Porosity Effects on the Dispersion Relation of Water Waves Through Dense Array of Vertical Cylinders

Journal of Marine Science and Engineering

Article Porosity Eﬀects on the Dispersion Relation of Water Waves through Dense Array of Vertical Cylinders

Joﬀrey Jamain 1, Julien Touboul 1 , Vincent Rey 1 and Kostas Belibassakis 2,*

1 Université Toulon, Aix Marseille Université, CNRS/INSU, IRD, MIO UM 110, Mediterranean Institute of Oceanography, 83130 La Garde, France; joﬀ[email protected] (J.J.); [email protected] (J.T.); [email protected] (V.R.) 2 School of Naval Architecture and Marine Engineering, National Technical University of Athens, Zografos, 15773 Athens, Greece * Correspondence: kbel@ﬂuid.mech.ntua.gr; Tel.: +30-210-7721138

Received: 19 October 2020; Accepted: 21 November 2020; Published: 24 November 2020

Abstract: There is growing interest for water-wave flows through arrangements of cylinders with application to the performance of porous marine structures and environmental flows in coastal vegetation. For specific few cases experimental data are available in the literature concerning the modification of the dispersion equation for waves through a dense array of vertical cylinders. This paper presents a numerical study of the porosity effects on the dispersion relation of water waves through such configurations. To this aim, the sloshing problem in a tank full of vertical cylinders intersecting the free surface is studied using the finite element method, and the influence of the porosity on the wave number is quantified. On the basis of numerical results, a new modification of a dispersion relation for porous medium is suggested based on a wide range of collected data. Moreover, the domain of validity of this new dispersion relation is examined considering the number of cylinders and the extrapolation to the infinite medium.

Keywords: water waves; porous medium; dispersion relation; FEM analysis

1. Introduction For many years, the scientific community dedicated significant efforts to the understanding and modeling of the wave–structure interaction; see, e.g., Mei [1] and Molin [2]. Among the structures, many studies are focused on cylinders, since they constitute a basic component of rigs and structures employed in ocean and coastal engineering applications. Many experimental works (e.g., Goda [3]), mathematical models and numerical methods (e.g., John [4], Yang and Ertekin [5]) are developed to understand the reference problem of water waves interacting with a single, bottom mounted or floating, surface piercing, vertical cylinder; see also Young [6], Sabunku and Calisal [7]. A review can be found in McNatt [8]. However, with the development of complex ocean structures, such as very large floating structures (Wang and Tay [9]), arrays of devices for the outtake of marine renewable energies (Falnes [10], Balitsky et al. [11]), and for porous coastal and harbor defenses (Arnaud [12]), the problem has been expanded to more complex geometrical configurations, where dense arrangements of vertical cylinders are involved. Similarly, there is growing interest concerning similar problems in environmental flows in vegetation, such as the propagation of water waves in mangrove forests (e.g., Massel [13], Mei et al. [14]). The above configurations, of course, lead to a large increase of the difficulty, which has been addressed by using various strategies. An important body of the literature is dedicated to the study of water waves propagating through dense arrays of cylinders. If it is not possible to be exhaustive,

J. Mar. Sci. Eng. 2020, 8, 960; doi:10.3390/jmse8120960 www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2020, 7, x FOR PEER REVIEW 2 of 17 one should emphasize the studies of Linton and Evans [15], Maniar and Newman [16], McIver [17] or Kagemoto et al. [18]; see also Belibassakis et al. [19]. From the point of view of numerical methodology, the direct calculation of water waves through dense arrays of cylinders in complex configurations is computationally demanding. Of course, recent improvementsJ. Mar.both Sci. Eng.in 2020computational, 8, 960 resources and in numerical optimization2 oftechniques 17 offer possibilities to solve this problem when the degree of complexity remains reasonable (Fu et al. [20]). one should emphasize the studies of Linton and Evans [15], Maniar and Newman [16], McIver [17] or Thus, another Kagemotopossible et approach al. [18]; see also is Belibassakis to rely on et al. ex [19tensions]. of the potential flow theory, whereby the cylinder domain isFrom treated the point as ofa homogeneous view of numerical methodology,porous medium. the direct The calculation dispersion of water equation, waves though, is modified to takethrough the extra dense inertial arrays of and cylinders dissipative in complex terms configurations into account. is computationally Such formulations, demanding. for instance, Of course, recent improvements both in computational resources and in numerical optimization are suggested techniquesby Sollitt off ander possibilities Cross [21], to solve Madsen this problem [22] when or the Yu degree and of Chwang complexity remains[23]. These reasonable modifications of the dispersion( Fuwere et al. [20used]). Thus, recently another possibleto model approach the is reflection, to rely on extensions transmission of the potential (Arnaud flow theory, et al. [24]) and whereby the cylinder domain is treated as a homogeneous porous medium. The dispersion equation, diffraction (Arnaudthough, et is modifiedal. [25]) to of take water the extra through inertial and an dissipative idealized terms porous into account. medium Such formulations, of various porosity and specific surface.for instance, are suggested by Sollitt and Cross [21], Madsen [22] or Yu and Chwang [23]. In the meantime,These modifications by implementing of the dispersion an wereexperiment used recentlyal approach to model the relying reflection, on transmission the sloshing response (Arnaud et al. [24]) and diffraction (Arnaud et al. [25]) of water through an idealized porous medium of a tank filled ofwith various homogeneous porosity and specific vertical, surface. surface piercing cylinders, Molin et al. [26] questioned the formulation of theIn thedispersion meantime, by equation implementing to an be experimental considered approach and relying introduced on the sloshing a new response expression. An of a tank filled with homogeneous vertical, surface piercing cylinders, Molin et al. [26] questioned illustration of thethe formulation tank used of the in dispersion the above equation work, to be wh consideredich filled and introduced with a aregular new expression. pattern of vertical cylinders, is presentedAn illustration in ofFigure the tank 1. used Although in the above the work, sloshing which filledproblem with a regularfinds patternapplications of vertical to engineering mechanics, thecylinders, eigenvalue is presented problem in Figure of1 fluid. Although motion the sloshing within problem tanks finds containing applications arrays to engineering of surface piercing mechanics, the eigenvalue problem of fluid motion within tanks containing arrays of surface piercing obstacles providesobstacles also provides useful also information useful information for for environmental environmental maritime maritime problems. problems.

Figure 1. Sloshing tank used by Molin et al. [26]. Figure 1. Sloshing tank used by Molin et al. [26]. The present work is a numerical extension of the study by Molin et al. [26]. The resonant sloshing in tanks with internal (sloshing-suppressing) structures has been discussed by various authors; The presentsee, work e.g., Faltinsen is a numerical and Timohka extension [27]. Although of the the solution study of the by nonlinear Molin problem et al. [26]. is very The important resonant sloshing in tanks with internalstill the solution (sloshing-suppressing) of the linearized problem provides struct usefulures information has been concerning discussed the natural by various sloshing authors; see, modes and frequencies for generalizing the method (Timohka [28]). Since the structures (cylinders and e.g., Faltinsen tankand walls) Timohka contain vertical [27]. boundariesAlthough the 3Dthe problem solution can be of reduced the tononlinear 2D governed problem by the Helmholtz is very important still the solutionequation of onthe the linearized horizontal plane. problem An important provides novelty of useful the present information approach is that concerning it is an easily the natural applicable method to more general shapes of tanks, with arbitrary distributions and general cross sloshing modessection and characteristics frequencies of the for structures generalizing in the tank. the method (Timohka [28]). Since the structures (cylinders and tankA numericalwalls) contain solver, providing vertical the boundaries eigenmodes of the tank, 3D basedproblem on a finite can element be reduced approach, to 2D governed by the Helmholtzis developed, equation allowing on onethe to horizontal extend the analysis plane. of the An influence important of the porosity noveltyτ of the mediumof the (definedpresent approach is as the ratio of water volume in terms of the total volume of the domain) on the dispersion equation. that it is an easilyThe applicable paper is organized method as follows. to more In Sectiongeneral2, the shapes equations of of tanks, the problem with and arbitrary the FEM distributions and general crossnumerical section procedure characteristics for the solution of are the presented. structures A comparison in the oftank. the tool with experiments by Molin et al. [26] is also performed here, to provide an insight of the performance of the present model. A numericalNext, solver, Section3 focusesproviding on the analysisthe eigenmodes of the e ffect of of porosity the ontank, the equivalent based on dispersion a finite equation, element approach, is developed, allowingor equivalently one to theto equivalentextend the bulk analysis wavenumber ofk τtheof water influence oscillations of developedthe porosity in the tank.τ of the medium (defined as the ratio of water volume in terms of the total volume of the domain) on the dispersion equation. The paper is organized as follows. In Section 2, the equations of the problem and the FEM numerical procedure for the solution are presented. A comparison of the tool with experiments by Molin et al. [26] is also performed here, to provide an insight of the performance of the present model. Next, Section 3 focuses on the analysis of the effect of porosity on the equivalent dispersion equation, or equivalently to the equivalent bulk wavenumber 𝑘 of water oscillations developed in the tank. A modification of this equation is introduced. Finally, results are summarized in Section 4, where also conclusive remarks are provided.

