Validation of mooring model
In this section, simulations are undertaken with the aim to validate the static/quasi-static and dynamic mooring models developed here. The validation cases presented here follow the experimental setup of .Tohanning and Smith (2006); .Tohanning et al. (2007), provides measurements of mooring tensions under static and dynamic conditions. A catenary mooring line is attached to a floating buoy that is driven horizontally through forced motion, with its fairlead kept at a height of h = 2.651m from the seabed. The line has a length L = 6.98m, diameter do = 2.5e_3m, submerged weight wo = 1.036Nm_1, and axial stiffness EAo = 560kN. The characteristics of the cable used numerically here are the same as the experimental ones (see Table 2), with drag and added mass coefficients taken from typical studless chain values. As the water level is hmw = 2.8m, the cable is fully submerged at all times. A schematic representation of the setup is shown in Figure 17. Note that all numerical simulations presented in this section ran in serial mode on a laptop, due to the fact that the computationally intensive fluid solver is not involved.
5.2.1 Mooring statics
For the static analysis, the mooring line is held at 16 different surge positions and the resulting static cable tensions at the fairlead are recorded by .Tohanning et al. (2007). Numerically, the dynamic model uses 100 ANCF elements for the mooring line, and is driven from a fully stretched position (X = JL2 — li2 = 6.457m relative to the anchor) back to X = 5.5m at a
Table 2: Characteristics of mooring lines for Johanning et al. (2007) test case.
Figure 17: Setup for mooring dynamic validation case.
Figure 18: Mooring static analysis: Comparison of experimental data with static and dynamic models comparison. Та- axial tension; Тц: horizontal tension; Ту: vertical tension; d — xq: length of cable on seabed. Experimental data from Johanning et al. (2007).
rate of X = — 5mms-1 with time steps of At = le_4s. The motion is slow enough in order to minimise any dynamic effects. The static catenary model calculates tensions analytically for the same surge range as the dynamic model.
The results are presented in Figure 18, with tensions recorded every 0.01m for the dynamic and static models and compared to the 16 experimental points. The length of the mooring line on the seabed for the different surge positions according to the static model is also plotted for reference. The static equations and mooring dynamics model present a total agreement for the tensions at the fairlead across the whole surge range. This result is meaningful in terms of cross-validation because the static/quasi-static and dynamic models use two very distinct approaches: analytical catenary equations and FEM respectively. Both approaches present an excellent agreement for the partly lifted regime. Analytical and numerical results start diverging from recorded experimental tensions once the line is fully lifted (X > 6.3m). Note that this might be an inconsistency in the experiment as the vertical tension at the fairlead recorded from Johanning et al. (2007) is constant once the line is fully lifted, while it is expected to further increase. The reasons for this remain unknown, but this might be due to slight tilting of the device during the experiment when the line becomes fully lifted.
5.2.2 Mooring dynamics
After validation tests for static tensions, a surge oscillation is now prescribed at the fairlead, following a sinusoidal evolution in time for a total excursion of 0.1/imwi. The tests were carried out for different initial positions X° (with corresponding static pretension Тдо) and frequencies, as shown in Table 3. The experimentally recorded surge displacement is used as input for the prescribed motion of the fairlead. From this sinusoidal motion in surge, the dissipated energy of the horizontal tension Tjj is calculated from the area within the closed-loop (hysteresis) of the X-Тц curve. Note that a total of sixteen cases were run experimentally, but only nine cases are presented here due to the experimental data that was made available. The case with most significant nonlinear effects (number 16 in Table 3) is chosen to perform a cable mesh sensitivity analysis by keeping a constant timestep of At = lx 10_4s. Results for the error in energy dissipation compared to the most refined case are shown in Figure 19 where spatial convergence of order 1.9 is observed as the number of elements ns increases. Following these results, a discretisation of n£ = 100, and At = 1 x 10_4s is selected.
Indicator diagrams for each case are presented in Figure 20, showing the variation in horizontal tension over a single cycle of the surge excursion for the numerical simulation and the experiment. The horizontal tension calculated by the quasi-statics model for the same surge excursion is also included. Results show that the quasi-statics model is in good agreement with the mooring dynamics model regarding minimum and maximum horizontal tensions. Minimum experimental tensions are also in good agreement with the numerical results across all cases. However, for case 12 and any other subsequent cases with a higher
Table 3: Parameters and results for mooring line damping test cases (numbered following Johanning et al. (2007)).
Figure 19: Spatial sensitivity on cable dynamics with At = 1 x 10 4s on case 16.
pretension stress, numerical and quasi-static results show higher maximum tensions than the experiment. This discrepancy is probably associated with potential problems in the experimental layout when the mooring line becomes fully lifted, as in the static tension cases.
The hysteresis of the curves formed in the indicator diagrams are due to nonlinear effects induced by the cyclic motion of the fairlead and can be used to calculate the damping of the line. The energy dissipation caused by drag forces on the mooring line can be calculated as follows:
where Тц is the horizontal tension, T the period of oscillation, and X and X are surge displacement and velocity, respectively. The non-dimensional energy damping (with a being the amplitude of oscillation) is plotted for each case as a function of non-dimensional pre-tension in Figure 21. Relatively good agreement between experimental and numerical results show that nonlinear effects are well simulated, with low damping when pretensions are low, and an exponential increase in damping as pretensions become higher and the line is fully lifted. In case 2, 6, and 8, the damping is low both experimentally and numerically, meaning that there is barely any dissipated energy. This can be verified in the indicator diagrams where all curves for these cases are in relatively good agreement with the quasistatics model (which does not simulate any nonlinear effect). For case 10 and above, energy damping becomes significant. There is a slight inconsistency in the experimental results, again around the point where the line becomes fully lifted, with the damping of case 12 higher than the damping of cases 13 and 14, both of which have a higher pretension that should lead to higher damping under forced oscillation. This inconsistency is not present in the numerical model where any case with a higher pretension consequently yields a higher damping. The good agreement between numerical and experimental damping for different pretensions, has proven that the gradient deficient ANCF mooring model can successfully capture nonlinear behaviour that occurs in real-world applications.