Critical assessment of various particle Fokker–Planck models for monatomic rarefied gas flows

Abstract

In the past decade, the particle-based Fokker–Planck (FP) method has been extensively studied to reduce the computational costs of the direct simulation Monte Carlo method for near-continuum flows. The FP equation describes a continuous stochastic process through the combined effects of systematic forces and random fluctuations. A few different FP models have been proposed to fulfill consistency with the Boltzmann equation, but a comprehensive comparative study is needed to assess their performance. The present paper investigates the accuracy and efficiency of four different FP models—Cubic-FP, ellipsoidal-statistical FP (ES-FP), and quadratic entropic FP (Quad-EFP)—under rarefied conditions. The numerical test cases include one-dimensional Couette and Fourier flows and an argon flow past a cylinder at supersonic and hypersonic velocities. It is found that the Quad-EFP model gives the best accuracy in low-Mach internal flows, whereas the ES-FP model performs best at predicting shock waves. In terms of numerical efficiency, the Linear-FP and ES-FP models run faster than the Cubic-FP and Quad-EFP models due to their simple algebraic nature. However, it is observed that the computational advantages of the FP models diminish as the spatiotemporal resolution becomes smaller than the collisional scales. In order to take advantage of their numerical efficiency, high-order joint velocity-position integration schemes need to be devised to ensure the accuracy of FP models with very coarse resolution.