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A modiﬁcation of this equation is introduced. Finally, results are summarized in Section4, where also conclusive remarks are provided.

2. Mathematical Model

2.1. Mathematical Model for the Dispersion Relation The sloshing problem of water in a tank filled with water up to a level and containing the dense arrangement of vertical cylinders is considered using potential flow theory, in conjunction with the assumption of small amplitude waves. Since the flow is irrotational, the velocity u is obtained as the gradient of a 3D velocity potential Φ: u = Φ. (1) ∇3 Equation (1) in conjunction with the incompressibility of the water, leads to the Laplace equation:

∆3Φ = 0. (2)

We consider the condition of impermeability at the bottom of the tank (z = h): −

∂Φ = 0, (3) ∂z z = h − and the linearized free surface boundary condition at z = 0:

∂2Φ ∂Φ + g = 0. (4) ∂t2 z=0 ∂z z=0

In the case of harmonic motion at given frequency ω, exploiting the fact that the walls of cylinders and tank boundaries are vertical the evanescent modes are not excited and the wave field Φ is represented by: iωt Φ = f (z)φ(x, y)e− , (5) where f is the vertical profile of the potential field and φ the corresponding 2D velocity potential on the x–y plane. With this form of potential, Equation (2) leads to:

f 00 (z) ∆φ(x, y) = , (6) f (z) − φ(x, y) where a prime denotes diﬀerentiation with respect to the involved variable and ∆ stands for ∆φ the two-dimensional Laplace operator. This means that that the quantity is constant everywhere − φ 2 2 in the water domain. The latter is denoted by kτ , and thus, kτ represents the wave number on the horizontal plane. Consequently, Equation (6) is written:

2 f 00 = kτ f . (7)

Furthermore, from Equation (7) we obtain that the 2D potential φ satiﬁes the Helmholtz equation on the horizontal plane: 2 ∆φ = kτ φ, (8) − valid in the water domain. The solution of Equation (7) satisfying the boundary condition (3) is:

cosh(kτ(z + h)) f (z) = . (9) cosh(kτh) J. Mar. Sci. Eng. 2020, 8, 960 4 of 17

iωt which substituted in the potential Φ(x, y, z, t) = f (z)φ(x, y)e− and used in the free surface condition (4) leads to: 2 ω = gkτtanh(kτh). (10)

In the sequel kτ is considered constant everywhere in the domain, deﬁned as

2 ∆φ kτ = . (11) − φ

It is noticed here that the associated wavelength

2π λτ = , (12) kτ does not correspond to the water wavelength λ1, which for the ﬁrst sloshing mode is twice the tank length L: 2π λ1 = = 2L , (13) k1 as implied by the fulﬁllment of the side boundary conditions on the vertical tank walls. The objective of this study was to determine a relation between kτ and k1

k G(τ) = τ , (14) k1 in order to associate the water wavelength to its frequency in the porous medium

2 ω = gk1G(τ)tanh(k1G(τ)h) . (15)

In Section 2.2 a numerical sloshing tank was modeled by using FEM to establish the above relation. A ﬁrst set of calculation, presented in Sections 3.1 and 3.2, will be useful to optimize the cylinders arrangement of the numerical sloshing tank in order to cover the widest range of cases and reduce computation time. Then, many cases were tested covering a porosity of τ = 0.3 up to 0.95 and a novel formulation of G(τ) was proposed. The results were also compared against the modiﬁcation of the dispersion relation proposed by Molin et al. [26], which was also expressed by Equation (15) with G(τ) = τ1/2.

2.2. FEM Numerical Scheme The present numerical method was based on the triangulation of the horizontal domain on the free surface and application of standard first-order FEM to the solution of the eigenvalue problem. As an example of verification, we first considered the case of sloshing modes in a circular container with vertical cylinders (poles) for which analytical solutions were available; see e.g., Timokha [28]. A particular example is presented in Figure2 for a container of unit radius with a single eccentric circular pole of radius 0.5 m centered at x = 0.5 m, considered by Timohka ([28], Section 4 and Figure 3). A sketch of the triangulation corresponding to a mesh of 4693 elements, which was proved enough for numerical convergence, is shown in Figure2a. The lowest eigenfunctions corresponding to the symmetric and antisymmetric modes are plotted in Figure2b,c, respectively and were found in very good agreement with the above analytical solution. Finally, the result of calculation by the present FEM concerning the lowest eigenvalue corresponding to the symmetric mode in the same container for various values of the radius Rin of the inner circular pole is illustrated in Figure3. The latter quantity is reported to be not well predicted by the asymptotic formula provided by Faltinsen and Timohka ([27], Equation (4.185)). However, the present results are found in good agreement with analytical predictions obtained by the variational Trefftz method by Timohka ([28], Figure 2). J. Mar. Sci. Eng. 2020, 7, x FOR PEER REVIEW 4 of 17

does not correspond to the water wavelength 𝜆, which for the first sloshing mode is twice the tank length L: 2𝜋 𝜆 = =2𝐿 , (13) 𝑘 as implied by the fulfillment of the side boundary conditions on the vertical tank walls.

The objective of this study was to determine a relation between 𝑘 and 𝑘 𝑘 𝐺(𝜏) = , (14) 𝑘 in order to associate the water wavelength to its frequency in the porous medium 𝜔 =𝑔𝑘𝐺(𝜏) tanh(𝑘𝐺(𝜏)ℎ) . (15) In Section 2.2 a numerical sloshing tank was modeled by using FEM to establish the above relation. A first set of calculation, presented in Sections 3.1 and 3.2, will be useful to optimize the cylinders arrangement of the numerical sloshing tank in order to cover the widest range of cases and reduce computation time. Then, many cases were tested covering a porosity of τ = 0.3 up to 0.95 and a novel formulation of 𝐺(𝜏) was proposed. The results were also compared against the modification of the dispersion relation proposed by Molin et al. [26], which was also expressed by Equation (15) with G(τ) = τ1/2.

2.2. FEM Numerical Scheme The present numerical method was based on the triangulation of the horizontal domain on the free surface and application of standard first-order FEM to the solution of the eigenvalue problem. As an example of verification, we first considered the case of sloshing modes in a circular container with vertical cylinders (poles) for which analytical solutions were available; see e.g., Timokha [28]. A particular example is presented in Figure 2 for a container of unit radius with a single eccentric circular pole of radius 0.5 m centered at x = 0.5 m, considered by Timohka ([28], Section 4 and Figure 3). A sketch of the triangulation corresponding to a mesh of 4693 elements, which was proved enough for numerical convergence, is shown in Figure 2a. The lowest eigenfunctions corresponding to the symmetric and antisymmetric modes are plotted in Figure 2b,c, respectively and were found in very good agreement with the above analytical solution. Finally, the result of calculation by the present FEM concerning the lowest eigenvalue corresponding to the symmetric mode in the same container for various values of the radius Rin of the inner circular pole is illustrated in Figure 3. The latter quantity is reported to be not well predicted by the asymptotic formula provided by Faltinsen and J. Mar. Sci. Eng. 2020, 8, 960 5 of 17 Timohka ([27], Equation (4.185)). However, the present results are found in good agreement with analytical predictions obtained by the variational Trefftz method by Timohka ([28], Figure 2).

