The modelling approach discussed in the previous sections is applied to FSI case studies involving waves and moored floating bodies. Numerical simulations of floating bodies are validated against experiments in Section 5.1 with the following cases: free decay of a heaving sphere in Section 5.1.1, wave-induced rolling response of a rolling caisson in Section 5.1.2, and heave and roll response of a floating caisson to an extreme event (focused wave) in Section 5.1.3. The mooring model is validated for catenaries against experimental data in Section 5.2 for: static tensions for different positions of the fairlead relative to the anchor in Section 5.2.1, and dynamic mooring line damping under forced sinusoidal motion of the fairlead in Section 5.2.2. Finally, combining all the different validated aspects, a full-fledged 3D numerical simulation of a floating platform for offshore wind turbines moored with 3 catenary lines is compared to experimental data in Section 5.3. In this last section, simulation of the uncoupled mooring model is first presented in Section 5.3.2, followed by the simulation with all the models presented in this work coupled together in Section 5.3.3.
All parallel simulations presented below ran on an HPC cluster composed of AMD Interlagos Opteron with 2.3GHz core speed, 32 cores per node, and 64GB memory per node. The number of cores used for a simulation is case-dependent, and is therefore mentioned in each relevant subsection.
Validation of FSI for floating bodies
5.1.1 Free oscillation—heaving cylinder
The case presented here describes the free oscillation of a floating cylinder. The cylinder motion is restricted in all directions with the exception of vertical motion (heave motion). The domain setup is based on the experiment conducted by Ito (1977), where a floating cylinder with length L = 1.83m and diameter D = 15.24cm is placed horizontally across the width of the tank at the mean water level. The small clearance between the tank walls and the edges of the cylinder (1.27cm) makes it possible to simulate the 3D experimental setup in a 2D numerical domain. The density of the cylinder is half the density of water, making the cylinder draught at equilibrium equal to its radius. The water depth is hmv = 1.22m. The numerical tank used here has a total length of Ltank = 20m and height of //tank = 2.44m, with absorption zones of length La|)s = 9m, enough to dissipate outgoing waves generated by the cylinder motion. The cylinder is placed in the middle of the tank with its centre at a distance of 10m from each boundary (see Figure 8).
Figure 8: Geometry of numerical domain for the simulation of free oscillation in heave of a floating cylinder. Solid line: initial position, dashed line: equilibrium position.
The numerical simulation is initialised with still water and the cylinder being lowered 2.54cm from its equilibrium position. The boundary conditions are ffee-slip at the numerical tank walls rwaii, zero pressure head at the atmosphere (top boundary of the tank ratm), and no-slip condition on the cylinder Ffns- The domain is discretised with triangular elements of minimum characteristic element size h°e constant up to a distance of ±2.54cm from the still water level and at the boundaries of the cylinder, with gradual coarsening of the mesh applied beyond this area (10% increase in neighbouring characteristic element size). To assess convergence of the coupled fluid and rigid body models, the simulation is repeated with different time-step, CFL numbers and mesh sizes. For all cases, the MBD solver uses a fixed time step of 1 x 10_5s, which is much smaller than the fluid time-step.
The results are presented in Figure 9a for different values of /г® and Figure 9c for different values of At. As shown in these figures, the numerical solution converges, and agreement
Figure 9: Time-series and sensitivity analysis of the free oscillation of a heaving cylinder.
with experimental data is already excellent for At = le-3 and h® = 0.02D. The case shown in Figure 9a had a mesh composed of 885,433 elements and took 9.4h running in parallel on 96 cores to produce 4s of data. The Root Mean Square (RMS) variation from the most refined simulations features a temporal order of convergence of 1, and a spatial order of convergence of 2, which is in accordance with the numerical schemes employed here.
5.1.2 Wave-induced motion-rolling Caisson
This case is based on the experiment conducted by .Tung et al. (2006), studying the rolling motion of a floating rectangular caisson. In the experiment, the floating caisson has a length of 0.3m, height of 0.1m and width of 0.9m. The caisson is mounted on the tank walls through a pair of bars and hinges sitting at the mean water level, aligned so that the axis of the joint goes through the centre of mass of the caisson. The roll moment of inertia has been measured experimentally as / = 0.236kgm2. Similar to the previous case, the experimental set-up can be simulated using a 2D numerical domain due to the small clearance from the flume walls.
