## Abstract

The three-dimensional water impact onto a blunt structure with a spreading rectangular contact region is studied. The structure is mounted on a flat rigid plane with the impermeable curved surface of the structure perpendicular to the plane. Before impact, the water region is a rectangular domain of finite thickness bounded from below by the rigid plane and above by the flat free surface. The front free surface of the water region is vertical, representing the front of an advancing steep wave. The water region is initially advancing towards the structure at a constant uniform speed. We are concerned with the slamming loads acting on the surface of the structure during the initial stage of water impact. Air, gravity and surface tension are neglected. The problem is analysed by using some ideas of pressure-impulse theory, but including the time-dependence of the wetted area of the structure. The flow caused by the impact is three-dimensional and incompressible. The distribution of the pressure-impulse (the time-integral of pressure) over the surface of the structure is analysed and compared with the distributions provided by strip theories. The total impulse exerted on the structure during the impact stage is evaluated and compared with numerical and experimental predictions. An example calculation is presented of water impact onto a vertical rigid cylinder. Three-dimensional effects on the slamming loads are the main concern in this study.

## 1. Introduction

The present analysis deals with the three-dimensional effects on water impact loads. Such loads are of concern for offshore structures subjected to steep and breaking wave impact [1], tsunami bore forces on coastal structures [2,3], interaction of dry-bed surges and broken waves with buildings [4,5]. Interaction of jets with obstacles is of concern in problems of violent sloshing in liquefied natural gas tanks of NO-96 type [6]. The inner surface of these tanks is manufactured with tongues of length about 3 cm perpendicular to the main flat surface. Impacts of sloshing waves and jets on these tongues may damage the inner structure of the tank. The loads exerted on structures by such water impacts are much higher than any loads associated with propagating gravity waves, but they have short duration. The geometrical configurations of these impact problems are complicated to analyse. The studies of the loads have been performed numerically by computational fluid dynamics (CFD) and smoothed particle hydrodynamics (SPH) methods [7,8], or by laboratory experiments [3].

Theoretical studies (see [1] for review) assume simplified profiles of the wavefront, either vertical, parallel to the surface of the structure, or slightly inclined. In the theoretical models, the flow caused by the impact is assumed to be potential, with viscous, gravity and surface tension effects neglected. It has been shown [1] that the shape of the free surface of an impacting liquid and the shape of the rigid boundaries far from the place of impact provide negligible contributions to the loads. This makes it possible to consider simplified impact configurations using the geometry near the site of impact with flat boundaries in many situations of practical importance. The simplest model relies on the concept of pressure-impulse [1,9]. In this model, the time-dependence of the size of the wetted area of the structure’s surface is not included. It is assumed that the structure is wetted instantly. Only the velocity of the impact and the shape of the impact region are important. This model is well validated against numerical and experimental results [1]. The model has been applied to practical problems in two-dimensional and three-dimensional formulations. However, the finite length of the structure was not included. The finite duration of the impact was taken into account by Korobkin [10] within the two-dimensional Wagner model, where the size of the wetted part of the wall was calculated as part of the solution.

Impacts on a structure by steep, breaking and broken waves are the most dangerous types of impact. Such waves may entrain significant amounts of air before the impact. The water in the impact region should be modelled as an aerated fluid [11]. However, the pressure impulse has been shown to be independent of the air fraction in water [12]. It has also been shown that the maximum stresses in an elastic wall impacted by a breaking wave are also independent of the air fraction in the impact region [13]. Truly three-dimensional problems of water impact are still complicated to study, even within pressure-impulse theory.

Three-dimensional unsteady flow of water impact, in which the impact region is expanding in time, has been solved only for lower half-spaces, with circular and elliptic contact regions on the flat boundary of the flow region, of infinite extent [14,15]. The added mass of a rectangular plate descending suddenly into a flat free surface was computed by Meyerhoff [16]. Three-dimensional impact problems are complicated owing to the mixed type of the governing boundary value problems. Mixed boundary value problems require Neumann and Dirichlet conditions to be satisfied on different portions of the boundary. Analytical studies conducted in three dimensions exist only for special cases of violent impact, i.e. water entry problems [14,15] and for ideal geometries such as elliptic paraboloids [17]. The major problem comes from an additional unknown, which is the time-dependent contact region—the zone in which the liquid is in contact with the structure. It is in the contact region, where a Neumann boundary condition is imposed and outside it a Dirichlet condition. The solutions of mixed boundary value problems are singular at the contact line—the curve that separates the parts of the boundary with different types of boundary condition.

