Calcul des efforts de deuxième ordre à très haute fréquence sur des plates-formes à lignes tendues
Computing High-Frequency Second Order Loads on Tension Leg Platforms
Institut Français du Pétrole
Le problème considéré ici est celui de l'évaluation des efforts excitateurs de deuxième ordre (en mode somme, c'est-à-dire prenant place aux sommes deux à deux des fréquences de houle) sur des plates-formes à lignes tendues. Ces efforts sont tenus pour responsables de comportements résonnants (en roulis, tangage et pilonnement) observés lors d'essais en bassin et pourraient réduire sensiblement la durée de vie en fatigue des tendons. Des résultats sont tout d'abord présentés pour une structure simplifiée, consistant en 4 cylindres verticaux reposant sur le fond marin. L'intérêt de cette géométrie est que tous les calculs peuvent être menés à terme de façon quasi analytique. Les résultats obtenus permettent d'illustrer le haut degré d'interaction entre les colonnes, et la faible décroissance du potentiel de diffraction de deuxième ordre avec la profondeur. On présente ensuite des résultats pour une plate-forme réelle, celle de Snorre.
Tension Leg Platforms (TLP's) are now regarded as a promising technology for the development of deep offshore fields. As the water depth increases however, their natural periods of heave, roll and pitch tend to increase as well (roughly to the one-half power), and it is not clear yet what the maximum permissible values for these natural periods can be. For the Snorre TLP for instance, they are only about 2. 5 seconds, which seems to be sufficiently low since there is very limited free wave energy at such periods. Model tests, however, have shown some resonant response in sea states with peak periods of about 5 seconds. Often referred to as springing , this resonant motion can severely affect the fatigue life of tethers and increase their design loads. In order to calculate this springing motion at the design stage, it is necessary to identify and evaluate both the exciting loads and the mechanisms of energy dissipation. With the help of the French Norwegian Foundation a joint effort was started in 1989 between IFP and NTH (Norwegian Institute of Technology ) with the goal of reaching a better understanding of the physical phenomena at hand, and of quantifying some of them, namely second-order loading and viscous damping. This article deals only with IFP's contribution, which is the calculation of second-order loading. Even though other processes are candidates for resonant loading (slamming, viscous effects, hull-tethers dynamics interaction), there is some indication that second-order phenomena, as predicted by potential theory, make the main contribution. According to this second-order theory, two independent wave components with frequencies omega1 and omega2 will contribute some loading at the sum frequency omega1 + omega2. Part of this loading can be obtained straightforwardly once the linearized (or first-order) diffraction problem has been solved at both frequencies. The remaining part, due to the second-order diffraction potential, can be expressed as an integral over the whole free surface, the integrand of which is the product of two terms :(a) The right-hand side of the free surface equation satisfied by the second-order diffraction potential. (b) The (linearized) radiation potential (for the considered degree of freedom) at the sum frequency omega1 + omega2. Even though the theory now appears to be well established, the numerical difficulties involved with its implementation are tremendous :(a) The right-hand side of the free surface equation contains double space derivatives (see Eq. 6) which are difficult to evaluate numerically. (b) The convergence of the free surface integral is very slow with increasing radial distances. (c) At the high wave frequencies that we are interested in, quite refined meshes of the hull and free surface are required, with several thousands of panels. Part of these difficulties have been overcome by taking advantage of the fact that, at these high frequencies, only the upper part of the TLP is reached by the first-order wave field (note that this statement does not apply to the secondorder wave field which penetrates much more deeply into water column). It is therefore legitimate to assimilate a TLP to an infinitely-deep four columns structure. For this type of structure there are quasi-analytical methods for solving first-order diffraction and radiation problems. Here we have made use of the Linton and Evans method (Ref.  ). As a result the first part of second-order loading (that does not involve the second-order diffraction potential) and the righthand side of the free-surface equation could be evaluated economically and with controlable accuracy. For the Snorre TLP (which we took as the exercise case) the auxiliary radiation potentials were obtained numerically, by running the BOLANG code. At the highest sum frequency considered (2. 8 rad/s), the hull was divided into 13888 panels, and the radiation potential was subsequently evaluated at 94616 points over the free surface (see Figs. 4 and 5). Calculations were also performed for an idealized deep-draft concrete TLP (such as planned for Heidrun), consisting of the upper 65 meters of 4 vertical columns standing on the seafloor. Thanks to an extension of Linton and Evans' theory, the corresponding radiation potentials could then be obtained semianalytically. The same axis-to-axis distance (76 meters) and the same column diameter (25 meters) were taken for both structures, which made possible some cross-checking of the results. For instance Figure 9 shows the surge damping, as obtained by the numerical model for the Snorre TLP and by the semianalytical one for the idealized TLP. Agreement appears to be quite good, which gives confidence in the convergence of the numerical calculations. Fig. 11 shows the second-order loads for surge, at omega1 + omega2 = 2. 65 rad/s, versus the difference frequency omega1 - omega2. Again agreement is quite good between both structures, but Figure 12 shows that this is not true for the pitch loads. From Fig. 13 it can be inferred that this difference is due to the second-order vertical loading that is applied at the column bases and on the pontoons. Lasly, and rather discouragingly for the designer it appears that second-order loads vary very quickly with the sum frequency (see Fig. 10) and also with the heading (Fig. 14), features that can be attributed to the interaction effects between the columns, which become more and more pronounced as the frequency increases.
© IFP, 1991