Solving the kinetic equation for all Knudsen numbers

Abstract

Starting from the Boltzmann kinetic equation, a system is constructed which allows one to calculate flow fields with strong deviation from equilibrium and for Knudsen numbers 0<Kn0<∞. This is achieved by interlacing the path-integral form of the kinetic equation with the equations of conservation (the first five moments of the equation of transfer). Thus, at the lower level, we maintain the ability of the drifting part of the kinetic equation to drive the state toward nonequilibrium and, at the upper level, we keep the properties of the hydrodynamic (collisional) system to transport information on the grid system in physical (r) space in a nonmaterial way by waves. The distribution function fr,c,t is defined on an equally spaced Cartesian grid in r space and in velocity (c) space. An important step in the construction of the interlaced system is to interpolate the distribution function fr,c,t at interstitial points in r,c space. Approximations of the path-integral equation and different levels of approximation of the gain function (collision term) allow an adequate and efficient description of flows at any degree of deviation from equilibrium. Introducing a phase function in c space allows one to select those collisions that contribute to the gain function. The determination of collision terms then becomes very efficient, and calculating a time step is reduced by orders of magnitude. The problem of maintaining conservation of the lower moments, generally arising when the gain collision integral is to be determined at interstitial points, is solved by imposing on f a set of integral constraints. The system was developed and tested using the Riemann problem and steady planar Couette flow. Results demonstrate that the system is able to maintain conservation, describe nonequilibrium over a large range of Knudsen numbers and over orders of magnitude of density change. In the example of Couette flow, the system was tested for 0.0001<Kn0<50 and wall Mach numbers Maw⩽3.