The most realistic model of wave impact is that which considers plunging overturning breaking waves for which the current knowledge relies mainly on experiments [18,19]. This model includes too many parameters, making it not very practical from the point of view of load predictions. The essential parameters are the wave height, water depth and the speed of impact [1]. These parameters lead to a simplified approach known as *steep wave impact*, which assumes a vertical liquid front face at the time of the impact with the structure. Under these conditions, we expect the greatest hydrodynamic loads. In this study, it is suggested to exclude the water depth between the approaching wave and the structure from the model and consider a semi-infinite rectangular liquid region approaching the structure at a constant speed. That is, as a wave approaches the structure the wave trough descends exposing the face of the structure, and the forward face of the wave steepens to become parallel to the structure’s vertical front. Then, the structure surface is dry just before impact. Such a situation was observed in laboratory experiments by Mogridge & Jameison [20]. We assume that the assumption of zero water depth in front of the structure provides useful estimates of the maximum loads.

The most realistic situation that is included in the concept of steep wave impact is the *dam-break flow*, namely a volume of liquid originally at rest and confined by a vertical barrier which is suddenly removed, releasing the liquid. Theoretical studies on dam-break flows are usually performed in two dimensions, without taking into account the presence of a structure in the path of the flow [21–27]. Three-dimensional studies of dam-break flows, known also as dry-bed surges, rely on numerical methods. Abdolmaleki *et al*. [28] simulated the impact on a vertical wall resulting from a dam-break flow. A dam-break flow and its impact on a rectangular obstacle were studied also by Aurelli *et al*. [29]. Both studies were conducted using the Navier–Stokes solver FLUENT. Their common characteristic was that, when the flow met the structure, the flow was too shallow for impact. Yang *et al*. [30] used a three-dimensional numerical model based on the unsteady Reynolds equations to simulate near-field dam-break flows and estimate the impact forces on obstacles. Kleefsman *et al*. [7] applied a volume-of-fluid (VOF) method to simulate the impact of a dam-break flow on rectangular bodies. The cases considered in [7] resemble flood-like flows and they cannot be characterized as violent wave impacts. Ramsden [4] concluded that the dry-bed surges do not lead to impact forces in experiments. However, such forces of large magnitude and short duration were computed by Cummins *et al*. [8]. To the authors’ best knowledge, there have been no three-dimensional studies, even using numerical methods, for complicated convex geometries, such as circular cylinders.

The main idea of this study is to consider the most dangerous scenario of three-dimensional water impact, when the bottom around the structure is dry, or becomes dry before the impact (see Mogridge & Jamieson [20]), and a liquid mass of finite depth (and infinite extent in the horizontal directions) approaches the structure with a vertical front face moving at a constant speed (figure 1). The outlined condition is idealized, and the prediction of the impact loads should be carefully compared with the loads experienced by the structure in realistic conditions.

We will determine the impact loads acting on the vertical rigid plate and compare them with those provided by two-dimensional strip theories. The loads predicted by the strip theories are expected to approximate well the three-dimensional loads in two limiting cases: when the plate width is much larger or much smaller than the water depth. The present formulation approximates the three-dimensional impact on a vertical column of rectangular section, which has been previously studied by CFD and SPH methods. The method of this paper allows us to derive closed-form relations for the pressure-impulse and the total impulse exerted on the plate. This study models the impact conditions in terms of the aspect ratio of the wetted part of the structure, and makes clear the conditions in which the two-dimensional strip theories provide reasonable approximations of the loads.

The method is applied to the time-dependent problem of water impact onto a circular cylinder, where the vertical front of the water region initially just touches the cylinder. Then, the wetted area of the cylinder surface expands in time, finally wetting half of the cylinder. The boundary of the contact region is approximated by straight vertical lines, the positions of which are predicted by two-dimensional theories of water impact with (Wagner’s approach) and without (von Karman’s approach) account for the deformation of the vertical free surface due to the impact. The forces exerted on the cylinder are evaluated within the three-dimensional impact theory. The difference between the forces predicted by the strip theories and the forces from the three-dimensional model was found to be significant.

