Study On Filter-Matrix Model Of Lattice Boltzmann Method

The lattice Boltzmann method (LBM) has been developed as an alternative numericalapproach for solving various fluid flow problems and attracted tremendous amount ofattention from the computational fluid dynamics (CFD) community in many scientificdomains. Different from the conventional numerical methods for solving the Navier-Stokes(N-S) equations, the LBM is based on mesoscopic models and the kinetic theories, featuringinherent parallelization of the algorithm, easy implementation, and reliable numericalrobustness. To our knowledge, in the existing basic models of LBM, such as the latticeBhatnagar-Gross-Krook (LBGK) and multiple-relaxation-time (MRT) models, there still aresome issues to be improved:1) the inherent deviation terms exist in the force models for bothLBGK and MRT models, which need to be fixed following the theoretical analysis;2) theLBGK model suffers from its poor numerical accuracy and stability;3) although the MRTmodel is superior over the LBGK model in terms of accuracy and stability, it still experiencesdifficulty in seeking suitable relaxation rates to achieve excellent results in terms of bothaccuracy and stability. Thus, it would be much more beneficial to the CFD community if anenhanced model is established in such a way that the advantages of MRT are unaffected whilegreatly overcoming its inherent difficulty in selecting appropriate parameters targeted atsatisfactory numerical accuracy and stability. So, the main motivation of this thesis is, to buildsuch a new enhanced model, filter-matrix lattice Boltzmann (FMLB) model, and studying onthe extension and development of this model through the systematic theoretical analysis andnumerical investigations.The main researches of this thesis are described in detail as follows:1. A new filter-matrix lattice Boltzmann (FMLB) model is proposed for incompressible fluidflows. Different from the traditional lattice Boltzmann equation (LBE), the staggeredformulation for the time-space discretization of governing equation is adopted here toeliminate the discrete effects comes from the force term of LBE. The DnQb model based onthe Hermite expansionis found in the FMLB model. The fliter-matrix (or transform matrix)and its inverse matrix are constructed from the generalized Hermite polynomials, and theD2Q9&D3Q19models are presented. Numerical simulations are performed on thetwo-dimensional (2D) lid-driven square cavity flows, three-dimensional (3D) lid-driven cubiccavity flows, and3D square duct flows. The numerical results show that the present FMLB model is superior to the LBGK model in terms of both accuracy and stability. When comparedwith the MRT model, the similar accuracy and slightly enhanced stability can be obtained bythe present FMLB model. The advantage of the present model is that it no longer relies ondifficult selection of the free parameters requested by the MRT model; in addition, the forceterm is already included in the collision operator. Furthermore, FMLB model can be regard asa special kind of MRT model under specific situations, which is confirmed by the theoreticalanalysis of Chapman-Enskog expansion.2. The FMLB-D2Q9model is applied, simply employing uniform grids, to a series oflid-driven deep-cavity flow simulations, including the steady flows at various Reynoldsnumbers from0.01to5000in various cavity aspect ratios ranging from1.5to7, the unsteadyflows in various deep-cavitys, and the flow bifurcations from steady state to unsteady state,the first Hopf bifurcations. Through comparison against the relevant numerical resultsreported in the available literature, the present FMLB model further demonstrates itsadvantages in numerical accuracy and stability over the LBGK model, which can be furthervalidated on its usefulness, effectiveness and robustness in studying the complexincompressible flows. In particular, the selection of afree parameter in the FMLB solutionvector is carefully examined, resulting in some general suggestions that may render theFMLB stability consistently secured for simulations of different flow scenarios.3. Based on the Boussinesq approximation, the temperature can be regard as a passive scalar,so the present FMLB-D2Q9model can be extended to include incompressible thermal flowsby coupling a class of temperature-distribution function for evaluating the temperature field.Two differenttemperature-distribution functions are discussed, and two passive scalar (T2Q4&T2Q9) models are investigated. In the case of2Dnatural convection flow, the numericalresults of the two present models are almost the same, and both exhibit good agreement withthe benchmark solution. In addition, by introducing the large eddy simulation (LES)subgrid-scale (SGS), the novel thermal FMLB (TFMLB) modelis proposed for simulatingturbulent natural convection.The numerical simulations of2D natural convection in a squarecavity are performed at high Rayleigh number varying from107to1010. Here, the influencesof the higher-order terms upon the present results at high Rayleigh numbers are examined,taking Ra=107and108as the example, revealing that the proper minimization of thehigher-order terms can improve numerical accuracy of present model for high Rayleighconvective flow. For the turbulent convective flow, the time-averaged quantities in the medianlines are presented and compared against those available results from previous studies. Thegeneral structure of turbulent boundary layersis well predicted. All numerical results exhibit good agreement with the benchmark solutions available in the previous literatures.4. Inspired by the studies of the forefathers, the present FMLB-D2Q9modelis first applied tostudy the microchannel gas flows, in which a Bosanquet-type effective viscosity is usedtocapture the flow behaviors in the transition regime. A kinetic boundary condition, thecombined bounce-backand specular-reflection (CBBSR) scheme with the second-order slipscheme, is also designed for the FMLB model. By analyzing a unidirectional flow, the slipvelocity and the discrete effects related to the boundary condition arederived within theFMLB model, and a revised scheme is presented to overcome such effects, which have alsobeen validated through numerical simulations. To gain an accurate simulation in a wide rangeof Knudsen numbers, covering the slip and the entire transition flow regimes, a set of slipcoefficients with an introduced fitting function is adopted in the revised second-order slipboundary condition. The periodic and pressure-driven microchannel flows have beeninvestigated by the present model in this study. The numerical results obtained by thisextended FMLB model agree fairly well with the solutions of the linearized Boltzmannequation, the direct simulation Monte Carlo results, the experimental data, and the previousresults of the LBM-MRT model.5. To improve the computational performance for the serial codes of both FMLB-andLBGK-D3Q19model, the optimization program design on the single processer, includingpropagation optimization, the optimization of merging the collisionand propagation steps andthe cache optimization in read or write data, are investigated through a systemic numericaltesting which are implemented in Fortran language. The present study confirmed that thecomputational performanceof serial code can be improved beyond30%through merging thecollisionand propagation steps; in addition, computational performance of the present FMLBmodel is reduced about11.3%to18.7%when compared to the corresponding LBGK model,but compared to the improvements in numerical accuracy and stability brought by the FMLBmodel, it is totally worth it.6. The LES-based FMLB-D3Q19model is applied to make a systemic numerical study on thefull development turbulent channel flows at Reτ=180and395. In this study, the effects of thesize of computational domain, the resolution size of computational grid, with or withoutintroducing the Vreman-SGS model, are investigated by comparing the turbulent statisticalresults; besides, the present FMLB-D3Q19model is also applied to direct numericalsimulation (DNS) on the turbulent channel flows at Reτ=180; all comparisons between thenumerical results obtained by the present models, the DNS benchmark solutions and those obtained by the MRT-Vreman model in previous literatures, demonstrating the effectivenessand reliability of the present FMLB models for simulating the wall-bounded turbulent flows.