Evaluation of the non-Newtonian lattice Boltzmann model coupled with off-grid bounce-back scheme: Wall shear stress distributions in Ostwald-de Waele fluids flow

We present a comprehensive analysis of the non-Newtonian lattice Boltzmann method (LBM) when it is used to simulate the distribution of wall shear stress (WSS). We systematically identify sources of numerical errors associated with non-Newtonian rheological behavior of fluids in off-grid geometries. We implement the single relaxation time, Bhatnagar-Gross-Krook (BGK), and multiple relaxation time (MRT) collision operators and investigate flow in a two-dimensional channel aligned with lattice directions and off-grid Hagen-Poiseuille flow of Ostwald-de Waele (power-law) fluids. As for boundary conditions, we implement constant body force-driven and pressure-driven flows. These two boundary conditions have different numerical challenges, which include numerical stability, accuracy, mass conservation, and compressibility effects, which are inherent in the LBM method. Our results indicate that MRT, when the relaxation times are adequately tuned in the non-Newtonian case, significantly improves the WSS distribution accuracy and the numerical stability of the LBM. MRT also enhances the stability and accuracy for non-Newtonian fluids compared with the Newtonian case, meaning that it is questionable if a BGK collision operator is appropriate to use in a non-Newtonian case with off-grid boundaries. When analyzing the non-Newtonian LBM in the context of staircase walls and interpolated bounce-back (IBB) walls, a MRT collision operator with the appropriate choice of tunable relaxation times makes it possible to achieve numerically accurate results without a significant increase in grid resolution for matching to the analytical solution of WSS distributions. In analyzing the non-Newtonian flows, we show that the viscosity dependency of bounce-back walls in the BGK-LBM deviates from the results obtained under Newtonian assumptions. The power-law index further influences these discrepancies, and errors caused by the viscosity dependency of the bounce-back boundary conditions can be effectively mitigated by implementing the MRT procedure. Results show that non-Newtonian fluids, in contrast with the Newtonian assumption, encounter a greater mass imbalance when flowing through a periodic system with IBB walls. MRT can address this challenge, as it allows for independent adjustments of physical relaxation times and enhances mass conservation in the case of non-Newtonian fluids. In pressure-driven non-Newtonian flows, there is a significant impact of bulk viscosity. This aspect is often overlooked in Newtonian simulations but can significantly impact fluid adapting to rapid changes in local effective viscosity. One of our main conclusions is that the MRT collision operator with tuned relaxation times can effectively resolve numerical problems caused by non-Newtonian rheological properties and off-grid geometries. We also provide practical guidelines for selecting the most suitable simulation approach.