Study On The Compressible Lattice Boltzmann Method

Lattice Boltzmann Method(LBM)is a mesoscopic approach based on the kinetic theory of fluids.In LBM,a fluid is considered as a group of discrete particles.These particles are infinitesimal on macroscopic observation and infinite on microscopic observation.These particles move with some specific velocities.The probability which particles move with a certain velocity is expressed as distribution function.Through the evolution and statistics of distribution function,the macroscopic fluid properties can be obtained.Compared with the traditional method which bases on Navier-Stokes(NS)equations,LBM has low-level physical background and the government equation of LBM has simpler structure which means that LBM is easy to be programmed.More important,because LBM bases on the kinetic theory,LBM has advantages during the simulation of the multi-scale behavior in complex fluid systems.Under certain conditions,LBM can recover the full compressible NS equations.Over the past few decades,LBM has became an accurate and effective numerical approach in the computational fluid dynamics and has been successfully applied in many areas of fluid mechanics,such as turbulent flows,multi-phase multi-component flows,porous media flows,micro-scale flows,chemical reaction flows and particle suspension flows.After years of development,many achievements of LBM have been obtained.But there are still some problems and constraints.An important problem of LBM is that it is difficult to simulate compressible flows.Compressible flows are very common in the field of natural science and engineering,especially in the field of aeronautics and astronautics.At present,the study of using LBM to simulate compressible flows is at the stage of theoretical research.Many compressible LBM models are proposed,but none of them is a perfect solution.So the simulation of compressible flows using LBM has important theoretical significance and broad application prospect.Moreover,the introduction of finite difference method and finite volume method can further extend the application range of LBM.Computational aeroacoustics(CAA)is a branch of computational fluid dynamics.Its aim is to study the sound generation mechanism in unsteady flows and provide theoretical basis for the prediction and control of noise.At present,there are two major categories of CAA: hybrid method and direct method.In hybrid method,the flow field is firstly simulated to find the acoustic source and then the acoustic field is calculated using the acoustic analogy method.In direct method,the acoustic field is a natural part of the flow field and calculated simultaneously with the flow field.Compared with hybrid method,computational cost of direct method is high.However,direct method can not only provide the details of sound generating mechanisms but also show the sound-flow interactions.As a result,direct method is more suitable for mechanism study.The presentation of compressible LBM model makes the simulation of acoustic problems at high Mach number using LBM possible.In this thesis,the basic theories of LBM are studied systematically,and some innovations and improvements are made about the theory and application of LBM.On the basis of previous research,a new compressible LBM model,potential energy double-distribution-function model,is proposed.Potential energy double-distribution-function compressible LBM model has adjustable specific heat ratio and Prandtl number,and potential energy distribution function can be obtained directly from density distribution function.Through the Chapman-Enskog(CE)expansion,potential energy double-distribution-function compressible LBM model can recover the full compressible Navier-Stokes(NS)equations.In the calculation,the two-dimension-seventeen-velocity lattice and circle function based equilibrium distribution function are used.The government equations are solved by finite difference method or finite volume method.Numerical simulations of Sod shock tube,Colella explosion wave,Couette flow and flows around a NACA0012 airfoil prove that potential energy double-distribution-function compressible LBM model can effectively simulate compressible flows,even the strong shock wave problem which has an extremely large pressure ratio,100,000.It also proves that the curve boundary can be simulated using the body-fitted grid in LBM.In order to further improve the performance of potential energy double-distribution-function compressible LBM model,a multiple-relaxation-time potential energy double-distribution-function compressible LBM model is structured,and equilibria of moments are also given.Through Colella explosion wave,free parameters of multiple-relaxation-time potential energy double-distribution-function compressible LBM model are analyzed numerically,and values of these parameters are suggested.Then flows around a circular cylinder under high Mach number are simulated.Numerical results show that,compared with the single-relaxation-time model,accuracy and stability of the multiple-relaxation-time potential energy double-distribution-function compressible LBM model are improved.Finally,the single-relaxation-time potential energy double-distribution-function compressible LBM model is used to simulate acoustic problems.The results of plane pressure pulse and circular pressure pulse are agreed with the reference results which prove that the potential energy double-distribution-function compressible LBM model has the cability of aeroacoustics calculation.Then the subsonic flow around a circular cylinder and transonic flow around an airfoil are simulated and the corresponding sound fields are obtained at the same time.The buffer zone and absorbing boundary condition is employed to eliminate the non-physical reflecting.The generation and propagation of the sound produced by the vortex shedding are reappeared clearly.The obtained results increase our understanding of the characteristic features of the aeroacoustic sound.