(a) (b) (c)

FigureFigure 2. The 2. Thecase case of a of circular a circular container container of ofunit unit radius radius with with a asingle single eccentric eccentric circular circular pole pole of of radius 0.5 m centered0.5 m at centered x = 0.5 m at consideredx = 0.5 m consideredby Timohka by ([28], Timohka Section ([28 4).], ( Sectiona) Mesh, 4). (b ()a first) Mesh, symmetric (b) ﬁrst and symmetric (c) antisymmetric and mode,J. Mar.(c )Sci.as antisymmetric calculated Eng. 2020, 7 by, xmode, FORthe present PEER as calculated REVIEW FEM. by the present FEM. 5 of 17

FigureFigure 3. 3.The The lowest lowest eigenvalue eigenvalue corresponding corresponding to the to symmetric the symmetric mode inmode the case in ofthe a circularcase of containera circular ofcontainer unit radius of unit for variousradius for values various of thevalues radius of the Rin ra ofdius the Rin inner of the circular inner polecircul centeredar pole centered at x = 0.5 at mx = considered0.5 m considered by Timohka by Timohka ([28], Section ([28], Section 4) as calculated 4) as calculated by the presentby the present method. method. 2.2.1. Algorithm 2.2.1. Algorithm The present study requires FEM calculations using a large amount of meshes. Consequently, it was The present study requires FEM calculations using a large amount of meshes. Consequently, it necessary to automatize the mesh generation of a numerical sloshing tank. The mesh generator was was necessary to automatize the mesh generation of a numerical sloshing tank. The mesh generator implemented based on the Delaunay triangulation. Among other information, the algorithm was was implemented based on the Delaunay triangulation. Among other information, the algorithm was based on the following more important input data: based on the following more important input data: The tank size length and width (L and W); •• The tank size length and width (L and W); The number of cylinders (therefore Nx and Ny for the number in both direction); •• The number of cylinders (therefore Nx and Ny for the number in both direction); • The porosity of the medium; • The porosity of the medium; • The arrangement of the cylinders; • The arrangement of the cylinders; • TheThe level level of of reﬁnement. refinement. • Then,Then, the thealgorithm algorithm follows follows the thefollowing following steps: steps: 1.1. CalculateCalculate thethe cylindercylinder radius radius accounting accounting for for the the target target value value porosity porosity (if (if necessary); necessary); 2.2. CalculateCalculate the the position position of the of centerthe center of cylinders of cylinde accountingrs accounting for their number for their and number the target and arrangement the target; 3. Generatearrangement; an initial mesh; 4.3. OptimizeGenerate thean initial triangle mesh; quality by slightly moving nodes; 4. Optimize the triangle quality by slightly moving nodes; 5. Refine the mesh. Steps 4 and 5 are repeated as many times as chosen by the user with the input “level of refinement”.

2.2.2. Arrangements Four different arrangements are automatized as shown in Figure 4. The experimental arrangement Figure 4a corresponds to the tank used for the experimental campaign conducted in Marseille and described in Molin et al. [26]. For aligned and misaligned arrangements Figure 4b,c, the distance between two successive cylinders remains the same lengthwise and widthwise (not necessarily the same distance in both directions). Furthermore, the distance to a wall is half the distance between two cylinders. Finally, the random arrangement Figure 4d follows no rules except the cylinders cannot intersect each other.

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5. Reﬁne the mesh. Steps 4 and 5 are repeated as many times as chosen by the user with the input “level of reﬁnement”.

2.2.2. Arrangements Four diﬀerent arrangements are automatized as shown in Figure4. The experimental arrangement Figure4a corresponds to the tank used for the experimental campaign conducted in Marseille and described in Molin et al. [26]. For aligned and misaligned arrangements Figure4b,c, the distance between two successive cylinders remains the same lengthwise and widthwise (not necessarily the same distance in both directions). Furthermore, the distance to a wall is half the distance between two cylinders. Finally, the random arrangement Figure4d follows no rules except the cylinders cannot J.intersectJ. Mar. Mar. Sci. Sci. Eng. eachEng. 2020 2020 other., ,7 7, ,x x FOR FOR PEER PEER REVIEW REVIEW 6 6 of of 17 17

((aa)) ((bb))

((cc)) ((dd))

FigureFigure 4. 4.4. DifferentDifferentDiﬀerent arrangementsarrangements arrangements andand and mesh,mesh, mesh, beforebefore before refinementrefinement reﬁnement ((aa)) ( experimental;aexperimental;) experimental; ((bb)) ( misaligned;bmisaligned;) misaligned; ((cc)) aligned(alignedc) aligned and and and ( (dd)) (random drandom) random arrangement. arrangement. arrangement.

2.2.3.2.2.3. Repeatability Repeatability of ofof the thethe Pattern PatternPattern IfIfIf several severalseveral aligned alignedaligned or oror misaligned misalignedmisaligned meshes meshesmeshes are areare repe repeatedrepeatedated lengthwiselengthwise or or widthwise,widthwise, the the arrangementarrangement remainsremains the the same same as as shownshown inin FigureFigure5 5..5. InInIn particular,particular,particular, resultsresultsresults obtainedobtainedobtained forforfor aaa given givengiven mesh meshmesh can cancan be bebe extrapolatedextrapolated to to an an infinite inﬁniteinfinite medium. medium.

FigureFigure 5.5. AA 16 1616 ×× 444 misaligned misaligned mesh mesh repeated repeated lengthwiselengthwilengthwisese andand widthwise.widthwise. This This creates createscreates a aa 48 48 ×× 1212 × × misalignedmisaligned mesh. mesh.

2.3.2.3. Numerical Numerical Solver SolverSolver TheThe numerical numerical solver solver was was based based on on the the finite ﬁnitefinite element element method method (FEM) (FEM) to to solve solve the the HelmholtzHelmholtz EquationEquation (8); (8); see see e.g., e.g., Johnson Johnson [29]. [[29].29]. Let LetLet us usus considerconsider thethe weakweak form form of of the the Helmholtz Helmholtz Equation Equation (8) (8)

2 Δ𝜙𝑊Δ𝜙𝑊x ∆φ=−𝑘W=−𝑘i = 𝜙𝑊𝜙𝑊k x φW , , i ,(16)(16) (16) − Ω Ω wherewhere (𝑊(𝑊)),, are are the the linear linear pyramid pyramid functions functions equal equal to to 1 1 for for node node number number i, i, and and 0 0 elsewhere elsewhere and and ΩΩ is is the the water water domain domain on on the the horizontal horizontal plan plane.e. Denote Denote the the approximation approximation of of the the field field as as

𝜙𝜙==∑∑𝜙𝜙𝑊𝑊, , (17)(17) wherewhere the the nodal nodal values values in in the the form form of of a a global global vector vector are are 𝜙𝜙 𝑿=𝑿=⋮⋮ . . (18)(18) 𝜙𝜙 UsingUsing the the FEM FEM approximation approximation Equation Equation (17) (17) in in Equation Equation (16), (16), we we obtain obtain 𝜙𝜙(Δ𝑊(Δ𝑊)𝑊)𝑊=−𝑘=−𝑘𝜙𝜙𝑊𝑊𝑊𝑊, ,

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where (Wi)i=1,N are the linear pyramid functions equal to 1 for node number i, and 0 elsewhere and Ω is the water domain on the horizontal plane. Denote the approximation of the ﬁeld as

XN φˆ = φjWj, (17) j=1 where the nodal values in the form of a global vector are    φ1     .  X =  .  . (18)  .    φN

Using the FEM approximation Equation (17) in Equation (16), we obtain     XN XN   2   φ ∆W W  = k φ W W ,  j x j i −  j x j i j=1 Ω j=1 Ω which, in conjunction with Green’s theorem and the no-penetration boundary conditions on the solid walls, leads to the following     XN XN   2   φj Wj Wi  = k φj WjWi . (19) j=1 x −∇ ∇  − j=1 x  Ω Ω

Equation (19) is rewritten in the following matrix form

AX = k2BX, (20) − or 1 2 B− AX = k X, (21) − where A = W W , B = W W , (22) i,j − x ∇ j∇ i i,j x j i Ω Ω are, respectively, the coeﬃcients of matrix A and B. It must be noticed that k2 is an eigenvalue of matrix 1 − C = B− A. In this way the problem is solved by calculating the matrix C eigenvalues. The numerical 1 solver takes as input a mesh, builds the matrix C = B− A, and calculates its ﬁrst eigenvalues and corresponding eigenvectors.

