Three-dimensional steep wave impact on a vertical cylinder*

Ioannis K. CHATJIGEORGIOU， Alexander A. KOROBKIN， Mark J. COOKER

1. School of Mathematics， University of East Anglia， Norwich， NR4 7TJ， UK

2. School of Naval Architecture and Marine Engineering， National Technical University of Athens， 15773， Greece，

E-mail:chatzi@naval.ntua.gr



Three-dimensional steep wave impact on a vertical cylinder*

Ioannis K. CHATJIGEORGIOU1，2， Alexander A. KOROBKIN1， Mark J. COOKER1

1. School of Mathematics， University of East Anglia， Norwich， NR4 7TJ， UK

2. School of Naval Architecture and Marine Engineering， National Technical University of Athens， 15773， Greece，

E-mail:chatzi@naval.ntua.gr

In the present study we investigate the 3-D hydrodynamic slamming problem on a vertical cylinder due to the impact of a steep wave that is moving with a steady velocity. The linear theory of the velocity potential is employed by assuming inviscid，incompressible fluid and irrotational flow. As the problem is set in 3-D space， the employment of the Wagner condition is essential. The set of equations we pose， is presented as a mixed boundary value problem for Laplace's equation in 3-D. Apart from the mixedtype of boundary conditions， the problem is complicated by considering that the region of wetted surface of the cylinder is a set whose boundary depends on the vertical coordinate on the cylinder up to the free-surface. We make some simple assumptions at the start but otherwise we proceed analytically. We find closed-form relations for the hydrodynamic variables， namely the time dependent potential， the pressure impulse， the shape of the wave front （from the contact point to beyond the cylinder） and the slamming force.

3-D impact， violent slamming， impulse pressure， integral equations

Introduction

The study of violent slamming due to breaking wave impact is very important for assessing the hydrodynamic loading exerted on both coastal and offshore structures. Breaking waves are often approximated as steep waves， as that configuration is considered the worst case of loading. In slamming dynamics it is important to model the early stage of impact， during a very short period of time after the first contact. Even when this time is infinitesimal， and the change in the velocity field is abrupt， the analytical approximation is a very difficult task especially when the domain of the fluid velocity is fully 3-D.

Slamming in marine applications can be encountered in several forms， e.g.， steep wave and breaking wave impact， wet deck slamming， green water slamming and water entry. Water entry problems appear to be a popular subject of investigation. Relevant problems have been studied both experimentally［1-4］and theoretically. Theoretical studies related to wave impact， rely mainly on purely numerical procedures， say Navier-Stokes solvers［5，6］， and not on analytical approximations due to the complications associated with the particular problems. Nevertheless， numerical approximations of that kind require huge computer storage capacities to accommodate very fine space-time grids，they are very time consuming and are prone to errors in resolving the free-surface and the transient pressure field. In addition， the numerical treatment of slamming phenomena is going through a series of simplified problems that possess fundamental characteristics of behavior of the flow［5］. It is evident therefore that an approximate analytical treatment can supplement numerical results and help us to interpret the problem with resolution.

Steep or breaking wave slamming problems are not as popular as the problems associated with gravity waves. There is much more literature on gravity waves than on wave slamming. Especially for bottom fixed vertical cylinders in waves， the developed theoretical methodologies have been extended to the second- and third-order problems. In this context one must not forget to mention the classical studies of Kim and Yue［7，8］and Malenica and Molin［9］. By contrast， thereare few studies of 3-D steep or breaking wave impact due to the difficulty of treating mathematically such problems. The associated difficulties are as follows: （1）the liquid at the time of impact does not cover the complete control volume and therefore there is not a global formulation such as in gravity waves， and （2）these kinds of concepts result in mixed boundary value problems， which are difficult to solve compared with Dirichlet or Neumann problems. As an example it is mentioned that in English there are only three books devoted to 3-D mixed boundary value problems， those of Sneddon［10］， Fabrikant［11］and Duffy［12］.

