Turbulence Model for a Wave Breaking Simulation

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Article A Modiﬁed k – ε Turbulence Model for a Wave Breaking Simulation

Giovanni Cannata * , Federica Palleschi, Benedetta Iele and Francesco Gallerano

Department of Civil, Constructional and Environmental Engineering, “Sapienza” University of Rome, 18, 00184 Roma (RM), Italy; [email protected] (F.P.); [email protected] (B.I.); [email protected] (F.G.) * Correspondence: [email protected]; Tel.: +39-064-458-5062

Received: 8 October 2019; Accepted: 29 October 2019; Published: 31 October 2019

Abstract: We propose a two-equation turbulence model based on modiﬁcation of the k ε standard − model, for simulation of a breaking wave. The proposed model is able to adequately simulate the energy dissipation due to the wave breaking and does not require any “a priori” criterion to locate the initial wave breaking point and the region in which the turbulence model has to be activated. In order to numerically simulate the wave propagation from deep water to the shoreline and the wave breaking, we use a model in which vector and tensor quantities are expressed in terms of Cartesian components, where only the vertical coordinate is expressed as a function of a time-dependent curvilinear coordinate that follows the free surface movements. A laboratory test is numerically reproduced with the aim of validating the turbulence modiﬁed k ε model. The numerical results − compared with the experimental measurements show that the proposed turbulence model is capable of correctly estimating the energy dissipation induced by the wave breaking, in order to avoid any underestimation of the wave height.

Keywords: turbulence model; wave breaking; shock-capturing scheme

1. Introduction In the contest of hydraulic engineering, the simulation of hydrodynamic fields, turbulence, and concentration fields of suspended solid particles under wave breaking permits the analysis of the effects produced by the structures on sea bottom and shoreline modifications. The definition of turbulent closure relations under wave breaking is significant in order to adequately represent sediment particle suspension and settling phenomena. One of the most used approaches is based on depth-averaged motion equations [1–4], which are obtained by assuming a simplified distribution of the hydrodynamic quantities along the vertical direction (depth-averaged models). In order to quantify the dissipation mechanism in the surf zone, two strategies are considered in the literature. The first strategy is based on an eddy viscosity approach; in particular, Kazolea and Ricchiuto [5] used a turbulent kinetic energy equation to calculate a breaking viscosity coefficient. In the context of the second strategy, the extended Boussinesq equations [1–3] are used to simulate the propagation of waves in deep water, where dispersive effects are dominant; in the surf zone, the nonlinear shallow water equations are used to simulate shoaling and wave breaking, because the dispersive terms of the Boussinesq equations are switched off. In the surf zone, the nonlinear shallow water equations, integrated by a shock-capturing numerical scheme, are used to simulate the wave breaking, where the nonlinear effects are dominant [4]: the dissipation of the total energy obtained in a shallow water shock is used to implicitly model the energy dissipation due to the breaking. It is

Water 2019, 11, 2282; doi:10.3390/w11112282 www.mdpi.com/journal/water Water 2019, 11, 2282 2 of 13 evident that it is necessary to introduce a criterion to establish where and when to switch from one system of equations to the other. In the context defined by the second strategy, the turbulence models are used just in order to determine a distribution of the eddy viscosity coefficient that does not intervene in the simulation of the wave resolving depth-averaged velocity fields but intervene in the simulation of only the concentration fields of suspended solid particles to follow the sea bottom evolution. In this case, the turbulent closure relations that derive from the calculation of the eddy viscosity vertical distribution are carried out by following the line proposed by Fredsøe et al. and Deigaard et al. [6,7] and used by Rakha [8]. In particular, under breaking waves, the instantaneous eddy viscosity vertical distribution is calculated by taking into account the turbulence contribution due to the wave boundary layer and the current and wave breaking [3]. These models are not able to adequately represent the three-dimensional hydrodynamic quantities. The first three-dimensional models for free surface flows numerically solve the Navier–Stokes equations by adopting the volume of fluid technique (VOF) to track the location of the free surface [9,10]: in the VOF technique, the calculus cells are arbitrarily crossed by the vertical fluxes. Therefore, the pressure and kinematic boundary conditions are not correctly assigned on the free surface. Another category of models overcomes the above-mentioned difficulty by mapping the physical domain that varies in time along the vertical direction in order to follow the free surface elevation (σ-coordinate transformation) [11] in a computational domain, which has always a rectangular prismatic shape. In this way it is possible to correctly assign the pressure and kinematic boundary condition on the upper face of the top computational cell [12,13]. These two above-mentioned categories of models empirically introduce, in the surf zone, an appropriate energy dissipation, so that it is possible to represent the decrease of the wave height in the aforementioned zone. Consequently, the location of the initial wave breaking point is settled “a priori”. The definition of the point is where the empirical energy dissipation must be introduced, and therefore the identification of the initial wave breaking point is settled “a priori”. Shock-capturing methods make it possible to model the energy dissipation due to the breaking by the total energy dissipation obtained in a shallow water shock. The σ-coordinate shock-capturing methods locate wave breaking without requiring any “a priori” criterion. In this paper we adopt the new approach proposed by Gallerano et al. [14] in which the motion equations are expressed in terms of variables that are Cartesian based: only the vertical coordinate is expressed as a function of a time-dependent curvilinear coordinate that follows the free surface movements. In this model the motion equations are numerically solved, on a time dependent curvilinear coordinate system, where the vertical coordinate can vary over time, by using a finite-volume shock-capturing scheme, which uses an approximate HLL (Harten, Lax and van Leer)-type Riemann solver [15]. The shock-capturing scheme [14] does not need to use any “a priori” criterion to identify the location of the initial wave breaking point, and it is able to correctly assign the boundary conditions on the free surface. In this model, by using a computational domain that has a limited number of points along the vertical direction, the numerical wave height fits the experimental measurements before the breaking point and the wave breaking is tracked correctly. Ma et al. [12] demonstrated that in general the shock-capturing model, defined in the context of the σ-coordinate, underestimates the energy dissipation at the breaking point and consequently overestimates the wave height in the surf zone. Therefore, it is necessary to introduce turbulence models that are able to adequately represent the specific energy dissipation at the breaking point. The definition of turbulent closure relations under wave breaking implies the necessity to consider the production of turbulent kinetic energy associated with the breaking in order to represent sediment particle suspension and settling phenomena. Water 2019, 11, 2282 3 of 13

In order to develop a wave resolving model for nearshore suspended sediment transport, Ma et al. [16] implemented a k ε model in a three-dimensional σ-coordinate model to simulate the − velocity and particles concentration fields. In general, k ε standard models are located in the context of Reynolds-Averaged Navier–Stokes − equations (RANS), in which all the unsteady velocity fluctuations are expelled from the simulation of the Reynolds time-averaged velocity fields. In this approach the effects of all unsteady periodic vortex structures and unsteady stochastic velocity fluctuations are represented in terms of the total transfer of energy dissipation from the averaged motion to all the scales of turbulent motion, through the Reynolds tensor. In the k ε standard models, the Reynolds tensor is related to all the energy of the unsteady − periodic vortex structures and unsteady stochastic velocity fluctuations that are expelled from the representation of the Reynolds averaged motion. Consequently, these models are not able to directly simulate unsteady periodic vortex structures because they overestimate the Reynolds tensor and produce an excessive dissipation of energy and non-physical wave height reduction. The velocity field in the surf zone is very complex and is characterized by unsteady periodic vortex structures and unsteady stochastic velocity fluctuations. Outside the surf zone, the production of turbulent kinetic energy is limited to the oscillating wave boundary layer; inside the surf zone there is the production of turbulent kinetic energy in the oscillating wave boundary layer and the production of turbulent kinetic energy near the wave crest, and between the oscillating wave boundary layer and the wave crest, there is a phenomenon of mainly turbulent kinetic energy dissipation. In general, the stochastic turbulent fluctuations are superimposed on the periodic unsteady motion of the vortex structures. Bosch and Rodi [17] proposed the simulation of the velocity fields characterized by the above-mentioned unsteady vortex structures by decomposing the instantaneous flow quantities in a time mean component, in a periodic component, and in a turbulent fluctuating component. Following this approach, the sum of the time mean and the periodic part gives rise to the ensemble-average component of the flow quantities. The latter are calculated by the numerical integration of the ensemble-average continuity and momentum equations; the complete spectrum of the stochastic motion is simulated by a statistical turbulence model. The models coherent with the above-mentioned approach are named in the literature as Unsteady Reynolds-Averaged Navier–Stokes (URANS) models. In this paper, we propose, for the simulation of the wave motion and the turbulence under wave breaking, to expel the turbulent kinetic energy associated with unsteady stochastic velocity fluctuations from the ensemble-average motion. This model can be considered in the context of the Unsteady Reynolds-Averaged Navier–Stokes (URANS) models. The modifications to the k ε model with respect − to the standard one consist of the following:

The closure relation of the production of turbulent kinetic energy takes into account the nonlinear • terms, following the procedure of [18,19]; The closure relation of the destruction of the dissipation rate is expressed by means of a dynamic • procedure, used by Yakhot et al. and Derakty et al. [20,21]; The production of the turbulent kinetic energy dissipation is calculated by a dynamic procedure. • In the surf zone the production of turbulent kinetic energy dissipation is limited by the dynamic coefficient, related to the local derivative of the free surface. Consequently, differently from the k ε standard model, it is possible to limit the non-physical decrease of the wave height. − The paper is structured as follows: in Sections2 and3 are presented the motion equations and modified turbulent k ε model, in which vector and tensor quantities are expressed in Cartesian − components on a general boundary conforming curvilinear time dependent coordinate system that includes a time-varying vertical coordinate. In Section4 the results are shown and discussed. Conclusions are drawn in Section5. Water 2019, 11, 2282 4 of 13

2. Motion Equations

Water We2019 consider, 11, x FOR aPEER transformation REVIEW xi = xi(ξ1, ξ2, τ), from a Cartesian coordinate system (x1, x2, t)4, of to 13 a curvilinear coordinate system (ξ1, ξ2, τ) and the inverse transformation, ξi = ξi(x1, x2, t). WeIn this consider paper a we transformation adopt a particular 𝑥 =𝑥 transformation(𝜉,𝜉,𝜏), from from a Cartesian Cartesian coordinate to curvilinear system coordinates (𝑥,𝑥,𝑡), to in aorder curvilinear to follow coordinate the free surface system movements(𝜉,𝜉,𝜏) and [14 the]. inverse transformation, 𝜉 =𝜉(𝑥,𝑥,𝑡). In this paper we adopt a particular transformation from Cartesian to curvilinear coordinates in order to follow the free surface movements [14]. x2 + h ξ1 = x1; ξ2 = ; τ = t (1) 𝑥 +ℎH 𝜉 =𝑥; 𝜉 = ; 𝜏=𝑡 𝐻 (1) in which H(x1, t) = h(x1) + η(x1, t) is the total water depth; h is the undisturbed water depth and η is in which 𝐻(𝑥 , 𝑡) = ℎ(𝑥 )+𝜂(𝑥 ,𝑡) is the total water depth; ℎ is the undisturbed water depth the free surface1 elevation;1 ξ1 is the1 horizontal boundary conforming curvilinear coordinates and ξ2 and 𝜂 is the free surface elevation; 𝜉 is the horizontal boundary conforming curvilinear is the time varying vertical coordinate that spans from ξ2 = 0 (at the bottom) to ξ2 = 1 (at the free coordinatessurface). The and irregular 𝜉 is varyingthe time domain varying in vertical the physical coordinate space isthat mapped spans intofrom a regular𝜉 =0 ﬁxed(at the domain bottom) in tothe 𝜉 transformed =1 (at the space, free surface). as shown The in irregular Figure1. varying domain in the physical space is mapped into a regular fixed domain in the transformed space, as shown in Figure 1.

Figure 1. A particular representation of the mesh in which ξ = 0 represent the bottom and ξ = 1 Figure 1. A particular representation of the mesh in which 𝜉2 =0 represent the bottom and 𝜉2 =1 represent the free surface. (l) (l) (l) Let →c (l) = ∂→x /∂ξ(𝑙) and (𝑙)→c = (𝑙)∂ξ /∂→x (l = 1, 2) be the l th covariant and contravariant Let 𝑐⃗(𝑙) =𝜕𝑥⃗/𝜕𝜉 and 𝑐⃗ =𝜕𝜉 /𝜕𝑥⃗ (𝑙=1,2) be the 𝑙−𝑡ℎ− covariant and contravariant base vectors, respectively, and let C and B be the metric tensor and its inverse, respectively, base vectors, respectively, and let 𝐶̅ and 𝐵 be the metric tensor and its inverse, respectively, whose components are whose components are (l) (𝑙) (m)(𝑚) lm 𝑙𝑚 𝑐𝑙𝑚c=𝑐=⃗(𝑙)→c∙𝑐⃗(𝑚)→c c 𝑐= →c=𝑐⃗ →c ∙𝑐⃗ ( l , m =(𝑙,1, 2 𝑚) = 1,2) (2) lm (l)· (m) · The Jacobian of the transformation is given by √𝑐 = qdet(𝐶̅) =𝐻. The Jacobian of the transformation is given by √c = det C = H. The equations are expressed in terms of 𝐻𝑢⃗ e 𝐻, the spatial averaged values over volume elementsThe equationsdefined in are the expressed form in terms of H→u e H, the spatial averaged values over volume elements deﬁned in the form 1 Z 𝐻 = ∗ 1 𝐻𝑑𝜉 (3) =𝛥𝐿 ∗ H Hdξ1 (3) ∆L∗ 2 ∆L2∗ 1 𝐻𝑢⃗ = ∗ 𝐻𝑢⃗ 𝑑𝜉𝑑𝜉 (4) ∆𝐴 ∆∗ ∗ ∗ where 𝛥𝐿 =Δ𝜉 is the contour line of the control area Δ𝐴 =Δ𝜉 Δ𝜉 in the transformed space. The motion equations, expressed in integral form in a time-dependent curvilinear coordinate system, are as follows:

Water 2019, 11, 2282 5 of 13

Z 1 H→u = H→u dξ dξ (4) ∆A 1 2 ∗ ∆A∗ = = where ∆L2∗ ∆ξ1 is the contour line of the control area ∆A∗ ∆ξ1 ∆ξ2 in the transformed space. The motion equations, expressed in integral form in a time-dependent curvilinear coordinate system, are as follows:

2 → P R h i ∂H u = 1 →u →u →v c→ H d ∂τ ∆A ∆L + (α) ξβ − ∗ α=1 α∗ ⊗ − · R h i 2 R h i → → → → 1 P → u u v c(α)H dξβ ∆A + Gηc(α)H dξβ − ∆Lα∗ ⊗ − · − ∗ ∆Lα∗ − α=1 (5) R h i R ∂q G c→ H d 1 1 c→ Hd d ∆L η (α) ξβ ∆A ρ ∆A ∂ξr (r) ξ1 ξ2 − α∗ − − ∗ ∗ P2 R h i R h i 1 S c→ H d S c→ H d ∆A ∆L + 2νT (α) ξβ ∆L 2νT (α) ξβ − ∗ α=1 α∗ · − α∗ − · where →u is the ﬂuid velocity; →v is the velocity of the moving coordinate lines; is the outer product; h⊗i is the scalar product; and G, ρ, and νT are, respectively, the constant of gravity, the ﬂuid density, h·i and the turbulent eddy viscosity. q is the dynamic pressure; 2 νTS = R is the stress tensor and S is the strain rate tensor; the ∆Lα∗ = ∆ξβ (α, β = 1, 2 are cyclic) is the contour line of the area ∆A∗ on which ξα is constant and which is located at the larger and smaller value of ξα, respectively.

Z Z  ∂H 1  h i h i  +  →u →v c→ H dξ →u →v c→ H dξ  = 0 (6)  (α) 2 (α) 2 ∂τ ∆ξ1  ∆ξ + − · − ∆ξ − ·  2 2− where ξ2 is the contour line of area ∆A∗ on which ξ1 is constant and which is located at the larger and smaller value of ξ2, respectively. Equations (5) and (6) are numerically solved, on a time dependent curvilinear coordinate system, where the vertical coordinate varies in time, by a ﬁnite volume shock capturing scheme, which uses an approximate HLL-type Riemann solver [15].