2.4. Validation of the Solver This numerical solver has been tested first using meshes representing tanks filled with water but without cylinders (τ = 1) in the case of rectangular tank with vertical walls. In this case, the wavelength is twice the tank length for the first eigenmode. Consequently, for a given tank length L, we have π k1 = L . Calculations were run with a 1.2-m-long tank and a width of 0.3 m. Three different levels of refinement were tested as shown in Figure6, and the result concerning the computed field is presented in Figure7. Figure8 illustrates the convergence of the results concerning the wave number of the first eigenmode, and the corresponding numerical values are reported in Table1. J. Mar. Sci. Eng. 2020, 7, x FOR PEER REVIEW 7 of 17 which, in conjunction with Green’s theorem and the no-penetration boundary conditions on the solid walls, leads to the following J. Mar. Sci. Eng. 2020, 7, x FOR PEER REVIEW 7 of 17 ∑ ∑ 𝜙 ∬ (−∇𝑊∇𝑊) =−𝑘 𝜙 ∬ 𝑊𝑊 . (19) which, in conjunction with Green’s theorem and the no-penetration boundary conditions on the solid walls,Equation leads (19) to the is rewrittenfollowing in the following matrix form

∑ 𝜙 (−∇𝑊 ∇𝑊 ) =−𝑘 ∑ 𝜙 𝑊 𝑊 . ∬ 𝑨𝑿= −𝑘 𝑩𝑿, ∬ (19) (20) Equation (19) is rewritten in the following matrix form or 𝑨𝑿= −𝑘𝑩𝑿, (20) 𝑩 𝑨𝑿= −𝑘 𝑿, (21) or where 𝑩 𝑨𝑿= −𝑘 𝑿, (21) 𝐴, =− ∇𝑊∇𝑊 , 𝐵, = 𝑊𝑊 , (22) where are, respectively, the coefficients of matrix 𝑨 and 𝑩. It must be noticed that −𝑘 is an eigenvalue of 𝑪=𝑩𝑨 𝐴, =− ∇𝑊∇𝑊 , 𝐵, = 𝑊𝑊 , (22) matrix . In this way the problem is solved by calculating the matrix C eigenvalues. The numericalare, respectively, solver takes the coefficients as input ofa matrixmesh, 𝑨builds and 𝑩 .the It must matrix be noticed 𝑪=𝑩 that𝑨 −𝑘, and is ancalculates eigenvalue its of first eigenvaluesmatrix 𝑪=𝑩 and corresponding𝑨. In this way eigenvectors. the problem is solved by calculating the matrix C eigenvalues. The numerical solver takes as input a mesh, builds the matrix 𝑪=𝑩𝑨 , and calculates its first 2.4. Validationeigenvalues of andthe Solvercorresponding eigenvectors. This numerical solver has been tested first using meshes representing tanks filled with water but 2.4. Validation of the Solver without cylinders (τ = 1) in the case of rectangular tank with vertical walls. In this case, the wavelengthThis is numerical twice the solver tank haslength been for tested the first usingeigenmode. meshes representingConsequently, tanks for filled a given with tank water length but 𝐿, without cylinders (τ = 1) in the case of rectangular tank with vertical walls. In this case, the we have 𝑘 = . Calculations were run with a 1.2-m-long tank and a width of 0.3 m. Three different wavelength is twice the tank length for the first eigenmode. Consequently, for a given tank length 𝐿, levelswe of have refinement 𝑘 = . Calculations were tested were as shown run with in aFigure 1.2-m-long 6, and tank the and result a width concerning of 0.3 m. the Three computed different field is presentedlevels of refinementin Figure were7. Figure tested 8as illustrates shown in Figurethe convergence 6, and the result of theconcerning results theconcerning computed the field wave number of the first eigenmode, and the corresponding numerical values are reported in Table 1. J.is Mar. presented Sci. Eng. 2020in Figure, 8, 960 7. Figure 8 illustrates the convergence of the results concerning the wave8 of 17 number of the first eigenmode, and the corresponding numerical values are reported in Table 1.

(a) (b) (c) (a) (b) (c) Figure 6. Empty tank and mesh (a) with 61 nodes; (b) with 213 nodes (refined once) and (c) with 793 Figure 6. EmptyEmpty tank tank and and mesh mesh (a ()a with) with 61 61 nodes; nodes; (b) ( bwith) with 213 213 nodes nodes (refined (reﬁned once) once) and and (c) with (c) with 793 nodes (refined twice). 793nodes nodes (refined (reﬁned twice). twice).

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Table 1. Wave number of an empty tank and error against the theory.

Number of Wave Number Error Against Nodes (m−1) Theory

61 2.6239 2.3% Figure 7. First sloshing eigenmod213 e for tank without2.6198 cylinders calculated0.07% by FEM mesh with 213 FigureFigure 7. First 7. First sloshing sloshing eigenmod eigenmodee forfor tank tank without without cylinders cylinders calculated calculated by FEM by mesh FEM with mesh 213 nodes. with 213 nodes. The wave field 𝜙 is presented793 using a colorplot.2.6185 0.02% nodes.The The wave wave ﬁeld fieldφ is presented𝜙 is presented using a using colorplot. a colorplot.

FigureFigure 8. 8.FirstFirst sloshing sloshing eigenmode eigenmode for for tank tank without without cy cylinders.linders. Convergence Convergence ofof thethe FEMFEM solversolver (log- log(log-log scale). scale). Table 1. Wave number of an empty tank and error against the theory. 2.5. Numerical Results against the Experiment 1 Number of Nodes Wave Number (m− ) Error Against Theory After the validation of61 the FEM solver 2.6239 for rectangular tank 2.3% with vertical walls against the theoretical value of k for213 a tank without cylinders, 2.6198 the present 0.07% method was tested against the experimental value for a dense793 array of cylinders 2.6185 presented in Molin 0.02% et al. [26]; see Figure 9.

2.5. Numerical Results against the Experiment After the validation of the FEM solver for rectangular tank with vertical walls against the theoretical value of k for a tank without cylinders, the present method was tested against the experimental value for a dense array of cylinders presented in Molin et al. [26]; see Figure9.

(a) (b)

Figure 9. Sloshing tank with an arrangement of vertical cylinders: (a) Experimental set-up and (b) numerical tank and FEM model mesh (6913 nodes).

In the above experiments only the eigenfrequency was reported, whereas the numerical FEM solver provides the wave number. In order to compare both results, an eigenfrequency is associated to the wave number using the dispersion relation of linearized water waves

𝜔 =𝑔𝑘 tanh(𝑘ℎ) , (23) where ℎ = 0.235 m is the water depth of the experimental tank (kh = 0.615); see Figures 1 and 9a. Tables 2 and 3 associate wave numbers from the solver and the eigenfrequencies due to Equation (10). The results were compared for different numbers of cylinders from 48 to 120. For the first comparison presented in Table 2 the meshes contained an average of 2748 nodes. Then a second comparison has been made with finer meshes with an average of 10,254 nodes, see Table 3.

J. Mar. Sci. Eng. 2020, 7, x FOR PEER REVIEW 8 of 17

Table 1. Wave number of an empty tank and error against the theory.

Number of Wave Number Error Against Nodes (m−1) Theory 61 2.6239 2.3% 213 2.6198 0.07% 793 2.6185 0.02%

Figure 8. First sloshing eigenmode for tank without cylinders. Convergence of the FEM solver (log- log scale).

2.5. Numerical Results against the Experiment After the validation of the FEM solver for rectangular tank with vertical walls against the J.theoretical Mar. Sci. Eng. value2020, 8of, 960 k for a tank without cylinders, the present method was tested against9 ofthe 17 experimental value for a dense array of cylinders presented in Molin et al. [26]; see Figure 9.

(a) (b)

Figure 9. SloshingSloshing tank tank with with an an arrangem arrangementent of ofvertical vertical cylinders: cylinders: (a) (Experimentala) Experimental set-up set-up and and (b) (numericalb) numerical tank tank and and FEM FEM model model mesh mesh (6913 (6913 nodes). nodes).

