A Fokker-Planck model of the Boltzmann equation with correct Prandtl number

Julien Mathiaud (1, 2) , Luc Mieussens (3, 2)

1 CESTA - Centre d'études scientifiques et techniques d'Aquitaine
2 IMB - Institut de Mathématiques de Bordeaux
3 CARDAMOM - Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts

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. This is obtained by replacing the diffusion coefficient (which is the equilibrium temperature) by a non diagonal temperature tensor, like the Ellipsoidal-Statistical model (ES) is obtained from the Bathnagar-Gross-Krook model (BGK) of the Boltzmann equation. Our model is proved to satisfy the properties of conservation and a H-theorem. A Chapman-Enskog analysis and two numerical tests show that a correct Prandtl number of 2/3 can be obtained.

Mots clés: H-theorem; Prandtl number; Fokker-Planck model; Ellipsoidal-Statistical model

Domaines: Equations aux dérivées partielles [math.AP]; Mécanique des fluides [physics.class-ph]

Julien Mathiaud, Luc Mieussens. A Fokker-Planck model of the Boltzmann equation with correct Prandtl number. Journal of Statistical Physics, 2016, 162 (2), pp.397-414. ⟨10.1007/s10955-015-1404-9⟩. ⟨hal-01122440⟩