Triple-deck structure

Computers Fluids Vol. 20, No. 3, pp. 269-292, 1991 Printed in Great Britain. All rights reserved

0045-7930/91 $3.00+ 0.00 Copyright © 1991PergamonPress plc

TRIPLE-DECK STRUCTURE ALl H. NAYFEH Engineering Science and Mechanics Department, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, U.S.A. (Received 19 January 1988; in final revised form 3 July 1990; receivedfor pubfication 14 May 1991)

Abstract--The asymptotic foundation of two-dimensional steady triple-deck theory is reviewed. The total flow is assumed to be the sum of a basic-flow component Q(x, y) governed by the boundary-layer equations and a disturbance component q(x, y), resulting from a sudden streamwise change (localized disturbance), such as that caused by a hump or dip, a suction or blowing discontinuity, a heating or cooling discontinuity, a trailing edge etc. Substituting the total flow Q + q into the steady two-dimensional Navier Stokes equations and subtracting the basic-flow quantitites yields nonlinear equations describing the disturbance quantities q. The disturbance quantities are assumed to vary with the streamwise scale X = (x - l)Re ~, where x = x'Ix*, x* is the center of the localized disturbance, Re = Uo~x r * */v, U* is the freestream velocity and v is the kinematic viscosity of the fluid. Introducing the stretching transformation Y = yRe~,/3 > 0, and scaling ff as ff = ff Re -z, where q~is the streamfunction of the disturbance, we find that there are three distinguished limits when 0 < ~ < ½and hence three sets of least degenerate problems, resulting in a triple-deck structure. These limits correspond to fl = ~t, fl = ~ and fl = ~+ ]~. Matching the pressure in these decks demands that ct = ] and leads to the triple-deck structure. The upper- and middle-deck problems are linear, whereas the lower-deck problem is linear when ~, > 3 and nonlinear when X ~<3 When ~ = ½' there are two distinguished limits corresponding to f l -- _i" g The uPper"deck 2 an~t 3. problem is linear, whereas the lower-deck problem is linear when Z > ~ and nonlinear when ~( ~<~. The theory is self-consistent and does not depend on the problem being investigated. Application of these theories to the linear three-dimensional compressible stability of two-dimensional compressible boundary layers is described and limitations of the triple-deck theory are discussed.

1. I N T R O D U C T I O N The main physical ideas underlying triple-deck theory were laid d o w n by Lighthill [1] in an article on the interaction between a b o u n d a r y layer and a supersonic stream arising from expansive steady disturbances (either an incident wave or a departure o f the wall shape from straight) or from relatively weak compressive disturbances. He treated the interaction mathematically by perturbing a parallel flow and linearizing the N a v i e r - S t o k e s equations a r o u n d it. He produced a coherent self-consistent theory o f this interaction that divides the region o f interest into three parts: the inviscid flow outside the b o u n d a r y layer in which the perturbations are governed by the P r a n d t l - G l a u e r t equation; the displaced b o u n d a r y layer in which the perturbations are governed by the linearized compressible Euler equations; and an inner part close to the wall in which the perturbations are governed by the imcompressible boundary-layer equations. Using this theory, Lighthill [1] indicated how spontaneous changes in the b o u n d a r y layer could be set up which might lead to separation and that the length o f the interaction is the order o f x* Re -3/s, where x* is the location o f the disturbance, Re = U ~* x r */ v * , U * is the freestream velocity and v* is the fluid kinematic viscosity. In this theory, Lighthill assumed a streamwise length scale that is O ( x * Re-~/:), which is the same order as the boundary-layer thickness. G a d d [2] developed an approximate theory to extend the analysis o f Lighthill to the nonlinear case. In the middle part, he determined the velocity and pressure on any steamline using Bernoulli's equation and used an approximate m e t h o d to solve the nonlinear boundary-layer equations in the inner layer. His calculated pressure variation up to and beyond separation is in close agreement with available experimental results at that time. Taking over, without prior justification, the basic ideas o f Lighthill and G a d d , Stewartson and Williams [3] extended Lighthill's theory to nonlinear interactions and put the approximate m e t h o d developed by G a d d on a firm basis. The most important o f these ideas is that the interaction length scale is O ( x * Re-3/s). Thus, they used it as their scaling factor in the streamwise direction instead of the scaling factor x* Re-'/2 used by Lighthill. Moreover, they assumed that the flow is described by three decks or nested b o u n d a r y layers. The middle deck, the displaced Prandtl b o u n d a r y layer, CAV2o~3-G

269

270

A L I H . NAYFEH

has a thickness of the order x* Re- L,=and is characterized by rotational, inviscid disturbances. The fundamental perturbation in this deck is a bending of the streamlines withou t any pressure gradient. The upper deck has a thickness of the order x* Re -3/8 and is characterized by irrotational, inviscid disturbances, and the governing equation of the perturbation in this deck is the Prandtl-Glauert equation. The lower deck has a thickness of the order x* Re 5.8 and is characterized by viscous, rotational disturbances. The wall boundary conditions are satisfied by the lower-deck governing equations. This so-called inner boundary layer is completely incompressible in character and independent of the wall thermal conditions. The consistency of these assumptions was established a posteriori.

Neiland [4] arrived, independently, at the same scalings and the triple-deck structure for the problem of propagation of perturbations upstream of the interaction between a hypersonic flow and a boundary layer. Messiter [5] also arrived, independently, at the same scalings and the triple-deck structure for the flow near the trailing edge of a flat plate. He showed that the solution obtained by Goldstein [6] to the boundary-layer equations downstream from the trailing edge of a flat plate breaks down if the dimensinless distance x * / L * from the edge, where L* is the length of the plate, is O (Re 3..8). He obtained a second-order correction to Goldstein's solution that accounts for the pressure gradient induced locally in the external flow and found that it overtakes the first-order term when x * / L is O(Re-3:8). Alternatively, he assumed the streamwise length scale X = R e ~ x * / L * and investigated the limit of the Navier-Stokes equations as Re--* oo. Matching the resulting expansion downstream with Goldstein's solution, he found that ~ = 3 leads to a distinguished limit and consequently arrived at the same scalings and triple-deck structure. Stewartson [7] discovered the same triple-deck structure for the trailing edge problem. Using order of magnitude arguments, Hunt [8] and Smith [9] investigated the structure of an incompressible flow at high Reynolds numbers (Re) past a hump on an otherwise smooth surface. Hunt [8] investigated the case of a short hump of length comparable with the thickness of the oncoming boundary layer, whereas Smith [9] investigated the case of a long hump of length larger than the thickness of the boundary layer. Smith et al [10] showed that all dominant flow properties in both cases can be obtained as special or limiting solutions of the triple-deck problem. Application of triple-deck theory to the problem of boundary-layer separation was reviewed by Brown and Stewartson [11], Messiter [12, 13], Stewartson [14], Sychev [15], Smith [16, 17] and Ragab and Nayfeh [18]. Application of triple-deck theory to the trailing edge problem was reviewed by Stewartson [I 9] and Messiter [13]. Numerical solutions of the nonlinear triple-deck equations for the flow over a hump were obtained by Sykes[20], Napolitano et a/.[21], Ragab and Nayfeh[22] and Smith and Merkin [23]. In this paper, we review the asymptotic foundation of triple-deck theory. We start with the case of steady two-dimensional incompressible flows. Then, we discuss the case of steady two-dimensional compressible flows. Finally, we discuss the linear three-dimensional compressible stability of two-dimensional compressible boundary layers. For incompressible flows, we derive the equations describing perturbations of the conventional boundary-layer equations due to the presence of a localized disturbance (hump, dip, suction, blowing, trailing edge) at x*. We consider the general length scale X = (x* - x*)Re~/x * , introduce the normal stretching transformation Y = Re~y*/x* and scale the disturbance streamfunction as ~ = R e - ~ ( X , Y). As R e - , m , we find that when 1 I 0 < ~ < ~ there are three distinguished limits corresponding to /3 = e,/~ = ~ and /? = ~+~a. Matching the pressure in the three decks demands that ~ = 3. Moreover, when X > ], the interaction equations in the lower deck are linear, and when Z ~<3, they are nonlinear. When ~ = ½, there are two distinguished limits corresponding to ~ = ½and/~ = -~. When Z > ~ the lower-deck problem is linear and when )~ ~<~ it is nonlinear. Numerical solutions of the first- and second-order triple-deck problems are reviewed and their accuracy and usefulness are discussed. Then, the first-order triple-deck and two-deck problems are presented for the case of two-dimensional compressible flows. It is pointed out that the accuracy of the triple-deck theory deteriorates at finite Re and that the second-order problem produces results inferior to those of the first-order problem. Finally, we point out the limited validity of triple-deck theory to the three-dimensional stability of supersonic and hypersonic boundary layers.

