NASA Technical Memorandum 100205

ICOMP-87-7

Similar Solutions for Viscous Hypersonic

Flow Over a Slender Three-Fourths-Power

Body of Resolution

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Chin-Shun Lin

Institute for Computational Mechanics in Propulsion

Lewis Research Center

Cleveland, Ohio

December 1987

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SIMILAR SOLUTIONS FOR VISCOUS HYPERSONIC FLOW OVER

A SLENDER THREE-FOURTHS-POWER BODY OF REVOLUTION

C.S. Lin

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135

SUMMARY

For hypersonic flow with a shock wave, there Is a similar solution con-

sistent throughout the viscous and Invlscid layers along a very slender three-

fourths-power body of revolution. The strong pressure Interaction problem can

then be treated by the method of similarity. In the present study, numerical

calculations are performed In the viscous region with the edge pressure dis-

tribution known from the Invlscid similar solutions. The compressible laminar

g boundary-layer equations are transformed Into a system of ordinary differential

00 equations. The resulting two-point boundary value problem Is then solved by

^ the Runge-Kutta method with a modified Newton’s method for the corresponding

boundary conditions. The effects of wall temperature, mass bleeding, and body

transverse curvature are Investigated. The Induced pressure, displacement

thickness, skin friction, and heat transfer due to the previously mentioned

parameters are estimated and analyzed.

INTRODUCTION

At hypersonic speeds, flow Is decelerated by the work of compression and

viscous dissipation, therefore a high- temperature gas is produced In the bound-

ary layer. The density of the hot gas Is very low, so the mass flux in this

boundary layer is small. Because of this high temperature, the thickness of

the boundary layer on the body surface Increases and the streamlines in the

flow external to the boundary layer are displaced outward. The displacement

thickness may be comparable to or may even exceed the body thickness, so that

the effect of body transverse curvature is significant. The effective thick-

ening of the body can also Induce a large pressure, which Is transmitted Into

the external invlscid field along the Mach lines. These pressures are then

transmitted essentially without change through the boundary layer and. In turn,

govern the growth of the boundary layer. Along with the pressure interaction,

the vorticity Interaction may also occur because of the curved shock wave.

Thus, the boundary-layer structure will be governed not only by the pressure

gradient, but also by the vorticity at the edge of the boundary layer (ref. 1).

Another Important fact Is that high temperature can also cause the gas to

depart from the perfect-gas behavior. Thus the high-temperature gasdynamics

needs to be taken Into account. The viscous-lnviscid interaction and the

physical-chemical phenomena are more or less dependent on each other – a fact

that makes the theoretical investigation much more difficult. In the present

study, we assume that the perfect-gas relation holds and that the vorticity

interaction Is negligible (i.e., only the pressure Interaction is considered).

For hypersonic flow with a shock wave, a similar solution Is found to be

consistent throughout the viscous and invlscid layers along a very slender

three-fourths-power body of revolution. The strong pressure interaction prob-

lem can then be treated by the method of similarity. In the present study,

numerical calculations are performed in the viscous region with the edge pres-

sure distribution known from the inviscid similar solutions. The compressible

laminar boundary-layer equations are transformed into a set of ordinary differ-

ential equations, and a two-point boundary value problem results. The Runge-

Kutta method is then used with a modified Newton’s method to solve the

resulting simultaneous nonlinear equations for the corresponding boundary

conditions.

Although the thermodynamic and fluid dynamic phenomena associated with

flight at hypersonic speeds have been the subject of intensive research for the

past decades, only a few studies of the effects of body transverse curvature

and mass bleeding have been carried out (refs. 2 to 7). The purpose of this

study is to contribute to the investigation of the effects of wall temperature,

mass bleeding, and body transverse curvature on the strong pressure interaction

region during hypersonic flights. The induced pressure, displacement thick-

ness, skin friction, and heat transfer are then analyzed based on these

parameters.

GOVERNING EQUATIONS AND BOUNDARY CONDITIONS

For axial ly symmetric flow with body forces neglected, the compress-

ible laminar boundary-layer equations can be written as follows:

Continuity:

Momentum:

Energy: