Internal Wave Generation in a Non-Hydrostatic Wave Model

water

Article Internal Wave Generation in a Non-Hydrostatic Wave Model

Panagiotis Vasarmidis 1,* , Vasiliki Stratigaki 1 , Tomohiro Suzuki 2,3 , Marcel Zijlema 3 and Peter Troch 1

1 Department of Civil Engineering, Ghent University, Technologiepark 60, 9052 Ghent, Belgium; [email protected] (V.S.); [email protected] (P.T.) 2 Flanders Hydraulics Research, Berchemlei 115, 2140 Antwerp, Belgium; [email protected] 3 Department of Hydraulic Engineering, Delft University of Technology, Stevinweg 1, 2628 CN Delft, The Netherlands; [email protected] * Correspondence: [email protected]; Tel.: +32-9-264-54-89

Received: 18 April 2019; Accepted: 7 May 2019; Published: 10 May 2019

Abstract: In this work, internal wave generation techniques are developed in an open source non-hydrostatic wave model (Simulating WAves till SHore, SWASH) for accurate generation of regular and irregular long-crested waves. Two different internal wave generation techniques are examined: a source term addition method where additional surface elevation is added to the calculated surface elevation in a specific location in the domain and a spatially distributed source function where a spatially distributed mass is added in the continuity equation. These internal wave generation techniques in combination with numerical wave absorbing sponge layers are proposed as an alternative to the weakly reflective wave generation boundary to avoid re-reflections in case of dispersive and directional waves. The implemented techniques are validated against analytical solutions and experimental data including water surface elevations, orbital velocities, frequency spectra and wave heights. The numerical results show a very good agreement with the analytical solution and the experimental data indicating that SWASH with the addition of the proposed internal wave generation technique can be used to study coastal areas and wave energy converter (WEC) farms even under highly dispersive and directional waves without any spurious reflection from the wave generator.

Keywords: non-hydrostatic model; SWASH; Internal wave generation; source term addition method; spatially distributed source function

1. Introduction Numerical wave propagation models are commonly used as engineering tools for the study of wave transformation in coastal areas. The number of numerical models based on the Navier-Stokes equation has recently increased remarkably, since they offer detailed and accurate predictions of the wave field but at a very high computational cost. As a result, numerical models solving approximated equations usually averaged over the vertical—like Boussinesq and nonlinear shallow water equations—are an essential tool, especially when large domains and sea states of several hours are considered. Boussinesq-type models [1–3] account for both nonlinearity and frequency-dispersion of the waves by using high-order derivative terms in the equations. As an alternative, models based on the non-hydrostatic approach [4–6] can resolve the vertical flow structure and can improve their frequency dispersion by using additional layers rather than increasing the order of derivatives as in the case of Boussinesq-type models. A representative model of this latter category is SWASH [7], a phase-resolving wave propagation

Water 2019, 11, 986; doi:10.3390/w11050986 www.mdpi.com/journal/water Water 2019, 11, 986 2 of 18 model based on the nonlinear shallow water equations with added non-hydrostatic effects. In the field of wave transformation in coastal areas, SWASH has already reached a fairly mature stage since it can incorporate nonlinear shallow-water effects, such as the generation of bound sub- and super-harmonics and near-resonant triad interactions [8–11]. In order to simulate waves in the nearshore zone correctly, the generation and absorption of waves at the boundary of models need to be modelled accurately. In the SWASH model, incident waves are generated by prescribing their horizontal velocity component normal to the boundary of the computational domain over the vertical direction. Additionally, to absorb and to prevent re-reflections in front of the numerical wave generator, a weakly reflective wave generation boundary condition [12] is applied in which the total velocity signal is a superposition of the incident velocity signal and a velocity signal of the reflected waves [13]. Verbrugghe et al. [14] applied a similar method to create open boundaries within a Smoothed Particle Hydrodynamics (SPH) solver. However, this method is based on the assumption that the reflected waves are small amplitude shallow water waves propagating perpendicular to the boundary of the computational domain and hence this method is weakly reflective for directional, dispersive waves. Furthermore, Wei and Kirby [15] found that this type of radiation condition at the wave generator boundary can lead to numerical errors when long time simulations are performed. Recently, a generating absorbing boundary condition (GABC) has been developed which is an enhanced type of a weakly reflective wave generation boundary condition and can partially absorb dispersive and directional waves [16,17]. However, in this method the level of re-reflection strongly depends on the initial approximations since the characteristics of the reflected waves (i.e., wave angle, wave celerity) inside the numerical domain cannot be estimated a priori. In general, it is not possible to find practical boundary conditions that do the above task perfectly. On the other hand, models utilizing a sponge layer are very effective in absorbing reflected waves. However, this implies that the waves have to be generated inside the computational domain instead of on the boundary. This internal wave generation technique in combination with numerical wave absorbing sponge layers was firstly proposed by Larsen and Dancy [18] for Peregrine’s classical Boussinesq equations [19]. Later, Lee and Suh [20] and Lee et al. [21] achieved wave generation for the mild slope equations of Reference [22] and the extended Boussinesq equations of Nwogu [23], respectively, by applying the source term addition method. Lee et al. [21] have shown empirically that the velocity of disturbances caused by the incident wave can be properly obtained from the viewpoint of energy transport. Further, Schäffer and Sørensen [24] theoretically derived the energy velocity by adding the delta source function to the mass conservation type equation and integrated asymptotically the resulting equation at the generation point. However, Wei et al. [25] found that the source term addition method in a single source line may cause high frequency noise in case of non-staggered computational grid. To deal with this problem, Wei et al. [25] derived a spatially distributed (Gaussian shape) source function for internal wave generation, where a mass source is added in the continuity equation. Later, Choi and Yoon [26] and Ha et al. [27] used this technique in a Reynolds-averaged Navier-Stokes (RANS) equations model and a Navier-Stokes equations (NSE) model, respectively. However, in both cases they used directly the formula of [25], which was derived from the Boussinesq equations under the shallow water assumption. Thus, their model was not capable to accurately generating deep water waves and consequently high-frequency components in case of irregular waves. In the present paper, a source term addition method and a spatially distributed source function for internal wave generation are implemented in the non-hydrostatic model SWASH, in order accurately generate regular and irregular long-crested waves. In addition, the energy velocity will be derived for the governing equations of SWASH in case multiple layers are implemented. To the present authors’ knowledge, these internal wave generation techniques which are commonly used in Boussinesq models [3] and mild-slope wave models [28], have not been derived and used in a non-hydrostatic wave model before, due to the complexity of the governing equations. So, the main objective of the present work concerns the implementation of the internal wave generation in SWASH to accurately generate even highly dispersive and directional waves, while at the same time re-reflections Water 2019, 11, 986 3 of 18 at the position of the wave generator are vanishing. Moreover, SWASH has already been used in the field of marine energy by simulating the wave-induced response of a submerged wave-energy converter [29]. This kind of application requires a homogeneous wave field in the whole numerical domain. Thus, the implementation of internal wave generation is important as it introduces noteworthy improvements in the model, which can then make full use of its benefits for the study of WEC farms. The paper is structured as follows. The numerical model SWASH is described in Section2 where the governing equations are presented. The implemented internal wave generation techniques are derived in Section3. Section4 provides a detailed overview of the model results for regular and irregular long-crested waves. In addition, validation results are presented where the accuracy of the model is compared with experimental data. The last sections provide conclusions and a summary discussion of the present study.

