.- 

NASA Technical Memorandum 102128 

I 

1 ICOMP-89-15 

,- 

. The Upper-Branch Stability of Compressible 

Boundary Layer Flows 

J.S.B. Gajjar 

Iowa State University 

Ames, Iowa 

and Institute for Computational Mechanics in Propulsion 

Lewis Research Center 

Cleveland, Ohio 

and 

J.W. Cole 

Iowa State University 

Arnes, Iowa 

August 1989 

(NASA-TG10212d) THE UPPER-BRANCY STAuIAITY 

nF CUMPRESSI~LE BWNDARY LAY~Q ~~nlnls (VASA. 

Lewis Research Center) 36 n C5CL 703 

LEWISRWARCH CENTER 

CAPWESTERN 

RWRVE UNIVERSITY * 

N89-2q74d 

unclas 

G3/36 0225361 
The Upper-Branch Stability of Compressible 

Boundary Layer Flows 

J.S.B. Gajjar* 

Mathematics Department 

Iowa State University 

Ames, Iowa 50011 

and Institute for Computational Mechanics in Propulsiont 

Lewis Research Center 

Cleveland, Ohio 44135 

and 

J.W. Cole* 

Mathematics Department 

Iowa State University 

Ames, Iowa 50011 

ADSTRACT 

The upper-branch linear and nonlinear stability of coin1xessil)le boundary layer flows 

is studied using the approach of Smith SC Bodonyi (1983) for a similar incompressible 

problem. Both pressure gradient boundary layers and Blasius flow are considered with 

and without heat transfer. and the neutral eigenrelations incorporating compressibility 

effects are obtained explicitly. The compressible nonlinear viscous critical layer equations 

are derived and solved numerically and the results indicate some solutions with positive 

phase shift across the critical 1a.yer. Various limiting cases are investigated including the 

case of much larger disturbance amplitudes and this indicates the structure for the strongly 

nonlinear critical layer of the Benney-Bergeron (1969) type. Finally we show also how a 

match .with the inviscid neutral inflexional modes arising from the generalized inflexion 

point criterion, is achieved. 

*Permanent address: -Mathematics Department, Exeter University, Exeter EX4 4QE, England. 

work funded under Space Act Agreement C99066G. 
$1 Introduction. 

The motivation for the present work arises from the need to consider the stability of 

compressible boundary layer flows but within the modern high Reynolds number asymptotic framework. Many of the earlier classical approaches to the stability of incompressible 

boundary layer flows (see for example Reid (1965)) have now been largely superseded by the 

more self consistent asymptotic methods based on triple-deck or multi-deck ideas. Smith 

(1979a,b) was the first to a.pply such techniques to stability problems and he showed how 

the triple-deck structure governs the lower branch stability of boundary layer flows. This 

then allowed for weakly non-linear effects, strongly nonlinear effects, effects of the nonparallelism of the basic flow to be assessed systematically. Subsequent papers have applied 

the same basic concepts but with more complicated structures to consider the upper-branch 

stability of incompressible boundary layer flows, see Smith & Bodonyi (1980), Smith & 

Bodonyi (1982), Bodonyi, Smith & Gajjar (1983); the lower- and upper-branch stability 

of three-dimensional boundary layer flows, see Stewart & Smith (1987), Bassom & Gajjar 

(1988). 

As far as the stability of compressible boundary layer flows is concerned there have been 

several contributions by Lees & Lin (1946), Lees 8~ Reshotko (1962), Reshotlco (1962), 

albeit on classical lines. Pate & Schueler (1969) and Kendall (1975) present the relatively 

few experimental studies of compressible boundary layer transition. Most of this and subsequent work is extensively reviewed by Reshotlco (1976), and Mack (1984,1986). However 

an important distinction between compressible and incompressible boundary layer flows 

is the existence of unstable linear inviscicl modes satisfying the generalized inflexion point 

criterion, see Lees & Lin (1946), or Mack (1984). This would suggest that many of the 

linear results obtained by Lees & Lin are still valid. 

Our main objective in this paper is to extend the work of Smith & Bodonyi (1982) who 

considered the upper-branch stability of incompressible boundary layer flows, to compressible flows. We address upper-branch stability rather than lower-branch stability because 

the disturbance structure for upper-branch modes is more clcsely related to that for the 

inviscid modes than is the triple-deck structure. The importance of the upper-branch and 

inviscid modes is highlighted by the fact that these modes have shorter scales and higher 

2 
growth rates than the lower-branch modes. For a discussion of some aspects of the lowerbranch linear instability properties of compressible boundary layers see the work by Smith 

(1987). Malik (1987) presents some recent computational results for hypersonic boundary 

layers. 

There have been very few studies of nonlinear effects in the stability of compressible 

boundary layer flows and in this paper we address at least some aspects of the nonlinear 

theory relevant for modes on the upper-branch. We show that the neutral modes are now 

governed by the properties of the compressible critical layer equation. The present work 

provides the foundation for studying nonlinear inviscid modes of either the generalized 

inflexion point type or the Benney-Bergeron (190) type. In fact the scalings and structures 

for studying the latter can be deduced from the limiting cases that we consider. 

