A novel method for water waves propagating in the presence of vortical mean flows over variable bathymetry

Abstract

A coupled-mode model is developed for the wave–current–seabed interaction problem with application to wave propagation and scattering by non-homogeneous currents of general vertical structure in variable bathymetry regions. The formulation is based on a velocity representation defined by a series of local vertical modes containing the propagating and evanescent modes, able to analytically treat the continuity condition and the bottom boundary condition on sloping parts of the seabed. Using the above representation in Euler equations, a coupled system of differential equations on the horizontal plane is derived with respect to the unknown horizontal velocity modal amplitudes. Under the assumption of small-amplitude waves, the system is linearized, and the dispersion characteristics of the model are studied for arbitrary vertical distribution of the mean flow indicating very fast convergence of the modal series over the whole frequency band of interest. Furthermore, interesting cases of wave–seabed–current interactions are studied and numerical results are compared against available experimental data indicating a very good performance.

Appendices

Expansion of the momentum equations on the eigenfunctions basis

Starting from Eq. (21), we now introduce the expansions (7–8). Proceeding term by term, we obtain

$$\begin{aligned}&\frac{\partial u}{\partial t} = \sum _{n=0}^{\infty } Z_n^{(1)} (z;h,\eta _0) \frac{\partial u_n(x,t)}{\partial t}, \frac{\partial }{\partial x} \left( \int _z^{\eta _0} \frac{\partial w}{\partial t} \mathrm{d}\xi \right) \\&\quad = -\sum _{n=0}^{\infty } \left\{ \frac{\partial }{\partial x} \left[ \frac{\partial }{\partial x} \left( Z_n^{(3)} \frac{\partial u_n}{\partial t} \right) - Z_n^{(2)}(\eta _0) \frac{\partial \eta _0}{\partial x} \frac{\partial u_n}{\partial t} \right] \right\} , \\&\frac{\partial }{\partial x}\left[ U_0(\eta _0)\cdot u(\eta _0) \right] = \sum _{n=0}^{\infty }\frac{\partial }{\partial x} \left( U_0(\eta _0)\cdot u_n \right) , \\&\frac{\partial }{\partial x}\left[ W_0(\eta _0)\cdot w(\eta _0) \right] = - \sum _{n=0}^{\infty }\frac{\partial }{\partial x}\left[ W_0 \frac{\partial }{\partial x} \left( Z_n^{(2)}(\eta _0) u_n \right) \right] , \end{aligned}$$

where $Z_n^{(3)}$ is defined by

$$\begin{aligned} Z_n^{(3)}=\int _z^{\eta _0} Z_n^{(2)} (\xi ;h,\eta _0) \mathrm{d}\xi . \end{aligned}$$

(37)

The expressions of $\mathcal {E}_1$, $\mathcal {E}_2$, $\mathcal {F}_1$ and $\mathcal {F}_2$ involving the vorticity fields, we start by expressing these components on the basis. We obtain

$$\begin{aligned} \varvec{\omega }= & {} \left\{ \begin{aligned} \omega _1 \\ \omega _2 \\ \omega _3 \end{aligned} \right. = \left\{ \begin{aligned} 0\\&-(w_x-u_z) \\&0 \end{aligned} \right. \nonumber \\= & {} \left\{ \begin{aligned} 0\\&\sum _{n=0}^{+\infty } \left\{ \left( Z_n^{(0)} + Z_{n_{xx}}^{(2)} \right) u_n + 2 Z_{n_{x}}^{(2)} u_{n_x} + Z_n^{(2)}u_{n_{xx}} \right\} , \\&0 \end{aligned} \right. \nonumber \\ \end{aligned}$$

(38)

where $Z_n^{(0)}$ is defined by $Z_n^{(0)}=\partial Z_n^{(1)}/\partial z$. Thus

$$\begin{aligned} \mathcal {E}_1= & {} - \sum _{n=0}^{+\infty } \left\{ W_0 \left( Z_n^{(0)} + Z_{n_{xx}}^{(2)}\right) u_n + \left( 2 W_0 Z_{n_x}^{(2)}\right) u_{n_x}\right. \nonumber \\&\quad \left. + \left( W_0 Z_n^{(2)}\right) u_{n_{xx}} \right\} , \end{aligned}$$

(39)

