Unsteady hypersonic boundary layers for slender axisymmetric bodies with large injection rates

Int. J. Engng Sci. Vol. 30, No. 6, PP. 793-803, Printed in Great Britain. All rights reserved

0020s7225/92 $5.00+ 0.00 Copyright @ 1992 Pergamon Press plc

1992

UNSTEADY HYPERSONIC BOUNDARY LAYERS FOR SLENDER AXISYMMETRIC BODIES WITH LARGE INJECTION RATES S. ROY and G. NATHi Department

of Mathematics,

Indian

Institute

of

Science, Bangalore-560012,

India

Abstract-A semi-similar solution of an unsteady hypersonic laminar compressible boundary layer flow over a slender axisymmetric body with massive blowing has been obtained when the free stream velocity varies arbitrarily with time. The governing partial differential equations have been solved numerically by combining the implicit finite difference scheme with the quasi-linearization technique. The results have been obtained for (i) an accelerating/decelerating stream and (ii) a fluctuating stream. The skin friction responds to the fluctuation in the free stream more compared to the heat transfer. It is observed that the effect of large injection (blowing) rates is to move the viscous boundary layer away from the surface. The effect of the variation of the density-viscosity product across the boundary layer is found to be negligible for large blowing rates. Massive blowing reduces significantly the values of skin friction and heat transfer but the effect of the transverse curvature parameter is just reverse. Location of the dividing streamline increases as injection rate increases, but decreases with the increase of the transverse curvature parameter.

INTRODUCTION

The advanced re-entry technology studies have simulated a continuing interest to increase the flight speed which gives rise to several aerodynamic problems. The flow past a very slender body of revolution (here the term “very slender” means that the body radius r, is small and hence the boundary layer thickness 6 is not negligible compared to local radius of the body) has received a considerable attention due to its practical needs of high speed flights, because the use of a slender body reduces the drag and even produces sufficient lift to support the body in air, especially at hypersonic speeds. The flow nature on slender body is much characterized by its two surface curvatures, viz. the longitudinal one in the meridian plane (k,) and the transverse one in a plane normal to the axis of symmetry (k2) (Fig. 1). In the usual treatment of boundary layer analysis k, and (dk,/&) are assumed to be of orders 6 and a2, respectively, and are very small compared to unity. Therefore the effect of the longitudinal curvature in the boundary layer is negligible. In such cases attention is paid only to the transverse curvature kz which affects the boundary layer and the effect is similar to that of a favourable pressure gradient. Hence, it tends to delay the separation and transition in the flow field. At hypersonic speeds, the heat transfer problem becomes very important and the use of large injection on the surface considerably reduces the heat transfer. Also, at hypersonic speed of the spacecraft and missiles, the front of the body is enveloped by a shock wave across which a large difference of pressure, density and velocity occurs. Probstein and Elliott [l] have obtained boundary layer solutions including transverse curvature. Yasuhara [2] has found out the similarity solution consistent throughout the viscous and inviscid layers over 3/4 power law slender bodies at hypersonic conditions. Later, he [3] presented the similarity solutions for small to moderate values of transverse curvature with linear viscosity-temperature relation (p a T-‘, p a T where p, p and T are, respectively, the density, coefficient of viscosity and temperature) and zero mass transfer. Stewartson [4] has treated the hypersonic flow of a model gas over very slender bodies and has obtained results that are valid in the limit of strong interaction (i.e. for large transverse curvature). Ellinwood and Mirels [5] have extended the Stewartson’s solution to non-model gases (p D:T-l, p ccT” where w is the index in the power-law variation of viscosity). Further, Mirels and Ellinwood [6] 7 Author to whom correspondence should be addressed. 793

794

S. ROY and G. NATH

Fig. 1. Flow model and coordinate system.

