Acceleration Algorithms For The Unified Gas-kinetic Scheme

Multiscale flow problems are involved in many engineering applications,such as launch of rockets,reentry of spacecrafts,altitude control of satellites and flow around high-speed vehicles.Since the macroscopic computational fluid dynamics(CFD)method based on Navier–Stokes(NS)equations with Newton’s law of viscosity and Fourier’s law of heat conduction becomes invalid in the rarefied flow simulations,it is necessary to develop an accurate and efficient multiscale method from the gas-kinetic theory,which is capable of continuum and rarefied flow simulations.The unified gas-kinetic scheme(UGKS)is a multiscale method based on the direct modeling of flow physics on the numerical discretization scale.With the multiscale flux function constructed from the integral solution of kinetic model equation,the UGKS couples the particles’ free transport and collision in the evolution process so that it could be able to resolve the multiscale flow physics in all Knudsen number regimes without the requirement of mesh size and time step smaller than particles’ mean free path and collision time,respectively.With the multiscale property,UGKS is more efficient than the single scale method based on the splitting treatment of particles’ free transport and collision,and has great potentials in real engineering applications for solving multiscale transport problems.However,it suffers from the high memory requirement and huge computational cost due to the employment of discrete velocity distribution function,especially for three dimensional high speed flows.Therefore,it is a must to develop the acceleration algorithms and to improve the computational efficiency of the UGKS for further applications in the academic studies and engineering problems.The current study considers the acceleration techniques in the deterministic method and the stochastic method,and the main works are given as follows.(1)The multiscale implicit UGKS(IUGKS)for unsteady flow simulations is constructed by alternately taking implicit iterations for the macroscopic and the microscopic governing equations,which greatly improves the computational efficiency of the UGKS both in the continuum and rarefied flow.(2)The geometric multigrid method is employed to further accelerate the solving of the implicit system in the IUGKS,which enhances the IUGKS with one order of magnitude increment of convergence efficiency for steady state solutions.(3)Considering that the stochastic particle method,such as the direct simulation Monte Carlo method,has great advantages in high speed rarefied flow simulations due to the optimal adaptive property in the velocity space,a unified gas-kinetic particle(UGKP)method is constructed by recovering the multiscale transport process of the UGKS with simulation particles,which at its root overcomes the disadvantage of the UGKS in the aspect of high memory requirement and huge computational cost for rarefied high speed flow simulations.(4)Furthermore,according to the dynamic evolution,the equilibrium part of the simulation particles can be represented in an analytic way by macroscopic flow variables.As a result,a unified gas-kinetic wave-particle(UGKWP)method is constructed with a novel wave-particle adaptive formulation,which could automatically reduce to the hydrodynamic gas-kinetic NS solver in the continuum regime,and further improve the computational efficiency of the UGKP method in the continuum and transition regimes.The novelty of the current work lies in(1)An alternate implicit iteration method is used to solve the large implicit system formed by the coupled macroscopic and microscopic governing equations,and a highly efficient IUGKS is constructed for unsteady multiscale flow simulations.(2)The UGKWP method is constructed,recovering the multiscale transport process of UGKS.With the wave-particle adaptive description,it could become a macroscopic NS solver and a stochastic particle method with high efficiency,respectively in continuum flow and rarefied regimes.With the implicit algorithm,multigrid acceleration technique from deterministic method and the adaptive property in velocity space from the stochastic particle method,the UGKS is greatly enhanced in the aspect of computational efficiency and memory requirement,and becomes a very powerful tool to solve multiscale transport problems.