Development of an efficient fluid-structure interaction model for floating objects

This thesis gives an overview of the process that led to the development of a novel semi-implicit fluid-structure interaction model. The thesis is dedicated to the creation of a new numerical model that allows to study ship generated waves and ship manoeuvers in waterways for various vessel characteristics and speeds in different external current situations. A model like this requires a coupling between the fluid and the solid to generate the waves and the hydrodynamic forces on the hull. Since the horizontal dimensions are significantly larger than the vertical dimension, we started by employing the shallow water equations, which are based on the assumption of hydrostatic pressure. The discretization was carried out taking only the nonlinear advective terms explicitly while the pressure terms are discretized implicitly, which makes the CFL condition milder. The price to pay for this semi-implicit discretization is an increase in the algorithm complexity compared to a fully-explicit method, but it is still much simpler than a fully-implicit discretization of the governing equations. Indeed, the mass and momentum equations couple, and finding the unknowns involves solving a system of equations with dimensions equal to the number of cells. The grid supporting the discretization is staggered, overlapping and Cartesian. Since the aimed application domain is inland waterways, it is paramount to allow wetting and drying of the cells. This was achieved by acting on the depth function, the relationship between the free-surface elevation and the water depth in the cell. The main novelty of this research project is the two-way coupling of the PDE system for the water flow with the ODE system for the rigid body motion of the ship. The hull defines the ship region, and its shape can range from a simple box to an STL file of a real 3D ship geometry. Where the hull is in contact with the water, the cells are pressurized. This pressurized group of cells generates waves as it moves, and its motion is influenced by incoming external waves. This result is obtained by imposing an upper bound to the depth function, so that the water depth does not increase when it reaches the hull elevation, while the pressure is allowed to increase. This upper bound increases the nonlinearity of the system, which may have dry cells, wet free-surface cells and pressurized cells. The solution of this system is found by a single nested-Newton iterative solver of Casulli and Zanolli [36], in which with two separate linearizations the system is written in a sparse, symmetric, positive semi-definite form. This particular form allows us to employ a matrix-free conjugate gradient method, and efficiently get the unknown pressure. The integral of the pressure over the hull is applied for the hydrodynamic force and torque acting on the ship. After adding the skin friction and other external forces from the propeller or the rudder, the total force is inserted in the equation of motion of the rigid body. The ODE system is discretized with a second-order Taylor method, and it is solved for the six degrees of freedom (3 coordinates for the position vector of the barycenter and 3 rotation angles), providing the next position and orientation of the ship. The vertical translation of the rigid body is governed by the gravitational force and the restoring force from Archimedes' principle. As the ship oscillates up and down, the gravitational potential energy is partially transferred to the radiated free-surface water waves, damping and eventually stopping the motion. Also, the ship pushes and pulls the water around it, inducing the added mass force. All these elements constitute the ODE that was used for the verification of the vertical degree of freedom. The numerical simulation gave the expected results for the vertical motion. The horizontal translation, important for the manoeuvers, presented a numerical instability unseen in our previous test cases, which is connected to the relative motion between the ship and the grid. In each time step in which the ship enters a new cell, the pressure sharply increases and decreases at the ship bow. An oscillation can build up in time and create an unphysical void below the vessel. We implemented a few ideas to attenuate the oscillations. At the heart of all the following techniques is the reduction of the time derivative of the water depth, especially for those cells transitioning to a pressurized state. All these modifications were effective at controlling the oscillations, each with a different intensity, and simulations with a horizontal motion are much more stable than without these techniques. With the collaboration of the BAW research institute, we worked on the model validation. We used data from two separate experiments to compare the measurements with the numerical results. Specifically, we focused on the ship-generated wave height and the hydrodynamic forces on the hull. The comparison is satisfactory for the wave height. The force and torque prediction is plausible but underestimated compared to the measurements. The model seems to displace the water volume correctly during the ship passage, while the force and torque response might need additional work to be trusted in applications. Even though the hydrostatic assumption is mostly correct in our range of applications, the presence and the motion of a ship could generate strong vertical accelerations of the flow, which may not be negligible. For this reason, we implemented an algorithm that corrects the velocity field, introducing also dispersive effects due to a non-hydrostatic pressure. The correction consists of a higher-order Bousinnesq-type term in the momentum equation and the solution of the resulting system. The non-hydrostatic update has a small influence on the wave generation, while it alters significantly the reaction forces. The subgrid method implementation allowed to benefit from high-resolution bottom descriptions while keeping the grid size coarse. The same subgrid can also be used for a refined definition of the hull, which makes the volume computations more accurate. Furthermore, the subgrid introduces new possible states for the cells, as they can be partially dry or partially pressurized. These intermediate states translate into smoother transitions from one state to the other when the free-surface is close to the bathymetry or to the hull. Concerning the software implementation of the developed scheme, in order to improve the execution performance of the prototype script formulated initially in Matlab, the numerical method was rewritten as a Fortran program. Also, thanks to the domain decomposition technique and the MPI standard, each simulation can run in parallel on multiple CPUs, leveraging the computational power of supercomputers. The coupling of the PDE and ODE system, together with an appropriate redefinition of the depth function, proved to be a valuable method for studying fluid-structure interaction problems. The combination of efficient numerical techniques led to the development of a tool with a potential to be applied in the practice for the simulation of floating objects in wide domains.