have presented the extensive numerical results for the self-similar flow over 314 power law bodies. The above authors [7] have also considered the effect of mass transfer on the foregoing problem. However, they encountered certain difficulties in obtaining the solution for large injection rates. For large blowing rates, the structure of the boundary layer is considerably different from that of moderate or no blowing rate. In this case the boundary layer consists of two parts: a thick inner layer of essentially constant shear, temperature and composition; and a relatively thin outer viscous layer in which transition from the inner layer to the inviscid external flow takes place. Liu and Chiu {S] have developed an implicit finite difference scheme with non-centered or unequal spacing in combination with a quasi-linearization technique to study the effect of large blowing rates. The method is also found to be fast as well as stable and the rate of convergence is independent of blowing rates. The case of massive blowing is of interest in the design of the heat shield for a Jupiter atmospheric entry probe [9]. Moreover, for entry or re-entry flight, the spacecreaft is decelerated and thus the unsteadiness in the flow field is also important [lO]. AI1 the earlier studies pertain to steady flows. The study of the effect of large injection in the analogous unsteady case has not been reported so far. However, the effects of large blowing rates on the boundary layer flows of some different geometrical configurations have been studied by several authors [ll-171. Also, it may be remarked that Libby and Cresci [18] have shown experimentally that even for large blowing rates the inviscid flow is not affected and hence the boundary layer concept can be applied to study the effect of large blowing rates without introducing any appreciable error in the analysis. The aim of the present analysis is to study the effect of large blowing rates on unsteady axisymmet~c boundary layer flow of a compressible fluid with variable properties past a very slender body of revolutions. The semi-similar solution of the boundary layer equations has been obtained numerically using the method of quasi-linearization [8, 19f and an implicit finite difference scheme with non-centered or unequal spacing [8]. Particular cases of the present results have been compared with those of Refs f3, 6, 7, 143. It may be remarked that the boundary layer analysis is not strictly valid for high energy viscous-shock layer type of flow field requiring massive blowing for controlling the heat-load to the surface. This is because the velocity profiles do not attain their asymptotic values at the edge of the boundary layer due to the occurrence of strong shock (i.e. u -+ &h instead of u -+ U, where e and sh denote conditions at the edge of the boundary layer and behind the shock, respectively). However, there exists a shear layer edge within the viscous-shock layer where h -+ h, for perfect gas. Nevertheless, the present analysis is expected to exhibit most of the properties of the flow field with massive blowing.

195

Unsteady hypersonic boundary layers

GOVERNING

EQUATIONS

We consider the unsteady axisymmetric boundary layer flow of a compressible fluid with variable properties past a very slender body of revolutions. Let x and y are the distances measured along the meridian surface from the nose of the body and normal to the surface. If r is the distance from any point to the axis of symmetry, then r = r(x, y) and r,,, = r,,,(x) is the distance from any point on the body surface to the axis (Fig. 1). If the thickness of the boundary layer is small compared with the longitudinal radius of curvature l/k,, then at any point in the boundary layer r(x, y) = r&V) + y cos 8. We have considered the Prandtl number, Pr, as constant because in most of the atmospheric flight problems its variation in the boundary layer is small [20]. For the sake of simplicity, high temperature effects such as ionization, recombination etc. have not been considered here. We assume that the external flow is hornentropic and the surface is maintained at a constant temperature. Under the foregoing assumptions the governing boundary layer equations are ]3,71 Continuity : (a/L@)(p)

+ (dldy)(pvr)

+ (dldx)(pur)

= 0

(1)

Momentum : p[(du/dt)

+ u(aul&)

+ v(duldy)]

= -(dpldx)

+ r-‘(alay)(pr(aulay))

(2)

Energy : p[(dH/dt)

+u(dH/dx)

+ v(dH/dy)]

= (l/Pr)r-‘(alay)(pr(aH/dy)

- (1- Pr)pru(aulay)).

(3)

The relevant initial and boundary conditions are U(X, _Y,0) = ui(x, Y), u(x, 0, t) = 0,

v(x9 .Y, O) = vi(x, Y),

v(x, 0, t) = %(X,

u(x, @J,t) = u,(x, t),

t),

H(x,

H(x~ Y, O) = fi(x7

Y)t

0, t) = H, = constant,

H(x, ~0, t) = H, = constant.

