A Fokker–Planck Model of the Boltzmann Equation with Correct Prandtl Number

Mathiaud, Julien; Mieussens, Luc

doi:10.1007/s10955-015-1404-9

Cited by 61 publications

(58 citation statements)

References 20 publications

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“…in which, for clarity, the dependence of M vib on F and G has been omitted, and the dependence of e vib on T as well. Finding F 1 and G 1 is less simple than for the BGK model: however, the computations are very close to what is done in the standard monatomic Fokker-Planck model (see [14] for instance), so that we only give the main steps here (see appendix A for details). First, the decomposition is injected into (33) to get…”

Section: Proof For the Fokker-planck Modelmentioning

confidence: 79%

“…where M vib [F, G] can be defined by (14) with R = 1. Similarly, the relations (3)-(7) between the translational, internal, and total energies and the temperature, have to be read with R = 1 in non-dimensional variables.…”

Section: Hydrodynamic Limits For Reduced Modelsmentioning

confidence: 99%

“…These model equations have already been extended to polyatomic gases, so that they can take into account the internal energy of rotation of gas molecules. They contains correction terms that lead to correct transport coefficients: the ESBGK or Shakhov's models [10,11,12], and the cubic Fokker-Planck and ES-Fokker-Planck [9,13,14,15].…”

Section: Introductionmentioning

confidence: 99%

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BGK and Fokker-Planck Models of the Boltzmann Equation for Gases with Discrete Levels of Vibrational Energy

Mathiaud

Mieussens

2020

J Stat Phys

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We propose two models of the Boltzmann equation (BGK and Fokker-Planck models) for rarefied flows of diatomic gases in vibrational non-equilibrium. These models take into account the discrete repartition of vibration energy modes, which is required for high temperature flows, like for atmospheric re-entry problems. We prove that these models satisfy conservation and entropy properties (H-theorem), and we derive their corresponding compressible Navier-Stokes asymptotics.

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Section: Proof For the Fokker-planck Modelmentioning

confidence: 79%

Section: Hydrodynamic Limits For Reduced Modelsmentioning

confidence: 99%

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BGK and Fokker-Planck Models of the Boltzmann Equation for Gases with Discrete Levels of Vibrational Energy

Mathiaud

Mieussens

2020

J Stat Phys

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“…The Fokker-Planck (FP) equation with advection-diffusion collision operator is widely used in modeling dynamic systems such as neutral molecule [1,2,3], plasma [4,5,6], photonics [7,8], and even biological [9], economic [10], and social [11] systems. The first FP equation for molecule system is derived from Boltzmann equation in gas kinetic theory when counting the gazing effect of molecule collisions [12].…”

Section: Introductionmentioning

confidence: 99%

Conservative discrete-velocity method for the ellipsoidal Fokker-Planck equation in gas-kinetic theory

et al. 2019

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A conservative discrete velocity method (DVM) is developed for the ellipsoidal Fokker-Planck (ES-FP) equation in prediction of non-equilibrium neutral gas flows in this paper. The ES-FP collision operator is solved in discrete velocity space in a concise and quick finite difference framework. The conservation problem of discrete ES-FP collision operator is solved by multiplying each term in it by extra conservative coefficients whose values are very closed to unity. Their differences to unity are in the same order of the numerical error in approximating the ES-FP operator in discrete velocity space. All the macroscopic conservative variables (mass, momentum and energy) are conserved in the present modified discrete ES-FP collision operator. Since the conservation property in discrete element of physical space is very important for numerical scheme when discontinuity and large gradient exist in flow field, a finite volume framework is adopted for the transport term of ES-FP equation. For nD-3V (n < 3) cases, a nD-quasi nV reduction is specially proposed for ES-FP equation and the corresponding FP-DVM method, which can greatly reduce the computational cost. The validity and accuracy of both ES-FP equation and FP-DVM method are examined using a series of 0D-3V homogenous relaxation cases and 1D-3V shock structure cases with different M ach numbers, in which 1D-3V cases are reduced to 1D-quasi 1V cases. Both the predictions of 0D-3V and 1D-3V cases match well with the benchmark results such as analytical Boltzmann solution, direct full-Boltzmann numerical solution and DSMC result. Especially, the FP-DVM predictions match well with the DSMC results in the M ach 8.0 shock structure case, which is in high non-equilibrium, and is a challenge case of the model Boltzmann equation and the corresponding numerical methods.

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“…Note also that our approach with the help of SDE technique differs from other approaches for obtaining gas dynamics equations connected to application of deterministic equations for distribution function in phase space [15,16,17,18] as well as other hierarchical models [19]. The models like ours with the coefficients a and σ depending on velocity are in use by the physics theoreticians for the study of phenomena in turbulent flows (f. e. [20]).…”

Section: Introductionmentioning

confidence: 99%

Meso - Macro Models for a Hard Sphere Gas

Богомолов¹,

Esikova²,

Kuvshinnikov³

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. Macroscopic system of gas dynamic equations, differing from Navier -Stokes and quasi gas dynamic ones, is derived from a stochastic microscopic model of a hard sphere gas in a phase space. The model is diffusive in velocity space and valid for moderate Knudsen numbers. The main peculiarity of our derivation is more accurate velocity averaging due to analytical solving stochastic differential equations with respect to Wiener measure which describe our original meso model. It is shown at an example of a shock wave front structure that our approach leads to larger than Navier -Stokes front widening that corresponds to reality. The numerical solution is performed by a (well suited to high performance computer applications) special "discontinuous" particle method.3121

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A Fokker–Planck Model of the Boltzmann Equation with Correct Prandtl Number

Cited by 61 publications

References 20 publications

BGK and Fokker-Planck Models of the Boltzmann Equation for Gases with Discrete Levels of Vibrational Energy

BGK and Fokker-Planck Models of the Boltzmann Equation for Gases with Discrete Levels of Vibrational Energy

Conservative discrete-velocity method for the ellipsoidal Fokker-Planck equation in gas-kinetic theory

Meso - Macro Models for a Hard Sphere Gas

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