(School of Marine Science and Technology,Sun Yat-sen University,Zhuhai 519000,China)

Abstract:A two-dimensional rectangular barge floating in variable bathymetry region is considered.The variable bathymetry is decomposed into a succession of steps and the fluid domain can be divided into a series of rectangular sub-domains.The eigenfunction expansion is used for the velocity potential defined in each region.The solution of velocity potential can be obtained based on the continuity hypothesis.The convergence of computational model is discussed and the parameters in the eigenfunction expansion method are suggested.The hydrodynamic coefficients of the barge under both the flat bottom and inclined bottom conditions are calculated and compared with the results in the literature.And motion responses of the barge in irregular waves are verified by the experimental results.

Key words:eigenfunction expansion method;variable bathymetry;barge

0 Introduction

The motion response of a floating body in variable bathymetry regions is a mathematically interesting issue attracting many researchers.A specific case is the project of Liquefied Natural Gas(LNG)terminals and LNG floating storage units.The wave-induced loads and motions of floating bodies in waters with a restricted depth are important for the design of mooring system and for insuring that under-keel clearance is sufficient.The estimation of motions and loads can be given by the solution of classical wave-body-seabed hydrodynamic interaction problems.

When the depth variation is small or the horizontal dimension of floating body is small in comparison to the bottom variation length,the water depth can be assumed to be constant.Otherwise the variations of bathymetry may have a significant effect on hydrodynamic behavior of the floating body.Under the assumption of slowly varying bathymetry,the mild-slope equations have been used for studying the floating body motions[1].And the method was extended in Ref.[2]to study steeper bathymetric variations.A linear hybrid model of mild-slope equation and boundary element method was developed to study the wave propagation around floating structures in coastal zones[3].The timedependent mild-slope equation model was adopted for a single and multiple wave energy converters based on the overtopping principle in Ref.[4]and Ref.[5].

The step method was developed for the wave transmission in variable bathymetry regions,which was extended for studying the reflection and transmission of normal incident waves by twodimensional trenches and shoals in Ref.[6].The three-dimensional bathymetric anomalies with gradual transitions in depth was studied in Ref.[7]and Ref.[8].The bathymetric anomaly was represented by a series of steps,which can approximate uniform or non-uniform slopes.Experimental study of the effect of variable bathymetry on the slow-drift wave response of floating bodies was given by Ref.[9].A rectangular barge moored at different positions along an inclined beach and submitted to irregular beam seas was studied and the results were compared with the solutions of the Newman’s approximation and the step method.

The hydrodynamic coefficients of a two-dimensional truncated rectangular floating structure in extremely shallow water was studied in Ref.[10]based on the method of matched asymptotic expansions.The radiation problem was solved under assumptions of linear wave theory,by matching two outer flows with the inner flow near the structure edge.The hydrodynamic coefficients of a three-dimensional floating truncated vertical cylinder was given by Ref.[11].

The simplicity of the Rankine sources method in conjunction with appropriate representations of the wave field in the exterior infinite domains was adopted for floating body in variable bathymetry regions[12].The fluid domain was divided into three sub-domains and the near field of floating body was solved by the Rankine sources method.Similarly a coupling method between the Boussinesq equations and the integral equation method with application to numerical wave tank was presented in Ref.[13].In order to solve the roll responses of ship-hull sections,the vortex particle method for the generation of vorticity in the boundary layer was combined with the boundary element method for the non-linear waves in Ref.[14].

In this paper,the eigenfunction expansion method was built for simulating rectangular barge in variable bathymetry regions.The fluid domain was divided into a series of rectangular sub-domains and eigenfunction expansions were used for the velocity potential.The particular solution was proposed for the radiation velocity potential.The convergence of eigenfunction expansion method was verified for different evanescent modes and sub-domains.Hydrodynamic coefficients of barge in variable bathymetry regions were discussed and compared with those calculated by boundary element method.The motion responses of the barge in irregular waves were calculated and compared with experimental results measured in the basin.

1 Mathematical model

As shown in Fig.1,the two-dimensional rectangular barge floats in waves with variable bathymetry.The wave field is excited by incident waves propagating from the left-hand side semi-infinite sub-domain.The assumption of an incompressible,inviscid and irrotational flow is adopted and the linearized potential flow theory is used for the hydrodynamic computations of the floating barge.The coordinate systemoxzis located on the free surface at rest withz-axis pointing upwards.

Fig.1 2D sketch of a barge floating in waves with variable bathymetry

The fluid domainDis decomposed into three partsD(m)(m=1,2,3).D(1)is the sub-domain characterized byx≤awhere the depth is constant and equal toh1;D(3)is the sub-domain characterized byx≥bwhere the depth is constant and equal toh3;D(2)is the variable bathymetry sub-domain(a＜x＜b)which contains the floating body.If the rigid bottom is defined byz=-h(x),the expression of function can be shown as follows:

The velocity potential field in time harmonic can be represented by:

whereωis the angular frequency of incident wave.Here the velocity potentialφ(x,z)can be decomposed into the classical form:

whereAis the incident wave amplitude andgis the acceleration due to gravity.φI(x,z)denotes the incident wave potential andφD(x,z)is the diffraction wave potential due to the variable bathymetry and fixed body.φRk()x,z(k=1,3,5)represents the radiation potentials associated with the motions of floating body,i.e.surge(k=1),heave(k=3)and pitch(k=5).akdenotes the complex amplitudes of corresponding motions of floating body.

