Numerical solutions to the viscous shock-layer blunt-body problem with inert gas injection

Abstract

A set of laminar hypersonic viscous shock-layer equations are obtained for a non-reacting binary mixture. The equations are valid to second-order in the inverse square root of a Reynolds number for the entire shock layer. A set of slip-flow boundary conditions which include mass transfer at the body surface is derived in a form applicable as boundary conditions for the governing equations. A set of modified Rankine-Hugoniot shock conditions are used as boundary conditions at the shock. The governing equations are solved using an implicit finite-difference scheme applicable to parabolic differential equations. The governing equations in their full form are parabolic-hyperbolic in nature and approximations are made to put them in total parabolic form allowing solutions to be obtained by marching step-by-step downstream from the stagnation streamline. The approximations are eliminated by iterations and solutions to the complete equations are found. Flow conditions applicable to re-entry problems are chosen for example solutions. Solutions are found for a hyperboloid asymptotic to a cone of total interior angle of 45°. A free stream velocity of 20,000 fps. is used in every case. Solutions are found for free stream conditions found at altitudes from 300,000 ft. to 100,000 ft. For the free stream species, two air models are chosen. In one case it is assumed that the reaction rates for the air components in the shock layer are so slow that they are neglected and the air is assumed to be a mixture of oxygen and nitrogen molecules. In the other case, the air is assumed to be in equilibrium at the temperature and pressure at each point in the shock layer and equilibrium air properties are used. Solutions are found with the injection of argon, helium, and air across the body surface. A mass injection distribution that decays exponentially from the stagnation point is used. A variety of total mass injection rates are used. Comparisons are made of the heat transfer, wall shear stress, and shock stand-off distances for the two air models with different injectants and injection rates. Results show that at altitudes above 250,000 ft. injection is ineffective in reducing heat transfer and wall shear stress. At lower altitudes, relatively small injection rates can cause sizable reductions in heat transfer and wall shear stress. Helium is the most effective of the injectants studied. At sufficiently high injection rates for the distribution used, the stagnation point heat transfer is reduced to an extent that the maximum heat transfer occurs downstream of the stagnation point. In cases where the shock shape and shock stand-off distance are appreciably changed by the injection, the heat transfer and wall shear stress do not recover their no injection values downstream as the injection rate goes to zero back on the body but remains below the no injection values.