Triple-deck structure

271

2. S T E A D Y I N C O M P R E S S I B L E

FLOW

We consider the steady incompressible flow past a two-dimensional body with a steady disturbance centered at x*. To this end, we define the coordinate system (x*, y*) and the Reynolds number Re = Uo~x,/v . Moreover, we introduce the dimensionless variables U

--

u* U*

v* v-

'

,

U*

P

.

p* - p * p ., U . . 2

,

.

x

.

x* x * .'

y* y

x*'

where the subscript ~ denotes freestream conditions, u and v are the velocity components tangent and normal to the undisturbed surface, respectively, and p is the pressure. In the absence of the disturbance, the flow is governed by the boundary-layer equations: ~U

~P

ax

=0,

(l)

dP 1 82U dx q R e O y 2 '

(2)

U=I?=0

at

y=0

(3)

U~Ue

as

y~oo.

(4)

uOU

l?c~U t?--y-

+

and

Solving equations (1)-(4) yields the basic state

u*

v*

~--~= Re l/2V(x, yRel/2),

U* = U(x'yRel/2)'

p* - p * ~p . U ,

(5)

- P(x).

If a small disturbance, such as a hump, dip, suction or blowing discontinuity, heating or cooling discontinuity, an edge or a corner is introduced, then the flowfield in its neighborhood is perturbed and can be expressed as U* u = U* = U(x, y R e 1/2)+ fi(x, y),

(6)

/)* v = U-~ = Re-mV(x' yRel/2) + z3(x, y)

(7)

and p* - p * p-

,

p Uo~

(8)

= P(x) + p(x, y).

Substituting equations (6)-(8) into the Navier-Stokes equations and subtracting the basic-flow quantities, we obtain 0fi

c3x+V--~-y + u

a/~ 1 /632~ 632t~h . 63~ +Re-'/2V~y+ - - -

8e oy C3fi

+ = = 0,

(9)

(lO)

and U~+

~-x +

- - - - -Re \Ox2 + ~__~yZ,]+ fi ~xx + Z3~yy= 0. ~--y-y+ Re- ~/2V~yy + -by

(11)

We assume that the effect of the disturbance extends over a streamwise distance O(x* Re-'), where c~ > 0, which is small compared with the development length scale x* of the basic boundary layer, so that the disturbance quantities vary with the streamwise scale X = (x - 1)Ret

(12)

ALI H. NAYFEH

272

Then, fi, t3,p ~ 0 as X ~ __ ~ when c~ >0.

(13)

Moreover, the effect of the disturbance decays away from the wall; fi, f,/~ ~ 0 asy ~oo.

(14)

At the wall, the boundary conditions depend on the problem under consideration. It is convenient to combine equations (9)-(11) using the streamfunction @(x,y), defined according to a=--, 8y

e

(15)

= -Tx

Then, it follows from equations (10) and (11) that U3 (V2ff)d2U0ff Ox ~y z Ox

02U0ff

Re-i/2(d~V2+8_~y2~a._~

3x z Ox

\ vx

&y

+ Re-'/2V ~ (V2~) + ay

Ox

(V2~) -

o (V2~) = Re-'V4~. ~x~yy

(16)

The boundary conditions (13) and (14) become Ox and

--,0 as x --* +_m when e > 0

(17)

and

d~ 0 8---~and ~&y

(18)

as y--* ~ .

2. I. Distinguished Limits Any straightforward expansion of equation (16) for large Re breaks down because it cannot satisfy all the boundary conditions at the wall. To determine a uniform expansion, one can use the method of matched asymptotic expansions [24-26] and introduce the stretching transformation Y = y R e B, f l > 0 .

(19)

Because the problem is nonlinear, we scale the streamfunction as =~k Re-Z,

Z>0.

(20)

Using equations (12), (19) and (20), we rewrite equation (16) as

O ( URe'~

a2@ Re 2p

~ 2, O:~,'~ +

a2U ,9~0 ~

-

~ , O2U Oq, r~e c~x2

\

- Re#- ,:2 ~ _ f f _ ~ + R e 3 y ~ j o y + R e , - l / 2 V ~ a¢ a ( _ 2a a2~°

- 2, a2q"~

(l~.4fl~4'~ +2fl e41/j {~41/J~ = Re- '~,..~ d y4 + 2Re2' dX2c3y------7 ¢ Re'~cgX4j,

~Ke ~--y-~--I-Re 2~OX2] aq, a ( ~ 2a a2q'

- 2, a2¢'~

(21)

where Ym= Y Rel/2.

(22)

Triple-deck structure

273

To determine the limit of equation (21) as R e ~ o o , we note that 2Y Re I/2-fl iffl > 1/2 lim U(1 + X R e -~, Y R e - ~ + I / 2 ) = ~ U ( 1 , I'm) iffl = 1/2 R~ (U~(1) iffl < 1/2,

(23)

where ~U 2=~--G-, (1,0) tlm

and

Ue(1)= lim U(I, Ym). Ym~OO

(24)

Letting Re ~ ~ in equation (2 1), we find one distinguished limit [24-26] when ct = 0, corresponding to fl = ½; two distinguished limits when 0t =½, corresponding to fl =½ and fl =~; and three distinguished limits when 0 < ~ < ½, corresponding to fl = ~t, fl = ½ and fl = 1+ g~t. 1 The first case leads to the conventional boundary-layer case and hence it will not be discussed further. The last two cases lead to viscous/inviscid interaction and will be discussed next, starting with the first case. 2.2. The Case ~t =

In this case, the length scale in the direction along the surface is the same order as the boundary-layer thickness. There are two distinguished limits when ~ = ½corresponding to fl = ½and fl = 32-.These limits lead to two decks or nested boundary layers. Lower deck

In this case, fl = 2, YI = ReZ/3 and @~= Re-X@,(X, Y~) + ' " ".

(25)

The distinguished limit of equation (21) depends on the value of X- When X > ~ and R e ~ o o , equation (21) tends to

03@1 2Y, - -

oxov

041//1 --=0.

(26a)

When X = 65-,as Re--,oo, equation (21) tends to

y~ 03@1

O@l 03@1

or, oxer?

0¢103@1 04@1 0 YI4 = O. exov

(26b)

Letting @(X, YI) = ~ (x, y) + Re- 5/6@1(X, Yl), where O~/Oy = U(x, y), we rewrite equation (26b) as

0@ 03@ o@o3@ 0'@ --~0, OY, OXOY~ OXOY~ OY~

(27)

which governs nonlinear two-dimensional incompressible boundary layers. Middle deck

In this case, fl =½, Ym = y R e 1/~ and @m= Re-X@m(X, Y m ) + ' " ".

(28)

As Re~oo, equation (21) tends to ( 03@m

u(1, r )\OXOY:m +

03@m'~ 02U 1

0@~

( , Vm)T£ =0,

(29)

with is Lighthill's equation. Matching the lower and middle expansions yields

@m(X, O) = lim @l()(, Yi)Yl~oc

(30)

274

ALl H. NAYFEH

Outside the boundary layer, U(I, Ym)~U~(1) and equation (29) simplifies to C31//m o-xo

0 3l//m +

(31)

= 0,

whose solution provides a boundary condition for equation (29) as Ym~OC. The decks and scalings for the linear case in this section correspond to those of Lighthill. 2.3. The Case 0 < ~ < ~

In this case, the length scale in the direction along the surface is small compared with L* but somewhat larger than the boundary-layer thickness. There are three distinguished limits corresponding to fl = ~, fl = ½and fl = i + 3~. 1 These limits lead to three decks or nested boundary layers. Lower deck

In this case, fl = ½+ ~ , Yl = Re~ and ~k~(X, Yt; R e ) = Re-Z~b,(X, Y~)+-... Again, the distinguished limit of equation (21) as R e ~ Z >½+2~, as R e ~ o o , equation (21) tends to ).Y~

,~3~, 1 8)(8 Y~

(32)

depends on the value of X. When

0401 - - = O. 8 Y~

(33)

When X = ½ + ~ , as R e p o t , equation (21) tends to =- 0 31~/1 01~/1 0 31]Jl z fl ~ + 0 Y, 8)(8 Y~

001031//i OX 0 YI

0 41~/1 0 Y~ = O.

