Unified gas-kinetic wave-particle methods I: Continuum and rarefied gas flow

Abstract

The unified gas-kinetic scheme (UGKS) provides a framework for simulating multiscale transport with the updates of both gas distribution function and macroscopic flow variables on the cell size and time step scales. The multiscale dynamics in UGKS is achieved through the coupled particle transport and collision in the particle evolution process within a time step. In this paper, under the UGKS framework, we propose an efficient multiscale unified gas-kinetic wave-particle (UGKWP) method. The gas dynamics in UGKWP method is described by the individual particle movement coupled with the evolution of the probability density function (PDF). During a time step, the trajectories of simulation particles are tracked until collision happens, and the post-collision particles are evolved collectively through the evolution of the corresponding distribution function. The evolution of simulation particles and distribution function is guided by the evolution of macroscopic variables, and guarantees the conservation of the scheme in the final wave-particle formulation. A new concept of multiscale multi-efficiency preserving (MMP) method is introduced. Multiscale preserving means UGKWP method preserves the flow regime from collisionless regime to hydrodynamic regime without requiring the cell size and time step to be less than the mean free path and collision time. Multi-efficiency preserving means the computational cost of the scheme including the computational time and memory cost is on the same level as the most efficient method in the corresponding regime, such as the particle methods in the rarefied regime and hydrodynamic solvers in continuum regime. The UGKWP method is shown to satisfy the MMP requirement. The UGKWP method is specially efficient for hypersonic flow simulation in all regimes in comparison with the discrete ordinate methods, and presents a much lower stochastic noise in the continuum flow regime in comparison with the particle-based Monte Carlo methods. Numerical tests for flows over a wide range of Mach and Knudsen numbers are presented. The examples include the hypersonic flow passing a circular cylinder at Mach numbers 20 and 30 and Knudsen numbers 1 and 10−4, low speed lid-driven cavity flow, laminar boundary layer, shock structure, and shock tube problems. These results validate the accuracy, efficiency, and multiscale and multi-efficiency property of UGKWP method.