduced by introducing a

further region where z ,.., k 1/3 << 1. The details are identical to those of Region IV in

Section 3; again we find we need P_ --_ 0 as 9 _ 0, resulting in an expression for

fo (_4go2 g + g0)2 + -_ao = 0

which can be solved for complex _; after an elementary contour integration and a little

manipulation we obtain

- exp(--T-) (_.14)

= \ 37rgl/4 ]

which corresponds to to an unstable mode of (4.37).

Thus we have obtained the structure of the vorticity mode in the strong interaction

region at small values of the wavenumber. We postpone any further discussion of our

results until the following section.

33
6 Discussion

We have investigated the instability of flat plate hypersonic boundary layers to the

vorticity mode of instability. This inviscid mode is associated with the generalized inflection point of the basic flow and is thought to be the most dangerous mode of instability

of a high Much:number flow. When the m0(le is neutral the wave propagates downstream

with the speed of the fluid at the generalized inflection point. At wavenumbers smaller

than the neutral value the mode is unstable and the growth rate attains it's maxim_lm

value at a finite value of the wavenumber. In the small wavenumber limit the growth rate

approaches zero and for the non-interactive boundary layer at sufficiently small wavenum'

ber the vorticity mode spreads out towards the lower boundary and reduces to an acoustic

mode at a countable infinite set of wavenumbers. We believe that a similar process happens in the strong interaction case since there the acoustic mode is correctly described by

a quasi:parallel theory there. We did:no_ pursue that caicuiuifonhere because it would be

essentially unchanged from that of Section 3 except that it would be made somewhat more

complicated by the necessity of treating the case _ = 4/3 as a special case in the strong

interaction zone.

We believe that the results we have presented in Section 4 are the first which show

the effect of a leading edge shock on any form of hydrodynamic instability. Interestingly

enough the shock does not have a direct influence on the vorticity mode; thus the main

effect of the shock is to restructure the boundary layer in the leading edge region and

thereby influence the susceptibility of the flow to inviscid disturbances.

The vorticity mode eigenvalue problem was formulated in the interactive region along

the plate at O(1) values of _. However to the authors' knowledge the basic flow in this

regime has not yet been calculated; the numerical problem was set up by Bush (1966) but is

sufficiently difficult to have remained unsolved. Thus we were unable to solve the eigenvalue

problem in this regime and therefore choose to consider the strong interaction regime where

a similarity solution for the basic state is available. An alternative to that limit would have

been to consider the weak-interaction problem, Bush and Cross (1967), where a different

similarity structure holds. We choose to concentrate on the strong interaction limit because

the growth rates there are bigger and if the flow is indeed unstable there the stability of

the flow further downstream is possibly of less relevance.

Unfortunately we are unaware of any experimental observations or other theoretic_,l

work which we could compare with our results for the strong interaction regime.

34
In Section 3 we showed how the acoustic inviscid mode emerges from the small

wavenumber description of the vorticity mode. Again it is not possible for us to compare our work with that of previous authors since it appears that the finite Mach number

calculations available, mostly due to Mack, have either being carried out using a Chapman viscosity law or a combination of Sutherland's and Chapman's law. In fact Mack's

calculations were carried out using a combination of the different laws so as to efficiently

model the viscosity-temperature structure of the fluid. The fact that the calculations of

Cowley and Hal1(1988) and Smith and Brown (1989) agree so well with Mack's calculations

suggests that over the part of the flow where instability took place Chapman's law was

being used; in the case of the vorticity mode this is clearly a bad approximation because

the mode locates itself in the layer where the basic temperature field varies rapidly.

35
REFERENCES

Balsa, T.F., & Goldstein, M.E. 1989 On the instabihties of supersonic mixing layers: a

high Mach number asymptotic theory. Submitted to J.Fluid Mech.

Brown, S.N., & Stewartson, K. 1975 A non-uniqueness of the hypersonic boundary layer.

Q.J.A.M, XXVIII, 75-90.