Here u, v are the velocity components in the x and y-directions, respectively; t is the dimensional time; p is the static pressure; H is the total enthalpy; V, is the normal velocity at the wall; the subscript “i” denotes initial conditions; and the subscript “w” denotes the conditions at the surface. Substituting the pressure gradient term in equation (2) from the inviscid flow solution and using the following transformations [3,5-71

u = u&h

t*) = us(x)@(t)*F(rl,t*),

u, = u&)$(t*), H=h+

0

f u2,

t* = (du,/dx)t,

LA% t*) = F(% t*) h = c,T,

H = H,G(rl,

t*),

v%YJ 9 = cx)“2f(~~t*M(t*), PU =

ES 30:6-H

r-‘(Wldy),

pv

=

-r-l

[(W/ax> +[4+/W dy1,

796

S. ROY and G. NATH

The equations (2) and (3) become (NRF’)‘+~~F~+B$(G-F’)+BL~{(G-EF’)-(I-E)F}-(l-E)F:I=O (NRC’)’ + Pr #fG’ - 2(1-

Pr)E(NRFF’)’

(4)

- p(1 - E)Pr CT = 0

(5)

where for slender bodies (u, = Z.L) [3,5-71,

,=$=*+A ‘(G-EF’)dn, I w .‘l;,

AJY-l)uJz ___my

PJ2,

( )’ = a/&j.

w w

Here r] is the transformed coordinate; t* is the dimensionless time; U, is the value of U, at t* = 0; v and f are the dimensional and dimensionless stream functions, respectively; F (or f’) is the dimensionless velocity component in x-direction; G is the dimensionless total enthalpy; h, cp and y are, respectively, specific enthalpy, specific heat at a constant pressure and the ratio of specific heats; m(=u,2/2H,), p and A are the dissipation, pressure gradient and transverse curvature parameters, respectively; N is the ratio of the density-viscosity product across the boundary layer, r#~is an arbitrary function of time t*; the subscript “w” denotes the value at freestream. We have also considered for the gas with variable properties modelled realistically, that is [217 221 pah-‘,

pmh”

and

Pr = constant,

where o = 0.5 represents high-temperature flows, 0 = 0.7 is appropriate for low temperature flows and w = 1 represents constant density-viscosity product simplifications [21]. The boundary conditions imposed on the set of equations (4) and (5) at any particular time t* 10 are F = 0,

G = G,

at rj=O

F+l,

G-+1

as n--+= I

(6)

where

f=JV F drl+fiw I1

fw= al+(t*)

and (y = _ (PNv(2E)1’2 Pdw4~2w We

limit ourselves

to flows where

the body grows as A?‘~, i.e. r, mx3’4 and pw a~-“~, and the mass

(pv)w ccx-3’4 [3,5-71 for the semi-similar solution so that A and (Yare constants

797

Unsteady hypersonic boundary layers

transfer parameter CY>O or cy< 0 according as there is a suction or injection. The initial conditions are given by the steady state equations which can be obtained from equations (4) and (5) by putting t* = 0, 9 = 1, $: = F: = Gr = 0. The steady state equations are +fF’ I- /3(G - F*) = 0

(NW’)

(NRC’)’ + PrfG’ - 2(1-

(7)

Pr)m(NRFF’)’

=0

(8)

Equations (7) and (8) satisfy the boundary conditions (6) with t* = 0. The important quantities are given by [3,5-71 skin friction coefficient:

physical

w-l -

4k

p’

p,u,A

w

0-l

we

where

The heat transfer coefficient in terms of Stanton number is: x-l

St=

(%w >

qw

pmum(He - H,) =Pr pmum(He - H,)

GV

2Pe = p,u,r,A

Pr(1 - E)( 1 - G,,,)

where G;=

G:. Pr( 1 - E)( 1 - G,)

= p,u,r,A

G;

0-l

and q,,, is the heat transfer rate to the wall.

RESULTS

AND DISCUSSIONS

Equations (4) and (5) with boundary conditions (6) and the initial conditions given by the steady state equations (7) and (8) have been solved nume~cally using an implicit finite difference scheme with the quasi-linearization method. For massive blowing cases, quasi-linear finite difference scheme with variable stepsize has been used to get stable results (i.e. very big stepsize near the wall and decreasing it as it approaches to the outer edge of the boundary layer) whereas the quasi-linear finite difference scheme with constant stepsize is able to give stable results only for 0 2 (Yzz -5. Since the method is described in complete detail in [8,19], its detailed description is not presented here. However, for the sake of completeness, its outline is given here. The non-linear coupled partial differential equations (4) and (5) were first linearized using quasi-linearization method [8,19]. The resulting linear partial differential equations were expressed in difference form (see Appendix). The equations were then reduced to a system of linear algebraic equations with a block tri-diagonal structure which is solved using Varga’s ~go~thm [23]. To ensure the convergence of finite difference scheme to the true solution, the stepsizes A?I and At* have been optimized and the results presented here are independent of the stepsizes at least up to the fourth decimal place. For zero to moderate injection (0 2 (Y2 -5) cases, the stepsizes An and At* have been taken as 0.02 and 0.02, respectively. The variable stepsize has been used for large injection case (-5 > ar 2 -30) in the ~-erection starting with the initial stepsize ho = 0.20 and reducing uniformly as it moves away from the wall to the final stepsize

