COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 16 (1978) 281-289 0 NORTH-HOLLAND PUBLISHING COMPANY

HIGH SPEED FLOW PAST A CONE WITH LARGE WALL INJECTION VELOCITIES

Maurice HOLT University of California, Bekeley, California, USA

and

Thomas D. TAYLOR The Aerospace Corporation, Los Angeles, California

Received 27 February 1978

In this paper a model is proposed for calculating the flow past a cone with massive blowing at the surface. The model consists of an inner flow layer separated from an outer flow by a contact surface. The inner flow is assumed to be inviscid and the method of integral relations is employed to solve the equations. The pressure on the contact surface is assumed to be Newtonian.

The model allows different blowing velocity distributions at the cone surface. Sample calculations are presented for a constant and linearly decreasing injection velocity. The results indicate a multivalued solution at the leading edge of the cone. Also the cone surface pressure is very sensitive to the initial slope of the contact surface.

The initial slope of the contact surface is an unknown parameter and must be determined by prescribing a down- stream flow condition. The surface pressure at the rear of the cone is chosen as the downstream condition.

1. Introduction

The problem of high speed flow past a cone when gas is injected through the surface at moderate or large velocities has recently received some attention, but at present it is only partially under- stood. In this paper the problem is investigated theoretically using an inviscid model. The method of solution is simple in concept and is used to determine the pressure distribution on the cone sur- face and the shape of the blowing layer in a number of cases.

Strong injection into the boundary layer has been investigated using a viscous model by Smith [ 1, 21 and Smith and Stewartson [ 3,4]. These authors have provided valuable information on the qualitative aspects of the phenomenon, using the triple deck method. To obtain quantitative data using the viscous model, more calculations are needed. A survey of the viscous model work so far completed is given in [ 51.

The inviscid model appears to be justified if the wall injection velocity exceeds the limits of boundary layer theory. In this case, the shearing stress near the wall is small and the viscous terms can be neglected near the cone. The problem then reduces to one of fitting an internal flow origi- nating at the body with an external, originally uniform stream. The two parts of the flow field are separated by a viscous layer of revolution across which the pressure is essentially constant and the normal velocity component is of the order of that in a boundary layer. When the viscous layer has thickness small compared to that of the blowing layer, it may be treated as a constant surface across which the pressure is continuous and the normal velocity is zero. The flow model then be- comes fully inviscid (fig. 1). The data for the model consist of (1) the conditions in the undisturbed

282 M. Holt and T.D. Taylor, High speed flow past a cone with krge wall injection velocities

Fig. 1. Model flow field.

uniform stream, (2) the cone angle, (3) the velocity of the injectant at each point along the cone surface and the surface temperature.

A conical solution to the inviscid model has been derived by Aroesty and Davis [6] in the case of an injectant with constant velocity normal to the cone surface and constant enthalpy. These authors find that a supersonic external flow can be matched to a subsonic internal flow, provided that the contact surface is conical and inclined at a unique angle to the axis of symmetry. If the internal flow is incompressible, this angle is independent of the injection velocity; if it is com- pressible, the angle is very weakly dependent on this velocity. Emanuel [ 71 extended this work to deal with constant oblique injection and found analogous results.

The disadvantage of the conical flow solution is that, since the slope of the contact discontinuity is fixed by the conical flow requirement, the value of the pressure on the solid cone surface is also fixed. In general, this rules out the possibility of attaining a prescribed pressure far downstream along the cone, a condition which is usually to be satisfied in practical problems.

To remove this restriction, the problem is treated in this paper by a method not based on as- sumption of conical flow. The injectant velocity is assumed to be normal to the solid cone surface, but its speed and enthalpy may be general functions of distance along the cone. Correspondingly, the contact discontinuity surface may be a general pointed body of revolution.

