water

Article 2-D Characteristics of Wave Deformation Due to Wave-Current Interactions with Density Currents in an Estuary

Woo-Dong Lee 1 , Norimi Mizutani 2 and Dong-Soo Hur 1,* 1 Department of Ocean Civil Engineering, Gyeongsang National University, Tongyeong 53064, Korea; [email protected] 2 Department of Civil and Environmental Engineering, Nagoya University, Nagoya 464-8603, Japan; [email protected] * Correspondence: [email protected]; Tel.: +82-55-772-9122



Received: 15 October 2019; Accepted: 2 January 2020; Published: 9 January 2020


Abstract: In this study, numerical simulations were conducted in order to understand the role of wave-current interactions in wave deformation. The wave-current interaction mechanisms, wave reflection and energy loss due to currents, the effect of incident conditions on wave-current interactions, the advection-diffusion characteristics of saltwater, and the effect of density currents on wave-current interactions were discussed. In addition, the effect of saltwater–freshwater density on wave-current interactions was investigated under a hypopycnal flow field via numerical model testing. Turbulence was stronger under the influence of wave-current interactions than under the influence of waves alone, as wave-current interactions reduced wave energy, which led to decreases in wave height. This phenomenon was more prominent under shorter wave periods and higher current velocities. These results increase our understanding of hydrodynamic phenomena in estuaries in which saltwater–freshwater and wave-current pairs coexist.

Keywords: wave-current interaction; hypopycnal flow; estuarine hydraulics; estuarine currents; Navier–Stokes solver

1. Introduction In the ocean, various continually interacting physical external forces. In particular, estuaries are characterized by freshwater flows from land and saltwater flows from the ocean, which meet and form a major pathway for material transportation. Estuaries are complex, therefore it is difficult to investigate their hydrodynamic characteristics. Thus, it is critical to analyze the dynamic interactions between complex estuary features, including wave-current interaction mechanisms. Furthermore, it is necessary to understand the effect of a density current, which is generated by density differences between saltwater and freshwater, on wave-current interactions. In the estuary that exhibits highly complex hydraulic characteristics, the circulation type varies significantly according to vertical salinity distribution, tide, and wave [1,2]. However, if two fluids differing in density meet at the estuary, three types of flow and material transport patterns are largely exhibited [3,4]. As shown in Figure1, homopycnal flow occurs in the case of no density di fference, hypopycnal flow occurs when the influx flow has a lower density, and hyperpycnal flow occurs when the influx flow has a higher density. Hypopycnal flow generally occurs in the estuary when fresh river water flows in to ocean and interacts with ocean waves. However, most studies that attempted to analyze the wave-current interaction mechanism at the estuary region did not consider the density discrepancy between freshwater and saltwater, and were performed under homopycnal flow conditions. Meanwhile, due to the developments of measurement equipment, wave transformation, wave set-up,

Water 2020, 12, 183; doi:10.3390/w12010183 www.mdpi.com/journal/water Water 2020, 12, 183 2 of 30 Water 2020, 12, x FOR PEER REVIEW 2 of 30 waveflow rate, set-up, salinity, flow andrate, temperature salinity, and at temperature the estuary are at beingthe estuary observed are [being5–9]. Inobserved some studies, [5–9]. In a modelsome studies,of the estuary a model was of composedthe estuary in was a three-dimensional composed in a three-dimensional experimental tank experi to measuremental tank wave to variationmeasure wavebased variation on wave-current based on interaction wave-current and interaction vertical salinity and vertical distribution salinity influenced distribution by density influenced current by densitybehavior current at the estuary behavior [10 ,11at]. the Onsite estuary investigations [10,11]. orOnsite transport investigations of three-dimensional or transport experiment of three- to dimensionalthe estuary requires experiment significant to the time estuary and money requires and significant only allows time for limitedand money information and only to beallows recorded for limitedat measurement information points. to be recorded at measurement points.

(a)

(b) (c)

Figure 1. ThreeThree types types of of estuarine estuarine hydrodynamics hydrodynamics based based on on the the density density differences differences between river and sea/lakesea/lake water (Modified (Modified from Bates [3] [3] and Boggs [4]). [4]). ( (aa)) Homopycnal Homopycnal flow; flow; ( (bb)) hypopycnal hypopycnal flow; flow; (c) hyperpycnal flow.flow.

Therefore, numericalnumerical simulations simulations are frequentlyare freque performedntly performed to analyze to the analyze hydraulic the characteristics hydraulic characteristicsof the estuary. of Most the ofestuary. studies Most have of used studies the model have basedused the on themodel depth-averaged based on the modeldepth-averaged [12–14] or modelthe 3-D [12–14] model or based the 3-D on σ model-coordinate based system on σ-coordinate [15,16]. This system numerical [15,16]. model This numerical cannot directly model simulate cannot directlywave behavior simulate due wave to a largebehavior discrepancy due to a between large discrepancy the calculation between grid andthe time.calculation To consider grid and the wavetime. Toreaction, consider the wavethe wave radiation reaction, stress estimatedthe wave from radi Simulatingation stress WAves estimated Nearshore from (SWAN Simulating [17]) orWAves WAve NearshoreModel (WAM (SWAN [18]) are[17]) substituted or WAve Model into the (WAM momentum [18]) are equations substituted [14, 19into,20 ].the momentum equations [14,19,20].Various theoretical, experimental and observational studies on waves propagating with currents haveVarious previously theoretical, been conducted experimental in the and field observational of coastal and studies ocean on engineering waves propagating [21–25]. with In particular, currents havethere previously have been manybeen conducted studies on in the the e fffieldects of co waveastal kinematics and ocean (changesengineering in the [21–25]. wave In number particular, and therefrequency have duebeen to many shoaling studies and refraction)on the effects and of dynamics wave kinematics (changes in(changes wave steepness in the wave and number wave-action and frequencyconservation) due accordingto shoaling to and wave-current refraction) and interaction dynamics [26 (changes–29]. Wave in wave kinematics steepness include and thewave-action effects of conservation)depth and current according in wavelength to wave-current variation interactio and dispersionn [26–29]. (Doppler Wave kinematics shift) according include to the opposing effects orof depthfollowing and current.current Wavein wavelength dynamics variation include energy and disper and actionsion (Doppler conservation shift) andaccording the variations to opposing in wave or followingheight and current. water levelWave [ 24dynamics,30]. Umeyama include energy [31] and an Leed action et al. conservation [32] explored and the the vertical variations distribution in wave heightof flow and and water turbulence level [24,30]. structures Umeyama under wave-current [31] and Lee et interactions al. [32] expl inored order the to vertical analyze distribution wave-current of flowinteraction and turbulence mechanisms. structures Smith et under al. [10 wave-current] investigated waveinteractions deformation in order by wave-currentto analyze wave-current interactions interaction mechanisms. Smith et al. [10] investigated wave deformation by wave-current interactions in a 3-D experimental basin with an estuary model. However, these experimental studies were not able to define the mechanisms of wave deformation and energy loss due to wave-current interactions.

Water 2020, 12, 183 3 of 30 in a 3-D experimental basin with an estuary model. However, these experimental studies were not able to define the mechanisms of wave deformation and energy loss due to wave-current interactions. Numerical studies have reproduced past experimental results, but have focused mostly on wave deformation, neglecting to analyze the mechanisms behind wave-current interactions. In addition, Umeyama [33] measured and analyzed the flow velocity and trace of water particles by combining them with the particle image velocimetry (PIV) and particle tracking velocimetry (PTV) when the wave and current have the same direction. Chen et al. [34] studied the trace of water particles using the electronic hydrometer when the wave and current have opposite directions. However, none of the experimental studies conducted so far describe the wave deformation phenomenon and the mechanism of energy decrease clearly. The abovementioned studies focus on the motion of water particles from the wave-current interaction instead of the characteristics of wave deformation. Fernando et al. [35] discussed the topographic change resulting from the wave-current crossing angle, and the characteristics of local and vertical distribution of flow velocity. However, it is difficult to extend this study to analyze the effect of the hydraulic pattern. Lim and Madsen [36] analyzed the vertical flow velocity structure and shear velocity for wave-current crossing angles of 30◦, 60◦, and 90◦ for conditions of smooth and uniform beds of fixed roughness. The results show that the fluid motion affects some topographic changes depending upon the wave-current crossing angle. Wave reflection by the opposing current is known to be weakened under wave-current interactions. Longuet-Higgins and Stewart [37] built a theoretical background for wave reflection by currents upon which many researchers have expanded [38]. These theoretical backgrounds remain insufficient for defining reflection characteristics under wave-current interactions. Lee et al. [39] did not discuss wave reflection, although a partial standing wave field was formed by the wave-current interactions in their experiments. Rey et al. [40] measured wave reflection using a 3-D experiment basin in which the wave and current coexist, using the incidence and reflected wave separation method proposed by Drevard et al. [41]. Lee et al. [32] investigated the effect of turbulence on the reduced wave height in a river channel. They analyzed the coefficient of wave reflection due to the current in the partial standing wave field. However, a structure was installed in their experimental water tank, so the net wave reflection due to the wave and current cannot be known. Currently, advances in computer technology have enabled 3-D numerical model experiments [42–45] using Navier–Stokes solvers with high calculation loads. In addition, Liu et al. [46] applied the homotopy analysis method (HAM) to the analysis of wave-current interaction to identify the utility and show the potential of solving the issue of strong wave-current interaction. Previous studies [32] were performed under homopycnal flow conditions without considering the density discrepancy at the estuary. It is expected that hydraulic characteristics according to wave-current interaction and a decreased mechanism of wave height will show a difference at the actual estuary. In this study, it was necessary to improve the existing numerical wave water tank in order to analyze density current caused by temperature or salinity difference. Afterwards, the measured wave height distribution and density current propagation processes according to wave-current interaction in the existing experiment were compared and verified in order to check the validity and effectiveness. Subsequently, the wave height decrease mechanism, wave reflection, average water level, and turbulence energy caused by wave-current interaction were analyzed numerically. Further, from the numerical simulation model considering hypopycnal flow, the influence of density current, that is exhibited in the case of density difference between the river and ocean, on wave-current interaction was thoroughly investigated.

2. Numerical Model The mechanism and hydrodynamic characteristics of the interactions between waves and currents were analyzed accounting for the densities of freshwater and seawater. For this purpose, a 3-D numerical analysis model [47,48] based on the existing 3-D numerical wave tank (NWT), LES-WASS-3D [49–51] was introduced to analyze density currents in this study. Water 2020, 12, 183 4 of 30

LES-WASS-3D, which is based on the 3-D Navier–Stokes momentum equations, uses the volume of fluid (VOF) method to reproduce complicated free surfaces, including wave breaking, and the porous body model (PBM) to reflect energy dissipation by a permeable structure; the model was based on Hur and Mizutani [52] and Hur et al. [53]. In addition, to analyze sub-grid scale (SGS) turbulence, Smagorinsky turbulence model (STM [54]) for the large eddy simulation (LES) was adopted in the NWT [55]. By considering inertia [56], turbulence [57,58], and laminar flow resistance [58,59] according to porous medium characteristics (diameter, porosity, and shape), the model can directly analyze interactions among waves, structures, and the seabed [49–51]. The water density state equation (refer to AppendixA[ 60]) and kinematic viscosity coefficient (refer to AppendixB[ 61]) were applied herein to analyze the density current in view of salinity and temperature. Salinity and temperature were quantitatively calculated by simultaneously applying a 3-D advection-diffusion equation. Due to flow separation around structures, present LES turbulence models do not provide accurate calculations of turbulence, which significantly influences the diffusion coefficient applied in the advection-diffusion equation. Thus, a dynamic modeling procedure involving the application of the dynamic eddy viscosity model suggested by Germano et al. [62] and Lilly [63] was applied to re-evaluate Smagorinsky constant (Cs).

