Current effects on nonlinear wave scattering by a submerged plate Lin, Hong Xing; Ning, De Zhi; Zou, Qing-Ping; Teng, Bin; Chen, Li Fen

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1 Heriot-Watt University Heriot-Watt University Research Gateway Current effects on nonlinear wave scattering by a submerged plate Lin, Hong Xing; Ning, De Zhi; Zou, Qing-Ping; Teng, Bin; Chen, Li Fen Published in: Journal of Waterway, Port, Coastal and Ocean Engineering DOI:.1/(ASCE)WW Publication date: 01 Document Version Peer reviewed version Link to publication in Heriot-Watt University Research Portal Citation for published version (APA): Lin, H. X., Ning, D. Z., Zou, Q. P., Teng, B., & Chen, L. F. (01). Current effects on nonlinear wave scattering by a submerged plate. Journal of Waterway, Port, Coastal and Ocean Engineering, (), [001]. DOI:.1/(ASCE)WW General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

2 Current effects on nonlinear wave scattering by a submerged plate Hong-Xing Lin 1, De-Zhi Ning, Qing-Ping Zou, Bin Teng, Li-Fen Chen Abstract: Based on a time-domain higher-order boundary element method (HOBEM), a two-dimensional (D) fully nonlinear numerical wave flume (NWF) is developed to investigate the nonlinear interactions between a regular wave and a submerged horizontal plate in the presence of uniform currents. A two-point method is used to discriminate bound (i.e., nonlinearly forced by and coupled to free waves) and free harmonic waves propagating upstream and downstream from the structure. The proposed model is verified against experimental and other numerical data for wave-current interaction without obstacles and nonlinear wave scattering by a submerged plate in the absence of currents. A first-order analysis shows that the reflection coefficient increases in the following current (i.e., current in the same direction as the incident wave) and decreases in the opposing current (i.e., current in the opposite direction as the incident wave). Moreover, the plate length for the maximum reflection to occur is not sensitive to the current. A second-order analysis indicates that downstream from the plate, the current has a stronger influence on the secondary free mode than on the first free mode. The energy transfer between the fundamental wave and the higher harmonics is intensified by a following current but weakened by an opposing current. The second free harmonic wave amplitude is affected more by the opposing current than the following current. Author keywords: Submerged plate; Wave-current interaction; Bound wave; HOBEM; Fully nonlinear numerical wave flume; Wave scattering. Introduction The submerged plate, used as a breakwater device, is less dependent on the bottom topography, more economical and can assure open scenic views. It allows seawater to exchange freely between the sheltered region and the open sea to prevent stagnation, pollution, transport of sediment to maintain the general partition of the natural seabed. It has been applied as an efficient breakwater in coastal and offshore zones (Guevel et al., 1, Graw, 1, Rahman et al., 00, Peng et al., 01). The submerged plate breakwater can be supported by a solid stem fixed on the bottom, mooring chains fastened to the seabed, tethered float with ballasts, etc. It reduces the wave height without blocking the downstream flow region, preventing destruction of the breakwater by large waves, and beach or waterfront erosion. Also, because there is a pulsating flow opposite to the wave propagation direction below the plate, which can drive a rotating water turbine, the submerged plate can be used as a nearshore wave energy converter 1 Postgraduate Student, State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, China. sunwind0@1.com Associate Professor, State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, China. dzning@dlut.edu.cn (Corresponding author). Assistant Professor, Department of Civil and Environmental Engineering, University of Maine Orono, ME, USA. qingping.zou@maine.edu Professor, State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, China. bteng@dlut.edu.cn Postgraduate Student, State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, China. chenlifen@1.com 1