(34)

If follows from equations (15) and (32) that u ~ = R e ~ ~88~(X, Y 0 + "

(35)

vl=_Re ~ z~(X,Y~)+.-..

(36)

and

Then, the lower limits of equations (10) and (11) lead to p l = Re-Z+ i,.2pl(X ) + ..

(37)

Middle deck

In this deck, fl = ½, Ym = Y Rel/2 and @m(X, Ym; Re) = Re ¢ ZqJm(X, Ym) -t-" ",

(38)

where ~ needs to be determined from matching the expansions in the upper and lower decks. It turns out that ~ = ½~. As R e ~ o o , equation (21) tends to . . 03~/m U(1, Xm)0x---8--17~

02U 01~¢m ~mm(1, Y m ) ~ - = 0 .

(39)

The solution of equation (39) can be expressed as ~1m = A ( X ) U ( I ,

Ym),

(40)

where A ( X ) ~ O as X ~ + oo.

(41)

Triple-deck structure

275

Matching the expansions in the lower and middle decks and using equations (23), we find that ReCA(X)2 Yi Re -'/3 = lim ~kI(X, Yl). Yl~oo

Hence, ~ = ~1 ,

~/m(fl(, rm; Re) = Re-Z+'/3A(X)U(I, Ym) + ' ' "

(42)

lim ~9, (X, Y, ) = 2 Y~.4 (X).

(43)

and Yl~c

If follows from equations (15) and (42) that dU u m= Re~"+½- ZA(X) ~ + ' "

(44)

and U

dd

m

:

__

R e - ~ + 4~/3 - - - ( X ) U ' " .

(45)

dX

Matching the pressure in the lower and middle decks demands that pm(X, 0) = Re -x+

'/2p.(X).

(46)

Then, the middle limit of equations (10) and (11) yields Pm(X, Ym) = Re-Z+

'/2P,(X) -k-'".

(47)

In contrast with Lighthill's formulation that uses a streamwise length scale O(Re-~..z), the general solution of the equations governing the perturbations in the middle deck is given in enclosed form. However, the perturbation in this deck is merely a bending of the streamlines without any pressure gradient compared with Lighthill's formulation which involves pressure gradients.

Upper deck In this deck, fl = ct, Yu = Y Re" and flu( X, Yu, Re) = Re°-Z~k,(X, Yo) + . . . ,

(48)

where 0 needs to be determined from matching the expansions in the upper and middle decks. As Re~, equation (21) tends to

t3----~~ - ~ + ~3X2} = 0,

(49)

which is the same as equation (31) obtained in the case of two decks in the region outside the basic boundary layer. Matching the expansions (42) and (48) in the middle and upper decks demands that Re°~k,(X, 0) = Re'/3A (X) U~(1). H e n c e , 0 = .i~ ,

~°(X, Yu; Re) = Re-X+'/3~ko(X, Y,) + . . .

(50)

g,.(x, o) = A (X)Uo(I).

(51)

and

The solution of equation (49) that satisfies equation (51) and the condition that ~g,u/~X~O as

Y u ~ is

0¢° _ uo_(l) ('~ c3X

rt

A'(~)r.

J_~(X-~)z+y2

de.

(52)

276

ALl

H. NAYFF.H

Hence, it follows from equations (15) and (50) that ~:u=

Re /+4~3Ue([)li" .

A'(~)Y~ (x-

)

2ru

~d£ + . . . -

(53)

and utl~

A'(¢)(X-~) . . .

Re z~4~,,3U¢(1) £/ ~- .

d~

(54)

The upper limit of equation (10) yields ~ bl u

(~p u

{55)

u~(l) ~=_,,x+ ~-x =o. Then, it follows from equations (54) and (55) that p ° = Re z-4~'3U~(l)f' x A ' ( ~ ) ( X - ~ )

d~ + •. ".

(56)

Matching the pressures (47) and (56) in the middle and upper decks demands that ct = 83-and p,(x)=U~(1)rt f:'~x-~A'(~) dG,

(57a)

which can be inverted to give dA dX

_

,31 f~ P'(~d~.

7zCe(l)

.~ .~--~

(57b)

Consequently, letting

O(X, y) = t/'t(X. y) q- Re-3/4OI(X, El), where O ~ t d y = U ( x , y ) , we can rewrite equation (34) as ~Y~,X~Y~

~,XOY~

?~y4

= o,

(58)

which governs nonlinear two-dimensional incompressible boundary layers. We note that the asymptotic expansion based on using the scale X = (x* - x * )Rei/:/x * , which is the order of the thickness of the basic boundary layer, is more general than the aymptotic expansion based on the scale X = (x* - x*)Re3/8x * used to develop the triple-deck theory. Equations (26) and (27) governing the disturbance near the wall obtained in the former case are the same as equations (33), (34) and (58) obtained using triple-deck theory. However, equations (39) and (49) governing the disturbance in the middle and upper decks are degenerate forms of equation (29) obtained by using the former scaling. Consequently, the results obtained by triple-deck theory are expected to be less accurate than those obtained using a streamwise scaling that is the order of the thickness of the basic boundary layer. The problem concerning viscous flows over a flat plate with a small hump situated downstream of its leading edge was first analyzed in the context of triple-deck theory by Smith [9]. He considered the first-order terms in the theory and linearized the lower-deck equations for very small heights of the hump in the lower-deck scaling. He obtained an analytic solution of the linearized problem by using Fourier transforms. Numerical solutions of the nonlinear triple-deck problem for the flow over a hump were obtained by Sykes [20], Napolitano et al. [21], Ragab and Nayfeh [22] and Smith and Merkin [23]. The accuracy of the finite-difference scheme in the work of Sykes [20] is secondorder in the mesh sizes Ax and Ay, whereas the finite-difference scheme of Napolitano et al. [21] is first-order accurate in Ax and second-order accurate in Ay. Ragab and Nayfeh [22] presented a second-order accurate scheme in Ax and Ay for solving the first- and second-order triple-deck problems and the interacting boundary-layer equations (IBL) for a quartic hump at Re = 2 × 105.

Triple-deck structure

277

For a hump whose dimensionless height h = 2.0, there is a small separation bubble on the downward side of the hump. The results show that the first-order triple-deck theory underpredicts the size of the separation region. However, their agreement with the solution of the IBL equations is satisfactory. Including the second-order terms in the triple-deck equations yields results that are in better agreement with those obtained with the IBL equations. For a hump with h = 2.4, the separation bubble is larger than that in the case h = 2.0. The minimum pressure predicted by the first-order triple-deck theory is higher than that obtained with the IBL equations; the reverse is true in the case of the second-order triple-deck theory. The maximum wall shear is underpredicted by both the first- and second-order triple-deck theories. However, the agreement between the triple-deck theory and the IBL equations is definitely improved by including the second-order terms, in contrast with the supersonic case [18], which is discussed in the next section. As the height of the hump increases, one would expect a larger deviation between the results obtained with the triple-deck theory and those obtained with the IBL equations. A second practical problem where triple-deck theory was very useful is that of the stability of flows over bodies with suction through porous strips [27-32]. Napolitano and Messick [27] developed a closed-form solution for the linearized steady lower-deck equations and calculated the wall pressure and shear stress for the case of blowing. Nayfeh et al. [28] obtained a solution for the linearized steady triple-deck equations for one finite-length suction strip and then appealed to the linearity of the problem and used superposition to obtain a closed-form solution for any number of strips. The results obtained with the linearized triple-deck theory are in excellent agreement with those obtained with the IBL equations for suction levels that have been proposed for laminar flow control. Reed and Nayfeh [29, 30] used the closed-form solution for a number of suction strips and developed a numerical-perturbation scheme for determining the stability of flows over flat plates and axisymmetric bodies, respectively. The scheme was used to optimize the suction strip configurations. The numerical-perturbation results are in excellent agreement with the detailed experimental stability results of Reynolds and Saric [31] and Saric and Reed [32].

