Supersonic laminar boundary layer separation in a concave corner

Acta Astronautica. Vol. 2, pp. 409-429. Pergamon Press 1975. Printed in the U.S.A.

Supersonic laminar boundary layer separation in a concave corner

M A U R I C E H O L T AND T S A I AN L U t

University of California, Berkeley, CA 94720, U.S.A.

(Received 11 July 1974)

Abstract--Separation and reattachment of a supersonic laminar boundary layer in a concave corner, due to a free interaction process, are investigated by the Method of Integral Relations. Effects of wall cooling and suction are included.

A criterion is established to fix the position of the initial point of interaction between the boundary layer and outer inviscid flow. This is determined by the condition that the integral curve for the pressure passes smoothly through a saddle point just downstream of reattachment.

Velocity profiles, pressure, temperature, and skin friction distributions are calculated for typical values of free stream Mach number, Reynolds number, Prandtl number, wall temperature, and wall suction. In one case comparison is made with experiment and good agreement can be obtained in pressure distributions if the usual interaction equation is modified.

Introduction

SEPARATION OF laminar boundary layers has been the subject of several recent investigations. For flow along a continuous wall separation is always associated with regions of rising pressure. In contrast to flow in accelerated regions the pressure distribution outside retarded boundary layers may not be determined in advance, before viscous effects are considered, but must be calculated simultaneously with the boundary layer growth. As a result, most of the current methods for calculating laminar boundary separation are based on interaction between flow in the boundary layer and recirculating region beyond separation, on the one hand, and inviscid flow in the stream outside the boundary layer, on the other. Up to the present, attention has been concentrated on separation in supersonic flow, since, in this case, the pressure in the outer inviscid flow depends only on local changes in boundary layer geometry. On the other hand, in subsonic flow the outer inviscid flow depends on the whole flow field.

The three principal methods for treating inviscid-viscous interaction in supersonic flow are (i) pure finite difference methods, (ii) triple deck matched expansion methods, (iii) integral methods. Methods (i) are still being developed; calculations of flow in a compression corner have been completed by Dwoyer (1973) using a technique described in Werle and Bertke (1972). Very recently a more sophisticated finite difference technique has been proposed and applied to model problems by Klineberg and Steger (1974). Method (ii) is described in detail in Stewartson and Williams (1969) and is an outcome of Lighthill's early interaction theory (1953). The integral methods are of two types; the first may be referred to as the Lees-Reeves technique (Lees and Reeves, 1964, Klineberg and

tMechanical Engineering Department.

409

410 M. HOLT and T. A. LU

Lees, 1969); it uses Falkner-Skan, Cohen-Reshotko, and Stewartson similarity profiles to represent velocity variations across the boundary layer. The second integral method, called the Hol t -Nielsen technique, is based on an extension of Dorodnitsyn's method of Integral Relations (Dorodnitsyn, 1960) to retarded and separated boundary layers. Its early application to the compression corner problem is described in Nielsen et al. (1966), and Holt (1966, 1967, 1968). A good description of the integral methods is to be found in Brown and Stewartson (1969).

The advantage of the Hol t -Nielsen integral technique is that flow variables across the boundary layer are represented by algebraic relations and when these are introduced the boundary layer equations reduce to a system of ordinary differential equations with algebraic coefficients. The equations are in a form convenient for numerical integration and can be systematically analyzed in the neighborhood of any singular points [such as those arising in wake flows, see Holt and Meng (1970)].

The application of the Method of Integral Relations to the problem of supersonic laminar boundary layer separation on a fiat plate is dealt with in a series of papers by Holt. In the first, Holt (1966), the effect of a small increase in pressure on uniform laminar flow past an adiabatic flat plate is investigated. The introduction of a very small pressure jump initiates a strong coupling between the outer inviscid flow and flow in the boundary layer. At a point just downstream of the pressure jump the boundary layer thickens, causing an increase in the effective slope of the interface between the inviscid and viscous regions. This leads to a further jump in pressure and, a little further downstream, continued thickening of the boundary layer. The close connection between pressure increase and boundary layer thickening causes the boundary layer to separate at a very short distance downstream of the point of initial pressure jump. The process causing this rapid separation is called free interaction and is distinct from the process when separation is dependent on details of body geometry. The second paper, Holt (1967), deals with the same problem when the fiat plate is cooled to different temperatures and shows that the cooling reduces the extent of separated

PRE-SE PARATION POST-SEPARATION

~ REGION ~ REGION

EDGE OF BOUNDARY LAYER-

DIVIDING STREAMLINE

u = O LINE--

" / I I / / I / ] I I I I . / I I I I / I I I I I / I I I /

L

REATTACHMENT REGION~

/ R

-×

Fig. 1. Schematic representation of supersonic laminar boundary-layer separation and reattachment at a concave corner.

Supersonic laminar boundary layer separation 411

regions. The third paper, Holt (1968), is the first attempt to calculate flow in a concave corner and gives details of the recirculating flow right up to reattachment for a series of positions of points where pressure increase is first applied.

