Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 341385, 15 pages http://dx.doi.org/10.1155/2014/341385

Research Article Second-Order Time-Dependent Mild-Slope Equation for Wave Transformation

Ching-Piao Tsai,1 Hong-Bin Chen,2 and John R. C. Hsu3,4

1 Department of Civil Engineering, National Chung Hsing University, Taichung 402, Taiwan 2 Department of Leisure and Recreation Management, Chihlee Institute of Technology, New Taipei 220, Taiwan 3 Department of Marine Environment & Engineering, National Sun Yat-Sen University, Kaohsiung 804, Taiwan 4 School of Civil & Resources Engineering, University of Western Australia, Crawley, WA 6009, Australia

Correspondence should be addressed to Ching-Piao Tsai; [email protected]

Received 16 October 2013; Accepted 1 May 2014; Published 4 May 2014

Academic Editor: Efstratios Tzirtzilakis

Copyright © 2014 Ching-Piao Tsai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This study is to propose a wave model with both wave dispersivity and nonlinearity for the wave field without water depth restriction. A narrow-banded sea state centred around a certain dominant wave frequency is considered for applications in coastal engineering. A system of fully nonlinear governing equations is first derived by depth integration of the incompressible Navier-Stokes equation in conservative form. A set of second-order nonlinear time-dependent mild-slope equations is then developed by a perturbation scheme. The present nonlinear equations can be simplified to the linear time-dependent mild-slope equation, nonlinear longwave equation, and traditional Boussinesq wave equation, respectively. A finite volume method with the fourth-order Adams-Moulton predictor-corrector numerical scheme is adopted to directly compute the wave transformation. The validity of the present model is demonstrated by the simulation of the Stokes wave, cnoidal wave, and solitary wave on uniform depth, nonlinear wave shoaling on a sloping beach, and wave propagation over an elliptic shoal. The nearshore wave transformation across the surf zone is simulated for 1D wave on a uniform slope and on a composite bar profile and 2D wave field around a jetty. These computed wave height distributions show very good agreement with the experimental results available.

1. Introduction mild-slope equation was in an elliptic type, it was not efficient in performing direct numerical calculation and also Accurate estimation of design waves, especially wave heights in treating the open boundary conditions. The conventional and their spatial distributions, is essential for the plan- schemes also required large computer storage and computing ning and installation of coastal structures, calculation of time. Later, numerical models, based on an elliptic form of longshore sediment transport, and prediction of nearshore the mild-slope equation, were proposed to compute the wave topographical changes. In practical engineering applications, field for a large coastal area2 [ –5]. These models obtained a narrow-banded sea state centred on a certain dominant the solution for the governing equations by iterations using wave frequency has remained popular for several decades, powerful and sophisticated algorithm. despite the use of random wave theories in more recent time. The mild-slope equation was also cast in a parabolic Because wave transformation from deep water into shallow form to promote computational efficiency [6] and to handle region involves nonlinear process due to shoaling, combined the problem with weak wave reflection7 [ ]. However, the refraction and diffraction, and reflection and breaking, var- parabolicapproximationmethodwasfoundinadequatefor ious forms for mild-slope equations have been developed. determining the wave field in the presence of a highly reflec- Previously, wave transformations involving refraction and tive structure. Later, Copeland [8] expressed the mild-slope diffraction had been calculated separately for simple beach equation as a pair of first-order hyperbolic equations, without topography, until Berkhoff [1] who presented a linear mild- thelossofthereflectedwave.WatanabeandMaruyama[9] slope equation for the combined effects. Because the original extended the set of time-dependent mild-slope equations 2 Mathematical Problems in Engineering including a wave energy dissipation factor for modelling 2. Nonlinear Wave Equations the nearshore wave field. Tsai et al. [10]incorporatedboth the nonlinear shoaling factor and the breaking-wave energy 2.1. Governing Equations. Consider a three-dimensional dissipation factor into the time-dependent mild-slope equa- wave motion in an incompressible, inviscid, and homoge- tionstoimprovetheaccuracyofthecalculationofwave neous fluid, the principle of conservation of mass gives the transformation across the surf zone. Taking large bottom following continuity equation: gradient into the mild-slope equation, the so-called extended 𝜕𝑤 mild-slope equation or the modified mild-slope equation ∇⋅𝑢+⃗ =0 (1) were also developed for waves over rapid variation of bottom 𝜕𝑧 [11, 12]. On the other hand, the complementary mild-slope equation was derived in terms of a stream function vector and the incompressible Navier-Stokes equations in conserva- rather than in terms of a velocity potential [13, 14]. All of these tive form lead to Euler’s equations as are in the family of linear mild-slope equations. 𝜕𝑢⃗ 𝜕𝑢⃗ 1 Based on the second-order Stokes wave theory, the + 𝑢⋅∇⃗ 𝑢+𝑤⃗ =− ∇𝑝, (2) nonlinear mild-slope equation with parabolic type was devel- 𝜕𝑡 𝜕𝑧 𝜌 oped [15, 16]. These theories are adaptable to the relatively 𝜕𝑤 𝜕𝑤 1 𝜕𝑝 deeper water. Belibassakis and Athanassoulis [17]proposed + 𝑢⋅∇𝑤+𝑤⃗ =− −𝑔 𝜕𝑡 𝜕𝑧 𝜌 𝜕𝑧 (3) a coupled-mode theory, by using a local mode expansion of the wave potential in conjunction with Luke variational principle, to extend the second-order Stokes theory for the in the horizontal and vertical directions of a Cartesian 𝑢⃗ = (𝑢, V) 𝑤 case of a generally shaped bottom profile connecting two half- coordinates system, respectively. In (1)–(3), and strips of constant depths; their model was then extended with are the Eulerian velocity components in the horizontal plane 𝑝 𝜌 application to nonlinear water waves propagating in variable and vertical direction, respectively; is the pressure, is the 𝑔 𝑡 bathymetry regions [18]. On the other hand, Boussinesq- density of the fluid, is the acceleration due to gravity, is the ∇ = (𝜕/𝜕𝑥, 𝜕/𝜕𝑦) type equations developed by Peregrine [19] were extended to time, and is the horizontal vector operator, 𝑥 𝑦 𝑧 cover varying depth adaptable to the shallow water [20]. The where , ,and are the Cartesian coordinates. improved Boussinesq-type equations can provide the wave The fluid motion described by(1)–(3)hastosatisfy dispersion and shoaling characteristics from shallow water to boundary conditions at the free surface and on the seabed. water depth with 𝑘ℎ =40,where𝑘 isthewavenumberand First, the dynamic and kinematic boundary conditions at the ℎ is the water depth [21]. Recently, Li [22] derived nonlinear free surface are wave equations based on combining the second-order Stokes 𝑝=0 𝑧=𝜂(𝑥, 𝑦,) 𝑡 , wave theory with the Boussinesq theory. (4) Many investigators have tried to develop nonlinear wave 𝜕𝜂 equations adaptable to arbitrary water depth. For instance, a 𝑤= + 𝑢⋅∇𝜂⃗ 𝑧=𝜂(𝑥,𝑦,𝑡) (5) 𝜕𝑡 weakly nonlinear and fully dispersive model was introduced by Nadaoka et al. [23], who generated a set of equations and the bottom boundary condition on an impermeable that rendered a single time-dependent nonlinear mild-slope seabed is equation for water waves [24]. The numerically simulated cnoidal waves and the second-order Stokes waves were found 𝑤=−𝑢⋅∇ℎ⃗ 𝑧=−ℎ(𝑥,𝑦) (6) in excellent agreement with theoretical solutions. However, their nonlinear model was only simulated to unidirectional 𝜂(𝑥, 𝑦, 𝑡) ℎ(𝑥, 𝑦) waves, rather than a two-dimensional wave field involving in which is the water surface elevation and waves combining refraction-diffraction and breaking. the local water depth. In this paper, a new nonlinear wave equation is developed for modelling wave field covering the entire range from 2.2. Derivation of the Nonlinear Wave Equation. To develop deep to shallow water and involving wave breaking. It starts a nonlinear wave equation, we perform depth-integration on from the depth-integration of the continuity equation and the governing equations (1)–(3) separately and then substitute nonlinear momentum equation. A set of fully dispersive the boundary conditions (4)–(6) into the appropriate resul- and weakly nonlinear wave equations is then developed by tant forms. First, upon integrating (1)verticallyfromseabed introducing a perturbation technique. The theoretical deriva- tothefreesurfaceandsubstitutingtheboundaryconditions tion is outlined in Section 2.InSection 3,weintroducethe given by (5)and(6), it yields finite volume method with a fourth-order Adams-Moulton 𝜂 predictor-corrector technique, rather than the usual finite 𝜕𝜂 +∇⋅∫ 𝑢𝑑𝑧=0.⃗ (7) difference and finite element schemes, for directly computing 𝜕𝑡 −ℎ the nonlinear equations. Finally, the present model is verified against Stokes wave and cnoidal wave propagation on uni- Second, after integrating the momentum equation (3)in form depth and nonlinear wave shoaling on a sloping beach. the vertical direction, from an arbitrary depth 𝑧 to the Examples for wave propagation on an elliptic shoal and wave free surface, and inserting the boundary condition for the transformation across the surf zone are also presented. pressure at the free surface, (4), it produces Mathematical Problems in Engineering 3

