Abstract
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 underkeel 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 solidlike 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 multiphase viscousflow solver developed, verified and validated exclusively for maritime applications. As such, it was originally developed for Newtonian fluids only. Since mud exhibits a nonNewtonian rheology, the `step zero' of this research was to implement the HerschelBulkley model, which allows to numerically simulate two important flow features of mud, i.e. its shearthinning 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 nonNewtonian rheology of mud was correct. This was done by using the Method of Manufactured Solutions (MMS), which allows to rigorously verify the code against userdefined exact solutions. The verification exercises showed that the code performs as intended for both single and twophase flows of HerschelBulkley 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 HerschelBulkley model was correctly implemented, obtaining fullyconverged solutions for realistic nonNewtonian problems may still be difficult. The nonNewtonian solver has thus been tested on the laminar flow of HerschelBulkley fluids around a sphere, as the latter is the simplest threedimensional flow exhibiting features that are typical of the flow around ships, such as boundary layer development and flow separation. Although obtaining a fullyconverged 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 HerschelBulkley) 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 overpredict 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 nonNewtonian fluid is how accurate are standard ReynoldsAveraged NavierStokes (RANS) models, which are developed for Newtonian fluids, when applied to nonNewtonian 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 nonNewtonian fluids and for high Reynolds numbers. In addition, a new RANS model, labelled SSTHB, 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 internalflow 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 fasttime simulators for research and commercial purposes, but also for pilots training.
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 solidlike 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 multiphase viscousflow solver developed, verified and validated exclusively for maritime applications. As such, it was originally developed for Newtonian fluids only. Since mud exhibits a nonNewtonian rheology, the `step zero' of this research was to implement the HerschelBulkley model, which allows to numerically simulate two important flow features of mud, i.e. its shearthinning 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 nonNewtonian rheology of mud was correct. This was done by using the Method of Manufactured Solutions (MMS), which allows to rigorously verify the code against userdefined exact solutions. The verification exercises showed that the code performs as intended for both single and twophase flows of HerschelBulkley 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 HerschelBulkley model was correctly implemented, obtaining fullyconverged solutions for realistic nonNewtonian problems may still be difficult. The nonNewtonian solver has thus been tested on the laminar flow of HerschelBulkley fluids around a sphere, as the latter is the simplest threedimensional flow exhibiting features that are typical of the flow around ships, such as boundary layer development and flow separation. Although obtaining a fullyconverged 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 HerschelBulkley) 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 overpredict 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 nonNewtonian fluid is how accurate are standard ReynoldsAveraged NavierStokes (RANS) models, which are developed for Newtonian fluids, when applied to nonNewtonian 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 nonNewtonian fluids and for high Reynolds numbers. In addition, a new RANS model, labelled SSTHB, 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 internalflow 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 fasttime simulators for research and commercial purposes, but also for pilots training.
Original language  English 

Qualification  Doctor of Philosophy 
Awarding Institution 

Supervisors/Advisors 

Thesis sponsors  
Award date  23 Feb 2023 
Print ISBNs  9789464589542 
DOIs  
Publication status  Published  2023 
Keywords
 CFD
 Ship
 Mud
 Numerical Modelling
 NonNewtonian
 Code verification
 Validation