Solution of the square lid-driven cavity flow of a Bingham plastic using the finite volume method

TL;DR

This paper evaluates the finite volume method for simulating Bingham plastic flows in a lid-driven cavity, highlighting convergence challenges and improvements with multigrid techniques, and comparing results with existing methods.

Contribution

It demonstrates the effectiveness of multigrid-accelerated SIMPLE algorithm for viscoplastic flow simulation and provides comparative results at high Bingham numbers.

Findings

Multigrid methods improve convergence for Bingham flows.

Convergence deteriorates with increasing Bingham number using standard SIMPLE.

Results align well with other numerical approaches at high Bingham numbers.

Abstract

We investigate the performance of the finite volume method in solving viscoplastic flows. The creeping square lid-driven cavity flow of a Bingham plastic is chosen as the test case and the constitutive equation is regularised as proposed by Papanastasiou [J. Rheol. 31 (1987) 385-404]. It is shown that the convergence rate of the standard SIMPLE pressure-correction algorithm, which is used to solve the algebraic equation system that is produced by the finite volume discretisation, severely deteriorates as the Bingham number increases, with a corresponding increase in the non-linearity of the equations. It is shown that using the SIMPLE algorithm in a multigrid context dramatically improves convergence, although the multigrid convergence rates are much worse than for Newtonian flows. The numerical results obtained for Bingham numbers as high as 1000 compare favourably with reported results of other methods.