A similar derivation can be performed for $\mathcal {E}_2$, and we obtain

$$\begin{aligned} \mathcal {E}_2 = \sum _{n=0}^{+\infty } \left\{ \left( \Omega _{0_2} Z_{n_x}^{(2)} \right) u_n + \left( \Omega _{0_2} Z_{n}^{(2)} \right) u_{n_x} \right\} , \end{aligned}$$

(40)

where $(\Omega _{0_1}, \Omega _{0_2}, \Omega _{0_3})$ refer to the mean flow vorticity, and are provided by

$$\begin{aligned} \varvec{\Omega _0} = \left\{ \begin{aligned} \Omega _{0_1} \\ \Omega _{0_2} \\ \Omega _{0_3} \end{aligned} \right. = \left\{ \begin{aligned}&0 \\&-\left( W_{0_x} - U_{0_z} \right) . \\&0 \end{aligned} \right. \end{aligned}$$

(41)

The scalar functions $\mathcal {F}_1$ and $\mathcal {F}_2$ are also given by

$$\begin{aligned} \begin{aligned} \mathcal {F}_1&= \sum _{n=0}^{+\infty } \left\{ \left[ \int _z^{\eta } \left( U_0\left( Z_n^{(0)} + Z_{n_{xx}}^{(2)}\right) \right) \mathrm{d}\xi \right] u_n \right. \\&\quad + \left[ \int _z^{\eta } \left( 2 U_0 Z_{n_{x}}^{(2)} \right) \mathrm{d}\xi \right] u_{n_x} + \left. \left[ \int _z^{\eta } \left( U_0 Z_{n}^{(2)} \right) \mathrm{d}\xi \right] u_{n_{xx}} \right\} , \end{aligned} \end{aligned}$$

(42)

and

$$\begin{aligned} \mathcal {F}_2 = \sum _{n=0}^{+\infty } \left\{ \left[ \int _z^{\eta } \Omega _{0_2} Z_n^{(1)} \mathrm{d}\xi \right] u_n \right\} . \end{aligned}$$

(43)

Detailed expression of the coefficients

In the present work, this system is investigated in the framework of two-dimensional flows. Under this hypothesis, the system is much more readable, and reduces to

$$\begin{aligned}&\begin{aligned}&\sum _{n=0}^M \left\{ A1_{mn}^{\text{2D }} \frac{\partial u_n}{\partial t} \right. + B1_{mn}^{\text{2D }} \frac{\partial ^2 u_n}{\partial t \partial x} + C1_{mn}^{\text{2D }} \frac{\partial ^3 u_n}{\partial t \partial x^2} \\&\quad + D1_{mn}^{\text{2D }} u_n + E1_{mn}^{\text{2D }} \frac{\partial u_n}{\partial x} + F1_{mn}^{\text{2D }} \frac{\partial ^2 u_n}{\partial x^2} \\&\quad + \left. G1_{mn}^{\text{2D }} \frac{\partial ^3 u_n}{\partial x^3} \right\} + H1_{m}^{\text{2D }} \frac{\partial \tilde{\eta }}{\partial x} = 0, \qquad m=0,1, \ldots , M, \text{ and } \end{aligned} \end{aligned}$$

(44)

$$\begin{aligned}&\frac{\partial \tilde{\eta }}{\partial t} + A2^{\text{2D }} \frac{\partial \tilde{\eta } }{\partial x} + \sum _{n=0}^M \left\{ B2_n^{\text{2D }} u_n + C2_n^{\text{2D }} \frac{\partial u_n}{\partial x}\right\} =0. \end{aligned}$$

(45)

