2.1 Drag and Inertial Resistances

Wave energy dissipation in porous media occurs dominantly via the turbulence of high speed waters and also by the friction of waters through porous media. These are called the turbulent and laminar drag resistances, respectively. The energy dissipation also occurs due to unsteadiness of water waves in porous media so-called inertial resistance. The momentum equation in porous media can be given as

where λ is the porosity, U = (u, v, w) is the seepage velocity vector, p is the pore pressure, ρ is water density, g is the gravitational acceleration, D is the drag resistance term, and I is the inertial resistance term, and ∇₃≡(∂/∂x, ∂/∂y, ∂/∂z) is the gradient operator. There are several ways to define the drag resistance term. Ergun (1952) defined the drag resistance term using a volume-averaged discharge velocity U'(= λU). In the presents study, we use Ergun’s definition of D in terms of the seepage velocity U instead:

where α_l and αt are coefficients representing the laminar and turbulent flow resistances, respectively, ν is the kinematic viscosity of water, and d is the size of the solid material. Engelund (1953) also used the Forchheimer type to define the drag resistance term:

where α_lE and α_tE are coefficients which represent the laminar and turbulent flow resistances, respectively, recommended by Engelund. The inertial resistance term I in Eq. (1) is given by

where κ is the added mass coefficient.

Substitution of Eqs. (2) and (4) into the momentum equation (1) gives

where the inertial resistance coefficient β is

and the drag resistance coefficient α is

We use the CADMAS-SURF (CDIT, 2001) for the Navier-Stokes equations model. The CADMAS-SURF was developed by the Coastal Development Institute of Technology, Japan. In the CADMAS-SURF, the momentum equation can be expressed as

where β_c = λ + (1 − λ)(1 + κ) and α = α_lE[(1 − λ)³/λ²]ν/d² + α_tE[(1 − λ)/λ]|U|/d are the inertial and drag resistance coefficients (Engelund, 1953), respectively, α_lE and α_tE are the laminar and turbulent drag resistance coefficients, respectively, suggested by Engelund. The inertial and drag resistance coefficients between eqs. (5) and (8) are related as

2.2 Governing Equations

The extended Boussinesq equations for waves in one porous layer derived by Vu et al. (2018) are given by

where u(= (u, v)) is the depth-averaged horizontal seepage velocity vector, γ(= 1/18) is a parameter to improve the dispersion relation in deeper water. For the water layer (i.e., β = 1 and α = 0), the momentum equation (11) becomes Madsen and Sørensen’s (1992) equation given by

The extended Boussinesq equations for waves in two porous layers derived by Huynh et al. (2017) are given by

where γ1(= 1/18), γ2(= 1/18) are parameters to improve the dispersion relation in deeper waters, the subscripts 1 and 2 imply the associated variables are in the upper and lower layers, respectively. If β₁ = ₁, α₁ = 0, λ1 = 1, that is, the upper layer is filled with water, then Eqs. (13) to (15) become Cruz et al.’s (1997) Boussinesq equations. Cruz et al.’s equations can be applied for waves on porous beds and submerged breakwaters. The present equations can be applied further for waves in two porous layers with different porosities.

The continuity and momentum equations of the CADMAS-SURF (CDIT, 2001) can be expressed as

where S_ρ and Su are source terms in the continuity and momentum equations, respectively. At the free surface, the transport equation of F is used as

where F is the volume of fluid and S_F is the source term of F.

3.1 Numerical Schemes

We use the finite-difference method to solve the extended Boussinesq equations (10) and (11) for one porous layer and the equations (13)~(15) for two porous layers. The time derivative terms are discretized with the Adam-Bashforth-Moulton predictor and corrector scheme following the FUNWAVE (Kirby et al., 1998). In the present study, we simulate wave propagation in horizontally one-dimensional case.

For waves in one porous layer, Eqs. (10) and (11) can be rewritten as

where

The subscripts in Eqs. (19)~(23) imply that the term is taken derivative with respect to the subscript. First, to discretize Eqs. (19) and (20) in time, we use the third-order Adams-Bashforth predictor scheme given by

where the superscripts n and n + 1 denote the present and the next time steps, respectively. The variable uⁿ⁺¹ which is included in Uⁿ⁺¹ is calculated using the LU decomposition method. Second, to discretize Eqs. (19) and (20) in time, we use the fourth-order Adams-Moulton corrector scheme given by

For waves in two porous layers, Eqs. (13)~(15) can be rewritten as

where

First, to discretize Eqs. (28)~(30) in time, we use the third-order Adams-Bashforth predictor scheme given by