798

S. ROY

and

G. NATH

hp = 0.02, but constant stepsize At* = 0.02 has been used in t*-direction. Computations have been carried out on a VAX-8810 digitial computer for various values of A (0 5 A 5 lOO), (~(0>(~2--30), w (0.5~~~1.0), G, (0.20~G,,~0.60) and m (0~~~~0.5). In all numerical computations Pr has been taken as 0.72. The edge of the boundary layer q_ is taken between 6 and 225 depending upon the blowing rate (LY)and curvature parameter (A). q_ increases with the increase of blowing rate as well as curvature parameter. It is found that for a given injection rate, r], changes very rapidly as A increases from 0 to 10 after which the change is small. For example, for & = -30, qx = 95 when A = 0; but qZ = 210 when A = 10; and q-_= 225 when A = 100. The unsteady freestream velocity distributions considered here are given by f$(t*) = 1 f &t*2,

(E >(I)

and

$(t*)

= [l + E, cos(o*t*)]/(l

+ F, j

which represent, respectively, the constantly accelerating/decelerating tIow and fluctuating flow and w* is the frequency parameter. Such a distribution of freestream velocity may be relevant in space flight, flow over rotating helicopter blades or about fluttering wing sections, etc. It may be remarked that steady state equations (7) and (8) were solved numerically for o = 1.0, m = 1.0 and A #O without mass transfer by Yasuhara [3] using finite difference technique, for w = 1.0, m = 1.0 and A # 0 with small injection (0 2 (t 2 -0.5) by Mirels and Ellinwood [6], for moderate injection (0 2 [Y2 -4.5) with w f 1 by the above authors [7] using finite difference technique and for w # 1, m = 0 and A = 0 with large injection at the surface by Krishnaswamy and Nath [14] using implicit finite difference scheme with the quasi-linearization technique. In order to test the accuracy of our method, we have compared our skin friction and heat transfer parameters for steady state case with those of Refs [3, 6, 7, 141. The results are found to agree at least up to the 3rd place of the decimal with those of Refs [6, 7, 141 whereas some small deviations are observed in .C and G:, for G, = 1.0 with Yasuhara’s results [3]. The maximum difference is found to be 1 .l% in the heat transfer parameter (;I, Some of the comparisons are shown in Tables 1-3. The results for accelerating flow ($(t*) = 1 + etch, t‘ > 0) are presented in Figs 2-5, for decelerating flow ($(t*) = 1 - Et**, E > 0) in Figs 6 and for oscillating flow (4J(t*) = (1 + E, cos(w*t*))/(l + E,)) in Fig. 7. The effects of mass injection ((u < 0) and curvature parameter (A 2 0) on velocity and enthalpy profiles (F, G) have been exhibited in Fig. 2. It is noted that the increase of injection and curvature parameters enhance the boundary layer thickness qx, thereby delays the rate at which the boundary layer approaches the free stream. Moreover the presence of the curvature parameter A increases the curvature of the profiles (F, G) considerably more than those without curvature parameter. Also it is clear from Fig. 2 that for large blowing rates in the absence of curvature parameter, there is a thick inner layer close to the surface where the Table

1. Comparison

of skin friction and heat transfer parameters (fz,, GL,) with the results for Pr = 0.70, (Y= 0, w = 1.0 and m = 1.0

0

0.470 (0.470)1

1

0.469 (0.470) 0.620 (0.621) 1.261 (1.265)

0.346 (U. 3S6)

of Yasuhara

0.638 (0.638)

0.468 (0.469)

0.791 (0.783)

0.572 (0.573)

-0.065 (-0.068) 0.362 (0.363)

1.244 (1.255) 0.755 (0.753)

--0.226 (-0.229) 0.453 (0.450)

1.825 (1.846) 0.876 (0.874)

--0.365 (-0.360) 0.530 (0.529)

-0.103 (-0.105)

2.148 (2.170)

-0.213 (-0.217)

2.874 (2.901)

-0.407 (-0.401)

0

0 1 1

tValues

inside the parentheses

are obtained

by Yasuhara

[3].

