Numerical study of the properties of the central moment lattice Boltzmann method

Abstract

SUMMARYCentral moment lattice Boltzmann method (LBM) is one of the more recent developments among the lattice kinetic schemes for computational fluid dynamics. A key element in this approach is the use of central moments to specify the collision process and forcing, and thereby naturally maintaining Galilean invariance, an important characteristic of fluid flows. When the different central moments are relaxed at different rates like in a standard multiple relaxation time (MRT) formulation based on raw moments, it is endowed with a number of desirable physical and numerical features. Because the collision operator exhibits a cascaded structure, this approach is also known as the cascaded LBM. While the cascaded LBM has been developed sometime ago, a systematic study of its numerical properties, such as the accuracy, grid convergence, and stability for well-defined canonical problems is lacking, and the present work is intended to fulfill this need. We perform a quantitative study of the performance of the cascaded LBM for a set of benchmark problems of differing complexity, viz., Poiseuille flow, decaying Taylor-Green vortex flow, and lid-driven cavity flow. We first establish its grid convergence and demonstrate second-order accuracy under diffusive scaling for both the velocity field and its derivatives, that is, the components of the strain rate tensor, as well. The method is shown to quantitatively reproduce steady/unsteady analytical solutions or other numerical results with excellent accuracy. The cascaded MRT LBM based on the central moments is found to be of similar accuracy when compared with the standard MRT LBM based on the raw moments, when a detailed comparison of the flow fields are made, with both reproducing even the small scale vortical features well. Numerical experiments further demonstrate that the central moment MRT LBM results in significant stability improvements when compared with certain existing collision models at moderate additional computational cost.

Fluid flow in wall-driven enclosures with corrugated bottom

“…With the use of BGK collision approximation, for very large Reynolds numbers, the algorithm becomes numerically unstable, that is when τ is very close to 0.5. Hence, other discrete versions of the collision operators are available in the literature [37,38] which could then be applied instead of BGK approximation.…”

“…In order to simulate very high Re flows, finer grid resolution is necessary for numerical stability. If one employs the BGK approximation, a large number of lattice points are required (cell Reynolds number for BGK model is around 15) or advanced collision models are encouraged to be used [37,38,42].…”

Multi-Well Stimulation Optimization in Shale Formation: Modeling and Application

“…Fluid flow in fracture Numerical model of fluid flow in fractures mainly contains continuity equation, momentum equation, and mass balance equation (Ning et al, 2015c). The continuity equation of fluid flow in fractures is as follows,…”

Alternative formulation to incorporate forcing terms in a lattice Boltzmann scheme with central moments

“…(21) and those provided by our numerical experiments, respectively. Then, the relative error is computed as err = r an − r num r an (22) and is depicted in Fig. 1 as a function of the grid dimension.…”

“…This particular one-way coupling implies that the collision is configured as a structured hierarchical pyramidal sequence of relaxation in an ascending order of moments, hence the name "cascaded" LBM. These models are characterized by very high properties in terms of accuracy, stability, and convergence [18][19][20][21][22][23][24][25].…”