Analyses Of Typical Lattice Boltzmann Models And Their Macroscopic Reconstructions

Numerical simulations of incompressible flow have wide applications in engineering.As a kind of mecroscopic numerical method,owing to its fundamental physical description and simple implementation,the lattice Boltzmann method(LBM)obtained increasing attention.It has been widely applied to simulate various non-equilibrium physical problems,such as fluid flows,heat and mass transfer.However,LBM has some intrinsic drawbacks,which mainly include:(1)It is limited to uniform mesh in general,and the implementation in a non-uniform mesh is complicated.(2)The time step is coupled with the mesh space,which complicates the implementation of an adaptive mesh or a multi-block mesh.(3)It needs extra memory space to store distribution functions.These intrinsic drawbacks prevent LBM from being widely applied to simulate practical engineering problems.To overcome these drawbacks,a number of works have been made.However,due to the limitation of the characteristics of LBM,these works can only partially overcome the drawbacks mentioned above in general.On the other hand,by using the conventional Chapman-Enskog(C-E)expansion analysis,the standard lattice Boltzmann equation(LBE)can recover the second-order continuity and momentum equations.If the macroscopic equations recovered from the conventional C-E expansion analysis(MEs-CE)can be solved directly,the three drawbacks of LBM can be overcome easily.However,it is found from some numerical tests that directly solving MEs-CE suffers from serious instability,which shows that MEs-CE cannot explain the good stability of the standard LBE.In the present paper,by using Taylor expansion analysis,the more actual macroscopic equations(MAMEs)recovered from the standard LBE are derived at first.The discretized MAMEs can stably simulate incompressible flow.Compared with MEs-CE,MAMEs retain some additional terms.It is found from some numerical tests that these additional small terms are essential to stabilizing computation,however,do not have an obvious effect on numerical accuracy.Based on MAMEs,a finite difference method(FDM)solver,which can overcome the three darwbacks of LBE,is constructed.Since C-E expansion analysis cannot recover the additional terms correctly,the present paper analyzes the drawbacks of the conventional C-E expansion analysis,and proposes the improved C-E expansion analysis.By using the improved C-E expansion analysis,MAMEs can be correctly recovered.Based on Taylor expansion and the improved C-E expansion analysis,the present paper derives the macroscopic equations of some typical LB models,which include two incompressible LB models,thermal LB model and three simplified LB models,and investigates the stability and accuracy of these macroscopic equations through theoretical analyses and numerical tests.Considering that finite volume method(FVM)has been widely used in engineering applications owing to its flexibility in discretization,the present paper proposes a FVM solver based on the macroscopic equations recovered from He-Luo model by using the improved C-E expansion analysis,and simplifies the lattice Boltzmann flux solver(LBFS).These works build a foundation for applying the derived macroscopic equations to simulate engineering problems.