A BGK model for high temperature rarefied gas flows

Abstract

High temperature gases, for instance in hypersonic reentry flows, show complex phenomena like excitation of rotational and vibrational energy modes, and even chemical reactions. For flows in the continuous regime, simulation codes use analytic or tabulated constitutive laws for pressure and temperature. In this paper, we propose a BGK model which is consistent with any arbitrary constitutive laws, and which is designed to make high temperature gas flow simulations in the rarefied regime. A Chapman-Enskog analysis gives the corresponding transport coefficients. Our approach is illustrated by a numerical comparison with a compressible Navier-Stokes solver with rotational and vibrational non equilibrium. The BGK approach gives a deterministic solver with a computational cost which is close to that of a simple monoatomic gas.

BGK and Fokker Planck Models for thermally perfect gases

“…It is interesting to compare our new model to the work of [23,24] and [19]: in these recent papers, the authors also proposed, independently, BGK and ES-BGK models for temperature dependent δ, like in the case of vibrational energy. However, they are not based on an underlying discrete vibrational energy partition, and the authors are not able to prove any H-theorem.…”

“…When Kn is very small, if all the time and space derivatives of F and G are O(1) with respect to Kn , then ( 20 The same analysis can be applied for the reduced Fokker-Planck model ( 22)- (24).…”

A Note on the Steady Navier–Stokes Equations Derived from an ES–BGK Model for a Polyatomic Gas

“…Since these Boltzmann equations are very awkward to deal with, simpler kinetic models have been proposed, mainly of Bhatnagar-Gross-Krook (BGK) type, replacing integral Boltzmann operators by proper relaxation operators driving distribution functions towards the Maxwellian equilibrium configuration. Various BGK models have been proposed for polyatomic gases, possibly also involving mixtures of monoatomic and polyatomic particles and simple chemical reactions [6][7][8][9][10][11][12][13]. Unfortunately, classical BGK approximations are not able to fit the correct Prandtl number [1].…”

Convergence estimates of a semi-Lagrangian scheme for the ellipsoidal BGK model for polyatomic molecules

“…The importance of developing polyatomic versions of the BGK model has been recognized soon after the inception of the model -which is very natural since most of the gas molecules consists of several atoms -and the several attempts to derive polyatomic version of the BGK model have been proposed in the literature. The polyatomic generalization of the BGK model can be realized in various manners such as the introduction of new variables describing the internal energy due to the inner configuration of the molecules [2,4], vibrational excitation [3] , and reformulation into the gas mixture framework [20,29]. In this paper, we are interested in the polyatomic BGK model obtained from the so called ellipsoidal BGK model [2,8,18] (Polyatomic ES-BGK model):…”

“…The parameter I ∈ R + is related to internal energy ε due to rotation and vibration ε(I) = I 2 δ , where δ > 0 represents the number of degrees of freedom for the internal motion of the molecules such as the rotation and vibration. Our independent variables x and v belong to phase space (x, v) ∈ T d ×R 3 , with T d ≡ R d /Z d , and t ≥ 0 denotes the time. The Knudsen number κ > 0 is the ratio between the mean free path of the gas molecules and the macroscopic length scale of the problem.…”