The internal flow is calculated by the method of integral relations. Introducing Cartesian coor- dinates x and y along and normal to the cone surface, respectively, the continuity equation, two momentum equations and the energy equation are written in divergence form. These are then in- tegrated between the cone and contact discontinuity on the assumption that certain combinations of the flow variables are linear functions of the y coordinate. The resulting system of ordinary dif- ferential equations can be integrated in the x direction to determine the cone surface pressure and density and the flow variables along the contact surface. The ordinate on the free surface is left as a variable and can only be determined by matching with the outer flow.

The pressure for the outer flow is calculated using a modified Newtonian approximation. In all cases the shape of the contact discontinuity surface is determined by matching the pressures on the external and internal sides. This is done step by step.

In order to determine the flow uniquely within the inner region, the initial slope of the contact

M. Halt and T.D. Taylor, High speed flow past a cone with large wall injection velocities 283

surface or a prescribed downstream boundary condition is required. In this paper the initial slope of the contact surface is chosen as a parameter of the problem. Under these restrictions the flow equations describing the inner flow are integrated numerically for two prescribed surface injection velocities. The cases studied correspond to constant injection velocity and linearly decreasing in- jection velocity, respectively. In all cases the surface pressure is found to be extremely sensitive to the initial contact surface slope.

The results obtained from the general method take account of both vorticity and compressibility. These include special cases where such effects are negligible, and, near the leading edge, numerical results can be compared with an anaiytical solution.

2. Description of numerical method

The interior flow past the cone is described by the conservation equations written in divergence form

?I$ +?$+p=o,

where

F” = pur, F’ = (p + pu*)r, F* = puvr,

(2.1)

F3 =FO

3 '

GO=pvr, G’=F*, G* = (p +pv*)r, G3 = F3 Go/F0 ,

K” = 0, K’ = -psin8, P = -pcose, K3 = 0.

In these relations p is the pressure, p is the density, u and v are velocity components in the x and y directions, r = x sin 8 + y cos 0, 8 is the cone half-angle, and 7 is the ratio of specific heats cP/c,. These equations are supplemented by the equation of state

p=pRT (2.2)

and the following boundary conditions: On the cone surface the temperature and velocity are pre- scribed:

T, = T,(x) , (2.3)

u. = 0, v, = vo(x) . (2.4)

Across the contact surface the pressure is continuous and the normal velocity component is zero, leading to the conditions

Pl = pm + C’p, V, sin*a = p. , (2.5)

284 M. Holt and T.D. Taylor, High speed flow past a cone with large wall injection velocities

DG/Dt = 0, (2.6)

In these equations 00 denotes the free stream values, C* is a correction coefficient in the Newtonian pressure formula, OL is the angle between the cone surface and the tangent to the con- tact surface at a given x, 6(x) is the inner layer thickness, and the suffices 1 and 0 refer to the contact surface and the cone surface, respectively. The equations of motion are integrated numer- ically by applying the method of integral relations in the first approximation (see [ 83). Each of eqs. (2.1) is integrated with respect to y between 0, and 6 and the unknown integrands are repre- sented by linear functions of the form

F’ = F; + (Ff - F;) (y/s> (2.7)

Eqs. (2.1) are then reduced to the approximating system of ordinary differential equations

(2.8)

These equations are integrated numerically to satisfy the boundary conditions (2.3)-(2.6). Note that no attempt is made to reduce these expressions to a form in which the unknowns are deter- mined explicitly. In the conservation form represented by eq. (2.8) the leading edge singularity can be handled numerically without the need to develop expansions of the equations in series. This is accomplished by introducing an implicit numerical method to start the integration - a step made necessary since the derivatives of 6(Fi + Ff)/2 vanish at x = 0. The value of d6/dx = tan (Y = GP/Ff’ at x = 0 is also unknown and must be determined by prescribing the downstream pressure. In addition it is important to observe that dS/dx at x = 0 is not necessarily infinite, i.e. the contact surface is not normal to the cone surface at x = 0.

In general, the determination of dd/dx at x = 0 requires an interation to converge on the pre- scribed surface pressure at the cone base. From the experimental results of Hartunian and Spencer [9] this pressure appears to be close to the wake base pressure. The data are insufficient, however, to determine the downstream pressure accurately. Therefore, in this study the initial value of d6/dx = Gy/F,O is prescribed and a range of downstream pressures are generated. The completed details of the numerical integration are available in [ 101.

