When investigating the effect of muddy seabeds on marine vessels using Computational Fluid Dynamics (CFD) software, one challenge is to adequately describe the complex non-Newtonian fluid behaviour of mud. Although a number of rheological models have been proposed in the past, mud sediments are often simply regarded either as highly viscous Newtonian fluids or as Bingham fluids in many engineering applications. In this study, we investigate the accuracy of the Bingham model for numerical predictions of the viscous forces on a plate moving through fluid mud in laminar regime. In this context, a plate could be regarded as the flat bottom of a ship hull. The aim is to provide CFD practitioners with information about the accuracy of the Bingham model for the prediction of the frictional resistance of a ship sailing through fluid mud. This work presents a comparison of experimental and numerical data on the resistance of a plate moving through fluid mud from the Europoort area (Netherlands). Results suggest that the regularised Bingham model can be a reasonable compromise between simplicity and accuracy for CFD simulations to investigate the effect of muddy seabeds on marine vessels. A comparison between CFD data and analytical formulas is also presented.@en

The increasing size of today's ships is a major concern for navigation in confined waters. In order to ensure safe manoeuvres, port authorities prescribe, among others, a minimum under-keel clearance that must be maintained by the ships during navigation. However, the seabed of ports situated at the estuaries or along rivers is often covered by mud as a result of sedimentation. Hence, while the position of a solid bottom is clearly defined and can be easily detected by sonar techniques, the presence of deposited sediments makes the definition of "bottom" and "depth" less clear. This also poses some questions on the optimal dredging strategy to adopt to minimise maintenance costs while ensuring the required safety. For practical reasons, port authorities define the (nautical) bottom as the level where the mud reaches either a critical density or a critical yield stress (i.e. the shear stress below which the fluid behaves as a solid-like material). However, an optimal choice that minimises dredging activities while preserving the required safety shall also take into account the behaviour of ships. As the understanding of the link between mud rheology and ships' controllability and manoeuvrability with muddy seabeds is rather limited, this research project was started. With the rapidly increasing power of today's computers, Computational Fluid Dynamics (CFD) has become a viable option to study this problem. The CFD code selected for this research is a multi-phase viscous-flow solver developed, verified and validated exclusively for maritime applications. As such, it was originally developed for Newtonian fluids only. Since mud exhibits a non-Newtonian rheology, the `step zero' of this research was to implement the Herschel-Bulkley model, which allows to numerically simulate two important flow features of mud, i.e. its shear-thinning and viscoplastic behaviour. Other rheological characteristics, such as thixotropy, were not considered in this study as they are deemed of minor importance at this stage. The next step was concerned with ensuring that the modification of the flow solver to account for the non-Newtonian rheology of mud was correct. This was done by using the Method of Manufactured Solutions (MMS), which allows to rigorously verify the code against user-defined exact solutions. The verification exercises showed that the code performs as intended for both single- and two-phase flows of Herschel--Bulkley fluids. The illustrated procedure can be readily adapted to verify the correct implementation of other rheological models that may be implemented in the future. In this case, it is recommended to examine, in addition to the grid convergence of velocity and pressure, also the grid convergence of the apparent viscosity as the latter is particularly sensitive to coding mistakes related to the implementation of the new rheological model. While code verification ensured that the Herschel--Bulkley model was correctly implemented, obtaining fully-converged solutions for realistic non-Newtonian problems may still be difficult. The non-Newtonian solver has thus been tested on the laminar flow of Herschel-Bulkley fluids around a sphere, as the latter is the simplest three-dimensional flow exhibiting features that are typical of the flow around ships, such as boundary layer development and flow separation. Although obtaining a fully-converged solutions was indeed challenging, it was possible to replicate data from the literature with good accuracy. This provided confidence to employ the CFD code to simulate ships sailing through fluid mud. The verification of the CFD code was followed by validation of the mathematical model. The problem of a ship sailing through fluid mud was simplified into a simpler one, i.e. a plate moving through homogeneous mud as to mimic a portion of the hull penetrating the mud layer. The objective was to investigate the accuracy of the (regularised) Bingham model (which is a special case of Herschel-Bulkley) to predict the frictional forces on a plate moving through mud. The comparison between experimental and numerical data showed that the ideal Bingham model well captures the relative increase in the resistance due to the increase in the mud concentration but, at low speed, it tends to over-predict the resistance. On the other hand, choosing a lower regularisation parameters seem more favourable, both from the numerical and physical perspective. In fact, this research showed that better predictions at low speed were achieved by using lower regularisation parameters that were determined from the first points in the mud flow curves. It should be noted, however, that the thixotropy of mud and possible deflections of the plate during the experiments may prevent drawing definitive conclusions. Finally, one question arising when simulating a ship sailing through a non-Newtonian fluid is how accurate are standard Reynolds-Averaged Navier-Stokes (RANS) models, which are developed for Newtonian fluids, when applied to non-Newtonian flows. In the last step of this dissertation, the accuracy of three RANS models was assessed against published Direct Numerical Simulations (DNS) data for pipe flows. From this study it was concluded that, among the three tested Newtonian RANS models, the SST model produced the best predictions and it is reasonably accurate for weakly non-Newtonian fluids and for high Reynolds numbers. In addition, a new RANS model, labelled SST-HB, has been developed. The new model showed good agreement with DNS of pipe flows in the mean velocity, average viscosity, mean shear stress budget and friction factors. However, the new RANS model was calibrated and tested for pipe flows only, a relatively simple internal-flow problem. Hence, the applicability of the new model to complex external flows, such as the flow around a ship, still requires further investigations. Furthermore, RANS simulations with some realistic mud conditions predicted laminar flow in the mud layer. In this case, the use of the standard SST model is recommended. The developed and tested CFD code, together with other insights provided by this research, can be used in the future to both numerically investigate the effect of mud on ships and to obtain the hydrodynamic coefficients for manoeuvring models. These models could then be used in real- and fast-time simulators for research and commercial purposes, but also for pilots training. @en