In the aboveabove experimentsexperiments only the eigenfrequencyeigenfrequency was reported, whereas the numericalnumerical FEM solver providesprovides thethe wavewave number.number. In In order order to to compare compare both both results, results, an an eigenfrequency eigenfrequency is associatedis associated to theto the wave wave number number using using the the dispersion dispersion relation relation of linearizedof linearized water water waves waves

𝜔 =𝑔𝑘2 tanh(𝑘ℎ) , (23) ω = gkτtanh(kτh) , (23) where ℎ = 0.235 m is the water depth of the experimental tank (kh = 0.615); see Figures 1 and 9a. whereTables h2 =and0.235 3 associatem is the wave water numbers depth of from the experimentalthe solver and tank the (eigenfrequencieskh = 0.615); see Figuresdue to 1Equation and9a. Tables(10). The2 and results3 associate were compar wave numbersed for different from the numbers solver and of cylinders the eigenfrequencies from 48 to 120. due to Equation (10). The resultsFor the were first comparedcomparison for presented diﬀerent numbersin Table of2 the cylinders meshes from contained 48 to 120. an average of 2748 nodes. Then a second comparison has been made with finer meshes with an average of 10,254 nodes, see Table 2. Numerical eigenfrequencies with an average of 2748 nodes. Table 3. 1 Number of Cylinders Numerical Wave Number (m− ) Associated Eigenfrequency (rad/s) 48 2.52 3.62 60 2.49 3.59 84 2.44 3.52 120 2.38 3.44

Table 3. Numerical eigenfrequencies with an average of 10,254 nodes.

1 Number of Cylinders Numerical Wave Number (m− ) Associated Eigenfrequency (rad/s) 48 2.49 3.59 60 2.46 3.55 84 2.39 3.46 120 2.32 3.37

For the ﬁrst comparison presented in Table2 the meshes contained an average of 2748 nodes. Then a second comparison has been made with ﬁner meshes with an average of 10,254 nodes, see Table3. The results were very encouraging. Indeed, on one hand numerical results were close to the experiment with an error systematically under 4.5%; see Table4. On the other hand, increasing the level of refinement improved the accuracy of the results, see Figure 10. Unfortunately, this arrangement with cylinders very close to the tank wall was costly in computation time, and no better meshes were tested.

Table 4. Numerical eigenfrequencies against the experiment presented in Molin et al. [26].

Diﬀerence with Diﬀerence with Experimental Eigenfrequencies the Numerical the Numerical Number of Cylinders (rad/s) Eigenfrequencies (Average Eigenfrequencies (Average of 2748 Nodes) of 10,254 Nodes) 48 3.55 2.10% 1.21% 60 3.50 2.55% 1.39% 84 3.41 3.17% 1.49% 120 3.30 4.31% 2.08% J. Mar. Sci. Eng. 2020, 7, x FOR PEER REVIEW 9 of 17

Table 2. Numerical eigenfrequencies with an average of 2748 nodes.

Number of Numerical Wave Associated Eigen- Cylinders Number (m−1) Frequency (rad/s) 48 2.52 3.62 60 2.49 3.59 84 2.44 3.52 120 2.38 3.44

Table 3. Numerical eigenfrequencies with an average of 10,254 nodes.

Number of Numerical Wave Associated Eigen Cylinders Number (m−1) Frequency (rad/s) 48 2.49 3.59 60 2.46 3.55 84 2.39 3.46 120 2.32 3.37

The results were very encouraging. Indeed, on one hand numerical results were close to the experiment with an error systematically under 4.5%; see Table 4. On the other hand, increasing the level of refinement improved the accuracy of the results, see Figure 10. Unfortunately, this arrangement with cylinders very close to the tank wall was costly in computation time, and no better meshes were tested.

Table 4. Numerical eigenfrequencies against the experiment presented in Molin et al. [26].

Difference with the Difference with the Experimental Numerical Numerical Number of Eigenfrequencies Eigenfrequencies Eigenfrequencies Cylinders (rad/s) (Average of 2748 (Average of 10,254 Nodes) Nodes) 48 3.55 2.10% 1.21% 60 3.50 2.55% 1.39% 84 3.41 3.17% 1.49% 120 3.30 4.31% 2.08% J. Mar. Sci. Eng. 2020, 8, 960 10 of 17

eigen frequency eigen

J. Mar. Sci. Eng.Figure 2020,Figure 7, x10. FOR Comparison 10. PEERComparison REVIEW between between thethe numerical numerical model model and theand experiment. the experiment. 10 of 17 3. Estimation of the Porosity Eﬀect on the Dispersion Relation 3. Estimation of the Porosity Effect on the Dispersion Relation

3.1. Independence of G(τ) against the Number of Cylinders 3.1. Independence of 𝐺(𝜏) against the Number of Cylinders A ﬁrst set of calculations concerns the relation between the number of cylinders and the wave A first set of calculations concerns the relation between the number of cylinders and the wave number for a given value of porosity. Three diﬀerent meshes were tested with respectively 4, 36 and 100number cylinders, for a given as shown value in of Figure porosity. 11. TheThree quantities different that meshes remained were thetested same with in allrespectively variants were 4, 36 and 100 cylinders, as shown in Figure 11. The quantities that remained the same in all variants were The porosity τ; • LengthThe porosity and width; τ; • Length and width; Nx/Ny ratio. • Nx/Ny ratio.

(a) (b) (c)

Figure 11. Porosity τ = 0.50.5 with with ( (aa)) 4 4 cylinders; cylinders; ( (bb)) 36 36 cylinders cylinders and and ( (cc)) 100 100 cylinders cylinders in the tank.

Porosity valuesvalues ofofτ τ= =0.3, 0.3, 0.5, 0.5, 0.7 0.7 and and 0.9 0.9 were were tested tested and and results results are listedare listed in Table in Table5. For 5. all Forcases, all thecases, influence the influence of the number of the number of cylinders of cylinders was found was to befound negligible. to be negligible. As a consequence, As a consequence, further calculations further werecalculations undertaken were with undertaken a number with of cylinders a number as smallof cylinders as possible as small in order as possible to reduce in the order computation to reduce time. the Indeed,computation meshes time. with Indeed, a small meshes number with of cylinders a small require number a smaller of cylinders number require of nodes a smaller to coincide number with allof thenodes details to coincide of the mesh. with all the details of the mesh.

Table 5. Effect of the number of cylinders. Table 5. Effect of the number of cylinders. Wave Number with Wave Number with Wave Number with Max Deviation Porosity τ Mean (m 1) Wave1 Wave 1 Wave 1 − to the Mean 4 Cylinders (m− ) 36 Cylinders (m− ) 100 Cylinders (m− ) Max Number with Number with Number with Porosity0.3 τ 1.6968 1.7110 1.7122Mean 1.7067 (m−1) Deviation 0.58% to 0.54 2.0772 Cylinders 36 Cylinders 2.0996 100 Cylinders 2.1009 2.0926 0.74% the Mean 0.7 2.2946(m−1) 2.3030(m−1) (m 2.3037−1) 2.3004 0.25% 0.9 2.5033 2.5052 2.5052 2.5046 0.05% 0.3 1.6968 1.7110 1.7122 1.7067 0.58% 3.2. Scaling0.5 Effect 2.0772 2.0996 2.1009 2.0926 0.74% 0.7 2.2946 2.3030 2.3037 2.3004 0.25% A second0.9 set of calculations2.5033 concerns2.5052 the study of scaling2.5052 eff ects. An initial2.5046 pattern corresponding0.05% to the misaligned arrangement was repeated until reaching a tank length of L = 1.8 m; see Figure 12. 3.2. Scaling Effect A second set of calculations concerns the study of scaling effects. An initial pattern corresponding to the misaligned arrangement was repeated until reaching a tank length of L = 1.8 m; see Figure 12.

(a) (b) (c)

Figure 12. Porosity value τ = 0.5 with a tank of length of (a) L = 0.6 m and 2 cylinders; (b) L = 1.2 m and 4 cylinders and (c) L = 1.8 m and 6 cylinders.

The objective was to quantify the deviation of the non-dimensional wave number when the pattern was repeated. Obviously, this quantity did not remain the same and the results listed in Table 6 demonstrated that the deviation was not very important except for small porosities. Nevertheless, the deviation to the mean for small value of porosity remained a good approximation for general studies. Consequently, further computations are run with the smallest repeatable pattern.

