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

Abstract

We analytically and numerically investigate the boundary slip, including the velocity slip and the temperature jump, in immersed boundary-thermal lattice Boltzmann methods (IB-TLBMs) with the two-relaxation-time collision operator. We derive the theoretical equation for the relaxation parameters considering the effect of the advection velocity on the temperature jump of the IB-TLBMs.The analytical and numerical solutions demonstrate that the proposed iterative correction methods without the computational cost of the sparse matrix solver reduce the boundary slip and boundary-value deviation as effectively as the implicit correction method for any relaxation time. Because the commonly used multi-direct forcing method does not consider the contributions of the body force to the momentum flux, it cannot completely eliminate the boundary slip because of the numerical instability for a long relaxation time. Both types of proposed iterative correction methods are more numerically stable than the implicit correction method. In simulations of flow past a circular cylinder and of natural convection, the present iterative correction methods yield adequate results without the errors of the velocity slip, the temperature jump, and the boundary-value deviation for any relaxation time parameters and for any number of Lagrangian points per length. The combination of the present methods and the two-relaxation-time collision operator is suitable for simulating fluid flow with thermal convection in the multiblock method in which the relaxation time increases in inverse proportion to the grid size.

Slip velocity boundary conditions for the lattice Boltzmann modeling of microchannel flows

“…For the planar wall, Section 4, whose surface is defined by the outward unitary normal vector ⃗ n = n z ′ ⃗ e z ′ , we employ the coordinate transformation ⃗ e y = cos 𝜃 ⃗ e y ′ − sin 𝜃 ⃗ e z ′ and ⃗ e z = sin 𝜃 ⃗ e y ′ + cos 𝜃 ⃗ e z ′ so that 𝛿 y = − sin 𝜃 𝛿 z ′ and 𝛿 z = cos 𝜃 𝛿 z ′ , and c qy = − sin 𝜃 c qz ′ and c qz = cos 𝜃 c qz ′ , which permits writing the closure relation, Equation (42), as follows:…”

“…To the best of the authors' knowledge, those MR slip schemes 21,22 are the only existing pathway in LBM to reproduce Equation (1) within a parabolic level of accuracy, for any arbitrary shaped walls. Alternative LBM slip strategies either support the parabolic accuracy limited to lattice‐aligned surfaces 34,36‐39 or, otherwise, exhibit a degraded accuracy (lowering from second‐ to first‐order) when applied to nonmesh aligned walls 40‐44 . Still, despite the superior accuracy of the MR‐based slip boundary schemes, 21,22 they carry a few points worthwhile improvement, namely: (i) nonlocality of implementation, for example, requiring at least two nodes to accommodate arbitrarily rotated parabolic solutions; (ii) inadequacy of the scheme to operate on edge/corner nodes due to the lack of neighboring nodes; and (iii) inherent difficulty to independently prescribe normal/tangential conditions in a linkwise manner.…”

“…While in the LBM context this benchmark has been largely studied for 2D channel geometries, 21,[36][37][38][39][40][41]44 its extension to 3D ducts has received considerable less attention, apart from a few exceptions that modeled the slip condition on circular tubes 43 and rectangular ducts. 43,49 Unlike for no-slip walls, the assessment of convergence rates and other accuracy measures have been scarcely reported for slip-flow problems, particularly when the wall does not align with the LBM uniform Cartesian mesh, 21,41,42 much seldom in 3D domains. We address this question here by considering the discretization of distinct channel cross-sections with rectangular, triangular, circular and annular shapes.…”

Numerical simulations of flows in cerebral aneurysms using the lattice Boltzmann method with single- and multiple-relaxation time collision models

“…A collision model having a fewer number of free parameters would be of use in practice. Therefore the applicability of the two-relaxation time (TRT) collision model (Ginzburg et al, 2008(Ginzburg et al, , 2010Seta et al, 2014Seta et al, , 2018, in which one relaxation parameter is used to determine the fluid viscosity and the other is a free parameter used to improve numerical stability and accuracy, is examined in this section.…”