The variables u₁ⁿ⁺¹ and u2n+2 which are included in both u₁ⁿ⁺¹(u1, u2) and U₂ⁿ⁺¹(u₁, u₂) are calculated using the LU decomposition method. Second, to discretize Eqs. (28) - (30) in time, we use the fourth-order Adams-Moulton corrector scheme given by

3.2 Determination of Optimal Values of Drag and Inertial Resistance Coefficients

To simulate wave propagation and energy dissipation in porous media, we need to determine values of inertial and drag resistance coefficients. In the inertial resistance coefficient β given by Eq. (6), we use the value of added mass coefficient as κ = 0.34 which was suggested by Lara et al. (2012) for waves in a porous breakwater. In simulating the Boussinesq equations, Vu et al. (2018) found optimal values of the laminar and turbulent drag resistance coefficients α_l = 800 and αt = 3 which give minimum root-meansquared relative errors between the transmission coefficients by the Boussinesq equations and those of Vidal et al.’s (1988) hydraulic experimental data. In simulating the Boussinesq equations for waves in one and two porous layers, we use Vu et al.’s suggested values of the laminar and turbulent drag resistance coefficients.

In the present study, to simulate the CADMAS-SURF for waves in one and two porous layers, we find optimal values of the laminar and turbulent drag resistance coefficients α_lE and α_tE following Vu et al.'s approach. Vidal et al. conducted hydraulic experiments to measure transmission coefficients of solitary waves passing through a porous breakwater. The experimental conditions of Vidal et al. are given in Table 1. We calculate the root mean squared relative error Er of the transmission coefficients given by

Table 1.

Experimental conditions of Vidal et al. (1988) for solitary waves in a porous breakwater

where K_tCAD and K_{t exp} are the transmission coefficients obtained by the CADMAS-SURF and the Vidal et al.’s data, respectively, and N is the total number of the transmission coefficients in each case. We compare averaged values of Er for the 6 total cases and then choose optimal values of α_lE and α_tE which yield the smallest averaged error. Table 2 shows the root mean squared relative errors with the different drag resistance coefficients of α_lE = 1000, 2000, 3000 and α_tE = 0, 1, 2. Finally, the drag resistance coefficients of α_lE = 2000 and α_tE = 0.7 are found to yield the smallest averaged error.

Table 2.

Root mean squared relative errors of transmission coefficients of the CADMAS-SURF model against those of Vidal et al.’s experiment

Fig. 2 shows the variation of transmission coefficients with wave nonlinearity for models with α_l = 800, αt = 3 (Boussinesq equations), and α_lE = 2000 and α_tE = 0.7 (CADMAS-SURF). The CADMAS-SURF with α_lE = 2000 and α_tE = 0.7 as well as the Boussinesq equations with α_l = 800, αt = 3 yields close solutions to the hydraulic experimental data. Hence, optimal values of the drag resistance coefficients are determined as α_lE = 2000 and α_tE = 0.7 in the CADMAS-SURF.

Fig. 2.

Variation of transmission coefficients with nonlinearity for models with α_l = 800, α_t = 3 (Boussinesq equations) and α_lE = 2000 and α_tE = 0.7 (CADMAS-SURF).

We also investigate the energy dissipation coefficient for waves in a porous breakwater. The energy dissipation coefficient can be defined as

Fig. 3 shows the variation of energy dissipation coefficients with wave nonlinearity with the optimum values of the drag resistance coefficients. It is noticeable that, even without experimental data, numerical solutions of the energy dissipation coefficient are close to each other between the Boussinesq equations and the CADMAS-SURF. It is interesting that the energy dissipation coefficient decreases slightly with the increase of wave nonlinearity. Figs. 2 and 3 show that, as wave nonlinearity increases, the increase of the reflection coefficient is more significant than the decrease of the transmission coefficient and thus the energy dissipation coefficient would decrease.

Fig. 3.

Variation of energy dissipation coefficients with wave nonlinearity for models with α_l =800, α_t = 3 (Boussinesq equations) and α_lE = 2000 and α_tE = 0.7 (CADMAS-SURF).

3.3 Numerical Simulation of Overtopping of Cnoidal Waves on a Porous Breakwater

The surface elevation of cnoidal waves can be expressed as

where ζ_tr is the elevation of wave trough, H is the height of incident waves, Cn is the Jacobian elliptic function, K₁ and K2 are the complete elliptic integrals of the first and second kind, respectively, and m is the modulus to determine the wave shape.