Nowadays， 2-D wave impact problems on vertical structures， e.g.， walls， are well treated in the literature. For the global theory of the water impact onto vertical walls that relies on the pressure impulse concept， the reader is referred to studies of Cooker and Peregrine［13］and Peregrine［14］. The research has been extended to more complicated configurations such as impact on an elastic wall［15］， breaking wave impact［16］，breaking wave impact on permeable barriers［17］and wave impact on perforated structures［18］. It is evident that 2-D approaches are relatively efficient for vertical rectangular structures. Nevertheless， 2-D approximations are insufficient for realistic three dimensional geometries， such as axisymmetric bodies and vertical cylinders with an elliptical uniform cross-section.

Clearly， for vertical cylinders subjected to steep wave impact a Wagner boundary condition must be taken into account. This is due to the fact that in reality the wave front that impacts the structure obtains a curved configuration from the time varying contact line as first suggested by Wagner［19］. This is an additional complexity associated with the problem at hand. It is also true that there are few analytical studies on 3-D wave impact on structures in the literature. Among the very few that consider water entry problems， are those due to Scolan and Korobkin［20，21］and Korobkin and Scolan［22］.

In the present study we consider a different 3-D problem of wave impact. The subject of investigation is a vertical circular cylinder that experiences the impact of an incident wave that before impact was moving with constant velocity. The linear theory of a velocity potential is employed by assuming inviscid，incompressible fluid and irrotational flow. As the problem is set in 3-D space， the employment of the Wagner condition is essential. The set of equations we pose is presented as a mixed boundary value problem for Laplace's equation in 3-D. Apart from the mixedtype of boundary conditions， the problem is complicated by considering that the region of wetted surface of the cylinder is a set， whose boundary depends on the vertical coordinate on the cylinder， up to the free-surface. We make some simple （but valid） assumptions at the start but otherwise we proceed analytically. The most important assumption we make is that the boundary curve of the wetted surface on the cylinder's contact line does not vary significantly in the vertical direction: it moves laterally around the cylinder. This assumption is employed mathematically by assuming that the first （and accordingly the second） derivative with respect to the vertical coordinatez is very small compared with unity. This allows us to treat a set of 2-D problems， see Fig.1. We find closed-form relations for the hydrodynamic variables， namely the timedependent potential， the pressure impulse， the shape of the wave front （from the contact point to beyond the cylinder） and the slamming force.

1. The boundary value problem

Our purpose is to investigate the steep wave impact on a cylinder taking into account the impacted width that varies with time and the vertical coordinate. The assumption is made that the wetted region of the cylinder is a vertical strip that widens while the impact advances， starting as a line segment and broadening as time increases.

The problem at hand is considered with the aid of the schematic in Fig.1. The steep wave impact， can be approximated by the equivalent “reversed” problem of the cylinder moving against a stationary wave （rectangular） front with constant， given velocityV . At the beginning of time t=0， the wave front （steep wave）nearly touches the outer surface of the cylinder. As t→0+the cylinder penetrates the fluid. The steady advance of the wave face （Von Karman's approach）assumes the contact point lies atx=0without taking into account the cylinder's influence on the position of the free-surface［x=-η（y， z； t），y=±b（ z； t）］ast→0+. From Fig.1 the Wagner contact point is denoted byy=b（ z； t），x=-η（b， z； t）where bis one of the unknowns of the problem.

Fig.1 Sketch for notations. Plan view. Thez axis increases towards the viewer. The free-surface asymptotes to the y -axis

We investigate the short period of time after initial impact between the wave and the cylinder， i.e.，t→0+. In thex，yframe of reference of Fig.1， the fluid is at rest at infinity. The circular cylinder moves vertically downward at constant speedValong the positive x -axis. The free-surface approaches the yaxis asy→∞. The boundary value problem is symmetric with respect to the x-axis so we treat the quarter plane shown in Fig.2. During the initial stage of the impact， when the displacement of the wave front is small compared with the size of the wetted area， the boundary conditions of the hydrodynamic model can be approximately linearized and imposed on the initial position of the liquid boundary.

Fig.2 Quarter-plane linearized fluid domain （-1　＜　z＜0）

The reduction of the half-plane of Fig.1 to the quarter-plane of Fig.2 is realized mathematically by requiring that the y-component of the velocity of the liquid vanishes aty=0. The governing hydrodynamic problem is described by the following system.