The study is structured as follows: in §2, we formulate the three-dimensional impact problem and its two-dimensional approximations; the three-dimensional solution of that problem is derived in §3, whereas relevant numerical results and comparisons with two-dimensional strip theory approximations are provided in §4; §5 is dedicated to the calculation of the total impulse exerted on the structure owing to the impulsive pressure. The solution method is extended in §6 to tackle the three-dimensional time-dependent impact problem for a circular cylinder. Time-varying hydrodynamic loads exerted on the circular cylinder are calculated. In §7, the results predicted by the developed method are compared with reported numerical and experimental data. The conclusions are drawn in §8 and the supplementary material of the study is provided in appendix A.

## 2. The hydrodynamic boundary value problem and its two-dimensional versions

The interaction between a liquid volume of thickness *H* and a vertical rigid plate of width 2*L* is considered in the Cartesian coordinate system *x*, *y*, *z* with the origin at the centre of the plate at the water level (figure 1). The liquid volume moves towards the plate at a uniform speed *V* . The liquid occupies the semi-infinite region *x*>0, −*H*<*z*<0 at the time of impact, *t*=0, the time when the forward front face of the wave has met the plate. The plane *z*=−*H* is the rigid bottom and the plane *z*=0 is the horizontal free surface of the liquid before impact. The vertical boundary of the liquid region, *x*=0, consists of the contact region, |*y*|<*L*, between the liquid and the rigid plate, and the liquid free surface, where |*y*|>*L*. The liquid is assumed to be inviscid and incompressible. Gravity and surface tension effects are neglected. During the initial stage of impact, the displacements of the liquid particles are small. This makes it possible to linearize the boundary conditions and impose them on the initial position of the liquid boundary. This approximation is known as pressure-impulse theory [9]. The formulated problem is linear with mixed boundary conditions on the vertical boundary of the flow region. The velocity of the flow caused by impact is described by a velocity potential that changes from *ϕ*_{b} before impact to *ϕ*_{a} after impact [9]. The pressure-impulse is a function of *x*, *y*, *z*, which is defined to be the time-integral of the pressure over the short time interval [0,Δ*t*] of impact.

For small duration of the impact stage, Δ*t*, the pressure-impulse is defined by
*ρ* is the water density. For a uniform flow of speed *V* before impact, *ϕ*_{b}=−*V* *x* and *ϕ*_{a}∼−*V* *x* as *L* to increase with time but for now, we treat the instantaneous flow.

The problem is treated in non-dimensional (tilde) variables, *ϕ*(*x*,*y*,*z*)=*ϕ*_{a}(*x*,*y*,*z*)−*ϕ*_{b}(*x*,*y*,*z*) has the form
*h*=*H*/*L*, which is the aspect ratio of the rectangular plate. Condition (2.3) follows from the linearized dynamic condition on the liquid free surface and the initial condition that the potential *ϕ* of the flow induced by the impact is zero before the impact. Condition (2.5) on the plate, accounts for the fact that the liquid hits the plate with a non-zero normal velocity component, but subsequently cannot penetrate the plate, and stays in sliding contact with the plate. The top and the side views of the plate at the instant of impact are shown in the sketch of figure A1 of the electronic supplementary material.

For a narrow plate, *h*≫1, the potential is approximately independent of the vertical coordinate *z* far from the upper free surface. Correspondingly, for a very wide plate, *h*≪1, the potential is independent of the transverse coordinate *y* far from the edges of the plate. These approximations correspond to the strip theory solutions, *ϕ*^{(h)}(*x*,*y*) and *ϕ*^{(v)}(*x*,*z*), where the superscripts (h) and (v) indicate that the strips are respectively horizontal and vertical. According to strip theory, the three-dimensional plate is discretized vertically or horizontally by narrow strips perpendicular to the much longer dimension of the plate. Thus, the *z*- or the *y*-variation of the potential can be omitted, and the problem is reduced to two dimensions. The potential *ϕ*^{(h)}(*x*,*y*) on the plate, within the strip theory approximation, is given by [16,31]
*z*=0, where the potential should approach zero as specified by condition (2.3). The following vertical strips approximation of the potential, *ϕ*^{(v)}(*x*,*z*), on the plate, *x*=0, −*h*<*z*<0, has been given in references [9,32]; it is
*y*=±1.