2. SWASH Model

2.1. Mathematical Formulation SWASH is an open source non-hydrostatic wave model [7]. The governing equations of the model are based on the nonlinear shallow water equations with added non-hydrostatic eﬀects, which are derived from the incompressible Navier-Stokes equations. The numerical domain is bounded vertically by the free-surface z = η(x, t) and the bottom z = d(x), where t is time, d is the still water depth − and x and z are the Cartesian coordinates, where z is directed positive upwards. The equations for the 2D vertical domain are ∂u ∂w + = 0 (1) ∂x ∂z ∂u ∂uu ∂wu 1 ∂ ph + pnh ∂τ ∂τ + + = + xz + xx (2) ∂t ∂x ∂z −ρ0 ∂x ∂z ∂x ∂w ∂uw ∂ww 1 ∂p ∂τ ∂τ + + = nh + zz + zx (3) ∂t ∂x ∂z −ρ0 ∂z ∂z ∂x where u(x, z, t) and w(x, z, t) are the horizontal and vertical velocities, ph and pnh are the hydrostatic and non-hydrostatic pressures, ρ0 is density and τxx, τxz, τzx, τzz are the turbulent stresses. The kinematic boundary conditions are

∂η ∂η w = + u (z = η) (4) ∂t ∂x ∂d w = u (z = d) (5) − ∂x − The dynamic boundary conditions at the surface are a constant pressure and no surface stresses. The free surface elevation η is determined by considering the continuity for the entire water column

Z η ∂η ∂ + udz = 0 (6) ∂t ∂x d − At the bottom boundary, a bottom stress is included based on a quadratic friction law

U U τ = c | | (7) b f h where h = η + d is the total water depth, U is the depth averaged velocity and cf is a dimensionless friction coeﬃcient. At the outlet of the domain a sponge layer can be employed to minimize reﬂections. The sponge layer formula as described in Reference [30] is used, where the free surface elevation η and the velocity component u are relaxed at every time step. Water 2019, 11, 986 4 of 18

In SWASH the full process of wave breaking is not simulated but instead a breaking wave is considered analogous with a hydraulic bore. However, a high vertical resolution is needed to resolve accurately this process. Hence, Smit et al. [31] proposed a breaking formulation which can reproduce wave breaking with a coarse vertical resolution. A hydrostatic pressure distribution is assumed p when ∂η/∂t > a gh, where g is the gravitational acceleration and a is the maximum wave steepness before breaking. The numerical implementation is based on an explicit, second order finite difference method for staggered grids, where the mass and momentum are strictly conserved at a discrete level. In the horizontal direction rectilinear or orthogonal curvilinear grid can be applied, while in the vertical direction the computational domain is divided into a fixed number of layers. A more detailed overview of the numerical methods and equations are given in Reference [7].

2.2. Wave Generation in SWASH In the SWASH model, incident waves are generated by prescribing their horizontal velocity component normal to the boundary of the computational domain over the vertical direction. To prevent re-reflections in front of the numerical wave generator, a weakly reflective wave generation boundary condition is adopted [12]. For the case of one layer the imposed depth-averaged horizontal velocity component is given by r ω g u(t) = η + (η η ) (8) kd t d t − i where ω is the angular frequency, k is the wave number, d is the still water depth, g is the gravitational acceleration, ηt is the target surface elevation and ηi is the instantaneous surface elevation. In case of multiple layers, a hyperbolic cosine distribution is assumed for the horizontal velocity component. As it can be noticed from the second term of Equation (8), this method is valid only in case that the reflected waves are shallow water waves propagating perpendicular to the boundary of the computational p domain, since a shallow water phase velocity, C = gd is used. Hence, it is assumed that the high frequency energy gets dissipated inside the numerical domain before arriving to the wave generation boundary [31]. In addition, a linear superposition of the incident and reflected wave is assumed and thus can only be applied to small amplitude waves.

3. Internal Wave Generation In cases of directional and dispersive waves an internal wave generation technique can be applied to avoid re-reflections due to the weakly reflective wave generation boundary. Two different internal wave generation techniques are proposed here and are implemented in the SWASH model. The first one is a source term addition method proposed by Lee et al. [21], while the second one is a spatially distributed source function proposed by Wei et al. [25].

3.1. Energy Velocity At ﬁrst, a new energy velocity for the system of SWASH equations is mathematically derived following the methodology of Reference [24] who derived the energy velocity for Nwogu’s Boussinesq equations. In this paper, the derivation for the case of two equidistant vertical layers is presented, while the extension to more layers is straightforward. Bai and Cheung [32] showed that the governing Equations (1)–(3) can be converted into a Boussinesq form by expressing the non-hydrostatic pressure and vertical velocity component in terms of surface elevation and horizontal velocity component. Adding a point source to the continuity equation, the linearized governing equations on a constant depth for the case of two equidistant vertical layers read ∂η ∂u + d = Λ(t)δ(x) (9) ∂t ∂x Water 2019, 11, 986 5 of 18

∂u ∂η 5 ∂3u 1 ∂3uˆ + g d2 d2 = 0 (10) ∂t ∂x − 16 ∂x2∂t − 8 ∂x2∂t ∂uˆ 1 ∂3u 1 ∂3uˆ d2 d2 = 0 (11) ∂t − 8 ∂x2∂t − 16 ∂x2∂t where u is the depth integrated horizontal velocity component, uˆ is the inter-layer velocity variations, δ is the Dirac delta function and Λ is the source function applied to a single point. From Equation (9) it can be noticed that the ∂u/∂x term must be the one balancing the Dirac delta function and thus u and uˆ have a discontinuity. Taking the time derivative of Equation (10) and eliminating η from Equation (9) gives

∂2u ∂2u 5 ∂4u 1 ∂4uˆ ∂δ gd d2 d2 = g Λ (12) ∂t2 − ∂x2 − 16 ∂x2∂t2 − 8 ∂x2∂t2 − ∂x Eliminating the second and third term of Equation (12) using Equation (11) to get

∂2u g 1 ∂2uˆ 5 ∂2uˆ 1 ∂4uˆ ∂δ 8 uˆ + gd + d2 = g Λ (13) ∂t2 − d 2 ∂x2 − 2 ∂t2 32 ∂x2∂t2 − ∂x Taking the x derivative two times and eliminating the ﬁrst term using Equation (11) gives

1 ∂2uˆ g ∂2uˆ 1 ∂4uˆ ∂4uˆ 1 ∂6uˆ ∂3δ 8 8 + gd 3 + d2 = g Λ (14) d2 ∂t2 − d ∂x2 2 ∂x4 − ∂x2∂t2 32 ∂x4∂t2 − ∂x3 Integrating 4 times Equation (14) from x = ε to x = +ε, with the limit ε 0 and requiring − → the integrals of uˆ to be continuous at x = 0, we get

