Numerical Simulation of the Shock-wave Structure with Different Kinetic and Continuum Models

Gas flows in the transitional regime between continuum and free-molecular flow are widely present in nature and can be often encountered in various practical applications. Noticeable attention has been recently paid to modeling that sort of flows owing to intensive development of microand nanotechnologies and design of miniature technical devices. The Navier-Stokes equations, even with the velocity slip and temperature jump boundary conditions, are known to be inadequate for the description of transitional flows. At present time, the most common approach is the direct statistical simulation based on solving the Boltzmann equation with the Monte Carlo method. Owing to the stochastic character of the DSMC (direct simulation Monte Carlo) method, however, there arise a number of difficulties in modeling unsteady and low-speed flows with this method. The approach based on the direct deterministic solution of the distribution-function kinetic equations is free of that kind of difficulties. On other hand, when solving the Boltzmann equation, an accurate calculation of multidimensional integrals in the collision term is very resource-consuming. A more efficient approach is to solve relaxation-type kinetic equations, which are approximations of the Boltzmann equation, such as the Bhatnagar-Gross-Krook (BGK) equation, the Shakhov model, or the ellipsoidal statistical BGK model. Many features of transitional flows are properly described by that sort of approximating equations [1,2]. Another possible approach for simulating the transitional gas flows is to use higherorder (in terms of the Knudsen number) continuum models such as the Burnett equations [3]. In present paper, several kinetic and continuum models are used to study the stationary shockwave structure problem, which is a classical example of the flow including strong non-equilibrium phenomena. Being on the one hand a simple stationary one-dimensional problem, on the other hand it allows one to study the essential characteristics of the shock. The relaxation-type kinetic models are solved by a finite-difference code. The discrete ordinate method is used in the velocity space, and the high-resolution shock-capturing WENO (weighted essentially non-oscillatory) scheme is employed to approximate spatial derivatives in the coordinate space. Time integration is carried out using the second order Runge-Kutta TVD method.