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.
An immersed boundary–simplified lattice Boltzmann method (IB-STLBM) is proposed in this paper for the simulation of incompressible thermal flows with immersed objects. The fractional step technique is adopted to resolve the problem in two successive steps. In the predictor step, the simplified thermal lattice Boltzmann method (STLBM) is utilized to resolve the intermediate flow variables without considering the immersed objects. The STLBM is advantageous over the conventional thermal lattice Boltzmann method (TLBM) in memory cost, boundary treatment, and numerical stability. In the corrector step, the boundary condition-enforced immersed boundary method (IBM) is used to give correction values of velocity and temperature for accurate interpretation of the Dirichlet boundary conditions on the surface of the immersed objects. Based on the present IBM, some novel strategies can be applied in the evaluation of hydrodynamic forces or thermal parameters of the immersed objects. Five numerical examples are presented for comprehensive validation of the accuracy and robustness of IB-STLBM in various two- and three-dimensional thermal flow problems.
The recently developed multiphase lattice Boltzmann flux solver (MLBFS) overcomes the limitations in the multiphase lattice Boltzmann method (MLBM), such as the coupled time step and mesh step, uniform meshes, and complex distribution functions (DFs) treatment at the boundary. Unlike the original MLBFS deduced from the standard lattice Boltzmann method, an improved multiphase lattice Boltzmann flux solver (IMLBFS) is proposed based on the Chapman–Enskog analysis of the MLBM which has a source term stemming from the density contrast and surface tension force. In this way, the surface tension force is considered when reconstructing the numerical interface fluxes, which gives the present method stronger physical basis. As a result, the IMLBFS is more stable than the MLBFS. Moreover, the IMLBFS simplifies the process of reconstructing interface fluxes and avoids the complicated calculation of the source term in the MLBM. Some moments of the DFs and source terms are directly given as macroscopic variables to avoid additional computations and storage. This strategy ensures that the IMLBFS even has higher computational efficiency than the MLBFS. To test the proposed IMBFS for large-density-ratio flows, complex interfacial changes and high Reynolds number (up to 10 000), several typical problems are studied, including the static Laplace law, the droplet spreading on a flat surface, the unsteady Rayleigh–Taylor instability, the bubble rising under buoyancy, and the droplet splashing on a thin film. Simulations suggest that the present method predicts smaller spurious velocities, and it is more stable and efficient than the original MLBFS.
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Highlights • Four slip boundary conditions are presented for liquid flow. • The proposed schemes surmount the barrier of limited slip length. • The proposed schemes are specified by the slip length. • Two slip boundary conditions are suitable for curved walls.
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