2 + 2 ! 1 + 1 2 ∂ uˆ ∂ uˆ − gd uˆ uˆ − + d = g Λ (15) 2 − 32 ∂t2 − ∂t2 −

In Equation (15), uˆ is an odd function and thus the left and right contributions are identical. In SWASH an approximation of the exact linear dispersion relation, which depends on the number of vertical layers, is used (see AppendixA). For the case of two equidistant layers it is given by

1 + 1 (kd)2 ω2 = gk2d 16 (16) 3 2 1 4 1 + 8 (kd) + 256 (kd)

In addition, at the wave generation boundary, apart from the progressive waves, evanescent modes are included as well. These evanescent modes are a general characteristic of the Equations (9)–(11). Hence, Equation (16) can be rewritten as

2 2 2 ω2 1+ 1 (kd) 1+ 1 (k d) ω d = ed ( )2 16 = ( )2 16 e g g kd 3 2 1 4 ked 3 2 1 4 (17) → 1+ 8 (kd) + 256 (kd) 1+ 8 (ked) + 256 (ked) where k and ke are the wave numbers for progressive waves and evanescent modes, respectively. Equation (17) yields two solutions

1+ 1 (kd)2 (k d)2 (kd)2, (k d)2 16 (18) e e 1 5 ( )2 → → − 16 + 256 kd Furthermore, away from the source point, x = 0, the solutions of Equations (9)–(11) can be written as η = η exp[i(ωt kx)] + η exp[i(ωt kex)] (19) p0 − e0 − u = u exp[i(ωt kx)] + u exp[i(ωt kex)] (20) p0 − e0 − uˆ = uˆ exp[i(ωt kx)] + uˆ exp[i(ωt kex)] (21) p0 − e0 − Water 2019, 11, 986 6 of 18 where the subscripts p and e stand for progressive and evanescent modes, respectively. By using Equations (10) and (11), the inter-layer velocity amplitude uˆ p0 can be expressed in terms of surface elevation amplitude ηp0 1 gk (kd)2 uˆ p0 = η (22) −8 ω 3 2 1 4 p0 1 + 8 (kd) + 256 (kd)

In addition, the double integral of uˆ with respect to x gives an odd function and it must vanish at the source point, x = 0 and thus from Equation (21) we can write

2 ke uˆ e0 = uˆ p0 (23) − k2

Finally, Equation (21) at x 0+ is imported in Equation (15) and eventually uˆ and uˆ are → e0 p0 going to be replaced using Equations (22) and (23) respectively. As a result, Equation (15) becomes

Λ = 2Ceηp (24) where ηp = ηp0 exp[iωt] (25)   2 2 ! 2 ! 1 gk  (kd)  ke 1 1 ω d Ce = d  1 (26) 8 ω  3 2 1 4  − 2 2 − 32 g 1 + 8 (kd) + 256 (kd) k

By eliminating ω and ke using Equations (16) and (18), respectively, it can be noticed that the energy velocity Ce is equal to the group velocity Cg as it can be calculated from the approximated dispersion relation by taking the derivative of ω with respect to k

p 2 4 64 dg(256 + 32(kd) + 5(kd) ) ∂ω Ce = q = = Cg (27) 3 ∂k (16 + (kd)2)(256 + 96(kd)2 + (kd)4)

Similarly, the energy velocity (group velocity) Ce can be derived for any number of vertical layers. In Figure1, the normalised energy velocities Ce/Cg Airy for one, two and three vertical layers are plotted as a function of dimensionless depth kd. In addition, the range of dimensionless depth kd as a function of number of vertical layers where the relative error in the normalised energy velocities Ce/Cg Airy stays below 3% is given in Table1. It can be observed that by increasing the number of layers, a better ﬁt is achieved with the exact linear solution Ce/Cg Airy = 1, fact that makes the model applicable for higher kd values.

Table 1. Range of dimensionless depth kd as a function of number of vertical layers

and the corresponding relative error in the normalised energy velocities Ce/Cg Airy.

Number of Layers kd (-) Error (%) 1 0.55 3 ≤ ≤ 2 6.00 3 ≤ ≤ 3 12.50 3 ≤ ≤ Water 2019, 11, x FOR PEER REVIEW 7 of 18 Water 2019, 11, 986 7 of 18

Figure 1. Comparison of normalised energy velocities Ce/Cg Airy for diﬀerent number of vertical layers. Figure 1. Comparison of normalised energy velocities C⁄C for different number of vertical 3.2. Sourcelayers. Term Addition Method

In the source term addition method proposed by Reference [21], additional surface elevation η∗ is addedTable with 1. the Range desired of dimensionless energy to the depth calculated kd as surface a function elevation of numberη at theof vertical wave generation layers and linethe for C ⁄C each timecorresponding step and isrelative given error by in the normalised energy velocities . I Ce∆t Numberη of∗ =Layers2η kdcos (-) θ Error (%) (28) x 1 ∆ ≤0.55 ≤3 where ∆x is the grid size in the x-axis, ∆t2is the time step,≤6.00θ is the angle≤3 of the incident wave ray from I x-axis, η is the water surface elevation of3 incident waves≤12.50 and Ce is≤3 the energy velocity. 3.3. Spatially Distributed Source Function 3.2. Source Term Addition Method In the spatially distributed source function proposed by Reference [25], a spatially distributed In the source term addition method proposed by Reference [21], additional surface elevation η∗ mass is added in the continuity equation. Hence, the spatially distributed source function is applied on is added with the desired energy to the calculated surface elevation η at the wave generation line for an area in contrast to the source term addition method which is applied on a line as it is demonstrated each time step and is given by in Figure2. For a single wave component, the source function can be deﬁned as follows C Δt η∗ =2η cos θ (28) f(x, y, t) = g(x)DcosΔx(ωt kysinθ) (29) − where Δx is the grid size in the x-axis, Δt is the time step, θ is the angle of the incident wave ray where D is the source function amplitude and g(x) is the shape of the source function which can be from x-axis, η is the water surface elevation of incident waves and C is the energy velocity. arbitrarily chosen. Here, a smooth Gaussian shape has been applied. Wei et al. [25] derived the source function3.3. Spatially amplitude Distributed D for Source the extended Function Boussinesq equations of [23] and is given by

2 4 3 In the spatially distributed source function2η (ω propα1gkosedd ) cosby Referenceθ [25], a spatially distributed D = 0 − (30) mass is added in the continuity equation. Hence,ωkI(1 theα spatially(kd)2) distributed source function is applied on an area in contrast to the source term addition− method which is applied on a line as it is wheredemonstratedα = 0.390, in Figureα = 2. Forα + a1 single/3 and wave I is the compon integralent, deﬁned the source by function can be defined as follows − 1 f(x, y,t)r=g(x)Dcos(ωt − kysin2  θ) (29) π  (kcosθ)  I = exp  (31) where D is the source function amplitude andβ g(x−) is the4β shape of the source function which can be arbitrarily chosen. Here, a smooth Gaussian shape has been applied. Wei et al. [25] derived the source function amplitude D for the extended Boussinesq equations of [23] and is given by

