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

Abstract

We propose an extension of the Fokker-Planck model of the Boltzmann equation to get a correct Prandtl number in the Compressible Navier-Stokes asymptotics for polyatomic gases. This is obtained by replacing the diffusion coefficient (which is the equilibrium temperature) by a non diagonal temperature tensor, like the Ellipsoidal-Statistical model is obtained from the Bathnagar-Gross-Krook model of the Boltzmann equation, and by adding a diffusion term for the internal energy. Our model is proved to satisfy the properties of conservation and a H-theorem. A Chapman-Enskog analysis shows how to compute the transport coefficients of our model. Some numerical tests are performed to illustrate that a correct Prandtl number can be obtained.

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

“…Now Combining equations (15) and (16), one gets that there exists four real numbers A, B, C, D and one vector E ∈ R 3 , independent of v and ε, such that:…”

“…It is difficult to derive a Fokker-Planck model for the distribution function f with discrete energy levels. We find it easier to directly derive a reduced model, by analogy with the reduced BGK model (21)(22) and by using our previous work [15] on a Fokker-Planck model for polyatomic gases. We remind that the original Fokker-Planck model for monoatomic gas can be derived from the Boltzmann collision operator under the assumption of small velocity changes through collisions and additional equilibrium assumptions (see [8]).…”

“…Note that both models do not provide a correct Prandtl number, which can lead to errors for the computation of fluxes in numerical simulations. This is a usual problem with single parameter models like BGK or Fokker-Planck: this can be corrected by a modification of the models like with the ES-BGK or ES-FP approaches, as it has been done for polyatomic gases (see [11,15] for instance). Then we have to approximate σ(F 1 ) and q(F 1 , G 1 ) up to O(Kn ).…”

“…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].…”

A BGK model for high temperature rarefied gas flows

“…We point out that this article is a first step towards a correct computation of the parietal heat flux: since we use a BGK model, it is clear that our model does not have a correct Prandtl number, as usual. This might be solved by using the ES-BGK approach [3,22,23] to capture the correct relaxation times for energy and fluxes [19].…”