Starting with the Smith & Bodonyi (1982) paper, hereafter referred to SB, the most 

obvious strategy for studying the effects due to compressibility is to gradually increase 

the Mach number until significant changes from the incompressible theory occurs. For 

general pressure gradient boundary layers we find that the first major contributions due 

to compressibility occur when the Mach number becomes O( l), for flows with and without 

heat transfer. In the outermost potential flow region the controlling equation is now the 

Prandtl-Glauert equation. Also if there is heat transfer then the density and temperature 

disturbances become singular in the vicinity of the critical layer. This then leads to an 

additional contribution to the disturbances shearing stress which balances the usual contribution at the wall and gives rise to the dispersion relations. The total contribution to 

the disturbance shearing stress at the critical layer is now proportional to the quantity 

D(~BDUB) occurring in the generalized inflexion point criterion. Here D = a/aY, where 

Y is the boundary layer coordinate, and Un,p~ are the basic streamwise velocity and 

density respectively. For linear neutral modes to exist we require that the flow must be 

subsonic and that D(~BDUB) must be negative. Expressions for the neutral frequencies 

and wavenumbers are obtained explicitly for both the linear and nonlinear theories. In 

the nonlinear theory these neutral modes depend on the amplitude of the disturbance 

indirectly through the phase shift across the critical layer, as in SB, but now the phase 

shift has to be determined by solving a generalization of the Haberman (1972) equation 

3 
'- 

heat transfer. The nonlinear theory for pressure gradient boundary layers is considered in 

section 3 where we also derive the compressible critical layer equation. The properties of 

this equation are studied in section 4 and finally we conclude with some brief additional 

comments in section 5. 

PRECEDING PAGE BLANK NOT FILMED 

5 
§2 Basic Equations and Linear Theory. 

The Navier-Stokes equations are nondimensionalized as follows with the suffix 00 denoting local free stream values and an asterix denoting dimensional quantities. We introduce 

non-dimensional Cartesian coordinates (x*, y*) = L(r, y), corresponding velocity components (u*, v*) = Uoo(u, v), pressure p, = pooULp, time t, = (L/U,)t, density p* = p-p, 

temperature T, = T,T, coefficient of viscosity p* = pmp, coefficient of bulk viscosity 

pi = pkp’. This gives the non dimensional equations; 

(continuity:) 

l 

(2.la) 

~ (momentum equations:) 

(equation of st ate:) 

7@iP = PT, 

(energy equation:) 

(2. IC) 

where 

au, aup 

dxp ax, 

e,p = - + -. 

The nondimensional constants appearing in (2.1) are Re = U,L/vm the Reynolds 

number, op the Prandtl number, y the ratio of specific heats, M, = Uoo/q the local Mach 

Number with q being the speed of sound. The characteristic lengthscale L can be taken 

to be the local station, say 50. We also assume that the Chapman viscosity law p = CT, 

where C is Chapman’s constant, holds. The Reynolds number Re is taken to be large 

throughout and we set the Prandtl number to be equal to unity. 

6 
92.1 Pressure gradient boundary layer flows with heat transfer and insulated 

wall conditions. 

The basic boundary layer flow is given by, with y = Re-iY, 

' For general pressure gradient boundary layers the following properties of the basic flow 

also hold, 

where the coefficients XI, X2, etc. depend on xo and the Mach number. For insulated wall 

conditions R1 = SI = 0, and if there is heat transfer then SI # 0. The curvature term A2 

depends on the pressure gradient and heat transfer. For further details of the basic flow 

see Stewartson (1964). 

We next introduce infinitesimal disturbances of size 6 say, (6 << 1)) and consider the 

stability of the basic flow. Since our interest is centered on the upper-branch stability of 

compressible boundary layer flows, the most obvious starting point is the work of Smith & 

Bodonyi (1982). By gradually increasing the Mach number, we find that the first significant 

changes due to compressibility occur when M& becomes O( 1). This is true also for Blasius 

flow with heat transfer ( with zero pressure gradient but R1 # 0), although for Blasius flow 

with insulated wall conditions (R1 = A2 = 0) the significant changes due to compressibility 

arise much earlier. This special case is treated separately later in this section. 

7 
So taking the Mach number to be 0(1), it is appropriate to consider the scalings and 

disturbance structure for infinitesimal perturbations of the basic flow. The basic structure 

is as set out in SB, see also Fig. 1, and this continues to hold for compressible flows 

provided we consider wavenumbers of O(ReA) and frequencies of O(Rei), relevant to the 

upper-branch. There are five distinct zones 21-25; the outer potential flow region 21 of 

thickness O(Re-A), the main part of the boundary layer 22 , a thinner inviscid region 23 

of thickness O(Re-A) containing the critical layer; zone 24 the critical layer of thickness 

O(ne-g), and finally zone 25 the wall layer of thickness O(Re-A). As in SB we consider 

disturbances proportional to E = ezp[i(iiX - CT)] , so that 

i 

~ 

where c.c denotes the complex conjugate and 2 = e5X,t = e47, and with the wavenumber 

& and frequency 6 expanded as 

Here e = Re-h is a small parameter. For neutral stability (~0,oO are taken to be real. 

I The major differences here from the work of SB arise because of the additional contributions due to the density and temperature. The disturbances now have the expansions 

in the various regions as: 

- 

' [EUO + . . . , €Co + . . . , ~fi + . . . , EPO + . . . , €60 + . . .] 

in 21 where y = e51j, 

[uo + Ell1 + . . . , €210 + 2211 + . . . ,€Po + €2pl + . . . , 

Po + EPl + . . . ,Bo + E01 + . . .] 

[do) + 

[do) + . * . , €3d0) + . . . , €p) + . * . , Ep + . . . , 

in 22 where y = @Y, 

(2.6) + . . . , E221(0) + E321(') + . . . , ,p(O) + €y1) + . . . , 

,do) + E,#) + . . . , e(o) + €e(1) + . . . ] 

in 23 where y = E'T, 

+ . . . ] 

in 25 where y = c8$, 

8 
together with appropriate expansions for the basic flow using (2.2),(2.3). 

In the outermost potential flow region the leading order equation is now the PrandtlGlauert equation, 

Anticipating the requirement that M& < 1 for neutral solutions and applying a boundedness condition at ij = 00 gives 

with w = (1 - M&)i. Also Po is an unknown constant. 

Solutions in the other regions follow in a straightforward fashion and the important ones 

are given by 

Here we have put the wavespeed co = oO/crO, A0 is an unknown constant, and tf = (XI Y - 

co). Also dl) contains terms which are regular as tf + 0. From (2.8) it is seen that as 

9 
the critical level is approached (( --$ O+), the leading order solutions for the density and 

temperature fluctuations in zone 23 become singular if we have heat transfer, i.e. if # 0. 