Water 2019, 11, x FOR PEER REVIEW 8 of 18

2η (ω −α gkd)cosθ D= (30) ωkI(1−α(kd))

where α = −0.390, α = α+1⁄ 3 and I is the integral defined by Water 2019, 11, 986 8 of 18 π (kcosθ) I= exp − (31) β 4β where β = 20/W2 and W is the width of the source area. whereEquation β=20W (30)⁄ isand derived W is the for thewidth extended of the source Boussinesq area. equations of Nwogu and thus cannot be appliedEquation directly (30) to the is SWASHderived for equations the extended for the Boussi case ofnesq multiple equations layers. of However, Nwogu and Equation thus cannot (30) can be beapplied written directly in terms to ofthe the SWASH energy equations velocity as for the case of multiple layers. However, Equation (30) can be written in terms of the energy velocity as cos θ D cos= 2Cθ eη0 (32) D=2C η I (32) I Equation (32) can be implemented in SWASH by using the energy velocity that corresponds to Equation (32) can be implemented in SWASH by using the energy velocity that corresponds to the number of vertical layers used in the model. The energy velocity for the case of two vertical layers the number of vertical layers used in the model. The energy velocity for the case of two vertical layers has already been derived in Section 3.1. In this way, we overcome the limitation that [26,27] observed, has already been derived in Section 3.1. In this way, we overcome the limitation that [26] and [27] where their models, using Equation (30), were not applicable to deep water conditions. observed, where their models, using Equation (30), were not applicable to deep water conditions.

Figure 2. Definition sketch of the implemented internal wave generation methods (a) source term Figure 2. Definition sketch of the implemented internal wave generation methods (a) source term addition method and (b) spatially distributed source function. addition method and (b) spatially distributed source function. 4. Model Tests 4. Model Tests The internal wave generation techniques, implemented in SWASH as described in Section3, have beenThe usedinternal to generate wave generation regular and techniques, irregular long-crested implemented waves. in SWASH The obtained as described results are in validatedSection 3, againsthave been analytical used solutions to generate and experimentalregular and datairregu includinglar long-crested water surface waves. elevations, The obtained orbital results velocities, are frequencyvalidated spectraagainst and analytical wave heights. solutions and experimental data including water surface elevations, orbital velocities, frequency spectra and wave heights. 4.1. Regular Waves 4.1. Regular Waves Firstly, the applicability and accuracy of the internal wave generation techniques to generate linearFirstly, regular the waves applicability in transitional and andaccuracy deep of water the conditionsinternal wave are examined.generation Thetechniques numerical to basingenerate is similarlinear toregular the one waves in Figure in transitional2 with sponge and deep layers water at the conditions left and right are examined. boundaries The with numerical a width basin of 3 L is (wheresimilarL tois the waveone in length) Figure and2 with a flat sponge bottom. layers The at internal the left wave and generatorright boundaries is positioned with a at width a distance of 3L of(where 3L from L theis the sponge wave layer. length) Two and diff erenta flat wavebottom. conditions The internal are examined: wave generator one intransitional is positioned water at a (kd = π/6) and one in deep water (kd = π, highly dispersive wave). For dimensionless depth kd = π/6 one vertical layer is applied while for dimensionless depth kd = π two equidistant vertical layers are applied in order to keep the relative error below 3% (Figure1, Table1). For both cases, the grid cell size is chosen so that ∆x = L/50, while the time step is equal to ∆t T/350, where T is the wave period. ≈ Water 2019, 11, x FOR PEER REVIEW 9 of 18

distance of 3L from the sponge layer. Two different wave conditions are examined: one in transitional water (kd = π/6) and one in deep water (kd = π, highly dispersive wave). For dimensionless depth kd = π/6 one vertical layer is applied while for dimensionless depth kd = π two equidistant vertical layers are applied in order to keep the relative error below 3% (Figure 1, Table 1). For both cases, the Water 2019, 11, 986 9 of 18 grid cell size is chosen so that Δx = L/50, while the time step is equal to Δt ≈ T/350, where T is the wave period.

InIn Figure Figure3, the3, the normalised normalised water water surface surface elevation elevationη/η0 calculatedηη⁄ calculated using the using source the term source addition term methodaddition (blue method dashed (blue lines) dashed and the lines) spatially and the distributed spatially sourcedistributed function source (red function dashed lines) (red dashed is presented. lines) Theis presented. computed The results computed at a distance results of at 1 aL distancefrom the of internal 1L from generator the internal and generator at a time period and at ofa timet = 10 periodT to t of= 18t T= are10T compared to t = 18 toT theare resultscompared of the to analyticalthe results solution of the analytical (solid black solution lines) (solid for the black same lines) conditions. for the Thesame agreement conditions. is veryThe agreement good while is thevery behavior good while of the the twobehavior internal of the wave two generation internal wave techniques generation is similar,techniques with is a similar, small deviation with a small for the deviation case of thefor highlythe case dispersive of the highly wave dispersive (Figure3 c,d).wave For (Figure the latter 3c,d). case,For the spatiallylatter case, distributed the spatially source distributed function provides source function a slightly provides better ﬁt a with slightly the analytical better fit solution.with the analytical solution.

