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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 on non-Newtonian flows. In most cases, non-Newtonian models can be implemented in existing Newtonian solvers by relatively simple modification of the viscosity. However, due to scarcity of analytical solutions for non-Newtonian fluid flows and widespread use of regularisation methods, performing Code Verification to ensure that the code is free of mistakes 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 results of three verification exercises based on MMS: (i) steady single-phase flow; (ii) unsteady two-phase flow with smooth interface; (iii) two-phase flow with free surface. We show that, with the first and second exercise, rigorous verification of non-Newtonian fluid solver is possible both on single and two-phase flows. The third test case revealed that `spurious velocities' typical of free-surface calculations with the Volume-of-Fluid (VoF) model lead to `spurious viscosities' in the non-Newtonian fluid. The procedure is illustrated herein on a second-order finite-volume CFD code using the Herschel-Bulkley fluid model as an example. Nonetheless, the same methodology is applicable to any CFD code and it can be easily extended to all rheological models falling under the class of Generalised Newtonian Fluid (GNF) models.

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The ship’s resistance and manoeuvrability in shallow waters can be adversely influenced by the presence of fluid mud layers on the seabed of ports and waterways. Fluid mud exhibits a complex non-Newtonian rheology that is often described using the Herschel–Bulkley model. The latter has been recently implemented in a maritime finite-volume CFD code to study the manoeuvrability of ships in the presence of muddy seabeds. In this paper, we explore the accuracy and robustness of the CFD code in simulating the flow of Herschel–Bulkley fluids, including power-law, Bingham and Newtonian fluids as particular cases. As a stepping stone towards the final maritime applications, the study is carried out on a classic benchmark problem in non-Newtonian fluid mechanics: the laminar flow around a sphere. The aim is to test the performance of the non-Newtonian solver before applying it to the more complex scenarios. Present results could also be used as reference data for future testing. Flow simulations are carried out at low Reynolds numbers in order to compare our results with an extensive collection of data from the literature. Results agree both qualitatively and quantitatively with literature. Difficulties in the convergence of the iterative solver emerged when simulating Bingham and Herschel–Bulkley flows. A simple change in the interpolation of the apparent viscosity has mitigated such difficulties. The results of this work, combined with our previous code verification exercises, suggest that the non-Newtonian solver works as intended and it can be thus employed on more complex applications.