A solution algorithm for the fluid dynamic equations based on a stochastic model for molecular motion

• Published in: Journal of computational physics Vol. 229; no. 4; pp. 1077 - 1098
• Main Authors: Jenny, Patrick, Torrilhon, Manuel, Heinz, Stefan
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Inc 20.02.2010; Elsevier
• Subjects: ALGORITHMS; APPROXIMATIONS; Boltzman equation; BOLTZMANN EQUATION; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Computational techniques; Exact sciences and technology; FOKKER-PLANCK EQUATION; Kinetic gas theory; Mathematical methods in physics; MATHEMATICAL SOLUTIONS; MOLECULAR DYNAMICS METHOD; MONTE CARLO METHOD; NAVIER-STOKES EQUATIONS; NUMERICAL ANALYSIS; PDF methods; Physics; SIMULATION; Stochastic differential equations; STOCHASTIC PROCESSES; Kinetic gas theory; Boltzman equation; Monte-Carlo method; Stochastic differential equations; Fokker–Planck equation; PDF methods; Stochastic model; Boltzmann equation; Fluid dynamics; Statistical moment; Digital simulation; Molecular dynamics; Fokker-Planck equation; Thermodynamic equilibrium; Relaxation time; Calculation methods; Monte Carlo methods; Algorithms; Kinetic equations; Differential equations; Calculation; Kinetic theory; Stochastic equation
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2009.10.008

In this paper, a stochastic model is presented to simulate the flow of gases, which are not in thermodynamic equilibrium, like in rarefied or micro situations. For the interaction of a particle with others, statistical moments of the local ensemble have to be evaluated, but unlike in molecular dynamics simulations or DSMC, no collisions between computational particles are considered. In addition, a novel integration technique allows for time steps independent of the stochastic time scale. The stochastic model represents a Fokker–Planck equation in the kinetic description, which can be viewed as an approximation to the Boltzmann equation. This allows for a rigorous investigation of the relation between the new model and classical fluid and kinetic equations. The fluid dynamic equations of Navier–Stokes and Fourier are fully recovered for small relaxation times, while for larger values the new model extents into the kinetic regime. Numerical studies demonstrate that the stochastic model is consistent with Navier–Stokes in that limit, but also that the results become significantly different, if the conditions for equilibrium are invalid. The application to the Knudsen paradox demonstrates the correctness and relevance of this development, and comparisons with existing kinetic equations and standard solution algorithms reveal its advantages. Moreover, results of a test case with geometrically complex boundaries are presented.