Figure 3. Comparison between computed (dashed lines) and target (solid lines) normalised water Figure 3. Comparison between computed (dashed lines) and target (solid lines) normalised water surface elevation η/η0 for the case of linear waves with dimensionless depth (a,b) kd = π/6 (one layer) surface elevation ηη⁄ for the case of linear waves with dimensionless depth (a,b) kd = π/6 (one and (c,d) kd = π (two equidistant layers). layer) and (c,d) kd = π (two equidistant layers). In order to check the capability of the model for handling reflected dispersive waves In order to check the capability of the model for handling reflected dispersive waves a a computational domain with a sponge layer at the left boundary and a fully reflective wall at the right computational domain with a sponge layer at the left boundary and a fully reflective wall at the right boundary is used. The internal wave generator is positioned at the middle of the computational boundary is used. The internal wave generator is positioned at the middle of the computational domain which has a total length of 12L. The wave height is H = 0.02 m, the wave period is T = 3 s domain which has a total length of 12L. The wave height is H = 0.02 m, the wave period is T = 3 s and the water depth is d = 10 m. These wave conditions give a dimensionless depth of kd = 4.5 and the water depth is d = 10 m. These wave conditions give a dimensionless depth of kd = 4.5 and and thus two equidistant vertical layers are applied. The model is applied with a grid cell size of thus two equidistant vertical layers are applied. The model is applied with a grid cell size of Δx = 0.3 ∆x = 0.3 m and a time step of ∆t = 0.0125 s. m and a time step of Δt = 0.0125 s. Figure4 shows a snapshot of normalised water surface elevation η/η0 at t = 50T generated Figure 4 shows a snapshot of normalised water surface elevation ηη⁄ at t = 50T generated using the source term addition method (blue solid line) and the spatially distributed source function using the source term addition method (blue solid line) and the spatially distributed source function (red dashed line). The two computed profiles are identical, while their agreement with the analytical (red dashed line). The two computed profiles are identical, while their agreement with the analytical solution (black markers) is excellent. solution (black markers) is excellent. The waves that are generated at the internal wave generator propagate towards both ends The waves that are generated at the internal wave generator propagate towards both ends of the of the domain. The waves are fully reflected at the right boundary and then are fully absorbed domain. The waves are fully reflected at the right boundary and then are fully absorbed at the sponge at the sponge layer. This, together with the fact that the target wave is a linear wave and that the length layer. This, together with the fact that the target wave is a linear wave and that the length of the of the numerical domain is an integer multiple of the wave length, results in a profile that is a standing numerical domain is an integer multiple of the wave length, results in a profile that is a standing wave with perfect nodal points. In addition, the results show that the reflected waves pass through wave with perfect nodal points. In addition, the results show that the reflected waves pass through the generation area without distortion and that the sponge layer is capable of absorbing the highly the generation area without distortion and that the sponge layer is capable of absorbing the highly dispersive wave. It has to be mentioned that the weakly reflective wave generation (Section 2.2) cannot be applied in this case since the assumption that the reflected waves are shallow water waves p propagating with a phase velocity of C = gd is violated. Water 2019, 11, x FOR PEER REVIEW 10 of 18 dispersive wave. It has to be mentioned that the weakly reflective wave generation (Section 2.2) cannot be applied in this case since the assumption that the reflected waves are shallow water waves Water 2019, 11, 986 10 of 18 propagating with a phase velocity of C = gd is violated.

Figure 4. Snapshot of normalised water surface elevation η/η at t = 50T using two internal wave Figure 4. Snapshot of normalised water surface elevation ηη⁄ 0 at t = 50T using two internal wave generation techniques and the analytical solution (black markers) for the case of deep water linear generation techniques and the analytical solution (black markers) for the case of deep water linear waves (H = 0.02 m, T = 3.0 s, d = 10 m, two equidistant layers) with a sponge layer at the left boundary waves (H = 0.02 m, T = 3.0 s, d = 10 m, two equidistant layers) with a sponge layer at the left and a fully reflective wall at the right boundary. boundary and a fully reflective wall at the right boundary. In the proposed internal wave generation techniques, the horizontal velocity component is not prescribedIn the overproposed the vertical internal direction wave generation in contrast techni to theques, weakly the reflective horizontal wave velocity generation component (Section is 2.2not). prescribedHowever, as over it can the be vertical observed direction from Figure in contrast5 the horizontal to the weakly and vertical reflective velocity wave profiles generation are correctly (Section 2.2).calculated However, for both as it internal can be waveobserved generation from Figure techniques 5 the athorizontal a distance and of 1verticalL from thevelocity generation profiles point. are L correctlyThe wave calculated conditions for are both the same internal as in wave the previous generation test, techniques while ten equidistantat a distance vertical of 1 layers from have the generationbeen applied point. in order The towave achieve conditions a good agreementare the same with as thein the analytical previous hyperbolic test, while profile. ten equidistant verticalHere, layers it has have to bebeen mentioned applied in that order in case to achi of deepeve a watergood agreement waves, when with coarse the analytical vertical resolution hyperbolic is profile.applied, the weakly reflective wave generation technique needs calibration to generate the target wave height,Here, since it thehas hyperbolicto be mentioned profile that of the in horizontalcase of deep velocity water componentwaves, when cannot coarse be vertical accurately resolution described is applied,by a small the number weakly of reflective vertical layers. wave Ongeneration the other te hand,chnique the internalneeds calibration wave generation to generate techniques the target that waveare presented height, since in this the paper hyperbolic do not needprofile calibration, of the horizontal since the velocity source iscomponent directly linked cannot with be theaccurately surface describedelevation insteadby a small of the number velocity of component. vertical layers. On the other hand, the internal wave generation techniquesIn all thethat previous are presented tests in this this paper section, do linearnot need waves calibration, have been since examined. the source In is orderdirectly to linked study withthe behaviour the surface of theelevation internal instea waved generationof the velocity techniques component. under non-linear waves, waves with different waveIn heights all the areprevious tested. tests The internalin this section, wave generator linear waves is positioned have been at examined. the middle In of order the computational to study the behaviourdomain which of the has internal a total wave length generation of 32L with techniqu spongees layers under at non-linear the left and wave rights, boundarieswaves with anddifferent a flat wavebottom. heights The wave are tested. heights The are internalH = 0.1 wave m, 1 m,generato 2 m, 3r m, is positioned while the wave at the period middle is ofT =the6 scomputational and the water L domaindepth is dwhich= 10 m.has These a total wave length conditions of 32 withgive asponge dimensionless layers at depth the left of kd and= 1.3right and boundaries thus two equidistant and a flat H T bottom.vertical layersThe wave are applied. heights Theare model = 0.1 is m, applied 1 m, 2 with m, 3 a m, grid while cell sizethe wave of ∆x =period2.0 m is and a= time6 s and step the of d kd water∆t = 0.025 depth s. is = 10 m. These wave conditions give a dimensionless depth of = 1.3 and thus two equidistant vertical layers are applied. The model is applied with a grid cell size of Δx = 2.0 m and a Figure6 shows snapshots of normalised water surface elevation η/η0 at t = 50T for the different Δt timewave step heights. of Profiles = 0.025 generateds. using the spatially distributed source function are only presented ηη⁄ t = 50T hereFigure since the 6 sourceshows termsnapshots addition of methodnormalised becomes water unstable surface for elevation high wave heights. at FromFigure for the6, differentit can be observedwave heights. that by Profiles increasing generated the ratio usingH/d ,th thee spatially waves are distributed transforming source from function linear (Figure are only6a) presentedto non-linear here where since thethe wavesource crests term areaddition becoming meth sharperod becomes and theunstable wave for troughs high wave flatter. heights. From Figure 6, it can be observed that by increasing the ratio Hd⁄ , the waves are transforming from linear (Figure 6a) to non-linear where the wave crests are becoming sharper and the wave troughs flatter.

Water 2019, 11, x FOR PEER REVIEW 11 of 18 Water 2019,, 11, x 986 FOR PEER REVIEW 1111 of of 18 18

Figure 5. Comparison between computed (markers) and target (solid lines) maximum horizontal and FigureFigure 5. 5. ComparisonComparison between between computed computed (markers) (markers) and and target target (solid (solid lines) lines) maximum maximum horizontal horizontal and vertical velocity profiles using two internal wave generation techniques for the case of deep water verticaland vertical velocity velocity profiles proﬁles using using two two internal internal wave wave generation generation techniques techniques for for the the case case of of deep deep water water linear waves (H = 0.02 m, T = 3.0 s, d = 10 m, ten equidistant layers). linearlinear waves waves ( (HH = 0.020.02 m, m, T = = 3.03.0 s,s, dd= =10 10 m, m, ten ten equidistant equidistant layers). layers).

