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The focus of the present work is to provide an analysis for the acoustic and thermal properties of the energy-conserving lattice Boltzmann models, and a solution to the numerical defects and instability associated with these models in two and three dimensions. We discover that a spurious algebraic coupling between the shear and energy modes of the linearized evolution operator is a defect universal to the energy-conserving Boltzmann models in two and three dimensions. This spurious mode coupling is highly anisotropic and may occur at small values of wave number k along certain directions, and it is a direct consequence of the following key features of the lattice Boltzmann equation: (1) its simple spatial-temporal dynamics, (2) the linearity of the relaxation modeling for collision operator, and (3) the energy-conservation constraint. To eliminate the spurious mode coupling, we propose a hybrid thermal lattice Boltzmann equation (HTLBE) in which the mass and momentum conservation equations are solved by using the multiple-relaxation-time model due to d'Humières, whereas the diffusion-advection equation for the temperature is solved separately by using finite-difference technique (or other means). Through the Chapman-Enskog analysis we show that the hydrodynamic equations derived from the proposed HTLBE model include the equivalent effect of gamma=C(P)/C(V) in both the speed and attenuation of sound. Appropriate coupling between the energy and velocity field is introduced to attain correct acoustics in the model. The numerical stability of the HTLBE scheme is analyzed by solving the dispersion equation of the linearized collision operator. We find that the numerical stability of the lattice Boltzmann scheme improves drastically once the spurious mode coupling is removed. It is shown that the HTLBE scheme is far superior to the existing thermal LBE schemes in terms of numerical stability, flexibility, and possible generalization for complex fluids. We also present the simulation results of the convective flow in a rectangular cavity with different temperatures on two opposite vertical walls and under the influence of gravity. Our numerical results agree well with the pseudospectral result.

A three-dimensional lattice Boltzmann model with thirty two discrete velocity distribution functions for viscoelastic fluid is presented in this work. The model is based upon the generalized lattice Boltzmann equation constructed in moment space. The nonlinear equilibria of the model have a number of coupling constants that are free parameters. The dispersion equation of the model is analyzed under various conditions to obtain the constraints on the free parameters such that the model satisfies isotropy and Galilean invariance. The macroscopic equations are also derived from the lattice Boltzmann model through the dispersion equation analysis and the Chapman-Enskog analysis. We demonstrate that the dispersion equation analysis can be used as a general and effective means to derive hydrodynamic equations, excluding some nonlinear source terms, from the lattice Boltzmann model, to obtain conditions for its isotropy and Galilean invariance, and to optimize its stability. We show that the hydrodynamic behavior of the lattice Boltzmann model has memory effects, and that in the linear regime, it behaves as a viscoelastic fluid described by the Jeffreys model. Some numerical results to verify the theoretical analysis of the model are also presented.

A thirteen-velocity three-dimensional lattice Boltzmann model on a cubic grid is presented. The transport coefficients derived from the standard Chapman-Enskog expansion are given together with the conditions for isotropy and Galilean invariance. The different invariants of the model are discussed. The results of measurements of drag and torque on a free falling sphere in a cylinder are in good agreement with solutions of the Navier-Stokes equation. Comparison of the time evolution of a freely decaying Taylor-Green vortex computed by fast Fourier transform and by the present model is presented.