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Fokker–Planck model for computational studies of monatomic rarefied gas flows

Published online by Cambridge University Press:  31 May 2011

M. H. GORJI ,
M. TORRILHON  and
P. JENNY
M. H. GORJI*
Affiliation:
Institute of Fluid Dynamics, ETH Zentrum, Sonneggstrasse 3, 8092 Zürich, Switzerland
M. TORRILHON
Affiliation:
Department of Mathematics, RWTH Aachen University, Schinkestrasse 2, D-52062 Aachen, Germany
P. JENNY
Affiliation:
Institute of Fluid Dynamics, ETH Zentrum, Sonneggstrasse 3, 8092 Zürich, Switzerland
*
†Email address for correspondence: gorjih@ifd.mavt.ethz.ch
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Abstract

In this study, we propose a non-linear continuous stochastic velocity process for simulations of monatomic gas flows. The model equation is derived from a Fokker–Planck approximation of the Boltzmann equation. By introducing a cubic non-linear drift term, the model leads to the correct Prandtl number of 2/3 for monatomic gas, which is crucial to study heat transport phenomena. Moreover, a highly accurate scheme to evolve the computational particles in velocity- and physical space is devised. An important property of this integration scheme is that it ensures energy conservation and honours the tortuosity of particle trajectories. Especially in situations with small to moderate Knudsen numbers, this allows to proceed with much larger time steps than with direct simulation Monte Carlo (DSMC), i.e. the mean collision time not necessarily has to be resolved, and thus leads to more efficient simulations. Another computational advantage is that no direct collisions have to be calculated in the proposed algorithm. For validation, different micro-channel flow test cases in the near continuum and transitional regimes were considered. Detailed comparisons with DSMC for Knudsen numbers between 0.07 and 2 reveal that the new solution algorithm based on the Fokker–Planck approximation for the collision operator can accurately predict molecular stresses and heat flux and thus also gas velocity and temperature profiles. Moreover, for the Knudsen Paradox, it is shown that good agreement with DSMC is achieved up to a Knudsen number of about 5.

JFM classification

Mathematical Foundations: Computational methods Rarefied Gas Flow: Kinetic theory Micro-/Nano-fluid dynamics: Non-continuum effects
Type
Papers
Information
Journal of Fluid Mechanics , Volume 680 , 10 August 2011 , pp. 574 - 601
Copyright
Copyright © Cambridge University Press 2011

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