See also Lees & Reshotko (1062), and Mack (1084). For compressible flow the coefficient 

of the logarithmic term in the solution for v(') now has a contribution proportional to 

(Rl/Ro) arising from the variation of the basic density or temperature. It can be seen 

that the total contribution is now proportional to ay (pow) Jy=o. a 

Matching between zone 21 and 22 and requiring that the normal velocity fluctuation 

goes to zero at the wall then gives from (2.7),(2.8), 

and 

Hence we obtain the relation connecting the wavenumber and wavespeed 

If ML > 1 then with the solution for outgoing waves from the Prandtl-Glauert equation, the details in the other regions 22-25 remain largely unaffected. However when the 

matching is performed, because of an extra i fa.ctor now introduced in (2.9), this shows 

that no neutral solutions for AIL > 1, with these scalings, exist. With M& = 0 in (2.9) 

this reduces to the relation obtained by SB. 

The singularity in the density, temperature a.nd velocity fluctuations in zone 23 requires 

a thinner region the critical layer of thickness O(E 3 ). However for conciseness we summarize the main result that the solutions in 23 continue to hold for ( < 0 provided we take 

lnt 3 (In - ir) for ( < 0. The details of the linear critical layer may be deduced from 

the nonlinear theory in section 3 below as a limiting case for small amplitudes. The waII 

layer is essentially the same as that in SB and gives the condition 

2 

. 

10 
where now 

= (%)+ e-i% 

and p~g = p(S0) is the coefficient of viscosity at the wall. 

Matching with 23 therefore yields the second relation 

(2.10) 

Combining this with (2.9) we finally obtain the linear neutral eigenrelations for the upperbranch st ability of compressible boundary layer flows, 

and 

., [- ($ + 2) .I 1 ($$)". 

(2.1 1 a) 

(2.1 1 b) 

(2.1 IC) 

For the above relations to hold we require, in addition to the condition for subsonic flow, 

2x2 Rl 

Ai Ro 

that the quantity x = (- + -) be negative. 

If M& = 0, and we set p~g = & = 1,Rl = 0, then the results of SB are obtained. (The 

latter conditions are necessary for consistency with the work of SB where a uniform basic 

density and temperature field is taken.) We note that the (1 - M&)i factors cannot be 

rescaled out of our problem because R1 depends on M:. 

Some interesting limiting cases of the relations (2.11) can be examined further. Firstly if 

x -+ 0- with the Mach number fixed, then from (2.11ab) co - (-x)-3 and CYO N (-x)-i. 

Hence formally if x becomes O(e3 = Re-$) then since the effective wavenumber CY is 

1 

11 
a = c5a0 + . . . , we obtain 

This implies that the critical layer moves away from the wall and into the main part 

of the boundary layer. The limiting neutral structure is then precisely that of classical 

form with inviscid scalings, i.e. the streamwise and lateral scalings are comparable to the 

boundary layer thickness. Also the condition x -+ 0- implies that the above neutral modes 

match on to the neutral inflectional modes which come from the generalized inflexion point 

criterion. The above situation is similar to that occurring in the upper-branch stability 

of three-dimensional incompressible boundary layer flows, Bassom & Gajjar (1988). It 

should be noted however, (as a referee has kindly pointed out ), that by varying x the 

properties of the basic flow are being changed. In Bassom SC Gajjar the Rayleigh scalings 

were obtained by changing the orientation of the three-dimensional disturbance. 

As an example if we consider Blasius flow with heat transfer, then 

where D is some positive constant and sw is the ratio of the prescribed wall temperature to 

the recovery temperature, Stewartson (1964). The condition that x < 0 means therefore 

that the neutral modes exist only for subsonic flow with cooled walls (s, < 1). As we let 

s, + 1- then x + 0-, and the neutral structure shrinks and we go over to the inviscid 

scalings. If we include small pressure gradients then there are some values of s, (sw < s,, 

and swu, > 1) for which the neutral solutions exist for heated walls also. Again in the limit 

s, ---t sw,- we obtain the inviscid scalings. 

I The second limiting case concerns the limit Ad& -+ 1 - . From (2.11ab) we find 

(2.12) 

Formally if (1 - M;) becomes O(E'* = Re-') then again the critical layer moves away 

from the wall. But now the streamwise lengthscale increases and the frequency decreases 

to both become O(1). Since also from the solutions in the outermost region 21 €5y - 

12 
e5a,'(l - M&)-i becomes O(Rei), then this suggests a three tiered structure with a 

streamwise lengthscale of O(l), and regions of lateral extents O(ne-i), O(1) and O(Rei). 

The implications and importance of this transonic regime require further study. Professor 

F.T. Smith (private communication) has indicated that a similar situation arises in the 

transonic regime for the lower-branch modes, see also Smith (1987). 

Lastly we note that if we increase the disturbance size 6 and allow for nonlinear effects 

then, with O(1) Mach numbers, and as in SB the critical layer becomes fully nonlinear 

when S becomes O(ek). The structure and analysis for the nonlinear compressible critical 

layer is considered in section 3. 

Before that however we return to the case of Blasius flow with insulated wall conditions. 

It is clear from (2.11) that with A2 = = 0 the above analysis breaks down. The main 

reason is that the effects due to compressibility start coming in for much smaller Mach 

numbers, because the generalized inflexion point is in the main part of the boundary layer 

for O(1) Mach numbers but retreats to the wall as M& --+ 0. 

$2.2 Blasius flow with iiisuIated waI1 conditions. 