𝑝 𝜂 𝜕𝑤 𝜂 𝜕𝑤 =𝑔(𝜂−𝑧) + ∫ ( ) 𝑑𝑧 + ∫ (𝑢⋅∇𝑤+𝑤⃗ ) 𝑑𝑧. The flow velocity 𝑤 in the vertical direction 𝑧 can be 𝜌 𝑧 𝜕𝑡 𝑧 𝜕𝑧 obtained upon depth integration of (1)from−ℎ to 𝑧 and the (8) bottom boundary condition equation (6), rendering

𝑓 (𝑘ℎ,) 𝑧 Substituting (8)into(2) renders the momentum equation 𝑤(𝑥,𝑦,𝑧,𝑡)=−∇⋅[ 𝑈(𝑥,𝑦,𝑡)],⃗ 𝑘 for the 𝑥-and𝑦-axis in the horizontal directions, respectively, (12) as follows: 𝑘 (ℎ+𝑧) 𝑓 (𝑘ℎ,) 𝑧 = sinh . cosh 𝑘ℎ 𝜕⇀󳨀𝑢 𝜕𝑢⃗ +𝑔∇𝜂+(𝑢⋅∇⃗ 𝑢+𝑤⃗ ) 𝜕𝑡 𝜕𝑧 Inserting (11)into(7) gives the following depth-integrated (9) continuity equation: 𝜂 𝜕𝑤 𝜂 𝜕𝑤 =−∇∫ ( )𝑑𝑧−∇∫ (𝑢⋅∇𝑤+𝑤⃗ )𝑑𝑧. 𝑧 𝜕𝑡 𝑧 𝜕𝑧 𝜕𝜂 𝜂 +∇⋅[(∫ 𝐹𝑑𝑧)𝑈]=0.⃗ (13) 𝜕𝑡 −ℎ It is worth noting that the convective terms, (𝑢⋅∇⃗ 𝑢+⃗ 𝑤𝜕𝑢/𝜕𝑧)⃗ and (𝑢⋅∇𝑤+𝑤𝜕𝑤/𝜕𝑧)⃗ in (8)and(9), are transformed Substituting (11)and(12)into(8), and after laborious manip- 2 2 to ∇(𝑢⋅⃗ 𝑢+𝑤⃗ ) and 𝜕(𝑢⋅⃗ 𝑢+𝑤⃗ )/𝜕𝑧 in Nadaoka et al. [23], ulations, we obtain the pressure expression as based on the irrotational assumption before the integration process described above. However, such transformations 𝑝 𝜕𝑈⃗ 𝜕𝑈⃗ =𝑔(𝜂−𝑧)−𝑝̃ (∇ ⋅ )−𝑝̃ (∇ℎ ⋅ ) do not completely satisfy the irrotationality for variable 𝜌 𝑎 𝜕𝑡 𝑏 𝜕𝑡 depth, that is, 𝑢⃗𝑧 −∇𝑤=0̸ ,whichwasmentionedin[23]. Subsequently, the present equation (9)isretainedinthe − 𝑝̃ [(𝑈⋅∇)(∇⋅⃗ 𝑈)⃗ − (∇ ⋅ 𝑈)⃗ (∇ ⋅ 𝑈)]⃗ (14) original forms of the nonlinear convective terms; that is, we 𝑐 preserve, completely, the irrotational condition. It is noted ̃ ⃗ ⃗ 2 that Belibassakis and Athanassoulis [18] preserved this flow − 𝑝𝑑 [(∇ ⋅ 𝑈) (∇ℎ ⋅ 𝑈)] + 𝑂 [(∇ℎ) ], irrotationality by working with a potential formulation; they derived a nonlinear coupled-mode system of equations on the where horizontal plane, with respect to the free surface elevation and the unknown modal amplitudes, in conjunction with Luke 𝜂 𝑓 𝑝̃𝑎 = ∫ ( )𝑑𝑧, variational principle. 𝑧 𝑘 The horizontal velocity vector can be expressed in terms of a vertical distribution function and a corresponding 𝜂 𝜕 𝑓 𝑝̃𝑏 = ∫ ( )𝑑𝑧, horizontal velocity vector at the still water level, in the form 𝑧 𝜕ℎ 𝑘 of (15) 𝜂 𝑓 𝑝̃𝑐 = ∫ 𝐹( )𝑑𝑧, 𝑁 𝑧 𝑘 ⇀󳨀 cosh 𝑘𝑚 (ℎ+𝑧) ⃗ 𝑢(𝑥,𝑦,𝑧,𝑡)= ∑ 𝑈𝑚 (𝑥,𝑦,𝑡), (10) 𝑘 ℎ 𝜂 𝑚=1 cosh 𝑚 𝜕 𝑓 𝜕𝐹 𝑓 𝑝̃𝑑 = ∫ [𝐹 ( )− ( )] 𝑑𝑧. 𝑧 𝜕ℎ 𝑘 𝜕ℎ 𝑘 ⃗ where 𝑈𝑚(𝑥, 𝑦, 𝑡) is the 𝑚th component of the horizontal The last term on the right-hand-side (RHS) of(14)isthe velocity vector at 𝑧=0,and𝑘𝑚(𝑥, 𝑦) is the local wave number. The full dispersivity for a broad-banded wave field higher order term of the bottom slope which may be omitted can be attained by taking only a few terms [23]. However, under the assumption of mild-slope. Multiplying (9) by the vertical distribution function 𝐹 considering the fact that a narrow-banded sea state centred 𝑧=−ℎ 𝜂 around a certain dominant wave frequency may be described in (11) and then integrating it from to ,aswell with sufficient accuracy by the single-component model as after complex mathematical manipulations, produces a equations [24], that is, 𝑁=1,itbecomes depth-integrated expression for the momentum equation in the form of

⇀󳨀𝑢(𝑥,𝑦,𝑧,𝑡)=𝐹(𝑘ℎ,) 𝑧 𝑈(𝑥,𝑦,𝑡),⃗ 𝜕𝑈⃗ 𝛼̃ + 𝛼̃ 𝑔∇𝜂 + 𝛼̃ (𝑈⋅∇)⃗ 𝑈⃗ 1 𝜕𝑡 2 3 𝑘 (ℎ+𝑧) (11) 𝐹 (𝑘ℎ,) 𝑧 = cosh . ⃗ ⃗ ⃗ ⃗ cosh 𝑘ℎ + 𝛼̃4 (∇ ⋅ 𝑈) 𝑈+𝛼̃5 (∇ℎ ⋅ 𝑈) 𝑈

𝜕𝑈⃗ 𝜕𝑈⃗ This single-component model has remained popular and = 𝛼̃ ∇(∇⋅ )+𝛼̃ (∇ ⋅ )∇ℎ useful in practical coastal engineering applications. 6 𝜕𝑡 7 𝜕𝑡 4 Mathematical Problems in Engineering

𝜕𝑈⃗ + 𝛼̃ (∇ ⋅ )∇𝜂 Substituting (18)into(13)and(16)andtakingTaylorseries 8 𝜕𝑡 expansion on the still water level 𝑧=0,weobtaintheconti- nuity equation and the momentum equation, respectively, as 𝜕𝑈⃗ follows: + 𝛼̃9 (∇ℎ ⋅ ∇𝜂) 𝜕𝑡 0 𝜕𝜂 3 +∇⋅[(∫ 𝐹𝑑𝑧+𝜂)𝑈]=𝑂(𝜀⃗ ), (19) ⃗ ⃗ ⃗ ⃗ 𝜕𝑡 −ℎ + 𝛼̃10∇[(𝑈⋅∇)(∇⋅𝑈) − (∇ ⋅ 𝑈) (∇ ⋅ 𝑈)] 0 ⃗ 0 2 𝜕𝑈 + 𝛼̃ [(∇ ⋅ 𝑈)⃗ (∇ℎ ⋅ ∇) 𝑈]⃗ (∫ 𝐹 𝑑𝑧) +(∫ 𝐹𝑑𝑧)𝑔∇𝜂 11 −ℎ 𝜕𝑡 −ℎ

0 + 𝛼̃ [(𝑈⋅∇)(∇⋅⃗ 𝑈)⃗ − (∇ ⋅ 𝑈)⃗ (∇ ⋅ 𝑈)]⃗ ∇𝜂 1 2 2 12 + [ ∫ 𝐹(𝐹 −𝑓 )𝑑𝑧 2 −ℎ + 𝛼̃ [(∇ ⋅ 𝑈)⃗ (∇ℎ ⋅ 𝑈)⃗ ∇𝜂]( +𝑂[ ∇ℎ)2] 13 0 𝑘2𝑐4 −2 ∫ 𝐹𝑑𝑧 ]∇(𝑈⋅⃗ 𝑈)⃗ 2 (16) −ℎ 𝑔