The present paper contains the first complete treatment of the whole compression corner problem by the present method, tracing the growth of boundary layer and separated region from the beginning of free interaction to a point downstream of reattachment. Temperature effects are included and the representation of the enthalpy profile used in Holt (1968) is improved. The connection between the initial interaction point and reattachment is established. The former must be chosen so that the integral curves downstream of reattachment approach uniform, constant pressure conditions and the choice is found to be unique. The correct solution in a given case is found by iterating on the position of initial pressure jump, a process which converges quite rapidly.

Calculations are carried out for two compression corner angles, two values of free stream Reynolds numbers, three values of free stream Mach number, and three wall temperature conditions. The effects of all these parameters, and also wall suction are shown in typical cases. Comparison is made with experimental observations by Sfeir (1970). In an attempt to attain closer agreement with Sfeir's pressure distribution the usual interaction equation was modified with surprisingly encouraging results.

Mathematical formulation of the problem

Physical plane The two dimensional laminar compressible boundary layer equations have the

following form.

a(p~) F O(pO) = 0 ( 2 . 1 ) Ox Oy

_ a ~ + _ a a d p , + a a ~

+ pu pv Ty

- a [ a ( a 2 / 2 ) l

Here x and y are independent variables with x measured along the tangent to the body and y along the normal; a(x, y), ~3(x, y) are the velocity components along x and y axes; p(x, y), T(x, y) are the density and temperature of the ideal gas, respectively; p, is the pressure, which is governed by the Prandtl-Meyer relationship at the outer edge of the boundary layer;/z is the viscosity of the gas, assumed to be a linear function of the variable T; Cp, R, tr are, respectively, the specific heat at constant pressure, the gas constant, Prandtl number, all assumed given constants.


Equations (2.1)-(2.3) are supplemented by the equation of state,

Pe = p R T (2.4)

and viscosity-temperature relation

~ = C T No To (2.5)

where C is the Chapman-Rubesin constant. Solution of Eqs. (2.1)-(2.5) are sought which satisfy the following boundary conditions

a (x, o) = o

a(x , o~) = a , ( x )

lim C~T + = Vm~ y ~ 2

(2.6)

T(x, 0) = Tw = const.

As in Holt (1968), both the Stewartson and Dorodnitsyn transformations are applied to these equations. The transformed variables are defined as follows:

Stewartson plane The transformed coordinates in the Stewartson plane (Stewartson, 1949) may

be written as

fO x X = C p~a~ dx poao

y = fY plaz P dy 3o poao p~

(2.7)

which have been modified from the original forms by incorporating the Chapman-Rubesin constant, C, into the X-coordinate. The velocity components in this plane are

U ao al

1 po(ao)2ao__ fYp,a, e a y + V =-~-~ \ a , / 3x Jo poaop~

1 p o a o p _ v (2.8)

C p~al po

The suffixes 0 and 1 refer to conditions outside the boundary layer in the undisturbed and disturbed regions, respectively.


We may fur ther define a dimensionless total enthalpy function s as follows:

GT+ T s = 1 - - 2 (2.9)

C p T ~ 2

Dorodnitsyn plane In order to make all the variables dimensionless in the Stewar tson plane we

make a fur ther t ransformat ion, first introduced by Dorodni tsyn (1960), namely,

fo x U, dX ~= U o l

.0 = - ~ o / It \ 1-'o /

U Ul

(2.10)

, i 19 = - ~ l ~t \ V o /

In addition, v is replaced by w where

w = v + u.0 U1 (2.11)

The length l is a reference length, while U~ denotes differentiation with respect to

The resulting equations in the Dorodni tsyn plane are

Ou + Ow = 0 (2.12) a~ 0.0

OU'~-W~=(lws BU2)U~ 02u u-b-- ~ 0.02 (2.13)

(2.14)

where rn, = {(3' - 1)/2}M, 2, with boundary conditions

u (~:, O) = 0

u (~, ~ ) = 1 (2.15)

s ( ~ o ) = sw = const.


Three possible cases exist for sw (i) sw > 0, which corresponds to a cooled wall. In the special case where

sw = 1 the wall temperature is at absolute zero. (ii) Sw--0, which corresponds to a wall at the free stream stagnation

temperature. This is the case of an insulated plate combined with the condition tr = 1.

(iii) sw < 0, which corresponds to heating problems. This is not considered here.

The boundary layer equations, including heat transfer, in the Dorodnitsyn plane, together with the integral equations which follow, were first stated by Pavlovskii (1962). Suction effects were added by Lyu (1962).

