The Research Of Discrete Unified Gas Kinetic Scheme For Incompressible Navier-stokes Equations And Nonlinear Convection-diffusion Equation And Its Application

Incompressible Navier-Stokes equations and nonlinear convection-diffusion equation(NCDE)play important roles in the research of nature and industrial production.For example,in phase-field models for multiphase flow,the incompressible Navier-Stokes equations describe the fluid flow,and the Allen-Cahn equation(or Cahn-Hilliard equation)is the convection-diffusion equation;In Rayleigh-Bénard system,the incompressible Navier-Stokes equations also describe the fluid flow,and the temprature equation controls the distribution of the temprature.So,developing the efficient numerical methods of the two equations really means a lot to human being.With the develotpment of the computer technology,the mesoscopic numerical methods based on the discrete velocity Boltzmann equation(DVBE)are becoming more and more popular,because they are not only naturally parallel,but also easy to handle the complex boundary conditions and compute multi-field problems.From the end of the last century,the lattice Boltzmann method(LBM)develops rapidlly and has achieved many accomplishments.Then,some scholars use the LBM to solve the convection-diffusion problems,because the algorithms are easy and they are naturally parallel.However,some shortcomings have been found due to its collision-propagation principle,for example,LBM can not adopt non-uniform meshes,its stability is not very good,and so on.Though many researchers develop some modified models,such as multi-relaxation-time lattice Boltzmann(MRT-LB)model and rectangular lattice Boltzmann(RLB)model,the effect is limited.On the other hand,the DUGKS is a hybrid method with LBM and finite volume method,not only its stability is better than LBM,but also it adopt the non-uniform meshes easily.However,the research of incompressible Navier-Stokes equations and non-linear convection-diffusion equation is poor.So,we have carried out the following works:(1)We have developed a new DUGKS model for incompressible Navier-Stokes equations.Different from the previous work,we adopt a new equilibrium distribution function,and the pressure is independent of density.So,the equation of state is not needed to calculate the pressure,it is consistent with the macroscopic methods.Then,we prove that the present DUGKS can solve the incompressible Navier-Stokes equations through Chapman-Enskog analysis.To satisfy the Dirichlet boundary condition of pressure,we adopt a new mesh type different from previous works and a new non-equilibrium extrapolation method.Finally,the second order convergence rate can be demonstrated in the numerical solutions.In the meantime,we prove that our DUGKS can decrease the error by changing the Ma number.(2)To solve the NCDE,we develop a new DUGKS model and fill the gap of the research of DUGKS solving NCDE.Different from the incompressible Navier-Stokes equations,maybe some nonlinear functions of macro-quantities exist in the convection term or the source term.So,to keep the naturally parallel of DUGKS,we design three methods to fit the different problems.Then,we redesign the equation of III calculating the micro flux,cut down some extra gradient calculation.From the Chapman-Enskog expansion,we prove that the DVBE and the redesigned equation for micro flux can recover the NCDE in the same moment conditions.Finally,we carry some numerical experiments and demonstrate that the present DUGKS have second order convergence rate.In the meantime,compared with LBM,we can see that the DUGKS is more stable than LBM and the advantages of the non-uniform meshes.(3)We find that the present DUGKS models have some disadvantages,for example,they can not solve the anisotropic problems,the bounce-back boundary conditon may bring some extra errors.In order to solve these problems,we adopt the multi-relaxation-time collision operator,and develop a new multirelaxation-time discrete unified gas kinetic scheme(MRT-DUGKS).From the Chapman-Enskog expansion,we prove that the MRT-DUGKS can solve the incompressible Navier-Stokes equations and NCDE in different moment conditons.Different from single-relaxation-time(SRT)model,in MRT-DUGKS,every direction's distribution function is influenced by the distribution functions at all directions.So,the thought of solving pressure in SRT model can not be used in MRT-DUGKS.To get a general equation of calculating pressure,we observe many discrete velocity models and the inverse matrix of the collision matrix.Then,we give a method to calculate pressure ignoring the discrete velocity models.Finally,through some numerical experiments,we show the capacity of the MRT-DUGKS by two anisotropic problems,and research the influence of adjustable factor.(4)Based on the previous works,we develop a new DUGKS model to study the Rayleigh-Bénard convection system which coupled with the incompressible Navier-Stokes equations and the temperature equation.To overcome the curse of dimensionality,we use the GPU high-performance algorithms to speed up our new DUGKS,the method can increase the efficiency by 30 times.Then,through the nature convection,we demonstrate the capacity of the new DUGKS.Finally,we research some problems of the mixed heat boundary conditions in the Rayleigh-Bénard convection system,and the influence of the different position and area of the mixed heat boundary conditions.In conclusion,to solve incompressible Navier-Stokes equations and NCDE,we propose two new DUGKS models,the new models can adopt non-uniform meshes and are more stable than LBM.Then,to overcome the shortcomings like anisotropic problem,we adopt the multi-relaxation-time collision operator,develop the MRT-DUGKS.Finally,based on the previous works,we study the influence of the mixed heat boundary conditions in Rayleigh-Bénard convection system.