3. STEADY C O M P R E S S I B L E FLOW In this case, the basic state has the form

u*

v*

U ~ = U ( x , Ym), ~

= R e - i / 2 V ( x , Ym),

T* = T ( x , Ym),

p* --P* p*~ U f

1 yM~'

p* p----~-= Po (x, Ym),

(59)

where P0 T = 1 because the pressure is constant across the boundary layer. Again, if we perturb this basic state, substitute the result into the compressible Navier-Stokes equations and the equation of state for a perfect gas, and introduce the transformations (12) and (19), we find that there are two distinguished limits when ~ = l, corresponding to fl = ½and 3, and there are three distinguished limits when 0 < ~ < ½, corresponding to fl = ~, ½and ½+ ½~. Matching the pressure in the latter case yields ~ = 3. Next, we discuss these two cases separately, starting with the first. 3. I. The Case ~ = Lower deck

In this deck, fl = 3, Y] = YRe2/3 and (ill, /~1,/~1, ~1, ~1) = Re-z+2/3[ut, Re_l/6pl ' Pl, Tj] (X, Yi) + ' " '.

(60)

Then, the conservation-of-mass equation yields ~U 1

~V1

(61)

278

ALl H. NAYFEH

The y-momentum equation yields t3p~/t3Y~= 0 or Pl = P~(x) and the x-momentum equation yields: [ 0ul ] dp, o~2U, po(1,O) U ' ( I , O ) Y t ~ + U'(I,O)v, = - d - ~ + # ( l , )~-y-~z

(62)

where ;~ > ~; or f 0u, + p0(1,0) LU'(I' 0)Ytff~

cgu, 0 u , ] _ - ddp, O2u, U'(l,Olv,+u~-~+Vlc~y~]- ~ + p(1,0) 0y~

(63)

when g = ~. These equations must satisfy the boundary conditions at the wall and their solutions must match the solutions of the equations in the middle deck. We note that to this order the lower-deck equations are the same as those obtained in the case of incompressible flow and that the influence of the energy equation is of higher order.

Middle deck In this deck, fl = ½, Ym= Y ReL'2 and (/~m,~m,P^m, ~m ~r~) : Re Z+ l/2[Hm,Vm,Pm , Pm, Tm](X, Ym) + . , ".

(64)

Then, the conservation-of-mass equation yields (7.J[ ohm + U(I, Ym) ~-x0Pm+ OC~m[po(l, Ym)vm] = O. Po( i, Ym) ~qT..

(65)

The x-momentum equation yields OUm ddy~ (1, U(i, Ym)~f~-'bVrn

p0(1, Ym) I

m

Ym)] + ~opm -~=0;

(66)

the y-momentum equation yields pl(l,

y ~Um C3pm Ym)U(1, m)~ -'~-~mm =

0;

(67)

the energy equation yields p,(l, Ym) I

C3Tm aT 1 ] - ( 7 - 1 ) M 2 ~ U ( 1 , Ym)-~-=0; C3Pm U(l, Ym)~-~+Vm-~m(,Ym)

(68)

and the equation of state yields Tm Pm 7 MLpm = - ~ + - - . P0

(69)

Eliminating tim from equations (65), (68) and (69) and using the fact that P0 T = I, we have ~Vm (~Um ~ 2 - t~Pm ~Ym }- ~ + IVI~ U ~ = 0.

(70)

Eliminating um from equations (66) and (70), we obtain &'m

dU

Mz U2

t~pm --r)--b--T=0

or OYm

+

U2

OX

(71)

Eliminating um from equations (67) and (71), we have ~2pm d m(In M 2) c~pm- (M 2 02pm &Y2m d 63Yrn -- 1)-ff~-=O, where

M = M~U/xfT.

(72)

Triple-deck structure

279

Outside the boundary layer, U = 1, T = 1, M = Moo and equation (72) becomes

a2pm ay2rn

(

M2 a2Pm ~-l)~-f=0,

(73)

which is the Prandtl-Glauert equation. The solutions of equation (73) are expected to satisfy the boundary conditions at infinity. Then, they are used as initial conditions for equation (72) at a Ym = Ye, which is outside the basic boundary layer. The solutions of equations (72) must match the solutions of the lower-deck equations; i.e.

UI(X, 00) = vm(x, 0)

and

pm(X, 0) =pl(X).

(74)

The linear lower-deck equations (61) and (62) and middle-deck equation (72) are due to Lighthill [1]. From these equations he was able to produce a coherent self-consistent theory of an interaction between a weak shock and a boundary layer. He found that the interaction extends upstream to a distance that is the order of Re 3/8. 3.2. The Case ~ - 3 8 In this case, there are three decks corresponding to fl = ~, ½ and 38 [3, 4, 19]. Lower deck In this deck, fl = ~, Y1= ReS/8 and

(/jl, bl,/~l,/31, ~-I) = Re-Z+5/S[ul, Re

1/4

Vl' Re-l/8pl ' Pl, TI](X, Y~) + " ' -

(75)

Then, the lower-deck equations are given by either equations (61) and (62) when X > 43-or equations (61) and (63) when X = 3. Again, the influence of the energy equation is of higher order and the equations in this inner boundary layer are completely incompressible in character but the relative changes in velocity are substantial. Middle deck In this deck, fl = ½, Ym = Y Rel/2 and

(tlm, t3m,/om,pm, ~/'m)= Re z+s'S[um,Re-l/8vm, Re-l/Spm,Pm, Tm](X, Ym)+ "''.

(76)

Then, the middle-deck equations take the form 0Um ~ p,(1, Ym)-ff~- + V(1, Ym)

0 + ~ m [p0(1, rm)Vm] = 0,

(77)

Onm dU U(1, Y m ) - ~ " + U md-y- m (1 , Ym) = 0,

(78)

63pm = 0,

(79)

~rm dT m - t - V mdrmdT P0( 1, Ym) [ U(1, Ym)-~" - - ( 1 , rm) ] - ( 7 - 1)MZU( 1,

C~pm=0 Ym)~

(80)

and Tm Pm 7M~pm = "~- -t - - . P0 It follows from equations (77)-(81) that

um=A(X')U'(I, Ym), Vm=

dA d x U ( 1, Ym), pm=A(X)p'o( 1, Ym), Pro=Pro()(),

(81)

(82)

where A(X)--*O as X--*oe. Matching the lower- and middle-deck solutions, we have pm(X)=pl(X )

and

u~(X, YO--,A(X)U'(1,O)as Yr-,oe.

(83)

This deck constitutes the main part of the boundary layer in which the fundamental perturbation is merely a bending of the streamlines without affecting the pressure gradient.

280

4L) H NAYFEH

Upper deck In this deck, fl = 3 y. = v Re 3'8 and (hu, b",/~u tSo [c")= Re- ~ ""2[uo, G,Pu ,p., T~I(X, r,) + . . . .

(84)

Then, the upper-deck equations take the form 6u u

Op.

,

8Vu

~ y + ~ . ~ + yr~ = o,

(85)

?uo ?p° ~:~ + ) ~ = o,

(86)

(?G 8Pu ~ y + ,~y,~ = o,

(87)

I)M

(88)

8X

(7

ca

0

and 7M~pu = T. + p..

(89)

Thus, the perturbations in this deck are inviscid and irrotational. Equations (85)-(89) can be combined into the Prandtl-Glauert equation '~"2pu (M}, '?:P~ r---~- 1) ~ = O.

(90)

For a supersonic flow requiring the free interactions to be self-generating, we permit only downstream-facing waves. Then, the solution of equation (90) is given by

p,(X, Y~,) = F(X

x/~7, ---1ru).

(91)

Then, it follows from equations (85)-89) that

u,(X, I1.)

-F(X

x/ " M 2~:- 1 r.)

(92)

1F(X - x / M 2 - 1Y.).

(93)

--

and

G(X, Yu) = ~ -

Matching the upper- and middle-deck solutions, we have

F(X)=pm(X)

and

dAdA_ ~/M~2 _ I F ( X ) .