Governing equations and initial conditions

Integral equations After the compressible boundary layer equations have been transformed into

the Dorodnitsyn plane they are ready to be reduced to a system of ordinary differential equations by using the method of integral relations. First, a certain set of weighting functions f(u), which are continuous and differentiable and will be chosen later, are introduced. An equation of divergence form is obtained by multiplying Eq. (2.12) by f(u) and Eq. (2.13) by f'(u), and adding the results. Another equation in this form is obtained similarly by multiplying Eq. (2.12) by sf(u), Eq. (2.13) by sf'(u), Eq. (2.14) by f(u), and adding these three results. The two resulting equations are integrated across the entire boundary layer with respect to the independent variable 7- The variable of integration is then changed from ~ to u by introducing the velocity gradient

du Z = - - 0~7

which is proportional to the shear stress and is a function of u only. The final equations are

d folUf(u) du = f(0) Wo(~) + -~, joUI rl(1-u2)f'(U)duz --~,(J'fo 'sf'(u)du-f'(O)z°z z

- f~ f"(u)zdu (3.1) Jo

= sw/(0)w°(~) + U~, z -U~ , z d~ z

Io s, ,. a s m l

where f(u) belongs to a set of weighting functions which is complete and always


satisfies the condition

lim f ( u ) = 0

The simplest set of f ( u ) is proposed by Dorodni tsyn (1960),

f ( u ) = ( 1 - u ) , ( 1 - u) z . . . .

and can be used in separated as well as at tached flow. The number of functions f ( u ) needed depends on the order of approximation. In the n th approximat ion for at tached adiabatic flow Dorodni tsyn (1960)

represents both z / ( 1 - u ) and (1 - u ) / z as polynomials in u of degree n - 1, so that

n 1 z = ( 1 - u ) ( b o + b , u + . . . + b . ,u ) (3.3)

1 l / z = ) u -----S(1 - (Co + c , u + " • • + c . - , u " - ' ) (3.4)

The fac tor (1 - u) is included to take account of the exponential decay of (1 - u) with ~/at the outer edge of the boundary layer. These expressions have since been modified by Hol t (1966) for separated boundary layer flow. A factor (u + a ) 1~2 precedes (1 - u) in Eqs. (3.3) and (3.4), to account for the fact that d z / d u = ~ at the point of max imum reverse flow velocity u = - a , so that the z ' s are double valued when u < 0 in a separated region, as shown in Fig. 2. In the separated flow region each representat ion is matched at u = -o~ where z and other higher order derivatives of u with respect to r/ are continuous. The second order approximat ion is then found to be

1 ),/~(1 z = - - ( u + a - u ) z > o (3.5) Co

z = - l ( l + o t ) ( u + t ~ ) ~/2 z < 0 (3.6) Co

These expressions are used throughout the present analysis. Their advantage is that all the physical conditions in the problem are satisfied and the integral relations can be integrated across the entire separated layer without requiring the use of any patching technique. This approach has also been applied to the supersonic laminar wake problem by Holt and Meng (1970, 1971).

The formulat ion of the approximating expression for the enthalpy profile can follow the same line of physical and mathemat ical reasoning. Cohen and Reshotko (1956) found in their similarity solution that s is a monotonic function of ~ in both accelerating and decelerating flow. The total enthalpy profiles near a separat ion point are shown in Fig. 2. The distinctive feature of the total enthalpy profile in the s - u plane is that it has an infinite slope at u = - a . It must also be noted that the


1

8u 57

t

s

U

0u

T/

1 u

(0) PRE-SEPARATION

Fig. 2. Velocity, velocity gradient reattachment point.

1

u 1

I

u - ~ 1 u

Ou

u 1 u

s

u -o~ I u

(b) SEPARATION (c) POST-SEPARATION

and enthalpy profiles near a separation o r

wall en tha lpy flux is finite at the separat ion point. In this case

as 1 a u ~ (u + ~ ) ' ~

W h e n we integrate across the b o u n d a r y layer and take into accoun t the a sympto t i c behav io r at the outer edge of the b o u n d a r y layer, then

S -- (N/J1 + or] - N / [ U + o~1)

With the interpolat ion func t ion represen ted by a polynomia l in u, it then fol lows that the approximat ing express ion for the total en tha lpy profile, s, can be given as

s = (~v/[l + ~t] - ~ / [u + t~])(do + d l u + . • • + d , _ l u "-~) (3.7)

N o w , for second order approximat ion , all the approximat ing express ions to be


used in Eqs. (3.1) and (3.2) can be evaluated as follows:

Attached region

1 z = - V ( u + ~)(1 - u)

Co

1 Co

z = ~/(u + t~)(1 - u)

] = U s ( v ' [ l + a ] - V [ u + a ] ) x , " [ l + ~ ] - V a ) Co

Separated region

z < O z > O

Z ---- (1 + a)X/(u + a )

Co

1 z = - - N/(u + a ) ( 1 - u)

Co

1 Co 1 co

z (1 + a)X/(u + a ) z V ( u + a)(1 - u)

s=s~\ x/(l+a)+Va /, <0)

( s. d, ] S = ( ~ / [ l + t ~ ] - - X / [ U + a ] ) x / [ l + a ] + ~ / a ) k--(u+C~)c0 ( z > 0 )

