Wave Simulation in Truncated Domains for Offshore Applications

<Subject and main objective>There is a desire to assess extreme wave loads on offshore structures like Floating Production, Storage and Offloading (FPSO) vessels, either for design, or for evaluation when circumstances near the structure change. Design formulae for extreme wave loads are scarce and have limited validity for specific structures. Simplified theory, such as linear potential theory, which is often used for motion analysis of offshore structures, does not represent the hydrodynamics involved in a wave impact with sufficient accuracy. For this reason, extreme wave loads are often assessed in physical experiments at model scale. Extreme wave loads may also be simulated by means of detailed numerical modeling. ComFLOW is the name for a numerical method specifically developed for simulating wave impact events. Simulation of wave impacts on offshore structure with ComFLOW is the subject of this thesis.Simulations should represent wave interaction with the structure as if it were out at sea. For reasons of efficiency, the computational domain can not be much larger than the offshore structure it contains. Wave simulation in computational domains of limited size requires special measures to reduce spurious reflection of waves at the boundaries of the domain. The main objective of this thesis is to find or develop means to efficiently reduce spurious reflection from the boundaries in ComFLOW.<Numerical method>ComFLOW is based on the Navier-Stokes equations. For the derivation of the numerical method in ComFLOW, a finite volume discretization for Cartesian grids has been adopted. The discretization yields a skew-symmetric operator for the convective term and a symmetric operator for the viscous term in the momentum equation, resembling the symmetry properties of the analytical operators. Forward Euler time discretization is applied, in which the pressure term is evaluated implicitly. Substituting the discrete momentum equation in the discrete continuity equation gives a Poisson equation for the pressure which is solved iteratively by means of Successive Over-Relaxation (SOR) with an optimal choice for the relaxation coefficient in case the absorbing boundary condition derived in this thesis is not applied in the simulation. When the absorbing boundary condition is required, the Poisson equation (with additional terms resulting from the discretization of the absorbing boundary condition that do not fit the typical Poisson-stencil) is solved with a general sparse-matrix solver, in which the stabilized Bi-Conjugate Gradient (BiCGSTAB) iteration method is combined with an Incomplete Lower-Upper preconditioner (ILU(")).The free surface in ComFLOW is advected by means of an improved Volume of Fluid (iVOF) algorithm. The improvement consists of a local height function that aggregates fluid in a stencil of three cells in all spatial directions when fluxing fluid. The local height function reduces mass loss and so-called flotsam and jetsam, a numerical artifact consisting of disconnected droplets of fluid approximately one grid cell in size.<Spurious wave-energy dissipation>In coarse-grid simulations, a specific amount of grid-size-dependent numerical viscosity is applied to obtain a velocity field without spatial instabilities, effectively resulting in a first-order upwind discretization of the convective term in the momentum equation. Upwind discretization induces spurious wave energy dissipation, which is undesirable because it reduces the forces involved in a wave impact on the structure. As a result, the grid size needs to be chosen sufficiently fine to make sure that the structure in the simulation endures the full impact for the wave condition that was specified. A better discretization than first-order upwind for the convective term can reduce spurious wave-energy dissipation. With a better discretization the grid size can be chosen less fine, resulting in more efficient simulations.In this thesis, a Lax-Wendroff discretization with a min-max flux limiter was evaluated for the convective term in the momentum equation in simulations with standing waves. Standing wave simulations are an efficient means to investigate the effect of different discretizations on the reduction of wave energy dissipation, because all the physics of wave motion are included without disturbances from incoming wave or absorbing boundary procedures. The reduction of wave-energy dissipation with Lax- Wendroff discretization was limited compared to first-order upwind discretization of the convective term in the momentum equation.It was hypothesized and found that also the free surface displacement algorithm was a source for wave-energy dissipation, because VOF with piecewise-constant interface reconstruction (Simple Line Interface Construction, SLIC) includes a first-order upwind discretization of the fluid flux. The discretization of the fluid flux can be improved when Piecewise-Linear Interface Construction (PLIC) is adopted. The reduction of wave-energy dissipation of PLIC compared to SLIC was limited as well.A considerable decrease of the wave-energy dissipation was only obtained by combining PLIC in the free surface displacement algorithm with Lax-Wendroff discretization of the convective term in the momentum equation in propagating wave simulations. In these simulations, the combination of PLIC and Lax-Wendroff gives 2% waveenergy dissipation over two wave lengths, whereas the original combination of SLIC and first-order upwind in the convective term gives 18% wave-energy dissipation.<Wave generation>In ComFLOW, waves are commonly generated by specifying velocities as a Dirichlet boundary condition at the inflow. The velocities originate from wave theory. Wellknown wave theories for steep waves are Stokes 5th order theory and stream function theory (Rienecker-Fenton), for which analytical solutions are available. In this thesis, Rienecker-Fenton solutions are used to generate regular waves at the boundary. It was investigated how well ComFLOW represents these analytical solutions by means of a grid convergence study. We considered both the free surface elevation and the vertical profile of the horizontal velocity two wave lengths away from the inflow boundary. The free surface in wave crests was approximated reasonably well by ComFLOW for the finest grid that was considered; in wave troughs, however, there remained a considerable difference between ComFLOW and the analytical solution. The reason for the differences during wave troughs is not well understood and further research is required. Also for the velocity profile during a wave crest there were differences: not at the free surface, where we would have expected them because of the large gradients in the velocity, but the differences were mainly near the bottom. This is also part of continued research.Steep irregular waves were generated in ComFLOW by using velocities from linear wave theory with multiple frequency components, and