Kinetic-Theoretic Description of the Formation of a Shock Wave

Abstract

The flow of a gas in a shock tube is treated in the context of kinetic theory as an initial value problem for the one-dimensional Krook equation. The only simplifying assumption made is that the gas is one-dimensional. This corresponds to a gas with an adiabatic constant γ = 3. A finite difference method is proposed, utilizing the fact that, for sufficiently small time intervals, the gas experiences essentially free molecular flow, so that the collision effects can be treated as a first-order correction. The conservation laws are not used. The computed solution agrees excellently with the classical solution, but in addition, has shock structure, diffusion of the contact discontinuity, and dispersion of the expansion wave, all incorporated. The same procedure is used to calculate steady-state shock structure, to which the shock developed in the shock tube is compared. It is seen that for the strength of the shock calculated (Pressure ratio of tube 10:1, shock Mach number 1.43, γ = 3), the shock is essentially ``fully developed'' after about 20 mean collision times of the low-pressure gas. In conclusion, an exact reduction procedure is given by which problems for a monatomic three-dimensional gas (γ=53) are reduced to the solution of two simultaneous Krook equations, manageable by the present procedure without vastly increasing the need for computing capacity.