3 (Cater, 00, Carter et al., 00, Orer and Ozdamar, 00). During wave transformation over a submerged plate, higher bound and free harmonic waves generated by a nonlinear shoaling effect may affect sailing conditions. Therefore, it is of practical importance to account for effects of the plate on wave transformation, especially when waves and currents are both present. Extensive research has been performed using linear wave theory on wave scattering by a submerged plate. For example, based on the long wave approximation, Siew and Hurley (1) derived the reflection and transmission coefficients using the method of matched asymptotic expansions. Using the method of Siew and Hurley (1), Patarapanich (1) found that the reflection coefficient oscillates with the ratio of the plate length and the wavelength, due to the energy flux across different regions around the plate. Subsequently, Patarapanich and Cheong (1) investigated, both experimentally and numerically, the reflection and transmission of regular and irregular waves by a submerged plate. They derived the occuring conditions for the minimum transmission of the waves over a submerged plate. Recently there are some investigations on nonlinear scattering by a submerged horizontal plate. Brossard and Chagdali (001) performed experiments to study higher harmonics generation by waves over a submerged plate. One or two moving probes were adopted to separate bound and free harmonic waves based on the Doppler shift. It was found that small submergences enhance the transfer of energy from the fundamental mode to higher harmonic modes. Later, Brossard et al. (00) further analyzed nonlinear wave scattering by a submerged plate to study the resonant behavior of the wave over the plate. Liu et al. (00) numerically and experimentally studied the nonlinear interactions between non-breaking waves and a submerged horizontal plate. Their numerical results, predicted by the desingularized boundary integral equation method (DBIEM), are in good agreement with the experimental data. Current effects on wave scattering by a submerged plate, however, were not investigated in these studies. In coastal zones, waves and currents generally coexist and their interaction plays an important role in ocean dynamic processes. Rey et al. (00) experimentally investigated the effects of currents on wave diffraction at a submerged plate. They found that currents have a little effect on the locations of maxima and minima of the reflection coefficient. Chen et al. (1) used an enhanced Boussinesq model to study nonlinear interactions of shallow-water waves in the presence of currents. Their study shows that the ratio of the energy in higher-order modes to that in the fundamental mode is weakened by an opposing current, but enhanced by a following current. However, the effect of currents on higher bound and free waves has not been taken into account by these works. Vortices generated at the trailing edge of the obstacle have received more attention lately. The experimental results of Ting and Kim (1) show that viscous effects are significant on the transmission side but negligible on the incident side. The flow separation has little effect on the free surface (Huang and Dong, 1). Beji and Battjes (1) conducted experiments of waves propagating over a bar and found that wave breaking has little effect on the generation of higher harmonics and mainly contribute to wave energy dissipation. Furthermore, higher harmonics generated over a submerged plate can be predicted well by a numerical model based on potential theory (Liu et al., 00). In the presence of currents, Grue et al. (1) analytically studied the propulsion of a foil moving in water, assuming inviscid fluid except for the vortex wake region. Zaman et al. (00) proposed a numerical model using two empirical equations to account for the effects of the flow separation for waves propagating over a bottom-mounted obstacle in the presence of currents. Their model satisfactorily predicted the observed wave heights and the drop of the mean water levels in their experiments. Moreover, their model resulted in good agreement with the experimental data without invoking two empirical equations, when the current velocity was less than the critical velocity. More recently, Rey and Touboul (0) experimentally investigated the interactions between a submerged

4 plate and regular and irregular waves in the presence of currents. Their experimental results compared well with their extended analytical potential flow model. Therefore, potential flow theory is suitable for the wave-current-body interaction problems when the current velocity is small compared with the wave velocity, i.e., up to 1% of the incident wave velocity (Koo and Kim, 00). In the present numerical examples, the current velocity is kept small; e.g., less than 1% of the current-free incident wave velocity, so that the flow separation can be neglected in the numerical simulations. In this study, nonlinear wave-current-body interactions are investigated utilizing the proposed nonlinear higher-order boundary element method (HOBEM). The primary objective of this work is to investigate the effects of the currents on the fundamental and the second harmonic waves, and the effects of the submergence depth and the length of the plate on the nonlinear wave scattering in the presence of a uniform current. This paper is organized as follows. Section briefly describes the proposed HOBEM model, and the method of separating bound and free wave components in the travelling waves. In Section, the present model is verified against the available data, and the results of wave decomposition are presented and discussed. Finally, the conclusions are presented in Section. Mathematical formulations Fig. 1 Schematic diagram of a two-dimensional numerical wave-current flume The influence of waves and a uniform current on a submerged plate in a two-dimensional fluid domain is shown in Fig. 1. The flow is assumed to be inviscid and incompressible with irrotational flow, such that there exists a velocity potential in the fluid domain. A Cartesian coordinate system oxz is chosen. Its origin is on the still water level at the left end of the domain, and the z-axis is positive upward. As shown in Fig. 1, h denotes the static water depth, h s the submergence, B the plate breadth, and W the plate thickness. The total velocity potential can then be expressed as Ux ( x, z, t), where U is the steady uniform current velocity whose strength varies with the local flume depth as required by the mass conservation and ( x, z, t) is the perturbation potential due to waves. The Laplace equation is the governing equation for this boundary value problem, which is satisfied by both Φ and in the computational domain Ω. Given the boundary conditions, the unknown velocity potential on the impermeable surface and unknown normal velocity on the free surface can be determined by solving the following boundary integral equation based on the Green s second identity (Brebbia and Walker, ; Ligget and Liu,