Hence,

dA dX-

4

M2 ~ - Ipm(X)'

which, upon using equation (83), becomes dA dX

_ _

_

_

M

2

--

lp,(x).

(94)

Stewartson and Williams [3] rescaled all the variables to remove the multiplicative constants from the equations, boundary conditions, and matching conditions, thereby reducing the lower-deck problem to a universal form.

3.3. Critique of Triple-deck Theory We note that the equations governing the inner boundary layer (lower deck) in the triple-deck case (~ = 3) have the same form as those obtained in the two-deck case (a = ½). Moreover, the equations governing the upper and lower decks in the triple-deck case are degenerate forms of the

Triple-deck structure

281

equations governing the middle deck in the two-deck case. However, the equations governing the perturbations in the middle deck of the two-deck case are complicated and their solutions need to be obtained, in general, numerically. On the other hand, the equations governing the perturbations in the middle deck in the triple-deck case have a closed-form solution. For supersonic flow past a compression ramp, Jenson et al. [33] obtained numerical solutions to the first-order triple-deck problem. More details were given by Rizzetta [34] for the first-order problem and for other flow situations. Burggraf et al. [35] presented a comparison of calculations based on the triple-deck theory and calculations based on the interacting boundary-layer model for Re ranging from 104 to 109. Their results at Re = 104 are in good agreement with experimental data. They also indicate that the triple deck gives the correct qualitative trends but is quantitatively accurate only at very high Re (agreement is reached at Re = 109). Moreover, Stewartson and Williams [3] remarked that "although internal consistency and rationality are essential for the ultimate success of a theory comparison with good experimental work is also very important. On the present score the present theory is satisfactory as the shape of the pressure curve is concerned but less satisfactory in its estimate of the pressure rise at separation and the scale of the interaction in the x-direction, differences of up to about 20% being found." Moreover, Katzer [36] analyzed the interaction of an oblique shock wave with a laminar boundary layer on an adiabatic flat plate by numerically solving the Navier-Stokes equations for Mach number (M) ranging from 1.4 to 3.4 and Re ranging from 105 to 6 x 105. His numerical results agree well with experimental data but not with triple-deck theory. For finite Re the triple deck tends to overestimate the length scale substantially and the discrepancy increases with increasing M. Stewartson and Williams [3] argued that the theory presented here can be regarded as the uniformly valid first term in an asymptotic expansion in powers of Re-~/8 and retaining more terms might improve the correlation of theory and experiment. Brown and Williams [37] and Ragab and Nayfeh [18] found that this is not the case and including the second term in the expansion worsens rather than improves the correlation in the case of supersonic flows. As pointed out, the governing equations in the inner boundary layer are completely incompressible in character and the influence of the energy equation is of higher order. Hence, the first-order tripledeck problem is independent of the thermal conditions at the wall; they apply to adiabatic as well as heated or cooled walls. To account for the thermal wall conditions, Brown and Williams [37] extended the analysis of Stewartson and Williams [3] and Neiland [4] to second-order for the case of an adiabatic wall. They determined the pressure and its gradient at the point of separation in the free-interaction problem and found that including the second-order terms worsens rather than improves the correlation between triple-deck theory and experiments. Ragab and Nayfeh [18] developed a second-order triple-deck theory for the case of nonadiabatic walls. They also developed a second-order acurate finite-difference scheme for solving the second-order problem. They used "it to calculate the flow past a compression ramp with a small separation bubble. For the case of an adiabatic wall, comparing the results obtained by using the first- and second-order triple-deck problems with results obtained by using the full Navier-Stokes equations [38, 39] or the IBL equations [40], Ragab and Nayfeh [18] found that including the second term in the expansion worsens rather than improves the correlation. Moreover, for the case of cooled walls, they found that the second term in the expansion may produce negative densities, depending on the Re and the degree of cooling, indicating a breakdown of the triple-deck theory. 4. LINEAR STABILITY OF COMPRESSIBLE B O U N D A R Y LAYERS The earliest attempt at formulating a compressible stability theory was made by Kfichemann [41] who neglected viscosity, the mean temperature gradient and the curvature of the mean velocity profile. Lees and Lin [42] and Lees [43] were the first to derive the basic equations for the linear parallel stability analysis of compressible boundary layers. This theory was extended by Dunn and Lin [44], Reshotko [45] and Lees and Reshotko [46]. These early theories were asymptotic or approximate in nature and proved to be valid only up to low supersonic M values. The use of direct computer solutions to exploit the full compressible stability equations was initiated by Brown [47] and Mack [48]. An extensive treatment of the parallel stability theory for compressible flows is given by Mack [48-50]. As M increases, the dissipation terms become important and a three-dimensional

282

AH H. NAYFEH

disturbance cannot be treated by an equivalent two-dimensional method as is usually done for the incompressible case. Mack found that neglecting the dissipation terms can lead to a 10% error in the disturbance amplification rate. Reviews of the compressible stability problem are given by Mack [48-50] and Nayfeh [51]. The most important feature of the stability of supersonic laminar boundary layers is that there can be more than one mode of instability contributing to the growth of the disturbance. The first mode is similar to the Tollmien-Schlichting instability mode of incompressible flows, while the second and higher unstable modes are unique to compressible flows. Mack [48-50] found that there are multiple values of wavenumbers for a single disturbance phase velocity whenever there is a region of supersonic mean flow relative to the distrubance phase velocity. We will refer to these modes as Mack modes. For imcompressible flows, higher modes are associated with higher wavenumbers at different phase speeds. It is an interesting facet of compressible two-dimensional boundary layers that the most unstable first-mode wave need not be parallel to the freestream as M approaches one. At supersonic speeds the most unstable first-mode wave is oblique or three-dimensional. In contrast with the first mode, the most unstable second (Mack) mode is two-dimensional. As M increases to the hypersonic regime, the second mode displays growth rates that are higher than those of the three-dimensional first mode. However, the maximum growth rate is less than that of the first mode at M. In this section, we consider the linear three-dimensional compressible stability of the basic compressible steady state given by equations (59). To this end, we superimpose small unsteady three-dimensional disturbances on the basic state and obtain the total-flow quantities

O(x, y, z, t) = qo(X, y) + O(x, y, z, t), where q stands a disturbance equations and equations and

for u, v, w, p, p, T and #, the subscript zero denotes a basic state and the hat denotes state. Substituting the total-flow quantities into the compressible Navier-Stokes equation of state, subtracting the basic-flow quantities, linearizing the resulting dropping the hat for ease of notation, we obtain

?t + ~x (P°U + puo) + ~0' (poV + pro)+~T~z (P°W)= 0,

(95)

/~u ~. ~Uo ~. ~Uo', ( 0 u 0 ~.0~ po~-~ + ~°~ +"-~x + ~'°G +'-o>, ..)+ P/\"oG + ~o=-/cy j _

~_1)'c_~_~ [ ( BUr+m=+m&'Ow~l ' " 1/ Ou° ~vo'-] ~)-t-(#tr-~x+m~Y)A

Op + R e ( ~ x 0x

~o

~

cy

(3,,0

po ~ + .O yx + ~ ~8v° + vOvv.. + V ~>' I + p Uo~

+ v0-7--/ ~y J

+oX)J

.

+

z+ ~

#o m ~ x + r f f f y + m ~ z ) + l a t m ~ x + r o y j j + # O ~ z z

' (97)

(~,, ~w P° - & - + u ° g - x x + V ° ~ ) .

+~o~?

(m~xx+ , . ~,,y

b(,.0 + ~

+ r - ~ -z

rn

+m

~:v)

,

(98)

283

Triple-deck structure

aT aTo aT po T i + UT;x + UOTx

E

= (7 -- 1)ML

aT° +vo~--y +p Uo ap ap + Re 40 +u°-~xx+V°ay

+

~ePr -~x

~,o ~

+ ~,

-~x) (99)

+ ~ t,~'°~ + ~ ' ~ ) + ~'°az~J and p

T

-

Po

p

t

To

(100)

Po'

where 40 is the perturbation dissipation function defined by

2(ou e, VOuo Ovo]{ L[¢O.oy

(0v0V1

.

a.00,0

la.o a,o]2t

+ \-~y+~x)t~y +axjj+#L L\ ax) +\ay) _]+Zm~x~y+t-~y+~x )j"

(101)

The constants r and m are given by r=~(e+2)

and

m=-~(e-1),

(102)

where e = 0 corresponds to the Stokes hypothesis. The Reynolds number Re and Prandtl number Pr are given by Re=

U'L* oo ~ v*~

and

"* c*

Pr= I~ p ~ '

(103)

where v*, ~* and x * are the freestream kinematic viscosity, dynamic viscosity and thermal conductivity, respectively, and c* is the dimensional specific heat at constant pressure. We assume that the dynamic viscosity is a function of temperature only so that (104)

/~ = T ~-~-~ r0" The boundary conditions at the wall are u = v = w = T =0aty

= 0.

