Inviscid Hypersonic Flow Over Blunt-Nosed Slender Bodies

Abstract

At hypersonic speeds the drag/area of a blunt nose is much larger than the drag/area of a slender afterbody, and the energy contained in the flow field in a plane at right angles to the flight direction is nearly constant over a downstream distance many times greater than the characteristic nose dimension. The transverse flow field exhibits certain similarity properties directly analogous to the flow similarity behind an intense blast wave found by G. I. Taylor and S. C. Lin. Conditions for constant energy show that the shape of the bow shock wave R(x) not too close to the nose is given by R/d = K_1 (γ)(d/c)^(1/2) for a body of revolution, and by R/d = K_0(γ) (x/d)^(2/3) for a planar body, where d is nose diameter, or leading-edge thickness. A comparison with the experiments of Hammitt, Vas, and Bogdonoff on a flat plate with a blunt leading-edge at M_∞ = 13 in helium shows that the shock wave shape is predicted very accurately by this analysis. The predicted surface pressure distribution is somewhat less satisfactory. Energy considerations combined with a detailed study of the equations of motion show that flow similarity is also possible for a class of bodies of the form r_b ~ x^m, provided that m' ≤ m ≤ 1, where m' = 3/4 for a planar body and m' = (3/2(γ+1))/(3γ + 2) for a body of revolution. When m < m' the shock shape is not similar to the body shape, and except for the constant energy flows the entire flow field some distance from the nose must depend to some extent on the details of the nose geometry. Be again again utilizing energy and drag considerations one finds that at hypersonic speeds the inviscid surface pressures generated by a blunt nose are larger than the pressures produced by boundary layer growth on a flat surface over a distance from the nose of order ℓ, where ℓ/d ≃ 1/15 ((Re_d)/M_∞^2))^3 (Here Re_d is free-stream Reynolds number based on leading-edge thickness.) Thus at M_∞ = 15 the viscous interaction effects should be important for Re_d < 10^3, but somewhere in the range 1500 < Re_d < 2000 the inviscid effects must spread rapidly over the plate surface, and certainly for Re_d > 3000 the inviscid pressure field is dominant and determines the boundary layer development, skin friction and heat transfer over the forward portion of the body. These rough estimates are in qualitative agreement with the experimental results of References 7 and 9.

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