Unified gas-kinetic wave-particle methods IV: multi-species gas mixture and plasma transport

In this paper,we extend the unified gas-kinetic wave-particle(UGKWP)methods to the multi-species gas mixture and multiscale plasma transport.The construction of the scheme is based on the direct modeling on the mesh size and time step scales,and the local cell’s Knudsen number determines the flow physics.The proposed scheme has the multiscale and asymptotic complexity diminishing properties.The multiscale property means that according to the cell’s Knudsen number the scheme can capture the non-equilibrium flow physics when the cell size is on the kinetic mean free path scale,and preserve the asymptotic Euler,Navier-Stokes,and magnetohydrodynamics(MHD)when the cell size is on the hydrodynamic scale and is much larger than the particle mean free path.The asymptotic complexity diminishing property means that the total degrees of freedom of the scheme reduce automatically with the decreasing of the cell’s Knudsen number.In the continuum regime,the scheme automatically degenerates from a kinetic solver to a hydrodynamic solver.In the UGKWP,the evolution of microscopic velocity distribution is coupled with the evolution of macroscopic variables,and the particle evolution as well as the macroscopic fluxes is modeled from a time accumulating solution of kinetic scale particle transport and collision up to a time step scale.For plasma transport,the current scheme provides a smooth transition from particle-in-cell(PIC)method in the rarefied regime to the magnetohydrodynamic solver in the continuum regime.In the continuum limit,the cell size and time step of the UGKWP method are not restricted by the particle mean free path and mean collision time.In the highly magnetized regime,the cell size and time step are not restricted by the Debye length and plasma cyclotron period.The multiscale and asymptotic complexity diminishing properties of the scheme are verified by numerical tests in multiple flow regimes.

Unified Gas-Kinetic Wave-Particle Methods VI:Disperse Dilute Gas-Particle Multiphase Flow

A coupled gas-kinetic scheme(GKS)and unified gas-kinetic wave-particle(UGKWP)method for the disperse dilute gas-particle multiphaseflow is proposed.In the two-phaseflow,the gas phase is always in the hydrodynamic regime and is fol-lowed by GKS for the Navier-Stokes solution.The particle phase is solved by UGKWP in all regimes from particle trajectory crossing to the hydrodynamic wave interac-tion with the variation of particle’s Knudsen number.In the intensive particle colli-sion regime,the UGKWP gives a hydrodynamic wave representation for the particle phase and the GKS-UGKWP for the two-phaseflow reduces to the two-fluid Eulerian-Eulerian(EE)model.In the rarefied regime,the UGKWP tracks individual particle and the GKS-UGKWP goes back to the Eulerian-Lagrangian(EL)formulation.In the tran-sition regime for the solid particle,the GKS-UGKWP takes an optimal choice for the wave and particle decomposition for the solid particle phase and connects the EE and EL methods seamlessly.The GKS-UGKWP method will be tested in allflow regimes with a large variation of Knudsen number for the solid particle transport and Stokes number for the two-phase interaction.It is confirmed that GKS-UGKWP is an efficient and accurate multiscale method for the gas-particle two-phaseflow.

Convergence proof of the DSMC method and the Gas-Kinetic Unified Algorithm for the Boltzmann equation

This paper investigates the convergence proof of the Direct Simulation Monte Carlo(DSMC) method and the Gas-Kinetic Unified Algorithm in simulating the Boltzmann equation.It can be shown that the particle velocity distribution function obtained by the DSMC method converges to a modified form of the Boltzmann equation,which is the equation of the gas-kinetic unified algorithm to directly solve the molecular velocity distribution function.Their convergence is derived through mathematical treatment.The collision frequency is presented using various molecular models and the local equilibrium distribution function is obtained by Enskog expansion using the converged equation of the DSMC method.These two expressions agree with those used in the unified algorithm.Numerical validation of the converging consistency between these two approaches is illustrated by simulating the pressure driven Poiseuille flow in the slip transition flow regime and the two-dimensional and three-dimensional flows around a circular cylinder and spherical-cone reentry body covering the whole flow regimes from low speed micro-channel flow to high speed non-equilibrium aerothermodynamics.

A discrete unified gas-kinetic scheme for multi-species rarefied flows

A discrete unified gas kinetic scheme(DUGKS)is developed for multi-species flow in all flow regimes based on the Andries-Aoki-Perthame(AAP)kinetic model.Although the species collision operator in the AAP model conserves fully the mass,momentum,and energy for the mixture,it does not conserve the momentum and energy for each species due to the inter-species collisions.In this work,the species collision operator is decomposed into two parts:one part is fully conservative for the species and the other represents the excess part.With this decomposition,the kinetic equation is solved using the Strang-splitting method,in which the excess part of the collision operator is treated as a source,while the kinetic equation with the species conservative part is solved by the standard DUGKS.Particularly,the time integration of the source term is realized by either explicit or implicit Euler scheme.By this means,it is easy to extend the scheme to gas mixtures composed of Maxwell or hard-sphere molecules,while the previous DUGKS[Zhang Y,Zhu L,Wang R et al,Phys Rev E 97(5):053306,2018]of binary gases was only designed for Maxwell molecules.Several tests are performed to validate the scheme,including the shock structure under different Mach numbers and molar concentrations,the Couette flow under different mass ratios,and the pressure-driven Poiseuille flow in different flow regimes.The results are compared with those from other reliable numerical methods based on different models.And the influence of molecular model on the flow characteristics is studied.The results also show that the present DUGKS with implicit source discretization is more stable and preferable for gas mixture problems involving different flow regimes.

A Gas-Kinetic Unified Algorithm for Non-Equilibrium Polyatomic Gas Flows Covering Various Flow Regimes

In this paper,a gas-kinetic unified algorithm(GKUA)is developed to investigate the non-equilibrium polyatomic gas flows covering various regimes.Based on the ellipsoidal statistical model with rotational energy excitation,the computable modelling equation is presented by unifying expressions on the molecular collision relaxing parameter and the local equilibrium distribution function.By constructing the corresponding conservative discrete velocity ordinate method for this model,the conservative properties during the collision procedure are preserved at the discrete level by the numerical method,decreasing the computational storage and time.Explicit and implicit lower-upper symmetric Gauss-Seidel schemes are constructed to solve the discrete hyperbolic conservation equations directly.Applying the new GKUA,some numerical examples are simulated,including the Sod Riemann problem,homogeneous flow rotational relaxation,normal shock structure,Fourier and Couette flows,supersonic flows past a circular cylinder,and hypersonic flow around a plate placed normally.The results obtained by the analytic,experimental,direct simulation Monte Carlo method,and other measurements in references are compared with the GKUA results,which are in good agreement,demonstrating the high accuracy of the present algorithm.Especially,some polyatomic gas non-equilibrium phenomena are observed and analysed by solving the Boltzmann-type velocity distribution function equation covering various flow regimes.