(105)

The boundary conditions on the velocity fluctuations u and w represent the no-slip conditions and the boundary condition on the velocity fluctuation v represents the no-penetration condition. For a gas flowing over a solid wall, the temperature remains at its mean value unless the frequency is small (i.e. stationary or near stationary disturbances). The boundary conditions as y--*oo are

u(y), v(y), w(y),p(y) and

T(y) are bounded as y ~ o o .

(106)

As will be described later, neutral subsonic disturbances decay to zero as y--. oo, whereas neutral supersonic disturbances do not vanish as y---*oo. The basic flow given by equations (59) varies with the streamwise coordinate x = x*/x~* as well as the transverse coordinate y. However, at high Re, the wavelength 2" of the instability wave is short relative to the development length x,* of the boundary layer. This assumption is called the quasi-parallel assumption and results in a major simplification in the linearized Navier-Stokes equations; namely, V0 and all streamwise derivatives of the basic state can be neglected. Consequently, in addition to being independent of t and z, the coefficients of equations (95)-(101) are independent of x and hence they have normal-mode solutions. For a treatment of the nonparallel case, we refer the reader to E1-Hady and Nayfeh [52] and Nayfeh [53].

284

ALl H. NAYFEH

4.1. Orr-Sommerfeld Approach In most stability calculations, one usually assumes the wavelength of the instability wave to be the same order as the boundary-layer thickness and hence uses the length scale ,~,~' = , v * / x * U* instead of x* to normalize all coordinates and 6*/U* instead of ~X'*~/U*r~ tO normalize the time. Therefore, the dimensionless coordinates and time used in the stability analysis arc x*-x*=x*XRe

i,'2 z * - z * = x * Z R e

1:

v*=x*~;,,Re

':

U*t*/x*=-t Re ~2

(107)

Using these dimensionless variables, we express the disturbance quantities in the normal-mode torm

[u, v,p, T, w] = [& ~,fi, T, ~]exp[i(~X + flZ - 02t)],

(108)

where ~, g,/~, T and k are functions of Ym, ~ and fl are the wavenumbers in the streamwise and spanwise directions and 02 is the frequency. For temporal stability, ~ and fl are real and 02 is complex. For spatial stability, 02 is real but ~ and fl are complex. For the general case, ~, fl and 02 are complex. Substituting equations (108) into equations (95)-(101), (105) and (106), using equation (104), eliminating p and dropping the tilde for ease of notation, we obtain the eigenvalue problem

Dv + i ~ u - T DT0 o v +i(~Uo-02) ( ? M ~ p - ~oo) T + ifiw = 0, i(ocUo - 02)u + vDUo + i~Top

~

To

(109)

[-/~0(r~ 2 + flZ)u

- ~fl#0(m + l)w + i(m + l)~#oDToDv + u~DuDTo

+io~#'ovDTo+l~oDZu + D(u'oDUo)T +p~DUoDT] = 0 , i(~Uo - 02)v + ToDp

(110)

x /To ~ [i(m + 1)~#0Du

+ irn~p~uDTo - (o~2 + fl2)#oV + rl~'oDToDv +imfll~'owDTo+i~#'oDUoT+r#oDZv +i(m + 1)fl/~oDw] = 0.

To

i(~Uo- 02)w + iflTop

~

,/Re

(111)

[i(rn + l)~fl/~ou

+iflp'ovDTo+i(m + 1)fl#0Dv - #o(~2 + rfl2)w + #~DToDw + #0D2w] = 0 ,

(l 12)

i(~Uo - co)T + vDTo - i(7 - 1)To M2~ (~Uo - co)p - (7 - 1)M~ T--~°~[2#oDU0(Du + i~v) + #~(DU0)2T]

,/Re

To

- Pr~-~

[ - #o (~ 2 + fl 2)T + D(#o D T) + D(# ~D To T)] = 0, u=v=T=w=Oat

Ym=0

( 1 13)

(114)

and u, v, w, p, T are bounded as Y m ~ ,

(1 15)

where D = d/d Ym. Given the basic flow and four of the parameters ~t,, ~ , fl,, fl~, 02, and 02~, where the subscripts r and i refer to the real and imaginary parts, one can solve the eigenvalue problem to determine the remaining two. There are a number of numerical techniques for determining solutions to the eigenvalue problem including shooting and finite-difference methods[51]. Moreover, there are asymptotic solutions of this eigenvalue problem, one of which is triple-deck theory [54, 55]. As mentioned earlier, the other asymptotic solutions are found to be valid up to low supersonic M values. The triple-deck theory will be shown to be more limited in its validity.

Triple-deck structure

285

In contrast with the case of incompressible boundary layers, compressible boundary layers, even on flat plates, have an inviscid instability, which increases with increasing M [48, 50]. The inviscid instability is governed by equations (109)-(115) with Re being set equal to infinity; i.e. it is governed by

(

Dv+i~u-

+i(~Uo-o9) ?M~P-~o ° +iflw=O,

i(~Uo-m)T+v

(116)

i(~Uo - tn)u + v DUo + i~Top = O,

(117)

i(~tUo - og)v + ToDp = O,

(118)

i(~Uo - ~o)w + iflToP = O,

(119)

D T 0 - i(7 - 1)ToM~(~tUo-cO)P = 0 , u=v=w=

T = O a t Ym=O

(120) (121)

and u, v, w,p, T are bounded as Y m ~ .

(122)

Equations (116)-(120) can be combined into the following second-order differential equations governing p and v: D2p - D(ln Mr2) Dp - k2(1 - M~)p = 0

(123)

l n l _ M~ M ~ j "] Dr" - k2(1 - M~)( = 0,

(124)

( ~ U o - ~o)M~ kx/~0 '

(125)

and D2(+D where k 2 _ ~2 ..[_ f12,

Mr -

v ~ = ctG - co'

We note that M, can be interpreted as the local Mach number of the mean flow in the direction of the wavenumber vector k = ~i + flj relative to the phase velocity ~o/k. In general, M r is complex and it is only real for neutral disturbances. Outside the boundary layer, To = 1 and U0 = 1 are constant and hence Mr = M r is a constant and equations (123) and (124) reduce to D2p - k2(1 - M})p = 0

(126a)

D2( - k 2(1 - M})( = 0,

(126b)

and

where M r = (~ - ~o)M~/k.

(127)

We note that equations (126) and (127) agree with the unsteady linearized Euler equations in the freestream. Moreover, we note that M~ may be < 1 even for two-dimensional cases although M~ > 1. It is clear from equations (126) that neutral (~, fl and 09 are real) disturbances decay outside the boundary layer iff M~ < l; these disturbances are termed subsonic disturbances. When M } > l, neutral disturbances do not vanish outside the boundary layer and they represent Mach or sound waves o f the relative flow; they are termed supersonic waves and may be outgoing or incoming waves. When M~ = l, the neutral disturbances are called sonic waves. These classifications are due to Lees and Lin [42]. For the case of two-dimensional inviscid waves in a two-dimensional boundary layer, Lees and Lin [42] established a number of conclusions for the case of temporal waves. First, the necessary and sufficient condition for the existence of a neutral subsonic wave is the presence of a generalized inflection point Ys > Y0 in the boundary layer at which D(p0DU0) = 0, where U0(y0) = l - 1/M~. CAF 203--H