(3.8)

(3.9)

Flow inclination p F r e e interaction process Oswatitsch and Wieghardt (1948) were among the first to study the process of

interaction between an outer inviscid high-speed flow and an inner viscous layer. Later , Chapman et al. (1956) established the " f ree interaction" concept. It was observed that in supersonic flow the aft-body geometry and the mechanism that induces separation play no dominant or direct role upon fore-body flow characteristics. In this context , it may then be further determined that the viscous-inviscid interaction starts at the point where a small pressure increment causes the boundary layer to grow. Due to the growth of this boundary layer the outer edge streamline will be deflected, which in turn will cause further pressure increases. When a supersonic stream turns into a corner all compression waves coalesce into an oblique shock far away from the surface. Therefore , the flow field close to the surface but outside the boundary layer can be assumed to be an irrotational isentropic field. The streamline inclination can then be formally related to the Prandt l -Meyer relation to determine the pressure distribution.

Since there is no meaningful way of defining boundary layer thickness, the variation in its displacement thickness is used to calculate the local flow

418 M. H O L T and T. A. L U

inclination. The interaction equation can then be stated as follows

da* dH= dx dx tan 0 (3. lO)

where H(x) is the height of the body surface and 0 is the angle of local flow inclination. In a steady two-dimensional supersonic flow, the relation of the Prandtl-Meyer function u to the flow inclination 0 is defined by

0 = ~,(M=)- u(M) (3.11)

together with the displacement thickness

o r

6* = poaoUol 1 [ p,a, U, N/(R0) (1 fo' fo' u(1

+m,) ( 1 - U ) d u + m , - U ) d u Z Z

(3.12)

The complete interaction process can then be represented analytically. This provides a means of numerical forward integration without requiring specification of the pressure distribution a priori. Futhermore, this free interaction process eliminates the singularity at the separation point so often encountered by previous investigators.

Final form of governing system of ordinary differential equations Introducing the expressions (3.8) and (3.9) into the integral equations (3.1),

(3.2) and the interaction equation (3.10), we obtain a system of ordinary differential equations, which may be written

21 A22 A23 A24[ " b2 31 A~2 A33 A34| | / J~ b,

A~, A~2 A o, A~4] L Jdl = b (3.13)

where A~j and bi are algebraic expressions, depending on Co, a, m,, d,, sw, Ro, and g. with

2 a )1/2 _ g,(a) = 2(n + 1) [(1 + ng, l(a)]

go(a) = 2[ ( I + a ) '12 - a m|


Ro is the Reynolds number based on free stream conditions and distance from the leading edge to the corner. The expressions for these coefficients in the attached flow region are

A11= gl

Alz =cog;

1 [ ( CoSw ' 2 x / ( l + a ) A t 3 = ~ co(go+g,)-2 x / ( l + a ) + x / a + d , ) l n x / ( l + a ) + X / a

- & ( l _ X,/(1 + a ) g o ) [

A~4 = 0

A21 = ( g l - - g2)

A= = co(g~ - g '2)

A23 =77 - co (go -gz ) - v ' ( l+a ) go+d,g,

CoSw

A24 = 0

A31 ~¢/(1 + a ) - V'a X/ (1 + a ) g , -

[ CoSw ( g, X/(1 + a)gl q g, 1 ) A,2= x/(l-~g)--V~ 2x/(l+~) + 2x/~ 4V~x/(I+~)

dig= ] + 2~/(1 + a) ~- N/(1 + ct)dlg~

l{w,+o,r cos .2,] U,, ['k/(1 - ( - ~ - "V'a (go + g,) + d,(g, + A33

3 CoSw 5 ] - -}(.k/(l+ot)_X,/a)+gd,

CoS~ [ 4 x / ( l + a ) l n 2 v ' ( l + a ) ) ] - ('V'(1 + a ) - N/or) 2 "V'(1 + a ) + "k/a - go

( 2swd, a,='~ - ~ ¢ / ( l + a ) - ~ / a + 2 c o ]

2X,/(1 + a ) x 4V'(1 + a) In (.X/(1 + a) + .X,/a)

A34 = (x/(1 + ot)g., - 1 )

2(1 + a ) g o - g , + 2X/(1 + ~)]}

A,,= go+l-T-~m g,-2 X/(I+~)-X/~ int~/(l+~)+~/~lJ


r , m t , C o S w ( 2 ~ ¢ / ( 1 + o L ) ) ] A,, = / C o g o + . . ~ cog~ In

t l ÷ m , X / a x / ( l + a ) ( x / [ l + a l - x / a ) \X/(1 + or) + X/or

1 ~" 2m, [cogo+cog,-2t' CoSw d l ) l n [ 2~v/(l+oe) "~ A43 =-~, [ ~ \ k / ( l + a ) - X/a + \%/(1 + a) + X/a }

