Three-dimensional steep wave impact on a vertical plate with an open rectangular section
Graphical abstract
(a) The leading order potential during the impact of a steep wave on a vertical plate with an open rectangular section, (b) The first-order potential during the impact of a steep wave on a vertical plate with an open rectangular section.
Introduction
It is crucial to know the effects of violent wave impact on structures that are built in or are operated in sea waves. We are concerned with the forces and flows which occur when waves break against structures such as harbour walls, moored or fixed offshore platforms and seagoing vessels in head seas. We focus on violent breaking wave impact and this should be distinguished from the effects induced by regular waves. Steep and breaking waves can exert forces many times greater than non-breaking waves such as standing waves adjacent to a monolithic vertical harbour wall. Standing waves have pressure and velocity fields that varying in a time-harmonic manner within a continuous dynamic process. By contrast, a violent wave impact is a discontinuous process in time, with a sudden collision between a volume of water in the incident breaking wave and the structure. The duration of the typically huge pressure associated with impact is very short, and the pressure exerts a huge short-lived net hydrodynamic load on the impacted structure. In order to form a mathematical theory of impact it is important to study and understand in detail the early stages of impact, when the largest hydrodynamic load occurs.
There are other facets of the problem that distinguish wave impact and make it particularly difficult to study. Much work has been done in theoretical and controlled experimental investigations in two space dimensions (2D). However, a major challenge, to bring the theory closer to realistic sea wave conditions, is to understand wave impact in three space dimensions (3D). The mathematical theory of breaking-wave impact involves satisfying nonlinear boundary conditions on a free surface. The position of the free surface is one of the unknowns of the problem, and it has to be found as part of the solution – it is a so-called free-boundary problem. The problems should be structured as boundary value problems with mixed conditions; so called Mixed Boundary Value Problems (MBVPs). An additional difficulty arises from the fact that in the general case of 3D wave impact, the impacted wetted region on the structure is one of the problem's unknowns, whose boundary should be determined, based on specific assumptions regarding the pressure and velocity of the liquid on the contact line where the free-surface meets the structure [1], [2], [3]. Nevertheless, we will not go pursue that discussion here as the theoretical model used in this paper requires the wetted region to be prescribed. Another difficulty is the nonlinear boundary conditions, and this aspect is simplified by using linearisations of the boundary conditions.
Judging from the published literature, wave impact has been studied theoretically much less than the class of problems associated with the entry of a body into initially static water – so called water-entry problems. In wave impact problems, the region of liquid that impacts the structure has free surfaces on two sides, the top surface of the wave and the steep wave front that hits the structure. In contrast, in water entry problems, which also lead to slamming, the single free surface has a simpler geometry. The more complicated shape and behaviour of the free surfaces during wave impact is harder to model than for a water-entry problem.
Clearly, the proper formulation of a wave impact problem should be set in 3D. The difficulties associated with 3D descriptions and the solution methodologies that should be followed, discourages investigation. In the review paper of Peregrine [4] the author recognises that ‘In the three-dimensional world at the edge of an ocean, many aspects of the fluid dynamics may differ, so we reconsider some of our assumptions’. Nevertheless, the literature associated with analytical studies in wave impact problems has so far focused almost exclusively on 2D formulations and approximations, relevant to the geometric settings of the most violent impacts. The 2D theory has reached some maturity. Although they consider mainly impact on vertical walls, they have introduced additional model features that could make the results more realistic. In this context, Wood et al. [5] used pressure-impulse theory to investigate the effect of trapped air in breaking wave impact. Using the same method Wood and Peregrine [6] extended their work to study wave impact on a porous berm. Violent breaking wave impact was investigated in a series of papers by Bullock et al. [7] and Bredmose et al. [8], [9]. Bullock et al. [7] presented experimental measurements of pressure from breaking wave tests on vertical and inclined walls. Brendmose et al. [8] considered the effect of the trapped air in the cavity formed by a breaking wave, while in the last study of the same series of papers, Brendmose et al. [9] took into account possible aeration of the liquid. Cooker [10] studied the interaction of a breaking wave with a permeable vertical wall. Experiments on breaking wave impacts on vertical walls were performed by Cuomo et al. [11]. They generated waves that steepened due to a sloped bed that ran up to the wall. The shoaling conditions were arranged to ensure that the steepening of the waves led to impacts as close to 2D as possible. Examples of studies that rely on numerical methods to solve 2D problems of breaking wave on vertical walls are those due to Rafiee et al. [12] and Carratelli et al. [13]. Both studies apply the method of Smoothed Particle Hydrodynamics (SPH), a method that appears to be more flexible than other Computational Fluid Dynamics (CFD) solvers for impact problems.