Figure 6. Snapshot of normalised water surface elevation η/η at t = 50T using the spatially Figure 6. Snapshot of normalised water surface elevation ηη⁄0 at t=50T using the spatially Figure 6. Snapshot of normalised water surface elevation ηη⁄ at t=50T using the spatially distributeddistributed source source functionfunction wavewave generationgeneration technique for ( (aa)) HdH⁄/d == 0.010.01 (linear), (linear), (b ()b )HdH⁄ / d = =0.1,0.1, (c) distributed source function wave generation technique for (a) Hd⁄ = 0.01 (linear), (b) Hd⁄ = 0.1, (c) (cHd)H⁄ / d = 0.2= 0.2 and and (d) ( dHd)H⁄ / =d 0.3.= 0.3. Hd⁄ = 0.2 and (d) Hd⁄ = 0.3.

Water 2019, 11, x FOR PEER REVIEW 12 of 18 Water 2019, 11, 986 12 of 18

4.2. Irregular Waves 4.2. Irregular Waves We consider two test cases of uni-directional irregular waves fitting a JONSWAP (Joint North Sea WaveWe consider Observation two testProject) cases spectrum, of uni-directional with a significant irregular wave waves height, fitting H a JONSWAP = 0.5 m, a water (Joint depth, North dSea = 10 Wave m, a Observation peak enhancement Project) factor, spectrum, γ = 3.3 with and a significanttwo different wave peak height, waveH periods,s = 0.5 m,T a = water 12 s and depth, T =d =8 10s. m,The a peakfrequency enhancement range is factor, confinedγ = 3.3 between and two 0.5 diffferent and peak 3f wave. Theperiods, internalT pwave= 12 sgenerator and Tp = 8is s. positionedThe frequency at the range middle is confined of the computational between 0.5fp domainand 3fp and. The sponge internal layers wave are generator placed at is the positioned left and rightat the boundaries. middle of the For computational both test cases, domain the model and sponge is applied layers with are placed a grid atcell the size left andof Δx right = 1.0 boundaries. m and a timeFor both step test of cases,Δt = 0.025 the model s. However, is applied for withthe case a grid of cellT size = 8 s of two∆x equidistant= 1.0 m and vertical a time step layers of ∆ havet = 0.025 been s. appliedHowever, to forcorrectly the case describe of Tp = the8 s twohigh-frequency equidistant verticalpart of layersthe spectrum have been in appliedcontrast totocorrectly the case describeof T = 12the s high-frequencywhere one layer part is enough. of the spectrum in contrast to the case of Tp = 12 s where one layer is enough.

InIn Figure Figure 77,, aa comparisoncomparison isis mademade betweenbetween thethe targettarget frequencyfrequency spectrumspectrum ((SSt)) and and the the simulated simulated frequencyfrequency spectra spectra generated generated using using the the two two internal internal wave wave generation generation techniques techniques for the for cases the casesof T of=

12Tp s= (Figure12 s (Figure 7a) and7a) andT =T p8 =s (Figure8 s (Figure 7b).7b). The The surface surface elevations elevations η ηatat the the electronic electronic wave wave gauges, gauges, which are are positioned positioned at at a a distance distance of of 3 3LLp (where(where Lp is is thethe wavewave length corresponding to to the the peak peak = = period) from the internal wave generator, areare recordedrecorded fromfromt t= 25Tp to tt= 300300TTp withwith a a sampling sampling intervalinterval of of 0.2 0.2 s. The The recorded recorded data data are are processed in in segments of 2048 points per segment. A A taper taper window and an overlap of 20% are used for smoother and statistically more significant signiﬁcant spectral estimates.estimates. TheThe resultingresulting frequency frequency spectra spectra agree agree very wellvery withwellS twithapart S from apart the onefrom that the corresponds one that correspondsto the source to term the source addition term method addition for method the case for of theT pcase= 12 of s,T where = 12 s, high-frequency where high-frequency noise can noise be canobserved. be observed. In addition, In addition, it is it found is found that that the the magnitude magnitude of of this this noise noise strongly strongly depends depends onon the grid resolutionresolution sincesince aa coarsercoarser resolution resolution leads leads to to less less noise. noise. Moreover, Moreover, the the small small diﬀ erencedifference around around the peak the peakof the of frequency the frequency spectrum spectrum in Figure in 7Figureb could 7b be co causeduld be by caused numerical by numerical dissipation dissipation since the electronicsince the wave gauges are positioned at a distance of 3L from the internal wave generator. electronic wave gauges are positioned at a distancep of 3L from the internal wave generator.

FigureFigure 7.7. ComparisonComparison between between the the frequency frequency spectra spectra resulting resulting from using from two using internal two wave internal generation wave

generationtechniques andtechniques the target and frequency the target spectrum frequencySt for spectrum irregular S waves for irregular with Hs = waves0.5 m, dwith= 10 H m (a =) 0.5Tp =m,12 d s =(one 10 m layer) (a) T and = 12 (b )Ts (onep = 8layer) s (two and equidistant (b) T = 8 layers). s (two equidistant layers).

4.3.4.3. Oblique Oblique Waves Waves in in a a Basin Basin with with Constant Constant Depth Depth InIn this this section, section, oblique oblique waves waves are are generated generated in in a a basin basin with with constant constant dept depthh by by using using the the spatially spatially distributeddistributed source source function function wave wave generation generation techni technique.que. The The numerical numerical basin basin is 190 is m 190 long, m long,90 m wide 90 m

