A numerical investigation of irregular water waves and their statistical properties

The work presented in this thesis is based on two main objectives. First, a numerical model for the simulation of water waves based on fully nonlinear potential ow theory in two horizontal dimensions is developed, and second, this model is used to study the statistical properties of irregular wave fields. The diffculties that must be overcome when developing such a numerical model are primarily related to computational expenses and temporal stability. An accurate numerical description of irregular wave fields requires on the order of a few million grid points in the horizontal dimensions alone, and combined with the fact that substantial changes in the wave fields' statistical properties take place over roughly a few hundred characteristic wave periods or more, it becomes clear that a numerical method must be very cost efficient to be useful in practice. Moreover, for the final results to be considered reliable, the time integration of the numerical method must of course be stable, and in an attempt to achieve the best possible combination of efficiency and stability, two ideas for solving the governing equations are tested in this thesis. While these ideas are both based on the akharov-formulation of the free surface boundary conditions and integration of these in time using the same method of lines-approach, they solve the Laplace problem inherent in the potential ow formalism in fundamentally different ways.

The first idea is to Taylor expand the velocity potential in the vertical coordinate around the still water level such that the Laplace problem is translated into a set of equations relating the surface potential to the potential at the still water level. While this idea is also used in the classical high-order spectral (HOS) method, a new method is developed in this thesis to solve this set of equations numerically exactly. Not surprisingly, the proposed method is capable of obtaining a significantly more accurate solution to the Laplace problem than the HOS method. The proposed method, however, also turns out to be very sensitive to small inaccuracies in the surface quantities, and to obtain similar stability properties in the time integration as the HOS method it must therefore be used with a smaller truncation order of the underlying Taylor series. For that reason, the range of applicability of the proposed method becomes roughly the same as that of the HOS method, and since the latter is superior in terms of efficiency, the development of the proposed method is not pursued in greater detail.

The second idea is to solve the Laplace problem by resolving all three spatial dimensions numerically, and two different methods are developed based on this idea. Motivated by the fact that several authors have claimed that aliasing errors may lead to unstable time integration (meaning that either a dealiasing procedure or an artificial damping scheme is required to maintain stability during simulations), the first of these methods discretizes the spatial part of the governing equations using a spectral Galerkin method which is inherently free of these errors. The method is shown to be capable of solving the Laplace problem for essentially all water depths and values of the wave steepness to any desired accuracy, and below a certain threshold steepness it in fact offers stable time integration. Above the threshold steepness the time integration, however, becomes unstable, and the method thus provides clear evidence that temporal instabilities in volumetric methods are not necessarily due to aliasing errors. The cause of the instability remains unknown, but a stability analysis is developed which can be used to predict the threshold steepness very accurately. Despite the instability, the method is found to be substantially more robust and accurate than the HOS method, but its computational cost implies that it cannot be used to simulate large irregular wave fields within a practical time span.

Precisely because aliasing errors are found not to be the cause of the temporal instability, the second method resolving all spatial dimensions of the Laplace problem is based on a pseudospectral discretization. In this method nothing is done to prevent the aliasing phenomena, and the method turns out to have accuracy and stability properties that are very similar to those of the method based on the Galerkin discretization. Moreover, it is much more efficient, not only because the expensive zero-padding technique for dealiasing is avoided, but also because it utilizes an artificial boundary condition with which a given accuracy can be reached using substantially fewer grid points in the vertical dimension. Although the method (like seemingly all other methods) needs artificial damping in order to integrate steep waves in time, it is both much more accurate than the HOS method and efficient enough that large simulations can be carried out. For that reason, this method is used for the investigations of irregular wave fields.

In this thesis, three investigations of large irregular wave fields are presented. In all of these, the water is assumed to be infinitely deep, but otherwise the investigations are quite different. In the first investigation, only short-crested wave fields with a single specific directional spreading are considered and the effect of the wave steepness on the statisitcal properties of the surface elevation and the uid kinematics are studied. The accuracy of a number of theoretically derived probability density functions (PDFs) for the surface and crest elevation are tested, before the PDFs of the uid velocities and accelerations at the surface are determined. Finally, the statisitcal properties of the uid velocities and accelerations that accompany very large crests are investigated, both in terms of their magnitude and where they are expected to occur relative to the wave crest. In the second investigation, both the steepness and the directional spreading of the wave fields are varied and it is studied how the wave-induced force on a vertical circular cylinder, as predicted by the Morison equation, changes under these variations. While the numerical simulations provide the basis for the investigation, a series of new analytical results are also derived, and with these it is possible to conclude how the different components of the force are affected by nonlinearity. A main conclusion of the investigation is that the wave-induced force can be accurately estimated from first-order theory in the limit where the ratio between the cylinder radius and the characteristic wavelength is large compared to the wave steepness. In the third investigation, the depth-decay of the mean and variance of the Eulerian and Lagrangian velocities in the main direction of the wave field are studied. Second-order potential ow theory is used to derive expressions for these quantities in terms of improper integrals, for which very simple yet accurate analytical approximations are found in the limit of large depth. The theoretical results are compared to those of numerical calculations. It is shown that the theoretical results compare well to those of the numerical simulations even for the cases where the numerical simulations are known to be affected substantially by third-order effects.