5 ): G( q, p) ( q) ( p) ( p) ( ( q) G( q, p) ) d, p (1) n n where represents the entire computational boundary, p and q are the source point (x 0, z 0 ) and the field point (x, z), respectively, and α is the solid angle coefficient determined by the surface geometry of a source point position. G is a simple Green function, considering the image of the Rankine source about the seabed, and can be written as G( p, q) (ln r1 ln r) /, where and r ( x x ) ( z z h). 0 0 r ( x x ) ( z z ) Mixed initial and boundary conditions On the instantaneous free surface, both the fully nonlinear kinematic and dynamic boundary conditions are satisfied and a mixed Eulerian-Lagrangian method is used to describe the time-dependent free surface with moving nodes. A damping layer at the end of the numerical flume is added to gradually absorb the wave energy in the direction of the wave propagation. At the frontal damping zone, i.e., damping zone 1, the damping scheme is designed to damp only the waves reflected from the obstacle, while preserving the original incident waves. Extra damping terms are added to both the kinematic and dynamic free-surface boundary conditions. Therefore, the free-surface boundary conditions can be written as follows: Dx U on F () Dt x D v1( x)( e) v( x) on F () Dt z D 1 g v1( x)( e ) v( x) on F () Dt where e and e are the reference values in the absence of the obstacle under the same computational conditions, which can be determined by the second-order analytical solution derived by Baddour and Song (). Damping parameters v ( x ) and v ( ) 1 x are given as follows: v i x = a dω x x i L b (x<x 1 for i=1;x>x for i=.) 0 otherwise () where x 1 and x are the starting positions of damping zones 1 and, respectively, d is the damping coefficient, L b is the length of the damping zone, and ω is the fundamental wave angular frequency. In this study, =1.0 and L b is twice the incident wavelength (see Tanizawa, 1). d The boundaries at the tank bottom and the body, B, are considered impermeable. At the right end of the wave flume, i.e., the outlet boundary, O, the no-flux condition is imposed for the unsteady velocity potential. Therefore, the normal velocities at these boundaries are as follows: 0 on B and O () n For the inlet boundary, I, a designated fluid particle velocity is applied as a feeding function as follows:

6 I I on I () n n x gae cosh k( z h) cosh k( z h) I sin( kx t) Ae( ku ) sin ( kx t) () ku cosh kh sinh kh where I is the incident velocity potential, g is the gravitational acceleration, ω is the angular frequency, k is the wave number, and A e is the current-affected wave amplitude. Based on the conservation of wave action (Bretherton and Garrett, 1), A e satisfies the following relation: ku Cg 0 Ae A0 () C where C g is the group velocity and A 0 and C g0 denote the current-free wave amplitude and group velocity, respectively. The wave number is determined by the modified dispersion relation as follows: g ku gk tanh kh () To solve the above boundary value problem in the time domain, initial conditions are required as follows: t0 t0 0 () Numerical solution In the present study, the boundary surface is discretized using three-node line elements. The geometry of each element is represented by quadratic shape functions, thus the entire curved boundary can be approximated by higher-order elements. Within the boundary elements, physical variables are also interpolated by the same shape functions, i.e., the elements are isoparametric. Then boundary integral equation Eq. (1) can be discretized as follows (Ning and Teng, 00): N 1 N G( p, q( )) 1 ( q( )) ( p) ( p) h ( ) ( ) (, ( )) ( ) 1 i ji J d G p q J d 1 j 1 i 1 n (1) j1 n where N is the number of the discretized elements on the whole boundary, denotes the node number per element, ξ is the local intrinsic coordinate, J(ξ) is the Jacobian matrix relating the global coordinates to the local intrinsic coordinates within the corresponding element, and hi ( ) denotes the shape function. The solid angle coefficient α is computed using an indirect method, i.e., the constant potential method (Ning et al., 0). The integrals are calculated using a four-point Gauss quadrature method. The derivatives of the velocity potential on the free surface are contained in the free-surface boundary conditions, Eqs. () and (). As the higher-order boundary element is used, they can be obtained from the following equation: 1 x z x (1) nx n z z n where n=(n x, n y ) is the unit normal vector. Finally, the entire set of equations can be expressed in matrix form, and the unknowns can be moved to the left-hand side as follows: 1 X A F (1) where X is the vector of unknown potential and normal velocity, A is the influence matrix and F is the vector obtained from integration in terms of the known potential and velocity on the boundary. To solve

7 the resulting matrix Eq. (1), the preconditioned Generalized Conjugate Residual (GCR) (Saad and Schultz, 1) is used. In the GCR, the iteration is stopped when the module of the relative residual is smaller than -. Because the initial calm boundary conditions (i.e., t0 t0 0 ) are given on the free surface, including two damping layers, the free-surface boundary conditions in Eqs. ()-(), considered as the ordinary differential equations for, x and η, are obtained by solving Eq. (1) and advanced in time using the fourth-order Runge-Kutta (RK) scheme for the new variables (, x and η) at the next time- step (Ning and Teng, 00). Then the fluid domain is remeshed and enters a new computational loop. Decomposition of harmonic waves As waves propagate over a submerged plate, the shoaling effect will induce higher harmonic waves, which are released as free waves when they enter the deep water. The two-point method of Grue (1) is modified to discriminate bound and free higher harmonic waves upstream and downstream from the structure in the presence of a uniform current. The free-surface elevation at a point x at the lee side of the obstacle can be expressed as follows: (1) (1) (1) ( n) (1) ( n) TF TF TB TB n ( n) ( n) ( n) atf cos[ k x nt) TF ] n ( x, t) a cos( k x t ) a cos[ n( k x t) ] where a is the amplitude and the subscripts TB and TF denote transmitted bound and transmitted free waves, respectively. The superscript n denotes the nth harmonic wave, ω denotes the fundamental angular frequency, k (1) and k (n) are the wave numbers for the first and the nth free harmonics, respectively, and φ is the wave phase. The wave numbers for free waves can be determined by the following modified dispersion equation: ( ) ( ) ( ) n k n U gk n tanh k n h (1) n = 1,, (1) The Fourier transform is introduced as follows: ( n) ˆ ( x) ( x, t)exp( int ) dt n = 1,, (1) 0 The Fourier transform is applied to the time series of the wave elevation of two points (i.e., x 1 and x, where x =x 1 +Δx) downstream from the structure, then the amplitudes of the free and bound harmonics are given by: ( n) ( n) (1) ˆ ( x ˆ ( ) 1) ( x1 x)exp( ink x) n af n =,, (1) ( n) (1) sin(( k nk ) x / ) a ( n) B ˆ ( x ) ˆ ( x x)exp( ik x) ( n) ( n) ( n) 1 1 ( n) (1) sin(( k nk ) x / ) n =,, (1) For the reflected bound and free waves, the amplitudes are determined by the wave records of two points upstream from the structure in a similar way. The wave numbers of reflected wave, k (n) R, can be

8 determined by the following modified dispersion equation (see Grue (1) for details) n k U gk tanh k h n = 1,, (0) ( ) ( ) ( ) R R R The reflection and transmission coefficients of the fundamental wave are defined as follows: (1) (1) ar atf R ; T (1) a (1) (1) I ai where a (1) R and a (1) I are the reflected and incident fundamental wave amplitudes, respectively. Results and discussion Model Validation Waves in a flat-bottom flume with a uniform current The model is used to simulate nonlinear wave-current interactions in a flat-bottom flume and compared with the experimental results. The experiments were conducted in a wave flume located in the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, PR China. The glass-walled wave flume is m long, m wide and 1. m deep. The flume is equipped with a hydraulic piston-type wavemaker at one end, and a slope beach is deployed to diminish wave reflection at the other end. In the experiments, the wave period T= s, the still water depth h =0. m, and the current velocity U=0, ±0. m/s are used. Initial numerical tests were carried out to assess the convergence of the adaptive spatial and temporal discretization for the case with current-free incident wave amplitude A 0 =0.0 m and U=-0. m/s. Fig. depicts the free-surface time history at the distance x=0. m from the incident boundary, obtained using a fine mesh (0 meshes per wave length, i.e., x=λ/0); an intermediate mesh (0 meshes per wave length, i.e., x=λ/0) and a coarse mesh (0 meshes per wave length, i.e., x=λ/0). In all cases, meshes were used across the depth of the numerical wave flume. The results obtained using the fine and intermediate meshes are identical, indicating that mesh convergence was achieved using the intermediate mesh. Similar tests for three time steps (T/0, T/0 and T/0) using the intermediate mesh indicate that t=t/0 is sufficient. Fig. shows the time series of the nonlinear free-surface wave elevation at the distance x=0. m from the initial wavemaker position in zero, following and opposing- currents. Overall, the model results are in good agreement with the experimental data. The wave height in the opposing current is larger than that in the following current. Fig. shows the distribution of the crest elevation, η crest, with different current-free incident wave amplitudes (A 0 ) in the presence of zero, following and opposing- currents. The comparisons of the experimental data, linear analytical solution and proposed model results are also shown. The figure illustrates that both numerical and experimental results agree well with each other, but they deviate from the linear analytical solutions as A 0 increases. This means that the linear analytical solution under-predicts the real values when strong nonlinearity is present. Fig. Comparison of wave elevations at the distance x=0. m from the initial wavemaker position.

9 Fig. Time series of nonlinear free-surface wave elevation for the following, opposing and zero- currents with A 0 =0.0 m, x=0. m: (a) U=0. m/s, (b) U=-0. m/s, (c) U=0.0 m/s.