The Laplace equation，

in the linearized fluid domain. Note that z=-1denotes the bottom andz=0the upper free-surface. The linearized dynamic condition on the free-surface， namely

and

The kinematic condition on the flat bottom

Assuming a small penetration depth， the linearized kinematic condition on the cylinder is

The symmetry kinematic condition on y =0is

The Wagner condition is that the free-surface meets the surface of the cylinder. This must be satisfied on y=b（ z； t ）which is expressed as

and finally the condition at infinity

The free-surface behaviour of the wave front （see Fig.1） is obtained by employing the free-surface kinematic condition on the wave face posed at x=0

The initial conditions of the boundary value problem of Eqs.（1）-（9） at t =0are φ=η=b=0.

2. Solution of mixed boundary value problem

2.1 An approximate solution

Using separation of variable solutions of Eq.（1）we can write an expression forφ， which satisfies Eqs.（2）， （4）， （6） and （8）， as follows

where

and ξn（u） are functions to be determined. Note that the unknown expansion coefficients Bnare functions oft. In general，ξnis also a function oft.

The conditions remaining to be satisfied are Eqs.（3）， （5） and the Wagner condition， Eq.（7）. Next，we will focus on Eq.（3） and Eq.（5）， whilst the Wagner condition will be considered at the end of the analysis. Introducing Eq.（10） into Eqs.（3） and （5）， yields

Both Eqs.（12） and （13）， are valid in the interval -1＜z＜0and this notation will be omitted in the following for brevity.

Equations （12） and （13）， are next recast using Abramowitz and Stegun［23］

where Jvis the Bessel function of the first kind with fractional orderv . Belowvis equal to -1/2. Thus Eqs.（12） and （13）， imply the following pair of equations

Equations （15）， （16） form a combined dual Fourier Bessel-trigonometrical series. There have been studies in the past that considered dual Fourier Bessel series［24］or dual trigonometrical series［25-27］but to the best of our knowledge the complicated form of Eqs.（15）， （16）was never studied. For a review of dual trigonometrical series or dual Fourier-Bessel series， the reader can refer to the books of Sneddon［10］and Duffy［12］.

Progress in solving the system of Eqs.（15）， （16）can be taken forward by first satisfying one of the two relations. The analysis should be performed with extreme caution as a particular selection for the function of a ξn（u） does not necessarily guarantee the existence of the integrals on u∈［0，∞）in Eqs.（15）， （16）.

First we satisfy Eq.（16）. To this end we exploit a useful relation found in Gradshteyn and Ryzhik［28］， or Watson［29］， Eq.（1）

In Eqs.（17a）， （17b） we have converted the original relation to comply with our symbols. The only requirement for the validity of Eqs.（17a）， （17b） is Re（µ）= Re（ v）=-1/2＞-1. In order to take into account all possible solutions of Eqs.（17a）， （17b） and accordingly to construct the functionξn（u）， we specifically chooseµto be integer and positive. Next， we assume

where Cmare arbitrary expansion coefficients. Since b is a function ofzwe must assume that its derivatives （first and second） with respect tozare negligible. That ensures that Eq.（10） still satisfies Laplace's equation.

Further， Eq.（18） is substituted into Eq.（16） which after rearranging terms yields

where we assume that Anm=BnCm. Indeed Eq.（19） is valid due to Eq.（17） and accordingly Eq.（16） of the mixed boundary value problem has been satisfied.

The next step is to substitute Eq.（18） into Eq.（15）. Doing so， one gets

We do not have a closed-form expression for the integral of Eq.（20） so we calculate this integral numerically. The integral is divergent for m =0and m=1. This can be easily verified by taking the limit values of the Bessel functions for large arguments. Hence in order to have a conservative and convergent solution we exclude these terms from our solution by making An0=An1=0in Eqs.（19）， （20）. For orders m≥1the integral decreases very fast for increasing m and forall modes λn. To show this we provide Fig.3 that depicts the values of this integral for the first ten modes λn（see Eq.（11））， for m=2， 3 and 4 and a specific value of b =0.1095.