Bush, W.B. 1966 Hypersonic strong interaction similarity solutions for flow past a flat

plate. J.Fluid Mech. 25, 51-64.

Bush, W.B., & Cross, A.C: 1967 Hypersonic weak interaction similarity solutions fer

flow past a flat plate. J.Fluid Mech. 29, 349-359.

:i

Cowley, S.J. & Hall, P. 1988 On the instability of flow past a wedge. ICASE REPORT

88-72 and to appear in J.Fluid Mech..

Hall, P. & Fu, Y. 1989 GSrtler vortices at hypersonic speeds. Theor. Comp.Fluid Mect....

1, 125-134.

Jackson, T.L. & Grosch, C.E. 1989 Inviscid spatial instability of a compressible mixing

layer. J.Fluid Mech 208 609-638.

Lee, R.S. & Cheng, H.K. 1969 On the outer edge problem of a hypersonic boundary

layer. J.Fluid Mech. 38, 161-179.

Lees, L. & Lin, C.C. 1946 Investigation of the stability of the laminar boundary layer

in a compressible fluid. NACA Tech. Note No. 1115

Luniev, V.V. 1959 On the similarity of hypersonic viscous flows around slender bodies

PMM 23, 193-197.

Mack, L.M. 1984 Boundary-layer linear stability theory. In Special Course on Stability

and Transition of Laminar Flow. AGARD Rept 709.

Mack, L.M. 1987 Review of linear compressible stability theory. In Stability of Time

Dependent and Spatially Varying Flows, eds. D.L. Dwoyer &. M.Y. Hussaini. SpringerVerlag.

36
Papageorgiou,D.T. 1989 Linear instability of supersonic plane wakes. ICASE REPORT

89-66.

Reshotko, E.

311-350.

1976 Boundary-layer stability and transition. Ann. Rev. Fluid Mech. 8,

Seddougui, S.O., Bowles, R.I. & Smith, F.T. 1989 Surface-cooling effects on compressible

boundary layer instability. ICASE REPORT 89-95.

Smith, F.T. 1979a On the non-parallel flow stability of the Blasius boundary layer. Proc.

R. Soc. Lond. A366, 91.

Smith, F.T. 1979b Nonlinear stability of boundary layers for disturbances of various sizes.

Proc. R. Soc. Lond. A368, 573 (also corrections 1980, A371, 439-440).

Smith, F.T. 1989 On the first-mode instability in subsonic, supersonic or hypersonic

boundary-layers. J.Fluid Mech 198 127-154.

Smith, F.T. & Brown, S.N. 1989 The inviscid instability of a Blasius boundary layer at

large values of the Mach number. J.Fluid Mech. submitted for publication.

Stewartson, K., 1964. Theory of laminar boundary layers in compressible fluids. Oxford

University Press.

Stewartson, K. 1955 On the motion of a flat plate at high speed in a viscous compressible

fluid - II. Steady motion. J. Aero. Sci. 22 303-309.

37
0

_r_

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CIO

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c_

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38
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39
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41
IV

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0(_ -') I

III O(&;-)'

/

o(a_))

/

II o0)

/\

_m

wall layer

/

\ \ \

Figure (3.3) The different regions which emerge in the small & limit.

42
Figure (3.4) The first three acoustic mode eigenfunctions. Modes 1,2,3 have

0,1,2 zeros in (0, oo) respectively.

0.5

0.0

-0.5

-1.0

I I I I I I I

0 2 4 6 8 10 12 _"

43
O_t_

1.0

0.8

0.6

0.4

0.2

0.0

1 I I I I

-4 -2 0 2 4

h

Figure (3.5a) The real part of the solution of (3.17) for -5 < &l < 5.

44
0.0

-0.2

-0.4

-0.5

-0.8

-1.0

-1.2

B

J

I t I t I

-4 -2 0 2 4

Figure (3.5b) The imaginary part of the solution of (3.17) for -5 < &_ < 5.

45
C_

0.998

0.997

0.996

0.995

0.994

0.993

0.992

E

I I I I I I I

100 t50 200 250 300 350 400

L

Figure (3.6a) The real part of the wavespeed co as a function of L for

100 < L < 500 and the case of an insulated wall.

46
CO_

0.0008

0.0006

0.0004-

0.0002

0.0000

I

IO0

I t 1 I t I

150 200 250 300 350 400 L

Figure (3.6b) The imaginary part of the wavespeed co as a function of L

for 100 < L < 500 and the case of an insulated wall

47
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\
Report Documentation Page

1. Report No.

NASA CR-18205 ]

ICASE Report No. 90-40

4. Title and Subtitle

2. Government Accession No.

ON THE INSTABILITY OF HYPERSONIC FLOW PAST

A FLAT PLATE

7. Author(s)

Nicholas Blackaby

Stephen Cowley

Philip l/all

9. Performing Organ_ation Name and Address

Institute for Computer Applications in Science

and Engineering

Mail Stop 132C, NASA Langley Research Center

Hampton, VA 23665-5225

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Langley Research Center

Hampton, VA 23665-5225

3. Recipient's Catalog No.

5. Reloo. Date

May 1990

6. Performing Organization Code

8. Performing Organization Report No.

90-40

10. Work Unit No.

505-90-21-01

11. Contract or Grant No.

NASI-18605

13. Type of Report and Period Covered

Contractor Report

14. Sponr_'ing Agency Code

15. Supplernen=_ Notes

Langley Technical Monitor:

Richard W. Barnwell

Final Report

Submitted to J. Fluid Mechanics

16. Abmrom The instability of hypersonic boundary-layer flows over flat plates is considered. The viscosity of the fluid is taken to be governed by $utherland's law, which gives a much more accurate repreaeacation of the temperature dependence of fluid viscosity ar hypersonic speeds rhea Chapman's approximate liuear law; although at lower speeds the temperature vsriarion of the mean srate is less

pronounced so chat the Chapman law can be used with some confidence. Attention is focussed on the

so-called "vorclclty" mode of instability of the viscous hypersonic boundary layer. This is thought

Co be the fastest growing Inviscld disturbance at hypersonic speeds; it is also believed to have an

aaymprotlcally larger growth rate than any viscous or centrifugal instability. As a starting point

we inveetlgate =he instability of the hypersonic boundary layer which exists far downstream from the

leadi_ edge of the plate. In this regime the shock that is attached to the leading edge of the

place plays no role, so that the basic boundary layer is non-interactive. It is shown that the vortlclty mode of instability of this flow operates on a significantly different lengthscale than that

obtained if a Chapman viscosity law is assumed (see Smith and Brown, 1989). In particular, we find

cha_ the growth rate predicted by a linear viscosity law overestimates the size of the growth rate by

O(M ). Neat, the development of the vorciclty mode as the wavenumber decreases is described, and

it is e_own that acoustic modes emerge when the wsveuumber has decreased from it's 0(1) initial value

to O(M-_). Finally, tile lnvlscid instability of the boundary layer near tile leading edge in the interaction zone is discussed and particular attention is focussed on the strong interaction region

which occurs sufficiently close co the leading edge. We find that the vortictty mode in this regime

is again unstable, and that it is concentrated in the transition layer at the edge of the boundary

layer where the temperature adjusts from its large, O(M2), value in the viscous boundary layer, to

trs 0(1) free stream value. The existence of the shock indirectly, but significantly, influences

the Instability problem by mpdlfylng the basic flow structure in this layer.

17. Key Words (Suggested by Author($}l 18. Diltn'b_ion Statement

02 - Aerodynamics

instability, transition, hypersonic 34 - Fluid Mechanics and Heat Transfer

19. Security Clusit (of this report}

Unclassified

NASA FORM 1(_S OCT 8B

Unclassified - Unlimited

_. S_ur_ C_=Lit (o! _ page) 21. No. of _ges

Unclassified 52

22. Price

A04

NASA-Langley,|W0