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

Abstract

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.

GKS and UGKS for High-Speed Flows

“…With the discrete velocity distribution function, the compatibility condition cannot be naturally satisfied due to the numerical quadrature error, so several conservative kinetic methods [54,[118][119][120][121][122][123] have been constructed, which could maintain the compatibility condition in the discrete form and show that the conservation of collision term can slightly release the strict requirement of velocity space discretization, and gain better convergence for implicit calculations.…”

A direct relaxation process for particle methods in gas kinetic theory

“…This 0-Dimensional (0-D) homogenous case is used to examine the validity and accuracy of the DR process. The initial distribution consists of two Maxwellian distributions to mimic the high non-equilibrium distribution in the normal shock wave 56 . The macroscopic variables of the two Maxwellian distributions are as follows:…”

A multi-scale kinetic inviscid flux extracted from the gas-kinetic scheme for simulating incompressible and compressible flows

“…The implied mechanism is that: initially, the two parts of distribution when combining together, deviate from the equilibrium Maxwellian distribution function, and stress and heat flux are generated; after sufficient collision, both stress and heat flux are decayed into zero, and ḡ is achieved; since the Maxwellian distribution has the locally maximum entropy, ḡ can be viewed as being totally thermalized. This merging process can finish during about 10 molecular mean collision time [34,35]. Therefore, the TTT actually describes the behaviors of molecules that experience sufficient collisions.…”