A modal discontinuous Galerkin method for simulating dusty and granular gas flows in thermal non-equilibrium in the Eulerian framework

Abstract

A modal discontinuous Galerkin method was developed for computing compressible rarefied gaseous flows interacting with rigid particles and granular medium. In contrast to previous particle-based models that were developed to handle rarefied flows or solid phase particles, the present computational method employs full continuum-based models. This work is one of the first attempts to apply the modal discontinuous Galerkin method to a two-fluid model framework, which covers a wide range of gas and solid phase regimes, from a continuum to non-equilibrium gas, and from dusty to collisional regimes. The rarefaction effects were taken into account by applying the second-order Boltzmann-Curtiss-based constitutive relationship in a two-fluid system of equations. For the dust phase, computational models were developed based on the kinetic theory of the granular flows. Due to the orthogonal property of the basis functions in the method, no specific treatment of the source terms, commonly necessary in the conventional finite volume method, was required. Moreover, a high-fidelity approach was selected to treat the non-strictly hyperbolic equations of a dusty gas. This allows the same inviscid numerical flux functions to be applied to both the gaseous Euler and solid pressureless-Euler system of equations. Further, we observed that, for the discretization of the viscous fluxes in multiphase cases, the local discontinuous Galerkin is superior to the first method by Bassi and Rebay. After a verification and validation study, the new computational model was used to simulate the impingement of an underexpanded jet on a dusty surface in a rarefied condition. A surface erosion model based on viscous erosion associated with aerodynamic entrainment was implemented at a solid surface. Simulation cases in the near-field of the nozzle flow were tested to evaluate the capabilities of the present computational model in handling the challenging problems of multi-scale multiphase flows.