Fig.3 Variation of the infinite integral in Eq.（20） against λn（n =1-100）for m =2， 3 and 4

It is evident that the values of the infinite integral in Eq.（20） for m=3，4，…are practically negligible compared to the results withm=2. Consequently we can simplify the functionξn（u） retaining only the term m=2. The radical reduction of the series has an additional side-effect， to avoid possible numerical inaccuracies associated with the very small values of the integral asnincreases. Thus the potential of Eq.（10）can now be written as

where we assumed µ=2whilst Anis literally An2. Also， Eq.（20） is rewritten as

Our next goal is to calculate the expansion coefficients An. Clearly， the eigenfunctions sin（λnz）are orthogonal in the interval ［-1，0］. However， the associated orthogonality relation cannot be employed in the current form of Eq.（22） asbis also a gradually varying function ofz . In the end we will employ the orthogonality relation satisfied by sin（λnz）， but only after manipulating Eq.（22）. First we normalizeyby b letting y=br，0≤r≤1. Thus Eq.（21） is transformed into

Accordingly， both sides of Eq.（23） are multiplied by rv+1and the final products are integrated with respect tor in the interval ［0，1］. That is， we do not satisfy Eq.（23） for each point in the interval［0，1］but this equation is satisfied only in some global sense as it is explained above. Thus， we find

The integration on the right-hand side yields unity as v=-1/2. Using of Gradshteyn and Ryzhik［28］we find that the inner integral on the left-hand side of Eq.（24） is

Equation （25） is valid for all v＞-1and we have alreadyv=-1/2. Substituting Eq.（25） into Eq.（24）leads to

Equation （26） refers to the interval -1＜z＜0. Nevertheless， the orthogonality relation is still inapplicable and accordingly Eq.（26） must be simplified to a certain extent. To this end we assume that the wetted part of the cylinder has a half-widthb（ z； t）， which varies slowly with respect toz， for allt， in the following way

where the coefficient b0（ t ）is explicitly determined by the von Karman's approach （no free-surface spreading）（see Fig.1）. For a circular cylinder of radiusR the von Karman's section b0（ t ）is given by

Also，b1/2in the right hand side of Eq.（26） is approximated by

Recall thatεis a function ofz alone and b0is a function oft alone. The occurrence of a small scaling factor in the analysis suggests assuming a perturbation series expansion for the unknown coefficients Anas well. Thus Introducing Eq.（27） and Eqs.（30）-（33） into Eq.（26）and equating ε0and ε1powers one gets forand）

respectively. We can now carry out a Fourier analysis on z∈［0，1］for Eqs.（34）， （35）. It is noted that the orthogonality constant for sin（λnz）is 1/2， whilst

Accordingly the expansion coefficientsandsatisfy the pair of relations

The derivation of the expansion coefficientsanddoes not depend onz . The dependence onthe vertical coordinatez is involved via the perturbation expansion of Eq.（33） where the scaling factor εis a function ofz . However，andare still functions of time. In order to employ the Wagner condition of Eq.（7）， we must first determine the free-surface displacement through Eq.（9） which suggests that

In other words， the numerical implementation of the outlined solution process requires the calculation of the expansion coefficientsandat discrete time steps in the interval［0，t］in order to derive scalar functions and then employment of a numerical method for the integral involved in Eq.（39）. In more detail the free-surface displacement at y=b（ z； t）as required by the Wagner condition （7） will be given by

Equation （40） combined with the Wagner condition of Eq.（7）， forms a nonlinear equation to be solved in terms of the unknown scaling factor ε（z）. The steps of the numerical solution are: （1） for a given timet we compute the Von Karman section b0（ t）via Eq.（28），（2） the interval［0，t ］is discretized into a sufficient number of temporal nodes t， （3） for each of these nodes （the time steps） we evaluate the expansion coefficientsandthrough Eqs.（37）， （38） determining two scalar functions of timet， （4） these are carried into the nonlinear equation that is formed by combining the Wagner condition of Eq.（7） with Eqs.（27） and （40）， （5） this equation is solved numerically in terms of the scaling factor ε（z）for discretised values ofzthat lie in the interval ［-1，0］. Note that the integrals involved in Eq.（40） are treated numerically as well.

