Second-Order Compressible Boundary-Layer Theory with Application to Blunt Bodies in Hypersonic Flow,

Viscous hypersonic flow near the nose of a blunt body is considered on the basis of the Navier-Stokes equations. Conventional boundary layer theory is embedded in a systematic expansion scheme. The general theory of the second approximation is developed. Seven second-order effects are identified: longitudinal curvature, transverse curvature, slip, temperature jump, entropy gradient, stagnation enthalpy gradient, and displacement. Their evaluation for a blunt body is outlined, and numerical results given for the stagnation region of a cooled sphere at infinite Mach number. In that example the increase in heat transfer due to the entropy gradient is reduced one-third by the other second-order effects. INTRODUCTION Ferri & Libby (Ref. l) first pointed out that the boundary layer on a blunt body in supersonic flow is influenced by the external vorticity generated by the bow shock wave. Theories of this "vorticity interaction" have since been developed by (among others) Hayes and Probstein (Ref. 2) and Ferri, Zakkay and Ting (Ref. 3 ) . Unfortunately, these two theories differ by a factor of more than 5 in their predictions of the increase in heat transfer due to external vorticity. Furthermore, two objections have been raised against both theories. First, Rott and Lenard (Ref. k) point out that the effect of external vorticity is only one of a number of second-order effects in the boundary layer, all of which should Presented at ARS International Hypersonics Conference, Cambridge, Massachusetts, August ΐβ-ΐ8, I96I; this work was carried out under Contract AF49(638)-965 with the Air Force Office of Scientific Research. •̂ Department of Aeronautical Engineering. 57 HYPERSONIC FLOW RESEARCH logically be considered concurrently· Second, study of incompressible flow (Van Dyke in Ref. 5) indicates that matching of the boundary layer with the outer rotational flow has not previously been carried out correctly. The present study aims to clarify this situation by calculating the complete second approximation for the boundary layer near the nose of a blunt body in hypersonic flow, and in a typical case to compare the contributions to heat transfer and skin friction of all the second-order effects. CONTINUUM FLOW PAST A BLUNT BODY IN THE VISCOUS HYPERSONIC LIMIT The author considers a symmetric plane or axisymmetric blunt body in a steady uniform hypersonic stream as indicated in Fig. 1. The body has a nose radius a, and is assumed to be analytic at least past the limiting characteristic that reaches the subsonic region. The gas is taken to be perfect with constant specific heats and Prandtl number. The viscosity is assumed to depend only on temperature, in this section as its ω power. Dimensional analysis shows that for a given body, gas and dimensionless surface temperature condition, the flow depends upon only the free stream Mach number and nose Reynolds number M R = 5»i ( 1 . 1 ) OO ^oo The applicability of the Navier-Stokes equations are considered when both these parameters are large or, more formally, as they both tend to infinity at rates whose relationship is to be deduced. The inviscid stagnation temperature is needed, which is given by the energy equation as 1 + ̂ M 2 « M 2 as M « ( 1 2 ) oo (The meaning of the various subscripts is indicated in Fig. 1.) This is true provided y does not simultaneously approach unity; and one may definitely exclude that unrealistic Newtonian limit, with its attendant non-uniformities· Thicknesses of Shock Wave and Boundary Layer It is known from various theories that at infinite Mach number the thickness Δ of the full shock layer is some fraction of the nose radius