When M& becomes O(I2e-h) then the neutral eigenrelations for the upper-branch of 

Blasius flow with insulated wall conditions will be significantly modified from the incompressible case. A rough argument to show this is as follows. If we consider the neutral 

modes satisfying the generalized point criterion, for Blasius flow with insulated wall conditions, then these satisfy the condition 

dY (pB$$) =o, UB = c, at Y = Yc, (2.13) 

where c is the phase speed of the wave, and p~, VB are the basic density and streamwise 

component of velocity. 

The basic velocity and density are given by 

(2.14a) 

13 
where f(fs) is the Blasius function satisfying 

f"' + ff" = 0, f(0) = f'(0) = 0, f'( 00) = 1, 

and fj is related to Y by the Dorodonitsyn Howarth transformation 

For small fs, (2.13) yields 

where 

Hence 

Now for small c the critica 

fj-Oor 

ayer posit 

(2.14b) 

(2.14~) 

(2.15a) 

(2.15b) 

(7 - 1)X;M& qnJ- 6x4 

on is given by Y - c/XI, and since on the upperbranch of incompressible Blasius flow c - O(Re-h), Smith &. Bodonyi (1980), this suggests looking at the scaling M& - O(l2e-k). 

If we set €1 = Re-&, and M& = a&el, then the basic flow has the following properties, 

-1 

and close to the wall with y = e;'Y, 
where the constants aij, bij can be obtained explicitly from (2.14),(2.15), and in particular 

The linear disturbance structure for incompressible flows is set out in Smith & Bodonyi 

(1980), see also Fig.2, and this work can be extended to compressible flows in a straightforward fashion, using the properties of the basic flow as given above. For brevity however 

only the main points are outlined. The wavenumber and frequency now expand as 

9 9 8 

CY = Rem& - Rem60 +Rem& +. . . , 

Q = ~~$6 nebo + ... , 

respectively, and co = ao/ao is the phase speed. We set 5 = c:Xl and t = and look 

for disturbances proportional to El = ei(ErX1-trl). Then in the main part of the boundary 

layer, the disturbances have the expansions 

.ii = uo + €1211 + . . . 

tj = ElVO + €1V1 + . . . 

II = e1po + €:PI + . . . ) 

P = ElPO + ":p1 +. . *, 

s" = €100 + 

2 

+ . . . , 

Substitution into the Navier-Stokes equations produces the leading order solutions 

where &, Po are constants. By writing down similar expansions for the outermost potential 

flow region and matching with the solutions above gives the relation, 

15 
as in the incompressible case. 

Next in the thinner inviscid region which is part of the main boundary layer containing 

the critical layer, with y = ei'v, the disturbances expand as 

The leading order solutions for u(O), v('),p('), are as in SBl and produce the relation 

after requiring that the normal velocity fluctuation tends to zero at the wall. The solution 

for the density disturbance yields, 

Again as for the previous case studied, the density and temperature fluctuations become 

quite large in the vicinity of the critical layer. The higher order solutions can be obtained 

by successively solving the equations at each order. As in SB1 it is necessary to proceed 

to O(E~) in the expansions for 6, where we find that 

n 

16 
where PLT contains algebraic terms which are regular as Y + co/all, and 

The F1 contribution to the logarithmic term is as in the incompressible case but the 

additional contribution due to the FO term, arises from compressibility effects. 

The critical layer needs to be introduced to smooth out the singularities in the density, 

temperature and velocity fluctuations, but the ensuing result is the familiar '--i7r' jump in 

the logarithmic term across the critical layer. The leading order wall layer solutions are as 

in SB1. After completing the matching, we finally obtain the eigenrelations, 

and 

Substituting for b32,a43 and a44, from (2.16), in terms of the scaled Mach number gives, 

(2.18) 

Hence for the neutral modes to exist we require also from (2.18) that 

The eigenrelations are plotted in Fig. 3a. If a& = 0 then (2.17),(2.18) reduce to the 

expressions obtained by Smith & Bodonyi (1980). Clearly the neutral solutions exist only 

for a certain range of wavenumbers and frequencies. If A?& becomes large, then from 

(2.18) we get 

(2.19) 

where 0 < c,, t << 1. Hence for increasing Mach number the critical layer moves away from 

the wall. When A?& becomes O(Reh) formally, i.e. the unscaled Mach number becomes 

17 
0(1), and from (2.17) and (2.19) it is seen that the wavespeed becomes O(1), and the 

wavenumber and frequency become O( Rei). So again the inviscid scalings are recovered. 

F'rom (2.15a), (2.19) it is clear that in this limit a match with the neutral inflexional mode 

1 for low O(1) Mach numbers is achieved. If we replace EO by c and M by M0 in (2.19) for 

O(1) Mach numbers, then the predicted asymptote agrees well with the computed phase 

shift c from the generalized inflexion point criterion, see Fig. 3b and also Fig. 9.2 in Mack 

I ' (1984). 

18 
53 Nonlinear compressible critical layer equations. 

As in SB when the disturbance size S becomes O(ef) then nonlinear effects in the 

critical layer become important and cannot be neglected. For Blasius flow with insulated 

wall conditions the scalings and disturbance size to produce a nonlinear critical layer are 

somewhat different and are not considered here, but see Cole (1989) for a discussion of 

this case. 

Since we are interested in wavenumbers of O(ReA) and frequencies of O(Rei) we set 

-4 d 

+--E (aofEa1 +...)E, d 

at 

- 

where as in SB the slower a/dx variations do not play any significant role, and can be 

ignored. The constants ao, al, CTO, a1 . . . ) are taken to be real, so that we are considering 

neutral disturbances and the wavespeed co = ao/ao. 

Before proceeding with the critical layer analysis, some brief details of the solutions in 

the other regions are necessary. 

In zone 22, the main part of the boundary layer, with y = @Y, the total expansions for 

the flow quantities now take the form, 

Substitution into the Navier-Stolces equations then produces the leading order solutions, 

19 
The unknown disturbance function i(X) will be set equal to A0 COSX later, but for convenience we assume the more general form for now. 

