Low Mach number compressible multiphase particle-in-cell method for viscous flow problem

Recently, Tian et al. developed a new compressible multiphase flow model, termed CMP-PIC (compressible multiphase particle-in-cell) method [1], for the simulation of full flow patterns ranging from dilute to dense flows of gas-particle systems. Within this model, the governing equations for the gas phase are formulated in the Eulerian framework, and a five-equation transport model for multi-material compressible flows is adopted. For the particle, it is tracked in the Lagrangian coordinate system based on the particle-in-cell (PIC) method. The collisions among particles are simulated with a coarse-grained discrete element method (DEM). A two-way coupling model is derived based on a comparative study with the Baer and Nunziato (B-N) model. In this work, we adopt the Weiss-Smith local preconditioner to the density-based CMP-PIC method to solve compressible multiphase flows at low Mach numbers. A third-order Runge-Kutta method, which uses an inner iteration of pseudo-time within the physical time, is then derived to recover the time accuracy impaired by the preconditioner. The AUSM⁺-up scheme, characterized by improved accuracy and low numerical dissipation across a wide range of Mach numbers, including low-speed flows, is introduced to the CMP-PIC method for the computation of the convective fluxes of the immiscible two-material compressible flow. Then, we derive a discretization scheme for the nozzling term that is compatible with the AUSM⁺-up flux scheme. The governing equation for the gas phase is extended from the Euler equation to the Navier-Stokes equation for the simulation of viscous flows. The proposed method is first validated against a series of classical benchmarks, including a stationary contact discontinuity interacting with particles, the Rogue test, and the Gresho vortex. Its accuracy in resolving low-Mach-number viscously coupled systems is further established through the particle-laden Gresho vortex and the steady incompressible lid-driven cavity problems. Subsequently, the method is employed to investigate the transient flow dynamics within a complex multiphase lid-driven cavity, with a specific focus on the parametric sensitivity of particle density, particle diameter, and lid velocity.