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

Jenny, Patrick; Torrilhon, Manuel; Stefan, Heinz G.

doi:10.1016/j.jcp.2009.10.008

Cited by 124 publications

(122 citation statements)

References 28 publications

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“…Galilean invariance, conservation of mass, momentum and energy have already been shown by Jenny et al (2010). Moreover, the H-theorem and accordingly the second law of thermodynamics can be proved for the Fokker-Planck equation with A i being time-independent (see Shiino 1987).…”

Section: Modelling Aspectsmentioning

confidence: 81%

“…It has been shown by others that a Fokker-Planck (FP) approximation of the Boltzmann collision operator can be used up to quite large Knudsen numbers and that it in terms of computational cost it is much more attractive than DSMC (Jenny et al 2010). Here, we generalize the simple linear model with the objective to fix inconsistencies regarding the Prandtl number.…”

Section: Fokker-planck Approximationmentioning

confidence: 99%

“…Following the line of arguments by Jenny et al (2010), a method with the following properties will be devised.…”

Section: Solution Algorithmmentioning

confidence: 99%

“…Numerical scheme Since (2.23) and (2.24) form a nonlinear stochastic system, we cannot seek for an analytical solution like for the linear Langevin model discussed by Jenny et al (2010). However, in the following, we present a novel numerical scheme, which will minimize the numerical errors of the whole algorithm.…”

Section: Solution Algorithmmentioning

confidence: 99%

“…In comparison with BGK-based models which describe jump-type stochastic processes for evolution of the molecular velocity distribution, the proposed solution algorithm, on the other hand, employs a diffusion-type process, whereas much larger time steps and coarser spatial discretization become possible. The current paper is an extension of the work by Jenny et al (2010) with the objective to fix the problem with the wrong Prandtl number. This was achieved by introducing a drift term which is nonlinear with respect to the molecular velocity.…”

Section: Introductionmentioning

confidence: 99%

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

2011

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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 FokkerPlanck 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.

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Section: Modelling Aspectsmentioning

confidence: 81%

Section: Fokker-planck Approximationmentioning

confidence: 99%

“…Following the line of arguments by Jenny et al (2010), a method with the following properties will be devised.…”

Section: Solution Algorithmmentioning

confidence: 99%

Section: Solution Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%