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.