Free oscillation and decay tests were first performed with the fluid initially at rest, and introducing a roll displacement of 15° from the equilibrium position of the caisson, with no initial velocity. For the numerical set-up tank length, Ltank = 5m with absorption zones of length, Labs = 2m on either end of the tank are used. Boundary conditions are similar to the previous case, with free-slip condition on tank walls rwaii, atmospheric condition along the top boundary of the tank ratm, and no-slip condition on the boundaries of the caisson Ffns- The numerical simulation runs for at least ATsim = 4s in order to record the oscillating signal for the same length of time as Jung et al. (2006). The fluid domain is spatially discretised with h°e = 0.005m around the mean water level and up to a distance of 0.45m from the barvcentre of the floating body, with a gradually coarsened mesh used in the same manner as described in the previous case. The two-phase flow solver uses a timestep of At = 5 • 10_3s while the rigid body solver uses a fixed time step of 1 • 10_5s. Note that although friction from the hinges acting on the rolling motion of the caisson is not mentioned in Jung et al. (2006), it is argued in the literature that friction energy losses due to this support system were probably present during the experiment, as raised by Calderer et al. (2014); Bihs and Kamath (2017); Chen et al. (2016) and confirmed herein. A linear dissipation term Сш proportional to the angular velocity is therefore added to the equation of motion of the caisson in order to represent these losses from the experimental results, and was set to Сш = 0.275 after trial and error, similarly to Bihs and Kamath (2017). Note that this test case was also presented in de Lataillade et al. (2017) on an older implementation of the present FSI framework and without taking the friction into account.
Results of free-oscillations tests are presented in Figure 10, along with experimental data and the results from the Particle-In-Cell (PIC) model from Chen et al. (2016). It is observed that the natural period of the rolling motion is very well predicted numerically, regardless of the usage of a dissipation term. For the amplitude decay, however, a large numerical underestimation is observed when Сш = 0 but this disagreement with the experiment exists for both Chen et al. (2016) and the current model. Very good agreement with the experiment and the current model can be recovered with Сш = 0.275, confirming that it is likely that friction occurred during the experiment.
The response of the rolling motion against regular waves is subsequently investigated. Using the same numerical setup, the leftmost boundary and relaxation zone are adapted for regular wave generation boundary, and the length of the numerical domain is adjusted according to the wavelength Л of the generated wave, as shown in the schematic representation of the numerical domain in Figure 11. The RAO of the caisson is calculated using different wave conditions, as shown in Table 1. Each case runs for a time equivalent to 30 generated wave periods, and the amplitude of the response is averaged over the last 10 periods. Run-
Figure 10: Extinction curves of free oscillation in roll.
Figure 11: Geometry of numerical domain for the simulation of free roll oscillation of a floating caisson.
ning in parallel on 160 cores, it took 4h to complete the case with the sortest wave period (T = 0.6s), and 48h for the case with the longest wave period (T = 1.4s). The response of the rolling caisson is shown in Figure 12, and compared with the PIC model from Chen et al. (2016), experimental data from Jung et al. (2006) and linear potential flow theory. Results with the calibrated friction coefficient Сш = 0.275 from the free-decay tests are also plotted in Figure 12, while the PIC model does not consider friction losses. In general terms,
Table 1: Wave characteristics and resulting roll response for computing Response Amplitude Operator (RAO) of Figure 12.
Figure 12: RAO of rolling caisson under regular wave loads. Analytical (linear theory) curve and experimental points digitised from .Tung et al. (2006); monolithic model results (PICIN) digitised from Chen et al. (2016).
results are in good agreement with experimental data along the tested frequency range. It is worth noting that the two CFD models (Proteus and PIC) present better agreement than the potential flow theory in the longest wave regime (lo ^ wjy), as nonlinear effects are taken into consideration in both models. Close to resonance, results without the dampening coefficient are overestimating the measured response, while the inclusion of the friction terms dampens the rolling oscillations noticeably. This suggests that the calibrated value
Сш = 0.275 obtained from free oscillations may not be valid for the forced oscillation case, as the dissipation term may be nonlinear. The results presented here show the capabilities of the model to reliably predict the response of a floating body to different wave loads.
5.1.3 Response to extreme event—floating structure
This validation case is based on the experiment of Zhao and Hu (2012), featuring a 2- DOFs (heave and roll) inverted T-shaped floating structure under extreme wave loads. The experiments were performed in a numerical tank with a flat bottom. The lower platform of the structure has length and height of 0.500m and 0.123m, respectively. In order to prevent any overtopping water from being transmitted behind the structure, a superstructure of length 0.200m and height 0.250m was mounted on the lower platform. The clearance between the side walls and the floating structure is 5mm on either side, such that the wave processes can be considered two-dimensional, as in the previous cases. The water depth at rest was /imwl = 0.4m and the structure was allowed to heave and roll by attaching the structure to a support system consisting of a cylindrical joint and a heaving rod. A detailed account of the physical model including body properties (mass, moment of intertia etc.) is given in Zhao and Hu (2012). The domain is represented schematically in Figure 13.