10 Fig. Comparisons of crest elevations from experimental data, linear analytical solutions and model results for different A 0 : (a) U=0. m/s, (b) U=-0. m/s, (c) U=0.0 m/s. Wave scattering by a submerged plate without a current To further validate the present model, the developed numerical wave flume (NWF) is then used to study waves propagating over a submerged plate in the absence of a current, as shown in Fig. 1. In this case, the current-free incident wave amplitude is A 0 =0.01 m, the still water depth h=0. m, the length of the plate B=0. m, the thickness of the plate W=0.01 m, and the submergence of the plate h s =0. m. As before, convergence tests were performed to determine the optimal mesh size on the plate surfaces as x=b/0 and z=w/. Fig. shows snapshots of the wave profiles at t=0t and T, in which x=0

11 indicates the initial wavemaker position and the vertical solid line denotes the starting position of damping zone. There is a good match between the two wave profiles, indicating that numerical stability has been achieved. The initially periodic and regular wave profiles at the weather side of the plate become distorted downstream from the plate, which means that a number of short waves are introduced into the flow after the flow passes the plate. Additionally, the transmitted wave energy is completely absorbed in damping zone, indicating the effectiveness of the damping scheme Fig. Snapshots of wave elevation at t=0t and T. The vertical solid line denotes the starting position of damping zone. Fig. shows the comparisons of the reflection and transmission coefficients of the fundamental wave predicted by the present model with the experimental data of Brossard et al. (00) and the numerical results of Liu et al. (00) for a range of B/L s (where L s denotes wavelength above the plate). The figure shows good agreement for the reflection coefficient. Both the numerical models slightly overestimate the observed transmission coefficient. However, the variation of the transmission coefficient with B/L s is captured well by the models. The present model is based on potential theory, thus does not account for viscous effects or flow separation at the trailing edge of the plate, and the resulting energy dissipation. This results in an overestimation of the transmission coefficient compared with the experimental data. The results indicate that the viscous effects are prominent only in the transmission process, and have little effect on the reflection process (Ting and Kim, 1). The predicted second free harmonic wave amplitude downstream from the plate is compared with the experimental data of Brossard et al. (00) and the numerical results of Liu et al. (00) vs. B/L s in Fig.. The data show that the second free harmonic wave amplitudes are well-predicted by the present model, even though the model neglects viscous effects. This suggests that energy dissipation occurs primarily in the fundamental waves. Another related case is carried out by comparison with the experimental and numerical results of Liu et al. (00) to test the robustness of the present model. In this example, the current-free incident wave amplitude is A 0 =0.01 m, the still water depth h=0. m, the length of the plate B=0. m, the thickness of the plate W=0.01 m, and the current velocity U=0 m/s. Fig. shows that the predicted time series of the free-surface displacement downstream from the plate is in good agreement with the data collected by a probe at. m from the plate center ( Liu et al. 00). Fig. shows that the predicted reflection and transmission coefficients of the fundamental wave as function of B/L s compare well with the results of Liu et al. (00). The transmission coefficients are also slightly over-predicted by the numerical models for the same reasons stated above. As shown in Fig., the second bound and free harmonic wave amplitudes at the lee side of the plate show good agreement with those of Liu et al. (00). The second free harmonic wave amplitudes are in general much larger than the second bound harmonic wave amplitudes.

12 Fig. Comparisons of the (a) reflection and (b) transmission coefficients of the fundamental wave predicted by the present model with the experimental data of Brossard et al. (00) and the numerical results of Liu et al. (00). Fig. Comparisons of the second free harmonic wave amplitude downstream from the plate predicted by the present model with the experimental data of Brossard et al. (00) and the numerical results of Liu et al. (00).

13 Fig. Comparison of the simulated free-surface wave elevation downstream from the plate predicted by the present model with the experimental data of Liu et al. (00). The distance between the probe and the center of the plate is.0 m, T=1.1 s, h s =0.1 m. Fig. Comparisons of the (a) reflection and (b) transmission coefficients of the fundamental wave predicted by the present model with the experimental and numerical results of Liu et al. (00), h s =0. m. 1