2.2 Further simplifications

The numerical procedure briefly outlined above，can be further simplified by employing additional assumptions. In particular we will assume that the expansion coefficientsanddo not change much for small values oftand hence they can be considered constant in the interval［0，t→0+］. The relative validity of this assumption has been verified through extensive numerical experiments. Furthermore，the present analysis considers only the initial stage of the steep wave impact ast→0+. Accordingly the von Karman sectionb0（ t）can be approximated by a series expansion as

from which only the first term will be eventually retained. The velocity potential can now be written as

Expanding the infinite integral for 0≤y≤band x= 0， （Abramowitz and Stegun［23］， Watson［29］） Eq.（42）becomes

Accordingly the velocity at x =0is

Finally the free-surface displacement of the wave front is obtained from the t -integral of Eq.（44） in which η（y， z；0）=0.

The definite time integral in Eq.（45） allows the calculation of a solution in closed-form， provided that b0is approximated by the first term of the expansion of Eq.（41）. Before doing that， we note that the definiteintegral in Eq.（45） can be written aswhere αis independent oft. This can be evaluated in closed-form with respect to Anger， Weber and Lommel functions. The expression is complicated but the complexity is lessened using the fixed value of µ=2. Thus， the definite integral in Eq.（45） becomes

The above expression is relatively easy to handle numerically. We now turn to satisfying the final condition on the problem， namely the Wagner condition of Eq.（7）. Using all the above this is

Again， the Wagner condition of Eq.（47） is a nonlinear equation with only one unknown， that is the scaling factor ε（z）. The solution of this equation for allzin the interval［-1，0］can be obtained using trivial methods of numerical analysis. Here we employed

Matlab's dedicated function fsolve.

2.3 Potential， pressure and free surface displacement The potential and the pressure are calculated on the structure at x = 0 and within the interval 0 ≤ y ≤b（z；t） . Using the employed simplifications and the assumptions made， the velocity potential from Eq.（43）obtains a compact form. In particular， in our case the infinite integral of Abramowitz and Stegun［23］， with μ = 2 simplifies the expression for the velocity potential to

Accordingly， using Eqs.（45）， （46） the free-surface displacement x=η（y， z； t）of the free-surface for b（ z；t）＜yis

The above expression for the free-surface displacement x=η（y， z； t）can be simplified using formulae found in Watson［29］and Gradshteyn and Ryzhik［28］，respectively:

whereK is the modified Bessel function of the second kind and as always v=-1/2. Accordingly thefree-surface displacement x=-η（y， z； t）is simply

Equation （52） implies that the free-surface displacement of the wave decreases to zero exponentially as one moves away from the Wagner contact point y=b towardsy→∞. In fact， the previous remarks and expectations， are in compliance with the Wagner model of the free-surface's attachment to the body.

Finally the pressure on the cylinder also has an elegant form. The pressure distribution is obtained from

where ρis the constant density of the water. Taking the time derivative of Eq.（43） with µ=2and evaluating the integrals， it can be shown that the pressure is

The expressions （48） and （54） for the potential and the pressure have the property that each one vanishes at the boundary of the wetted region of the cylinder on z =0， and on the Wagner contact curve at y=b. The force exerted on the cylinder is calculated by integrating the pressure over the wetted region of the cylinder.

3. Numerical results

In this section we present some indicative calculations using the solutions outlined above that concern steep wave impact on a vertical cylinder. In particular only one cylinder has been considered， with radius R =0.4 mand a fixed water depth equal to 1 m. The wave velocity was taken equal to V=3m/s， i.e.，close toV=（gh）1/2. We provide numerical predictions for the contact curvey=b（ z； t）， the velocity potential， the pressure impulse， the impact force and the displacement of the wave face due to the cylinder，from the Wagner contact point into the far-field.