We shall obtain the solution of the three-dimensional problem (2.2)–(2.6), using series comprising trigonometric and Mathieu functions. Afterwards, we compare the potential *ϕ*(0,*y*,*z*) on the vertical plate with approximations (2.7) and (2.8) in order to find the ranges of the parameter *h*, in which these approximations can be used. In §6, we use these results to solve the problem of water impact on a vertical circular cylinder, in which the half-width *L* of the wetted region increases with time in a way that depends on the shape and curvature of the body being hit by the water front.

## 3. Solution of the three-dimensional problem of wave impact

The solution of the problem (2.2)–(2.6) is sought in the form
*ϕ*_{n}(*x*,*y*) satisfy the following mixed boundary value problem of modified Helmholtz type, in which subscripts *x* and *y* denote partial derivatives:

To find the solution of (3.2)–(3.5), we consider the interval *x*=0, −1<*y*<1, as a degenerate ellipse with semi-axes *a* and *b*, where *a*→1 and *b*→0. A sketch of the plate represented by a degenerate elliptical cylinder is shown in figure A2 of the electronic supplementary material.

We use two-dimensional elliptical coordinates (*u*,*v*), defined by *u*_{0}<*u*, 0≤*v*≤*π*, where *u*=*u*_{0} is the surface of the elliptical plate. For a plate of zero thickness, we consider the limit as *a*→1,*b*→0,*c*→1 and *u*_{0}→0. Equation (3.2) has the following form:
*ϕ*_{n}(*u*,*v*)=*U*_{n}(*u*)*V* _{n}(*v*), equation (3.6) provides two differential equations, for *U*_{n}(*u*) and *V* _{n}(*v*)
*μ*_{n}=−(λ_{n}/2)^{2} and *σ* are the separation constants of the Mathieu equations. The former is known as the Mathieu parameter. The boundary condition (3.3) provides *ϕ*_{n}(*u*,0)=*ϕ*_{n}(*u*,*π*)=0 and yields the boundary conditions for equation (3.8), *V* _{n}(0)=*V* _{n}(*π*)=0. Non-trivial solutions of the periodic Mathieu equation (3.8) with zero boundary conditions exist only for certain values of *σ* known as the characteristic values of (3.8) [34]. These non-trivial solutions are known as the odd periodic Mathieu functions *se*_{2m+1}(*v*,−*q*_{n}), where *q*_{n}=−*μ*_{n} and *m* is a non-negative integer. The corresponding characteristic values are denoted by *σ*, and decaying as *u* denotes partial derivative. To satisfy (3.9), we write
*u*-derivative. Multiplying (3.11) by *se*_{2k+1}(*v*,−*q*_{n}), integrating from *v*=0 to *v*=*π*, and using the orthogonality relation of the periodic Mathieu functions ([34]; 20.5.3), we arrive at the following formula for the expansion coefficients *π*] and [0,*π*] intervals.

The integral in (3.12) is evaluated using the series expansion ([35], 8.611.3)
*q*_{n}. Then
*ϕ*_{n}(*u*,*v*) are given by
*u*=0, where

Note that *q*_{n} is related to *h* by *q*_{n}=[*π*(2*n*−1)/(4*h*)]^{2}. Taking the limits as *h*→0, we exploit the asymptotics of *q*_{n}→0 and

## 4. Numerical results

The double series in (3.16) are truncated, 0≤*m*≤*M* and 1≤*n*≤*N*, and evaluated using purpose-made routines for computing the coefficients

Figure 2 shows the vertical distribution of the velocity potential at the vertical section *y*=0.5, given for different values of *h*. The associate curves at the vertical section *y*=0 are shown in figure A4 of the electronic supplementary material. The potential is symmetric with respect to *y*=0. The three-dimensional solution tends to the two-dimensional solution (2.7) for *h*>100. Also, figure 2 shows the velocity potential calculated by the second two-dimensional approximation (2.8), valid for *h*≪1. Here, only the case *h*=0.5 is shown. It is seen that (2.8) provides a good approximation of the potential for *h*≪1 and far from the plate edges. Near the plate edges, three-dimensional effects are important for any *h*.