The method is applied to determine the flow past a 10” half-angle cone when the injection velocity is (A) constant, (B) decreases linearly.

3. Numerical results and discussion

To establish the general nature of flouj about a cone with extreme surface injection, two prob- lems are considered. Each has the following flow conditions in common:

Cone half-angle = 10”

c’ = 1

M. Holt and T.D. Taylor, High speed flow past a cone with large wall injection velocities 285

0.6 -

Fig. 2. Effect of initial slope on contact surface (case A).

U, = 7000 ft/sec ,

T, = lOOO”K,

T, = 250°K ,

Y = 1.4.

The problems are then separated into cases A and B by the boundary conditions

Case A: v,/U, = 0.08, corresponding to uniform injection velocity,

Case B: v,/U, = 0.08 (1 - x/l), corresponding to linearly decreasing injection velocity (1 is the cone length).

Results for these two cases are shown in figs. 2-7. Case A is considered for values of initial angle of the contact surface varying between 48” and 60” (note: this angle is measured from the cone surface). When the contact surface is straight, the angle between the axis of symmetry and the contact surface is found to be 58” 3 1’ (i.e. 48”3 1’ from the cone surface). This agrees closely with the result of Aroesty and Davis [6] as well as that of Emanuel [7]. From fig. 3 it is evident that, as the initial angle of the contact surface increases, the pressure on the cone surface de- creases. At the angle of 59”45’ the pressure at the rear of the cone is approximately zero. This solution corresponds closely to flow over a finite cone followed by expansion to a low base pres- sure. In fig. 2 the shapes of the contact surface for different initial angles are shown. The surface densities are shown in fig. 4. Since the surface temperature is constant, these are proportional to the pressure.

286 M. Holt and T.D. Taylor, High speed flow past a cone with large wall injection velocities

0.7 a,,= 4E031’CONICAL SOLUTION

CASE A

0.6 f$ =0.08

0.5 r a0 = 55O48’ P 2 en

35 ao= 4E031’ CONICAL SOLUTION

t

CASE A

3o k.008 Uco .

25

0 0.2 0.4 0.6 0.8 1.0

Fig. 4. Surface density variation (case A).

Fig. 3. Surface pressure variation (case A).

The results show that the surface pressure and hence the surface density are extremely sensitive to small variations in the initial slope of the contact surface. Even though the inner flow is sub- sonic, there appears to be a jump in the pressure between the contact surface and body surface at the apex.

Case B is studied less extensively for a range of initial angles between 59”46’ and 6 1”O’. Results are given in figs. 5-7 and show that, for a given initial contact surface slope, the surface pressure for a decreasing injection velocity falls less rapidly than that for constant injection (compare results corresponding to CX,, = 59”45’). Further, the layer thickness is correspondingly larger. The density is shown in fig. 7.

CASE 8

Fig. 5. Effect of initial slope on contact surface (case B).

M. Holt and T.D. Taylor, High speed flow past a cone with large wall injection velocities 281

CASE 0 I

Fig. 6. Surface pressure variation (case B).

25 CASE B

2O _ $ =O.O8(I-x/l)

I OO

I I I I I I I I I 0.2 0.4 0.6 0.6 1.0

X/f

Fig. 7. Surface density variation (case B).

The general method does not determine the surface conditions at the leading edge. This can be seen from eq. (2.8), where all terms are factored by 6 and r, which vanish at the apex.

In further applications of the method a more detailed calculation of the external flow field should be used to determine the pressure on the dividing streamline. It would then be possible to consider contact surfaces with apex angles smaller than the conical value and to refine the treat- ment of flows with superconical apex angles.

It is of interest at this point to compare the leading edge contact surface angles just found with those predicted by an irrotational analysis.

288 M. Halt and T.D. Taylor, High speed flow past a cone with large wall injection velocities

4. Analysis near the leading edge

In special cases, when compressibility and vorticity are neglected, the flow field near the leading edge can be represented by a potential 4 satisfying Laplace’s equation, which is conveniently ex- pressed in polar coordinates Y, the radial distance from the cone apex 0, and 0, the angle between a radius vector and the cone axis.