131

799

Unsteady hypersonic boundary layers Table 2. Comparison of skin friction and heat transfer parameters (fz, CL) with the results of Ellinwood and Mirels [7] for Pr = 0.70, fi = l/14, o = 1.0 and G,. = 0.0 A=1

A = 10

A=5

--(Y

f::

G:,

f::

G:,

fi

GL

1

0.3502 (0.3500)?

0.2574 (0.2570)

0.4446 (0.4443)

0.3270 (0.3274)

0.5470 (0.5482)

0.4053 (0.4048)

5

0.1515 (0.1513)

0.1218 (0.1217)

0.1974 (0.1972)

0.1607 (0.1609)

0.2472 (0.2475)

0.2032 (0.2036)

10

0.0865 (0.0861)

0.0755 (0.0754)

0.1131 (0.1133)

0.1020 (0.1018)

0.1427 (0.1430)

0.1305 (0.1304)

20

0.0445 (0.0440)

0.0438 (0.0435)

0.0582 (0.0580)

0.0603 (0.0599)

0.0731 (0.0734)

0.0771 (0.0775)

40

0.0211 (0.0207)

0.0241 (0.0238)

0.0275 (0.0271)

0.0334 (0.0333)

0.0346 (0.0342)

0.0434 (0.0436)

tValues inside the parentheses

are obtained by Ellinwood and Mivels [7].

changes in the values of velocity and enthalpy are very small and a thin outer viscous layer where the transition from the inner layer to the inviscid external stream takes place rapidly. The effects of o (which characterizes the variation of density-viscosity product across the bounary layer), injection and dissipation parameters (LY,m) on skin friction and heat transfer (Ek, GL) are displayed in Figs 3 and 4. At any time t* 2 0, injection decreases both the skin friction and heat transfer. The effect of o becomes less pronounced as the blowing rate increases and for large blowing rates, the effect is almost negligible. To show the phenomenon more clearly, some of the results are presented quantitatively. For example, for t* = 1.0, w = 0.5, skin friction and heat transfer reduce approximately by 83 and 85%, respectively, as the injection rate increases from -1 to -5. The physical reason for this behaviour is that the injection increases both velocity and thermal boundary layer thicknesses which results in the decrease of their gradients. Due to the increase of the dissipation parameter m, EL increases but G; decreases and as the dissipation parameter increases, Gh increases remarkably with the increase of time t*. This behaviour is in support of the common fact that the viscous dissipation affects the thermal boundary layer more than the momentum boundary layer. The effects of injection (a: < 0) and curvature parameter (A 2 0) on dividing streamline

Table 3. Comparison of skin friction and heat transfer parameters (f:, GL) for Pr = 0.72, fi = 0.50, m = 0.0, A = 0 and G, = 0.6 with those of Krishnaswamy and Nath 1141 w=o.70

w=o.50 ff

f&z

w=l.OO

G:.

f;

G:.

R

G:,

-1

0.2970 (0.2966)t

0.0346 (0.0344)

0.3039 (0.3036)

0.0356 (0.0352)

0.3130 (0.3131)

0.0362 (0.0362)

-2

0.1505 (0.1505)

0.0021 (0.0020)

0.1505 (0.1505)

0.0015 (0.0015)

0.1504 (0.1504)

0.0009 (0.0009)

-3

0.1000 (0.1000)

0.0 (0.0)

0.1000 (0.1000)

0.0 (0.0)

0.1000 (0.1000)

0.0 (0.0)

-5

0.0604 (0.0608)

0.0 (0.0)

0.0604 (0.0608)

0.0604 (0.0608)

0.0 (0.0)

-10

0.0300 (0.0303)

(Zo”)

0.0300 (0.0303)

0.0300 (0.0303)

(8::)

tValues inside the parentheses

0.0 (0.0)

are obtained by Krishnaswamy and Nath [14].