An important simplification is to treat the face of the breaking wave as parallel to the wall at the instant of impact. For a vertical wall the wave front is considered everywhere vertical, thus ensuring maximum hydrodynamic load. In this context, Korobkin and Malenica [14] studied analytically steep wave impact on an elastic wall, and recently Noar and Greenhow [15] applied the steep wave impact concept to rectangular geometries using pressure-impulse theory. Again, both studies were conducted in 2D. For 2D wave impact problems involving non-classical boundary conditions, such as those determined by porous or perforated surfaces, see the short review paper of Korobkin [16].
By contrast, there are few studies in 3D and they have only employed numerical solvers. One method for generating a wave that collides violently with a structure is that of the so-called dam-break flow. That is, the wave generated by a liquid domain which originally is confined by a barrier and released suddenly. Examples of 3D studies on the subject are those due to Kleefsman et al. [17] and Yang et al. [18]. The former study applied a Volume-of-Fluid (VOF) method to simulate the impact of a dam-break flow on perfectly rectangular bodies. Yang et al. [18] used the unsteady Reynolds equations to simulate near-field dam-break flows, and to estimate the impact forces on obstacles. The cases considered resemble flood-like flows and they cannot be characterized as violent wave impact.
Dam-break flows have also been studied using Smoothed Particle Hydrodynamics (SHP) methodologies. Examples are the studies of Gómez-Gesteira and Dalrymple [19] and Cummins et al. [20] who examined the impact of a single wave, originating from a dam-break, with a tall coastal structure. In both studies, the structure was a vertical rectangular column. To the authors' best knowledge, there have been no 3D studies, even using numerical methods, for more complicated convex geometries, such as circular cylinders. The main difficulty arises from the fact that the impacted wetted region is not known explicitly and must be treated as one of the problem's unknowns. Clearly, relevant difficulties are not encountered when the impacted structure is rectangular with the front face of the wave parallel to one of the plane vertical faces of the rectangular column. It is evident that in this case the boundary of the region hit by the wave is known in advance. The same is true for the contact lines between the liquid and the body, as these coincide with the column's vertical edges.
The present study is a contribution towards 3D approaches on wave impact problems. The physical scenario is a vertical plate, which is subjected to impact by a steep wave which before impact was propagating towards the plate with a constant uniform velocity. The problem is complicated by assuming that the plate has a rectangular opening. The opening (or gap) extends horizontally to the vertical edges of the plate. The positions of the upper and lower edges of the rectangular gap are fixed in time, and we investigate the influence of different positions. At impact the liquid facing the gap is free to discharge through it. Our aim is to find the influence of the gap dimensions on the total force on the structure and the sudden flow through the gap. There are several motivations for having a gap in the plate. One is to extend our earlier work [21] with a single rectangular plate, to treat more realistic structures that may be cracked or contain a designed slot. Another context is a plunging wave impact that captures a thin bubble of constant-pressure air adjacent to the structure, in which of the forward face of the wave is in contact with the structure only above and below the air pocket.
Mathematically, the complexity of the problem originates from the fact that two mixed boundary value problems (MBVP) should be considered in the two directions of the forward face of the wave. To solve simultaneously both problems, the plate is approximated by a degenerate elliptical cylinder with zero semi-minor axis. Having properly formulated one of the two MBVPs, the second (in the vertical direction) yields a one-dimensional MBVP involving triple trigonometric series. It should be mentioned that in contrast to dual trigonometric series, triple series have been only a little studied. The solution provided is based on the transformation of the triple trigonometric series into triple series of integral equations. The solution method allows the derivation of analytical expressions for the velocity potential. One goal is to arrive at expressions that relate the 3D solution to 2D approximations. A second goal of the work is to use pressure-impulse theory to estimate the total impulse on the plate due to the impact.