286

AL! H. NAYFEH

The phase velocity Cs = co/~s of this neutral wave is U0(ys), the mean velocity at the generalized inflection point Ys, which is larger than 1 - 1/M~. Second, a sufficient condition for the existence of an unstable subsonic wave is the presence of a generalized inflection point at y~ > Y0; its phase velocity c > 1 - I/M~. Compressible boundary-layer flows over insulated flat plates always have inflection points and hence they are unstable to inviscid disturbances. Third, there is a neutral subsonic wave having the wavenumber ~ = 0 and the phase velocity c = co. Fourth, when M~ < 1 everywhere in the boundary layer, there is a unique wavenumber ~s corresponding to the phase velocity cs for the neutral subsonic wave. Using extensive numerical calculations, Mach [49, 50] established the existence of an infinite sequence of discrete wavenumbers ~n, corresponding to an infinite sequence of discrete modes when M~ > 1 somewhere in the boundary layer. He referred to the modes that are additional to the mode found by Lees and Lin [42] as higher modes and we will refer to them as the Mack modes. In contrast with the first mode whose existence depends on the presence of a generalized inflection point, the Mack modes exist whenever M~ > 1, irrespective of the presence or absence of a generalized inflection point. Later, Mack developed a simple theory that provides an approximation to the infinite sequence of wavenumbers ~n and Nayfeh [51] improved on this approximation by using turning-point analysis. The lowest M at which the Mack modes exist in the boundary layer on an insulated flat plate is 2.2. It turns out that this is the lowest M at which subsonic higher-mode disturbances exist. The lowest of the subsonic modes is called the second mode and it is the most amplified of the Mack modes. We note that the inviscid solution breaks down in at least two regions where viscous effects are expected to be important: near the critical layer (i.e. location where Mr = 0), where the inviscid solution is singular, and near a rigid boundary, where the inviscid solution cannot satisfy the no-slip boundary conditions. To determine a uniform expansion everywhere, 0 ~
Motivated by the fact that the typical dimensional wavelength of the neutrally stable modes along the lower branch in the incompressible boundary layer on a flat plate decreases proportionally to Re 3~ as Re--+ vo, Smith [54] developed an asymptotic expanson of the instability problem by using a streamwise length scale X defined by x * - x * = x * X R e - 3~

insteadof *

*

*

x* - x* = x* X Re

~:

1,4

He also assumed that the time scales as U ~ t /x~ = t Re instead of U~ t*/x* = t Re ~2 This is a very critical assumption as discussed below. Consequently, he argued that as Re-+ ~ , the modes of instability take on the triple-deck form along and near the lower branch of the neutral stability curves. Later, Smith [55] argued that the three-dimensional first modes in supersonic boundary layers take on the triple-deck form at large Re along and near the lower branch of the neutral stability curve. He used the scale z* - z* = z* Z Re -a/s in addition to the above scalings. With these scales, the instability problem takes on the following triple-deck form. Middle deck

In this deck, Y~ = y * R e l a / x * and (Urn

Urn, W m,

T m, T m, pro) = (Urn,

Re-i/8

Um'

Re-l/8

Wm '

Re-1iSprn ,

T m , Pm)

exp[i(~2X + f l Z -- &t)] + . . . ,

(128)

Triple-deck structure

287

where ~ and fl are the streamwise and spanwise wavenumbers and 05 is the frequency. Owing to the new scalings, = ~ R e -1/8, f l = / ~ R e -x/8 and

c o = o S R e ~/~,

(129)

where e, fl and co are the parameters used in the preceding section. Hence, the dimensionless wave velocity is O(Re -~/8) and tends to zero as Re--, oc, in disagreement with the inviscid results. With these scalings, the middle-deck problem is Drn

Vm"Jr i ~ u

Drn To

(130)

i~Uou m q- vmDmUo = 0,

(131)

Dmprn = 0,

(132)

iYtUoWm + iflToPm = 0

(133)

iotUoTm + vrnDmTo = 0,

(134)

-

To

Urn

i~ Uo rrn

- 0,

m -- -

-

-

To

and where D m= d/dyrn. Hence, um=ADrn~,

Vm=-i~AUo,

pm = const

(135a)

and Tm=ADmTo,

flToprn ~U0 '

Win--

(135b)

where A and Pm are constants to be determined from matching. Equations (130)-(134) can be obtained by substituting equations (128) and (129) into the inviscid equations (116)-(120) and keeping the dominant terms. Consequently, these middle-deck equations are degenerate forms of the inviscid equations. We note that time does not appear explicitly in the middle-deck equations due to the different scalings of time and streamwise coordinates; namely x* - x * = x* X Re 3..8 and

U~t*/x* = t Re -1'4.

Moreover, the middle-deck solution is regular except at the wall, implying that the critical layer merges with the wall viscous layer, in disagreement with all known theoretical and experimental results. Upper deck

In this deck, Y. = y Re 3/8 and (uU, vU, w~,pU, TU, p~)--Re-l/8(u~,Vu, W u , p . , T . , p u ) e x p [ i ( ~ X + f l Z - t h t ) ] +

"...

(136)

The factor Re ~/sin equation (136) is required for matching the upper- and middle-deck expansions. Then, the upper-deck problem is Duv u + i~u, + i~(TM~pu - Tu) + ifiwu = 0,

(137)

uu +Pu = 0,

(138)

i~vu + Dopo = 0,

(139)

0~Wu+/~Pu = 0

(140)

and T,-(7-

1

) M ok p o = 0 .

(141)

Again, the upper-deck equations (137)-(141) can be obtained by substituting equations (i 29) and (136) into the inviscid equations (116)-(120) and keeping the dominant terms. As in the middle deck, time does not appear explicitly in this deck due to the different scalings of time and streamwise distances.

288

ALI H. NAYFEH

Equations (137)-(141) can be combined into D2uPu_ (~: + f12 _ ~2M2)p ~ = 0.

(142)

It is clear that according to this theory, temporal waves (irrespective of whether they are stable or unstable) decay in the freestream iff~ 2 +/~2 > ~2M2 or fl/~ > x//--~ - 1; i.e. they must be threedimensional and be directed outside the local wave-Mach-cone direction. Based on this, Smith [55] concluded that "Any unstable three- and two-dimensional supersonic waves that are less obliquely inclined are found to suffer from strong nonparallel-flow effects because their length scales are much greater, comparable with the development length of the basic compressible boundary layer." This conclusion is at variance with the results of the inviscid linearized Navier-Stokes equations discussed in the preceding section. It is a consequence of the time scaling being different from that of the streamwise length scales. Had we used the same scaling for both time and lengths, we would have obtained DT,, pu _ [~2+fi2 _ ( ( ¢ _ a~)-M~]pu ~ , 2 = 0 in place of equation (142), which is the linearized form of the Euler equations in the freestream Then neutral disturbances decay in the freestream iff ~ z + f12> ( a _ o3):M~ or M~ < 1, where M s = (02 -og)Mo~/x//~T-+fl 2 is the local Mach number of the freestream in the direction of the wavenumber vector ~i +/qj relative to the phase velocity. This condition is the same as that found in the inviscid case. It can be satisfied even for two-dimensional waves. The decaying solution of equation (142) can be expressed as P u = B e x p [ - ( d 2 + 8 2 _ a M ~ ) - 22 ,:2yu],

(143)

where B is a constant and the real part of the radical is positive. Then, it follows from equation (139) that . (~2 ..{_ 1~2 -- Ra2"lt ar21Vlm},l/2

Vu = --l

B e x p [ - (o22+ 82 - ~2M2)1"2 Yu].

(144)

Matching the pressure in the middle and upper decks yields Pm = B,

(145)

whereas matching the transverse velocities in these decks yields ~2A = B(~2 +/~2 _ ~ZM~),/2"

(146)

Then, it follows from equations (145) and (146) that ~ZA =pm(~ 2 + 82 -- &2M2)'/2.

(147)

We note that according to this theory, there is no inviscid instability because time does not appear explicitly in the upper- and middle-deck problems. If one enforces the inviscid conditions that v m and T mare zero at the wall, one finds from equations (135) that A = 0 and hencepm = B = 0 according to equations (145) and (147). In other words, there are no inviscid disturbances, which is a serious limitation of this theory. Moreover, as mentioned earlier, the middle-deck problem is regular except at the wall and hence the critical layer merges with the wall viscous layer, contrary to all known theoretical and experimental results. Therefore, to determine the instability, one must include the viscous effects by considering the lower-deck problem. L o w e r deck

In this deck, Yl = YRe~/8 and (u l, v I, wl, pL, T 1, pl) = (ul, Re -':4 v,, w,, Re- 1/8pl, T~, pOexp[i((tX + I~Z - o3t)] + . . . .