14 Fig. Comparisons of the second bound and free harmonic wave amplitudes downstream from the plate predicted by the present model with the experimental and numerical results of Liu et al. (00), h s =0. m. By holding some input parameters (A 0 =0.01 m, h=0. m, i.e., H 0 /h=0.1) constant, and introducing a uniform current into the flow, the effects of wave period, length and submergence of the plate, and current velocity, on the reflection coefficient and the second free harmonic wave amplitude are investigated in the following sections. To illustrate the capability of the proposed model for nonlinear waves, the nonlinearity parameter ε=h e /h s and the Ursell number, U r =H e L s /h s (i.e., the ratio of the nonlinear parameter, H e /h s, and the dispersion parameter (h s /L s ) ), are defined, where H e =A e represents the incident current-affected wave height, h s the immersion depth of the plate, and L s is the wavelength above the plate obtained from Eq. () by replacing h with h s. Fundamental waves In this section, currents are imposed on the wave field to examine wave-current-body interactions with special attention to the reflection coefficient. Fig. shows the computed reflection coefficient of the fundamental wave vs. the wave period, T, with a plate length B=0. m and two immersion depths h s, for input parameters ε=(0.-0.) and U r =(.-.). The data in the figure show that the reflection coefficient increases to its maximum and then decreases to a smaller value with increasing wave period. Moreover, the reflection for different currents exhibit the same trend but the corresponding magnitude is larger for a following current than that for zero current, and vice versa for an opposing current. This behavior can be predicted from linear theory. When incident waves are against a current, the incident wave height will increase, and the reflected waves will be reduced by the same current, resulting in a reduced reflection coefficient. The maximum reflection coefficients for different currents correspond to almost the same wave period. This suggests that the current has a weak influence on the occurring condition of the maximum reflection coefficient, but affects the magnitude of reflection. This result is consistent with the experimental results of Rey et al. (00). This phenomenon demonstrates that the model configuration designed for a particular range of frequencies and zero current should also be adequate in the presence of currents. The data in Fig. (a) and (b) show that the reflection coefficient decreases with increasing immersion depth. 1

15 Fig. Computed reflection coefficient of the fundamental wave vs. wave period, T, with plate length B=0. m. C g0 is equal to the current-free group velocity at T=0. s: (a) h s =0.0 m, (b) h s =0. m. Fig. 1 shows the predicted reflection coefficient of the fundamental wave vs. plate length, B, with wave period T=0. s and submergence h s =0. m. The reflection coefficient oscillates between its maximum and minimum values as the plate length, B, changes. The total reflection comprises the reflections from the leading edge of the plate, the trailing edge of the plate and below the plate (Patarapanich, 1). When the reflections from the two edges of the plate are in phase at the leading edge, then the interference is constructive. Consequently, the fundamental wave is at resonance, and the reflection is at its maximum. If the two reflections are out of phase at the leading edge, the total reflection will decrease to zero. Therefore, the interference accounts for the oscillation of the reflection coefficient. The location of the maximum and minimum, shown in Fig. 1, presents similar characteristics as those in Fig.. Taking an opposing current as an example, the incident wavenumber is increased whereas the reflected wavenumber is decreased by the Doppler effects of current. Consequently, the phase difference at the leading edge between the reflected waves from the two edges of the plate is close to that in the case of zero- current. Also, the occurrence of the maximum and minimum is hardly affected by the current. Fig. 1 shows the computed reflection coefficient of the fundamental wave vs. the ratio of plate length, B, and wavelength, L s, with submergence h s =0. m in the absence of a current. The data show that the line of the reflection coefficient shifts upward as the wave period, T, increases. Also, the reflection 1

16 coefficient at resonance becomes larger as the wave period increases, which is shown more clearly in Fig. 1. The wave period at which the maximum reflection occurs in Fig. (a) is different from that at which the first-order resonance occurs (i.e., the phase difference between the primary reflected waves at the two edges of the plate being degree) in Fig. 1. This result is similar to as shown in Fig. of Brossard et al. (00). The deviation between these two wave periods, however, is not entirely due to damping as explained in Brossard et al. (00). For example, for wave period T=0. s (i.e., B/L s =0. in Fig. (a) corresponding to the maximum reflection), the reflection coefficient at resonance is 0., as shown in Fig. 1, which is larger than the maximum reflection, i.e., 0., in Fig. (a). Moreover, the length of the plate does not correspond to that of the resonant condition, which indicates that the reflections from the two edges of the plate are not in phase at the leading edge. The wave period at resonance in Fig. (a) (i.e., B=0. m, h s =0. m) is about 0. s obtained both from the experiment in Fig. of Brossard et al. (00) and the proposed numerical model. Therefore, the condition of the maximum reflection in Fig. (a) is not the same as that for resonance. The reflection coefficient of a longer wave not at resonance may be larger than that of a shorter wave at resonance. We believe this is the reason for the forementioned deviation between the wave period for maximum reflection and resonance. Fig. 1 presents the effects of the current velocity on the reflection coefficient with wave period T =0. s, plate length B=0. m and two immersion depths, h s, in which parameters ε=(0.1-0.) and U r =(-.). The data show that the reflection coefficients increase approximately linearly as the dimensionless current velocity increases. As the submergence, h s, increases, the reflection coefficient decreases. Fig. 1 Computed reflection coefficient of the fundamental wave vs. the plate length, B, with wave period T =0. s and submergence h s =0. m. 1