0 𝐹−𝐹2 𝜕𝑈⃗ in which =(∫ 𝑑𝑧) ∇ (∇⋅ ) 2 −ℎ 𝑘 𝜕𝑡 𝜂 𝜂 𝜂 𝛼̃ = ∫ 𝐹2𝑑𝑧, 𝛼̃ = ∫ 𝐹𝑑𝑧, 𝛼̃ = ∫ 𝐹3𝑑𝑧, 0 𝜕 1−𝐹 𝜕𝑈⃗ 1 2 3 +[∫ 𝐹 ( )𝑑𝑧](∇⋅ )∇ℎ+𝑂(𝜀3). −ℎ −ℎ −ℎ 2 −ℎ 𝜕ℎ 𝑘 𝜕𝑡 𝜂 2 (20) 𝛼̃4 =−∫ 𝐹𝑓 𝑑𝑧, −ℎ These two equations are accurate to the second-order of 𝜀. 𝜂 𝜂 𝐹 𝑓 2 𝜕𝐹 𝜕 𝑓 Substituting the expressions of and givenin(11)and 𝛼̃5 = ∫ [𝐹 −𝑘𝐹𝑓 ( )] 𝑑𝑧, 𝛼̃6 = ∫ 𝐹𝑝̃𝑎𝑑𝑧, −ℎ 𝜕ℎ 𝜕ℎ 𝑘 −ℎ (12)into(19)and(20), we obtain the final expression of the continuity equation and momentum equation as follows: 𝜂 2 𝛼̃7 =2∫ 𝐹𝑝̃𝑏𝑑𝑧, 𝜕𝜂 𝑐 −ℎ +∇⋅[( +𝜂)𝑈]⃗ = 0, 𝜕𝑡 𝑔 (21) 𝜂 𝜕𝑝̃ 𝜂 𝜕𝑝̃ 𝑎 𝑏 2 4 𝛼̃8 = ∫ 𝐹 𝑑𝑧, 𝛼̃9 = ∫ 𝐹 𝑑𝑧, 𝜕𝑈⃗ 1 𝑘 𝑐 −ℎ 𝜕𝜂 −ℎ 𝜕𝜂 𝑛 +𝑔∇𝜂+ (1 − 3 )∇(𝑈⋅⃗ 𝑈)⃗ 𝜕𝑡 2 𝑔2 𝜂 (22) 2 𝛼̃10 = ∫ 𝐹𝑝̃𝑐𝑑𝑧, 1 𝑐 (1−𝑛) 𝜕𝑈⃗ −ℎ = ∇[ (∇ ⋅ )] 𝑐2 𝑘2 𝜕𝑡 𝜂 𝜂 𝜕𝑝̃ ̃ 𝑐 𝛼̃11 =2∫ 𝐹𝑝𝑑𝑑𝑧, 𝛼̃12 = ∫ 𝐹 𝑑𝑧, in which −ℎ −ℎ 𝜕𝜂 2 𝑔 𝜂 𝑐 = tanh 𝑘ℎ, (23) 𝜕𝑝̃𝑑 𝑘 𝛼̃13 = ∫ 𝐹 𝑑𝑧. −ℎ 𝜕𝜂 1 2𝑘ℎ (17) 𝑛= [1+ ]. (24) 2 sinh (2𝑘ℎ) ⃗ ⃗ The momentum equation, (16), retains a full nonlinear The coefficient to the third term ∇(𝑈⋅𝑈) on the left- form under the mild-slope condition. As it is difficult to solve hand-side of (22) consists of two parts. The first part, 1/2,is derived from the nonlinear convective terms of the horizontal (16) directly by a numerical scheme; a perturbation technique 2 4 2 is then proposed. The surface displacement and velocity field velocity;andthesecondpart,−3/2(𝑘 𝑐 /𝑔 ), is the nonlinear 2 4 2 are expressed by effect of the vertical velocity. The value of (𝑘 𝑐 /𝑔 ) tends to 1 in deep water, implying that the nonlinear variation of vertical 𝜂=𝜀𝜂 +𝜀2𝜂 +⋅⋅⋅, velocityisinthesameorderasthehorizontalvelocityandit 1 2 cannot be neglected, whereas it approximates reasonably to (18) 𝑐 𝑈=𝜀⃗ 𝑈⃗ +𝜀2𝑈⃗ +⋅⋅⋅ , zero in the very shallow water. It is worth noting that the and 1 2 𝑘 can be computed for a prescribed wave period and a local depth based on the dispersion relationship shown in (23). The where 𝜀 is the dimensionless nonlinearity parameter of wave dispersion characteristics are essentially the same as in [23] amplitude, and the subscriptions 𝑖 (𝑖 = 1, 2, ..) in (18) denote because both models are approximated to the second-order the 𝑖th order approximation. of 𝜀. Mathematical Problems in Engineering 5

2 Equations (21)and(22) together become the so-called The term of 𝑂[(∇ℎ) ] canbeomittedbytheassumption second-order time-dependent mild-slope equation, because of mild-slope bottom. Upon inserting (29)into(28), the theycanbeusedtorepresentthewavefieldsthatareweakly nonlinear equation developed in this study is reduced to nonlinear and fully dispersive and on the basis of the mild- ⇀󳨀 slopeassumption.Differingfrom[23], the present equations 𝜕𝑄 𝑐2 + ∇(𝑛𝜂)=0. (30) (21)and(22) can be directly used to calculate the wave trans- 𝜕𝑡 𝑛 formationofwavefieldinplanebecausetheverticalvelocity has been embedded in the nonlinear convective terms into Equations (25)and(30) are identical to the so-called time- a simpler wave model. The consistency of the equation can dependent mild-slope equation in Copeland [8]. Eliminating ⃗ be simplified into several well-established theories, such as flow rate 𝑄 from (25)and(30) renders the elliptic mild-slope linear mild-slope equation, nonlinear long wave equation, equation in Berkhoff1 [ ]. and Boussinesq equation. The process is demonstrated below. 2.4. Nonlinear Long Wave Equation. For long wave theory in 2.3. Linear Mild-Slope Equation. Upon dropping the nonlin- shallow water, wave celerity (𝑐),groupvelocity(𝑐𝑔), and flow ear term in (21) for the sake of linear (Airy) wave theory, this rate (𝑄⃗)areusuallygivenby equationcanberewrittenas 𝑐=𝑐 = √𝑔ℎ, 𝑄⃗ = (ℎ+𝜂) 𝑈⃗, (31) 𝜕𝜂 ⇀󳨀 𝑔 +∇⋅𝑄=0, (25) 𝜕𝑡 where 𝑈⃗ is the horizontal velocity identical to mean flow 𝑄⃗ velocity in shallow water condition. By dropping the nonlin- where is the linear flow rate of the waves defined by ear terms in the vertical direction in (22) and introducing (31), 0 2 a system of nonlinear long wave equations can be written in 𝑐 ⃗ 𝑄⃗ = ∫ 𝑢𝑑𝑧=⃗ 𝑈⃗. (26) terms of the mean flow velocity 𝑈,givenby −ℎ 𝑔 𝜕𝑈⃗ 1 +𝑔∇𝜂+ ∇(𝑈⋅⃗ 𝑈)⃗ = 0. (32) On the other hand, (22)canberewrittenforthelinearwaves 𝜕𝑡 2 as For an irrotational flow, the term ∇(𝑈⋅⃗ 𝑈)/2⃗ canbereplacedby 𝜕𝑈⃗ 1 𝑐2 (1−𝑛) 𝜕𝑈⃗ 𝑛 +𝑔∇𝜂= ∇[ (∇ ⋅ )] . (𝑈⋅∇)⃗ 𝑈⃗, thus rendering the usual expression for the nonlinear 𝜕𝑡 𝑐2 𝑘2 𝜕𝑡 (27) long wave equation.

Substituting (26)into(27), then we obtain 2.5. Boussinesq Equations. Boussinesq equations account for the effect of wave nonlinearity but weak dispersion, that the 𝜕𝑄⃗ 𝑐2 1 (1−𝑛) 𝜕𝑄⃗ (1−𝑛) 𝜕𝑄⃗ + ∇𝜂 = ∇ [ ∇⋅ − ∇𝑐2 ⋅ ] . wave celerity and group velocity may be approximated by the 2 2 2 𝜕𝑡 𝑛 𝑛 𝑘 𝜕𝑡 𝑐 𝑘 𝜕𝑡 first two terms in the Taylor series, (28) 1 𝑐2 =𝑔ℎ(1− 𝑘2ℎ2), In the right-hand side of (28), 3 𝑐 (33) 𝑔 1 2 2 1 (1−𝑛) 𝜕𝑄⃗ 1 (1−𝑛) 𝜕2𝜂 𝑛= =(1− 𝑘 ℎ ). ∇[ (∇ ⋅ )] = − ∇[ ] 𝑐 3 𝑛 𝑘2 𝜕𝑡 𝑛 𝑘2 𝜕𝑡2 2 Substituting (33)into(21)and(22) and replacing 𝑘 𝑈⃗ by 1 = ∇[𝑐2 (1−𝑛) 𝜂] −∇(∇⋅𝑈)⃗ for the irrotational flow in shallow water condition, 𝑛 the present nonlinear mild-slope equation gives (1−𝑛) = 𝜂∇𝑐2 𝜕𝜂 ℎ3 𝑛 +∇⋅[(ℎ+𝜂)𝑈]⃗ + ∇2 (∇ ⋅ 𝑈)⃗ = 0, 𝜕𝑡 3 𝑐2 𝑐2 (34) + ∇𝜂 − ∇(𝑛𝜂), 𝜕𝑈⃗ 1 𝑛 𝑛 +𝑔∇𝜂+ ∇(𝑈⋅⃗ 𝑈)⃗ = 0. 𝜕𝑡 2 1 (1−𝑛) 𝜕𝑄⃗ (1−𝑛) 𝜕𝑄⃗ − ∇[ ∇𝑐2 ⋅ ]=− ∇𝑐2 (∇ ⋅ ) 𝑛 𝑐2𝑘2 𝜕𝑡 𝑛𝑐2𝑘2 𝜕𝑡 Equations (34) are the standard form of traditional Boussi- nesq equations. +𝑂[(∇ℎ)2] 3. Numerical Methods (1−𝑛) =− 𝜂∇𝑐2 +𝑂[(∇ℎ)2]. 𝑛 3.1. Finite Volume Method. A finite volume method is used to (29) calculate the wave heights based on the governing equations 6 Mathematical Problems in Engineering given by (21)and(22). These two equations are first proceeded And the equations of the corrector-step are expressed by using a volume integration over a specific control volume ∀, as follows: 𝑚+1 𝑚 Δ𝑡 𝑚+1 𝑚 𝑚−1 𝑚−2 𝜕𝜂 𝑐2 𝜂 =𝜂 + (9Φ + 19Φ −5Φ +Φ ) , (39) ∫ ( )𝑑∀+∫ [∇ ⋅ ( +𝜂)𝑈]⃗ 𝑑∀ =0, 24 ∀ 𝜕𝑡 ∀ 𝑔 𝑚+1 𝑚 {[𝐴] 𝑈}⃗ ={[𝐴] 𝑈}⃗ 𝜕𝑈⃗ 1 3𝑘2𝑐4 ∫ (𝑛 )𝑑∀+∫ [𝑔∇𝜂 + (1 − )∇(𝑈⋅⃗ 𝑈)]⃗ 𝑑∀ 2 ∀ 𝜕𝑡 ∀ 2 𝑔 Δ𝑡 + (9Ψ𝑚+1 + 19Ψ𝑚 −5Ψ𝑚−1 +Ψ𝑚−2), 24 1 𝑐2 (1−𝑛) 𝜕𝑈⃗ = ∫ [∇ ( ∇⋅ )] 𝑑∀. (40) 2 2 ∀ 𝑐 𝑘 𝜕𝑡 (35) where By the divergence theorem, (35) renders 𝜕𝜂 1 𝑐2 1 2 𝑐2 + 𝑐2 − ( ) =− ∫ [𝑁⋅(⃗ +𝜂)𝑈]⃗ 𝑑𝑆 +⃗ ⃗ −⃗ ⃗ Φ=− ∑ {𝑆𝐾 ⋅[( +𝜂)𝑈] − 𝑆𝐾 ⋅[( +𝜂)𝑈] }, 𝜕𝑡 ∀ 𝑠 𝑔 ∀ 𝑔 𝑔 𝑚 𝐾=1 𝐾 𝐾 𝜕 1 1 𝑐2 (1−𝑛) {𝑛𝑈−⃗ ( ) ∫ 𝑁⋅[⃗ (∇ ⋅ 𝑈)⃗ 𝐼]⃗ 𝑑𝑆} 𝑔 2 𝜕𝑡 ∀ 𝑐2 𝑘2 Ψ=− ∑ [𝑆+⃗ ⋅(𝜂)+ 𝐼−⃗ 𝑆−⃗ ⋅(𝜂)− 𝐼]⃗ 𝑚 𝑠 𝑚 𝐾 𝐾 𝐾 𝐾 ∀ 𝐾=1 1 =− 𝑔 ∫ 𝑁⋅(𝜂𝐼)⃗ 𝑑𝑆 ∀ 1 1 3𝑘2𝑐4 𝑠 − [ (1 − )] ∀ 2 𝑔2 1 1 𝑘2𝑐4 − [ (1 − 3 )] ∫ 𝑁⋅[(⃗ 𝑈⋅⃗ 𝑈)⃗ 𝐼]⃗ 𝑑𝑆 2 ∀ 2 𝑔 𝑠 2 𝑚 + − × ∑ [𝑆+⃗ ⋅(𝑈⋅⃗ 𝑈)⃗ 𝐼−⃗ 𝑆−⃗ ⋅(𝑈⋅⃗ 𝑈)⃗ 𝐼]⃗ , (36) 𝐾 𝐾 𝐾 𝐾 𝐾=1 in which subscription 𝑚 denotes the mean quantities refer- ∀ 𝑆 𝐼⃗ 2 2 + ring to the centre of the volume , is the control surface, is 1 1 𝑐 (1 − 𝑛) + 𝑁⃗ 𝑆 [𝐴] 𝑈=𝑛⃗ 𝑈−⃗ ( ) ∑ [( ) ⋅(∇⋅𝑈)⃗ 𝐼⃗ the unit vector, and is the normal vector of .Byknowing ∀ 𝑐2 𝑘2 𝐾 𝐾=1 𝐾 the quantities at a starting time step (denotes as 𝐺𝑚), the integral form of (36) can be used to calculate the change in 2 − 𝑐 (1−𝑛) − 𝐺𝑚 within a time increment, and thus update the 𝐺𝑚 values −( ) ⋅(∇⋅𝑈)⃗ 𝐼]⃗ , 𝑘2 𝐾 of the next time step. 𝐾

Regarding the computational mesh, it is envisaged that 2 1 + − thegridsizeshouldbefineenoughtoresolveimportant ∇⋅𝑈=⃗ ∑ [𝑆+⃗ ⋅(𝑈)⃗ − 𝑆−⃗ ⋅(𝑈)⃗ ] 𝐾 𝐾 𝐾 𝐾 characteristic details yet coarse enough to keep storage and ∀ 𝐾=1 computational time requirements reasonable. For instance, a (41) fine mesh is used in the area where the wave transformation is significant, such as the region around the structure or near wavebreaking.Fortheefficientandaccuratecomputationfor in which superscript 𝑚 represents the computed values of 𝜂, an irregular geometric boundary, the boundary-fitted mesh 𝑈⃗, Φ,andΨ at the 𝑚th time iteration step, and superscript system is obtained in the present computation according to a 𝑚+1denotes the predicted values at the 𝑚+1th time method based on Poisson equations [25]. step. The subscript 𝐾 in (41)representsthetwodirectionsin 𝑥-𝑦 plane, in which superscript “+” denotes the quantities 3.2. Adams-Moulton Predictor-Corrector Schemes. The intheupstreamofcontrol-volumeand“−”forthosein +⃗ −⃗ fourth-order Adams-Moulton predictor-corrector technique the downstream. The section vectors denoted as 𝑆𝐾 and 𝑆𝐾 is introduced to compute (36). The discrete equations of the represent opposing faces of the middle volume. The definition predictor-step are given by of variables in the finite volume method is illustrated in Δ𝑡 Figure 1. In the discrete continuity equations, that is, (37) 𝜂𝑚+1 =𝜂𝑚 + (55Φ𝑚 − 59Φ𝑚−1 + 37Φ𝑚−2 −9Φ𝑚−3), 24 and (39), the new values can be computed explicitly. But the (37) implicit process is required to obtain the new values for the momentum equations, (38)and(40), because the dispersive 𝑚+1 𝑚 {[𝐴] 𝑈}⃗ ={[𝐴] 𝑈}⃗ term involves the time and spatial derivatives simultaneously. That is, the surface elevation 𝜂 canbesolvedexplicitly,but Δ𝑡 𝑈⃗ + (55Ψ𝑚 − 59Ψ𝑚−1 + 37Ψ𝑚−2 −9Ψ𝑚−3). the velocity is solved by an implicit numerical scheme. The 24 successive over-relaxation iteration method (SOR) is adopted (38) for the implicit scheme. Mathematical Problems in Engineering 7

→ → where + + + S2 , U2 , 𝜂2

⃗ ⃗ ⃗ 𝐴1 ={[(𝑆1) ⋅(𝑅𝑖+1,𝑗+1 + 𝑅𝑖+1,𝑗) → → 𝑖+1,𝑗+1 + + + S1 , U1 , 𝜂1 +(𝑆⃗ ) ⋅(𝑅⃗ + 𝑅⃗ )] 1 𝑖+1,𝑗 𝑖+1,𝑗−1 𝑖+1,𝑗 → → → S−, U−, 𝜂− U, 𝜂 1 1 1 −[(𝑆⃗ ) ⋅(𝑅⃗ + 𝑅⃗ ) 1 𝑖−1,𝑗+1 𝑖−1,𝑗+1 𝑖−1,𝑗

+(𝑆⃗ ) ⋅(𝑅⃗ + 𝑅⃗ )] 1 𝑖−1,𝑗 𝑖−1,𝑗−1 𝑖−1,𝑗 → → − − − S2 , U2 , 𝜂2 +[(𝑆⃗ ) ⋅(𝑅⃗ + 𝑅⃗ ) 2 𝑖+1,𝑗+1 𝑖+1,𝑗+1 𝑖,𝑗+1 Figure 1: Definition sketch of the variables in finite volume method. +(𝑆⃗ ) ⋅(𝑅⃗ + 𝑅⃗ )] 2 𝑖,𝑗+1 𝑖−1,𝑗+1 𝑖,𝑗+1

−[(𝑆⃗ ) ⋅(𝑅⃗ + 𝑅⃗ ) → → 2 𝑖+1,𝑗−1 𝑖+1,𝑗−1 𝑖,𝑗−1 i−1, j+1 ( ) ( ) S2 i, j+1 S2 i+1, j+1 i,j+1 i+1,j+1 +(𝑆⃗ ) ⋅(𝑅⃗ + 𝑅⃗ )]} , 2 𝑖,𝑗−1 𝑖−1,𝑗−1 𝑖,𝑗−1 → ( ) → S1 ( ) 𝐴 ={[(𝑆⃗ ) ⋅(𝜂 +𝜂 ) 𝐼⃗ i−1,j+1 S1 2 1 𝑖+1,𝑗+1 𝑖+1,𝑗+1 𝑖+1,𝑗 i+1,j+1 → U, 𝜂 i, j +(𝑆⃗ ) ⋅(𝜂 +𝜂 ) 𝐼]⃗ i−1, j i+1, j 1 𝑖+1,𝑗 𝑖+1,𝑗−1 𝑖+1,𝑗 ⃗ ⃗ → −[(𝑆1) ⋅(𝜂𝑖−1,𝑗+1 +𝜂𝑖−1,𝑗) 𝐼 → (S ) 𝑖−1,𝑗+1 ( ) 1 S1 i+1,j i−1,j +(𝑆⃗ ) ⋅(𝜂 +𝜂 ) 𝐼]⃗ 1 𝑖−1,𝑗 𝑖−1,𝑗−1 𝑖−1,𝑗 i−1, j−1 i+1, j−1 +[(𝑆⃗ ) ⋅(𝜂 +𝜂 ) 𝐼⃗ 2 𝑖+1,𝑗+1 𝑖+1,𝑗+1 𝑖,𝑗+1 → i, j−1 ( ) → S2 ( ) i,j−1 S2 i+1,j−1 +(𝑆⃗ ) ⋅(𝜂 +𝜂 ) 𝐼]⃗ 2 𝑖,𝑗+1 𝑖−1,𝑗+1 𝑖,𝑗+1

−[(𝑆⃗ ) ⋅(𝜂 +𝜂 ) 𝐼⃗ Figure 2: Mesh arrangement and corresponding variables around 2 𝑖+1,𝑗−1 𝑖+1,𝑗−1 𝑖,𝑗−1 the centre of a control volume. +(𝑆⃗ ) ⋅(𝜂 +𝜂 ) 𝐼]}⃗ , 2 𝑖,𝑗−1 𝑖−1,𝑗−1 𝑖,𝑗−1 ⃗ ̈ ̈ ⃗ 𝐴3 ={[(𝑆1) ⋅(𝑈𝑖+1,𝑗+1 + 𝑈𝑖+1,𝑗) 𝐼 As shown in Figure 2, the quantities of variables 𝜂 and 𝑖+1,𝑗+1 𝑈⃗ are set in the centre of the control volume. Then the flux +(𝑆⃗ ) ⋅(𝑈̈ + 𝑈̈ ) 𝐼]⃗ terms, Φ and Ψ,in(37)to(40),canbeexpressedas 1 𝑖+1,𝑗 𝑖+1,𝑗−1 𝑖+1,𝑗 ⃗ ̈ ̈ ⃗ −[(𝑆1) ⋅(𝑈𝑖−1,𝑗+1 + 𝑈𝑖−1,𝑗) 𝐼 1 𝑖−1,𝑗+1 Φ𝑖,𝑗 =− ⋅𝐴1, 2(∀ +∀ +∀ +∀ ) +(𝑆⃗ ) ⋅(𝑈̈ + 𝑈̈ ) 𝐼]⃗ 𝑖,𝑗 𝑖+1,𝑗 𝑖,𝑗+1 𝑖+1,𝑗+1 1 𝑖−1,𝑗 𝑖−1,𝑗−1 𝑖−1,𝑗 𝑔 Ψ𝑖,𝑗 =− ⋅𝐴2 +[(𝑆⃗ ) ⋅(𝑈̈ + 𝑈̈ ) 𝐼⃗ 2 𝑖+1,𝑗+1 𝑖+1,𝑗+1 𝑖,𝑗+1 2(∀𝑖,𝑗 +∀𝑖+1,𝑗 +∀𝑖,𝑗+1 +∀𝑖+1,𝑗+1) 1 (42) +(𝑆⃗ ) ⋅(𝑈̈ + 𝑈̈ ) 𝐼]⃗ − 2 𝑖,𝑗+1 𝑖−1,𝑗+1 𝑖,𝑗+1 4(∀𝑖,𝑗 +∀𝑖+1,𝑗 +∀𝑖,𝑗+1 +∀𝑖+1,𝑗+1) ⃗ ̈ ̈ ⃗ −[(𝑆2) ⋅(𝑈𝑖+1,𝑗−1 + 𝑈𝑖,𝑗−1) 𝐼 3𝑘2𝑐4 𝑖+1,𝑗−1 ×(1− ) ⋅𝐴 , 𝑔2 3 𝑖,𝑗 +(𝑆⃗ ) ⋅(𝑈̈ + 𝑈̈ ) 𝐼]}⃗ , 2 𝑖,𝑗−1 𝑖−1,𝑗−1 𝑖,𝑗−1 8 Mathematical Problems in Engineering

𝑐2 Incident waves 𝑅⃗ =( 𝑖,𝑗 +𝜂 ) 𝑈⃗ , 𝑖,𝑗 𝑔 𝑖,𝑗 𝑖,𝑗

󳰀 𝛼inc Δr 𝑈̈ = 𝑈⃗ ⋅ 𝑈⃗ =𝑈2 +𝑉2 . B B 𝑖,𝑗 𝑖,𝑗 𝑖,𝑗 𝑖,𝑗 𝑖,𝑗 𝛼out (43) Outgoing waves 3.3. Initial and Converge Conditions. The condition of zero flow or cold start has been set as the initial condition for most numerical simulations. In this case, an unsteady fluctuation Figure 3: Sketch of the offshore open boundary. will commonly be produced in the initial computing stage, but it will become negligible when the computation time is long enough, and finally a steady state can be obtained. In the explicit iteration for solving the surface elevation, the where the subscriptions inc and out represent the quantities of incoming and outgoing waves, respectively, Δ𝑟 is the length Courant stability condition has to be satisfied at each time 󸀠 󸀠 of 𝐵𝐵 , 𝐵 is the phantom point of 𝐵,and𝛼 is the angle step, that is, 󸀠 out between outgoing wave and 𝐵𝐵 ,asshowninFigure 3. ∀ The second type is for the side or lateral boundary of Δ𝑡 ≤𝑐 𝑓 ⋅ min (󵄨 󵄨) (44) 󵄨𝑐⋅⃗ 𝑆󵄨⃗ the computation domain, which is generally a virtual open 󵄨 󵄨 boundary. It is treated as having no wave reflection at this boundary, hence, allowing waves to pass freely through them. in which 𝑓𝑐 is the safe coefficient (commonly taken as 0.6). In This can be represented by the computations, several iteration cycles are usually required toachieveasteadyresult.Thecomputedresultsareregarded 𝑡 𝑡−𝜏 (𝜂) 󸀠 =(𝜂) , as reaching a steady state when the iterative errors of the 𝐵 𝐵 𝑡 𝑡−𝜏 (48) computed wave height are less than 2%. As for the velocity (𝑈)⃗ =(𝑈)⃗ . in the horizontal plane, 𝑈⃗, each time interval in the SOR 𝐵󸀠 𝐵 iteration method needs to satisfy the following converge And the third is the internal boundary. If an impermeable condition: structure exists, where full reflection from the structure 󵄨 ⃗ 𝑚+1 ⃗ 𝑚 󵄨 surface is assumed for the internal boundary associated with 󵄨𝑈 − 𝑈 󵄨 −4 Δ𝑒⇀󳨀 = (󵄨 󵄨)≤10 𝑈 max 󵄨 󵄨 (45) wave reflection, it can be set as 󵄨 𝑈⃗ 𝑚+1 󵄨 𝑡 𝑡 (𝜂) 󸀠 =(𝜂) , Δ𝑒⇀󳨀 𝐵 𝐵 in which 𝑈 denotes the absolute maximum error. 𝑡 𝑡 (𝑈⃗ ) =(𝑈⃗ ) , 𝑆𝑡 𝐵󸀠 𝑆𝑡 𝐵 (49) 3.4. Boundary Specifications. Three types of boundary con- ditions are specified for computing the wave field. These are ⃗ 𝑡 ⃗ 𝑡 (𝑈𝑆𝑛) 󸀠 =−(𝑈𝑆𝑛) , appliedusingphantompointexteriorfromthecomputational 𝐵 𝐵 mesh. The first one is an offshore boundary which isthe where subscriptions St and Sn represent the tangent and incident wave conditions. In general, there exist not only normal components at the boundary, respectively. incident waves but also outgoing wave components across the offshore boundary due to the reflection from structures or the shore. Referring to [9, 26], such outgoing waves should be 4. Numerical Results transmitted freely through the boundary. Thus the offshore Validation of the present model is made by comparing the boundary is set as numerical results with analytical solutions and/or experi- (𝜂)𝑡 =(𝜂 )𝑡 +(𝜂 )𝑡−𝜏 mental data for Stokes wave and cnoidal wave propagation 𝐵󸀠 inc 𝐵󸀠 out 𝐵 on uniform depth, one-dimensional wave shoaling, and two- 𝑡 𝑡 𝑡−𝜏 (46) (𝑈)⃗ =(𝑈⃗ ) +(𝑈⃗ ) , dimensional wave transformation across an elliptic shoal. 𝐵󸀠 inc 𝐵󸀠 out 𝐵 Wave transformation involving energy dissipation due to wave breaking is finally simulated for wave propagation on where auniformslopebeachandabar-typebeachandwavearound (𝛼 ) ajetty. 𝜏=Δ𝑟⋅cos out , 𝑐 4.1. Stokes Waves. The first case of validation is the numerical (𝜂 )𝑡 =(𝜂)𝑡 −(𝜂 )𝑡 , (47) out 𝐵 𝐵 inc 𝐵 simulation of Stokes waves to the second-order. The com- 𝑘𝐻 𝑡 𝑡 𝑡 puted results for two different values in intermediate (𝑈⃗ ) =(𝑈)⃗ −(𝑈⃗ ) , ℎ/𝐿 = 0.2 𝐻 out 𝐵 𝐵 inc 𝐵 depth ( ) are depicted in Figure 4,inwhich is Mathematical Problems in Engineering 9