3. Turbulence Model The turbulence model proposed for the simulation of the wave motion is a modiﬁed k ε. − The governing equations are written in a boundary conforming curvilinear coordinate system in which vector and tensor quantities are expressed in Cartesian components. This form of equation does not exist in the literature. In an original conservative form, the transport equations of turbulent kinetic energy k and the dissipation rate ε are expressed in terms of conservative variables Hk and Hε and in integral form in a curvilinear coordinate system. Let us deﬁne the following cell-averaged values in the transformed space. Z 1 Hk = kH dξ dξ (7) ∆A 1 2 ∗ ∆A∗ Z 1 Hε = εH dξ dξ (8) ∆A 1 2 ∗ ∆A∗ The transport equation of k is

P2 R h i R h i ∂Hk = 1 k →u →v c→ Hd k →u →v c→ Hd + ∂τ ∆A ∆L + (α) ξβ ∆L (α) ξβ − ∗ α=1 α∗ − · − α∗ − − · 2 1 P R νT ∂k R νT ∂k (9) + ν + r c→ c→ Hdξ ν + r c→ c→ Hdξ + ∆A ∆L + σk ∂ξ (r) (α) β ∆L σk ∂ξ (r) (α) β ∗ α=1 α∗ · − α∗ − · 1 R + ∆A (P ε) H dξ1dξ2. ∗ ∆A∗ − Water 2019, 11, 2282 6 of 13

The transport equation of ε is

P2 R h i R h i ∂Hε = 1 →u →v c→ Hd →u →v c→ Hd + ∂τ ∆A ∆L + ε (α) ξβ ∆L ε (α) ξβ − ∗ α=1 α∗ − · − α∗ − − · 2 R R (10) 1 P νT ∂ε → → νT ∂ε → → + ∆A + ν + σ r c(r) c(α)Hdξβ ν + σ r c(r) c(α)Hdξβ + ∗ ∆Lα∗ ε ∂ξ · − ∆Lα∗ ε ∂ξ · α=1 R − 1 ε ( ) + ∆A k C1εP C2εε H dξ1dξ2. ∗ ∆A∗ − The kinematic viscosity of water is ν and the turbulent eddy viscosity is calculated by

k2 ν = C . (11) T µ ε

In Equations (9)–(11) σk, σε, and Cµ are constant coeﬃcients given by

σk = 0.72; σε = 0.72; and Cµ = 0.09. (12)

The novelties of the k ε model with respect to the standard one consist of the following: − The coeﬃcient C is determined by a dynamic procedure proposed by Yakhot et al. [20]: • 2ε 3 Cµζ (1 ζ/4.38) C2ε = 1.68 + − (13) 1 + 0.012ζ3 p k where ζ = ε 2SS. The closure relation that limits the production of dissipation rate is proposed as a function of the • local derivative of the free surface.

C = 2.15 (2.15 1.42)B(x) (14) 1ε − − C varies between (1.42 2.15); B(x) is a linear function and it is related to the steepness of the 1ε ÷ wave front (∂η/∂t):  p  0 se ∂η/∂t 0.15 gh0  p ≤ p B(x) =  0 1 se 0.15 gh < ∂η/∂t 0.3 gh (15)  0 0  − ≤p  1 se ∂η/∂t > 0.3 gh0

The closure relation of the production of turbulent kinetic energy takes into account the nonlinear • terms and is given by Z ∂→u P = R : c→ Hdξ dξ (16) − ∂ξ ⊗ (α) 1 2 ∆A∗ α where : is the scalar inner product between the second order tensors. The Reynolds stress R h i is calculated by a nonlinear model and is expressed in a boundary conforming curvilinear grid, in which vector and tensor quantities are given in terms of Cartesian components: " # 2 T 3 k ∂→u → ∂→u → k ∂→u → ∂→u → R = Cd c( ) + c( ) + kI C1 2 c( ) c( ) − ε ∂ξα ⊗ α ∂ξα ⊗ α − ε ∂ξα ⊗ α · ∂ξα ⊗ α T# ∂→u ∂→u ∂→u ∂→u + c→( ) c→( ) c→( ) c→( ) I ∂ξα ⊗ α · ∂ξα ⊗ α − ∂ξα ⊗ α · ∂ξα ⊗ α (" T# " T# ) (17) k3 ∂→u ∂→u 1 ∂→u ∂→u C2 c→( ) c→( ) c→( ) c→( ) I − ε2 ∂ξα ⊗ α · ∂ξα ⊗ α − 2 ∂ξα ⊗ α · ∂ξα ⊗ α (" T # " T # ) k3 ∂→u ∂→u 1 ∂→u ∂→u C3 c→( ) c→( ) c→( ) c→( ) I − ε2 ∂ξα ⊗ α · ∂ξα ⊗ α − 2 ∂ξα ⊗ α · ∂ξα ⊗ α Water 2019, 11, x FOR PEER REVIEW 7 of 13

