In this paper, we present a unified gas-kinetic wave-particle (UGKWP) method on unstructured mesh for the multiscale simulation of continuum and rarefied flow. Inheriting from the multiscale transport in the unified gas-kinetic scheme (UGKS), the integral solution of the kinetic model equation is employed in the construction of the UGKWP method to model the flow physics on the scales of cell size and time step. A novel wave-particle adaptive formulation is introduced in the UGKWP method to describe the flow dynamics in each control volume. The local gas evolution is constructed through the dynamical interaction of the deterministic hydrodynamic wave and the stochastic kinetic particle. To model the gas dynamics on the scales of cell size and time step, the decomposition, interaction, and evolution of the hydrodynamic wave and the kinetic particle depend on the ratio of time step to local collision time. In the rarefied flow regime, the UGKWP method recovers the nonequilibrium flow physics by discrete particles and performs as a stochastic particle method. In the continuum flow regime, the UGKWP method captures the flow behavior solely by macroscopic variable evolution and becomes a gas-kinetic hydrodynamic flow solver, the same as the gas-kinetic scheme, for viscous and heat-conducting Navier–Stokes solutions. In the transition regime, both kinetic particle and hydrodynamic wave contribute adaptively in the UGKWP to capture the peculiar nonequilibrium flow physics in a most efficient way. In different flow regimes, the Sod shock tube, lid-driven cavity flow, laminar boundary layer, and high-speed flow around a circular cylinder are computed to validate the UGKWP method on unstructured mesh. The UGKWP method obtains the same UGKS solutions in all Knudsen regimes. However, with an automatic wave-particle decomposition, the UGKWP method becomes very efficient. For example, at Mach number 30 and Knudsen number 0.1, the UGKWP has several-order-of-magnitude reductions in computational cost and memory requirement in comparison with UGKS. Overall, the UGKWP can capture the gas dynamics in all flow regimes efficiently and accurately from the free molecular transport to the Navier-Stokes flow evolution.

The unified gas kinetic scheme (UGKS) is based on direct modeling of gas dynamics on the mesh size and time step scales. With the modeling of particle transport and collision in a time-dependent flux function in a finite volume framework, the UGKS can connect the flow physics smoothly from the kinetic particle transport to the hydrodynamic wave propagation. In comparison with the direct simulation Monte Carlo (DSMC) method, the current equation-based UGKS can implement implicit techniques in the updates of macroscopic conservative variables and microscopic distribution functions. The implicit UGKS significantly increases the convergence speed for steady flow computations, especially in the highly rarefied and near continuum regimes. In order to further improve the computational efficiency, for the first time, a geometric multigrid technique is introduced into the implicit UGKS, where the prediction step for the equilibrium state and the evolution step for the distribution function are both treated with multigrid acceleration. More specifically, a full approximate nonlinear system is employed in the prediction step for fast evaluation of the equilibrium state, and a correction linear equation is solved in the evolution step for the update of the gas distribution function. As a result, convergent speed has been greatly improved in all flow regimes from rarefied to the continuum ones. The multigrid implicit UGKS (MIUGKS) is used in the non-equilibrium flow study, which includes microflow, such as lid-driven cavity flow and the flow passing through a finite-length flat plate, and high speed one, such as supersonic flow over a square cylinder. The MIUGKS shows 5–9 times efficiency increase over the previous implicit scheme. For the low speed microflow, the efficiency of MIUGKS is several orders of magnitude higher than the DSMC. Even for the hypersonic flow at Mach number 5 and Knudsen number 0.1, the MIUGKS is still more than 100 times faster than the DSMC method for obtaining a convergent steady state solution.

Discrete unified gas-kinetic scheme (DUGKS) is a multiscale numerical method for flows from continuum limit to free molecular limit, and is especially suitable for the simulation of multiscale flows, benefiting from its multiscale property. To reduce integration error of the DUGKS and ensure the conservation property of the collision term in isothermal flow simulations, a conserved-DUGKS (C-DUGKS) is proposed. On the other hand, both DUGKS and C-DUGKS adopt Cartesian-type discrete velocity space, in which Gaussian and Newton-Cotes numerical quadrature are used for calculating the macroscopic physical variables in low-speed and high-speed flows, respectively. However, the Cartesian-type discrete velocity space leads to huge computational cost and memory demand. In this paper, the isothermal C-DUGKS is extended to the nonisothermal case by adopting coupled mass and inertial energy distribution functions. Moreover, since the unstructured mesh, such as the triangular mesh in the two-dimensional case, is more flexible than the structured Cartesian mesh, it is introduced to the discrete velocity space of C-DUGKS, such that more discrete velocity points can be arranged in the velocity regions that enclose a large number of molecules, and only a few discrete velocity points need to be arranged in the velocity regions with a small amount of molecules in it. By using the unstructured discrete velocity space, the computational efficiency of C-DUGKS is significantly increased. A series of numerical tests in a wide range of Knudsen numbers, such as the Couette flow, lid-driven cavity flow, two-dimensional rarefied Riemann problem, and the supersonic cylinder flows, are carried out to examine the validity and efficiency of the present method.