In this paper, a simplified lattice Boltzmann method (SLBM) without evolution of the distribution function is developed for simulating incompressible viscous flows. This method is developed from the application of fractional step technique to the macroscopic Navier-Stokes (N-S) equations recovered from lattice Boltzmann equation by using Chapman-Enskog expansion analysis. In SLBM, the equilibrium distribution function is calculated from the macroscopic variables, while the non-equilibrium distribution function is simply evaluated from the difference of two equilibrium distribution functions. Therefore, SLBM tracks the evolution of the macroscopic variables rather than the distribution function. As a result, lower virtual memories are required and physical boundary conditions could be directly implemented. Through numerical test at high Reynolds number, the method shows very nice performance in numerical stability. An accuracy test for the 2D Taylor-Green flow shows that SLBM has the second-order of accuracy in space. More benchmark tests, including the Couette flow, the Poiseuille flow as well as the 2D lid-driven cavity flow, are conducted to further validate the present method; and the simulation results are in good agreement with available data in literatures.

In this paper, an improved discrete velocity method (DVM) is developed for efficient simulation of fluid problems in all flow regimes. Compared with the conventional explicit DVM, the present scheme could effectively remove its drawbacks of low accuracy and efficiency in continuum flow regime with no deterioration of its performance in rarefied flow regime. One of the novel strategies adopted in the new method is to introduce a prediction step for solving the macroscopic governing equation. By using the prediction step, the equilibrium state is first estimated before solving the discrete velocity Boltzmann equation (DVBE). As a result, the collision term in the DVBE can be discretized implicitly to improve the stability and efficiency of the conventional explicit DVM. Another contribution of the new method is to physically reconstruct numerical flux at the cell interface by incorporating the collision effect into the process. To maintain simplicity and efficiency of the conventional DVM, in the present scheme, the collision effect in the flux reconstruction at the cell interface is considered through the solution of the macroscopic governing equation. This can effectively control the effect of numerical dissipation in the process of updating the macroscopic flow variables in the continuum flow regime. Analyses indicate that the prediction step does not contribute to the evolution of distribution functions in the highly rarefied flow regime while dominating the solutions in the continuum flow regime. Accordingly, the improved scheme automatically converges toward the conventional explicit DVM in the free molecular flow regime and approaches the Navier-Stokes solver in the continuum flow regime.

In this paper, a simplified multiphase lattice Boltzmann method (SMLBM) is developed for modeling incompressible multiphase flows with large density ratios and complex interfaces. SMLBM is derived from reconstructing solutions to macroscopic equations recovered from the multiphase lattice Boltzmann model through Chapman-Enskog expansion analysis and resolved in a predictor-corrector step. The Cahn-Hilliard equation is utilized as the interface tracking algorithm, the solution of which is also reconstructed within the lattice Boltzmann framework. The resultant formulation of SMLBM reflects a direct update of macroscopic variables instead of distribution functions, which could remarkably save virtual memory and facilitate implementation of physical boundary conditions. Numerical tests also indicate that our scheme could effectively simulate challenging cases with large density ratios (up to 1000) and complex interfaces, and at high Reynolds numbers.