2.1. Governing Equations The basic set of calculations consists of the continuity equation (Equation (1)), which includes the source term that generates waves and currents without reflection in 3-D incompressible and viscous fluids, and a modified Navier–Stokes momentum equation (Equation (2)), in which the fluid resistance of the permeable structure is applied. ∂νi γi = q∗ (1) ∂xi " !# ∂νi ∂νi 1 ∂p ∂ ∂νi ∂νj γν + γiνj = γν + γi(ν + νt) + Qi Ri γν gi Ei (2) ∂t ∂xj − ρ ∂xi ∂xj ∂xj ∂xi − − − − where xi is the Cartesian coordinate system (x, y, and z); t is time; νi is the velocity components (u, ν, and w); ρ is the fluid density; p is the fluid pressure; ν is the kinematic viscosity coefficient; νt is the eddy viscosity coefficient from the turbulence model; γν and γi are the volume and surface porosities, as shown in Figure2; Ri is the fluid resistance term for the porous media; gi is the acceleration of gravity; E refers to the wave energy damping term (= βw; where β is the damping factor which i − equals 0 except for the added damping zones) for the vertical velocity only; Qi refers to the wave and 2ν ∂q∗ current source terms (= ); and q∗ is the source term required to generate waves and currents, 3 ∂xk and it is defined as:  q/∆xs : x = xs q =  (3) ∗   0 : x , xs where q is the flux density; ∆xs is the grid size at the source position (x = xs). As suggested by Brorsen and Larsen [64], the flux density of the wave and current generation sources q in Equation (3) is gradually increased during three wave periods starting with wave and current generation, as shown in Equation (4).  2V0ζ[1 exp(2t/Ti)] : t/Ti 3 q =  − ≤ (4)  2V0ζ : t/Ti > 3 where V0 is the horizontal velocity determined by incident condition (in case of wave, based on the 3rd-order Stokes wave theory); and ζ is the intensity factor at the wave source (= (ηs + h)/(η0 + h)); ηs is the water surface elevation at the source position, η0 is the water surface elevation estimated using the 3rd-order Stokes wave theory, and h is the water depth). The constant “2” accounts for two flows Water 2020, 12, 183 5 of 30 propagating to the left and right sides in NWT. The depth integrated quantity of q is adjusted by ζ to achieveWater 2020 the, 12,same x FOR quantityPEER REVIEW in the non-reflection condition [65]. 5 of 30

(a) (b)

Figure 2.2. DefinitionDefinition of the volume and surface porosities.porosities. Each cell is flaggedflagged as one of three typestypes γ γ = according to the γνvvalue: value: an an empty empty cell cell occupied occupied by by fluidfluid (γ( ν v= 11),), anan impermeableimpermeable cellcell withoutwithout fluid flow (γν == 0 ), and a permeable cell containing both solid and fluid (0<<<γ γν < 1), where Vsolid fluid flow ( v 0 ), and a permeable cell containing both solid and fluid ( 01v ), where Vsolid Vsolid and Asolid are the volume and surface area of solids in a cell, respectively. (a) γν = 1 ; and A are the volume and surface area of solids in a cell, respectively.− ∆xi∆x j∆(xak ) solid Asolid (b) γi = 1 . − ∆Vxj∆xk A γ =−1 solid ; (b) γ =−1 solid . v ΔΔ Δ i ΔΔ F is the volumexijkxx of fluid (VOF) in eachx jkx cell [66], which can be expressed as fluid conservation by applying an assumption of fluid incompressibility and a function volume based on PBM, as in EquationAs suggested (5). by Brorsen and Larsen [64], the flux density of the wave and current generation ∂F ∂F sources q in Equation (3) is graduallyγ increased+ γ ν during= threeFq wave periods starting with wave and(5) ν t i i x ∗ current generation, as shown in Equation (4).∂ ∂ i where F is the VOF function. In the numerical simulation, each cell is assigned as one of three types by the VOF function value: fluid cell (Fζ= 1),−≤ empty() cell (F = 0), and free surface cell (0 < F < 1). 21exp2/:/3VtTtT0 ii q =  (4) ζ > 2.2. Advection-Diffusion Equations 2:/3VtT0 i In density current analyses, quantitative decisions about influential factors are important in order towhere accurately V0 is calculatethe horizontal the fluid velocity density determinedρ and kinematic by incident viscosity condition coeffi (incient caseν, whichof wave, are based substituted on the into3rd-order the governing Stokes equations.wave theory); The state and equation ζ is used the to calculateintensity thefactor density at andthe kinematic wave source viscosity ( coefficient of water is a function of temperature T and salinity S. Therefore, the 3-D advection-diffusion =+()()ηηhh +); η is the water surface elevation at the source position, η is the water equationss adopted0 hereins include Equation (6) for temperature and Equation (7) for salinity.0 surface elevation estimated using the 3rd-order Stokes wave theory, and h is the water depth). The ! constant “2” accounts for two flows∂C propagating∂C ∂to the left∂C and right sides in NWT. The depth γν + γiνi γiεi = 0 (6) integrated quantity of q is adjusted∂t by ζ∂x ito− achieve∂xi ∂thexi same quantity in the non-reflection condition [65]. ( νt : horizontal direction εi = (7) F is the volume of fluid (VOF) νint/ eachσc :cell vertical[66], which direction can be expressed as fluid conservation by applying an assumption of fluid incompressibility and a function volume based on PBM, as in where C is temperature T or salinity S, and ε is the horizontal and vertical diffusion coefficients. In Equation (5). i this study, σc, which is the Prandtl/Schmidt number, has a value of 1.0, as estimated from experimental ∂∂ results [67] and ocean observations [68,69γγ]. FF+=* viivFq (5) ∂∂tx 2.3. Turbulence Model i whereIn general,F is the toVOF reproduce function.complete In the numerical turbulence, simulation the calculation, each cell is area assigned must as be one larger of three than types the = = << representativeby the VOF function scale of value: the fluid, fluid and cell the( F grid1), must empty be cell set smaller( F 0 than), and the free minimum surface cell turbulence ( 01F scale.). It is essential to consider turbulence when modeling a 3-D phenomenon, therefore the grid number of Re2.2.9/ Advection-Diffusion4 (Re; Reynolds Number) Equations is required. However, it is practically impossible conduct numerical In density current analyses, quantitative decisions about influential factors are important in order to accurately calculate the fluid density ρ and kinematic viscosity coefficient v , which are substituted into the governing equations. The state equation used to calculate the density and kinematic viscosity coefficient of water is a function of temperature T and salinity S . Therefore, the 3-D advection-diffusion equations adopted herein include Equation (6) for temperature and Equation (7) for salinity.

Water 2020, 12, 183 6 of 30 analysis using this number of grids. Thus, the use of a turbulence model is practical for reproducing turbulence. To this end, the STM [54] was adopted in this study. However, a model constant (Cs) must be applied in the eddy viscosity model for the fluid. For this, the appropriate Cs can be calculated through the dynamic modeling procedure suggested by Germano et al. [62] and Lilly [63]. The dynamic modeling method expresses the physical quantity included in the analytical grid as functions of time and space. This method does not require prior designation of unknown model constants or fine tuning and can accurately predict laminar flow and asymptotic behavior near walls. In addition, the model constant has a negative value showing energy backscattering, as the model constant Cs is applied temporally and spatially.

2.3.1. Eddy Viscosity Model

In the STM, length and speed are expressed as L = Cs∆ and V = L S , respectively. The speed | | calculation assumes that the pure dissipation equals the energy transferred from the large-scale to small-scale analytical grids, and that the eddy viscosity coefficient νt formula is expressed as:

2 νt = (Cs∆) S (8) | |  1/2 S = 2S S (9) | | ij ij ! 1 ∂νi ∂νj Sij = + (10) 2 ∂xj ∂xi

∆ = (∆x ∆y ∆z)1/3 (11) · · where Sij is the strain tensor for the grid size, and ∆ is the filter length scale.

2.3.2. Dynamic Eddy Viscosity Model

In the LES grid, the stress τij of the sub-grid scale can be modeled via Equation (12) and Equation (13) using STM. τ = ν ν ν ν (12) ij i j − i j δ ij 2 τ = τ = 2Cs∆ S S (13) ij 3 kk | | ij where S and ∆ have the same definition in the STM. | | Likewise, the stress τij of the secondary stress can be modeled as Equations (14) and (15) by applying a secondary filter. T = νˆ νˆ ν ν (14) ij i j − i j δ ij 2 Tij = T = 2Cs∆ˆ Sˆ Sˆij (15) 3 kk A secondary filter (the test filter) with width ∆ˆ larger than ∆ is introduced. It is generally assumed that the width of the test filter is two times of that of the grid filter and that the Cs of the two filters are the same. These assumptions are adopted in this study. Tij and τˆij are related by the Germano identity.

L T τˆ = ν ν νˆ νˆ (16) ij ≡ ij − ij i j − i j

Substituting the sub-grid scale stress τij from Equation (13) and secondary filter stress Tij from Equation (15) into Equation (16) produces relationships such as those seen in Equation (18).

δ   ij 2 2 Lij L = 2Cs ∆ˆ Sˆ Sˆij ∆ S Sij = 2CsMij (17) − 3 kk − − | | − Water 2020, 12, 183 7 of 30

2 Mij = α Sˆ Sˆij S Sij (18) − | | where α is the ratio of the test and grid filters (= ∆ˆ /∆). There is only one unknown quantity, Cs, in Equation (17). Both Lij and Mij are tensors, so Equation (17) is an overdetermined equation system. Applying a least-squares technique to Equation (17) produces a least error value for Cs. LijMij Cs = (19) −2MijMij

A critical problem exists in the determination of the space-temporal function Cs. If Cs exhibits severe fluctuations or negative values, numerical instability is induced, and it ultimately diverges. To prevent such numerical instability, a space mean in uniform direction of numerator and denominator is required locally for every calculation. In addition, a clipping of Cs is also required so that ν + νt can alwaysWater 2020 be, 12 a, positivex FOR PEER (+ REVIEW) value. On following the above process, most of the limitations of STM can8 of be30 overcome. Germano et al. [62] proposed 1/2 as the optimal α value in flow calculation while applying 2.4. Non-Reflected Boundary the dynamic eddy viscosity model. We also apply α = 1/2 in this study. Figure 3 shows a schematic diagram of a numerical water basin with a variable grid system and damping factors β in an added damping zone forLij Mnon-reflectedij wave generation [70]. Each grid Cs = (20) cell in the damping zone increases its size by a −factor2MijM 1.03ij with respect to the previous cell as they approach the open boundaries, as shown in Equation (21). This approach serves to make the analysis 2.4.zone Non-Reflected independent Boundary from the influence of the numerical water basin boundaries. In addition, wave φ reflectionFigure is3 controlled shows a schematic by setting diagram the horizontal of a numerical differences water in physical basin with quantities a variable i grid, such system as velocity and dampingand the VOF factors function,β in an to added zero at damping the numerical zone for wa non-reflectedter basin boundary, wave generation as shown [ 70in]. Equation Each grid (22). cell in the damping zone increases its size by a factor 1.03 with respect to the previous cell as they approach 1.03Δx − : positivedamping zone the open boundaries, as shownΔ=x in Equationi (21).1 This approach serves to make the analysis zone i  Δ (21) independent from the influence of the1.03 numericalxi+1 : water negativedamping basin boundaries. zone In addition, wave reflection is controlled by setting the horizontal differences in physical quantities φi, such as velocity and the VOF function, to zero at the numerical water basin∂φ boundary, as shown in Equation (22). i = 0 ∂ (22) ( xi 1.03∆xi 1 : positive damping zone ∆xi = − (21) For the boundary condition of 1.03the ∆impermeablexi+1 : negative slope damping in a quadrilateral zone mesh system according to spatial discretization based on finite difference method, the impermeable condition is adopted in ∂φi the normal direction and slip condition is adopted in= the0 tangential force for the inclined structure (22) by ∂x applying the reasonable boundary [48,51] based oni Petit et al. [71].

Figure 3. Schematic diagram of grid system and damping factor in a numerical wave basin.

2.5. Numerical Water Basin Stability Figure 4 presents the computed water surface profiles in the numerical water basin during one wave period. Figure 3 shows that the wave spatial envelope is gradually attenuated in both of the added damping zones by the numerical dispersion caused by the coarse grid system and the wave damping term included in the momentum equation; the wave spatial envelope vanishes after x/Li = 2. An added damping zone with a length over 2 Li was applied in this study. From numerical result such as those in Figure 3, one can conclude that the present non-reflected system works efficiently to absorb waves in the damping zone and release energy out of the open boundaries.

Water 2020, 12, 183 8 of 30

For the boundary condition of the impermeable slope in a quadrilateral mesh system according to spatial discretization based on finite difference method, the impermeable condition is adopted in the normal direction and slip condition is adopted in the tangential force for the inclined structure by applying the reasonable boundary [48,51] based on Petit et al. [71].

2.5. Numerical Water Basin Stability Figure4 presents the computed water surface profiles in the numerical water basin during one wave period. Figure3 shows that the wave spatial envelope is gradually attenuated in both of the added damping zones by the numerical dispersion caused by the coarse grid system and the wave damping term included in the momentum equation; the wave spatial envelope vanishes after x/Li = 2. Water 2020, 12, x FOR PEER REVIEW 9 of 30 An added damping zone with a length over 2 Li was applied in this study. From numerical result such as those in Figure3, one can conclude that the present non-reflected system works e fficiently to absorb Water 2020, 12, x FOR PEER REVIEW 9 of 30 waves in the damping zone and release energy out of the open boundaries.

Figure 4. Computed wave profiles for each time step and envelope in a numerical water basin. Figure 4. Computed wave profiles for each time step and envelope in a numerical water basin. 2.6. NumericalFigure 4. Model Computed Validation wave profiles for each time step and envelope in a numerical water basin. 2.6. Numerical Model Validation 2.6.2.6.1. Numerical Wave-Current Model ValidationInteraction 2.6.1. Wave-Current Interaction The hydraulic model experiment in Iwasaki and Sato [72] was simulated to verify the 2.6.1. Wave-Current Interaction characteristicsThe hydraulic of wave model deformation experiment by in wave-current Iwasaki and Sato interactions [72] was simulated in an open to channel. verify the Figure characteristics 5 shows theof wave numericalThe deformationhydraulic water modeltank by wave-current simulate experimentd by interactions Iwasakiin Iwasaki and in Sato anand open [72].Sato channel.Wave [72] andwas Figure currentsimulated5 shows generation theto verify numerical sources the characteristicswater(without tank reflection) simulated of wave were by deformation Iwasaki deployed and atby Sato each wave-current [72 end]. Wave of the and interactionsanalysis current domain, generation in an andopen sourcesadditional channel. (without Figuredamping reflection) 5 shows zones thewere numerical deployedincluded water atat eachboth tank endends. simulate of theFor analysisdverification, by Iwasaki domain, anda numerical andSato additional [72]. calculationWave damping and currentwas zones conducted generation were includedwith sources an atx x (withoutdirectionboth ends. reflection) grid For size verification, of were 1 cm, deployed a az numericaldirection at each grid calculation end size of theof 0.5 analysis was cm, conducted and domain, a time withand increment additional an direction of 1/1000 damping grid s. sizezones of were1 cm, included a z direction at both grid sizeends. of 0.5For cm,verification, and a time a incrementnumerical of calculation 1/1000 s. was conducted with an x direction grid size of 1 cm, a z direction grid size of 0.5 cm, and a time increment of 1/1000 s.

Figure 5. The numerical water basin is modelled after the experimental campaign by Iwasaki and Figure 5. The numerical water basin is modelled after the experimental campaign by Iwasaki and Sato Sato [72]. [72]. FigureFigure 5.6 Theshows numerical the wave water height basin distribution is modelled after while th thee experimental wave propagates campaign over by anIwasaki opposing and Sato current, Figure 6 shows the wave height distribution while the wave propagates over an opposing where[72]. (a) shows a current velocity of Vc = 6.3 cm/s and (b) shows a current velocity of Vc = 19 cm/s. current,Here, the where black (a) circle shows ( a) current and blue velocity triangle of V (Nc )= indicate6.3 cm/s and the experimental(b) shows a current results, velocity and black of Vc = line 19 (cm/s.—)Figure and Here, blue the6 shows line black (— circlethe) represent wave () and height the blue modelled distribution triangle values. (▲ )while indicate Wave the the heightwave experimental propagates damping results, due over to and wave-currentan opposingblack line current,(interactions━) and where blue is observedline(a) shows (━) represent ina current the experimental thevelocity modelled of results Vc =values. 6.3 and cm/s calculationsWave and height(b) shows shown damping a current in Figure due velocity to6. wave-current In this of V study,c = 19 ▲ cm/s.interactionsthe experimental Here, the is observed black results circle in are the( reproduced) experimentaland blue triangle well results by ( the )and indicate calculations. calculations the experimental Furthermore, shown in Figureresults, the inclination 6. and In thisblack study, of line the (the━) experimentaland blue line results (━) represent are reproduced the modelled well by values. the calculations. Wave height Furthermore, damping duethe inclinationto wave-current of the interactionswave height is distribution observed in is thequite experimental similar in the results calculations and calculations and the experime shown ntsin Figure for wave 6. Indeformations this study, thewith experimental coexisting wave-current results are reproduced fields. This indicateswell by the that calculations. the ratios of Furthermore,wave height damping the inclination due to ofwave- the wavecurrent height interaction distribution are similar. is quite These similar results in the were calculations used to and determine the experime the validitynts for andwave effectiveness deformations of withthe numerical coexisting model wave-current for wave fields. transformation This indicates duri thatng an the interaction ratios of wave with height an opposing damping current. due to wave- current interaction are similar. These results were used to determine the validity and effectiveness of the numerical model for wave transformation during an interaction with an opposing current.

Water 2020, 12, 183 9 of 30 wave height distribution is quite similar in the calculations and the experiments for wave deformations with coexisting wave-current fields. This indicates that the ratios of wave height damping due to wave-current interaction are similar. These results were used to determine the validity and effectiveness of the numerical model for wave transformation during an interaction with an opposing current. Water 2020, 12, x FOR PEER REVIEW 10 of 30

Water 2020, 12, x FOR PEER REVIEW 10 of 30

(a)

(a)

(b)

Figure 6. Comparison between measured and calculated wave heights under wave-current Figure 6. Comparison between measured and calculated wave heights under wave-current interactions. (a) Vcinteractions.= 6.3 cm/s; ( (ab) )VVc =c 6.3= 19 cm/s; cm /(s.b) Vc = 19 cm/s. (b) Figure 6. Comparison between measured and calculated wave heights under wave-current 2.6.2.2.6.2. Salinity Salinity Advection Advection interactions. (a) Vc = 6.3 cm/s; (b) Vc = 19 cm/s. FigureFigure7 shows 7 shows a numerical a numerical water water tank tank based based on on the the hydraulic hydraulic model model experiment experiment inin HuppertHuppert and 2.6.2. Salinity Advection Simpsonand Simpson [73]. The [73]. water The tankwater was tank 960 was cm 960 long cm long and and 27 cm27 cm wide, wide, and and the the waterwater depth depth was was 10 10 cm. cm. A A compartment of 30 cm long (x0) was installed in the water tank and filled with the density of compartmentFigure of 30 7 cmshows long a numerical (x ) was installedwater tank in based the wateron the tankhydraulic and filledmodel withexperiment the density in Huppert of saltwater 03 saltwaterand Simpson (ρs) of3 [73].1.0115 The g/cm water. The tank remaining was 960 cm area long was and filled 27 cm with wide, the and density the water freshwater depth was (ρf) 10of cm.0.9999 3 (ρs) of 1.01153 g/cm . The remaining area was filled with3 the density freshwater (ρf) of 0.9999 g/cm . g/cmA compartment. The density of difference 30 cm long (Δ ρ(x) 0was) was 0.0116 installed g/cm in theand water the reduced tank and gravity filled withg’ (= Δρtheg/ densityρf) was of 11.4 3 2 The density difference (∆ρ) was3 0.0116 g/cm and the reduced gravity g’(= ∆ρg/ρ ) was 11.4 cm/s . cm/ssaltwater2. The grid (ρs) ofsize 1.0115 for verificationg/cm . The remaining was Δx = area 1 cm, was Δ yfilled = 1 cm,with and the densityΔz = 0.25 freshwater cm, and ( theρf f) of numerical 0.9999 The gridcalculationg/cm size3. forThe was verification density carried difference out was for∆ Δ(xΔt ρ==) 1/10001was cm, 0.0116 ∆s.y = g/cm1 cm,3 and and the∆z reduced= 0.25 cm,gravity and g’ the (= Δρ numericalg/ρf) was 11.4 calculation was carriedcm/s2 out. The for grid∆ tsize= 1 for/1000 verification s. was Δx = 1 cm, Δy = 1 cm, and Δz = 0.25 cm, and the numerical calculation was carried out for Δt = 1/1000 s.

Figure 7. The numerical water basin is modeled after the experimental campaign by Huppert and FigureSimpson 7.FigureThe [73].7. numerical The numerical water water basin basin is is modeled modeled after the the experimental experimental campaign campaign by Huppert by Huppert and and SimpsonSimpson [73]. [73].

Water 2020, 12, 183 10 of 30 Water 2020, 12, x FOR PEER REVIEW 11 of 30

FigureFigure8 8shows shows the the temporal temporal evolution evolution of the of salinitythe salinity front front due to due the to density the density di fference. difference. The upper The pictureupper picture shows theshows experimental the experimental results andresults the and lower the picture lower picture shows theshows calculated the calculated results. results. The figure The alsofigure shows also theshows saltwater the saltwater shapes shapes when (a)whent = 4.4(a) t s, = (b)4.4t s,= (b)6.8 t s,= (c)6.8 ts,= (c)9.7 t = s, 9.7 and s, (d)andt =(d)17.5 t = 17.5 s after s after the partitionthe partition was was removed removed from from thecompartments. the compartments.

(a)

(b)

(c)

(d)

FigureFigure 8.8. ComparisonComparison betweenbetween experimentalexperimental (upper(upper picture)picture) andand numericalnumerical (lower(lower picture)picture) resultsresults showingshowing thethe evolutionevolution of of advection-di advection-diffusionffusion of of a a saltwater saltwater mass. mass. ( a()a)t t= =4.4 4.4 s; s; ( b(b))t t= = 6.86.8 s;s; ((cc)) tt == 9.7 s; ((dd))t t== 17.5 s.

Water 2020, 12, 183 11 of 30

Water 2020, 12, x FOR PEER REVIEW 12 of 30 3. Numerical Results 3. Numerical Results 3.1. Wave-Current Interactions without Density Current 3.1. Wave-Current Interactions without Density Current 3.1.1. Description of Numerical Water Basin and Incident Conditions 3.1.1. Description of Numerical Water Basin and Incident Conditions A numerical wave tank was defined as shown in Figure9 in order to numerically investigate the mechanismsA numerical of wave-current wave tank was interactions defined as according shown in to Figure the velocity 9 in order of the to incident numerically wave investigate and current the in mechanismsan open channel. of wave-current To generate wavesinteractions based according on 3rd-order to the Stokes velocity waves of andthe incident currents wave without and reflection, current inwave an andopen current channel. sources To generate were installed waves atbased each endon 3rd-order of the analytical Stokes domain;waves and damping currents zones without were reflection,added at both wave ends and to current absorb wavesources energy. were Theinstalled length at of each the analyticalend of the domain analytical was domain; 5 Li of the damping incident zoneswavelength, were added the width at both was ends 30 cm,to absorb and the wave water energy. depth The was length 30 cm. of the In addition,analytical thedomain length was of 5 the Li ofdamping the incident zones wavelength, was set to morethe width than doublewas 30 cm, the incidentand the water wavelength depth towas absorb 30 cm. wave In addition, energy andthe lengthminimize of the reflection. damping zones was set to more than double the incident wavelength to absorb wave energyWave-current and minimize interaction reflection. numerical simulations were performed for 63 different cases involving threeWave-current incident wave interaction heights (H numericali = 3, 5, and simulations 7 cm), three were incident performed wave periodsfor 63 different (Ti = 1.2, cases 1.5, involving and 1.8 s), threeand seven incident current wave velocities heights ( (HVic == 3,0, 5, 5, and 10, 7 15, cm), 20, three 25, and incident 30 cm /waves). The periods wave celerity(Ti = 1.2, ( C1.5,i) is and based 1.8 ons), and3rd-order seven Stokes current wave velocities theory. (V Thec = 0, calculation 5, 10, 15, 20, grids 25, wereand 30∆x cm/s).= 2 cm, The∆y wave= 2 cm, celerity and ∆ (zC=i) is1 cm.based In on all 3simulations,rd-order Stokes the wave wave theory. was generated The calculation after current grids generation were Δx = in2 ordercm, Δy to = stabilize2 cm, and the Δ flowz = 1 field.cm. In all simulations, the wave was generated after current generation in order to stabilize the flow field.

FigureFigure 9. SketchSketch of of a a numerical numerical water water basin basin for for simulating simulating wave-current wave-current interactions.

3.1.2. Distribution of Wave Heights 3.1.2. Distribution of Wave Heights Figure 10 shows the wave height spatial distributions for various current velocity conditions when Figure 10 shows the wave height spatial distributions for various current velocity conditions the incident wave (Ti) period was 1.2 s. As shown in Equation (23), the wave height was calculated when the incident wave (Ti) period was 1.2 s. As shown in Equation (23), the wave height was using surface elevations (Equation (24)) from three periods and described via the non-dimensional calculated using surface elevations (Equation (24)) from three periods and described via the non- wave height (H/Hi) after the flow and wave field stabilized. dimensional wave height (H/Hi) after the flow and wave field stabilized.

Z +t+3Ti 11 tT3 i   HH()(xxtxtdtx) =−= max,min,[max{}ηηη()(x, t) min η{}(x,()t) ] dt (23)(23) 3Ti t t − 3Ti

k=kmax 1 X=−− kkm ax 1 η(x, t) = Fk∆zk h (24) η ()x,tFzh=Δ−− (24) k=2 kk k =2 where k = 2 and k = kmax 1 represent the bottom and top of the NWT, and ∆z is the grid size in the − k z direction.= =− Δ where k 2 and kkmax 1 represent the bottom and top of the NWT, and zk is the grid size in the z direction.

Water 2020, 12, 183 12 of 30 Water 2020, 12, x FOR PEER REVIEW 13 of 30

(a)

(b)

(c)

Figure 10. Spatial distributions of wave heights for an incident wave period of 1.2 s according to Vcc.. (a) Hi == 33 cm, cm, TTi i== 1.21.2 s,s, HHi/Li/iL =i 0.017,= 0.017, UrUr = 3.5;= 3.5; (b) ( bH)i H= 5i =cm,5 cm, Ti =T 1.2i = s,1.2 Hi s,/LHi =i /0.028,Li = 0.028, Ur=5.8;Ur =(c5.8;) Hi (=c )7H cm,i = T7i cm,= 1.2T is,= H1.2i/Lis, = H0.04,i/Li Ur= 0.04, = 8.1.Ur = 8.1.

In Figure 10,10, for all the cases, the wave height is higher than thethe incidentincident wavewave heightheight ((HHii) at x/Lii = 0.0. The The wave wave height height gradually gradually decreases decreases with with wave wave propagation. propagation. This This phenomenon phenomenon is is more more obvious as the ratio ((VVc//Cii)) between between current velocity (Vc)) and and incident wave celerity (Cii) increases. A partial standing wave field field is generally observed due due to to wave reflection reflection by the wave-current interactions. interactions. These results are in agreement with the experimentalexperimental results [[32,39,74].32,39,74]. In addition, an intersecting point with a wave height of H/Hi = 1 is observed in (a–c); this point moves farther from x/Li = 0 as Hi increases because Ci increases, thus decreasing Vc/Ci.

Water 2020, 12, 183 13 of 30

point with a wave height of H/Hi = 1 is observed in (a–c); this point moves farther from x/Li = 0 as Hi increasesWater 2020, because12, x FOR CPEERi increases, REVIEW thus decreasing Vc/Ci. 14 of 30 Furthermore, as Vc/Ci increases, the partial standing wave wave-height distribution is marginally formed.Furthermore, In this case, as theVc/C wavei increases, height the is partial also relatively standing high wave at wave-heightx/L = 0 in comparison distribution to is the marginally point at whichformed. the In partial this case, standing the wave wave height forms. is Thisalso phenomenonrelatively high was at x also/L = reported0 in comparison by Lee etto al. the [39 point], who at which the partial standing wave forms. This phenomenon was also reported by Lee et al. [39], who found Hi = 5 cm, Ti = 1.0 s, and Vc = 60 cm/s. In other words, shorter wave periods, lower wave heights, andfound higher Hi = 5 current cm, Ti = velocities 1.0 s, and lead Vc = to60 lower cm/s. waveIn other orbital words, motion shorter velocity. wave periods, Therefore, lower this wave phenomenon heights, appearsand higher to becurrent a result velocities of the predominance lead to lower wave of wave orbital energy motion loss velocity. over wave Therefore, reflection. this Nevertheless,phenomenon furtherappears study to be isa requiredresult of tothe verify predominance this mechanism. of wave energy loss over wave reflection. Nevertheless, further study is required to verify this mechanism. Figure 11 shows the spatial distribution of the H/Hi according changes in the Vc under Ti = 1.5 s. Figure 11 shows the spatial distribution of the H/Hi according changes in the Vc under Ti = 1.5 s. This figure shows a tendency similar to that in Figure 10, which has Ti = 1.2 s. However, the Vc/Ci This figure shows a tendency similar to that in Figure 10, which has Ti = 1.2 s. However, the Vc/Ci is is small compared to the case with Ti = 1.2 s; thus, the effects caused by a decrease in wave height duesmall to compared decreases to in the interactions case with decrease, Ti = 1.2 s;and thus, changes the effects in the caused wave by height a decrease rate become in wave weaker. height due For to decreases in interactions decrease, and changes in the wave height rate become weaker. For this this reason, points (H/Hi = 1) generally move far from x/Li = 0. The wave height distribution becomes reason, points (H/Hi = 1) generally move far from x/Li = 0. The wave height distribution becomes narrower because the range over which Vc/Ci fluctuates is relatively narrow. narrower because the range over which Vc/Ci fluctuates is relatively narrow.

(a)

(b)

Figure 11. Cont.

Water 2020, 12, 183 14 of 30 Water 2020,, 12,, xx FORFOR PEERPEER REVIEWREVIEW 15 of 30

(c)

FigureFigure 11.11. Spatial distributions of wave heights for an incident wave period of 1.5 ss accordingaccording toto VVcc... ((aa)) HHiii == 33 cm, cm, Tiii == 1.51.5 s,s, Hii///Lii == 0.013,0.013,0.013, UrUr == 6.1;6.1;6.1; (( (bb)) HHii i === 555 cm,cm, cm, TTii i=== 1.51.51.5 s,s, s, HHii//iL/Lii i=== 0.021,0.021,0.021, UrUr === 10.2;10.2;10.2; ((c ()c )HHii =i= = 77

7cm, cm, TiiT ==i 1.5=1.51.5 s,s, H s,iiH//Liii/ =L= i0.03,0.03,= 0.03, Ur Ur== 14.2.14.2.= 14.2.

FigureFigure 12 showsshows the the wave wave height height spatia spatiall distribution distribution with with changes changes in the in the Vcc underunderVc under Tii == T1.81.8i = s.s. 1.8 ThisThis s. Thisfigure figure also shows also shows a tendency a tendency similar similar to those to thosein Figures in Figures 8 and8 9, and and9, also and shows also shows prolonged prolonged wave- wave-currentcurrent interaction interaction period period characteristics. characteristics. As the T Asii increases,increases, the Ti increases, thethe rangerange the overover range whichwhich over Vcc/ whichCii fluctuatesfluctuatesVc/Ci fluctuatesdecreases, decreases,resulting in resulting a narrower in awave narrower height wave distribution. height distribution. In addition, Inas the addition, incident as wave the incident period waveand wave period height and waveincrease, height the increase,Cii increasesincreases the andandCi increases thethe decreasesdecreases and the inin decreaseswavewave heightheight in wave weaken,weaken, height resultingresulting weaken, inin resultinggentler inclinations in gentler inclinationsin the wave inheight the wave distribution. height distribution.

(a)

(b)

Figure 12. Cont.

Water 2020, 12, 183 15 of 30 Water 2020, 12, x FOR PEER REVIEW 16 of 30

(c)

FigureFigure 12.12. SpatialSpatial distributionsdistributions of of wave wave heights heights for for an an incident incident wave wave period period of 1.8of 1.8 s according s according to V toc.( Vac). H(ai)= H3i = cm, 3 cm,Ti =Ti1.8 = 1.8 s, Hs,i /HLi/L=i 0.01,= 0.01,Ur Ur= =9.3; 9.3; (b ()bH) Hi =i =5 5 cm, cm,T Ti i= = 1.81.8 s,s, HHii/Lii = 0.017,0.017, UrUr == 15.5;15.5; (c (c) )HHi i== 7 7cm, cm, TTi =i =1.81.8 s, s,HiH/Lii/ L= i0.024,= 0.024, Ur Ur= 21.7.= 21.7.

ConsideringConsidering thethe aboveabove discussiondiscussion ofof FiguresFigures8 8–10–10,, the the wave wave orbital orbital motion motion velocity velocity and and current current velocityvelocity have the the same same direction direction under under the the wave wave trough trough and and different different directions directions under under the wave the wave crest crestwithin within the wave-current the wave-current interaction. interaction. As confirme As confirmedd in the inexisting the existing experiment, experiment, more intense more intense wave- wave-currentcurrent interactions interactions occur in occur the wave in the crests wave than crests in thanthe wave in the troughs. wavetroughs. This leads This to wave leads reflection to wave reflectionand wave andheight wave decay. height decay.

3.1.3.3.1.3. Characteristics of Turbulent KiniticKinitic EnergyEnergy (TKE)(TKE) The mean phase-averaged TKE (K) is the average turbulence over ten wave periods, as shown in The mean phase-averaged TKE ( K ) is the average turbulence over ten wave periods, as shown Equation (25). This TKE is calculated using Equations (26) and (27), as suggested by Christensen [75]. in Equation (25). This TKE is calculated using Equations (26) and (27), as suggested by Christensen Equation (26) can be used to calculate the TKE of a grid scale (KGS) using the flow velocity calculated [75]. Equation (26) can be used to calculate the TKE of a grid scale (K GS ) using the flow velocity by the numerical model, and Equation (27) can be used to calculate TKE of a sub-grid scale (KSGS) by applyingcalculated the by values the numerical calculated model, by LES. and The Equation center (27) of the can analysis be used zone to calculate is the measuring TKE of a sub-grid point. scale

( K SGS ) by applying the values calculated by LES. The center of the analysis zone is the measuring Z t+10T point. 1 i K = (KGS + KSGS) dt (25) 10Ti t

+ 1 tT10 i  =+1 2 ()2 2 KKGS = u0 + KKdtν0GS+ w0 SGS (26)(25) 2t 10Ti  ν 2 K = t (27) SGS C ∆ =++1 ′′22s ′ 2 KGS ()uvw (26) where u0,ν0, and w0 are turbulence components; 2νt is the eddy viscosity coefficient estimated using STM; Cs is a model constant calculated by the dynamic eddy viscosity model; and ∆ is the filter length scale. 2 Figure 13 shows the vertical distribution of theKvaccording to the ratio of current velocity and = t wave celerity (Vc/Ci). This figure shows thatK theSGS value of the mean phase-averaged TKE is larger with(27) C Δ wave-current interactions than with waves alone. The intensitys generally increases with higher Vc/Ci values. In particular, still water with higher velocity due to wave orbital motion shows higher TKE. where u′ , v′ , and w′ are turbulence components; v is the eddy viscosity coefficient estimated Therefore, wave height decreases with increased TKE. Thet relationship between TKE and wave height Δ decayusing STM; in the coexistingC s is a model wave-current constant calculated field will be by comprehensively the dynamic eddy discussed viscosity later, model; along and with waveis the energyfilter length characteristics. scale. Figure 13 shows the vertical distribution of the K according to the ratio of current velocity and wave celerity (Vc/Ci). This figure shows that the value of the mean phase-averaged TKE is larger with wave-current interactions than with waves alone. The intensity generally increases with higher Vc/Ci values. In particular, still water with higher velocity due to wave orbital motion shows higher TKE. Therefore, wave height decreases with increased TKE. The relationship between TKE and wave height decay in the coexisting wave-current field will be comprehensively discussed later, along with wave energy characteristics.

Water 2020, 12, x FOR PEER REVIEW 17 of 30 Water 2020, 12, 183 16 of 30 Water 2020, 12, x FOR PEER REVIEW 17 of 30

Figure 13. Vertical distributions of mean phase-averaged turbulent kinitic energy (TKE) according to

variationsFigure 13. inVertical Vc. distributions of mean phase-averaged turbulent kinitic energy (TKE) according to Figure 13. Vertical distributions of mean phase-averaged turbulent kinitic energy (TKE) according to variations in Vc. variations in Vc. Figure 14 shows the vertical distribution of the K according to the incident wave period (Ti). Figure 14 shows the vertical distribution of the K according to the incident wave period (Ti). The mean phase-averaged TKE increases with increasing Hi, Hi/Li, and Ursell number (Ur). This The meanFigure phase-averaged14 shows the vertical TKE increasesdistribution with of increasingthe K accordingH , H /L to, andthe incident Ursell number wave period (Ur). This(Ti). increment is larger near still-water levels. In addition, with longeri i incidenti periods, the mean phase- Theincrement mean phase-averaged is larger near still-water TKE increases levels. with In increasing addition, withHi, H longeri/Li, and incident Ursell number periods, ( theUr). meanThis averaged TKE increases in sections, except for those near the still-water level. Under the influence of incrementphase-averaged is larger TKE near increases still-water in levels. sections, In addi excepttion, for with those longer near incident the still-water periods, level. the mean Under phase- the wave-current interactions, TKE is closely related with velocity due to the orbital motion of waves and averagedinfluence TKE of wave-current increases in interactions, sections, except TKE for is closely those near related the with still-water velocity level. due Under to the orbitalthe influence motion of of current velocity. That is, as the incident wave height and period increase, the velocity of water wave-currentwaves and current interactions, velocity. TKE That is closely is, as the related incident with wave velocity height due and to the period orbital increase, motion the of waves velocity and of particles increases, and thus the TKE is increased by wave-current interactions. currentwater particles velocity. increases, That is, andas the thus incident the TKE wave is increased height byand wave-current period increase, interactions. the velocity of water Nevertheless, it is difficult to directly correlate TKE and the aforementioned wave height decay particlesNevertheless, increases, and it is dithusfficult the toTKE directly is increased correlate by TKEwave-current and the aforementioned interactions. wave height decay because they are not proportional. The wave energy conditions differ, therefore wave energy decay becauseNevertheless, they are not it is proportional. difficult to directly The wave correlate energy TKE conditions and the diaforementionedffer, therefore wavewave energyheight decay decay and wave height decay are not proportional to TKE. This relationship is closely related to the wave becauseand wave they height are not decay proportional. are not proportional The wave toenergy TKE. conditions This relationship differ, therefore is closely wave related energy to the decay wave energy described later and will be discussed in more detail in the following section. Wave height andenergy wave described height laterdecay and are will not bepropor discussedtional in to more TKE. detail This inrelationship the following is closely section. related Wave heightto the wave decay decay is caused primarily by turbulence in the coexisting wave-current field, therefore the energyis caused described primarily later by and turbulence will be indiscussed the coexisting in more wave-current detail in the field, following therefore section. the mechanism Wave height of mechanism of wave attenuation will be analyzed by exploring changes in TKE, wave energy, and decaywave attenuationis caused willprimarily be analyzed by turbulence by exploring in changesthe coexisting in TKE, wavewave-current energy, and field, wave therefore height decay. the wave height decay. mechanism of wave attenuation will be analyzed by exploring changes in TKE, wave energy, and wave height decay.

(a)

Figure(a 14.) Cont.

Water 2020, 12, 183 17 of 30 Water 2020, 12, x FOR PEER REVIEW 18 of 30

(b) (c)

FigureFigure 14.14. Vertical distributionsdistributions ofof meanmean phase-averagedphase-averaged TKE TKE ( K( K) according) according to variationsto variations in Hini.( Hai.) (Tai) =Ti 1.2= 1.2 s; (s;b )(bT)i T=i =1.5 1.5 s; s; (c ()cT) iT=i =1.8 1.8 s. s. 3.1.4. Wave Energy Loss Characteristics 3.1.4. Wave Energy Loss Characteristics Wave energy (E) lost due to wave-current interactions is calculated by applying the simulated Wave energy ( E ) lost due to wave-current interactions is calculated by applying the simulated data in Equation (28) after a stable wave-current coexisting field is attained. In Equation (29), the mean data in Equation (28) after a stable wave-current coexisting field is attained. In Equation (29), the mean energy loss (EL) is calculated for three periods from the energy ratio of the prior and following waves. energy loss ( EL ) is calculated for three periods from the energy ratio of the prior and following Subsequently, the average EL of five wavelengths is used to calculate the wavelength-averaged energy losswaves. (E ).Subsequently, the average EL of five wavelengths is used to calculate the wavelength- L Z ( ) η u2 + ν2 + w2 p averaged energy loss ( EL ). E = ρ + + gz dz (28) h 2 ρ − η 22 2 ++R t+3Ti R 1L =++ρ 1uvw1 pEdtdx Egzdz3Ti 0.5L t 0.5L (28) E = 1−h (29) L R2t+3T R 0.5Lρ − 1 1 i Edtdx 3Ti 0.5L t 0

+ Figure 15 shows the E according to the11V /C betweentT31i current L velocity and wave celerity under L c i Edtdx various wave conditions. Wave energy loss30.5TL increasestL as Vc0.5/Ci increases. This phenomenon can be =− i explained with the fact that TKE increasesEL 1 as Vc/Ci increases,+ which results in increased energy loss.(29) 11 tT30.5i L The energy decay rate was expected to increase because theEdtdx TKE becomes larger as the incident 30.5TLt 0 wave height increases. The energy decay duei to turbulence increases because higher wave energies accompany higher wave heights, but the general decay rate is relatively low compared to that for low Figure 15 shows the E according to the Vc/Ci between current velocity and wave celerity wave heights. L underThe various aforementioned wave conditions. analytical Wave results energy for waveloss increases height distribution, as Vc/Ci increases. TKE, and This wave phenomenon energy show can c i thatbe explained the TKE increaseswith the withfact that increasing TKE increasesVc/Ci rate. as V This/C leadsincreases, to wave which energy results and in height increased decay. energy This numericalloss. The energy mechanism decay can rate be was used expected to understand to increase wave because height the decay TKE due becomes to wave-current larger as interactions.the incident wave height increases. The energy decay due to turbulence increases because higher wave energies accompany higher wave heights, but the general decay rate is relatively low compared to that for low wave heights.

Water 2020, 12, 183 18 of 30 Water 2020, 12, x FOR PEER REVIEW 19 of 30

(a)

(b) (c)

Figure 15. Averaged energyenergy loss loss according according to to variations variations in Vinc .(Vac.) (Ha)i =Hi3 = cm; 3 cm; (b) H(bi)= H5i cm;= 5 cm; (c) H (ic=) H7i cm= 7. cm. 3.1.5. Wave Reflection Characteristics ThePartial aforementioned standing waves analytical due to results wave-current for wave interactions height distribution, are also found TKE, and in the wave aforementioned energy show wavethat the height TKE distribution.increases with Equation increasing (30), V whichc/Ci rate. was This proposed leads to by wave Healy energy [76], was and used height to calculate decay. This the numericalreflection coemechanismfficient with can the be wave-current used to understa interactions.nd wave In addition, height decay the values due calculatedto wave-current for five interactions.partial standing wave field wavelengths were averaged to produce the average reflection coefficient.

3.1.5. Wave Reflection Characteristics Hmax Hmin KR = − (30) Hmax + H Partial standing waves due to wave-current interactionsmin are also found in the aforementioned wherewave heightHmax anddistribution.Hmin are Equation the wave (30), heights which at thewas node proposed and the by anti-nodeHealy [76], in was a partial used to standing calculate wave the reflectionfield, respectively. coefficient with the wave-current interactions. In addition, the values calculated for five partialFigure standing 16 shows wave the field wavelength-averaged wavelengths were averaged reflection to coe producefficient the (KR average) according reflection to the ratiocoefficient. (Vc/Ci) between current velocity and wave celerity underHH various− wave conditions. The wave reflection V C K = max min rate generally increases as c/ i increases. WaveR reflection+ due to current is generated by a strong(30) interaction under the wave crest, where the waveHH orbitalmax motion min velocity and current velocity oppose each other. Therefore, the reflection rate increases as the current velocity increases. Meanwhile, (a–c), where Hmax and Hmin are the wave heights at the node and the anti-node in a partial standing whichwave field, feature respectively. the same incident wave height, show lower reflection rates with shorter incident wave periods. The wavelength is not long in waves with shorter periods, therefore the exposure time under Figure 16 shows the wavelength-averaged reflection coefficient ( KR ) according to the ratio

(Vc/Ci) between current velocity and wave celerity under various wave conditions. The wave

Water 2020, 12, x FOR PEER REVIEW 20 of 30

reflection rate generally increases as Vc/Ci increases. Wave reflection due to current is generated by a strong interaction under the wave crest, where the wave orbital motion velocity and current velocity Wateroppose2020 , each12, 183 other. Therefore, the reflection rate increases as the current velocity increases.19 of 30 Meanwhile, (a–c), which feature the same incident wave height, show lower reflection rates with shorter incident wave periods. The wavelength is not long in waves with shorter periods, therefore thethe waveexposure crest time decreases. under Therefore,the wave crest the wave-current decreases. Therefore, interaction the time wave-current becomes shorter interaction under time an oppositebecomes flowshorter velocity, under resultingan opposite in aflow lower velocity, reflection resu rate.lting in In a addition, lower reflection the current rate. byIn theaddition, incident the wavecurrent height by the does incident not appear wave to height have strong does not effects appear on the to wavehave reflectionstrong effects rate, on which the iswave similar reflection to the reflectiverate, which wave is similar characteristics. to the reflective wave characteristics. WaveWave reflectionreflection byby currents,currents, whichwhich waswas reportedreported byby anan existingexisting theoreticaltheoretical studystudy toto bebe insignificantinsignificant [ 37[37],], is is therefore therefore confirmed, confirmed, and and the the mechanism mechanism is elucidatedis elucidated above. above. The The reflection reflection rate rate is upis up to0.068 to 0.068 under under the incidentthe incident conditions conditions applied applied herein. herein. Such a Such wave a reflection wave reflection rate cannot rate be cannot ignored be whenignored analyzing when analyzing hydraulic hydraulic phenomena phenomena in coexisting in coexisting wave-current wave-current fields. fields.

(a)

(b) (c)

FigureFigure 16.16. Wavelength-averagedWavelength-averaged reflectionreflection coefficients coefficients according according to to variations variations in in VVc.c (.(a)a H) iH =i 3= cm;3 cm; (b) (bH)i H= i5= cm;5 cm; (c) H (ci) =H 7i =cm.7 cm.

3.1.6.3.1.6. MeanMean WaveWave LevelLevel CharacteristicsCharacteristics FigureFigure 17 17 shows shows thethe spatialspatial distributiondistribution ofof thethe meanmean waterwater levellevel accordingaccording toto thethe ratioratio betweenbetween the current velocity and wave celerity, V /C . As shown in Equation (31), the mean water level the current velocity and wave celerity, Vc/cCi.i As shown in Equation (31), the mean water level was wascalculated calculated using using surface surface elevations elevations (Equation (Equation (24)) (24))from from three three periods periods and anddescribed described via the via non- the non-dimensional mean levelη (η/Hi) after the wave and flow field stabilized. dimensional mean level ( / Hi ) after the wave and flow field stabilized. Z t+3Ti η = η(t) dt (31) t Water 2020, 12, x FOR PEER REVIEW 21 of 30

+ tT3 i Water 2020, 12, 183 ηη= ()tdt 20(31) of 30 t

In Figure 17, The mean water level also increases when Vc/Ci increases because the wave energy In Figure 17, The mean water level also increases when Vc/Ci increases because the wave energy decreases due to wave-current interactions, which show a tendency to increase because the wave decreases due to wave-current interactions, which show a tendency to increase because the wave energy decreases further when Vc/Ci increases. Meanwhile, these types of wave level distributions energy decreases further when Vc/Ci increases. Meanwhile, these types of wave level distributions create currents due to wave level differences; currents, in turn, increase the initial current velocity, create currents due to wave level differences; currents, in turn, increase the initial current velocity, which will accelerate wave height dissipation. To analyze this further, it is necessary to study the which will accelerate wave height dissipation. To analyze this further, it is necessary to study the interactions between the flow field and the wave field. interactions between the flow field and the wave field.