The coefficients are thus provided by

$$\begin{aligned} A1_{mn}^{\text{2D }}= & {} \left\langle \left( Z_n^{(1)} - Z_{n_{xx}}^{(3)} \right) ,Z_m^{(1)} \right\rangle + \left\langle 1,Z_m^{(1)} \right\rangle \frac{\partial }{\partial x} \left( Z_n^{(2)}(\eta _0)\frac{\partial \eta _0}{\partial x} \right) , \nonumber \\ B1_{mn}^{\text{2D }}= & {} - \left\langle 2 Z_{n_x}^{(3)}, Z_m^{(1)} \right\rangle + \left\langle 1,Z_m^{(1)} \right\rangle \left( Z_n^{(2)}(\eta _0)\frac{\partial \eta _0}{\partial x} \right) , \nonumber \\ C1_{mn}^{\text{2D }}= & {} - \left\langle Z_{n}^{(3)}, Z_m^{(1)} \right\rangle , \nonumber \\ D1_{mn}^{\text{2D }}= & {} \left\langle 1,Z_m^{(1)} \right\rangle \frac{\partial }{\partial x} \left( U_0(\eta _0) Z_n^{(1)} (\eta _0) -W_0(\eta _0) Z_{n_x}^{(2)}(\eta _0) \right) \nonumber \\&\qquad +\left\langle \left( W_0 \left( Z_n^{(0)} + Z_{n_{xx}}^{(2)}\right) \right) , Z_m^{(1)} \right\rangle - \left\langle \left( \Omega _{0_2} Z_{n_x}^{(2)} \right) , Z_m^{(1)} \right\rangle \nonumber \\&- \left\langle \frac{\partial }{\partial x}\left[ \int _z^\eta \left( U_{0} \left( Z_n^{(0)} + Z_{n_{xx}}^{(2)} \right) \right) \mathrm{d}\xi \right] , Z_m^{(1)} \right\rangle \nonumber \\&- \left\langle \frac{\partial }{\partial x} \left[ \int _z^\eta \Omega _{0_2} Z_n^{(1)} \mathrm{d}\xi \right] ,Z_m^{(1)} \right\rangle ,\nonumber \\ E1_{mn}^{\text{2D }}= & {} \left\langle 1,Z_m^{(1)} \right\rangle \left( U_0(\eta _0) Z_n^{(1)} (\eta _0) -W_0(\eta _0) Z_{n_x}^{(2)}(\eta _0) \right. \nonumber \\&\left. -\frac{\partial }{\partial x}\left( W_0(\eta _0) Z_{n}^{(2)}(\eta _0) \right) \right) \nonumber \\&+\left\langle \left( 2 W_0 Z_{n_x}^{(2)}\right) , Z_m^{(1)} \right\rangle - \left\langle \left( \Omega _{0_2} Z_{n}^{(2)}\right) , Z_m^{(1)} \right\rangle \nonumber \\&- \left\langle \left( \int _z^\eta \Omega _{0_2} Z_n^{(1)} \mathrm{d}\xi \right) ,Z_m^{(1)} \right\rangle \nonumber \\&- \left\langle \left( \int _z^\eta \left( U_0 \left( Z_n^{(0)} + Z_{n_{xx}}^{(2)} \right) \right) \mathrm{d}\xi \right. \right. \nonumber \\&\left. \left. + \frac{\partial }{\partial x}\left[ \int _z^\eta \left( 2 U_{0} Z_{n_x}^{(2)} \right) \mathrm{d}\xi \right] \right) , Z_m^{(1)} \right\rangle , \nonumber \\ F1_{mn}^{\text{2D }}= & {} - \left\langle 1,Z_m^{(1)} \right\rangle \left( W_0(\eta _0) Z_n^{(2)}(\eta _0) \right) +\left\langle \left( W_0 Z_{n}^{(2)} \right) , Z_m^{(1)} \right\rangle \nonumber \\&\quad - \left\langle \left( \int _z^\eta \left( 2 U_{0} Z_{n_x}^{(2)} \right) \mathrm{d}\xi + \frac{\partial }{\partial x}\left[ \int _z^\eta \left( U_{0} Z_{n}^{(2)}\right) \mathrm{d}\xi \right] \right) , Z_m^{(1)} \right\rangle ,\nonumber \\ G1_{mn}^{\text{2D }}= & {} - \left\langle \left( \int _z^\eta \left( U_0 Z_{n}^{(2)} \right) \mathrm{d}\xi \right) , Z_m^{(1)} \right\rangle , \nonumber \\ H1_{m}^{\text{2D }}= & {} g \left\langle 1,Z_m^{(1)} \right\rangle , \end{aligned}$$

(46)

and

$$\begin{aligned} A2_{n}^{\text{2D }}= & {} U_0, \nonumber \\ B2_{n}^{\text{2D }}= & {} Z_n^{(1)}(\eta _0) \frac{\partial \eta _0}{\partial x} + Z_{n_x}^{(2)}(\eta _0), \nonumber \\ C2_{n}^{\text{2D }}= & {} Z_n^{(2)}(\eta _0). \end{aligned}$$

(47)