A finite element discrete Boltzmann method for high Knudsen number flows

Abstract

Simulations of the discrete Boltzmann Bhatnagar-Gross-Krook (BGK) equation are an important tool for understanding fluid dynamics in non-continuum regimes. Here, we introduce a discontinuous Galerkin finite element method (DG-FEM) for spatial discretization of the discrete Boltzmann equation for isothermal flows with Knudsen numbers (Kn ∼ O(1)). In conjunction with a high-order Runge-Kutta time marching scheme, this method is capable of achieving high-order accuracy in both space and time, while maintaining a compact stencil. We validate the spatial order of accuracy of the scheme on a two-dimensional Couette flow with Kn = 1 and the D2Q16 velocity discretization. We then apply the scheme to lid-driven micro-cavity flow at Kn = 1, 2, and 8, and we compare the ability of Gauss-Hermite (GH) and Newton-Cotes (NC) velocity sets to capture the high non-linearity of the flow-field. While GH quadrature provides higher integration strength with fewer points, the NC quadrature has more uniformly distributed nodes with weights greater than machine-zero, helping to avoid the so-called ray-effect. Broadly speaking, we anticipate that the insights from this work will help facilitate the efficient implementation and application of high-order numerical methods for complex high Knudsen number flows.

Acoustic Simulation of Hydraulic Turbine Top Bolt Fatigue Crack Detection Based on DG-FEM

“…The Discontinuous Galerkin method (DG-FEM) has the features of both Finite Element Method (FEM) and Finite Volume Method (FVM) , and it is increasingly widely used. [6][7][8][9]. Just like the general finite element method, the DG method uses the element polynomial space as the approximate solution and test function space.…”