𝐿 𝑎 =𝐻/2 kH = 0.40, h/L = 0.2 the wave height, is the wavelength and 𝑜 .The 1.5 temporal and spatial resolutions are Δ𝑡 = 𝑇/60 and Δ𝑥 = 𝐿/30, respectively, where 𝑇 is the wave period. It can be seen 1.0 that the numerical results are in good agreement with the 0.5 o a theoretical solution of the Stokes waves to a second-order. The / 0.0 𝜂 minor discrepancy in phase for large 𝑥/𝐿 is possibly caused −0.5 by the truncation error in the discrete numerical process −1.0 which affects the solution of the phase velocity. Nevertheless, −1.5 the verification indicates the present model is capable of 012345 producing identical results to the Stokes waves in water of x/L kH = 0.50, h/L = 0.2 finite depth. 1.5 The vertical distribution of the horizontal velocity can be 1.0 calculated from (10) once the solution of 𝑈⃗ is obtained. The 0.5

maximum nondimensional horizontal velocity calculated o a 𝑧/ℎ / 0.0 from (10) is plotted against dimensionless depth in 𝜂 Figure 5. Good agreement is found upon comparing the −0.5 results with the experimental data of Kim et al. [27]. −1.0 −1.5 0 12345 4.2. Cnoidal Waves. It is well known that cnoidal wave x/L theory is preferred than Stokes wave theory in shallow water. Figure 6 demonstrates the simulated wave profiles of cnoidal Figure 4: Simulated Stokes waves in intermediate water depth waves in relative depth of ℎ/𝐿 = 0.05 for two different 𝐻/ℎ compared with analytical solution (circles: present solution, line: values. The temporal and spatial resolutions are Δ𝑡 = 𝑇/40 2nd-order Stokes wave theory). and Δ𝑥 = 𝐿/20 in this computation. The present numerical solution is also found in good agreement with the theoretical 0.4 solution of cnoidal waves. 2 In very shallow water region, cnoidal waves approach the h/(gT ) = 0.050 limit of a solitary wave. Since it is not possible to define a H/h = 0.277 finite wavelength for a solitary wave, as it is infinite in theory, 0.2 the simulation of the solitary wave is often made by taking a very small value of ℎ/𝐿.Hereℎ/𝐿 is taken as 1/50. Figure 7 ℎ/𝐿 illustrates the simulated wave profiles in relative depth of 0 =0.02for𝐻/ℎ =0.20and0.30,respectively.

From the comparisons above for simulations of nonlinear z/h waves on uniform depth, it reveals that the present model is −0.2 adequate for the entire range of water depth from deep to very shallow. The following simulations will cover nonlinear wave shoaling and two-dimensional wave field involving combined wave refraction-diffraction, even wave breaking. −0.4

4.3. Wave Shoaling on Uniform Slope. Shuto [28]proposedan −0.6 equation for nonlinear wave shoaling prior to breaking along 0 0.1 0.2 0.3 0.4 a uniform slope based on a K-dV formulation. His results U/(gh)0.5 were found to agree reasonably well with the experimental Present solution dataforawiderangeofwavesteepness[29]. The validity Experiment, Kim et al. (1992) of the present nonlinear model is further tested against the results of Shuto [28] by comparing the shoaling coefficients Figure 5: Vertical distribution of the computed maximum horizon- 𝐻/𝐻𝑜 versus relative water depth ℎ/𝐿𝑜 for wave steepness talvelocitycomparedwithexperimentaldata. 𝐻𝑜/𝐿𝑜 ranging from 0.001 to 0.08, on uniform beach slope of 1/30, in which 𝐻𝑜 and 𝐿𝑜 are the wave height and wavelength in deep water. As seen in Figure 8,theresults computed from the present model (in solid circles) are in the shoaling coefficient increases as wave steepness increases. good agreement with the nonlinear shoaling theory of Shuto However, the solution computed by the linear wave shoaling [28], despite small fluctuations about the simulated curves is independent of the wave steepness 𝐻𝑜/𝐿𝑜 which apparently within some range of ℎ/𝐿𝑜.Tsaietal.[10]incorporated underestimates any shoaling wave in shallow water compared a nonlinear shoaling factor into the linear time-dependent with the present nonlinear shoaling model, but has the same mild-slope equations; their results (circles in Figure 8)are value between the nonlinear and linear wave shoaling as ℎ/𝐿𝑜 also in good agreement with Shuto [28]. At a particular ℎ/𝐿𝑜, greater than 0.15. 10 Mathematical Problems in Engineering

H/h = 0.10, h/L = 0.05 3.5 2.0 1.5 3.0 Ho/Lo 1.0 0.001

o 0.5 0.002 a

/ 2.5 𝜂 0.0 0.005

−0.5 o 2.0 −1.0 0.01 H/H −1.5 0.02 012345 1.5 x/L 0.04 0.08 H/h = 0.20, h/L = 0.05 1.0 2.0 1.5 0.5 1.0 0.005 0.01 0.02 0.05 0.1 0.2

o 0.5 a h/Lo /

𝜂 0.0 −0.5 Shuto (1988) Tsai et al. (2001) −1.0 Linear solution Present solution −1.5 012345 Figure 8: Comparison on the wave height from the present nonlinear wave shoaling solution with other solutions. x/L

Figure 6: Simulated wave profiles for cnoidal waves compared favourably with analytical solution (circles: present solution, line: better resolve the wave transformation in the computation. Cnoidal wave theory). The simulation results for wave height distribution behind the shoal (seen in Figure 10) exhibit significant contribution of

ao/ho = 0.20, h/L = 0.02 the combined effect of refraction-diffraction. 0.5 The simulation results for wave height distribution along 0.4 the eight transects indicated in Figure 11 are compared with 0.3 the experimental data and the numerical model of Li and o h

/ 0.2 Anastasiou [3], who derived an improved solution using an 𝜂 0.1 elliptic form of the linear mild-slope equation. The com- 0.0 parison in Figure 11 demonstrates that the present nonlinear model can yield the wave height oscillations spatially dis- −0.1 012345tributed due to combined refraction-diffraction effect behind 𝐵 𝐸 x/L an elliptical shoal, particularly on transects to . a /h = 0.30, h/L = 0.02 0.5 o o 4.5. 1D and 2D Wave Transformation across Surf Zone. More- 0.4 over, the momentum equation used in the present model can 0.3 be applied to simulate wave transformation across the surf o h / 0.2 zone, upon incorporating a mixing term, 𝑀𝐷.Inthisway, 𝜂 0.1 (22)becomes 0.0 𝜕𝑈⃗ 𝑔 1 𝑘2𝑐4 −0.1 + ∇𝜂 + (1 − 3 )∇(𝑈⋅⃗ 𝑈)⃗ 012345 𝜕𝑡 𝑛 2𝑛 𝑔2 x/L (50) 1 𝑐2 (1−𝑛) 𝜕𝑈⃗ Figure7:Simulatedwaveprofileforsolitarywavesinveryshallow = ∇[ (∇ ⋅ )] + 𝑀 . 𝑛𝑐2 𝑘2 𝜕𝑡 𝐷 water compared with analytical solution (circles: present solution, line: solitary wave theory). Following Kabiling and Sato [31], the mixing term due to energy dissipation can be expressed by

