On the stability and accuracy of TRT Lattice-Boltzmann method for non-Newtonian Ostwald-de Waele fluid flows

Abstract

This paper brings a numerical analysis of the TRT modeling for the Lattice-Boltzmann method when solving flow for dilatant and pseudoplastic power-law fluids. Firstly, the method was reviewed to describe the required simulation parameters and the numerical methodology. Secondly, a mathematical procedure was performed to identify the characteristic relaxation frequency as a function of both flow and fluid properties and to work as a guide parameter for LBM operation. Then, a simple shearing Poiseuille flow was employed so its characteristic shear rate could be calculated as a function of fluid properties, given the flow was characterized by the Reynolds and Mach numbers. For this test case, convergence was then explored for a broad range of parameters, and its non-monotonic dependency on the Mach number for a given convergence criterion was shown. Then, stability maps were constructed based on the characteristic relaxation frequency, which showed a strong dependency between consistency and flow index so the simulation could converge. This was explored against the results from the converged tests, which pointed out the usefulness of the characteristic relaxation frequency in predicting stable solutions. Finally, quantitatively, it was shown that for this power-law fluid flow, the $\left|\right|L_2\left|\right|$ relative error depends on the Mach number to the power of $2(2-n)$ being now a function of the flow index, extending the previously reported dependency of the Mach number to the power of 2 for plane-Poiseuille flow.