𝑘 𝜕𝑢⃗ 𝜕𝑢⃗ 1 𝜕𝑢⃗ 𝜕𝑢⃗ ̅ −𝐶 ⨂𝑐(⃗) ∙ ⨂𝑐(⃗) − ⨂𝑐()⃗ ∙ ⨂𝑐()⃗ 𝐼 𝜀 𝜕𝜉 𝜕𝜉 2 𝜕𝜉 𝜕𝜉 𝑘 𝜕𝑢⃗ 𝜕𝑢⃗ 1 𝜕𝑢⃗ 𝜕𝑢⃗ ̅ (17) −𝐶 ⨂𝑐()⃗ ∙ ⨂𝑐()⃗− ⨂𝑐()⃗ ∙ ⨂𝑐()⃗𝐼 Water 2019, 11, 2282 𝜀 𝜕𝜉 𝜕𝜉 2 𝜕𝜉 𝜕𝜉 7 of 13 ̅ where 𝑇 expresses the transposed terms; 𝐼 is the identity tensor; 𝐶 , 𝐶1 , 𝐶2 , and 𝐶3 are empirical coefficients: where T expresses the transposed terms; I is the identity tensor; Cd, C1, C2, and C3 are empirical coeﬃcients: 𝐶 = 2 1 𝐶 = 1 .Cd= C1 =. 3 7.4+2Smax 185.2+3Dmax (18) C = 1 C = 1 (18) 2 + D2 3 +D2 𝐶 =− − 58.8 2 max 𝐶 =370.4 3 max . . where ( ) ( ) where k ∂ui k ∂ui Smax = max Dmax = max . (19) ε ∂xi ε ∂xj 𝑆 = 𝑚𝑎𝑥 𝐷 = 𝑚𝑎𝑥 . (19) The boundary conditions on the bottom concern the turbulent kinetic energy and the dissipation rate: The boundary conditions on the bottom concern2 the turbulent3 kinetic energy and the dissipation u∗ u∗ rate: k = p ε = (20) Cµ Ky ∗ ∗ 𝑘= 𝜀= (20) where u∗ represents the friction velocity: ∗ where 𝑢 represents the friction velocity: u 1 u ∗ = ln ∗ . (21) ∗ u K∗ ks/30 = ln . (21) ⁄ K = 0.4 is the von Karman constant and ks = 0.0002 is the roughness. 𝐾=0.4 is the von Karman constant and 𝑘 = 0.0002 is the roughness. 4. Results 4. Results In this section the numerical results obtained by the Gallerano et al. [14] numerical model are comparedIn this with section the the experimental numerical measurementsresults obtained by by Ting the and Gallerano Kirby [ 21et ].al. The [14] test numerical conducted model by Ting are comparedand Kirby with [21] consistedthe experimental of realizing measurements a spilling breakingby Ting and wave Kirby on a[21]. sloping The test beach. conducted It is possible by Ting to andexamine Kirby the [21] wave consisted energy dissipationof realizing in a thespilling surf zonebrea dueking to wave the turbulence on a sloping by meansbeach. ofIt theis possible turbulence to examinemodel. Thethe experimentalwave energy wavedissipat tankion was in 40them surflong, zone0.6 mduewide, to the and turbulence1.0 m deep by and means consisted of the of turbulencea beach with model. a constant The experimental slope equal wave to 1 : tank 35 connected was 40 m to long, a region 0.6 with m wide, a constant and 1.0 depth m deep equal and to consistedh = 0.4 m ;ofthe a beach wavemaker with a wasconstant placed slope at the equal left to end 1: of35the connected wave tank. to a Inregion this paper,with a inconstant order to depth save equalcomputational to ℎ = 0.4 time, m; the we numericallywavemaker reproducedwas placed at only the the left ﬁnal end eighteen of the wave meters tank. of theIn this experimental paper, in orderarrangement to save showncomputational in Figure time,2, and we the numerically abscissa x =reproduced0 m indicates only the the toe final of the eighteen slope wheremeters the of stillthe experimentalwater depth is arrangement equal to h = 0.38shown m. in Figure 2, and the abscissa 𝑥=0 m indicates the toe of the slope where the still water depth is equal to ℎ = 0.38 m.

FigureFigure 2. Spilling2. Spilling breaking breaking wave wave test [ 21test]. Computational[21]. Computational domain domain adopted adopted for the numerical for the numerical simulations. simulations. The computational grid adopted for all the numerical simulations consisted of 13, 728 grid cells in theThe horizontal computational direction grid in adopted which x forspanned all the fromnumerical1 m simulationsto 17 m and consisted the grid of spacing13,728 grid was 1 − − cells∆x1 =in 0.025the horizontalm. In the direction vertical direction,in which 13𝑥 gridspanned cells from were −1 used. m Asto −17 shown m inand Figure the grid3, the spacing mesh was reﬁned at the top and bottom boundaries. For this test, cnoidal waves with period T = 2 s and wave height H = 0.125 m were imposed as input boundary conditions. In the swash region, a wetting–drying treatment was used. Water 2019, 11, x FOR PEER REVIEW 8 of 13 was Δ𝑥 = 0.025 m. In the vertical direction, 13 grid cells were used. As shown in Figure 3, the mesh was refined at the top and bottom boundaries. For this test, cnoidal waves with period 𝑇= 2 Water s and2019 wave, 11, 2282 height 𝐻 = 0.125 m were imposed as input boundary conditions. In the swash8 of 13 region, a wetting–drying treatment was used.

Figure 3. Zoomed in view of the numerical mesh used for the simulations. Figure 3. Zoomed in view of the numerical mesh used for the simulations. 4.1. Spilling Breaking Test without Turbulence Model 4.1. Spilling Breaking Test without Turbulence Model In this subsection we present briefly the numerical results obtained without the turbulence modelIn this [22]. subsection we present briefly the numerical results obtained without the turbulence model Figure[22]. 4 shows the distribution of the wave crest and wave trough in the cross-shore direction, togetherFigure with 4 shows the mean the distribution water level, of obtained the wave by thecres Galleranot and wave et trough al. [14] numericalin the cross-shore model in direction, which the togetherturbulence with model the mean was water switched level, off .obtained The mean by waterthe Gallerano level and et the al. wave[14] numerical crest and model trough, in shown which in theFigure turbulence4, were computedmodel was by switched performing, off. respectively,The mean water a time-average level and andthe awave phase-average, crest and overtrough, five shownwave periodsin Figure of the4, datewere extracted computed from by the performing, numerical simulation,respectively, once a thetime-average simulation and reached a phase-average,a quasi-stationary over five state. wave In theperiods same of figure, the date the extracted experimental from the results numerical by Ting simulation, and Kirby once [21] the and simulationthe numerical reached results a quasi-stationary by Ma [12] are state. also In shown. the same In figure, the shoaling the experimental zone (x < 6.4resultsm) theby Ting numerical and Kirbyresults [21] obtained and the bynumerical the Gallerano results et by al. Ma [14 [12]] numerical are also modelshown. fit In quite the shoaling well with zone the experimental(𝑥<6.4 )m theresults, numerical while results after the obtained initial wave by the breaking Gallerano point et (xal.> [14]6.4 mnumerical) the numerical model resultsfit quite overestimated well with the the experimentalexperimentalWater 2019 results,, ones.11, x FOR Aswhile PEER shown REVIEW after in the Figure initial4, similar wave breaking numerical point results (𝑥 were > 6.4 obtained) m the numerical by Ma et9 al. resultsof [1312 ]. overestimated the experimental ones. As shown in Figure 4, similar numerical results were obtained by Ma et al. [12].