17 Fig. 1 Computed reflection coefficient of the fundamental wave vs. the ratio of the plate length, B, and the wavelength, L s, with submergence h s =0. m and current velocity U=0 m/s. Fig. 1 Computed reflection coefficient of the fundamental wave at resonance vs. the wave period, T, with submergence h s =0. m and current velocity U=0 m/s Fig. 1 Computed reflection coefficient of the fundamental wave vs. the ratio of current velocity, U, and the current-free group velocity, C g0, with wave period T =0. s and plate length B=0. m. Second free harmonics As waves pass over a submerged plate, the shoaling effect due to water depth reduction will introduce short waves into the flow. To understand the nonlinear deformations of waves in the presence of a current, the current effects on the second free modes are investigated. The computed second free harmonic wave amplitude downstream from the plate is plotted in Fig. 1 vs. the wave period, T, with plate length B=0. m and submergence h s =0.0 m and 0.1 m (the subscript 0 in the figure and hereafter denotes zero current). Fig. 1 shows that the maximum amplitude increases in the presence of an opposing current, and reduces in the presence of a following current, which may be because the incident fundamental wave amplitude is increased by the opposing current and decreased by the following current. The position of the peak value also varies with the current. This indicates that the current effects on the second free harmonic waves are more significant than on the fundamental waves. In addition, the second free harmonic wave amplitude is larger at a smaller submergence because of the stronger shoaling effect. The computed second free harmonic wave amplitude downstream from the plate is shown in Fig. 1(a) vs. the plate length, B, with wave period T=0. s and submergence h s =0. m. The amplitude exhibits an oscillation with the variation of plate length, B. The peak value increases with a following current and decreases with an opposing current, contrary to the feature shown in Fig. 1. This is because that the maximum energy transfer from the first-order mode to the higher-order modes for a following current is larger than that for an opposing current. That is, a following current can enhance the energy exchange 1

18 between the fundamental wave and the second free harmonic wave, and vice versa for an opposing current. The plate length for the maximum second free harmonic wave amplitudes vary with the currents, but the trends of second free harmonic wave amplitude vs. plate length for the following, zero and opposing currents are similar. This feature is obviously different from that of the reflection coefficient in Fig. 1. Fig. 1(b) illustrates that the ratio of plate length, B, and the second beat length, L () s (L () s =π/(k () -k (1) )), is a key parameter to characterize the secondary free mode. The data in Fig. 1(b) show that current effects on the second peak at a longer plate are more significant than on the first peak at a shorter plate. Similar to the resonance behavior of the reflected fundamental wave in Figs. 1 and 1, the phenomenon of the second free harmonic wave amplitude downstream from the plate obtaining its maximum with the variation of plate length, B, for a fixed wave period, is defined as second-order resonance in Fig. 1. This suggests that the peak value, in Fig., is not the value at second-order resonance. For example, from Fig. 1, for wave period T=0. s (i.e., B/L s =0. in Fig. corresponding to the maximum), the non-dimensional second free wave amplitude at second-order resonance is 0., which is larger than the peak value 0.1 in Fig.. From Fig. 1, the second free harmonic wave amplitude at second-order resonance increases with wave period, T. This is because the second free harmonic wave amplitude is dependent on the second beat length (L () s =π/(k () -k (1) )) for a given plate length, and the longer second beat length leads to larger energy exchange between the fundamental wave and higher harmonics (Chen et al., 1). Fig. 1 shows the distribution of the second free harmonic wave amplitude downstream from the plate vs. the ratio of current velocity, U, to current-free group velocity, C g0, with wave period T =0. s and plate length B=0. m. Compared with the zero current case, the second free harmonic wave amplitude attains a maximum at U/C g0 =-0.0 and 0 for an immersion depth hs=0.0 and 0.1 respectively and decays with the deviation of current velocity from these values. The amplitude increase as immersion depth decreases and is more sensitive to the opposing current than to the following current. Moreover, the current has a stronger impact on the secondary mode for smaller immersion depths. 1

19 Fig. 1 The computed second free harmonic wave amplitude downstream from the plate vs. wave period, T, with plate length B=0. m. C g0 is equal to the current-free group velocity at T=0. s: (a) h s =0.0 m, (b) h s =0. m. 1

20 Fig. 1 The computed second free harmonic wave amplitude downstream from the plate against (a) plate length, B, and (b) the ratio of plate length, B, and the second beat length, L () s, with wave period T =0. s and submergence h s =0. m. Fig. 1 The computed second free harmonic wave amplitude at second order resonance downstream from the plate vs. the wave period, T, with submergence h s =0. m and current velocity U=0 m/s. 1