In the inviscid region 23 containing the critical layer, with y = e7Y, the following 

expansions are implied from (3.1),(3.2): 

The suffix MF denotes mean flow terms (independent of X) which arise because of the 

interaction of the nonlinear critical layer with the outer flow. 

Substitution into the Navier-Stokes equation produces the solutions 

where the relation 

-- Qo dp(o) = Q0AXCO, 

RoX, ax 

is required in order that the do) solution tends to zero at the wall. The density disturbance 

p'"; becomes singular at the critical layer and this gives an additional contribution to the 

logarithmic term at higher order. In fact we have 

~ 

where [ = (XI? - co), A, is an unknown function of X and 

~ 

20 
The solution for u(l) can then be obtained from the continuity equation 

au(0) avo) +-- - 0. aY 

ad1) 

a0 - ax +Wdx‘ 

Hence in the critical layer 24, where y = 2 + EF~, we have from (3.3),(3.4) and (3.5)) 

Substitution into the Navier-Stokes equations then gives, 

At the next order, the governing equations a.re 

together with 

21 
and 

and finally, 

The pressure terms in (3.6) are independent of q to the order indicated. It is at this order 

that the first non-simple solutions are generated which smooth out the irregularities in p(O) 

and dl). Using the previous solutions we obtain after some manipulation the nonlinear 

compressible critical layer equations, 

where F(O) satisfies 

(3.7b) 

We note that for a more general viscosity law, the Chapman constant C in (3.7a) is replaced 

by dT ' 

+(SO) 

From (3.4)-(3.6) the boundary conditions are 

as 77 fw, 

Also matching with 23 gives 

do) = - cy0 Ax CO , 

so that if we set 2 = A0 cosX and introduce the normalizations 

(3.8a) 

(3.8b) 

22 
then (3.7),(3.8) become 

an a2 n - J(ZE - AC &, a0 dR a2 R 2- + sinX- - Ac- - dX az az2 

and 

an an o2 n 

dX aZ 2-+sinX-- AcW = 0, 

(3.10a) 

(3.10b) 

with the boundary conditions 

cos A' 

z 0 N (1 - J)Z + - +Hf as Z+fm, (3.10~) 

(3.10d) 

Here the parameter J = R~A~/(~A~Ro + XlR1). The set (3.10a-d) constitutes an extension to compressible flow of the Haberman (1972) equation. The nonlinear compressible 

critical layer equation differs from the usual Haberman equation because of the extra forcing term on the right hand side of (3.10~~). The parameter J depends on R1, the basic 

density gradient at the wall and A2 the curvature of the basic flow. For problems involving 

heat transfer J is nonzero. Also as in the linear theory the above analysis applies equally 

to Blasius flow with heat transfer when J = $. 

cos x 

z rII-z+- +A' as Z+ztm. 

The important quantity as far as the eigenvalue problem is concerned is the velocity 

jump and phase shift 4, resulting from the set (3.10). 

If we put St = aU/aZ, then from (3.10c), 

22 

U I- (1 - J)T + H*Z + cos X In 121 + @(X) as 2 -+ foo. 

The phase shift 4 is then given by 

and as in Haberman (1972), SB, the following identity can be derived, 

4 + J~H = 2X,(H+ - H-). (3.11) 

Here ~H(X~) is the phase shift from the Haberman problem, with J = 0 above. 

23 
The problem (3.10) has to be solved numerically to determine 4, but as far as the 

eigenvalue problem is concerned we note firstly that In( + (In 1[1+ i4) for [ < 0, in (3.5). 

Then since the leading order solutions outside the critical layer are determined from linear 

equations, the eigenrelations can be obtained as in section 2 with now -T being replaced 

by 4. 

This then yields the relations for neutral stability, 

and 

(3.12a) 

(3.12b) 

(3.12~) 

Clearly for neutral stability we require the condition M$, < 1, i.e. subsonic flow and 

(3.13) 

Finally since the equation for the density disturbance TI is the Haberman equation, 

(A+ - A-) is non zero, and hence there is a jump in the mean density across the critical 

layer. As the parameter Ac goes to infinity we recover the linear results since 4 -+ -7r. 

24 
54 Results and Discussion. 

The problem (3.10) was solved numerically using a Fourier series for Il and C? and solving 

a truncated system using second order central differencing. The discrete equations were 

solved iteratively using relaxation especially for the small A, values. The usual checks were 

made on grid size independence, the number of Fourier terms used etc. The phase shift 

4 and the vorticity jump (H+ - H-) are shown in Fig. (4ab) for several values of J. For 

J = 0 our results agree with those of Haberman (1972) and SB. Solutions were obtained 

for both positive and negative values of J. For J becoming large and negative, 4 becomes 

much more negative than the J = 0 case. For J positive however, for a fixed value of 

A, the phase shift is larger than the equivalent Haberman value. Our calculations also 

indicate that there exists critical values of J = J, > 0 such that for J > J,, the phase shift 

becomes positive for some values of A,. 

Clearly J < 0 implies either R,/Ro > 0 and x < 0, or R1/& < 0 and x > 0. Our 

results indicate that no solutions are possible with the latter case since from (3.13) we 

also require 4 > 0. The former case implies a positive basic density gradient (and hence a 

negative temperature gradient) at the wall, and applies only for accelerated flows. 

If on the other hand J > 0, then we require that Rl/Ro > 0 and x > 0 and hence 4 > 0, 

or RI/Ro < 0 and x < 0 ( and 4 < 0). The numerical results show that there are solutions 

with positive 4 if J > J, > 0 ancl 0 < A, < Acrit(J), where is the value of A, such 

that #(J,X,) = 0. Since Rl/Ro > 0 this implies heated walls only. The other case with 

4 < 0 has solutions only for J < Jc,Acril < A, and requires cooled walls. 