J. Mar. Sci. Eng. 2020, 7, x FOR PEER REVIEW 10 of 17

3. Estimation of the Porosity Effect on the Dispersion Relation

3.1. Independence of 𝐺(𝜏) against the Number of Cylinders A first set of calculations concerns the relation between the number of cylinders and the wave number for a given value of porosity. Three different meshes were tested with respectively 4, 36 and 100 cylinders, as shown in Figure 11. The quantities that remained the same in all variants were • The porosity τ; • Length and width; • Nx/Ny ratio.

(a) (b) (c)

Figure 11. Porosity τ = 0.5 with (a) 4 cylinders; (b) 36 cylinders and (c) 100 cylinders in the tank.

Porosity values of τ = 0.3, 0.5, 0.7 and 0.9 were tested and results are listed in Table 5. For all cases, the influence of the number of cylinders was found to be negligible. As a consequence, further calculations were undertaken with a number of cylinders as small as possible in order to reduce the computation time. Indeed, meshes with a small number of cylinders require a smaller number of nodes to coincide with all the details of the mesh.

Table 5. Effect of the number of cylinders.

Wave Wave Wave Max Number with Number with Number with Porosity τ Mean (m−1) Deviation to 4 Cylinders 36 Cylinders 100 Cylinders the Mean (m−1) (m−1) (m−1) 0.3 1.6968 1.7110 1.7122 1.7067 0.58% 0.5 2.0772 2.0996 2.1009 2.0926 0.74% 0.7 2.2946 2.3030 2.3037 2.3004 0.25% 0.9 2.5033 2.5052 2.5052 2.5046 0.05%

3.2. Scaling Effect A second set of calculations concerns the study of scaling effects. An initial pattern J.corresponding Mar. Sci. Eng. 2020 to, 8the, 960 misaligned arrangement was repeated until reaching a tank length of L = 111.8 of m; 17 see Figure 12.

(a) (b) (c)

Figure 12.12. Porosity value τ = 0.5 with a tank of length of (a) L == 0.6 m andand 22 cylinders;cylinders; ((b)) LL == 1.2 mm and 44 cylinderscylinders andand ((cc)) LL == 1.8 m and 6 cylinders.

The objectiveobjective was was to quantifyto quantify the deviationthe deviation of the non-dimensionalof the non-dimensional wave number wave when number the pattern when wasthe repeated.pattern was Obviously, repeated. this Obviously, quantity didthis not quantity remain did the no samet remain and the the results same and listed the in results Table6 demonstratedlisted in Table that6 demonstrated the deviation that was the not deviation very important was not except very forim smallportant porosities. except forNevertheless, small porosities. the deviation Nevertheless,to the meanJ. Mar. for smallSci. Eng. value2020, 7, ofx FOR porosity PEER REVIEW remained a good approximation for general studies. Consequently, 11 of 17 the deviationJ. Mar. Sci. Eng. to 2020 the, 7 ,mean x FOR PEERfor smallREVIEW value of porosity remained a good approximation for 11 of general17 furtherstudies. computations Consequently, are further run with computations the smallest are repeatable run with pattern. the smallest repeatable pattern. Table 6. Effect of the number of cylinders. Table 6. Eﬀect of the number of cylinders. kτ/k1 with 2 kτ/k1 with 4 kτ/k1 with 6 Max kτ/k1 with 2 kτ/k1 with 4 kτ/k1 with 6 Max Porosity τ Cylinders Cylinders Cylinders Mean (m−1) Deviation to Porosity τ k /k withCylinders kCylinders/k with Cylindersk /k with Mean (m−1) DeviationMax Deviation to Porosity τ τ 1 τ 1 τ 1 Mean (m 1) (m−11) (m−1) 1 (m−1) 1 − theto Mean the Mean 2 Cylinders (m(m−−1)) 4 Cylinders(m−1 (m) − ) 6 Cylinders(m−1) (m− ) the Mean 0.30.3 0.61530.6153 0.64210.6421 0.6464 0.6464 0.6346 0.6346 3.1% 3.1% 0.50.5 0.76310.7631 0.78320.7832 0.7877 0.7877 0.7780 0.7780 1.9% 1.9% 0.70.7 0.85230.8523 0.86830.8683 0.8700 0.8700 0.8635 0.8635 1.3% 1.3% 0.90.7 0.94950.8523 0.95270.8683 0.8700 0.9532 0.8635 0.9518 1.3% 0.2% 0.9 0.9495 0.9527 0.9532 0.9518 0.2%

3.3. Estimation of G(τ𝐺)(in𝜏) the Case of Regular Arrangements 3.3. Estimation of 𝐺(𝜏) in the Case of Regular Arrangements 3.3.1. FEM Convergence and Validation 3.3.1. FEM Convergence and Validation In orderIn order to extend to extend the previous the previous results results and studyand stud theiry their property property of repeatability, of repeatability, a series a series of numerical of testsnumerical was performed tests was for performed a tank size for of La tank= 0.6 size m andof LW = 0.6= 0.3 m and m containing W = 0.3 m twocontaining cylinders two and cylinders using both alignedand and using misaligned both aligned arrangements. and misaligned One arrangemen hundredts. meshes One hundred were generatedmeshes were with generated the mesh with generator the presentedmesh generator in Section presented 2.2 for porosity in Section value 2.2 for ranging porosity from valueτ= ranging0.3 up from to τ= τ 0.95;= 0.3 up see to Figures τ = 0.95; 13 see and 14. In FigureFigures 15 ,13 the and calculated 14. In Figure results 15, were the plottedcalculated for results two different were plotted levels for of two refinement. different Forlevels the of whole refinement. For the whole range of porosity value from 0.3 to 0.95, no bigger difference than 0.23% rangerefinement. of porosity For value the whole from 0.3 range to 0.95, of porosity no bigger value difference from 0.3 thanto 0.95, 0.23% no bigger for the difference misaligned than arrangement 0.23% for the misaligned arrangement and 0.63% for the aligned arrangement was observed between the and 0.63%two levels for theof refinement. aligned arrangement Thus, it was was assumed observed that the between average the of two4186 levels nodesof was refinement. enough to Thus,exploit it was assumedtwo levels that the of refinement. average of Thus, 4186 nodesit was assumed was enough that tothe exploit average the of 4186 results. nodes was enough to exploit the results.

(a) (a) (b) (c) Figure 13. Different misaligned arrangements and meshes used for the numerical campaign with (a) FigureFigure 13. 13.Di ﬀDifferenterent misaligned misaligned arrangementsarrangements and and meshes meshes used used for the for numerical the numerical campaign campaign with (a) with 𝜏 = 0.3; (b) 𝜏 = 0.6; and (c) 𝜏 = 0.95. (a) τ 𝜏= =0.3; 0.3; ( b(b)) τ𝜏= = 0.6;0.6; and ( (cc)) 𝜏τ = 0.95.0.95.

(a) (a) (b) (c) Figure 14. Different aligned arrangements and meshes used for the numerical campaign with (a) 𝜏 = FigureFigure 14. Di 14.ﬀ Differenterent aligned aligned arrangements arrangements and and meshes meshes used used forfor thethe numericalnumerical campaign campaign with with (a()a 𝜏) τ= = 0.3; 0.3; (b) 𝜏 = 0.6; and (c) 𝜏 = 0.95. (b) τ0.3;= 0.6; (b) and = 0.6; (c) τand= 0.95.(c) = 0.95.

J. Mar. Sci. Eng. 2020, 7, x FOR PEER REVIEW 12 of 17 J. Mar. Sci. Eng. 2020, 7, x FOR PEER REVIEW 12 of 17 J. Mar. Sci. Eng. 2020, 8, 960 12 of 17 1 1 / k k / k k

Figure 15. Calculated non-dimensional wavenumber for different levels of mesh refinement, FigureFiguremisaligned 15. 15.Calculated arrangement.Calculated non-dimensional non-dimensional wavenumber wavenumber for for di differentfferent levels levels of of mesh mesh refinement, refinement, misalignedmisaligned arrangement. arrangement. 3.3.2. Simplified Dispersion Equation in the Studied Porous Medium 3.3.2. Simplified Dispersion Equation in the Studied Porous Medium 3.3.2.From Simplified the above Dispersion set of Equaticomputations,on in the FigureStudied 16 Porous was plotted Medium for both aligned and misaligned From the above set of computations, Figure 16 was plotted for both aligned and misaligned arrangements. In the same figure the analytical expression provided by Molin et al. [26], Equation arrangements.From the In above the same set figure of computations, the analytical Figure expression 16 was provided plotted by for Molin both et aligned al. [26], and Equation misaligned (16) (16) that was considered among other formulations was also plotted for comparison. thatarrangements. was considered In the among same other figure formulations the analytical was expression also plotted provided for comparison. by Molin et al. [26], Equation (16) that was considered among other formulations was also plotted for comparison. 1 1 0.9 0.9

1 0.8

1 0.8 / k / k

/ k / 0.7

k 0.7 aligned cylinders 0.6 alignedmisaligned cylinders cylinders 0.6 misalignedG( )= 1/2 cylinders 0.5 G( )= 1/2 0.50.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

FigureFigure 16. 16.Results Results of the of numericalthe numerical campaign campaign concerning concerningG(τ) shownG(τ) shown by dots. by The dots. expression The expression G(τ) = τG(1/2τ) = 1/2 consideredFigureτ considered 16. among Results among other of the formulationsother numerical formulations bycampaign Molin by Molin et concerning al. [ 26et] al. is also[26] G(τ is) shown shown also shown by by using dots. by the usingThe solid expression the line. solid line. G(τ ) = τ1/2 considered among other formulations by Molin et al. [26] is also shown by using the solid line. SeveralSeveral polynomial polynomial expressions expressions of ofG( G(τ)τ were) were proposed proposed based based on on a least-squarea least-square ﬁt, fit, as as obtained obtained usingusing theSeveral the constrained constrained polynomialG( τG() = τexpressions)1 =for 1 forτ = τ 1.= 1.of For For G( the τthe) alignedwere aligned proposed case case we we based obtained obtained on froma fromleast-square the the regression regression fit, as analysis obtained analysis forusingfor polynomial polynomial the constrained degree degree 2, G( 2,τ ) = 1 for τ = 1. For the aligned case we obtained from the regression analysis

for polynomial degree 2, 𝐺G(𝜏()τ=) = −0.0508𝜏0.0508τ2++ 0.5474𝜏0.5474τ + +0.5033, 0.5033, 0.3 < < τ 𝜏 < 0.95, < 0.95, (24)(24) − 𝐺(𝜏) = −0.0508𝜏 + 0.5474𝜏 + 0.5033, 0.3 < 𝜏 < 0.95, (24) and for polynomial degree 3, and for polynomial degree 3, and for polynomial degree 3, G(τ) = 0.9484τ3 2.0054τ2 + 1.7998τ + 0.2572, 0.3 < τ < 0.95. (25) −

J. Mar. Sci. Eng. 2020, 8, 960 13 of 17

Furthermore, for τ > 0.5, the curve looks linear enough so that a linear regression was also proposed as follows: G(τ) = 0.4651τ + 0.5349, 0.5 < τ < 0.95. (26)

The error between the expression of G(τ) and the numerically tested cases is presented in Table7 below:

Table 7. Comparison between diﬀerent polynomial models for the aligned arrangement.

Deg. of the Polynomial Max. Deviation to the Model Max. Deviation to the Model (%) 1 (τ > 0.5) 0.0075 0.97% 2 0.0357 3.57% 3 0.0150 2.39%

In the case of the misaligned arrangement, the following expressions were obtained, respectively:

G(τ) = 0.0586τ2 + 0.5822τ + 0.4764 , 0.3 < τ < 0.95 , (27) − G(τ) = 0.8044τ3 1.7165τ2 + 1.6445τ + 0.2676 , 0.3 < τ < 0.95 (28) − The error between the expression of G(τ) and the numerical case is presented in the Table8 below:

Table 8. Comparison between diﬀerent polynomial models for the misaligned arrangement.

Deg. Max. Deviation to the Mean Max. Deviation to the Mean (%) 1 (τ > 0.5) 0.0056 0.73% 2 0.0314 5.10% 3 0.0138 2.24%

Finally, for high porosity values, the two curves were very close, see Figures 17 and 18. This is not surprising because kτ/k 1 when τ 1. So, it was suggested to use the following expression 1 → → G(τ) = 0.4763τ + 0.5237, 0.5 < τ < 0.95, (29)

which is the average of Equations (26) and (28) and was plotted in Figure 19. We got very good results J.from Mar. Sci. this Eng. regression, 2020, 7, x FOR and PEER the deviationREVIEW did not become larger than 1.31%. 14 of 17

0.9 1 1 1

/ k / k / / k 0.8 k k k

0.7 aligned arrangement regression deg.3 0.6 0.2 0.4 0.6 0.8 1

(a) (b) (c)

FigureFigure 17. 17. PolynomialPolynomial regressions regressions for for 𝐺(𝜏)G(τ) in case of aligned aligned arrangements: arrangements: ( (aa)) polynomial polynomial degree degree 1; 1; ((bb)) polynomial polynomial degree degree 2 and ( c) polynomial degree 3. 1 1 1 / k / k / / k / k k k

(a) (b) (c)

Figure 18. Polynomial regressions for 𝐺(𝜏) in the case of a misaligned arrangement: (a) polynomial deg.1; (b) polynomial deg.2 and (c) polynomial deg.3. 1 / k k

Figure 19. Linear regression for 𝜏 > 0.5 for both aligned and misaligned arrangements.

In concluding this part we could observe that both the present results agreed well with the dispersion relation proposed by Molin et al. [26] for porosities above 0.7. If for porosity down to 0.3, a better fit was obtained through the polynomial approximation of degree 3, and for porosity above 0.5, a linear fit appeared appropriate.

3.4. Estimation of 𝐺(𝜏) in the Case of Random Arrangements It was demonstrated in Section 3.3 that the dispersion equation could be very well modeled in the case of regular arrangements of cylinders. Nevertheless, in nature the arrangement is usually random. In this section, how relevant it is to model the studied porous medium based on random arrangements and extending the previous expressions was check. Ten random arrangements were tested with 16 cylinders per mesh for given porosity value of τ = 0.6, 0.7, 0.8 and 0.9 and were

J. Mar. Sci. Eng. 2020, 7, x FOR PEER REVIEW 14 of 17

0.9

1 J. Mar. Sci. Eng. 2020, 7, x FOR PEER REVIEW1 1 14 of 17

/ k / k / / k 0.8 k k k

1 0.7 aligned arrangement 0.9 regression deg.3 0.6 1 1 1 0.2 0.4 0.6 0.8 1

/ k / k / / k 0.8

k k k (a) (b) 0.7 (c) aligned arrangement regression deg.3 0.6 Figure 17. Polynomial regressions for 𝐺(𝜏) in case of aligned arrangements:0.2 (a 0.4) polynomial 0.6 0.8 degree 1 1; (b) polynomial degree 2 and (c) polynomial degree 3. (a) (b) (c)

J. Mar. Sci.Figure Eng. 2020 17. Polynomial, 8, 960 regressions for 𝐺(𝜏) in case of aligned arrangements: (a) polynomial degree14 1; of 17 (b) polynomial degree 2 and (c) polynomial degree 3. 1 1 1 / k / k / / k / k k k 1 1 1 / k / k / / k / k k k

(a) (b) (c) (a) (b) (c) Figure 18. Polynomial regressions for 𝐺(𝜏) in the case of a misaligned arrangement: (a) polynomial 𝐺(𝜏) deg.1; Figure(bFigure) polynomial 18. 18.Polynomial Polynomial deg.2 regressions regressions and (c) forpolynomial forG (τ) in the in thedeg.3. case case of aof misaligned a misaligned arrangement: arrangement: (a) ( polynomiala) polynomial deg.1;deg.1; (b )(b polynomial) polynomial deg.2 deg.2 and and (c) ( polynomialc) polynomial deg.3. deg.3. 1 1 / k k / k k

Figure 19. Linear regression for 𝜏 > 0.5 for both aligned and misaligned arrangements.

