Lattice-Boltzmann simulation of creeping generalized Newtonian flows: Theory and guidelines

The accuracy of the lattice-Boltzmann (LB) method is related to the relaxation time controlling the flow viscosity. In particular, it is often recommended to avoid large fluid viscosities in order to satisfy the low-Knudsen-number assumption that is essential to recover hydrodynamic behavior at the macroscopic scale, which may in principle limit the possibility of simulating creeping flows and non-Newtonian flows involving important viscosity variations. Here it is shown, based on the continuous Boltzmann equations, that a two-relaxation-time (TRT) model can however recover the steady Navier-Stokes equations without any restriction on the fluid viscosity, provided that the Knudsen number is redefined as a function of both relaxation times. This effective Knudsen number is closely related to the previously-described parameter controlling numerical errors of the TRT model, providing a consistent theory at both the discrete and continuous levels. To simulate incompressible flows, the viscous incompressibility condition Ma(2)/Re << 1 also needs to be satisfied, where Ma and Re are the Mach and Reynolds numbers. This concept is extended by defining a local incompressibility factor, allowing one to locally control the accuracy of the simulation for flows involving varying viscosities. These theoretical arguments are illustrated based on numerical simulations of the two-dimensional flow past a square cylinder. In the case of a Newtonian flow, the viscosity independence is confirmed for relaxation times up to 10(4), and the ratio Ma(2)/Re = 0.1 is small enough to ensure reliable incompressible simulations. The Herschel-Bulkley model is employed to introduce shear-dependent viscosities in the flow. The proposed numerical strategy allows to achieve major viscosity variations, avoiding the implementation of artificial viscosity cut-off in high-viscosity regions. Highly non-linear flows are simulated over ranges of the Bingham number Bn is an element of [1, 1000] and flow index n is an element of [0.2, 1.8], and successfully compared to prior numerical works based on Navier-Stokes solvers. This work provides a general framework to simulate complex creeping flows, as encountered in many biological and industrial systems, using the lattice-Boltzmann method. (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

The accuracy of lattice-Boltzmann method is related to the relaxation time controlling flow viscosity. The two-relaxation-time model can recover the steady Navier-Stokes equations without restrictions on fluid viscosity. In order to simulate incompressible flows, the viscous incompressibility condition needs to be satisfied, and a local incompressibility factor can be defined to control the accuracy of flows involving varying viscosities.