The potential 4 is a linear combination of fundamental solutions of the form

@ = rm {A, Pm (cos 6) + B, Q, (cos 19)) , (4.1)

where Pm and Q, are Legendre functions of the first and second kind, and A, and B, are constants. When the injection velocity is of the form

ue = vormpl , (4.2)

we can represent 4 by a single term. In this case it can be shown that, to order r, the dividing stream-line is straight. The boundary conditions (2.4) and (2.6) then reduce to

A, Pm (cos a,> + B, Q, (cos 8,) = 0 , (4.3)

A, P,!&os 0,) + B, Q:, (cos 6,) = 0 ,

where primes denote differentiation with respect to 8, and 0, is measured here from the cone axis. To satisfy eqs. (4.3) and (4.4) for nontrivial values of A, and B, , we require that

Pm (cos 0, ) Q, (~0s ec 1 = 0. (4.5)

P;(cos e,) Q:,(COSes)

For given Bc and m eq. (4.5) determines 8,. The initial angle of the dividing streamline has been calculated from eq. (4.5) when 19, = 10” and m takes the three values 1, 2, 3. The results are

m es

1 56” 22’ ,

2 38”21’,

3 30”47’ or 84”41’ .

The first angle has previously been determined by Aroesty and Davis [6] and correspond to the conical flow, uniform pressure and constant injection velocity solution.

We recall that in case A of the general numerical solution, when the contact surface is straight, the compressible conical solution is obtained and the angle 8, is found to be 58”3 1’. Compressibil- ity therefore has a slight effect on the initial angle.‘This is because the magnitude of the blowing

M. Holt and T.D. Taylor, High speed flow past a cone with large wall injection velocities 289

velocity, if finite, has no effect on the initial angle when the contact surface is straight, a result observed by the authors as well as by Aroesty and Davis. The angle for m = 2 corresponds to a linearly increasing injection velocity with lower initial pressure. Numerical results are not avail- able for comparison in this case. There are two solutions of eq. (4.5) for m = 3, the first of which is physically realistic; the second must be discarded since it leads to a tangential velocity along the dividing streamline directed towards the cone apex. There are no solutions for m > 3.

The results of the leading edge analysis are of interest in showing that the initial contact sur- face angle is limited for irrotational flows. When rotational effects are included, however, this limitation is removed and a downstream boundary condition must be introduced to make the problem unique.

References

[l] F.T. Smith, Boundary layer flow near a discontinuity in wall conditions, J. Inst. Math. Applies 13 (1974) 127-145. [2] F.T. Smith, Laminar flow over a small hump on a flat plate, J. Fluid Mech. 57 (1973) 803-824. [3] F.T. Smith and K. Stewartson, On slot-injection into a supersonic laminar boundary layer, Proc. Roy. Sot. London A 332

(1973) l-22. [4] F.T. Smith and K. Stewartson, Plate-injection into a separated supersonic laminar boundary layer, J. Fluid Mech. 58 (1973)

143-159. (51 K. Stewartson, Multistructured boundary layers, Advan. Appl. Mech. 6 (1974) 145-239. [6] 3. Aroesty and S.H. Davis, Inviscid cone flows with surface mass transfer, AIAA .I. 4 (1966) 1830-1832. [ 71 G. Emanuel, Blowing from a porous cone or wedge when the contact surface is straight, AIAA J. 5 (1967) 534-538. [8] M. Holt, Numerical methods in fluid dynamics [Springer Series in Computational Physics 11 (Springer, Berlin, New York,

1977). [9] R.A. Hartunian and D.J. Spencer, Experimental results for massive blowing studies, AIAA J. 5 (1967) 1397-1401.

[lo] T.D. Taylor, Massive ablation of a cone in high temperature supersonic flow (Picatinny Arsenal, Dover, N.J., Tech. Rept. No. 35T1, Jun. 1967).