The nonlinear neutral curves will differ considerably depending on the above cases. Some 

sample curves of the nonlinear neutral amplitude against frequency are sketched in Fig. 5. 

It can be seen in Fig. 5a that if J < J, then the behaviour of A0 is qualitatively similar 

to that for the incompressible (SB) case. If however J, < J and 4 < 0 then (Fig. 5b) the 

amplitude reaches a peak and decreases to zero as the neutral frequency increases. Finally 

if J, < J and 4 > 0, then Fig. 5c indicates the possible behaviour. We note that these 

latter solutions are not present in the linear case as the amplitude approaches zero. There 

is a critical value of the neutral frequency below which the nonlinear solutions disappear. 

25 
In the limit J + 00, A, = 0(1), a limiting solution of (3.10) can be found in the form 

= JRi + . . . , 4 = J& + . . . , 

where R1 satisfies 

(4.la) 

with the boundary conditions, 

The phase shift from the solution of (4.1) is shown in Fig. 6 and this is in good agreement 

with the numerical solutions of the full equations, see Fig. 7. For IJI large the above 

predicts large mean vorticity gradients and phase shifts across the critical layer. 

The limit A, + 00, IJI + 00, J = O(A,), can be analyzed in the same manner as the 

linear limit with J = O(1). This then reproduces the linear result that 4 + -r as A, + 00, 

with J = O(A,). 

The last case which merits further consideration is the limit A, -+ 0 when the critical 

layer becomes strongly nonlinear. Our numerical results indicate and also the corresponding incompressible limit suggests expansions of the form, 

Q=fio+A,61 +..., n = I=lo + Acfil + .. . . (4.2) 

Since II satisfies the Haberman (1972) equation, with the same boundary conditions, the 

results of Haberman (1972), Brown & Stewartson (1978), SB, still hold and we get 

The function K(q+) is determined from a secularity condition at higher order. In fact 

where 

26 
Here No is a constant, and (4.4) expresses the fact that the density is constant inside the 

closed streamlines q* < 1, and is continuous across the boundaries, see SB for further 

details. 

After substituting (4.2) into (3.10) the equation for 60 produces 

where Go is an unknown function of q* but again determined from a viscous secularity 

condition at higher order. The equation for fi1 is 

dill dl71 d a60 

dX %* dq* drl* 

- = -JsinX- - -(,Z-), 

with 

= Nl rl* < 1, 

where N1 is a constant and IC1 is an unknown function of 7,. Hence using the above 

expression for 01, and applying a periodicity condition on fi1 gives for q* > 1, 

After using the boundary conditions as q* 3 00 this now gives 

1 

For r]* < 1 the approach adopted by SB, Brown-Stewartson (1978), and Goldstein & 

Hultgren (1988) gives GI, = 0 inside the closed streamlines. From (4.6) GI, becomes 

logarithmically singular at the boundaries of the closed streamlines as q* + 1+, and we 

find that 

27 
Viscous layers are thus necessary near the boundaries of the cat's eye to resolve the 

discontinuity in the vorticity gradient. Although the vorticity gradient is discontinuous 

also near the boundary q* = 1 in the incompressible case, here there is a much stronger 

discontinuity because of heat transfer. Details of the viscous layer calculation are omitted 

here, but in summary we find that the phase shift from the viscous layer alone is 

In16 1 3J 

16 2 -Xcr[J(- + 5) - (1 - -)] 

across each layer. The phase shift across the inviscid region, from q, = 1 to 77* = 00, above 

the cat's eye is 

C(1) 3J 27r 7r Xc[-(l - 2J) - n(1 - 3) + C(2)J - -J + -Jln(16)]. 2 d 3 16 

The same phase shift occurs below the cat's eye. Here C(*) is the constant appearing in 

the incompressible theory, see SB, and 

Our calculations give 

Thus the total phase shift is found to be given by 

The above calculation neglects the smaller contributions from the edges of the cat's eyes 

at X = 0,2r. 

For J = 0 the above asymptote agrees with that in SB. For J # 0, the last term in (4.8) 

represents a contribution from the viscous layers which does not cancel out, in contrast to 

the incompressible case. 

Comparisons of (4.8) with the full numerical calculations are encouraging and the general 

trend for different values for J is correctly predicted, see Fig. 8. 

28 
For fixed Mach number and as A, -+ 0 (3.12),(4.8) shows that the wavespeed and 

wavenumber increase and we recover exactly the scalings of SB and Bodonyi, Smith & 

Gajjar (1983) for the strongly nonlinear critical layer problem. 

Next for fixed properties of the basic flow, the numerical results show that there are 

cases when q5 + fO as A, + kAcpit(J), for O(1) values of Then (3.9) together with 

(3.12) shows that A0 - cyo scol and co - lq5l-i,cy0 - lq5l-i. Thus formally in the limit 

when co becomes O(E-') the disturbance amplitude (egAo) falls to become O(Re-b). In 

this regime the Rayleigh scalings are obtained but with a viscous nonlinear critical layer 

and with zero phase shift across the critical for the leading order stability problem. 

-1 

As x + 0, for A, = O(l), the situation is different from the linear case. Since x --+ 0 

implies IJI + 00 for Rl/Ro # 0, we have that xq5 + R,i/Ro to leading order. Fkom 

(3.13) we need RI/Ro positive and so this limit applies for heated walls only. In order to 

retreive the Rayleigh scalings of the linear theory the limiting process A, + 00, J -+ 00, 

but J = O(A,) is necessary. 

> 

Finally the last case concerns the limit M& + 1 - . Again for fixed A, = O(l), the 

wavespeed and the wavenumber become O( 1) when (1 - M&) N O( l/Re). But from (3.9) 

we have A N cyo scil, so that in the above limit eiA becomes O(1). Thus in the transonic 

regime, for the nonlinear problem, the disturbance amplitude becomes O( 1) comparable 

with the basic flow. The earlier comments regarding the three-tiered structure still hold 

-2 

and again this special case deserves further study. 