Fig.4 Time evolution of the wetted zone's boundary y　=　b（ z；t ）. Times shown: （0.001 s， 0.001 s， 0.01s）. Here R= 0.4，V=3

Fig.5 Contours of the potential in the impact zone -φ（0，y， z；t ）at t=0.01（R =0.4，V =3）. Number of modes in z ，N=10

The numerical results are shown in Figs.4-8. In Fig.4 we provide the impacted length bas a function of time and the vertical coordinatez. Each quantity is shown by snapshots taken in the interval t=［0.001，0.01］， i.e.， at the initial stages of impact， with a time step∆t=0.001. Clearly，bis nearly constant except near the top of the cylinder. Also，bis bigger than the Von Karman equivalent b0which was also expected. Indeed， the insignificant variation ofb in terms of the vertical coordinatez demonstrates the consistency of the flow with our assumptions that the derivatives ∂b/∂zand ∂2b/∂zare negligible compared with unity. Recall that this assumption was employed to ensure that Laplace's equation is satisfied. Nevertheless， it is evident that this is not the case at theupper part of the cylinder where bbends towards smaller values. Even in this case however， the change is not that big， whilst it must be said that the vertical variation of the impacted section complies with the intuition that the impacted area should decay in the vertical direction. At times greater thant=0.01（which is the last shown in Fig.4）， the variation ofbas a function ofz exhibits stronger fluctuations. Another point that must be highlighted is that although b appears to be constant alongz ， the truth is that there are some small variations which suggest that the boundary of the wetted region is sensitive to the Wagner condition.

Fig.6 Contours of the slamming pressure p　（0，y， z； t）/ρat t=0.01（R =0.4，V =3）. Number of modes inz，N=10

Figure 5 shows the contours of the velocity potential and Fig.6 shows the associated contours of the hydrodynamic pressure impulse. The pressure is normalized byρ. All figures correspond to the last investigated time instant，t=0.01. In the horizontal direction the contours are shown as a function of the node （here eachb（ z； t）has been discretized by 50 nodes） due to the fact thatbis a function ofz. It is interesting to observe that the contours of the potential differ from the contours of the pressure. Nevertheless，the pressure and the potential both tend to zero as the boundaries of the wetted （and impacted region） are approached: at the upper free-surface and the moving contact line of the wave front. This rational outcome，complies with the physics of the problem. It must also be observed that the spatial maximum of the potential lies aty=0， on the centre line of the impacted zone on the cylinder. The global maximum pressure in Fig.6 occurs on the centreline of the cylinder and near the surface. Four other local maxima are also visible on the cylinder at the same height.

The hydrodynamic force exerted on the cylinder is shown in Fig.7. The force starts at t=0+with a nonzero value and then decreases astincreases. The hydrodynamic impulsive pressure is inversely proportional to the time， so the maximum force should be expected to occur at the very early stages of impact.

Fig.7 Slamming force normalized by ρR2V2（R =0.4，V= 3）

Fig.8 Wave front displacement η（y， z； t）/h as a function of the scaled distance y/ b＞1at t =0.01（R =0.4，V= 3）， for several values ofz

We complete the discussion on the numerical results by providing some data on the variation of the free-surface rise of the wave front on the Wagner contact point and beyond. The results are shown in Fig.8. Here， the differences due to the cylinder size are not that important. What it is important to underline is that the free-surface tends to zero asy→∞. The free-surface displacement near the bed （z=-1）is about twice that near the upper free-surface（z= -0.1）-this accounts with the greater spreading of the wetted zone lower than nearz =0plotted in Fig.4.

4. Conclusions

We investigated the 3-D violent slamming induced by the steep wave impact on a vertical cylinder. Linear potential theory was employed while the freesurface displacement was approximated by the Wagner approach. The novelty of our study relies on the assumption that the instantaneous wetted surface of the cylinder during the impact varies both in time and height.

The mathematical model has been presented as a mixed boundary value problem the solution of whichresulted in a complicated dual Fourier-Bessel-trigonometrical series expansion. Several simplifications were employed that allowed the derivation of expressions for all hydrodynamic parameters， namely the potential，the impulse pressure and the free-surface displacement.

It was found that the instantaneous contact line is nearly vertical along the cylinder's height at the early stages of the impact. In addition， the contours of the potential and the pressure exhibit a very interesting pattern. It was also remarked that the maximum hydrodynamic loading occurs at the early stages of the impact followed by an exponential-like decay. Finally，the closed-form relations for the free-surface displacement to around the cylinder towards the far field，comply with Wagner's approach for the associated configuration.

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10.1016/S1001-6058（16）60657-1

September 17， 2015， Revised May 12， 2016）

* Biography: Ioannis K. CHATJIGEORGIOU （1966-）， Male，Ph. D.， Associate Professor

2016，28（4）:523-533