- d,(1 - X/(1 + a )go ) + \-~--L-~_ 1 / 1 + ml

[ ,,,, ( co.~ ) , . ~ .~ .+o~, ) x Cogo+l----~-~m cog,-2 x / ( l+a)_X/a+d, l n \ x / ( l + o t ) + X / a

- d , ( 1 - x / [ l + a l g o ) ] }

A , ~ = - [2" 2X/(I + or) a]}go)] m (.V/[1 + a ] + "V/a) + (1 - {X/[1 +

b, = Wo(~) + x/°~ Co

b2 = wo(~) + 2~/O~co 20 [(gl - g 9 + a (go - g,)]

~,:,~wo(,,+~o,lO+ (-~+ ,) 1 sw

X --4Co(X/[1 + a ] - X / a )

+ ~ [ { , ~ , + o , , ( , , - , ~ + o , o - o , , , - ~ (~+ o)]} , s,.. ,,~,,o(,~,, + o,_,~o,]+, (~_,)

+~- [2Co(X/[1 + a ] - %/a) Co Co

m l

b4 = I x/(Ro_______~) tan 0 C (1 + mo)

The corresponding coefficients in separated flow are listed in Lu (1970).

Ini t ia l cond i t i ons

B l a s i u s t ype For flow over a flat plate at zero angle of incidence, the velocity and

temperature components are functions of a single similarity variable

Therefore, in this case, au j~-11~ au

(3.14)


and ~ is constant when u is constant. We may therefore write

z, = - K . ~ -112, d. = ~a.,s;11~ (3.15)

where K, and A, are constants. Substituting into the given system of ordinary differential equation yields a set of algebraic relations under the conditions U, = 0 and & = 0. We obtain

1 112~ 112 CO = ' ~ o O~

(3.16) 1/2

K o 2 = g l a 2

with two algebraic equations

[(gl-gz)+a(go-gl)]---~- 1+ = 0 (3.17)

and

1 1 o" V ~ J

1 1~ Ko 3 = s - { ( l + l ) Ko-(gl-g2)~oo--~( 1+ ].~/a [x/(l+a)-g'+'~g2]}

Ko {1 1 1 + 1 ml 1

Solving (3.17), (3.16), (3.18) in succession, the initial values of a, co and dl can be determined.

Asymptotic suction type

The well-known asymptotic suction profile of incompressible flow can be applied in the present analysis of the adiabatic suction case. The profile on the Stewartson plane can be t ransformed to the Dorodni tsyn plane. A set of algebraic equations is obtained.

Co - (3.19) Wo

and

"V'(a + 1/2) f 2~/o~ = exp

X/or I-, (X/[1 + a] + v ' [ l / 2 + or]) X/(1 +~ ) [m (%/[1 + o r ] - ' x / [ l [2 + a] )

- In (~V/[ 1 + a ] + V a ) l (V[1 + ~ ] - x /o t ) J J

(3.20)


For given Wo, wall suction velocity, the initial values of Co and a are then determined.

Computational procedure

Numerical computation procedure The first order derivatives 60, &, U1 and dl are determined f rom Eq. (3.13)

using Cramer ' s rule. They are then integrated simultaneously by the Adams-Moul ton method with a step size of 10 -3 and truncation error limit of 10 5.

Since the governing boundary layer equations are of parabolic type there is no ups t ream influence to initiate the forward integration. An initial point of interaction is chosen arbitrarily. A very small per turbat ion is given to pJ/po to start the free-interaction process at this point, thus generating self-induced separation. The position of the initial point is varied until the integral curve so generated satisfies the free s t ream conditions far downstream; namely, that & and UI reach zero simultaneously.

The iteration process for determining the initial interaction point can be explained with reference to Fig. 3. The position of the actual compress ion corner is fixed at x/L = 1.0 and integrations are carried out for different values of the distance f rom this point to the initial interaction point. We may denote the reciprocal of this distance by to and different numerical values of to are at tached to the various integral curves downst ream of the corner. For all values of to shown the integral curves have the same form up to some distance ups t ream of reat tachment . Thereaf ter the behavior depends on the value of to. I f to is too small, the pressure reaches a maximum premature ly and rea t tachment is never reached. If to is increased to a value of approximate ly 2 rea t tachment takes place and the pressure increases for some distance in the at tached boundary layer flow. For slightly larger values of to the integral curves for the pressure are in two categories, (i) those for which the pressure reaches a max imum and then

2.4

2.2

2£

Po f.6

1 4 -

1.2

I I I I I I I I I I & _ ~ f 2 . 0 5 " - ~ / 20P9 ~,;o 203

-- ReL= 1.4 x 105 15284 ~''?--0~6

-- ~ = I I o R , ~ . ~ (~ = 2X315--~-2.01528~

_ / / ~ . 9 5 2.0

/ 1.9 ~ . " MI=(

y w 21.7

kc I I -4/ I I r I I I I 0 0 .2 0 .4 0 6 0 .8 1 0 x / L I 2 I 4 I 6 f 8 2 0 2.4

Fig. 3. Criterion for locating initial point of interaction region.