A focused wave is generated to represent interaction of an extreme event with the floating body, with an amplitude of aj = 0.06m, focus time t/ = 20s and the focus point of the wave coinciding with the location of the barvcentre of the body (x = 7m). The domain is discretised temporally with CFL = 0.1 and spatially with h°e = 0.005m, the latter being kept constant up to a distance of ±ay around the still water level and, similarly to previous cases, the mesh gradually coarsened beyond that. The boundary conditions are set similarly to the cases in the previous section and the focused wave is generated at the leftmost boundary, with the assistance of a generation zone of length 1A (the wavelength corresponding to peak frequency). Dirichlet boundary conditions for the free surface elevation and the fluid velocity profile were calculated using the .TONSWAP spectrum. Due to nonlinear wave- wave interaction, the actual focus point and time might be different from the analytical values. Using a trial-and-error approach, the wave components must therefore be assigned appropriately to chosen phases so that the wave focuses at the intended location. The following phase shift фпит is therefore applied numerically on all frequency components of the .TONSWAP spectrum, and differs with the analytical phase shift фапа as follows:
with к and / are the frequency and wavenumber of each component, respectively, Xf the focus point and tj the focus time of the wave. The focused wave is first generated in the
Figure 13: Geometry of numerical domain for the simulation of 2 DOFs floating caisson under extreme wave loads.
Figure 14: free surface elevation of focused wave over time at x = Xf.
numerical flume without the structure in order to validate it against the experimental wave. The free surface elevation obtained numerically over time at x = Xf is shown in Figure 14, along with the experimental data and the analytical solution,with the latter derived from linear wave theory. It is observed that the model is in good agreement with the experimental data. Note that the analytical solution presents deeper wave troughs and appears to focus at tf = 20.0s, while both the experimental and numerical results focus at tf = 20.3s, probably due to nonlinear interaction of wave components.
The structure is now introduced in the numerical domain and its interaction with the focus wave is simulated. With a total of 490,543 mesh elements, running in parallel on 196 cores, the simulation produced 30s of data in 57h. Snapshots of the numerical results around the focus time of the extreme wave are shown in Figure 15, and compared to photos from the experiment. These show a clear qualitative similitude of free surface profiles as well as floating body position and rotation between the numerical and experimental results. Furthermore, the overtopping events are well represented numerically with water hitting the superstructure at t = 20.3s (which corresponds to the actual numerical and experimental focus time), and retreating from the structure at t = 20.7s.
Quantitative results in terms of response of the floating body are shown in Figure 16 for heave and roll. The results are also compared to the numerical results from the PIC model of Chen et al. (2016) where a completely different approach is used for modelling the fluid and, most importantly, a monolithic scheme is used for fluid-structure coupling. Because the monolithic approach ensures an inherently stable simulation with no added mass effect, the response obtained by Chen et al. (2016) can be used for code-to-code validation against the added mass stabilised CSS scheme that is used here. In terms of heave, the experiment and the two numerical models predict the peak frequency accurately at /;f = 0.97Hz which is the one obtained experimentally, and both underestimate the amplitude of the heave response of the body (16.8% for the explicit scheme and 10.9% for the monolithic scheme). In terms of roll, the peak response frequency is underestimated with the explicit scheme by 6.67%, while the monolithic scheme and experimental results are in agreement with f° = 1.00Hz. The amplitude in roll is however better estimated with the explicit scheme with an overestimation of the response of 5.6%, against an overestimation of 22.5% for the monolithic scheme. Furthermore, it appears clearly on the time-series of Figure 16c that the response frequency of the explicit scheme is initially in phase with the experimental response but that phase shift occurs after the peak of the focused wave has passed (t > 21s).
It is worth noting that running this simulation without an added mass stabilisaton scheme for fully explicit partitioned schemes leads to an unconditionally unstable simulation. It is therefore demonstrated that both the explicit coupling scheme with the added mass stabilization is a viable approach, as results are in good agreement with experimental data
Figure 15: Snapshots of floating body hit by focused wave (tf = 20s) at 19.9s, 20.1s, 20.3s, 20.5s, and 20.7s as shown on the graph at the top. Left: snapshots from numerical model, right: photographs from the experiment of Zhao and Hu (2012).
Figure 16: Response in heave and roll of floating body to focused wave loads. Experimental results from Zhao and Hu (2012), numerical (PIC) results from Chen et al. (2016).
and the ones obtained by the inherently stable monolithic numerical scheme of Chen et al. (2016).