In concludingFigure 19. Linear this part regression we could for τ >observe0.5 for boththat aligned both the and present misaligned results arrangements. agreed well with the dispersionFigure relation19. Linear proposed regression by Molin for 𝜏 et > al. 0.5 [26] for for both porosities aligned above and misaligned0.7. If for porosity arrangements. down to 0.3, a betterIn concluding fit was obtained this part through we could the polynomial observe that approximation both the present of degree results 3, and agreed for porosity well with above Inthe concluding0.5, dispersion a linear fit relationthis appeared part proposed appropriate.we could by Molin observe et al. [26that] for both porosities the present above 0.7. results If for porosity agreed down well with the dispersionto 0.3, relation a better fit proposed was obtained by throughMolin et the al. polynomial [26] for porosities approximation above of degree 0.7. If 3, for and porosity for porosity down to 0.3, above3.4. Estimation 0.5, a linear of fit𝐺( appeared𝜏) in the Case appropriate. of Random Arrangements a better fit was obtained through the polynomial approximation of degree 3, and for porosity above 0.5, a linear3.4. Estimation fitIt was appeared demonstrated of G(τ) appropriate.in the Casein Section of Random 3.3 that Arrangements the dispersion equation could be very well modeled in the case of regular arrangements of cylinders. Nevertheless, in nature the arrangement is usually random.It was In demonstrated this section, inhow Section relevant 3.3 that it is theto model dispersion the studied equation porous could medium be very wellbased modeled on random in 3.4. Estimation of 𝐺(𝜏) in the Case of Random Arrangements thearrangements case of regular and arrangements extending the of previous cylinders. expressi Nevertheless,ons was check. in nature Ten the random arrangement arrangements is usually were random. In this section, how relevant it is to model the studied porous medium based on random It wastested demonstrated with 16 cylinders in Section per mesh 3.3 for that given the porosity dispersion value equationof τ = 0.6, could0.7, 0.8 beand very 0.9 andwell were modeled in arrangements and extending the previous expressions was check. Ten random arrangements were the casetested of withregular 16 cylinders arrangements per mesh forof givencylinders. porosity Ne valuevertheless, of τ = 0.6, 0.7,in 0.8nature and 0.9 th ande arrangement were compared is usually random.with In the this expression section, given how byrelevant Equation it (29).is to Results model concerning the studiedG(τ )porous are shown medium in Figure based 20 and on random arrangementspresented and a deviation extending in the the random previous arrangement expressi casesons aswas compared check. to Ten the regularrandom arrangement; arrangements were tested seewith also 16 Table cylinders9. It is observed per mesh that for the modelgiven tendedporosity to overestimate value of τ the = 0.6, non-dimensional 0.7, 0.8 and wave 0.9 and were number in most cases, still, however, the derived expression was useful in describing the macroscopic effect of porosity on the porous medium, especially for larger values of porosity. J. Mar. Sci. Eng. 2020, 7, x FOR PEER REVIEW 15 of 17 compared with the expression given by Equation (29). Results concerning G(τ) are shown in Figure 20 and presented a deviation in the random arrangement cases as compared to the regular arrangement; see also Table 9. It is observed that the model tended to overestimate the non- dimensional wave number in most cases, still, however, the derived expression was useful in describing the macroscopic effect of porosity on the porous medium, especially for larger values of J. Mar. Sci. Eng. 2020, 8, 960 15 of 17 porosity.

0.95

0.9 1

/ k 0.85 k 0.8

0.75 random simulations model 0.7 0.5 0.6 0.7 0.8 0.9 1

Figure 20. Validity test of the model against random cylinder arrangements. Figure 20. Validity test of the model against random cylinder arrangements. Table 9. Deviation from the model for non-determinist cases.

PorosityTableτ 9.Max. Deviation Deviation from to the the Model model Max. for non-determinist Deviation to the Model cases. (%) 0.6 0.0914 12.72%Max. 0.7 0.1139Max. 15.33% Deviation to 0.8Porosity τ 0.0773 Deviation to 9.34% 0.9 0.0785the 8.98% Model the Model (%) Concerning the extension of0.6 present results to0.0914 more general configurations12.72% it is seen that the number of cylinders was not important and the most significant parameters were the porosity and the cylinder 0.7 0.1139 15.33% arrangement. The latter were required for a direct application of these results without concern about the scale. Furthermore,0.8 if the tank or domain0.0773 length /width ratio9.34% is not restricted, the error is found to be acceptable in most0.9 cases, and the deviation0.0785 is less important8.98% for high porosity values and large number of cylinders. In the present work the discussion on the effect of the porosity, Concerningthe specific surface, the extension through the of number present of cylinders, results andto mo there arrangement general configurations of the cylinders on it the is local seen that the wavenumber within the porous media was carried out through sloshing conditions within a tank of number of cylinders was not important and the most significant parameters were the porosity and given length, but valid whatever the water depth. Through the eigenvalues of the problem of sloshing, the cylinderthe calculated arrangement. fundamental The (or latter 1st harmonic) were required wave frequency for a direct had a application wavelength λ ofpat these the scale results of without concernthe about porous the media, scale. given Furthermore, by λp = 2Lp. Calculation if the tank of theor eigenvaluesdomain length/width for harmonic oscillations ratio is not of order restricted, the error isn givesfound the to frequency be acceptable corresponding in most to λ cases,p = 2Lp an/n.d A the complete deviation information is less on important the relation betweenfor high porosity valuesthe and wavelength large number and the of frequency cylinders. on a In given the frequency present rangework can the be discussion achieved by on means the ofeffect the present of the porosity, numerical method for various tank lengths, and this study is planned for future work. the specific surface, through the number of cylinders, and the arrangement of the cylinders on the local wavenumber4. Discussion and within Conclusions the porous media was carried out through sloshing conditions within a tank of givenDifferent length, modifications but valid are whatever proposed the concerning water depth. the dispersion Through relation the through eigenvalues a dense of array the of problem of sloshing,cylinders the calculated considered asfundamental a porous medium (or 1st within harmonic) an enclosed wave domain. frequency The derived had expressions a wavelength were λp at the scale ofestablished the porous using media, different given regular by and λ irregularp = 2Lp. arrangements. Calculation As of already the eigenvalues demonstrated for experimentally harmonic by oscillations Arnaud et al. [24], it was observed that for a given porosity and for a given arrangement, the wavenumber of order n gives the frequency corresponding to λp = 2Lp/n. A complete information on the relation did not depend on the number of cylinders, or in other words, on the specific surface, defined as between the wavelength and the frequency on a given frequency range can be achieved by means of the solid–fluid contact surface per volume unit. According to the results presented, the modification of the presentthe dispersion numerical relation method was strongly for various dependent tank to theleng arrangement.ths, and this These study results is planned were consistent for future with work. the numerical results based on the integral matching method, which showed through an interference 4. Discussionprocess that and the Conclusions wavelength depended on the arrangement as shown when staggered or aligned Different modifications are proposed concerning the dispersion relation through a dense array of cylinders considered as a porous medium within an enclosed domain. The derived expressions were established using different regular and irregular arrangements. As already demonstrated

J. Mar. Sci. Eng. 2020, 8, 960 16 of 17 cylinders were considered (see Figure 17 of Rey et al. [30]). In the case of regular arrangements, the wavenumber evolved linearly to the value provided by the standard linear water-wave dispersion relation for porosity larger than 0.7. Then, for low porosity (lower than 0.5) the wavenumber in the porous medium suddenly fell away from the water-wave value and this finding was repeated for all tested configurations. The present results need to be validated before being applied to general arrangements and configurations, and this task, including investigation of a very high value of porosity that requires increased computation cost, is planned for future work. Finally, as concerns the sloshing problem in a tank with vertical walls and structures, if flow separation effects are considered weak (in case of slow motions), the linear approach provides a useful approximate solution. In this case the present FEM is a very efficient computational tool providing useful information concerning the natural sloshing modes applicable to more general shape of tanks, with arbitrary distributions of internal structures with general cross section, which could be further exploited in CFD techniques generalizing the solution to include non-linear and viscous flow effects.

Author Contributions: Conceptualization, K.B., J.T., V.R.; Formal analysis, J.J.; Investigation, J.J.; Project administration, K.B.; Resources, K.B.; Software, J.J.; Supervision, K.B.; Validation, J.J., V.R.; Visualization, J.J.; Writing—original draft, J.J., J.T., Writing—review and editing, K.B., J.T. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Acknowledgments: The authors would like to thank Angeliki Karperaki from the Lab. of Ship and Marine Hydrodynamics of NTUA for useful discussion concerning the implementation of the FEM code. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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