The range of validity of the approximate solution (2.8) is shown in figure 3. Figure 3 corresponds to the horizontal section at *z*/*h*=−0.5. The associate results for *z*/*h*=−1 are shown in figure A5 of the electronic supplementary material. The plots demonstrate that (2.8) can be used for *h*<1/10 over 80% of the length of the plate but is invalid near the plate edges.

## 5. Total impulse on the plate owing to the impulsive pressure

The velocity potential *ϕ* given by (3.16) is used in this section to evaluate the total impulse acting on the vertical rigid plate owing to water impact. We study the dependence of the total impulse on the plate dimensions for different aspect ratios. The total impulse on the plate arises from the integration of the impulsive pressures over the impacted area. Cooker & Peregrine [9] define the pressure-impulse as the time integral of the pressure, over the short time interval of impact. According to Bagnold [36], the pressure-impulse (at a given point) is approximately constant. This is partly confirmed by the measurements of Richert [37] and suggests that the pressure-impulse is a better physical quantity to model than the peak pressure. Cooker & Peregrine [9] define the pressure-impulse as
*p* is approximated by the time derivative of minus the velocity potential times the water density. In accord with (5.1), the pressure-impulse *P*(*x*,*y*,*z*)=−*ρϕ*(*x*,*y*,*z*), where *ϕ*(*x*,*y*,*z*) is the change in velocity potential brought about by the impact. The total impulse, *I*, is the integral over the contact region of the pressure-impulse. Hence
*ρV* *L*^{3}. The velocity potential on the plate in the three-dimensional problem is given by (3.16) and the two-dimensional solutions for *h*≫1 and *h*≪1 by (2.7) and (2.8), respectively. Substituting (2.7) and (2.8) successively in (5.2), we obtain the following asymptotic formulae for *I*(*h*):
*ξ*(3)=1.202056903. The superscript (h) in (5.3) indicates that this formula is provided by the horizontal strip theory. The superscript (v) in (5.4) indicates that the vertical strip theory is used.

Substituting the exact formula (3.16) in (5.2) and using the associated negative Mathieu parameter expressions ([34]; 20.8.4 and 20.8.5), we obtain the total impulse as a function of *h*:
*h*-dependence of *I*(*h*) in (5.5) comes also through the Mathieu parameter *q*_{n}. The total impulse (5.5) and the two-dimensional approximations (5.3) and (5.4) are plotted in figure 4 as functions of the aspect ratio *h*. The range of interest is 0<*h*<1. Figure 4 demonstrates that the small-*h* approximation (5.4) can be used only for 0<*h*<0.3. Both approximations, *I*^{(h)}(*h*) and *I*^{(v)}(*h*), exceed *I*(*h*) in 0<*h*<1 owing to the three-dimensional end-effect near the upper free surface for *I*^{(h)}(*h*) and the end-effect near the vertical edges of the plate for *I*^{(v)}(*h*). The graph for *h*>1 is shown in figure A6 of the electronic supplementary material. We note that *I*^{(h)}(*h*) and *I*(*h*) differ by a constant value of order *O*(1) owing to the three-dimensional end-effect near the upper free surface, which is not accommodated by the horizontal strip theory.

Expression (5.5) is complicated to evaluate, so we present polynomial approximations with a relative error of less than 2%, for small, medium and large *h*

## 6. Three-dimensional water impact onto a vertical circular cylinder

The results of the previous sections, which were obtained from the pressure-impulse theory, can be applied to the three-dimensional unsteady problem of water impact onto a vertical cylinder with constant and smooth horizontal cross sections. The configuration of the problem is the same as in figure 1 but with the plate replaced with a cylinder. A circular cylinder of radius *R* is considered here but the approach can also be applied to other smooth cylindrical shapes. In this section, the original dimensional variables are used.

The early stage of water impact on the circular cylinder of radius *R* is studied within both the von Karman and the Wagner [38] models of the water impact (figure 5). The models are valid during the initial stage, when the displacement of the wavefront along the *x*-axis, *Vt*, is much smaller than the radius *R*, where *V* is the speed of the water front. We shall first analyse the Wagner model. The von Karman model is then a simplification of the Wagner model.

We assume that the horizontal dimension of the wetted part of the cylinder which is of order *H*. Hence, *V* *tR*=*O*(*H*^{2}). Then, the condition *V* *t*≪*R* implies that *H*≪*R*. Therefore, the three-dimensional Wagner model can be used for vertical cylinders or panels of small curvature where the horizontal dimension of the contact region is of the same order as the water depth.

Within the Wagner model of water impact, the boundary conditions are linearized and imposed on the initial positions of the liquid boundaries at the instant of impact, *t*=0 [38]. The size of the wetted part of the cylinder, where the linearized body boundary condition, *ϕ*_{x}=*V*, is imposed, is governed by the Wagner condition. This condition implies that the vertical boundary of the water front is continuous at contact lines, *y*=±*b*(*z*,*t*), where *b*(*z*,0)=0. Note that, in general, the unknown function *b*(*z*,*t*) depends on the vertical coordinate *z*. That is, the contact region is bounded below by the bottom, *z*=−*H*, bounded above by the upper free surface, *z*=0 and by curvilinear boundaries *y*=±*b*(*z*,*t*) at the right- and left-contact lines.

Let *x*=*ζ*(*y*,*z*,*t*) be the displacement of the wavefront outside the contact region, where |*y*|>*b*(*z*,*t*). Then, the Wagner condition provides the equation for the unknown function *b*(*z*,*t*),
*ϕ*(*x*,*y*,*z*,*t*) in the flow region, the kinematic and dynamic boundary conditions on the free surface and the body boundary condition *ϕ*_{x}=*V* in the impact region, *x*=0, |*y*|<*b*(*z*,*t*), −*H*<*z*<0. The displacement of the initially vertical free surface, *x*=*ζ*(*y*,*z*,*t*), is given by the linearized kinematic boundary condition, *ϕ*_{x}=*ζ*_{t}, where *x*=0, |*y*|>*b*(*z*,*t*). The Wagner condition can be readily satisfied in two-dimensional [39] and axisymmetric [40] impact problems. In truly three-dimensional problems of water impact, this condition is difficult to satisfy [40]. Analytical solutions of the three-dimensional impact problems are known only for elliptic contact regions [14,17]. In the von Karman model, the displacement of the free surface, *ζ*(*y*,*z*,*t*), is not taken into account in the calculation of the size of the contact region. That is, the left-hand side in (6.1) is set zero that gives the width of the contact region independent of the vertical coordinate *z* within the von Karman model.

During the early stage, when *b*≪*H*, we assume that the size of the contact region can be well approximated by the corresponding two-dimensional solution (independent of *z*) far from the upper free surface. This two-dimensional solution suggests
*b*_{w}(*t*) satisfies the following equation [41]

The function *E* in (6.3) denotes the complete elliptic integral of the second kind. The approximation (6.3) is not valid near the upper free surface at *z*=0. The dynamic boundary condition, *ϕ*=0, on the upper free surface, *z*=0, implies that the horizontal components of the velocity, *ϕ*_{x} and *ϕ*_{y}, are zero there. They are also zero at the edge of the upper free surface, where *ϕ*=0, *x*=0, *ϕ*_{x}(0,*y*,*z*,*t*)=*ζ*_{t}(*y*,*z*,*t*) on the edge *z*=0 and *x*=0, implies *ζ*_{t}(*y*,0,*t*)=0 and the Wagner condition (6.1) yields

We assume that at each time instant *t* the function *b*(*z*,*t*) starts from the value (6.4) at *z*=0 and increases monotonically with the distance from the upper free surface approaching *b*_{w}(*t*) given by (6.3) as *z*→−*H*, if the water is deep enough. We cannot prove this statement at this stage. To prove it, one needs to solve the three-dimensional problem of water impact on a vertical cylinder, find the function *b*(*z*,*t*) as part of the solution and compare it with the function *b*_{w}(*t*) from (6.3).

