A Fokker–Planck based kinetic model for diatomic rarefied gas flows

Abstract

A Fokker–Planck based kinetic model is presented here, which also accounts for internal energy modes characteristic for diatomic gas molecules. The model is based on a Fokker–Planck approximation of the Boltzmann equation for monatomic molecules, whereas phenomenological principles were employed for the derivation. It is shown that the model honors the equipartition theorem in equilibrium and fulfills the Landau–Teller relaxation equations for internal degrees of freedom. The objective behind this approximate kinetic model is accuracy at reasonably low computational cost. This can be achieved due to the fact that the resulting stochastic differential equations are continuous in time; therefore, no collisions between the simulated particles have to be calculated. Besides, because of the devised energy conserving time integration scheme, it is not required to resolve the collisional scales, i.e., the mean collision time and the mean free path of molecules. This, of course, gives rise to much more efficient simulations with respect to other particle methods, especially the conventional direct simulation Monte Carlo (DSMC), for small and moderate Knudsen numbers. To examine the new approach, first the computational cost of the model was compared with respect to DSMC, where significant speed up could be obtained for small Knudsen numbers. Second, the structure of a high Mach shock (in nitrogen) was studied, and the good performance of the model for such out of equilibrium conditions could be demonstrated. At last, a hypersonic flow of nitrogen over a wedge was studied, where good agreement with respect to DSMC (with level to level transition model) for vibrational and translational temperatures is shown.

A kinetic model of the Boltzmann equation for non-vibrating polyatomic gases

“…where we take T * = 91.5 K and Z ∞ rot = 18 (Gorji & Jenny 2013). Figure 2 shows the profiles of the normalized density and rotational temperature.…”

“…In these models, the gain part of the BCO is modelled by the Gauss, ellipsoidal Gauss, and Gauss-Hermite polynomials, while the loss part describes the exponential decay of the distribution function with a rate independent of molecular velocity. Recently, Gorji and co-workers have also proposed a model replacing the BCO by the Fokker-Planck collision operator (Gorji, Torrilhon & Jenny 2011;Gorji & Jenny 2013), which models the drift and diffusion in velocity space. Although this model is faster than the DSMC method near the continuum-fluid regime, for microflow simulations it suffers the same slowness as the DSMC method because of its particulate nature.…”

A unified stochastic particle Bhatnagar-Gross-Krook method for multiscale gas flows

“…Since the seminal work of Jenny etc. [34], the stochastic particle method based on the Fokker-Planck model has been developed and applied widely [35][36][37][38][39][40]. The integral solution of the Fokker-Planck model naturally couples the molecular convection and collision, and hence theoretically its viscosity and thermal conductivity can satisfy the NS solutions at large time steps [37,40].…”

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

“…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].…”

“…It is therefore interesting to extend the model equations to take this vibrational modes into account. Several extended BGK models have been recently proposed to do so, for instance [16,17,18,19], and a recent Fokker-Planck model has been proposed earlier in [13].…”