Figure 17. Spatial distributions of mean water level under wave-current interaction according to Figure 17. Spatial distributions of mean water level under wave-current interaction according to variations in Vc. variations in Vc. 3.2. Wave-Current Interactions with Density Current 3.2. Wave-Current Interactions with Density Current The effect of current due to density differences between freshwater and saltwater must be considered,The effect as estuaries of current contain due bothto density freshwater differences and saltwater. between In thefreshwater past, most and studies saltwater on wave-current must be considered,interactions as did estuaries not consider contain density both currents. freshwater Therefore, and saltwater. wave-current In the interactions past, most understudies the on influence wave- currentof density interactions differences did were not consider numerically density simulated current usings. Therefore, a newly wave-current suggested numerical interactions model under in order the influenceto calculate of densitydensity currents.differences were numerically simulated using a newly suggested numerical model in order to calculate density currents. 3.2.1. Description of Numerical Water Basin and Incident Conditions 3.2.1. Description of Numerical Water Basin and Incident Conditions A numerical water basin was prepared as shown in Figure 18 to evaluate the effect of density currentA numerical on wave-current water basin interactions was prepared in an estuary. as shown To in generate Figure waves18 to evaluate without the reflection, effect of wave density and currentcurrent on sources wave-current were installed interactions at both in ends an estuar of they. simulation To generate domain, waves and without damping reflection, zones were wave added and currentto absorb sources the wave were energy.installed The at both analysis ends zoneof the featuredsimulation a length domain, of and 5 Li dampingof the incident zones were wavelength, added toa widthabsorbof the 50 wave cm,and energy. a water The levelanalysis of 30 zone cm; featured a 1:5 slope a length was installed of 5 Li of to the a waterincident depth wavelength, of 10 cm toa widthserve of as 50 the cm, current and a source. water level of 30 cm; a 1:5 slope was installed to a water depth of 10 cm to serve as theThe current simulations source. were performed under fixed incident wave and current conditions and variable waterThe density. simulations The detailswere performed are described under in fixed Table incide1. Twont wave major and cases current were conditions considered: and one variable with waterdensity density. differences The underdetails wave-current are described interactions, in Table 1. and Two another major without cases were density considered: differences one (ρw with= ρc). densityThe first differences major case under was divided wave-current into three interact sub-experiments,ions, and another which without involved: density high density differences water ( massρw = ρandc). The low first density major current case was flows divided (ρw > intoρc); three low density sub-experiments, water mass which and high involved: density high current density flows water (ρw mass< ρc); and and low water density mass current with density flows ( stratificationρw > ρc); low density and middle water density mass and current high flowsdensity (ρ wcurrent1 < ρc

Water 2020, 12, 183 21 of 30 Water 2020, 12, x FOR PEER REVIEW 22 of 30

Figure 18. Sketch of a numerical water basin for simulating wave-current interactions with included Figure 18. Sketch of a numerical water basin for simulating wave-current interactions with included density differences. density differences.

Table 1. IncidentIncident wave wave and and current current conditions conditions for simula for simulatingting wave-current wave-current interactions interactions with density with densitydifferences. differences.

WaveWave Current Current Reduced Reduced GravityGravity Height Period Salinity Density Velocity Salinity Density ∆ρg/ρ CASE Height Period Salinity Density Velocity Salinity Density Δρg/ρminmin CASE 3 Hi Sw (psu) ρw (g/cm ) 2 3 2 T (s) w w 3 V (cm/s ) Sc (psu) ρ (g/cm ) g’ (cm/s ) H(cm)i i S (psu) ρ (g/cm ) Vcc cρc i S S ρ ρ c 2 T (s) w1 w2 w1 w2 2 S (psu) 3 g’ (cm/s ) (cm) Sw1 Sw2 ρw1 ρw2 (cm/s ) (g/cm ) 1 0 0.9982 0 0 0.9982 1 2 0 350.9982 1.0276 28.860 -- 0 0.9982 20 2 3 35 1.027635 01.0276 0.9982 28.86 - - 20 3 4 035 351.0276 0.9982 1.02760 17.50.9982 1.0129 28.86 14.33 5 6 1.5 0 0.9982 20 0 0.9982 0 4 0 35 0.9982 1.0276 17.5 1.0129 14.33 6 5 1.0024 4.12 5 6 1.5 0 0.9982 20 0 0.9982 0 7 10 1.0066 8.25 6 8 5 151.0024 1.0108 12.374.12 7 9 6 1.5 0 0.9982 20 10 201.0066 1.0150 16.498.25 8 10 15 251.0108 1.0192 12.37 20.62 11 30 1.0234 24.74 9 6 1.5 0 0.9982 20 20 1.0150 16.49 12 35 1.0276 28.86 10 25 1.0192 20.62 13 30 1.0234 4.02 11 14 30 251.0234 1.0192 24.74 8.08 12 15 35 201.0276 1.0150 28.86 12.17 6 1.5 35 1.0276 20 13 16 30 151.0234 1.0108 16.294.02 17 10 1.0066 20.45 14 25 1.0192 8.08 18 5 1.0024 24.64 15 19 20 01.0150 0.9982 12.17 28.86 16 206 61.5 1.5 035 351.0276 0.9982 1.027620 2015 17.5 1.0108 1.0129 16.29 14.33 17 10 1.0066 20.45 3.2.2.18 Density Current Formation Characteristics 5 1.0024 24.64 19 Figure 19 shows the flow fields for CASEs 1–4 in Table1, in0 which0.9982 current was generated28.86 without20 considering6 1.5 waves.0 In35 Figure0.9982 19 ,1.0276 four different20 current17.5 shapes 1.0129 can be identified14.33 due to density differences, including (a) a uniform-depth current due to a lack of density difference, (b) an3.2.2. undercurrent Density Current due toFormation a high-density Characteristics current, (c) an upper current due to a low-density current, and (d) a middle current due to a middle-density current flowing into the density stratification. Based Figure 19 shows the flow fields for CASEs 1–4 in Table 1, in which current was generated without considering waves. In Figure 19, four different current shapes can be identified due to density differences, including (a) a uniform-depth current due to a lack of density difference, (b) an undercurrent due to a high-density current, (c) an upper current due to a low-density current, and

Water 2020,, 1212,, 183x FOR PEER REVIEW 2223 of 30

(d) a middle current due to a middle-density current flowing into the density stratification. Based on on these results, various currents may arise from various density difference structures. Therefore, these results, various currents may arise from various density difference structures. Therefore, the the consideration of differences in density will affect the wave-current interactions in co-existing fields. consideration of differences in density will affect the wave-current interactions in co-existing fields.

(a)

(b)

(c)

(d)

Figure 19. Four types of density density currents with varying density differences. differences. ( a) Depth-uniform Depth-uniform current

(ρw = ρρc);c); (b (b) )Undercurrent Undercurrent ( (ρρww << ρρc);c );(c ()c Upper) Upper current current (ρ (ρw w> >ρcρ);c ();d ()d Middle) Middle current current (ρ (wρ1w <1 ρ

3.2.3. Wave Height Variation Figure 20 shows the wave height variations generated by wave-cu wave-currentrrent interactions with density didifferences,fferences, includingincluding (a) (a) interactions interactions between between the th wavee wave and and four difourfferent different current current shapes shapes due to densitydue to didensityfferences, differences, (b) an interaction (b) an interaction between thebetween wave the and wave undercurrent and undercurrent (ρw < ρc), and(ρw (c)< ρ anc), interactionand (c) an betweeninteraction the between wave and the upper wave currentand upper (ρw current> ρc). (ρw > ρc). All cases cases in in Figure Figure 20a 20 ashow show decreases decreases in wave in wave height height due dueto wave to wave energy energy loss caused loss caused by wave- by wave-currentcurrent interactions. interactions. In particular, In particular, CASE 19 CASE (ρw > 19 ρc ()ρ showsw > ρc )considerable shows considerable wave height wave decay height because decay becausethe inflowing the inflowing current is current concentrated is concentrated near the near water the surface, water surface, where wherethe velocity the velocity caused caused by wave by waveorbital orbital motion motion is large. is large.Therefore, Therefore, there may there be may stronger be stronger interactio interactionsns near the near water the surface, water surface, which whichinduce induce wave height wave heightdecay due decay to duewave to energy wave energy loss. In loss. addition, In addition, the wave the height wave heightdecay seen decay in seen CASE in CASE12 (ρw < 12 ρc ()ρ andw < ρwavec) and height wave seen height in CASE seen in 20 CASE (ρw1 < 20ρc < (ρ ρww12)< areρc lower< ρw2 )than are lowerthose seen than in those CASE seen 5 (ρ inw CASE= ρc) because 5 (ρw = theρc) becausecurrent flows the current in the flowslower-middle in the lower-middle layers, which layers, feature which lower feature velocity lower caused velocity by causedorbital motion, by orbital lead motion, to less leadinteraction to less between interaction waves between and currents, waves and which currents, results which in less results wave height in less wavedecay heightthan that decay in CASE than that 5 (ρ inw = CASE ρc), in 5which (ρw = differencesρc), in which in didensityfferences are in not density considered. are not considered.

Water 2020, 12, 183 23 of 30 Water 2020, 12, x FOR PEER REVIEW 24 of 30

(a)

(b)

(c)

FigureFigure 20.20.Spatial Spatial distributions distributions of waveof wave height height under under various various interactions interactions between between waves and waves currents. and (currents.a) According (a) According to type of densityto type current;of density (b) current; According (b) to According the undercurrent to the undercurrent (ρw < ρc); (c) ( Accordingρw < ρc); ( toc) theAccording upper current to the upper (ρw > ρcurrentc). (ρw > ρc).

InIn FigureFigure 2020b,b, CASEsCASEs 6,6, 8,8, andand 1111 showshow thethe samesame phenomenonphenomenon andand featurefeature lessless wavewave heightheight dissipationdissipation thanthan in in CASE CASE 5, 5, which which has has a uniform-depth a uniform-depth current. current. Furthermore, Furthermore, CASEs CASEs 6, 8, and6, 8, 11 and show 11 moreshow wavemore height wave dissipationheight dissipation under smaller under dismallerfferences differences in salinity. in This salinity. is because This currentis because generation current isgeneration more similar is more to that similar under to that a uniform-depth under a unifor currentm-depth when current the diwhenfference the difference in density in is small.density Inis Figuresmall. In20 c,Figure there 20c, is more there wave is more height wave decay height in CASEs decay 13,in CASEs 15, and 13, 18 than15, and in CASE18 than 5. in The CASE cases 5. with The uppercases currentwith upper also showcurrent a tendency also show toward a tendency higher wave toward height higher decay wave as density height di ffdecayerences as increase,density althoughdifferences the increase, effect is although not very strong.the effect is not very strong. TheseThese resultsresults indicate indicate that that vertically vertically varying varying currents currents affect affect wave wave deformation deformation due to due wave-current to wave- interactions.current interactions. Therefore, Therefore, density currents density must currents be considered must be inconsidered order to e ffinectively order to analyze effectively wave-current analyze interactionswave-current in interactions areas with density in areas di withfferences. density differences.

3.2.4.3.2.4. Vertical DistributionDistribution ofof TKETKE Figure 21 shows the vertical distribution of the mean phase-average TKE (K) under various Figure 21 shows the vertical distribution of the mean phase-average TKE ( K ) under various wave-current interactions due to density differences, including (a) interactions between the wave and wave-current interactions due to density differences, including (a) interactions between the wave and four different current shapes due to density differences, (b) an interaction between the wave and an undercurrent (ρw < ρc), and (c) an interaction between the wave and an upper current (ρw > ρc).

Water 2020, 12, 183 24 of 30

four different current shapes due to density differences, (b) an interaction between the wave and an undercurrent (ρ < ρ ), and (c) an interaction between the wave and an upper current (ρ > ρ ). Water 2020, 12, x FORw PEER cREVIEW w 25c of 30

(a)

(b) (c)

FigureFigure 21. 21. VerticalVertical profiles profiles of of the the mean mean phase-averaged phase-averaged turbulence turbulence intensities intensities under under wave-current wave-current interactions.interactions. (a ()a According) According to to the the type type of of density density current; current; (b ()b According) According to to the the undercurrent undercurrent (ρ (ρw w< <ρcρ);c ); (c()c )According According to to the the upper upper current current (ρ (ρww >> ρρc).c).