In the variable bathymetry region and the down-wave region there is no gain in separatingφIandφDwhich are associated together under the form:

The waves are propagating from the left-hand side semi-infinite sub-domain along theoxaxis.In the up-wave region(x≤a),the velocity potential of regular incoming waves is

where the wavenumberk10satisfiesω2=gk10tanhk10h1.

In the variable bathymetry regions,the velocity potentialφsatisfies the Laplace equation,the free-surface boundary condition and the no-flow condition on the solid boundaries:

The radiation potentialsφRk,[k=1,3,5]verify the boundary value problem

where the generalized normal vectorfkon the wetted surface aref1=nx,f3=nz,f5=(z-zG)nx-(x-xG)nz.is the outer normal vector.

2 Computational model

The boundary value problem of velocity potential will be solved based on matching eigenfunction expansions in the fluid domain.The fluid domain limited by the rectangular barge and variable bathymetry is discretized as a series of horizontal steps and defined as a succession of rectangular sub-domains.And the eigenfunction is used for the expressions of velocity potential in each subdomain.

For the definition of sub-domains,thexaxis is discretized intoN-1 pointsx1,x2,…,xN-1and the fluid domain is divided intoNsub-domains.xIandxJare the left and right boundaries of the barge respectively.In the successive sub-domains(i=1,…,I)and(i=J+1,…,N)excluding the sub-domains below the barge,the velocity potentialφi(x,z)can be written as:

and in the successive sub-domains(i=I+1,…,J)which are below the barge,the velocity potential can be written as:

In Eq.(8),the coefficientsBi0are the transmitted wave amplitudes and the coefficientsCi0are the reflected wave amplitudes.The coefficientsBimandCimare the amplitude functions for the evanescent modes which decay exponentially with the distance from the boundary.The wave numberskimandλimverify:

wheredis the draft of the barge,andβis the initial propagation angle with respect to thexaxis.

For the radiation potentialsφRk,a particular solution is introduced for the velocity potential below the barge due to the normal velocity on boundaries of the barge.Here the particular solutionφpkiis given for surge,heave and pitch motions:

And the rest part of the radiation potential is the same as the potentialφi.

To obtain the unknown coefficientsBimandCim,the velocity potentialφiand horizontal derivative of velocity potential satisfy:

Using the orthogonal properties of eigenfunction and integrating along the connected lines,a linear system with unknownsBimandCimcan be built.For the radiation potentialφRk,the process of solution is the same.Due to the appearance of the particular potential,some constant terms appear in the linear system.The coefficient matrix can be preconditioned by an incomplete LU-factorization and the linear system is solved by the GMRES(generalized minimal residual method)iterative scheme.With the solution of the linear system,the velocity potential and hydrodynamic coefficients of the floating body can be given.

3 Convergence of model

A two-dimensional rectangular barge is considered,with non-dimensional breadthB/h=1.5 and draftd/h=0.5,whereh=(h1+h3)/2 denotes the mean depth[12].The center of gravity has been selected to coincide with the center of flotation.The linear shoal betweenx=aandx=bcharacterized by a bottom slope of 0.125 is adopted.

Firstly,the convergence of evanescent modes in the eigenfunction expansion method has been tested for the hydrodynamic coefficients of the barge.The added masses,damping coefficients and wave loads of barge under sway,heave and roll motions are shown in Figs.2-4.The wave length obtained from linear dispersion relation is divided by the mean water depth,which gives the values of abscissa.The non-dimensional added masses,damping coefficients and wave loads are calculated based on the following expressions:

whereHis the incident wave height.Different evanescent modes are compared for all the non-dimensional hydrodynamic coefficients.From Figs.2-4,the computational model has converged and the discrepancy of numerical results can be ignored.In the following sections,20 evanescent modes are adopted for all the numerical computations.

Fig.2 Convergence of evanescent modes for added masses(a11 and a33)and damping coefficients(b11 and b33)

Fig.3 Convergence of evanescent modes for added masses(a55 and a15)and damping coefficients(b55 and b15)

Fig.4 Convergence of evanescent modes and different steps for wave loads

Fig.5 Convergence of different steps for added masses(a11 and a33)and damping coefficients(b11 and b33)

The fluid domain has been divided into a succession of sub-domains and the convergence of number of sub-domains is studied.For the case named＜Step 1＞,all the 37 sub-domains are distributed in the fluid domain where 11 sub-domains on the left side of the barge,15 sub-domains under the barge and 11 sub-domains are on the right side of the barge.For the case named＜Step 2＞,all the 52 sub-domains are distributed in the fluid domain where 16 sub-domains are on the left side of the barge,20 sub-domains under the barge and 16 sub-domains on the right side of the barge.For the case named＜Step 3＞,all the 72 sub-domains are distributed in the fluid domain where 21 sub-domains are on the left side of the barge,30 sub-domains under the barge and 21 sub-domains on the right side of the barge.