29 
$5 Summary. 

We have shown how the work of SB,SBl can be extended to compressible flows. In 

particular the results of SB continue to hold for compressible flows with insulated wall 

conditions until the Mach number becomes O(l), but for Blasius flow with insulated wall 

conditions the effects of compressibility start coming in when the Mach number reaches 

an O(Re-&) value. In the compressible theory the density and temperature fluctuations 

play an important role in determining the properties of the nonlinear compressible critical 

layer and hence the neutral modes. 

Several results with possible important practical implications are that the density and 

temperature disturbances are quite large near the critical layer, and in the nonlinear theory 

the interaction of the critical layer with the outer flow produces mean flow distortions in 

both the temperature and density fields. Also with increased disturbance amplitudes the 

nonlinear critical layer takes on a form similar to that of incompressible flows with open 

and closed streamlines. 

We have seen also how the classical inviscid scalings for the neutral inflexional waves are 

retrieved as the quantity D(~BDUB) goes to zero. The nonlinear disturbance structure for 

this limit, and also that for the strongly nonlinear critical layer, is currently being investigated. This should provide further new results on the nonlinear aspects of compressible 

boundary layer stability. In the strongly nonlinear case, as in SB and Bodonyi, Smith 8z 

Gajjar (1983), the influence of the transverse pressure gradients across the critical layer is 

not negligible, and in addition the effects of the varying density and temperature also has 

to be taken into consideration. How this affects the nonlinear neutral states remains to be 

seen. 

Finally we emphasize that although equilibrium critical layers are a special case, the 

current work provides the scalings and structures for investigating the possibly more important problem of spatially and temporally growing modes. 

30 
Acknowledgements. We would both like to thank the referees for their many helpful 

suggestions regarding the presentation of the work, and also Dr. J.W. Elliott for his 

comments. J.C. is grateful to the Science and Engineering Research Council of Great 

Britain for financial support. J.G gratefully acknowledges the support of ICOMP, NASA 

Lewis, where part of this work was done, and he is also grateful to the Computation Center 

at Iowa State University for a grant which enabled the numerical work in the paper to be 

completed. 

31 
REFERENCES 

Bawm. A.P. & Gajjar, J.S.B., Proc. Roy. SOC. A417 (1988), 179. 

Hcnney. D.J. & Bergeron, R.F., Stud. Appl. Math. 48 (1969), 181. 

Bodonyi. R.J.. Smith, F.T. & Gajjar, J.S.B., I.M.A.J. Appl. Math. 30 (1983), 1. 

Brown. S.N. & Stewartson, K., Geophys. Astro. Fluid Dyn. 10 (1978), 1. 

Cole. J.W., Ph.D. Thesis, Univ. of Exeter (in preparation) (1989). 

Gajjar, J. S.B. & Smith, F.T., J. Fluid Mech. 157 (1985), 53. 

Goldstein, M.E.. Durbin, P.A. & Leib, S.J., J. Fluid Mech. 183 (1987), 325. 

Goldstein. M.E. & Hultgren, L.S., J. Fluid Mech. 197 (1988), 295. 

Habcman. R.. Stud. Appl. Math. 51 (1972). 139. 

Kcnclall. J.M.. A.I.A.A. Jour. 13 (1975), 290. 

Lech. L. & Lin, C.C.. NASA Tech. note 11 15 (1946). 

Lccs. L. & Reshotko E., J. Fluid Mech. 12 (1962), 555. 

Mack, L.M., Agard. Rep. no. 709 (1984). 

black, L.M., “Proc. Symp. on the stability of spatially-varying and time dependent flows, 

(ed D.L. Dwoyer and M.Y. Hussaini) Springer, 1986. 

Malik, M.R. A.I.A.A. pap. no. 87-1414 (1987). 

Pate. S.R. & Schueler, C.J.,, A.I.A.A. Jour. 7 (1969), 450. 

Reid, W.H., “Basic Developments in Fluid Dynamics,” (ed. M. Holt) vol. 1, Academic, 1965 

Rcshotko E.. NASA Tech. note D1220 (1962). 

Reshotlio E., Annual Review of Fluid Mech. 8 (1976), 311. 

Smith. F.T.. Proc. Roy. Soc. A368 (1979a), 573. 

Smith, F.T., Proc. Roy. SOC. A366 (1979b), 9 1. 

Smith. F.T. & Bodonyi, R.J., O.J.M.A.M. 33 (1980), 293. 

Yiiiitli. F.T. & Bodonyi, R.J., J. Fluid Mech. 118 (1982), 165. 

Smith. F.T.. United Tech. Report no. UTRC 87-52 (1987). 

Stewart. P.A. & Smith, F.T., Proc. Roy. SOC. A409 (1987), 229. 

Stcwartson, K., “The theory of laminar boundary layers in compressible fluids. ,” Oxford Univ 

Press. 1064. 

7, 

32 
. 

22 

o(E~) 

T 

FIGURE 1. - SKETCH OF THE VARIOUS REGIONS FOR GENERAL BOUNDARY LAYER 

FLOWS. 21 IS THE OUTER POTENTIAL FLOW REGION, 22 THE MAIN PART OF 

OF THE BOUNDARY LAYER, 23 THE THINNER ADJUSTMENT REGION CONTAINING 

THE CRITICAL LAYER 24. AND 25 IS THE WALL LAYER. 

i2 IT I I 

-./I ........................................ 25 4 4 

o(E~~~Y' k- OQ,% -4 

FIGURE 2. - SKETCH OF THE DIFFERENT REGIONS FOR BLASIUS FLOW WITH 

INSULATED WALL CONDITIONS. 21 IS THE OUT3 POTENTIAL FLOW REGION. 

E THE WIN PART OF THE BOUNDARY LAYER, 23 THE THINNER ADJUSTMENT 

REGION CONTAINING THE CRITICAL LAYER 24, AND E IS THE WALL LAYER. 