FigureFigure 4. Ting 4. Ting and and Kirby Kirby [21 [21]] spilling spilling breaking breakingwave wave testtest case. case. Mean Mean water water level level and and distribution distribution of of phase-averagedphase-averaged crest andcrest troughand trou elevationsgh elevations along thealong cross-shore. the cros Comparisons-shore. Comparison between experimentalbetween

data (emptyexperimental circles), data numerical (empty circles) results, numerical by Ma et al.results [12] by (dashed Ma et line)al. [12] and (dashed numerical line) resultsand numerical obtained by the Galleranoresults obtained et al. [ 14by] the without Gallerano the et turbulence al. [14] without model the (solid turbulence line). model (solid line).

4.2. Spilling Breaking Test with Turbulence Model Figure 5 shows the distribution of the wave crest and wave trough in the cross-shore direction, together with the mean water level, obtained by the Gallerano et al. [14] numerical model in which the turbulence model is the standard 𝑘−𝜀 proposed by Rodi [23]. From this figure, it is possible to notice that for this test case the standard 𝑘−𝜀 turbulence model produces an excessive non-physical decrease in the wave height with respect to the experimental results [21].

Figure 5. Ting and Kirby [21] spilling breaking wave test case. Mean water level and distribution of phase-averaged crest and trough elevations along the cross-shore direction. Comparison between experimental data (empty circles) and numerical model in which the turbulence model is the standard 𝑘−𝜀 proposed by Rodi [23] (solid line).

Figure 6 shows a snapshot of the instantaneous spatial distribution of the turbulent kinetic energy (a,b) and the turbulent eddy viscosity (c,d), in two different time steps, obtained by the Gallerano et al. [14] numerical model in which the turbulence model is the standard 𝑘−𝜀 proposed by Rodi [23]. From this figure, it can be noticed that the turbulent kinetic energy is generated in correspondence with the wave fronts in all the surf zone, also before 𝑥=6.4 , m where the initial wave breaking point is located according to the experimental measurements [21]. Figure 6 also shows that, by using the standard 𝑘−𝜀 turbulence model [23], the simulated wave fronts are very smooth, due to an excess of the turbulent eddy viscosity produced by the modelled turbulent kinetic energy.

Water 2019, 11, x FOR PEER REVIEW 9 of 13

Figure 4. Ting and Kirby [21] spilling breaking wave test case. Mean water level and distribution of phase-averaged crest and trough elevations along the cross-shore. Comparison between Water 2019experimental, 11, 2282 data (empty circles), numerical results by Ma et al. [12] (dashed line) and numerical9 of 13 results obtained by the Gallerano et al. [14] without the turbulence model (solid line). 4.2. Spilling Breaking Test with Turbulence Model 4.2. Spilling Breaking Test with Turbulence Model Figure5 shows the distribution of the wave crest and wave trough in the cross-shore direction, Figure 5 shows the distribution of the wave crest and wave trough in the cross-shore direction, together with the mean water level, obtained by the Gallerano et al. [14] numerical model in which the together with the mean water level, obtained by the Gallerano et al. [14] numerical model in which turbulence model is the standard k ε proposed by Rodi [23]. From this ﬁgure, it is possible to notice the turbulence model is the standard− 𝑘−𝜀 proposed by Rodi [23]. From this figure, it is possible to that for this test case the standard k ε turbulence model produces an excessive non-physical decrease notice that for this test case the− standard 𝑘−𝜀 turbulence model produces an excessive in the wave height with respect to the experimental results [21]. non-physical decrease in the wave height with respect to the experimental results [21].

Figure 5. TingTing and and Kirby Kirby [21] [21] spilling spilling breaking breaking wave wave test test case. case. Mean Mean water water level level and and distribution distribution of ofphase-averaged phase-averaged crest crest and and trough trough elevations elevations alon alongg the the cross-shore cross-shore direction. Comparison betweenbetween experimental datadata (empty (empty circles) circles) and and numerical numerica modell model in which in which the turbulence the turbulence model ismodel the standard is the kstandardε proposed 𝑘−𝜀 by proposed Rodi [23 ]by (solid Rodi line). [23] (solid line). − Figure6 6shows shows a snapshota snapshot of theof instantaneousthe instantaneous spatial spatial distribution distribution of the turbulentof the turbulent kinetic energykinetic (a,b)energy and (a,b) the and turbulent the turbulent eddy viscosity eddy viscosity (c,d), in two(c,d), di infferent two timedifferent steps, time obtained steps, byobtained the Gallerano by the etGallerano al. [14] numerical et al. [14] model numerical in which model the turbulence in which model the turbulence is the standard modelk εisproposed the standard by Rodi 𝑘−𝜀 [23]. − Fromproposed this figure,by Rodi it can[23]. be From noticed this that figure, the turbulent it can be kinetic noticed energy that the is generated turbulent in kinetic correspondence energy is withgenerated the wave in correspondence fronts in all the with surf zone,the wave also fronts before inx all= 6.4them surf, where zone, the also initial before wave 𝑥=6.4 breaking , m where point isthe located initial accordingwave breaking to the point experimental is located measurementsaccording to the [21 experimental]. Figure6 also measurements shows that, by[21]. using Figure the 6 standardalso showsk that,ε turbulence by using modelthe standard [23], the 𝑘−𝜀 simulated turbulence wave frontsmodel are [23], very the smooth, simulated due wave to an fronts excess are of − thevery turbulent smooth, eddydue to viscosity an excess produced of the turbulent by the modelled eddy viscosity turbulent produced kinetic energy. by the modelled turbulent kineticFrom energy. Figures 5 and6 it can be deduced that, for this test case, the standard k ε turbulence − model [23] produces an excess of turbulent kinetic energy and turbulent eddy viscosity, which entails very small simulated wave heights and very smooth wave fronts in the whole surf zone. Figure7 shows the distribution of the wave crest and wave trough in the cross-shore direction, together with the mean water level, obtained by the Gallerano et al. [14] numerical model in which the turbulence model is the modified k ε turbulence proposed in this paper. In the same figure, the above − numerical results are compared with the experimental results by Ting and Kirby [21] and the numerical results obtained by the Gallerano et al. [14] numerical model in which the turbulence model is switched off. From this figure, it is possible to notice that the initial wave breaking point is correctly predicted by the modified k ε turbulence model; the modified k ε turbulence model adequately represents − − the distribution of the wave crest and wave trough before and after the initial wave breaking point. Figure8 shows a snapshot of the instantaneous spatial distribution of the turbulent kinetic energy (a,b) and the turbulent eddy viscosity (c,d), in two different time steps, obtained by the Gallerano et al. [14] numerical model in which the turbulence model is the proposed modified k ε. From this − figure, it can be noticed that the turbulent kinetic energy is generated, in correspondence with the wave fronts, mainly after the initial wave breaking point (x > 6.4 m), where the wave turbulence processes are significant. Water 2019, 11, x FOR PEER REVIEW 10 of 13