In Figure 21a, CASE 5 shows more balanced K under the interactions between waves and a In Figure 21a, CASE 5 shows more balanced K under the interactions between waves and a uniform-depth current than do the other cases. CASEs 12, 19, and 20 show strong K generation at uniform-deptheach current point current generated than do bythe density other cases. differences. CASEs In12, particular, 19, and 20K showdueto strong the wave-upper K generation current at eachinteraction current appearspoint generated to be strong by de nearnsity the differences. free surface In particular, in CASE 19, K whichdue to features the wave-upper the highest current wave interactionheight decay appears (Figure to be18 ).strong In addition, near the CASE free surf 5, whichace in featuresCASE 19, a uniform-depthwhich features the current, highest possesses wave heighthigher decayK than (Figure CASE 18). 12, whichIn addition, features CASE undercurrent 5, which andfeatures less wave a uniform-depth height dissipation, current, and possesses CASE 20, higherwhich featuresK than aCASE middle 12, current. which features This turbulence undercurrent appears and to beless the wave major height factor dissipation, in the wave and energy CASE loss 20,and which height features dissipation a middle and current. can explain This the turbulence formation appears mechanisms to be the of themajor abovementioned factor in the wave wave energy height lossdistributions. and height Indissipation Figure 21b, and CASEs can explain 8 and 11 the feature formation stronger mechanismsK when larger of the salinity abovementioned differences existwave in heightthe lower distributions. layer. However, In Figure CASE 21b, 6, whichCASEs features 8 and the11 lowestfeature salinitystronger di ffKerence, when shows larger a Ksalinityvertical differencesdistribution exist similar in the to lower that inlayer. CASE However, 5; this reducesCASE 6, the which wave features energy, the resulting lowest salinity in the wavedifference, height distribution described above. In Figure 21c, CASEs 13, 15, and 18 possess stronger K under larger shows a K vertical distribution similar to that in CASE 5; this reduces the wave energy, resulting salinity differences near the free surface. Under smaller salinity differences, K vertical distributions in the wave height distribution described above. In Figure 21c, CASEs 13, 15, and 18 possess stronger similar to that in CASE 5, which features a uniform-depth current, are observed, which reduces wave Kenergy, under resulting larger salinity in the wave differences height distributionnear the free discussed surface. earlier.Under smaller salinity differences, K verticalOverall, distributions wave-current similar to interactions that in CASE in cases5, which with features density a diuniform-depthfferences appear current, different are observed, from those whichfound reduces in cases wave that doenergy, not consider resulting density in the wave differences. height distribution Therefore, consideration discussed earlier. of currents due to Overall, wave-current interactions in cases with density differences appear different from those found in cases that do not consider density differences. Therefore, consideration of currents due to density differences is strongly recommended during analysis of wave-current interactions in water bodies with density differences, such as estuaries.

Water 2020, 12, 183 25 of 30 density differences is strongly recommended during analysis of wave-current interactions in water bodies with density differences, such as estuaries.

4. Conclusions In the estuary, most of the laboratory experiments and numerical simulations for analysis of the wave-current interaction were performed under homopycnal flow conditions that do not consider the density discrepancy between the river and ocean. However, hypopycnal flow, which refers to river water spreading out to the upper layer of ocean, actually occurs in estuary. In this study, to simulate wave-current interaction under hypopycnal flow conditions, we have revised the NWT to reproduce the density current based on the existing numerical model. To confirm the effectiveness and validity of the improved numerical model, the model was compared with the results of hydraulic experiments [72,73]. These comparisons confirmed the validity and effectiveness of the newly suggested numerical model in wave-current interaction simulations in estuaries that contain density differences. This study addressed wave attenuation, wave reflection, and wave energy reduction under wave-current interactions, which have not been addressed in previous studies. In addition, this study numerically explored the effects of current due to density differences on wave-current interactions in an open channel. Wave height decay and wave reflection mechanisms were analyzed in the coexisting wave-current field in terms of TKE and wave energy. The hydraulic characteristics of the numerically simulated wave-current interactions can be summarized as follows:

When the ratio (Vc/C ) between current velocity and wave celerity increases, wave height decay • i increases. Wave reflection by currents forms a partial standing wave field.

When the Vc/C velocity ratio increases, the overall mean phase-average TKE increases. At higher • i incident wave heights, wave steepness, and Ursell numbers larger increases occur near still-water levels. Longer incident periods lead to larger increases, except near the still-water level.

As a result of turbulence caused by wave-current interactions, wave energy increases as the Vc/C • i increases. When the incident wave height is high, TKE increases and the wave energy decay rate decreases. Higher wave heights have higher wave energy, therefore the energy decay rate is relatively low.

The rate of wave reflection due to currents becomes larger as the Vc/C increases. With longer • i wave periods, the exposure time in wave troughs, in which the wave orbital motion velocity and current velocity are opposite in direction, is longer, and thus the wave reflection rate is increased. As a result, reflection rates of up to 0.068 were found under the incident conditions applied in this study.

The mean water level increases more due to wave-current interactions as the velocity ratio Vc/C • i increases, which further reduces wave energy. In conclusion, the flow velocity turbulence element is increased in the coexisting wave-current field by wave-current interactions, which decreases the wave energy. Consequently, the wave height decays and the water level increases. Wave reflection by currents occurs under the wave trough. Thus, the results provide an explanation wave height decay and wave reflection mechanisms due to wave-current interactions. The influence of currents caused by density differences on wave-current interaction in an estuary can be described as follows: Various currents are generated depending on the density difference between two fluids. • Wave height decay due to various current-wave interactions is highest in cases involving upper • currents, in which strong interactions occur near still-water levels. Cases involving middle current and undercurrent feature less wave height decay than do cases with uniform-depth currents, which feature no density current. The mean phase-average TKE vertical distribution is broad near each type of current. Greater • wave height decay values are accompanied by higher mean phase-average TKE. Water 2020, 12, 183 26 of 30

In general, simulated wave-current interactions caused by density differences show different tendencies than cases that do not account for density differences. Therefore, currents arising from density differences must be considered when analyzing wave-current interactions in water bodies with density differences, such as estuaries.

Author Contributions: W.-D.L. and N.M. designed the study concept; W.-D.L. and D.-S.H. developed the code and performed the model simulations; All authors analyzed the results; W.-D.L. wrote the paper. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2018R1C1B6007461). Conflicts of Interest: The authors declare no conflict of interest.

Appendix A. Fluid Density Equations To analyze density current, it is important to estimate the density of a fluid accurately; thus, Equation (A1) was applied to estimate density according to temperature and salinity, as suggested by Gill [60]. Here, ρ0 is the density of 4 ◦C freshwater. ∆ρT is the increase in density with temperature change and is expressed as Equation (A2). ∆ρS expresses the change in density with changing salinity and is expressed as Equation (A3). The empirical constants used in the density calculation are shown in Table A1. ρ = ρ0 + ∆ρT + ∆ρS (A1) 2 3 4 5 ∆ρT = a T a T + a T a T + a T (A2) 1 − 2 3 − 4 5     ∆ρ = a a T + a T2 a T3 + a T4 S + a + a T a T2 S3/2 + a S2 (A3) S 6 − 7 8 − 9 10 − 11 12 − 13 14 where T is the temperature (◦C) and S is the salinity (psu).

Table A1. Density estimation coefficients.

ρ = 0.999842594 g/cm3 a = 6.536332 10 12 a = 5.38750 10 12 0 5 × − 10 × − a = 6.793952 10 5 a = 8.244930 10 4 a = 5.72466 10 6 1 × − 6 × − 11 × − a = 9.095290 10 6 a = 4.089900 10 6 a = 1.02270 10 7 2 × − 7 × − 12 × − a = 1.001685 10 7 a = 7.643800 10 8 a = 1.65460 10 9 3 × − 8 × − 13 × − a = 1.120083 10 9 a = 8.246700 10 10 a = 4.83140 10 7 4 × − 9 × − 14 × −

Appendix B. Kinematic Viscosity Coefficient Equations The kinematic viscosity coefficient ν for the fluid is calculated using Equation (A4). Here, the calculated value of Equation (A1) is substituted for the density ρ, and the viscosity coefficient µ is calculated using Equation (A5), which considers the water temperature and salinity, as suggested by Riley and Skirrow [61]. ν = µ/ρ (A4)

µ = µ0 + ∆µT + ∆µS (A5) 2 3 ∆µT = b T + b T b T (A6) − 1 2 − 3 2 ∆µS = b4S + b5S (A7) where, µ0 is the viscosity coefficient for 4 ◦C freshwater. ∆µT indicates the change in the viscosity coefficient with a change in temperature and is expressed via Equation (A6). ∆µS indicates the change in the viscosity coefficient with a change in salinity and is expressed via Equation (A7). The empirical constants used for the water viscosity coefficient are expressed in Table A2. Water 2020, 12, 183 27 of 30

Table A2. Viscosity estimation coefficients.

µ = 1.802863 10 2 g/cm s b = 1.31419 10 5 b = 2.15123 10 5 0 × − · 2 × − 4 × − b = 6.108600 10 4 b = 1.35576 10 7 b = 3.59406 10 10 1 × − 3 × − 5 × −

References

1. Geyer, W.R.; MacCready, P. The estuarine circulation. Annu. Rev. Fluid Mech. 2014, 46, 175–197. [CrossRef] 2. Bolaños, R.; Brown, J.M.; Souza, A.J. Wave–current interactions in a tide dominated estuary. Cont. Shelf Res. 2014, 87, 109–123. [CrossRef] 3. Bates, C.C. Rational theory of delta formation. Bull. Am. Assoc. Pet. Geol. 1953, 37, 2119–2162. 4. Boggs, S. Principles of Sedimentology and Stratigraphy, 2nd ed.; Prentice Hall: Englewood Cliffs, NJ, USA, 1995; p. 774. 5. Kang, K.R.; Iorio, D.D. Depth- and current-induced effects on wave propagation into the Altamaha River Estuary, Georgia. Estuar. Coast. Shelf Sci. 2006, 66, 395–408. [CrossRef] 6. Zippel, S.; Thomson, J. Surface wave breaking over sheared currents: Observations from the Mouth of the Columbia River. J. Geophys. Res. Ocean. 2017, 122, 3311–3328. [CrossRef] 7. Tanaka, H.; Nagabayashi, H.; Yamauchi, K. Observation of wave set-up height in a river mouth. In Proceedings of the 27th International Conference on Coastal Engineering, Sydney, Australia, 16–21 July 2000; pp. 3458–3471. 8. Nguyen, X.T.; Tanaka, H.; Nagabayashi, H. Wave setup at river and inlet entrances due to an extreme event. In Proceedings of the International Conference on Violent Flows, Fukuoka, Japan, 20–22 November 2007. 9. Donnell, J.O. Observations of near-surface currents and hydrography in the Connecticut River plume with the surface current and density array. J. Geophys. Res. Ocean. 1997, 102, 25021–25033. [CrossRef] 10. Smith, J.M.; Seabergh, W.C.; Harkins, G.S.; Briggs, M.J. Wave Breaking on a Current at an Idealized Inlet, Coastal Inlets Research Program, Inlet Laboratory Investigations; Technical Report CHL-98-31; US Army Engineer Waterways Experiment Station: Vicksburg, MS, USA, 1998. 11. Ibrahim, Z.; Latiff, A.A.A.; Halim, A.H.A.; Bakar, N.A.; Subramaniam, S. Experimental studies on mixing salt wedge estuary. Malays. J. Civ. Eng. 2008, 20, 188–199. 12. Shi, F.; Dalrymple, R.A.; Kirby, J.T.; Chen, Q.; Kennedy, A. A fully nonlinear Boussinesq model in generalized curvilinear coordinates. Coast. Eng. 2001, 42, 337–358. [CrossRef] 13. de Brye, B.; de Brauwere, A.; Gourgue, O.; Kärnä, T.; Lambrechts, J.; Comblen, R.; Deleersnijder, E. A finite-element, multi-scale model of the Scheldt tributaries, River, Estuary and ROFI. Coast. Eng. 2010, 57, 850–863. [CrossRef] 14. Pascolo, S.; Petti, M.; Bosa, S. Wave–Current Interaction: A 2DH Model for Turbulent Jet and Bottom-Friction Dissipation. Water 2018, 10, 392. [CrossRef] 15. Sutherlanda, J.; Walstrab, D.J.R.; Cheshera, T.J.; van Rijn, L.C.; Southgate, H.N. Evaluation of coastal area modelling systems at an estuary mouth. Coast. Eng. 2004, 51, 119–142. [CrossRef] 16. Lesser, G.R.; Roelvink, J.A.; van Kester, J.A.T.M.; Stelling, G.S. Development and validation of a three-dimensional morphological model. Coast. Eng. 2004, 51, 883–915. [CrossRef] 17. Booij, N.; Ris, R.C.; Holthuijsen, L.H. A third-generation wave model for coastal regions, Part I, Model description and validation. J. Geophys. Res. 1999, 104, 7649–7666. [CrossRef] 18. The WAMDI Group. The WAM model—A third generation ocean wave prediction model. J. Phys. Oceanogr. 1988, 18, 1775–1810. [CrossRef] 19. Zhang, Q.-H.; Tan, F.; Han, T.; Wang, X.-Y.; Hou, Z.-Q.; Yang, H. Simulation of sorting sedimentation in the channel of Huanghua harbor by using 3d multi-sized sediment transport model of EFDC. In Proceedings of the 32nd International Conference on Coastal Engineering, Shanghai, China, 30 June–5 July 2010; pp. 1604–1614. 20. Liang, B.; Zhao, H.; Li, H.; Wu, G. Numerical study of three-dimensional wave-induced longshore current’s effects on sediment spreading of the Huanghe River mouth. Acta Oceanol. Sin. 2012, 31, 129–138. [CrossRef] 21. Baddour, R.E.; Song, S. On the interaction between waves and currents. Ocean Eng. 1990, 17, 1–21. [CrossRef] 22. Peregrine, D.H. Interaction of Water Waves and Currents. Adv. Appl. Mech. 1976, 16, 9–117. 23. Jonsson, I.G.; Skougaard, C.; Wang, J.D. Interaction between waves and currents. Coast. Eng. 1970, 1, 489–507. 24. Wolf, J.; Prandle, D. Some observations of wave–current interaction. Coast. Eng. 1999, 37, 471–485. [CrossRef] Water 2020, 12, 183 28 of 30