33 
0 1 2 3 

-2 6 M, A 1 (Y-1116 CBS I I 

(a) PLOT OF THE NEUTRAL WAVESPEED co AS A FUNCTION OF 

THE SCALED MCH NUMBER Mm. CBS IS THE VALUE OF Eo 

WHEN N, = 0. i.e. THE MITH B BODONYl (1980) CASE. 

THE DOTTED LINE IS THE INVISCID LIMIT FOR INCREASING 

h- , SEE SECTION (2.19). - 

-5 

M, 

(b) THE SOLID LINE IS THE NEUTRAL PHASE SPEED c FROM 

THE GENERALIZED INFLEXION POINT CRITERION AND THE 

DASHED LINE THE ASYMPTOTE (2.19) REWRITTEN FOR (1) 

MCH NUMBERS, SEE SECTION 2. 

FIGURE 3. 

-4--. '-,,, _c_c___- .... ................... ------ 

I 

'. \ 

.... ....... ........ 

........ -2.0 

-1.4 

-0.8 

0.0 

0.8 

1.4 

2.0 

----- 

--- 

- 

.-- 

.._..- 

.- .- 

(a) THE COMPUTED PHASE SHIFT AS A FUNCTION OF h, 

FOR VARIOUS VALUES OF THE PARAMETER J IN ERUATION 

HABERMAN (1972) CASE. 

(3.10). THE SOLID CURVE IS THAT FOR J = 0, i.e. THE 

(b) THE VORTICITY JUMP H*-H- AS A FUNCTION OF h , FOR 

THE SAME VALUES OF J AS IN FIGURE 4 . 

FIGURE 4. 

34 
R 

a. 

(a-c) A SKETCH OF THE POSSIBLE NEUTRAL CURVES FOR THE 

NONLINEAR THEORY. A. IS THE NEUTRAL ARPLITUDE. 

IS THE NEUTRAL FREQUENCY. L.R.BSG DENOTE THE LINEAR, 

RAYLEIGH. AND BODONYI. YITH 8 GAJJAR (1983) LIMITS 

(lo 

AND SCALINGS RESPECTIVELY. (a) J< J,. JC< J AND 

A,, < Ac.(c) J, < J AND o < A , < ~RIT. 

F16URE 5. 

35 
le9 r I 

\ 

\ _______-_-_ --- 

_____---- e--- 

I 

A, 

FIGURE 6. - THE PHASE SHIFT &1(h,) AND THE VORTICITY 

JW (B+-i-) FOR THE CASE OF 1 J I >> 1. SEE SECTION 4. 

3- 

2. 

0 

-2 

0 -4 

Ii" 

a 

0 -0.8 

0 -2.0 

... -2.0 

-6 t 

a J = 0.8 

0 1.4 

- 0 2.0 

J = 2 .. 

0 

-1 

0 .1 .2 .3 .4 .5 

h, 

FIGURE 8. - A COMPARISON OF THE CALCULATED PHASE SHIFT 

WITH THE ASYMPTOTE (4.8) FOR SMALL VALUES OF A,, FOR 

DIFFERENT VALUES OF J. THE DASHED CURVES ARE THE 

ASYMPTOTES AND THE SYMBOLS ARE THE PHASE SHIFTS FROM 

THE NLYERICAL C(WPUTATI0NS. 

36 
~~~~~ _______ 

National Aeronautcs and 

17. Key Words (Suggested by Author(@) 18. Distribution Statement 

Stability Unclassified -Unlimited 

Compressible Subject Category 34 

Boundary layer 

Space Administration 

I 

19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No of pages 22. Price' 

Unclassified Unclassified 38 A03 

Report Documentation Page 

2. Government Accession No. NASA TM-102128 1. Report No. 

ICOMP-89-15 I 

4. Title and Subtitle 

The Upper-Branch Stability of Compressible Boundary Layer Flows 

7. Author@) 

J.S.B. Gajjar and J.W. Cole 

9. Performing Organization Name and Address 

National Aeronautics and Space Administration 

Lewis Research Center 

Cleveland, Ohio 44135-3191 

12. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Washington, D.C. 20546-0001 

3. Recipient's Catalog No. 

5. Report Date 

August 1989 

6. Performing Organization Code 

8. Performing Organization Report No. 

E-49 16 

10. Work Unit No. 

505-62-2 1 

11. Contract or Grant No. 

13. Type of Report and Period Covered 

Technical Memorandum 

14. Sponsoring Agency Code 

15. Supplementary Notes 

J.S.B. Gajjar, Iowa State University, Mathematics Department, Ames, Iowa 50011 and Institute for Computational 

Mechanics in Propulsion, NASA Lewis Research Center (work funded under Space Act Agreement C99066G). 

Space Act Monitor: Louis A. Povinelli. J.W. Cole, Iowa State University. Permanent address of both authors: 

Mathematics Department Exeter University, Exeter EX4 4QE, England. 

16 Abstract 

The upper-branch linear and nonlinear stability of compressible boundary layer flows is studied using the 

approach of Smith & Bodonyi (1982) for a similar incompressible problem. Both pressure gradient boundary 

layers and Blasius flow are considered with and without heat transfer, and the neutral eigenrelations incorporating 

compressibility effects are obtained explicitly. The compressible nonlinear viscous critical layer equations are 

derived and solved numerically and the results indicate some solutions with positive phase shift across the critical 

layer. Various lihiting cases are investigated including the case of much larger disturbance amplitudes and this 

iridicates the structure for the strongly nonlinear critical layer of the Benney-Bergeron (1969) type. Finally we 

show also how a match with the inviscid neutral inflexional modes arising from the generalized inflexion point 

criterion, is achieved. 

*For sale by the National Technical Information Service, Springfield, Virginia 22161 NASA FORM 1626 OCT 86