by using velocities obtained from solutions of a Finite-Difference Finite-Element Method (termed FDFEM in this work), which is a non-linear potential flow solver. The two generation methods were compared as follows. The solution for the surface elevation from the FDFEM method at a certain location was used as a reference solution. Velocity output from the FDFEM solution was taken two wave lengths’ distance before the reference location and imposed onto the ComFLOW domain. The ComFLOW solution at the reference location was then compared to the reference solution. It was found that the ComFLOW solution for the free surface elevation agrees well with the FDFEM solution.A similar procedure was applied for wave generation with velocities obtained from linear theory, but now the surface elevation from the FDFEM solution some distance before the reference location was used. This surface elevation was decomposed into its Fourier-components. For each component, linear potential theory gives the velocities and the combination of all these velocities provides the total velocity signal that is imposed onto the ComFLOW domain as a boundary condition. Again, the ComFLOW solution for the free surface at the reference location was compared to the FDFEM solution. Now, it was found that there are differences between the two solutions that do not decrease with increasing grid resolution. It is therefore concluded that linear theory should not be used for irregular wave generation when the ComFLOW solution is to be compared with experimental results.<Reducing spurious wave reflection at domain boundaries>Waves reflect from computational domain boundaries when no special measures are taken to prevent reflection. Dissipation zones are often used near domain boundaries to induce a rapid decrease of wave energy so that when waves reach the boundary and reflection occurs, the wave height has reduced to such an extent that spurious reflection does not interfere with the processes near the structure.Dissipation zones need to be several wave lengths long to be effective. In many cases, the part of the computational domain taken up by dissipation zones exceeds the size of the domain in which the wave interaction with the structure actually occurs. Dissipation zones take up a considerable amount of the total computational effort required for the simulation. Local absorbing boundary conditions are a more efficient alternative to dissipation zones. In this thesis, a local absorbing boundary condition for long-crested irregular waves is derived that is more efficient and more effective than a two-wave-lengths-long dissipation zone.The absorbing boundary condition in this study includes an approximation of the dispersion relation which is accurate within a range of wave numbers (or, equivalently, frequencies). It also includes second-order vertical derivatives of the solution variables along the boundary. The differential equation for the absorbing boundary condition is discretized implicitly, combined with the discrete momentum equation and included in the Poisson equation for the pressure. Theoretically, the reflection coefficient, i.e. the ratio of outgoing wave amplitude and reflected wave amplitude, for this absorbing boundary condition can be as low as 2% for wave components within the range 0 < kh <= 6. In actual wave simulations with this boundary condition, the obtained reflection coefficient was 5%. There are two main reasons for the difference between the theoretical value for the reflection coefficient and the one that was obtained in simulations: 1. linear potential theory was used to derive the absorbing boundary condition, whereas the equations that ComFLOW solves for are non-linear; 2. the second-order vertical derivative cannot be solved near the free surface and needs to be approximated. The approximation used near the free surface is rather crude and requires attention in future research.<Validation in 3D>Experiments with a schematized model of a semi-submersible in waves were performed at MARIN. Waves were generated with a pivoting wave maker at one end of the basin. At the other end, a parabolic beach was installed to reduce spurious wave reflection. The model, which was approximately 1m long, was placed in the middle of the basin and was kept restrained during the experiment. Pressure sensors and wave gauges were placed at several locations around the columns of the model. Also the deck of the model was fitted with pressure sensors. During the experiments, there was violent wave interaction with the model: wave impacts occurred on both columns and many also reached the deck.In ComFLOW, a similar setup was created. The computational domain extended over the full width of the basin. The side walls of the domain (x-z planes) were configured to represent the side walls of the basin. The incoming-wave boundary (x-y plane) was positioned approximately one typical wave length in front of the structure. Here, waves were generated by specifying the kinematics obtained from an FDFEM simulation that contained the entire basin, including wave maker and spending beach. At the outflow end of the computational domain in ComFLOW, the absorbing boundary condition derived in this study was applied. For stability reasons, the FDFEM simulation was performed with a relatively coarse grid and with a smaller water depth than the basin. For this reason, the waves in this simulation could only compare reasonably well to the waves in the experiment. And there was another issue. The position of the model relative to the wave board was not recorded with sufficient precision. At multiple output locations, the free surface elevation in de FDFEM simulation was compared to the free surface in an undisturbed test (without structure). The position where the simulation results best resemble the measurements was selected as most likely location for the model in the simulation.Keeping in mind that the waves in the simulation were slightly different from the waves in the experiment - not only as a result of model errors, but also because the true water depth in the experiment could not be represented in the FDFEM simulation the comparison between ComFLOWand the experiment turns out rather well. Pressures and surface elevations in ComFLOW show good agreement with the measurements.The 3D validation of ComFLOW was also an important test for the absorbing boundary condition that was developed in this thesis. The simulations show that the boundary condition can be applied in practical simulations with good results. These simulations could not be performed with a dissipation zone on a desktop computer, because with a dissipation zone it did not fit in memory. It would have been interesting to compare results between a 3D simulation with an absorbing boundary condition and one with a dissipation zone. On the other hand, we can conclude that the absorbing boundary condition enables us to perform simulations that could not be run on a desktop PC before the boundary condition was there.