25. Gonzalez, F.I. A case study of wave-current-bathymetry interactions at the Columbia River Entrance. J. Phys. Oceanogr. 1984, 14, 1065–1078. [CrossRef] 26. Grant, W.D.; Madsen, O.S. Combined wave and current interaction with a rough bottom. J. Geophys. Res. Ocean. 1979, 84, 1797–1808. [CrossRef] 27. Fredsøe, J. Turbulent boundary layer in wave-current motion. J. Hydraul. Eng. 1984, 110, 1103–1120. [CrossRef] 28. Christoffersen, J.B.; Jonsson, I.G. Bed friction and dissipation in a combined current and wave motion. Ocean Eng. 1985, 12, 387–423. [CrossRef] 29. Soulsby, R.L.; Hamm, L.; Klopman, G.; Myrhaug, D.; Simons, R.R.; Thomas, G.P. Wave–current interaction within and outside the bottom boundary layer. Coast. Eng. 1993, 21, 41–69. [CrossRef] 30. Jonsson, I.G. Wave–Current Interactions; The Sea; Le Méhauté, B., Hanes, D.M., Eds.; Wiley: New York, NY, USA, 1990; Volume 9, pp. 65–120. 31. Umeyama, M. Reynolds stresses and velocity distributions in a wave-current coexisting environment. J. Waterw. Portcoast. Ocean Eng. 2005, 131, 203–212. [CrossRef] 32. Lee, W.D.; Mizutani, N.; Hur, D.S. Experimental analysis of wave deformation and bottom flow under wave-current interaction in the river mouth. Ocean Eng. 2017, 140, 169–182. [CrossRef] 33. Umeyama, M. Coupled PIV and PTV measurements of particle velocities and trajectories for surface waves following a steady current. J. Waterw. Portcoast. Ocean Eng. 2010, 137, 85–94. [CrossRef] 34. Chen, Y.Y.; Hsu, H.C.; Hwung, H.H. Particle trajectories beneath wave-current interaction in a two-dimensional field. Nonlinear Process. Geophys. 2012, 19, 185–197. [CrossRef] 35. Fernando, P.C.; Guo, J.; Lin, P. Wave-current interaction at an angle 1: Experiment. J. Hydraul. Res. 2011, 49, 424–436. [CrossRef] 36. Lim, K.Y.; Madsen, O.S. An experimental study on near-orthogonal wave–current interaction over smooth and uniform fixed roughness beds. Coast. Eng. 2016, 116, 258–274. [CrossRef] 37. Longuet-Higgins, M.S.; Stewart, R.W. The changes in amplitudes of short gravity waves on steady non-uniform currents. J. Fluid Mech. 1961, 10, 529–549. [CrossRef] 38. Dingemans, M.W. Water Wave Propagation over Uneven Bottoms; Advanced Series on Ocean Engineering 13; World Scientific: London, UK, 1997; p. 1016. 39. Lee, K.H.; Mizutani, N.; Komatsu, K.; Hur, D.S. Experimental study on wave-current interaction. In Proceedings of the 16th International Offshore and Polar Engineering Conference, San Francisco, CA, USA, 28 May–2 June 2006; pp. 600–606. 40. Rey, V.; Charland, J.; Touboul, J. Wave–current interaction in the presence of a three-dimensional bathymetry: Deep water wave focusing in opposing current conditions. Phys. Fluids 2014, 26, 096601. [CrossRef] 41. Drevard, D.; Rey, V.; Fraunie, P. Partially standing wave measurement in the presence of steady current by use of coincident velocity and/or pressure data. Coast. Eng. 2009, 56, 992–1001. [CrossRef] 42. Mizutani, N.; Lee, K.H.; Komatsu, K.; Hur, D.S. Fundamental study on wave-current interaction. Proc. Coast. Eng. Conf. Jpn. Soc. Civ. Eng. 2005, 21, 307–312. (In Japanese) 43. Mizutani, N.; Hur, D.S.; Maeda, Y. Numerical analysis of nonlinear wave-current interaction in side harbor. Proc. Coast. Eng. Conf. Jpn. Soc. Civ. Eng. 2002, 49, 51–55. (In Japanese) 44. Lee, K.H.; Ohori, F.; Mizutani, N.; Kuwabara, S. A study on interactions between effluent currents and incoming waves near an estuary. Annu. J. Civ. Eng. Ocean. Soc. Civ. Eng. 2008, 24, 897–902. (In Japanese) 45. Lee, W.D.; Mizutani, N.; Hur, D.S. Effect of crossing angle on interaction between wave and current in the river mouth. J. Jpn. Soc. Civ. Eng. Ser. B3 (Ocean Eng.) 2011, 67, 256–261. (In Japanese) 46. Liu, Z.; Lin, Z.; Tao, L.; Lan, J. Nonlinear wave–current interaction in water of finite depth. J. Waterw. Portcoast. Ocean Eng. 2016, 142, 04016009. [CrossRef] 47. Lee, W.D.; Jeong, Y.H.; Jeon, H.S. Groundwater flow analysis in a coastal aquifer with the coexistence of seawater and freshwater by using a non-hydrostatic pressure model. J. Coast. Res. 2019, 91, 121–125. [CrossRef] 48. Lee, W.D.; Yoo, Y.J.; Jeong, Y.M.; Hur, D.S. Experimental and numerical analysis on hydraulic characteristics of coastal aquifers with seawall. Water 2019, 11, 2343. [CrossRef] 49. Hur, D.S.; Lee, W.D.; Cho, W.C. Three-dimensional flow characteristics around permeable submerged breakwaters with open inlet. Ocean Eng. 2012, 44, 100–116. [CrossRef] Water 2020, 12, 183 29 of 30

50. Hur, D.S.; Lee, W.D.; Cho, W.C. Characteristics of wave run-up height on a sandy beach behind dual-submerged breakwaters. Ocean Eng. 2012, 45, 38–55. [CrossRef] 51. Hur, D.S.; Lee, W.D.; Cho, W.C.; Jeong, Y.H.; Jeong, Y.M. Rip current reduction at the open inlet between double submerged breakwaters by installing a drainage channel. Ocean Eng. 2019, 193, 106580. [CrossRef] 52. Hur, D.S.; Mizutani, N. Numerical estimation of the wave forces acting on a three-dimensional body on submerged breakwater. Coast. Eng. 2003, 47, 329–345. [CrossRef] 53. Hur, D.S.; Mizutani, N.; Kim, D.S. Direct 3-d numerical simulation of wave forces on asymmetric structures. Coast. Eng. 2004, 51, 407–420. [CrossRef] 54. Smagorinsky, J. General circulation experiments with the primitive equation. Mon. Weather Rev. 1963, 91, 99–164. [CrossRef] 55. Hur, D.S.; Lee, K.H.; Yeom, G.S. The phase difference effects on 3-d structure of wave pressure acting on a composite breakwater. Ocean Eng. 2008, 35, 1826–1841. [CrossRef] 56. Sakakiyama, T.; Kajima, R. Numerical simulation of nonlinear wave interacting with permeable breakwater. In Proceedings of the 23rd International Conference on Coastal Engineering, Venice, Italy, 4–9 October 1992; pp. 1517–1530. 57. Ergun, S. Fluid flow through packed columns. Chem. Eng. 1952, 48, 89–94. 58. van Gent, M.R.A. Wave Interaction with Permeable Coastal Structures. Ph.D. Thesis, Delft University, Delft, The Netherlands, 1995; p. 175. 59. Liu, S.; Masliyah, J.H. Non-linear flows porous media. J. Non-Newton. Fluid Mech. 1999, 86, 229–252. [CrossRef] 60. Gill, A.E. Atmosphere-Ocean Dynamics; Academic Press: New York, NY, USA, 1982; p. 662. 61. Riley, J.P.; Skirrow, G. Chemical Oceanography 3; Academic Press: London, UK, 1965. 62. Germano, M.; Piomelli, U.; Moin, P.; Cabot, W.H. A dynamic subgrid-scale eddy viscosity model. Phys. Fluids 1991, 3, 1760–1765. [CrossRef] 63. Lilly, D.K. A proposed modification of the Germano subgrid-scale closure method. Phys. Fluids 1991, 4, 633–635. [CrossRef] 64. Brorsen, M.; Larsen, J. Source generation of nonlinear gravity wave with boundary integral equation method. Coast. Eng. 1987, 11, 93–113. [CrossRef] 65. Ohyama, T.; Nadaoka, K. Modeling the transformation of nonlinear waves passing over a submerged dike. In Proceedings of the 23rd International Conference on Coastal Engineering, Venice, Italy, 4–9 October 1992; pp. 526–539. 66. Hirt, C.W.; Nichols, B.D. Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 1981, 39, 201–225. [CrossRef] 67. Mellor, G.L.; Yamada, M. Development of a turbulence closure model for geophysical fluid problems. Rev. Geophys. 1982, 20, 851–875. [CrossRef] 68. Gregg, M.C.; D’Asaro, E.A.; Shay, T.J.; Larson, N. Observations of persistent mixing and near-inertial internal waves. J. Phys. Oceanogr. 1986, 16, 856–885. [CrossRef] 69. Peters, F.; Gregg, M.C.; Toole, J.M. On the parameterization of equatorial turbulence. J. Geophys. Res. 1988, 93, 1199–1218. [CrossRef] 70. Hinatsu, M. Numerical simulation of unsteady viscous nonlinear waves using moving grid system fitted on a free surface. J. Kansai Soc. Nav. Archit. 1992, 217, 1–11. 71. Petit, H.A.H.; Tonjes, P.; ven Gent, M.R.A.; van den Bosch, P. Numerical simulation and validation of plunging breakers using a 2D Navier–Stokes model. In Proceedings of the 24th International Conference on Coastal Engineering, Kobe, Japan, 23–28 October 1994; pp. 511–524. 72. Iwasaki, T.; Sato, M. Energy damping of wave propagating against currents (2). Proc. Coast. Eng. Conf. Jpn. Soc. Civ. Eng. 1971, 18, 55–59. (In Japanese) 73. Huppert, H.E.; Simpson, J.E. The slumping of gravity currents. J. Fluid Mech. 1980, 99, 785–799. [CrossRef] 74. Sakai, S.; Saeki, H. Effects of opposing current on wave transformation on sloping sea bed. In Proceedings of the 19th International Conference on Coastal Engineering, Houston, TX, USA, 3–7 September 1984; pp. 1219–1232. Water 2020, 12, 183 30 of 30

75. Christensen, E.D. Large eddy simulation of spilling and plunging breakers. Coast. Eng. 2006, 53, 463–485. [CrossRef] 76. Healy, J.J. Wave damping effect of beaches. In Proceedings, Minnesota International Hydraulic; St. Anthony Falls Hydraulic Laboratory: Minneapolis, MN, USA, 1952; pp. 213–220.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).