(148)

Triple-deck structure

289

Then, the lower-deck problem is

i(tu! + DIV 1+ i~wl = 0,

(149)

i[02U;(1, 0)I:1 - &lu~ + U;(1, 0)vl + i~T0(1, 0)p~- T0(1, 0)#0(1, 0)D~ul = 0,

(150)

DIp1 = 0,

(151)

i[~U;(1, 0)YI - o)]Wl"~ i/~T0(1, 0)p, - r0(l, 0)#0(1, 0)D~ w, = 0

(152)

and UI =

UI =

W1=

0

at Yl =

O.

(I 53)

As in the steady case, the influence of the energy equation is of higher order. Matching the streamwise velocities in the middle and lower decks yields

uI~AU~(1, O) as YI~ oo,

(154)

whereas matching the pressure and spanwise velocities in these decks yields P, =Pm = :iZA(~2 +

]~2 __

o22M2)-1/2

(155)

and w:-o

:T0(1, 0)Pm as YI~ oo. ~U~(1, 0)YI

(156)

Equations (149)-(152) and the boundary conditions (153)-(156) complete the specification of the eigenvalue problem. This eigenvalue problem can be solved either analytically using the Airy function of the first kind or numerically. The numerical results obtained by Smith [55] using the triple-deck theory do not agree with those obtained by numerically solving the viscous stability problem given by equations (109)-(115) or the inviscid stability problem given by equations (116)-(122). He concluded that in the supersonic case first modes of instability must be three-dimensional and be directed outside the local waveMach-cone direction. Furthermore, he found that the dimensional length scale 2" of first-mode waves is of the order Re -3/8Moo 15/4, implying that 2* O(1) at M~ = O(Re~:°). Hence, when R e = 2 . 2 5 x 106, 2 " = O ( 1 ) at M ~ = 4 . 3 . As a consequence, he cast some doubt on the computations above M~ = 2 and cast much doubt on the many results based on equations (109)-(115) (Orr-Sommerfeld results) at higher M~ values up to 10 and beyond. Because the triple-deck theory is the first term in an asymptotic expansion of the Orr-Somerfeld equations (109)-(115) in powers of Re -~/8 with special scalings (129) for the streamwise and spanwise wavenumbers and the frequency, the discrepancy between the asymptotic results and the full solutions of the original equations should be interpreted to be a limitation of the triple-deck theory rather than the other way around. As discussed earlier, one of the reasons for this discrepancy is the difference in scalings of time and length scales; i.e. co = O(Re-~/4), ~t = O(Re -3/8) and fl = O(Re 3/8).These scalings imply that the phase velocity is O(Re -'/8) and hence, as Re--,oo, the phase velocity tends to zero and the critical layer merges with the wall viscous layer. Moreover, time does not appear explicitly in the upper- and middle-deck problems, which resulted in the conclusion that first modes of instability must be three-dimensional and be directed outside the local wave-Mach-cone directions. Solutions of the Orr-Sommerfeld equations (109)-(115) [48-51, 56, 57], which are the linearized Navier-Stokes equations with the limitation that the length scale of the instability waves is small compared with the development length of the basic boundary layer, and the full Navier-Stokes equations [58] do not support these conclusions. The calculated phase speeds and the streamwise wavenumbers do not tend to zero as Re---,oo. Moreover, the critical layer does not merge with the wall viscous layer. Even without the unsteadiness and the questionable scaling of time, Stewartson and Williams [3] concluded that the triple-deck theory is only satisfactory so far as the shape of the pressure curve is concerned but less satisfactory in its estimate of the pressure rise at separation and the scale of interaction in the streamwise direction. Also the interacting boundary-layer solutions of Burggraf et al. [35] indicate that the triple deck gives the correct qualitative trends but is quantitatively .~_

290

ALl H. NAYFEH

accurate only at very high Re, of the order of 10 9. Furthermore, Katzer [36] concluded from his comparisons of numerical solutions of the Navier-Stokes equations and results of the triple-deck theory that, for finite values of Re, the triple deck tends to overestimate the length scale substantially and that this discrepancy increases, with increasing M. Stewartson and Williams [3] suggested that the discrepancy between experimental results and triple-deck theory may be due to the Re being only 104, while Re must be large (about 108 is called for) for the theory to be valid. Since formally their theory can be regarded as the uniformly valid first term ira an asymptofi~ expansion in Re ~.8,Stewartson and Williams [3] suggested that keeping more terms might improve the correlation. However, Brown and Williams[37] and Ragab and Nayfeh[18] found thai. including the second term worsens rather than improves the correlation of the triple-deck theor~ with experimental data and solutions of the interacting boundary-layer equations. Moreover, ~)~ the case of cooled walls, Ragab and Nayfeh [18] found that including the second term in th,~" expansion may lead to negative densities. 5~ C O N C L U D I N G

REMARKS

Recapping, we note that there are three distinguished streamwise length scales: (a) a length scale comparable to the development length L* of the basic boundary layer; (b) a length scale that is small compared with L* but larger than the boundary-layer thickness 6"; and (c) a length scale that is comparable to 6". The first case corresponds to higher approximations of the conventional boundary-layer theory. The second case produces a triple-deck structure. The upper-deck problem is inviscid and irrotational, the middle-deck problem is inviscid but rotational, and the lower-deck problem is governed by the conventional boundary-layer equations but subject to novel boundary conditions. The third case produces two decks: an upper-deck problem governed by the Euler equations and a lower-deck problem governed by the conventional boundary-layer equations. Thus, the difference between the last two cases is in the equations governing the inviscid disturbances. The upper- and middle-deck problems in the triple-deck structure are degenerate forms of the Euler equations. However, the middle-deck problem can be solved in closed form and hence the triple-deck problem is much simpler than the two-deck problem, but the region of validity of the two-deck problem is larger than that of the triple-deck structure. A self-consistent theory is presented for triple-deck theory. The theory is general and the scalings are independent of the physical configuration that generates the structure. The streamfunction that governs the total flow is assumed to be the sum of a component 7j* that governs the basic boundary-layer flow and ~p* that governs the perturbations resulting from a sudden streamwise change at x* in the configuration. Substituting ~ * + ¢ * into the Navier-Stokes equations and subtracting the basic-flow terms yields a nonlinear partial differential equation governing tp* The streamwise extent of the interaction is assumed to be characterized by the scale X = ( x * - x * ) R e ' / x * , where Re = U *~ x , * / : v * with U*~ being the freestream velocity and v* being the fluid kinematic viscosity. Introducing the stretching transformation Y = y * R e ~ / x * and the scaling ~9* = Re-zU * x* @(X, Y; Re), we investigate the limit of the differential equation governing 0* as R e ~ o c . We find that when 0 < ~ < ~ there are three distinguished limits corresponding to /3 = ~,/3 = ½and/3 = ~l + ~l , resulting in a triple-deck structure. The equation describing the flow in the upper deck (corresponding to /3 = ~) is linear and the disturbance there is inviscid and irrotational. The equation describing the flow in the middle deck (corresponding to/3 = ~) is linear and can be solved in closed form and the disturbance there is inviscid but rotational. The equation describing the flow in the lower deck (corresponding to /3 = ½+ ½:t) is linear if Z > ~ + ~:~ and nonlinear if ct = ½+ ~a, and the disturbance there is viscous. The asymptotic expansions of the streamfunction and their matchings do not impose any restrictions on the value of ~. Using the momentum equations and the expansions of the streamfunction, we determine the forms and scalings of the expansions of the pressure in the three decks. Matching these expansions demands that ~ = 3. Moreover, the equations governing the disturbance in the lower deck are linear when Z > ¼and nonlinear when Z = 3. The value of ;( and hence the linear or nonlinear behavior of the disturbance in the lower deck depends on the strength of the localized disturbance that generates the triple-deck structure. work was supported by Office of Naval Research under Contract No. N00014-85-K-0011. NR 4324201. The author is grateful to Dr Jamal Masad for his comments on the manuscript.

Acknowledgements--This

Triple-deck structure

291

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