800

S. ROY and G. NATH

0 Fig. 2. Effects of injection and curvature parameters (a; A) on velocity and enthatpy profiles (F, G) t* =o, ---, t* = 1.5. for c#s(~*)= 1 + .sP2, E =0.25; G, =0.6, w = 1.0 and m = 0.3; -,

location (?J)~=@are shown in Fig. 5. As the blowing rate increases, the Iocation of the dividing streamline increases whereas it decreases with the increase of the curvature parameter as weif as time t*. For example, for t* = 1.5, A = 50,the location of the dividing streamline (Y,?)~+~ shifts from 9 = 12.4 to q = 49.8 as the rate of injection increases from (Y= -10 to LY= -30. The effects of curvature parameter (A)and total enthalpy at wall (G,) for decelerating flow on skin friction and heat transfer (EL, GL) are shown in Fig. 6. It is observed that for all time f* 2 0, both skin friction and heat transfer (FL, G;1) increase with curvature parameter. To be more specific, we find that for t* = 1.0 skin friction and heat transfer increase approximately by 66 and 71%, respectively, as the curvature parameter A increases from 10 to 15. Due to the

4.0

m=0.5 -...-.

-1 Fw

Fig. 3. Effects of injection and o on skin friction and heat transfer @A, G,?,,)for #(t*) = 1+ 6**, E = 0.15; w= 1; ___) G,=O.6, A= 10 and m=0.3; _, w=o.5.

Fig. 4. Effect of dissipation parameter (m) on skin friction and heat transfer (FL, Cl) for @(t*) = I + ct**, ~=0.15;G,=0.6_, w=0..5,Ar10andru=-I; -, FL; ---, G:,.

801

Unsteady hypersonic boundary layers

--__

--mm-

1l-l I”

0

40

____

A

i---

80

100

Fig. 5. Effects of injection and curvature parameters (a; A) on the location of dividing streamline (n),=, for #(t*) = 1 + .s1*‘, E = 0.25; G, =0.6, o = 1.0 and t* =o, ---, t* = 1.5. m = 0.3; -,

t Fig. 6. Skin friction and heat transfer results (&,, G;L) for +(t*) = 1 - Et**, ~=0.15; w=O.5 and m=0.5; -, G, =0.6, FL; ---, G, = 0.6, G;:; -. -, G, = 0.2.

of total enthalpy at wall (G,), both skin friction and heat transfer (EL, G’:) increase and the change of G‘:, with the increase of time t* is more pronounced when G,,, = 0.2 as compared to G, = 0.6. As the total enthalpy at wall G, increases the fluid near the wall becomes less dense. The less dense fluid gets accelerated which results in higher friction and heat transfer. Skin friction and heat transfer (FL, G‘:) decrease with the increase of time t*. For example, for (Y= -3, A = 10, skin friction and heat transfer decrease approximately by 37 and 6%, respectively, as the time t* increases from 0 to 1.5. The effect of oscillatory freestream velocity on skin friction and heat transfer (EL, GL) is presented in Fig. 7. It is clear that the skin friction (EL) responds more to the fluctuations in the free stream velocity than the heat transfer (G‘:). The reason for such behaviour is that the increase

Fig. 7. Skin friction and heat transfer results e,=O.lO, w*=5.6;~=0.5andm=0.3;

(& GL) for q%(P) = [l +~~cos(~*t*)]/(l -, G,=0.6;---, G,,,=O.2.

+ cl),

802

S. ROY and G. NATH

skin friction depends on the velocity gradient in the boundary layer which responds strongly to the freestream ~uctuations whereas the heat transfer is comparatively less sensitive to the free fluctuations. Injection reduces both the values of skin friction and heat transfer (FL, CL) as well as their fluctuations while transverse curvature does the reverse.