Water 2019, 11, 2282 10 of 13

The comparison between Figures6 and8 shows that, under breaking waves, the modified k ε − turbulence model produces lower values of turbulent kinetic energy with respect to the standard one. Figure9 shows the vertical time-mean cross-shore horizontal velocity component and the turbulent kinetic energy obtained(a by) the proposed modified k ε turbulence model(c (crosses),) in comparison − with the experimental results (empty circles) by Ting and Kirby [21] at x = 7.25 m. From this figure, it is possible to notice a good agreement between the numerical and the experimental results for both Water 2019, 11, x FOR PEER REVIEW 10 of 13 the quantities.

(b) (d)

Figure 6. Ting and Kirby [21] spilling breaking wave test case. Instantaneous spatial distribution of the turbulent kinetic energy (a,b) and the turbulent eddy viscosity (c,d) in two different time steps, obtained by the Gallerano et al. [14] numerical model in which the turbulence model is the standard 𝑘−𝜀 proposed(a) by Rodi [23]. (c)

From Figure 5 and 6 it can be deduced that, for this test case, the standard 𝑘−𝜀 turbulence model [23] produces an excess of turbulent kinetic energy and turbulent eddy viscosity, which entails very small simulated wave heights and very smooth wave fronts in the whole surf zone. Figure 7 shows the distribution of the wave crest and wave trough in the cross-shore direction, together with the mean water level, obtained by the Gallerano et al. [14] numerical model in which the turbulence model is the modified 𝑘−𝜀 turbulence proposed in this paper. In the same figure, the above numerical results(b) are compared with the experimental results by(d) Ting and Kirby [21] and the numerical results obtained by the Gallerano et al. [14] numerical model in which the turbulence FigureFigure 6. 6.Ting Ting and and Kirby Kirby [21 [21]] spilling spilling breaking breaking wave wave test test case. case. Instantaneous Instantaneous spatial spatial distribution distribution of ofthe model is switched off. From this figure, it is possible to notice that the initial wave breaking point is turbulentthe turbulent kinetic kinetic energy energy (a,b) and(a,b the) and turbulent the turbulent eddy viscosityeddy viscosity (c,d) in (c,d two) in di twoﬀerent different time steps, time obtainedsteps, correctly predicted by the modified 𝑘−𝜀 turbulence model; the modified 𝑘−𝜀 turbulence model byobtained the Gallerano by the etGallerano al. [14] numericalet al. [14] modelnumerical in which model the in turbulencewhich the modelturbulence is the model standard is thek ε 𝑘−𝜀 − adequatelyproposedstandard represents by Rodi proposed [ 23the]. distribution by Rodi [23]. of the wave crest and wave trough before and after the initial wave breaking point. From Figure 5 and 6 it can be deduced that, for this test case, the standard 𝑘−𝜀 turbulence model [23] produces an excess of turbulent kinetic energy and turbulent eddy viscosity, which entails very small simulated wave heights and very smooth wave fronts in the whole surf zone. Figure 7 shows the distribution of the wave crest and wave trough in the cross-shore direction, together with the mean water level, obtained by the Gallerano et al. [14] numerical model in which the turbulence model is the modified 𝑘−𝜀 turbulence proposed in this paper. In the same figure, the above numerical results are compared with the experimental results by Ting and Kirby [21] and the numerical results obtained by the Gallerano et al. [14] numerical model in which the turbulence model is switched off. From this figure, it is possible to notice that the initial wave breaking point is correctly predicted by the modified 𝑘−𝜀 turbulence model; the modified 𝑘−𝜀 turbulence model adequately represents the distribution of the wave crest and wave trough before and after the initial wave breaking point.

Figure 7. TingTing and and Kirby Kirby [21] [21 ]spilling spilling breaking breaking wave wave test test case. case. Mean Mean water water level level and and distribution distribution of phase-averagedof phase-averaged crest crest and and trough trough elevations elevations alon alongg the the cross-shore cross-shore direction. direction. Comparison Comparison between experimental datadata (empty (empty circles) circles) and and numerical numerical results results obtained obtained by the Galleranoby the Gallerano et al. [14 ]et numerical al. [14] numericalmodel with model (solid with line) (solid and without line) and (dashed without line) (dashed the modiﬁed line) thek modifiedε turbulence 𝑘−𝜀 model. turbulence model. −

Figure 7. Ting and Kirby [21] spilling breaking wave test case. Mean water level and distribution of phase-averaged crest and trough elevations along the cross-shore direction. Comparison between experimental data (empty circles) and numerical results obtained by the Gallerano et al. [14] numerical model with (solid line) and without (dashed line) the modified 𝑘−𝜀 turbulence model.

