Wave Forces on a Body in Confined Waters

In this paper, we study the interaction of gravitational water waves with stationary ocean structures near a vertical boundary or in a wave channel. The water depth is assumed finite implying that the interference effect of not only the vertical boundaries but also the horizontal bottom is considered. Presented here are a potential flow theory and the boundary-element method (BEM) towards solving this problem. The numerical results include 1and 2order wave exciting forces of a hemisphere, and are compared with available computations and published model test results. The theoretical findings based on a parametric study in this paper are associated with practical design considerations for a proposed deepwater ocean basin to be built at Jervoise Bay, Western Australia. Introduction In many ocean engineering applications, there is a need to study wave interaction with a body or body system, which is affected by the surrounding boundaries of the wave field. For example, the motion behaviour of a vessel in a wave channel or close to a quay may be quite different from the same vessel in the open sea. Part of the motivation for this study is related to a proposed deepwater ocean basin to be built in Western Australia. It is proposed that the basin will have a horizontal operational area of 50m x 50m. The maximum depth is 20m plus a 15m pit. The basin will be equipped with multi-flap wave generators on two adjoining sides. On the opposite sides of the wave generators, absorbing beaches or wave diffusers may be fitted. We need to answer questions about the horizontal dimensions, the necessity to install wave absorbing beaches and active wave absorption on wave generators and the necessity of using a removable bottom for water depth change etc. For instance, when conducting model tests by using only one side of vertical wave generators, the other side will perform as a reflecting wall if no active wave absorption is attempted. Thus, wall effects cannot be totally neglected, even in a wide ocean basin designed to avoid significant wave reflections from the basin boundaries. This paper attempts to shed light on these questions by numerically modelling the wave forces on a surface-piercing body in wave tanks of various configurations and comparing the results with open-sea situations. Potential theory is often used to describe hydrodynamic wavebody interaction problems when the wavelength is not much longer than the body dimensions. For arbitrary-shaped bodies, the boundary-element method based on Green function approaches is a powerful numerical method for solving the problems. Successful computation of wave-body interactions is to a large extent dependent on efficient evaluation of the Green function. Newman [6], among others, has developed efficient algorithms for some of the Green functions for open-sea problems, which resulted in practically applicable boundary-element method computer programs such as WAMIT [4]. The Green function for a horizontally semi-infinite wave field is the sum of the Green function for the open sea and its image about the vertical boundary, which does not add in too much computational burden. However, a Green function that models wave-body interactions in a wave channel with two parallel vertical walls, such as a towing tank, introduces far more complexity. In this case, the Green function satisfies the linearized free-surface condition and the non-penetration condition on a flat bottom and the sidewalls. It may be formally represented by an infinite series of mirror images of the Green function in the open-sea. The convergence of such a series, however, is known to be extremely slow. In order to effectively model the channel problems, many efforts have been made towards accurate and efficient computation of the Green function; see for example, [1, 3, 5, 7, 9]. In the recent advancement made by Xia [9], a consistent asymptotic analysis has been developed for fast evaluation of the complete image series of the open-sea Green function. The numerical scheme was successfully applied to the investigation of the wave-interference effects on a truncated cylinder in a channel and was verified with a semi-analytical study published in [12]. Xia and Ronalds [10] presented numerical results for a FPSO hull in a wave tank and compared them with the numerical solution in open-sea. In what follows, we will present an introduction to the potential theory and the boundary-element method for wave-body interactions in an open-sea, near a vertical wall and in a wave tank of parallel sidewalls. Computational results for the first-order wave exciting force in heave and the second-order mean drift force in surge of a hemisphere in a wave tank with parallel sidewalls will be presented and be compared with model testing results published in [13]. These results clearly demonstrate the sidewall effects in typical offshore hydrodynamic modelling by a towing tank and the robustness of the present numerical approach. The numerical method will then be applied to a parametric study to investigate the effects of a single reflecting wall and two parallel reflecting walls on wave force modelling of the hemisphere. Associated discussions will be given towards the design and operation issues of the proposed Australian ocean basin. Boundary Element Method Potential Flow Theory We consider the problem of a three-dimensional body of arbitrary geometry subjected to regular waves in a wave basin of ‘infinite’ length and width b . The water depth in the basin is denoted by h . Cartesian coordinate system o-xyz is defined with x-axis along the ’longitudinal’ direction towards the absorbing beach and coincident with the centre line of the basin, y-axis pointing to the side wave maker, z-axis positive upwards and the origin o placed on the undisturbed free surface. A regular wave of circular frequency ω sent by the end wave maker propagates in the positive x-direction. The body is assumed fixed in the water. Under the assumptions of a perfect fluid and irrotational flow, the motion of the fluid field can be described in terms of velocity potential in the form { } t i e z y x t z y x ω φ − = Φ ) , , ( Re ) , , , ( (1) with the spatial potential ) ( ) , , ( D I i gA z y x φ φ ω φ + = (2) where A is the incident-wave amplitude; I φ and D φ are normalized incident and diffracted wave potentials. The incident-wave potential I φ may be explicitly written as ikx I e kh h z k cosh ) ( cosh + = φ (3) where k is the wave number satisfying the dispersion relation kh gk tanh 2 = ω , with g being the gravitational acceleration. The governing equation for the diffraction potential D φ is the Laplace’s equation subject to the linearized free-surface boundary condition and the non-penetration boundary condition on the seabed, on the vertical walls 2 1 ,b b y = that model the sidewalls of a wave tank and on the body surface 0 S ,