Implicit-correction-based immersed boundary–lattice Boltzmann method with two relaxation times

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.

Analytical and numerical studies of the boundary slip in the immersed boundary‐thermal lattice Boltzmann method

“…These advantages over the SRT collision operator, however, do not contribute to reducing the velocity slip at high relaxation time as shown in Figure 2B. In our previous study, 21 we proved both numerically and theoretically that the TRT collision model is as effective in reducing the boundary slip as the MRT collision model. Based on the relation shown in Equation 22, the TRT multi-direct forcing method eliminates the velocity slip in Figure 2C.…”

“…There are numerous solutions from theoretical, experimental, and numerical approaches to compare with the present results. In our previous study, 21 the implicit correction method with the TRT collision operator yielded reasonable numerical results in fluid flows past a circular cylinder at high viscosity. Because the multi-direct forcing method cannot eliminate the velocity slip for any relaxation parameters, we verify the feasibility and applicability of the iterative velocity/force-correction methods for the numerical simulation of the fluid flows at high viscosity.…”

“…34 The relaxation time v − is given so as to reduce the velocity slip. 21 The velocity slip and the temperature jump are zero, if the relaxation times …”

“…Although Peskin 1 indicates that Δs should not be larger than 0.5 x to avoid holes in the structures, the TRT implicit correction method succeeded in preventing fluid penetration through the wall for Δs ≈ 2 x . 21 To demonstrate the validity of the proposed iterative correction methods, we use Δs = 100 x ∕157 for the cylinder, which makes Δs ≈ 2 x for all calculations. The relaxation times are 1.25, 2.0, and 3.5 for Re = 40, 20, and 10, respectively.…”

“…16,18 By solving systems of algebraic equations, the implicit velocity-correction method with the TRT collision operator succeeded in removing all boundary errors that depend on the relaxation time. 21 Although Zhang et al numerically demonstrated that the IB-LBM using the iterative procedure with the MRT collision operator also reduced the boundary errors, they did not completely eliminate the velocity slip in their numerical calculations. 9 For IB-LBMs using iterative schemes, it is necessary to obtain an optimal analytical solution of the relaxation parameters to reduce the velocity slip.…”