Water 2019, 11, 986 13 of 18

Water 2019, 11, x FOR PEER REVIEW 13 of 18 wide and 1 m deep. Sponge layers are placed at the left and right boundaries with a width of 50 m and 1 m deep. Sponge layers are placed at the left and right boundaries with a width of 50 m while whileWater periodic 2019, 11, boundariesx FOR PEER REVIEW are applied at the top and bottom of the domain. The internal13 of 18 wave periodic boundaries are applied at the top and bottom of the domain. The internal wave generator is generator is parallel to the y-axis and is positioned at a distance of 100 m (x = 0) from the left boundary. paralleland to1 m the deep. y-axis Sponge and islayers positioned are placed at aat distance the left and of 100right m boundaries (x = 0) from with the a leftwidth boundary. of 50 m while The wave The wave height is H = 0.01 m, the wave period is T = 4 s and the wave propagation angle is θ = 15 . heightperiodic is H =boundaries 0.01 m, the are wave applied period at the is top T and = 4 sbotto andm the of thewave domain. propagation The internal angle wave is θ= generator 15°. The is grid◦ The gridparallel cell to size the isy-axis chosen and sois positioned that ∆x = at a∆ distancey = 0.15 of 100 m. m In (x order = 0) from to obtain the left a boundary. steady state The wave wave ﬁeld, cell size is chosen so that Δx = Δy = 0.15 m. In order to obtain a steady state wave field, waves are wavesheight are generated is H = 0.01 form, the a duration wave period of 180 is T s with = 4 s and a time the stepwave∆ propagationt = 0.0125 angle s. is θ= 15°. The grid generated for a duration of 180 s with a time step Δt = 0.0125 s. Figurescell size 8is and chosen9 present so that comparisons Δx = Δy = 0.15 between m. In order the to computed obtain a steady normalised state wave water field, surface waves elevation are Figures 8 and 9 present comparisons between the computed normalised water surface elevation η/η generatedat t = 40 forT aand duration the corresponding of 180 s with a time analytical step Δt solution. = 0.0125 s. It can be observed that the computed ηη⁄ 0 at t = 40T and the corresponding analytical solution. It can be observed that the computed solution coincidesFigures 8 and with 9 thepresent analytical comparisons solution between except the for computed the region normalised of the sponge water surface layers. elevation The excellent solutionηη⁄ coincides at t = 40T with and thethe analyticalcorresponding solution analytical except solution. for the regionIt can be of observedthe sponge that layers. the computed The excellent agreement indicates that the spatially distributed source function implemented in SWASH is able to agreementsolution indicates coincides thatwith thethe analyticalspatially solutiondistributed except source for the function region of implemented the sponge layers. in SWASH The excellent is able to accurately generate directional waves. accuratelyagreement generate indicates directional that the spatially waves. distributed source function implemented in SWASH is able to accurately generate directional waves.

Figure 8. Comparison between (a) computed and (b) target normalised water surface elevation ηη⁄ Figure 8. Comparison between (a) computed and (b) target normalised water surface elevation η/η0 Figureat t8. = Comparison 40T for the case between of oblique (a) computedlinear waves and (H ( =b 0.01) target m, T normalised = 4.0 s, d = water1.0 m and surface θ = 15°)elevation with a ηη⁄ at t = 40T for the case of oblique linear waves (H = 0.01 m, T = 4.0 s, d = 1.0 m and θ = 15◦) with at t = 40T for the case of oblique linear waves (H = 0.01 m, T = 4.0 s, d = 1.0 m and θ = 15°) with a a spongesponge layer layer at at the the left left and and rightright boundaries and and periodic periodic boundaries boundaries at y–direction. at y–direction. sponge layer at the left and right boundaries and periodic boundaries at y–direction.

FigureFigure 9. Comparison 9. Comparison between between computed computed (dashed (dashed line) line) and and target target (solid (sol line)id line) normalised normalised water water surface

elevationsurfaceη/ elevationη at t =ηη⁄ 40 atT fort = 40T the case for the of case oblique of oblique linear linear waves waves (H (H= =0.01 0.01 m,m, TT = = 4.04.0 s, s, dd = =1.01.0 m 0 and θm= and15◦ ):θ (=a 15°):) cross (a) sectioncross section at y =at 45y = m, 45 (m,b) ( crossb) cross section section at at x x= = 00 m.m. Figure 9. Comparison between computed (dashed line) and target (solid line) normalised water

surface elevation ηη⁄ at t = 40T for the case of oblique linear waves (H = 0.01 m, T = 4.0 s, d = 1.0 m and θ = 15°): (a) cross section at y = 45 m, (b) cross section at x = 0 m.

Water 2019, 11, x FOR PEER REVIEW 14 of 18 Water 2019, 11, 986 14 of 18 4.4. Wave Propagation over a Shoal in a Three-Dimensional Numerical Basin 4.4. WaveFinally, Propagation a three-dimensional over a Shoal in aversion Three-Dimensional of the developed Numerical model Basin with the spatially distributed sourceFinally, function a three-dimensional wave generation version technique of the is developed applied to model study with regular the spatially waves propagating distributed source over a functionshoal. The wave experiment generation conducted technique by isReference applied [33] to study has served regular as wavesa standard propagating test case overfor validating a shoal. Thephase experiment resolving conducted wave models. by Reference [33] has served as a standard test case for validating phase resolvingThe wavebathymetry models. of the experimental setup consist of an elliptic shoal resting on a plane sloping seabedThe and bathymetry the entire of slope the experimental is turned at an setup angle consist of 20° with of an respect elliptic to shoal the x-axis. resting Detailed on a plane information sloping on the geometry can be found in Reference [33]. The numerical basin is 45 m long (−20 < y < 25) and seabed and the entire slope is turned at an angle of 20◦ with respect to the x-axis. Detailed information on20 the m wide geometry (−10 < can x < be 10). found The internal in Reference wave [ 33generator]. The numerical is parallel basinto the is x-axis 45 m and long is ( positioned20 < y < 25)at a − anddistance 20 m wideof 10 (m10 (y <= x−10< 10).m) from The internalthe bottom wave boundary. generator The is parallel remaining to the numerical x-axis and domain is positioned includes − attwo a distance sidewalls of 10at mx = (y −10= m10 and m) fromx = 10 the m and bottom two boundary. sponge layers The remainingat y = −15 m numerical and y = 20 domain m with includes a width − twoof 5 sidewalls m. at x = 10 m and x = 10 m and two sponge layers at y = 15 m and y = 20 m with − − a widthThe of wave 5 m. height is H = 0.0464 m, the wave period is T = 1 s and the water depth at the position of theThe internal wave height wave isgeneratorH = 0.0464 is d m, = the0.45 wave m. These period wave is T =conditions1 s and the give water a dimensionless depth at the position depth of ofkd the = internal1.9 and thus wave two generator equidistant is d = vertical0.45 m. layers These are wave applied. conditions The model givea runs dimensionless for 50 s without depth any of kdstability= 1.9 and issues thus since two the equidistant reflected waves vertical that layers are re areaching applied. the offshore The model boundary runs for are 50 absorbed s without by any the stabilitysponge issueslayer, which since theis positioned reflected waves behind that the areinte reachingrnal wave the generator. offshore The boundary model areis applied absorbed with by a Δx Δy thegrid sponge cell size layer, of which = is =positioned 0.05 m, while behind the time the internal step is auto wavematically generator. adjusted The model during is the applied simulation with adepending grid cell size on of the∆x CFL= ∆y (Courant–Friedrichs–Lewy)= 0.05 m, while the time step condition is automatically where adjusteda maximum during CFL the value simulation of 0.5 is dependingused. on the CFL (Courant–Friedrichs–Lewy) condition where a maximum CFL value of 0.5 HH⁄ H is used.Figure 10 shows the plan view of the normalised wave height (where is the local wave height and H is the wave height at the wave generation boundary) in the whole computational Figure 10 shows the plan view of the normalised wave height H/H0 (where H is the local wave domain where the diffraction and refraction patterns of the waves are visible. The wave heights of height and H0 is the wave height at the wave generation boundary) in the whole computational domain wherethe model the di areffraction obtained and by refraction averaging patterns those of of the the la wavesst ten arewave visible. periods The of wave the simulation heights of (from the model t = 40 ares to obtained t = 50 s). by averaging those of the last ten wave periods of the simulation (from t = 40 s to t = 50 s).