CONCLUSIONS The effect of massive blowing is to move the dividing streamline away from the surface whereas it comes nearer to the surface with the increase of curvature parameter. The effect of the variation of the density-viscosity product across the boundary layer (i.e. variable gas properties) is found to be negligibIe for large blowing rates. However, they are found to be significant for small to moderate blowing rates. It is noted that the curvature parameter steepens both the velocity and enthalpy profiles but injection does the reverse. The increase of either curvature or injection or both enhances the thickness of the boundary layer. The skin friction and heat transfer can be reduced remarkably by injecting a large amount of fluid into the boundary layer but the curvature parameter increase both the skin fracture and heat transfer. In the case of oscillating flows, the response of the skin friction to the fluctuation is more than the heat transfer.

REFERENCES R. F. PROBSTEIN and D. ELLIOTT, f. Aeronaut. Sci. ‘23,208 (1956). M. YASUHARA, J. Phys. Sot. Jpn 11, 878 (1956). M. YASUHARA, 1. Aerospace Sci. 29, 667 (1962). K. STEWARTSON, Phys. Fluids 7, 667 (1964). [5] .I. W. ELLINWOOD and H. MIRELS, J. Fluid Mech. 34, 687 (1968). [6] H. MIRELS and J. W. ELLINWOOD. AIAA 1. 6, 2061 (1968). [7] J. W. ELLINWOOD and H. MIRELS, AIAA .L 7, 2049 (1969). [S] T. M. LIU and H. H. CHILI, AfAA J. 14, 114 (1976). [9] M. TAUBER, J. Spacecrafr Rockets 6, 1103 (1969). [lo] S. I. PA1 and A. P. SCAGLIONE, Appl. Sci. Res. 22, 97 (1970). [ll] T. KUBOTA and F. L. FERNANDEZ, AZAA 1. 6, 22 (1968). [12] D. R. KASSOY, J. Fluid Mech. 48, 209 (1971). [13] P. A. LIBBY, AIAA J. 14, 1273 (1976). 1141 R. KRISHNASWAMY and G. NATH, Phys. Ffuidr 22, 1631 (1979). 11.51R. KRISHNASWAMY and G. NATH, frrt. J. Heat Mass Transfer 25, 1639 (1982). [16] R. VASANTHA and G. NATH, Int. 1. Engng Sci. 23,561 (19%). [17] R. VASANTHA and G. NATH, AlAA J. 24, 1874 (1986). [18] P. A. LIBBY and R. J. CRESCI, J. Aerospace Sci. 28, 51 (1961). [19] K. INOUYE and A. TATE, AIAA 1. 12, 558 (1974). [20] A. WORTMAN, H. ZIEGLER and G. SOO-HOO, Inr. J. Heat Mass Trunsfer 14, 149 (1971). [Zl] J. F. GROSS and C. F. DEWEY, Fluid Dynamics Transactions (Edited by W. FISZDON), Pergamon Press. Oxford (1965). [ZZ] H. SCHLICHTING, Boundary Layer Theory, Chap. XIII. McGraw-Hill, New York (1979). 1231 R. S. VARGA, Mar& ~nterat~ve Analysis, p. 194. Prentice-Hall, New York (1962). [l] [Z] (3) [4]

13 Augtcsr 1991; accepted

(Received

11 September

Vol. II, p. 529.

1991)

APPENDIX Finite difference formulae using constant stepsize for small to moderate injection case (05 (Y5: -5) and variable stepsize for large injection case (-5 < ty c: -30) considered here are, respectively, given by F”= (F,,,,, and

-2F,,,,

+ F,,,,

F’ = (F,z.n+l - F,,,,-,)/(2An)

,)/(An)‘,

F”=v_F,I I m,n I +YF n m.n+ yn+,Fm.n+I*

F’=%

,K,,,+, + 4X,,,, + 4,+1%,+1

where 2 .-v”-l=h,,(h,+h,_,)’

2

-2 v”=h,h,;

etc.

“‘*‘=h,(h,+h,_,)

etc.

Unsteady hypersonic boundary layers There is a constant stepsize in P-direction

go3

and the difference form using backward difference scheme is given by F: = AL., - F,_,,,)/Al*

etc.

Suppose there are N nodal points in the interval [0, nJ with big mesh-size near the wall and decreasingly small as it approaches to the boundary layer edge. Let ha and hJrcp denote the starting mesh-size and the difference of mesh-size between the lines of interval, respectively, then $, h, = V-

where

h, = h, - nhstep

N = Ph, - h,,,, - {Wr, - L,)’

Wtep

-

with

hh,,,,~"*l