No abstract

In the present paper, we verify the effectiveness of the two-relaxation-time (TRT) collision operator in reducing boundary slip computed by the immersed boundary-lattice Boltzmann method (IB-LBM). In the linear collision operator of the TRT, we decompose the distribution function into symmetric and antisymmetric components and define the relaxation parameters for each part. The Chapman-Enskog expansion indicates that one relaxation time for the symmetric component is related to the kinematic viscosity. Rigorous analysis of the symmetric shear flows reveals that the relaxation time for the antisymmetric part controls the velocity gradient, the boundary velocity, and the boundary slip velocity computed by the IB-LBM. Simulation of the symmetric shear flows, the symmetric Poiseuille flows, and the cylindrical Couette flows indicates that the profiles of the numerical velocity calculated by the TRT collision operator under the IB-LBM framework exactly agree with those of the multirelaxation time (MRT). The TRT is as effective in removing the boundary slip as the MRT. We demonstrate analytically and numerically that the error of the boundary velocity is caused by the smoothing technique using the δ function used in the interpolation method. In the simulation of the flow past a circular cylinder, the IB-LBM based on the implicit correction method with the TRT succeeds in preventing the flow penetration through the solid surface as well as unphysical velocity distortion. The drag coefficient, the wake length, and the separation points calculated by the present IB-LBM agree well with previous studies at Re = 10, 20, and 40.

In the present paper, we apply the implicit-correction method to the immersed-boundary thermal lattice Boltzmann method (IB-TLBM) for the natural convection between two concentric horizontal cylinders and in a square enclosure containing a circular cylinder. The Chapman-Enskog multiscale expansion proves the existence of an extra term in the temperature equation from the source term of the kinetic equation. In order to eliminate the extra term, we redefine the temperature and the source term in the lattice Boltzmann equation. When the relaxation time is less than unity, the new definition of the temperature and source term enhances the accuracy of the thermal lattice Boltzmann method. The implicit-correction method is required in order to calculate the thermal interaction between a fluid and a rigid solid using the redefined temperature. Simulation of the heat conduction between two concentric cylinders indicates that the error at each boundary point of the proposed IB-TLBM is reduced by the increment of the number of Lagrangian points constituting the boundaries. We derive the theoretical relation between a temperature slip at the boundary and the relaxation time and demonstrate that the IB-TLBM requires a small relaxation time in order to avoid temperature distortion around the immersed boundary. The streamline, isotherms, and average Nusselt number calculated by the proposed method agree well with those of previous numerical studies involving natural convection. The proposed IB-TLBM improves the accuracy of the boundary conditions for the temperature and velocity using an adequate discrete area for each of the Lagrangian nodes and reduces the penetration of the streamline on the surface of the body.

In this paper, the lattice Boltzmann method (LBM) is applied to simulation of natural convection in porous media using Brinkman-Forchheimer equation. The BrinkmanForchheimer equation is recovered from a kinetic equation for the density distribution function that has a forcing term and the equilibrium distribution function including the porosity. The temperature equation which neglects the compression work done by the pressure and the viscous heat dissipation is calculated by a kinetic equation for thermal energy distribution function. The velocity and temperature profiles of the LBM shows good agreement with those of the finite difference method (FDM) for the Poiseuille flow filled with a porous medium and with the analytical solutions for the porous plate problem. The stream lines and isothermal patterns show that the LB model is able to keep the same accuracy with the FDM. For various values of Darcy and Rayleigh numbers, and of porosities, the solutions of the LBM are compared with those of earlier studies. The numerical experiment shows excellent agreement for the Brinkmanextended Darcy model and for the Brinkman-Forchheimer model. This paper leads to the conclusion that the LBM can simulate natural convection in porous media in both Darcy and non-Darcy region at the representative elementary volume scale.

No abstract

No abstract

An immersed boundary-lattice Boltzmann method (IB-LBM) using a two-relaxation time model (TRT) is proposed. The collision operator in the lattice Boltzmann equation is modeled using two relaxation times. One of them is used to set the fluid viscosity and the other is for numerical stability and accuracy. A direct-forcing method is utilized for treatment of immersed boundary. A multi-direct forcing method is also implemented to precisely satisfy the boundary conditions at the immersed boundary. Circular Couette flows between a stationary cylinder and a rotating cylinder are simulated for validation of the proposed method. The method is also validated through simulations of circular and spherical falling particles. Effects of the functional forms of the direct-forcing term and the smoothed-delta function, which interpolates the fluid velocity to the immersed boundary and distributes the forcing term to fixed Eulerian grid points, are also examined. As a result, the following conclusions are obtained: (1) the proposed method does not cause non-physical velocity distribution in circular Couette flows even at high relaxation times, whereas the single-relaxation time (SRT) model causes a large non-physical velocity distortion at a high relaxation time, (2) the multi-direct forcing reduces the errors in the velocity profile of a circular Couette flow at a high relaxation time, (3) the two-point delta function is better than the four-point delta function at low relaxation times, but worse at high relaxation times, (4) the functional form of the direct-forcing term does not affect predictions, and (5) circular and spherical particles falling in liquids are well predicted by using the proposed method both for two-dimensional and three-dimensional cases.

A lattice Boltzmann method (LBM) for two-phase nonideal fluid flows is proposed based on a particle velocity-dependent forcing scheme. The resulting macroscopic dynamics via the Chapman-Enskog expansion recover the full set of thermohydrodynamic equations for nonideal fluids. Numerical verification of fundamental properties of thermal fluids, including viscosity, thermal conductivity, and surface tension, agrees well with theoretical predictions. Direct numerical simulations of two-phase phenomena, including phase-transition, bubble deformation and droplet falling and bubble rising under gravity are carried out, demonstrating the applicability of the model.