The study is structured as follows: in Section 2 we formulate the hydrodynamic problem and describe the solution method employed to account for the mixed conditions in the horizontal direction. The solution method relies on the expansion of the velocity potential in elliptical harmonics, as products of the radial and periodic Mathieu functions. Section 3 contains expansions in perturbations to formulate the triple trigonometrical series MBVP. That problem is further analysed in Section 4. In Section 5 the associated MBVP is transformed into a MBVP involving integral equations. The method of solution relies on the reduction of the triple series into a dual series. That is achieved in Section 6. Section 7 is dedicated to the solution of the dual series. Section 8 applies pressure-impulse theory to calculate the total impulse exerted on the plate. Relevant computations are presented in Section 8 followed by a discussion. Finally, the conclusions of the study are presented in Section 9.
Section snippets
The hydrodynamic problem
The fluid in the steep wave moves at constant velocity towards the plate that is situated at X = 0 (Fig. 1). The fluid domain is part of X ≥ 0. The width of the plate is 2L and the water depth is constant and equal to H. A Cartesian coordinate system is defined with its origin fixed on the plate at X = 0, in the centre line of the plate at Y = 0 and on the free-surface on Z = 0. The Z-axis is pointing in the gravity direction and the flat bottom is situated on Z = H. The plate has a rectangular
Expansion of the derivative of the radial Mathieu function in a series of perturbations
Eq. (28) is further elaborated using the series expansions of the modified Bessel functions that can be found in [22; Eqs. (9.7.1)–(9.7.4)]. In particular it holds that where μ = 4k2. Substituting Eqs. (35)–
Mixed boundary value problems involving triple trigonometrical series
Boundary value problems of mixed type have many substantial applications, which aside from hydrodynamics involve rolling mechanics, electrostatics, thermoelasticity etc. (e.g. [24], [25], [26]). Here we consider a special case of mixed boundary value problems, those which involve trigonometrical series. Although mixed boundary value problems leading to dual trigonometrical series are well treated in the literature, there are only a few studies on triple trigonometrical series. The first
Transformation of the trigonometrical series into a system of integral equations
Following the work of Williams [37], the idea of solving MBVPs involving triple trigonometrical series through transformation into integral equations was inspired by Tranter [35]. However, Tranter [35] considered only cases where in the intermediate interval, here indicated by b1 < x < a1, the multipliers of trigonometrical functions (both sine and cosine) have the form (n − 1/2)En. Although that seems insignificant, it does not allow the employment of Tranter's [35] method that was based on
Transformation of the triple integral equations into dual trigonometrical series
Systems like Eqs. (63) and (64) are usually processed by attempting to satisfy by default one of the involved equations in a specific interval and then substituting the assumed solution into the remaining relations. Our procedure exploits the following integral relation [41; Eq. (3)], [38; p.401] for p = ±1/2 and n = 1, 2, 3, ….
For u < a2 the integral yields nonzero values and in particular
Computation of the expansion coefficients of the dual trigonometrical series
Eqs. (72) and (73) form a one-dimensional MBVP that involves dual trigonometrical series. Systems of that kind are well treated in the literature and there have been several authors who provided complete solutions. For a summary the reader is referred to the classical book of Sneddon [32]. However, it should be mentioned that the suggested solutions were derived without being complemented by numerical computations. The solution provided in Sneddon [32] for instance, although accurate, does not
The velocity potential
The method of perturbations suggests taking the following form for the total velocity potential, calculated exactly on the surface of the plate for u = 0
Accordingly, use of Eqs. (19), (40) and (42) requires that
The above are valid in the intervals 0 < z < b, a < z < h, while the
Conclusions
This study dealt with the 3D hydrodynamic impact problem on a vertical plate with a rectangular opening due to the impact of a steep wave. The solution method employed linear potential theory while the hydrodynamic problem was formulated as a boundary value problem of mixed type. The model equations defined two MBVPs for the two directions of the plate. The first MBVP was tackled assimilating the plate as a degenerate elliptical cylinder with negligible semi-minor axis. That allows the explicit
Acknowledgement
The authors are grateful to the EU Marie Curie Intra European Fellowship project SAFEMILLS “Increasing Safety of Offshore Wind Turbines Operation: Study of the violent wave loads” under grant 622617.
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