Water 2019, 11, x FOR PEER REVIEW 11 of 13

Figure 8 shows a snapshot of the instantaneous spatial distribution of the turbulent kinetic energy (a,b) and the turbulent eddy viscosity (c,d), in two different time steps, obtained by the Gallerano et al. [14] numerical model in which the turbulence model is the proposed modified 𝑘− Water𝜀. From 2019 this, 11, xfigure, FOR PEER it can REVIEW be noticed that the turbulent kinetic energy is generated, in correspondence11 of 13 with the wave fronts, mainly after the initial wave breaking point (𝑥>6.4 ),m where the wave turbulenceFigure processes8 shows aare snapshot significant. of the instantaneous spatial distribution of the turbulent kinetic energyThe (a,b) comparison and the turbulentbetween eddyFigure viscosity 6 and Figure(c,d), in 8 twoshows different that, timeunder steps, breaking obtained waves, by thethe Galleranomodified et𝑘−𝜀 al. [14] turbulence numerical model model produces in which lower the valuesturbulence of turbulent model is kinetic the proposed energy withmodified respect 𝑘− to 𝜀the. From standard this figure, one. it can be noticed that the turbulent kinetic energy is generated, in correspondence with the wave fronts, mainly after the initial wave breaking point (𝑥>6.4 ),m where the wave turbulence processes are significant. The comparison between Figure 6 and Figure 8 shows that, under breaking waves, the Watermodified2019, 11 , 2282𝑘−𝜀 turbulence model produces lower values of turbulent kinetic energy with respect11 to of 13 the standard one.

(a) (c)

(b) (d)

Figure 8. Ting and Kirby [21] spilling breaking wave test case. Instantaneous spatial distribution of the turbulent kinetic energy (a,b) and the turbulent eddy viscosity (c,d) in two different time steps, obtained by the Gallerano et al. [14] numerical model in which the turbulence model is the proposed modified 𝑘−𝜀. (b) (d) Figure 9 shows the vertical time-mean cross-shore horizontal velocity component and the FigureFigure 8. 8.Ting Ting and and Kirby Kirby [ 21[21]] spilling spilling breaking breaking wavewave testtest case. Instantaneous Instantaneous spatial spatial distribution distribution of of turbulent kinetic energy obtained by the proposed modified 𝑘−𝜀 turbulence model (crosses), in thethe turbulent turbulent kinetic kinetic energy energy (a(a,b,b)) andand thethe turbulentturbulent eddy eddy viscosity viscosity (c,d (c,)d in) in two two different diﬀerent time time steps, steps, comparison with the experimental results (empty circles) by Ting and Kirby [21] at 𝑥 = 7.25. m obtainedobtained by by the the Gallerano Gallerano et et al. al.[ 14[14]] numericalnumerical model in in which which the the turbulence turbulence model model is isthe the proposed proposed From this figure, it is possible to notice a good agreement between the numerical and the modiﬁedmodifiedk 𝑘−𝜀ε. . experimental− results for both the quantities. Figure 9 shows the vertical time-mean cross-shore horizontal velocity component and the turbulent kinetic energy obtained by the proposed modified 𝑘−𝜀 turbulence model (crosses), in comparison with the experimental results (empty circles) by Ting and Kirby [21] at 𝑥 = 7.25. m From this figure, it is possible to notice a good agreement between the numerical and the experimental results for both the quantities.

(a) (b)

FigureFigure 9. 9.Ting Ting and and Kirby Kirby [21 [21]] spilling spilling breaking breaking wavewave testtest case.case. ( (aa)) Value Value of of the the vertical vertical time-mean time-mean cross-shorecross-shore velocity velocity component component at xat= 𝑥7.27 = 7.27m;(b m;) value(b) value of the of turbulentthe turbulent kinetic kinetic energy energy at x =at7.27 𝑥=m.

Comparison between experimental data (empty circles) and numerical results obtained by the proposed

modified k ε turbulence model (crosses). − (a) (b) 5. ConclusionsFigure 9. Ting and Kirby [21] spilling breaking wave test case. (a) Value of the vertical time-mean 𝑥 = 7.27 m 𝑥= Across-shore modified kvelocityε turbulence component model at was proposed; (b) value that isof ablethe turbulent to simulate kinetic the energy hydrodynamic at field − and the turbulent kinetic energy under a breaking wave. In order to numerically simulate the wave propagation from deep water to the shoreline and the wave breaking, a numerical model was used in which the vector and tensor quantities are expressed in Cartesian components, where only the vertical coordinate is expressed as a function of a time-dependent curvilinear coordinate that follows the free surface movements. It was demonstrated that the shock-capturing model, defined in the context of σ-coordinate, underestimates the energy dissipation at the breaking point and consequently overestimates the wave height in the surf zone. Water 2019, 11, 2282 12 of 13

In this paper, a modiﬁed k ε turbulence model was proposed based on a dynamic procedure in − order to contain the production of kinetic energy dissipation in the surf zone and limit the destruction of the turbulent kinetic energy dissipation. It was demonstrated that the proposed turbulence model is capable of correctly estimating the energy dissipation induced by the wave breaking, in order to avoid any underestimation of the wave height.

Author Contributions: Conceptualization and methodology, G.C.; Validation, investigation, writing and editing, F.P. and B.I.; Supervision, F.G. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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