The assumption *b*(0,*t*)≤*b*(*z*,*t*)≤*b*_{w}(*t*), where −*H*≤*z*≤0, allows us to estimate the hydrodynamic force *F*(*t*) exerted on the circular cylinder. It is assumed that
*F*_{0}(*t*) and *F*_{w}(*t*) are the forces obtained for the contact regions |*y*|<*b*(0,*t*) and |*y*|<*b*_{w}(*t*), respectively, where −*H*≤*z*≤0. Note that *b*(0,*t*) and *b*_{w}(*t*) (the von Karman and the Wagner dimensions of the contact region) are known functions of time from (6.3) and (6.4) such that *b*(0,0)=0 and *b*_{w}(0)=0.

It is convenient to introduce the non-dimensional variables
*B*(*t*)=*b*(0,*t*) or *B*(*t*)=*b*_{w}(*t*) and the wetted part of the cylinder is |*y*|<*B*(*t*), −*H*≤*z*≤0. The potential *h*=*H*/*B*(*t*). Note that the time *t* plays the role of a parameter in the linearized hydrodynamic problem with respect to the velocity potential *ϕ*(*x*,*y*,*z*,*t*). Therefore, the non-dimensional potential *F*_{B}(*t*) calculated within the linearized model with the width of the contact region being 2*B*(*t*) is given by
*ϕ* is zero along the lines *y*=±*B*(*t*), −*H*≤*z*≤0. The double integral in (6.7) is equal to −*I*(*h*), where the total impulse, *I*(*h*), introduced by (5.2) is given by (5.5). Therefore,
*h*=*H*/*B*(*t*). The total impulse *I*(*h*) was investigated in §5 and *I*^{′}(*h*) is found numerically. During the early stage, *B*(*t*)≪*H*, formula (5.3) implies *I*(*h*)∼*πh*/2 as *B*(*t*)*H* is the impact area at time *t*. The hydrodynamic force can be normalized as
*F*_{B,sc}(*t*) is given by the right-hand side of (6.9). Equation (6.10) shows that the non-dimensional force *h*. Here, *h* → 0, as follows from (5.4). The forces *B* = *b*(0,*t*) and *B* = *b*_{w}(*t*) provide bounds for the hydrodynamic force *F*_{B}(*t*)/*F*_{B,sc}(*t*) computed for the three-dimensional contact region within the Wagner approach. The function *h* → 0.

The two-dimensional approximation of the hydrodynamic force (6.9), normalized by (1/2)*πρV* ^{2}*HR* reads
*τ*=*V* *t*/*R* is the scaled time. Taking *B*=*b*(0,*t*) for the von Karman model, we find
*B*=*b*_{w}(*t*) we use (6.3) and, after some algebra, we obtain
*K*(*B*/*R*) denotes the complete elliptic integral of the first kind. Equations (6.12) and (6.13) provide the two-dimensional von Karman and Wagner approximations, respectively, of the force exerted on a circular cylinder. The force (6.13) was derived by Korobkin [41] (figure 4), with a different normalization. The two-dimensional approximations (6.12) and (6.13) are compared with the three-dimensional force (6.8). Using (6.8) and (6.10), it can be shown that the hydrodynamic force is
*ε*=*H*/*R* denotes the aspect ratio of the circular cylinder. The term in brackets is given by (6.12) and (6.13) for the von Karman and the Wagner models, respectively. Figure 7 compares the forces predicted by the two-dimensional approximation and the present three-dimensional method for *ε*=1, using both models of wave impact. As implied by (6.12) the two-dimensional force *f*_{B} from the von Karman model with *B*=*b*(0,*t*) decreases linearly from 2 with increasing *τ*. It is interesting to observe that the two-dimensional Wagner approach for *B*=*b*_{w}(*t*) leads to a nearly linear decay of the force. In both cases, the maximum non-dimensional force occurs at the first instant of impact and then decay, whereas the contact region widens. As expected, the Wagner approach leads to larger hydrodynamic loads than the von Karman approach owing to the larger area of the contact region.

The results obtained using the present method, in particular (6.14), show a notable difference between the two-dimensional and three-dimensional forces for both models employed. However, the two-dimensional approximations reproduce well the maximum loading that occurs at *τ* = 0. The three-dimensional forces are smaller and decay much quicker with time than their two-dimensional approximations. The forces from the two-dimensional strip approach overestimate considerably the actual loading exerted on the circular cylinder.

