Viscous Aerodynamic Characteristics in Hypersonic Rarefied Gas Flow

This paper considers the problem of calculating viscous aerodynamic characteristics of blunt bodies at hypersonic speeds and at sufficiently high altitudes where the appropriate mean free path becomes too large for the use of familiar boundary-layer theory but not so large that free molecule concepts apply. Results of an order-of-magnitude analysis are presented to define the regimes of rarefied gas flow and the limits of continuum theory. Based on theoretical and experimental evidence, the complete Navier-Stokes equations are used as a model, except "very close" to the free molecule condition. This model may not necessarily give the shock wave structure in detail but satisfies overall conservation laws and should give a reasonably accurate picture of all mean aerodjmamic quantities. In this "intermediate" regime there are two fundamental classes of problems: a "viscous layer" class and a "merged layer" class, the latter corresponding to a larger degree of rarefaction. For the viscous layer class there is a thin shock wave, but the shock layer region between the shock and the body is fully viscous, although the viscous stresses and conductive heat transfer are small at the shock wave boundary. Here, the use of the Navier-Stokes equations with outer boundary conditions given by the Hugoniot relations is justified. For the merged layer class, the shock wave is no longer thin, and the Navier-Stokes equations can be used to give a solution which includes the shock structure and has free-stream conditions as outer boundary conditions. A simpler procedure is presented for "incipient merged" conditions where the shock may no longer be considered an infinitesimally thin discontinuity but where it has not thickened sufficiently to entail the "fully merged layer" analysis. In this case we approximate the shock by a discontinuity obeying conservation laws which include curvature effects, viscous stresses, and heat conduction. For a sphere and cylinder it is shown that the Navier-Stokes equations can be reduced to ordinary differential equations for both the viscous and merged layer class of problems. Solutions of these equations, when used in connection with hypersonic flow problems, are in general only valid in the stagnation region. To illustrate the viscous layer solutions, numerical calculations have been performed for a sphere and cylinder with the assumption of constant density in the shock layer, which is a useful