4.4. Wave Transformation across an Elliptic Shoal. The 2 𝑀 = ] ∇ 𝑈,⃗ (51) hydraulic model test [30] for wave propagation over a 3D 𝐷 𝑒 elliptical shoal on a plane beach with uniform slope 1 : 50 has where ]𝑒 is the eddy viscosity related to an energy dissipation usuallybeenemployedforwavemodelverification.Thewave factor, 𝑓𝐷,duetobreaking, height distribution was measured along eight cross sections indicated in Figure 9.Anincidentwavewithaperiodof 𝑔ℎ 𝑔 𝑄 −𝑄 1.0 sec and wave height of 2.32 cm was used in the simulation. ] = 𝑓 ;𝑓=𝛼 𝛽√ ( 𝑚 𝑟 ). 𝑒 2 𝐷 𝐷 𝐷 tan (52) The finer rectangular grids are set in the shoal region to 𝜔 ℎ 𝑄𝑠 −𝑄𝑟 Mathematical Problems in Engineering 11

Outside shoal Shoal boundary Shoal thickness h = 0.45 − 0.02(5.84 − y󳰀) (x󳰀/4)2 +(y󳰀/3)2 =1 −0.3 + 0.5[1 − (x󳰀/5)2 −(y󳰀/3.75)2]0.5 10

5 Wave Shoal x󳰀 T = 1.0 s Hi = 2.32 cm y󳰀 H

(m) 0 G y

F

ABC DE −5 m 5

0.4 h = 0.4 0.3 0.2 0.15 0.25 0.35 0.1 0.05

−10 −10 −50 51015 x (m)

Figure 9: Geometry showing wave propagation over an elliptic shoal.

T=1s, H = 2.32 cm In (52), 𝑓𝐷 is the energy dissipation factor, 𝜔 (= 2𝜋/𝑇) 10 i is the wave frequency, tan𝛽 is an average bottom slope 2 𝛼 2 from shoreline to wave breaking point, D is a modification coefficient (usually taken as 2.5 as suggested in[32]), 𝑄𝑚 is the 3 2 𝑄 𝑄 5 2 amplitude of flow rate, and 𝑟 and 𝑠 are determined from 2 2 the following equations: 1 2 1 2 3 𝑄 = 0.4 (0.57 + 5.3 𝛽) √𝑔ℎ3, 3 𝑠 tan (m) 0 4 2

y 5 1 2 3 (53) 3 1 2 𝑄 = 0.135√𝑔ℎ . 3 𝑟 2 −5 2 1 In (52), the dissipation factor 𝑓𝐷 is set equal to zero outside 2 3 the surf zone and in any region where 𝑄𝑚 <𝑄𝑟. On the basis of shallow water assumption for wave 2 ⃗ 2 ⃗ −10 breaking, we can simplify ∇ 𝑈=−𝑘𝑈 and render 𝑀𝐷 = −10 −50 51015 ⃗ −𝑓𝐷𝑈,thus(50)becomes x (m) 𝜕𝑈⃗ 𝑔 1 𝑘2𝑐4 H/H + ∇𝜂 + (1 − 3 )∇(𝑈⋅⃗ 𝑈)⃗ + 𝑓 𝑈⃗ i 𝜕𝑡 𝑛 2𝑛 𝑔2 𝐷 (7) 2.4 (3) 1.2 (6) 2.1 (2) 0.9 (54) 1 𝑐2 (1−𝑛) 𝜕𝑈⃗ (5) 1.8 (1) 0.6 = ∇[ (∇ ⋅ )] . (4) 1.5 𝑛𝑐2 𝑘2 𝜕𝑡

Figure 10: Simulation results on wave height distribution behind an This modified momentum equation can now be used for the elliptic shoal. wave transformation after wave breaking. To estimate the 12 Mathematical Problems in Engineering

3.0 3.0 ) ) i i 2.5 2.5 H/H H/H 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 Relative wave height ( height wave Relative Relative wave height ( height wave Relative 0.0 0.0 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5−4−3−2−1012345 Distance y (m) Distance y (m) (a) (b) 3.0 3.0 ) ) i i 2.5 2.5 H/H H/H 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 Relative wave height ( height wave Relative Relative wave height ( height wave Relative 0.0 0.0 −5−4−3−2−1012345 −5−4−3−2−1012345 Distance y (m) Distance y (m) (c) (d) 3.0 3.0 ) ) i i 2.5 2.5 H/H H/H 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5

Relative wave height ( height wave Relative 0.0 ( height wave Relative 0.0 −5 −4 −3 −2 −1 0 1 2 3 4 5 012345678910 11 Distance y (m) Distance x (m) (e) (f) 3.0

3.0 ) ) i i 2.5 2.5 H/H H/H 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 Relative wave height ( height wave Relative Relative wave height ( height wave Relative 0.0 0.0 012345678910 11 012345678910 11 Distance x (m) Distance x (m)

Experiment, Berkhoff 1982( ) Experiment, Berkhoff 1982( ) Li and Anastasiou (1992) Li and Anastasiou (1992) Present solution Present solution (g) (h)

Figure 11: Simulation results of wave height distribution along eight transects shown in Figure 9 andcomparisonwithexperimentaldataand other numerical models. Mathematical Problems in Engineering 13

T = 1.19 s Ho = 2.0 cm H = 6.00 cm 5.0 T = 1.2 s o 𝛼 =60∘ Incident waves 10 o 4.0

(m) 3.0

x 2.0 5

(cm) 2.0 3.0 H 2.5 1.5 2.0 2.5 2.5 2.5 2.5 1.0 2.0 1.5 1.5 1.0 1.0 0 0.5 0.5 0.0 1 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 20 y (m) −5 Figure 14: The computed wave field in the vicinity of a jetty. 0123456 x (m)

Experiment, Nagayama (1983) Tsai et al. (2001) instead of (22), is then used for calculating the wave height Watanabe and Maruyama (1986) Present solution transformation after wave breaking. Figure 12: Comparisons among numerical solutions and laboratory Numerical verification on one-dimensional wave trans- experiment for wave transformation on uniform slope. formation across the surf zone is now considered for two different bottom configurations, a uniform slope in 1/20 and a composite profile with a bar in 1/20 slope, as shown T = 0.94 s in Figures 12 and 13, respectively. The numerical results of Ho = 7.00 cm the present nonlinear model are compared with the linear 10 solution of wave transformation [9] and the experimental data of Nagayama [34].TheresultsofTsaietal.[10]arealso included in these figures. The local wave heights calculated using the present nonlinear model are in better agreement 5 with the experimental data than the linear model. In Figure 12, the computed wave heights increases as it (cm) shoals, which ultimately approaches a maximum at breaking, H then decreases rapidly until reaching the shoreline. For the 0 resultantwaveheighttransformationonacompositeslope, 1 1 20 as in Figure 13,waveheightreachesamaximumpriorto 1 20 20 the bar crest where first breaking occurs on the leading −5 uniform slope of 1/20. Wave height recovers soon after it 01234567 passesthebartroughconnectedtothetrailinguniformslope x (m) on which the second breaking appears. Finally, the surviving Experiment, Nagayama (1983) Tsai et al. (2001) wave diminishes at the shoreline. For the region before the Watanabe and Maruyama (1986) Present solution breaking on a uniform slope (Figure 12)andpriortothe first breaking on a composite slope (Figure 13), the present Figure 13: Comparisons among numerical solutions and laboratory nonlinear solution and linear model agree with each other, experiment for wave transformation on a composite profile of bar otherwise the linear model is found to deviate from the type. experimental results. This confirms that the present nonlinear model is adequate for wave decay, recovery, and secondary breaking on a composite beach profile. location of wave breaking, a breaking criterion proposed by The example of two-dimensional wave transformation Goda [33]isusedasfollows: was conducted for wave field in the vicinity of a jetty, for which experimental and numerical results by Watanabe and 𝐻𝑏 ℎ𝑏 4/3 Maruyama [9] were available. The jetty in the experimental = 0.17 {1−exp [−1.5𝜋 (1+15tan 𝛽)]} (55) 𝐿𝑜 𝐿𝑜 tank was placed perpendicular to the shoreline on a beach with a slope of 1/50. The model jetty was extended to an in which 𝐻𝑏 and ℎ𝑏 are the wave height and water depth offshore distance of 4 m, where the water depth was 8 cm. The ∘ of the breaking wave, respectively, and 𝐿𝑜 is the deep water waves were incident with angle 60 to the contour lines, and wavelength. The computed wave height at any location based the wave height and period in deep water were 2.0 cm and on (21)and(22)isalwayscomparedwiththevaluecalculated 1.2 s, respectively. from the right-hand side of (55). If the computed wave height The computed wave height distribution around the jetty is reaches the breaking criterion, the momentum equation (54), shown in Figure 14.Aclapotis gaufré wave system is visible in 14 Mathematical Problems in Engineering

5 The resultant nonlinear mild-slope equation is computed y=3m by a finite volume method with a fourth-order Adams- 4 Moulton predictor-corrector technique, rather than the usual 3 finite difference or finite element scheme. The validity ofthe