## 7. Validation

The three-dimensional theory of water impact developed in this paper can be applied to structures with either flat or convex surfaces. To validate this theory, the theoretical total impulse is compared with that for the three-dimensional wave impact onto a rectangular column with the wavefront resembling the face of a steep wave, see [8]. The simulated case concerned a dam-break flow impact onto a rectangular column with square cross section 0.12×0.12 m. The distance between the column and the dam was 0.5 m. The water front arrived at the column about 0.25 s after the flow starts. Cummins *et al*. [8] studied this problem numerically by an SPH method and compared the forces and the total impulse with experimental results. The total impulse on the column was evaluated by
*F*(*τ*)| is the magnitude of the hydrodynamic load exerted on the column. Figure 5 in [8] shows a nearly vertical front face of the dam-break wave at the instant of impact, the height of which is estimated to be *H*=0.16 m which leads to the aspect ratio *h*≈2.6 in our theory. It is difficult to extract the velocity of the wavefront from both the numerical results and the experimental measurements presented in reference [8]. The velocity of the wavefront is assumed equal to the critical velocity *g* is the gravitational acceleration. Using the scale of the total impulse, 0.28 Ns, and approximation (5.6) for the non-dimensional total impulse, *I*(2.6)≈3.15, the dimensional total impulse is 0.88 Ns, whereas figure 8 in Cummins *et al*. [8] estimates the total impulse, up to the instant of the maximum loading, to be about 0.9 Ns. This favourable agreement verifies the efficacy of the developed three-dimensional impact theory of this paper.

## 8. Conclusion

The three-dimensional hydrodynamic loads exerted on a rigid plate of finite dimensions by the impact of a rectangular liquid region have been studied within the pressure-impulse theory. The three-dimensional distributions of pressure-impulse over the plate have been determined. The effect of the plate aspect ratio on the pressure-impulse distributions has been studied. Both the pressure-impulse and the total impulse (the time-integral of the hydrodynamic force) are strongly dependent on the three-dimensional effects. The two-dimensional approximations of the total impulse provided by the strip theory can be used with vertical strips if the water depth is one-tenth of the width of the plate, and with horizontal strips and a correction by an additive constant (of value about −1.12) if the water depth is three times larger than the width of the plate.

The developed three-dimensional pressure-impulse theory has been extended to tackle the three-dimensional unsteady problem of water impact onto a circular cylinder. Both the von Karman and the Wagner models of water impact have been employed. The new model of three-dimensional impact assumes straight vertical contact lines provided either by the von Karman or the Wagner two-dimensional models. The pressure distribution in such contact regions is given by the three-dimensional theory. This new three-dimensional model provides bounds on the hydrodynamic force. Closed-form relations have been derived for both two-dimensional and three-dimensional approaches using both models of wave impact. It has been shown that the three-dimensional hydrodynamic force exerted on the cylinder and normalized by the force provided by the two-dimensional strip theory approximation, is a function of only the horizontal dimension of the contact region. In addition, it has been shown that the two-dimensional approach significantly overestimates the hydrodynamic load on the cylinder. This finding concerns both models of impact. Further, it was found that the maximum loading occurs at the start of impact, and is well reproduced by the two-dimensional approaches. The three-dimensional forces decay with time much faster than their two-dimensional approximations.

This study of three-dimensional effects on impact loads is limited to simplified impact geometry. However, the methods and estimates obtained are expected to be valuable for practical applications such as tsunami bore, dry-bed surges, breaking and broken wave impact on coastal and offshore structures.

## Data accessibility

This work does not have any experimental data.

## Authors' contributions

M.J.C. and A.A.K. suggested the problem and conceived the mathematical formulation. All authors contributed to deriving the solutions. I.K.C. developed the computer code and made the computations. All authors gave final approval for publication.

## Competing interests

We declare we have no competing interests.

## Funding

This work was supported by the EU Marie Curie Intra-European Fellowships (FP7).

## Acknowledgements

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 no. 622617.

- Received December 12, 2015.
- Accepted July 1, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.