21 Fig. 1 The computed second free harmonic wave amplitude downstream the plate vs. the ratio of the current velocity, U, and current-free group velocity, C g0, with wave period T =0. s and plate length B=0. m. Current effects on resonance The first-order and second-order resonances are investigated to further assess the effects of the current on wave harmonics. First, the current effects on the reflection coefficients and second free harmonic amplitude at resonance are examined. The resonance is obtained by varying the plate length for different current velocities. Fig. 0 shows that the relative reflection coefficient is proportional to the ratio of the current velocity to the current-free group velocity, U/C g0, and the currents remarkably influence the wave reflection. The plate length for the first-order resonance to occur maintains the same value for different currents. This confirms that the current has little effect on the condition to achieve maximum reflection, as shown in Fig. 1. In Fig. 1, the relative second free harmonic wave amplitude also increases with U/C g0, reaches a maximum, then decreases with U/C g0. The plate length corresponding to the second-order resonance increases with U/C g0, indicating that the second free harmonic wave is detuned by the current. Finally, chosing the plate length equal to that of second-order resonance for zero-current (i.e., B=0. m), the ratio of the second free harmonic wave amplitude downstream from the plate with a current to that without a current is shown in Fig.. Similar to Fig. 1, the relative second free harmonic wave amplitude also increases with U/C g0, reaches a maximum value close to unity, then decreases with U/C g0. The second free harmonic wave amplitude is reduced by the current. 1 Fig. 0 The ratio of the reflection coefficient of the fundamental wave with a current to that without a current and plate length vs. the ratio of the current velocity, U, and current-free group velocity, C g0, at the first-order resonance, with wave period T =0. s and submergence h s =0. m. 0

22 Fig. 1 The ratio of the second free harmonic wave amplitude downstream from the plate with a current to that without a current and plate length vs. the ratio of the current velocity, U, and current-free group velocity, C g0, at the second-order resonance, with wave period T =0. s and submergence h s =0. m Fig. The ratio of the second free harmonic wave amplitude downstream from the plate with a current to that without a current, with wave period T =0. s, plate length B=0. m and submergence h s =0. m. Conclusions A numerical wave flume (NWF) is developed to investigate the nonlinear transformation and scattering of waves over a submerged horizontal plate in the presence of a uniform current. The NWF is based on potential theory and a higher-order boundary element method (HOBEM) with the mixed Eulerian-Lagrangian approach to update the instantaneous free surface. As a time-marching scheme, the fourth-order Runge-Kutta method is used to update the time integration combined with remeshing at each time step. Bound and free harmonics in reflected and transmitted waves are decomposed using a two-point method. The results show that the reflection coefficient is enhanced by a following current and reduced by an opposing current. The wave interference causes the reflection coefficient to oscillate with changing plate length. Occurrence of the maximum reflection is independent of the current because the Doppler effects of a current on the incident and reflected waves cancel each other. This result indicates that a submerged plate designed for a particular range of frequencies remains effective in the presence of a current. The reflection coefficient increases approximately linearly as the current velocity increases. Additionally, at the resonance condition, the reflection coefficient increases as the wave period increases, and the reflection coefficient of a longer wave at non-resonance can be larger than that of a shorter wave at resonance. The maximum second free harmonic wave amplitude downstream from the plate for varying wave period is increased by the opposing current and decreased by the following current, which is the opposite 1

23 to that for varying plate length. This occurs because, for a fixed wave period, the longer second beat length in the following current increases the maximum energy transfer from the fundamental wave to the higher harmonics. The second free harmonic wave amplitude downstream from the plate oscillates with the plate length and is correlated with the ratio of the plate length to the second beat length. Unlike the reflection coefficient, the plate length for the maximum and the minimum second free wave amplitudes to occur downstream from the plate is shifted by the current. The second free harmonic wave amplitude downstream from the plate decrease with the magnitude of the current. The second free harmonic wave amplitude is more sensitive to the opposing current than the following current. The second-order analysis shows that at resonance, the growth of second free harmonic wave amplitude downstream from the plate with wave period, is strengthened by the longer second beat length. The resonant second free harmonic amplitude downstream from the plate also increases with the current velocity increase, to a maximum value. Current effects on the second free harmonic waves downstream from the plate are stronger than those on the fundamental waves. Current effects on the secondary mode are complicated and the mechanism of the second-order resonance remains unclear. The second free harmonic wave amplitude downstream from the plate is correlated with the ratio of the plate length and the second beat length, and more work will be done to provide insights into the second-order resonance. Acknowledgements The authors gratefully acknowledge financial support from the National Natural Science Foundation of China (Grant Nos. 1,, 1), the National Basic Research Program of China ( Program, Grant No. 0CB0) and the Fundamental Research Funds for the Central Universities (DUT1YQ). The third author would like to thank Maine Sea Grant, NSF grant 1 and the start-up fund by University of Maine.

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