Code Verification of Non-Newtonian Fluid Solvers for Single- and Two-Phase Laminar Flows

The presence of complex fluids in nature and industrial applications combined with the rapid growth of computer power over the past decades has led to an increasing number of numerical studies of non-Newtonian flows. In most cases, non-Newtonian models can be implemented in existing Newtonian solvers by relatively simple modifications of the viscosity. However, due to the scarcity of analytical solutions for non-Newtonian fluid flows and the widespread use of regularization methods, performing rigorous code verification is a challenging task. The method of manufactured solutions (MMS) is a powerful tool to generate analytical solutions for code verification. In this article, we present and discuss the results of three verification exercises based on MMS: (i) steady single-phase flow; (ii) unsteady two-phase flow with a smooth interface; (iii) unsteady two-phase flow with a free surface. The first and second exercises showed that rigorous verification of non-Newtonian fluid solvers is possible both on single- and two-phase flows. The third exercise revealed that “spurious velocities” typical of free-surface calculations with the Volume-of-Fluid model lead to “spurious viscosities” in the non-Newtonian fluid. The procedure is illustrated herein on a second-order finite volume flow solver, using the regularized Herschel-Bulkley fluid model as an example. The same methodology is however applicable to any flow solver and to all the rheological models falling under the class of generalized Newtonian fluid models.