Supersonic laminar boundary layer separat ion 423

decreases, (to =2.015), and (ii) those for which the pressure increases continuously (to = 2.016). In case (i) the condition/f/~ = 0 is attained before & = 0, while this situation is reversed in case (ii). The correct integral curve corresponds to the simultaneous satisfaction of & = 0, JV/, = 0. With the present procedure this condition can be bracketed by values of to differing by less than 10 -6 .

The results in Fig. 3 show that the integral curves of Eq. (3.13) have a saddle point downstream of reat tachment (provided the latter point is reached). The present procedure is adequate to determine the position of this point accurately and passage of the integral curves through the saddle point could be achieved by the jumping procedure used in the Supersonic Blunt Body program by Belotserkovskii (1962). To treat the saddle point in greater detail we would need to use the expansion techniques applied to the laminar wake problem (Holt and Meng, 1970). This would require a great deal of algebra and is not really justified in the present problem.

Numerical results and discussion

All the variables shown in the figures have been conver ted back to the physical plane in order to have meaningful comparisons. The numerical results for pressure distribution under different physical conditions are shown in Figs. 4-7. The effect of a cooled wall is clearly demonstrated with sw = 0.125 and sw = 0.25. The length of the interaction region is markedly reduced as the degree of cooling of the surface is increased. Also, for an adiabatic wall the interaction region is shortened by the introduction of suction at the wall. From this evidence we may confidently expect that the separated region could be easily collapsed if we could apply wall cooling and suction simultaneously. We may also expect a shortened separation region if we increase the free stream Mach number, or reduce the

I t ~ I I I I I I I 1.7554

2.2 - o sw= 0 " s w = O- 125

o s w = 0 .25 2 . 0 - -

1 . 8 - - P._&

Po

1 . 4 - -

i.o I I 0 o .2 0 .4 0 .6 0 .8 L o 1.2 1.4 1.6 u8

x / L Fig. 4. Pressure distribution in the interaction region. Adiabatic and non-adiabatic cases.

424 M. H O L T and T. A. L U

Pt m Po

2 . 4

2 . 2

2 . 0

1 . 8

1 .6

1 .4

I I

M o = 3 . 0 8 _ Re L = 1.4x I 0 5

S w = 0

o W o = O

_ z~ W o = - I

1 . 2 - -

i .o I I 0 0.2 0.4

I I I I I I

0.6 0.8 LO 1.2 1.4 t.6 1.8 × / L

Fig. 5. Pressure distribution in the interaction region. Adiabatic cases with and without suction.

. . . . ' ' ' ~ " 1 . 9 2 8 ' / M 0 = 3 .45 Re L : L4 x I05 ~ ~ /

2.2 - - o M 0 = 3 .08 Re L = 1.4 x I05 R ~ 5 ~ 0 1 5 2 8 4

15283 2 . 0 - -

1 . 8 - - P_L

PO 1.6--

1.4--

S S '

Lo I ] 0 0.2 0.4 0.6 0.8 i.o 1.2 i_4 1.6 1.8 2.0

x / L

Fig. 6. In f luence of f ree s t r e a m M a c h n u m b e r on p r e s s u r e d i s t r ibu t ion .

Reynolds number based on plate length, as shown in Figs. 6 and 7. At the same time, the plateau pressure is increased with Mach number (for a given Reynolds number) and decreases with increasing Reynolds number (for given Mach number). The shearing stress distributions in the interaction region for fixed Mach and Reynolds numbers and three wall tempera ture conditions are shown in Fig. 8. They all exhibit a minimum point at or close to the corner. This behavior has also been observed experimental ly by Sfeir (1970). Figure 9 shows the variation of velocity parameters , Co and a, in the interaction region in the adiabatic case. The sign convent ion we choose here for tx requires that it be always positive. We can then have u = - a at the point of max imum reverse flow. Downs t ream of the


2.4

2.2

2.0

1.8 P_.._!_L no

I I I I I I I I

M o = 3 . 0 8 OJ = ~ 1 . 9 5 3 1 0

- - S w = 0 ~ . ~ W 0 = 0

I - - ReL ~ = LO x 105 R 1_95304 "#

_f 1.6- -

1 .4 - -

1 . 2 - - S i.o I I I I I I

0 O.Z 0.4 0.6 0,8 1,0 1.2 1.4 1.6 x/L

Fig. 7. Influence of Reynolds number on pressure distribution.