(cm) present nonlinear model is also satisfactorily verified with H 2 examples that cover a wide range of practical problems, such 1 as Stokes wave to a second-order, cnoidal wave propagation on uniform depth, nonlinear wave shoaling on a sloping 0 012345 6beach,andwavetransformationonanellipticshoal.By x (m) introducing an extra term representing energy dissipation 5 duetowavebreakingtothemomentumequation,thepresent y=4.8m nonlinear model is suitable for the wave transformation 4 withinthesurfzoneonauniformslopeandonacomposite 3 profile with a bar on which second wave breaking occurs after

(cm) wave recovering. The comparisons of computed wave height H 2 distributions around a jetty indicate the present nonlinear 1 solutions are in very good agreement with the experimental results. 0 012345 6 x (m) Conflict of Interests 5 y=5.2m The authors declare that there is no conflict of interests 4 regarding the publication of this paper. 3 (cm)

H 2 Acknowledgment 1 Thismaterialwasbasedupontheworksupportedbythe 0 Ministry of Science and Technology, Taiwan, under Grant no. 012345 6NSC 99-2221-E-005-116-MY3. x (m)

Measured (Watanabe and Maruyama 1986) References Linear solution (Watanabe and Maruyama 1986) Present solution [1] J. C. W. Berkhoff, “Computation of combined refraction- diffraction,”in Proceedings of 13th International Confererence on Figure 15: Comparisons among numerical solutions and experi- Coastal Engineering (ASCE ’72),vol.1,pp.10–1972,1972. mental results for distributions of wave height at sections 𝑦 = 3.0 m, [2] V. G. Panchang, B. R. Pearce, G. Wei, and B. Cushman-Roisin, 4.8 m, and 5.2 m shown in Figure 14. “Solution of the mild-slope wave problem by iteration,” Applied Ocean Research,vol.13,no.4,pp.187–199,1991. [3] B. Li and K. Anastasiou, “Efficient elliptic solvers for the mild-slope equation using the multigrid technique,” Coastal frontofthejettybythereflectionfromit.Ontheotherhand, Engineering,vol.16,no.3,pp.245–266,1992. the wave diffraction was observed in the lee of the jetty. The [4] Y. Zhao and K. Anastasiou, “Modelling of wave propaga- comparisons of the wave height distributions at cross-shore tion in the nearshore region using the mild slope equation sections 𝑦 = 3.0 m, 4.8 m, and 5.2 m are shown in Figure 15, with GMRES-based iterative solvers,” International Journal for which indicates the present nonlinear solutions are in better Numerical Methods in Fluids,vol.23,no.4,pp.397–411,1996. agreement with the experimental results. [5]F.S.B.F.OliveiraandK.Anastasiou,“Anefficientcomputa- tional model for water wave propagation in coastal regions,” Applied Ocean Research, vol. 20, no. 5, pp. 263–271, 1998. 5. Conclusions [6] A. C. Radder, “On the parabolic equation method for water- wave propagation,” Journal of Fluid Mechanics,vol.95,no.1,pp. In this paper, the development of a second-order time- 159–176, 1979. dependent mild-slope equation is reported, based on the [7] P. L.-F. Liu and T.-K. Tsay, “On weak reflection of water waves,” fundamental principles of conservation of mass and momen- Journal of Fluid Mechanics,vol.131,pp.59–72,1983. tum, together with a depth-integration procedure. After some [8]G.J.M.Copeland,“Apracticalalternativetothe“mild-slope” appropriate mathematical simplifications, the present nonlin- wave equation,” Coastal Engineering,vol.9,no.2,pp.125–149, ear equation can yield the usual form of linear mild-slope 1985. equation, nonlinear long wave equation, and Boussinesq [9]A.WatanabeandK.Maruyama,“Numericalmodelingof equation, respectively. We have also demonstrated that the nearshore wave field under combined refraction, diffraction present nonlinear model is applicable to the entire range of and breaking,” Coastal Engineering Journal,vol.29,pp.19–39, water depth from deep to very shallow, even wave breaking. 1986. Mathematical Problems in Engineering 15

[10] C.-P. Tsai, H.-B. Chen, and J. R.-C. Hsu, “Calculations of wave [30] J. C. W. Berkhoff, “Refraction and diffraction of water waves; transformation across the surf zone,” Ocean Engineering,vol.28, wave deformation by a shoal, comparison between computa- no. 8, pp. 941–955, 2001. tions and measurements,”Report on Mathematics Investigation [11] P. G. Chamberlain and D. Porter, “The modified mild-slope W 154-III, Delft Hydraulics Laboratory, Delft, The Netherland, equation,” Journal of Fluid Mechanics,vol.291,pp.393–407, 1982. 1995. [31] M. B. Kabiling and S. Sato, “Two-dimensional nonlinear disper- [12] T.-W. Hsu, T.-Y. Lin, C.-C. Wen, and S.-H. Ou, “A comple- sive wave-current and three-dimensional beach deformation mentary mild-slope equation derived using higher-order depth model,” Coastal Engineering in Japan,vol.36,pp.195–212,1993. function for waves obliquely propagating on sloping bottom,” [32] A. Watanabe and M. Dibajnia, “Numerical model of wave Physics of Fluids,vol.18,no.8,ArticleID087106,2006. deformation in surf zone,” in Proceedings of 21st International [13]J.W.KimandK.J.Bai,“Anewcomplementarymild-slope Conference on Coastal Engineering (ASCE ’88),pp.578–587,June equation,” Journal of Fluid Mechanics,vol.511,pp.25–40,2004. 1988. [14] Y. Toledo and Y. Agnon, “Three dimensional application of the [33] Y. Goda, “Irregular wave deformation in the surf zone,” Coastal complementary mild-slope equation,” Coastal Engineering,vol. Engineering in Japan,vol.18,pp.13–26,1975. 58,no.1,pp.1–8,2011. [34] S. Nagayama, Study on the change of wave height and energy in [15] D. K. P.Yue and C. C. Mei, “Forward diffraction of Stokes waves thesurfzone[Bachelorthesis], Yokohama National University, by a thin wedge,” Journal of Fluid Mechanics,vol.99,no.1,pp. Yokohama, Japan, 1983, (Japanese). 33–52, 1980. [16] P. L.-F. Liu and T.-K. Tsay, “Refraction-diffraction model for weakly nonlinear water waves,” Journal of Fluid Mechanics,vol. 141, pp. 265–274, 1984. [17]K.A.BelibassakisandG.A.Athanassoulis,“Extensionof second-order Stokes theory to variable bathymetry,” Journal of Fluid Mechanics,vol.464,pp.35–80,2002. [18] K. A. Belibassakis and G. A. Athanassoulis, “A coupled-mode system with application to nonlinear water waves propagating in finite water depth and in variable bathymetry regions,” Coastal Engineering,vol.58,no.4,pp.337–350,2011. [19] D. H. Peregrine, “Long waves on a beach,” Journal of Fluid Mechanics,vol.27,pp.815–827,1967. [20] P. A. Madseon and H. A. Schaffer, “A review of Boussinesq- type equations for surface gravity waves,” in Advanced Coastal and Ocean Engineering, P. L.-F. Liu, Ed., vol. 5, Word Scientific, Singapore, 1999. [21]P.A.Madsen,H.B.Bingham,andH.Liu,“AnewBoussinesq method for fully nonlinear waves from shallow to deep water,” Journal of Fluid Mechanics,vol.462,pp.1–30,2002. [22] B. Li, “A mathematical model for weakly nonlinear water wave propagation,” Wave Motion,vol.47,no.5,pp.265–278,2010. [23] K. Nadaoka, S. Beji, and Y. Nakagawa, “Afully dispersive weakly nonlinearmodelforwaterwaves,”Proceedings of the Royal Society A,vol.453,no.1957,pp.303–318,1997. [24] S. Beji and K. Nadaoka, “A time-dependent nonlinear mild- slope equation for water waves,” Proceedings of the Royal Society A,vol.453,no.1957,pp.319–332,1997. [25] J. F. Thompson, F. C. Thames, and C. Wayne Mastin, “TOMCAT—a code for numerical generation of boundary- fitted curvilinear coordinate systems on fields containing any number of arbitrary two-dimensional bodies,” JournalofCom- putational Physics,vol.24,no.3,pp.274–302,1977. [26] H.-B. Chen and C.-P. Tsai, “Computations of nearshore wave transformation using finite-volume method,” Applied Ocean Research,vol.38,pp.32–39,2012. [27] C. H. Kim, R. E. Randall, S. Y.Boo, and M. J. Krafft, “Kinematics of 2-D transient water waves using laser Doppler anemometry,” JournalofWaterway,Port,Coastal,&OceanEngineering,vol. 118, no. 2, pp. 147–165, 1992. [28] N. Shuto, “Nonlinear long waves in a channel of variable section,” Coastal Engineering in Japan,vol.17,pp.1–12,1974. [29] K. Horikawa, Nearshore Dynamics and Coastal Processes, Uni- versity of Tokyo Press, 1988. Copyright of Mathematical Problems in Engineering is the property of Hindawi Publishing Corporation and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.