A Foundamental Research On The Lattice Boltzmann Method And Its Application For Incompressible Flows

In general, the macroscopic dynamics is insensitive to the details of the underlyingmicroscopic interactions, therefore, the artificial micro-dynamical models, which ignoreas much irrelevant details as possible but retain the basic conservation laws, provideconvenient alternatives to the description of the fluid hydrodynamics. The lattice gasmodels and their derivative, the lattice Boltzmann method (LBM), belong precisely tosuch a class of models. In the past few decades, the LBM has been taken as a promisingalternative to conventional numerical methods for fluid dynamics because of itsadvantages of the simplicity in programming, the ease of parallelization, and thecapability to handle complex geometric boundaries. Particularly, microscopic physics,responsible for many complicated fluid phenomena, are allowed to be incorporated intothe LBM owing to its kinetic essence, which makes the LBM applicable to complexfluid systems. In this thesis, a systematic research on the theory of the LBM and itspractical application in the incompressible flows is provided. My main works are:(1) Theoretical derivations form the Boltzmann equation to the lattice Boltzmannequation, and from the lattice Boltzmann equation to the macroscopic N-S equations areprovided. Then, a procedure of lattice Boltzmann modeling is developed. The momentsof the distribution function are modified firstly according to the desired macroscopicequations. The continuous equilibrium distribution function (EDF) pertaining to themodified moments is then derived with the Hermite expansion. After discretization byapplying the relevant lattice model, the discrete EDF for the desired macroscopicequations is obtained.(2) An alternative lattice Boltzmann model for incompressible flows is proposed.Under the basic assumptions that the fluid density is a uniform constant and the fluidpressure is independent of density, the moments of the distribution function aremodified. The continuous EDF in tensor Hermite polynomials is derived using themoment expansion and then discretized with the discrete velocity vectors. Besides, analternative pressure formula, which represents the pressure as the diagonal part of themomentum flux tensor, is adopted in the present model. Moreover, two strategies areintroduced to enhance the stability of the present model: one is the incorporation of themulti-relaxation-time (MRT) collision operator; alternatively, in accordance with theRegularized lattice BGK scheme, an additional relaxation time pertaining to the non-hydrodynamic mode is added to the BGK collision operator.(3) A comparative study on the four lattice Boltzmann models, the standard model,the He-Luo model, the Guo model, and the present model, is conducted. Theoretically,the macroscopic equations derived from the involved LB models are compared by theChapman-Enskog analysis. Then, the analytical framework proposed in M. Junk’s workis applied to investigate the finite difference stencils and the equivalent moment systemspertaining to the concerned LB models. Besides, the steady cavity flow,backward-facing step flow and the unsteady flow past an impulsively started cylinderare adopted as benchmark tests to compare the numerical performances of theconcerned models.(4) An alternative scheme to implement the velocity Dirichlet boundary conditionfor curved boundary in the LBM is proposed. The distribution functions at the boundarynodes are represented by the first order Chapman-Enskog expansion. And themacroscopic variables, necessary for the expression of the distribution function, aredivided into two parts: the known part derived from the physical boundary conditionson the boundary wall with the first order Taylor expansion and the unknown partcalculated from the known distribution functions at the same boundary node. Essentially,the unknown distribution functions are obtained as liner combination of the known ones,and the coefficients of the combination depend on the macroscopic constraints on theboundary wall, the geometric information of the boundary nodes and the relaxationparameters.(5) To suppress the unphysical spurious velocities resulting from the S-C model, analternative scheme to attain the nonideal state equation is proposed. By modifyingdirectly the second order moment equation of the EDF, the nonideal state equation isincorporated into the lattice Boltzmann model. Then, the MRT multiphase modelpertaining to the present scheme is developed.Conclusions drawn from above investigations are:(1) The standard incompressible N-S equations for both steady and unsteady flowscan be derived from the proposed incompressible lattice Boltzmann model. Furthertheoretical comparisons with other models demonstrate that: despite of differentexpression of the pressure, the discretization of the pressure gradient in the macroscopicN-S equations is realized by the same finite difference stencil; the laplacian of the zeroorder moment of the distribution function has explicit effect on the accuracy of themodeled stress tensor and can be avoided by the present model. Besides, it is found from the numerical results that: the present model provides better accuracy in the regionof high deviatoric stress, and the time-dependent flow features obtained from thepresent model are more in line with the incompressibility condition.(2) The proposed boundary scheme is second order accurate, which is proved bothfrom the theoretical derivations and the numerical validations. Besides, the presentscheme is different from the previous ones in two aspects: the determination of theunknown distribution functions is local and no access of the information from theneighboring fluid nodes is introduced; in the present scheme, a local curvilinearcoordinate system associated with the curved boundary wall is introduced, and thegeometric positions of the boundary nodes are determined directly by the theircoordinates, instead of the intersected lattice links in the previous schemes.(3) The proposed scheme to implement the state equation has the advantage thatthe local momentum conservation on each lattice site is preserved, which is helpful toeliminate the spurious velocities around the interfaces. The Chapman-Enskog analysisdemonstrates that, under the low Ma limit, the macroscopic N-S equations withnonideal state equation are derived from the present scheme. And the simulation of thehomogenous cavitation numerically validates the incorporation of the state equation.However, the surface tension is not included in the MRT multiphase model pertaining tothe present scheme since the relevant pressure tensor is completely isotropic.