From Figs.4-6,the non-dimensional hydrodynamic coefficients are compared with different numbers of sub-domains.The numerical results are convergent except that small discrepancies can be found in added massa15and damping coefficientb15.In the following sections,the type of Step 3 is adopted for all the computations.

Fig.6 Convergence of different steps for added masses(a55 and a15)and damping coefficients(b55 and b15)

4 Hydrodynamic coefficients

The hydrodynamic coefficients of the barge with linear shoal is given based on boundary element method[12].The near field is represented by boundary integral representation involving Rankine sources.The far field is modelled by complete series expansions derived by separation of variables in the constant depth half-strips.In the following Figs.7-9,the results given in the paper[12](blue lines)are compared with that calculated by the eigenfunction expansion method(red lines).The results given in the paper[12]concerning two bottom profiles are shown by solid lines(flat bottom)and lines marked as star(slope of 0.25).And the results given by the eigenfunction expansion method are shown by solid lines(slope of 0.125),lines marked as star(slope of 0.25)and lines marked as circle(slope of 0.3).The comparisons of excited loads,added masses and damping coefficients are shown in Figs.7-9.From the comparisons,the eigenfunction expansion method shows an overall good agreement with boundary element method.

In fact,the velocity potential of incident wave is different for step method and boundary element method.In the step method,the velocity potential of incident wave in a constant depth is adopted for the left-hand side semi-infinite sub-domain.But the incident wave potential in the boundary element method is calculated by the means of the consistent coupled-mode model[15],which involves the variable bathymetry region.There are some differences in the values of excited loads,added masses and damping coefficients.For the values ofa15andb15,the significant discrepancy is due to the sensitivity of coefficients.The results should be verified by experimental data.

Fig.7 Comparison of wave excited loads between boundary element method and eigenfunction expansion method

Fig.8 Comparison of added masses(a11 and a15)and damping coefficients(b11 and b15)

Fig.9 Comparison of added masses(a33 and a55)and damping coefficients(b33 and b55)

For an infinite water depth,the added massesa13,a35and damping coefficientsb13,b35of rectangular barge should be equal to zero.Due to the effect of variable bathymetry regions,the values of added masses and damping coefficients should be considered.In Figs.10-11,the coupling effects of sway,heave and roll motions can be found.

Fig.10 Values of added mass a13 and damping coefficient b13

Fig.11 Values of added mass a35 and damping coefficient b35

5 Experimental comparisons

Fig.12 Comparison of sway motion between numerical and experimental results

The experiments were carried out in the BGO-First offshore tank located in la Seyne sur Mer in France.The experimental settings are shown in Ref.[9].The length of basin is 40 m and width is 16 m.In our experiments,the false bottom was raised and inclined at a slope of 5%,starting from a depth of 1.05 m by the wavemaker side and emerging by 15 cm at its other side.The rectangular barge model has a length of 2.47 m,a beam of 0.6 m,a depth of 0.3 m and its draft during the tests was 0.12 m.The center of gravity is located at 0.135 m above keel line and a roll radius of gyration equal to 0.19 m.In our comparisons,the water depth of barge’s position is 54 cm.The barge was submitted to irregular waves of Pierson-Moskowitz spectra with a peak period of 1.2 s and a significant waveheight of 20 mm.The sway,heave and roll motion responses of the barge were compared between the numerical and experimental results as shown in Figs.12-14.The response amplitude operator(RAO)of experimental results are calculated by the following equation:

whereSoutis the spectrum of the barge motion andSinis the spectrum of the incident wave.

Fig.13 Comparison of heave motion between numerical and experimental results

Fig.14 Comparison of roll motion between numerical and experimental results

As viewed from the figures,the numerical results calculated by our eigenfunction expansion method coincide well with experimental results for sway and roll barge motions.For the heave motion of the barge,the oscillation ofRAOin the low frequencies can be found in the experimental results.Due to the reflected waves generated from the inclined bottom,the superposition of incident and reflected waves appears in the position of the barge.Non-linear shallow-water wave theory should be applied.

6 Concluding remarks

The eigenfunction expansion method built in the paper is an effective semi-analytical approach for calculating motion responses of regular volumes in the frequency domain.The rectangular barge floating in variable bathymetry regions is solved based on matching eigenfunction expansions and the particular solution of radiation velocity potential is given.The convergence of the computational model is discussed and the parameters in the model are suggested.Added masses,damping coefficients and excited loads of the barge are calculated and compared with results in the literatures.The discrepancies of hydrodynamic coefficients of the barge under both the flat bottom and inclined bottom conditions are presented.The motion responses of the barge present an overall good agreement with the experimental results except for heave motion in low frequencies.Furthermore,nonlinear shallow water wave theory should be applied for nonlinear motions of the barge.

Acknowledgments

The authors would like to thank Fabien Remy and Bernard Molin in Ecole Centrale Marseille for their provision of experimental results,and the National Natural Science Foundation of China for its financial support.