For cnoidal wave overtopping, we get approximate solutions of the one-layer and two-layer Boussinesq equations and compare these with more accurate solutions of the CADMAS-SURF. Experimental conditions are water depth of h = 30 cm, breakwater width of b = 20 cm, porous material size of d = 1.43 cm, porosity of λ = 0.44. We simulate the wave heights passing through the breakwater for cnoidal waves in three cases such as high-crested breakwater with h_c = 60 cm, low-crested breakwater with h_c = 1 cm, and submerged breakwater with h_c = −7 cm (see Fig. 1). We get approximate solutions of wave overtopping for cnoidal waves with different conditions of nonlinearity a/h = 0.033, 0.067, 0.133 and breakwater width of b = 5 cm, 10 cm, 20 cm, 30 cm.

Fig. 4 shows simulated water surface elevations at t = 7 sec for the high-crested breakwater (h_c = 60 cm), the lowcrested breakwater (h_c = 1 cm), and the submerged breakwater (h_c = −7 cm) using the one-layer Boussinesq equations, the CADMAS-SURF, and the two-layer Boussinesq equations, respectively. The wave nonlinearity is a/h = 0.033 and the breakwater width is b = 5 cm. Wave overtopping and reflection have already happened at t = 7 s. High reflection and low transmission are seen for waves on the high-crested breakwater while low reflection and high transmission are seen for waves on the submerged breakwater.

Fig. 4.

Surface elevations of cnoidal waves overtopping on a porous breakwater at t =7s (a/h = 0.033, b = 5 cm): (a) high-crested breakwater, (b) low-crested breakwater, (c) submerged breakwater.

Fig. 5 compares the transmission coefficients of cnoidal waves with different nonlinearities among the one- and two-layer Boussinesq equations and the CADMAS-SURF. The transmission coefficients of waves overtopping a lowcrested breakwater (obtained by the CADMAS-SURF) are greater than those of waves passing through a high-crested breakwater (obtained by the one layer Boussinesq equations) and less than those of waves passing through a submerged breakwater (obtained by the two layer Boussinesq equations). For waves with lower wave nonlinearity, overtopping waves can be predicted using the one-layer more accurately than two-layer Boussinesq equations.

Fig. 5.

Variation of transmission coefficient with nonlinearity for cnoidal waves.

Fig. 6 compares the reflection coefficients of cnoidal waves among the one-layer and two-layer Boussinesq equations and the CADMAS-SURF. The reflection coefficients of waves overtopping a low-crested breakwater are less than those of waves passing through a high-crested breakwater and greater than those of waves passing through a submerged breakwater.

Fig. 6.

Variation of reflection coefficient with nonlinearity for cnoidal waves.

Fig. 7 compares the energy dissipation coefficients of cnoidal waves among the one-layer and two-layer Boussinesq equations and the CADMAS-SURF. The energy dissipation coefficients increase with the increase of wave nonlinearity. The energy dissipation coefficients from the CADMASSURF are close to the one-layer Boussinesq equations. However, the energy dissipation coefficients by the two layer Boussinesq equations are smaller than those by the one-layer Boussinesq equations and the CADMAS-SURF.

Fig. 7.

Variation of energy dissipation coefficient with nonlinearity for cnoidal waves.

We also compare approximate solutions of cnoidal wave overtopping with different breakwater widths (b = 5 cm, 10 cm, 20 cm and 30 cm). For this numerical experiment, we keep initial conditions as the same as Vidal et al.’s experiment with nonlinearity a/h = 0.064. Fig. 8 shows that the transmission coefficients of waves overtoppinga low-crested breakwater (obtained by the CADMAS-SURF) are greater than those of waves passing through a high-crested breakwater (obtained by the one layer Boussinesq equations) and less than those of waves passing through a submerged breakwater (obtained by the two layer Boussinesq equations). Fig. 9 shows that the reflection coefficients of waves overtopping are slightly less than those of waves passing through a high-crest breakwater and higher than those of waves passing through a submerged breakwater.

Fig. 8.

Variation of transmission coefficient with breakwater width for cnoidal waves.

Fig. 9.

Variation of reflection coefficient with breakwater width for cnoidal waves.

Fig. 10 compares the energy dissipation coefficients among the one-layer and two-layer Boussinesq equations and the CADMAS-SURF. As the breakwater width becomes narrower, the energy dissipation would be negligibly smaller and thus the transmission and reflection coefficients would be near to unity and zero, respectively.

Fig. 10.

Variation of energy dissipation coefficient with breakwater width for cnoidal waves.