1.8

0 . 4 I I I I I I I I |

\ \ \ o ~ .=o ~o°~.o~ 0 . 3 - - \ N ' ~ q x o ~ z~ SW = 0.125 ReL =1 .4x105

~ a s w = 0,25

O.2

°0 "1 - - 1

,~l~ 0

-0.1 --

- 0 . 2 -- I I I I I I I I

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 I_Co 1.8 x /L

Fig. 8. Shearing stress distribution in interaction region. Without suction.

separation point a increases monotonical ly to the beginning of the wedge. This agrees with the conclusion of Stewartson and Williams (1969). It fol lows from this result that the height of the point of maximum reverse flow ,/ . . . . . is also monotonical ly increasing. This property is clearly confirmed by the velocity profiles given at different positions inside the interaction region in Fig. 10. It is always necessary to verify whether Chapman's "free interaction" assumption has been fulfilled in fact. We calculate two models under the same free stream condition with different deflection angles, namely l l ° and 13 °, respectively. We then find agreement between the initial increasing portions of the pressure curves for two different deflection angles in Fig. 11, as predicted. Figure 11 also


Co

5

4

3

2

I

- O z

- 0 . .

-Q~

- 0 . 1 - -

- 0 0

I I

M o = 3 . 0 8 Re L = 1.4 x 105

o( - -

C O - - - - - -

I I 0.2 0.4

[

S w = 0 W 0 = 0

I I I I I

/ I

0.6 0.8 1.0 1.2 1.4 1.6 x / L

Fig. 9. Variation of velocity parameters with x/L.

1.8

Y L

0 . 0 5

0 . 0 4

0 . 0 5

0 . 0 2

0 ,01

0 - I o

I I I

~ - = 0.9966

/ ~ / 0 . 8 9 6 £ --

/ / / / ~ 0 . 7 9 6 3

~ 0.496;

~ I 1 I L I 2 3 4 - 1

u / o o

M o = 3 .08 5 ReL= 1.4 x I0

;,o~ :_ o {- : , .o9~

~/I.3477

~ L5993

Y I I I o I 2 3

Fig. 10. Velocity profiles in the interaction region.

Pl

Po

2.4

2 .2 - -

2 .0 - -

1 .8--

1 .6- -

1 .4- -

1 . 2 -

1.0 0

I I

M o = 3_ 0 8 Re L = 1.4 x I 0 5

S w = O Wo = 0

13 o

I 0.2

I I I I I

0.4 0.6 0_8 1.0 1.2 1_4 1.6 1.8 x / L

Fig. 1 l. The effect of flare angle.


demonstrates the effect of the strength of disturbances which are produced by each individual deflection angle. The larger deflection angle would cause the interaction to start earlier. Figure 12 shows the total enthalpy flux at the wall for s+ -- 0.25. The most interesting point is the appearance of a peak around the point of separation. For the same conditions Fig. 13 shows the shapes of the dividing streamline and displacement thickness boundary.

-0.12

-O. IC

o J, ~" - 0.08

- 0 . 0 6

-0.04

i [ i M 0 = 3.08 Re L = 1_4x 105 . o +

0.6 0 .8 I.O 1.2 1.4 X / L

Fig. 12. Enthalpy flux at the wall in the interaction region.

Y

L

0.1(3

0.08 --

O.OE

0.04

0.02

0 0.5

I I I I 1 I I I I

o DISPLACEMENT THICKNESS DIVIDING STREAMLINE /

O U=O Mo= 3.08 Sw= 0.25 / ~ " R;L= L4x i o ~ /

I I I ~ L i I I I 0.6 0.7 0.8 0 .9 I.O I. I 1_2 I. 3 1.4

X / L

Fig. 13. Growth of boundary layer in the interaction region.

1.5

Comparison with experiment Measurements and observations of laminar supersonic flow over a corner were

made by Sfeir (1970) in the Berkeley 6"× 6" Supersonic Wind Tunnel at Mach number 2.64, Reynolds number 1.4 x 105, and wedge angles of 11 ° and 13 °. The present method was applied to the same flow conditions and comparison between theoretical and experimental curves is shown in Fig. 14. Figure 14 also shows a pressure curve obtained by the present method with a modified interaction condition.

The outstanding feature of Sfeir's curve is the attainment of a plateau pressure shortly downstream of separation. The present calculations show a plateau but the plateau pressure is lower than Sfeir's value, while the calculated pressures are higher than measured near reattachment. The calculated positions of separation and reattachment points agree well with experiment.


I I I I I R I 2256 I ~. ~ / / 2 . 1 2 5

2 . C - - o L = l l o R ~ , - _ ~ o . o

Rep = 1.4 x I0 . , ~ / ~ o oJ =L255

/ / l / / / o i~

o , l . 6 - ~ /

po o , .4- , ~ o

S . . / o - ,, o Sfeir _ S ~ z - - . . . . "" - Lu, 14o11"

1.2 j ~ - Lu, Holt (modif ied)

J~s ko . ~ . / I I I I I I I

0.4 0.6 0.8 Lo L2 1.4 ~.6 L8 2.0 x / L

Fig. 14. Pressure distribution. Comparison between theory and experiment.