Figure 10. Normalised wave height H/H0 in the whole computational domain for the experiment of Berkhoﬀ [34]. Figure 10. Normalised wave height HH⁄ in the whole computational domain for the experiment of Berkhoff [34].

Water 2019, 11, 986x FOR PEER REVIEW 1515 of of 18

Figure 11 shows the comparison of normalised wave heights H/H between numerical model resultsFigure (red 11 lines) shows and the comparisonexperimental of data normalised (black wavecircles) heights alongH /eightH0 between measurement numerical transects. model Additionally,results (red lines) in order and experimental to evaluate the data model, (black the circles) root along mean eight square measurement error (RMSE) transects. and the Additionally, Skill factor forin order the normalised to evaluate wave the model, heights the of root each mean section square are error calculated (RMSE) as: and the Skill factor for the normalised wave heights of each section are calculated as: ∑( P −O) ∑( P −O) r r (33) RMSE = PN Skill2 = 1 − PN 2 N = (Pi Oi) ∑ O = (Pi Oi) = i 1 − = i 1 − (33) RMSE N Skill 1 PN 2 − i = 1 Oi where O and P indicate the observed and predicted values, respectively. Very good agreement betweenwhere O andthe numericalP indicate themodel observed and the and experimental predicted values, model respectively. is observed Very(Figure good 11 agreementand Table between2). the numerical model and the experimental model is observed (Figure 11 and Table2).

Figure 11. Comparison of normalised wave heights H/H0 between numerical model results (red solid lines) and experimental data (black circles) along different measurement transects. Figure 11. Comparison of normalised wave heights H/H between numerical model results (red solid lines)Table and 2. Root experimental mean square data error (black (RMSE) circle ands) along Skill factor different of the measurement normalised wave transects. heights for each section.

Table 2. RootSection mean square 1 error 2(RMSE) 3and Skill 4 factor of 5 the normalised 6 7 wave heights 8 for each section. RMSE 0.056 0.116 0.102 0.074 0.103 0.082 0.157 0.108 Skill 0.944 0.884 0.901 0.929 0.902 0.889 0.901 0.865 Section 1 2 3 4 5 6 7 8 5. ConclusionsRMSE 0.056 0.116 0.102 0.074 0.103 0.082 0.157 0.108 Skill 0.944 0.884 0.901 0.929 0.902 0.889 0.901 0.865 In the present paper new internal wave generation techniques have been developed in an open 5.source Conclusions non-hydrostatic wave model, SWASH, for accurate generation of regular and irregular long-crested waves. In the present paper new internal wave generation techniques have been developed in an open Initially, two different internal wave generation techniques have been developed and implemented source non-hydrostatic wave model, SWASH, for accurate generation of regular and irregular long- in SWASH model. The first one is a source term addition method based on the method proposed by crested waves. Lee et al. [21], while the second one is a spatially distributed source function based on the method Initially, two different internal wave generation techniques have been developed and proposed by Wei et al. [25]. These techniques need an extension of the numerical domain to implemented in SWASH model. The first one is a source term addition method based on the method accommodate the sponge layer and the source area contrary to the weakly reflective wave generation proposed by Lee et al. [21], while the second one is a spatially distributed source function based on

Water 2019, 11, 986 16 of 18 boundary. Thus, the computational cost increases 15–30%, where the lower values stand for larger computational domains. However, the main advantage of the internal wave generation is that in cases of directional and dispersive waves the reflected waves are absorbed by the sponge layer that is positioned behind the internal wave generator in contrast with the weakly reflective wave generation boundary, which is not valid for these wave conditions due to the limitations as described in Section 2.2. Hence, the internal wave generation has a big advantage when wave energy converter (WEC) farms and man-made structures (e.g., breakwaters, artificial reefs, artificial islands) are examined, since the radiated and reflected waves, respectively, cannot be estimated a priori. In the source term addition method additional surface elevation is added to the calculated surface elevation while in the spatially distributed source function a spatially distributed mass is added in the continuity equation. In both cases, the source term propagates with a velocity which is called the energy velocity which for the system of SWASH equations is mathematically derived in Section 3.1. Then, these wave generation techniques has been used to generate regular and irregular long-crested waves. The results indicate that the developed model is capable of reproducing the target surface elevations as well as the target frequency spectrum. The overall performance of the spatially distributed source function is better than the source term addition method since the latter becomes unstable for large wave heights and may cause high frequency noise. Finally,the developed model is also used to study wave transformation over an elliptic shoal (Berkhoff shoal experiment) where very good agreement is observed between the numerical model and the experimental results. The aforementioned observations reveal that the spatially distributed source function developed here in SWASH can be successfully used to study coastal areas and wave energy converter (WEC) farms even under highly dispersive and directional waves, while at the same time re-reflections at the position of the wave generator are vanishing.

Author Contributions: P.V. developed the fundaments of the internal wave generation implementation. P.V. set up the numerical experiments. P.V. compared the numerical experiments with experimental data. V.S., T.S., M.Z. and P.T. supervised, proofread the text and helped in structuring the publication. Funding: This research was funded by Research Foundation—Flanders (FWO), Belgium, grant number of Ph.D. fellowship 11D9618N. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Appendix A Derivation of Linear Dispersion Relation The approximation of the exact linear dispersion relation for the case of two vertical layers is derived here. For the derivation, the linearized equations of [32] are used

∂η ∂u + d = 0 (A1) ∂t ∂x

∂u ∂η 5 ∂3u 1 ∂3uˆ + g d2 d2 = 0 (A2) ∂t ∂x − 16 ∂x2∂t − 8 ∂x2∂t ∂uˆ 1 ∂3u 1 ∂3uˆ d2 d2 = 0 (A3) ∂t − 8 ∂x2∂t − 16 ∂x2∂t

The plane wave solutions of Equations (A1)–(A3) can be written as

η = η exp[i(ωt kx)] (A4) 0 − u = u exp[i(ωt kx)] (A5) 0 − uˆ = uˆ exp[i(ωt kx)] (A6) 0 − Water 2019, 11, 986 17 of 18

Substituting Equations (A4)–(A6) into Equations (A1)–(A3) we get

kd η = u (A7) 0 ω 0

ω 5 1 η = u + (kd)2u + (kd)2uˆ (A8) 0 gk 0 16 0 8 0

1 2 8 (kd) uˆ 0 = u0 (A9) − 1 2 1 + 16 (kd) Substituting Equation (A7) into Equation (A8) and eliminating the inter-layer velocity amplitude uˆ 0 using Equation (A9) the linear dispersion relationship can be obtained

1 + 1 (kd)2 ω2 = gk2d 16 (A10) 3 2 1 4 1 + 8 (kd) + 256 (kd)

In the same way, the linear dispersion relationship for one layer and three equidistant layers is given by 1 ω2 = gk2d (A11) 1 2 1 + 4 (kd) 1 + 5 (kd)2 + 1 (kd)4 ω2 = gk2d 54 1296 (A12) 5 2 5 4 1 6 1 + 12 (kd) + 432 (kd) + 46656 (kd)

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