Instead of maintenance dredging, an alternative option for port authorities is to adapt the PIANC's nautical bottom approach. For practical purposes, the nautical bottom is defined as the level at which the fluid mud reaches either a critical density or a critical yield stress (the shear strength). These values generally correspond to a level at which the mud undergoes a so-called "rheological transition", where the density and strength of the mud increase rapidly over a short distance. Below this level, the mud becomes more and more like solid ground and is therefore no longer navigable. Recently, new scientific and practical research has been conducted in order to gain additional knowledge on navigability in ports with fluid mud layers. In particular, a systematic rheological analysis was conducted to determine the critical limits of the yield stresses and density of fluid mud. Furthermore, a Computational Fluid Dynamics (CFD) model was developed to numerically investigate the ship-mud interaction. The model was applied to study the effects of muddy bottoms on the full-scale resistance of a modern oil tanker at speeds between 3 and 9 knots. It was confirmed that not only the density but also the yield stress of the fluid mud should be considered in the practical application of the nautical bottom. Finally, the paper discussed how the standard maintenance dredging methods can be used for producing navigable fluid mud layers.@en

Instead of maintenance dredging, an alternative option for port authorities is to adapt the PIANC's nautical bottom approach. For practical purposes, the nautical bottom is defined as the level at which the fluid mud reaches either a critical density or a critical yield stress (the shear strength). These values generally correspond to a level at which the mud undergoes a so-called "rheological transition", where the density and strength of the mud increase rapidly over a short distance. Below this level, the mud becomes more and more like solid ground and is therefore no longer navigable. Recently, new scientific and practical research has been conducted in order to gain additional knowledge on navigability in ports with fluid mud layers. In particular, a systematic rheological analysis was conducted to determine the critical limits of the yield stresses and density of fluid mud. Furthermore, a Computational Fluid Dynamics (CFD) model was developed to numerically investigate the ship-mud interaction. The model was applied to study the effects of muddy bottoms on the full-scale resistance of a modern oil tanker at speeds between 3 and 9 knots. It was confirmed that not only the density but also the yield stress of the fluid mud should be considered in the practical application of the nautical bottom. Finally, the paper discussed how the standard maintenance dredging methods can be used for producing navigable fluid mud layers.@en

The presence of mud layers on the bottom of ports and waterways can have negative effects on the hydrodynamic behaviour of marine vessels. This numerical study investigates the effect of muddy seabeds on the full-scale resistance of an oil tanker sailing straight ahead. The objective is to determine the influence of factors such as the densimetric Froude number, UKC and mud rheology at speeds between 3 and 9 knots. The numerical study is conducted using a finite-volume Reynolds-Averaged Navier–Stokes (RANS) flow solver combined with the Volume-Of-Fluid (VOF) method to capture the mud–water interface. At certain critical speeds, the presence of mud increased the ship’s total resistance by up to 15 times compared to the case with solid bottoms. The non-Newtonian rheology of mud was found to influence the ship’s resistance mainly at low speeds and when sailing through the mud layer. This article also shows that, when sailing through mud, the computed resistance at high speeds may be underestimated because of two effects, namely ‘water lubrication’ and ‘numerical ventilation’.@en

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.@en

This article presents a new turbulence closure based on the k-ω SST model for predicting turbulent flows of Herschel–Bulkley fluids, including Bingham and power-law fluids. The model has been calibrated with direct numerical simulations (DNS) data for fully-developed pipe flow of shear-thinning and viscoplastic fluids. The new model shows good agreement in the mean velocity, average viscosity, mean shear stress budget and friction factor. The latter compares well also against correlations from the literature for a wide range of Reynolds numbers. With the new model, improvements are also observed in the iterative convergence, which is often difficult for calculations with yield-stress fluids. Additionally, three eddy-viscosity models for Newtonian fluids, namely the k-ω SST, k-kL and Spalart–Allmaras model, have been tested on turbulent Herschel–Bulkley flows. Results show that (i) the new model produces the best prediction; (ii) the standard SST model may be considered for simulations of weakly shear-thinning/viscoplastic fluids at high Reynolds numbers; (iii) the k-kL and the Spalart–Allmaras models appear to be unsuitable for turbulent Herschel–Bulkley flows. The new model is simple and appealing for engineering applications concerned with turbulent wall-bounded flows and is presented in a formulation that can be easily adapted to other generalised Newtonian fluids.@en

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.@en