The discrepancy between the theoretical and observed plateau pressures may be due to the use of too crude an interaction condition. As stated in Eq. (3.10), the effective body determining the outer inviscid flow is taken to be the solid, sharp cornered wall pushed out by the displacement thickness. One consequence of this is the discontinuity in pressure curve slope at the corner itself. It can be argued that the sharp corner is shielded by the recirculation zone and that a more realistic effective body boundary is that following the dividing streamline, rather than the solid wall. On this basis the theoretical curve for 6 = 11 °, M = 2.64 and Re = 1.4 x 105 was recalculated, with the wall height H in Eq. (3.10) replaced by the height of the dividing streamline (the latter calculated using the original interaction equation). The results are shown in the modified curve in Fig. 14. The improvement in the plateau pressure is remarkable and the agreement with experiment in the recompression region is slightly better than in the unmodified calculation.

Theoretical and experimental values of shearing stress in the separated region were also compared. Here agreement was less satisfactory mainly due to the difficulty of measuring velocities accurately in low speed regions near the wall or the u = 0 line.

These results encourage further improvement of the interaction equation, in which the dividing streamline is recalculated in each iteration. This modification is to be incorporated into a current investigation here of the more general problem of three dimensional supersonic laminar separation.

Concerning comparison with other theoretical calculations, most of these were carried out for flows where plateau regions were absent, either because of high values of free stream Mach number or low values of Reynolds number. The present case was calculated by Dwoyer but his pressure distribution does not reproduce the expected plateau behavior.


Acknowledgments--The authors are indebted to Daryoush Modarress for carrying out the calculations shown in Fig. 14. This work was supported by the Air Force Office of Scientific Research.

References

Belotserkovskii, O. M. (1962). J. Comp. Math. & Math. Phys. Academy of Sciences, USSR 2, 1062-1085.

Brown, S. N. and Stewartson, K. (1969). Laminar separation, Ann. Rev. Fluid Mech. 1. Chapman, D. R., Kuehn, D. M. and Larson, K. L. (1958). NACA Report 1356. Cohen, C. B. and Reshotko, E. (1956). NACA Report 1293. Dorodnitsyn, A. A. (1960). General method of integral relations and its application to boundary layer

theory, in Advances in Aeronautical Sciences, Macmillan, N.Y., pp. 207-219. Dwoyer, D. L. (1973). Proc. AIAA Computational Fluid Dynamics Conference, Palm Springs,

California. Holt, M. (1966). Separation of laminar boundary-layer flow past a concave corner, AGARD Conf.

Proc. No. 4, Rhode-Saint Gen~se, Belgium. Holt, M. (1967). Separation of laminar boundary layers on cooled re-entry bodies, Proc. X V I I Int.

Astronautical Congress, Polish Scientific Publishers, 383-401. Holt, M. (1968). Laminar boundary separation and reattachment near a concave corner, Proc. X V I I I

Int. Astronautical Congress, Polish Scientific Publishers, 539-544. Holt, M. and Meng, J. C. S. (1970). The calculation of base flow and near wake properties by the

method of integral relations, Proc. X I X Int. Astronautical Congress, Pergamon Press and Polish Scientific Publishers, 385-397.

Holt, M. and Meng, J. C. S. (1973). Astronautica Acta 18, 91-98. Klineberg, J. M. and Lees, L. (1969). A I A A J. 7, 2211-2221. Klineberg, J. M. and Steger, J. L. (1974). Paper 74-94, AIAA 12th Aerospace Sciences Meeting,

Washington D.C. Lees, L. and Reeves, B. L. (1964). A I A A J. 2, 1907-1920. Lighthill, M. J. (1953). Proc. Roy. Soc. A 217, 478. Lu, T. A. (1970). Theoretical investigation of supersonic laminar separation in a concave corner, Univ.

of Calif., Berkeley, Report AS-70-2 (Ph.D. Thesis). Lyu, Shen'-Tsyuan' (1962). J. Comp. Math. and Math. Phys., Academy of Sciences, USSR 2,

1000-1020. Nielsen, J. N., Lynes, L. L. and Goodwin, F. K. (1966). Theory of laminar separated flow on flared

surfaces including supersonic flow with heating and cooling, in AGARD Separated Flows, Part I, 31-68.

Oswatitsch, K. and Wieghardt, K. (1948). NACA TM 1189. Pavlovskii, Y. N. (1962). J. Comp. Math. and Math. Phys., Academy of Sciences, USSR 2, 1021-1042. Sfeir, A. A. (1970). Interaction d'une couche limite laminaire avec une onde de choc dans un coin de

compression, R~v. Roum. Sci. Techn. M~c. Appl. 15, 1375-1391. Stewartson, K. (1949). Proc. Roy. Soc. A, 200, 84-1-0. Stewartson, K. and Williams, P. G. (1969). Proc. Roy. Soc. A, 312, 81-206. Werle, M. J. and Bertke, S. D. (1972). A I A A J. 10, 1250-1252.

Supersonic laminar boundary layer separation in a concave corner

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