A numerical study of the run-up and the force exerted on a vertical wall by a solitary wave propagating over two tandem trenches

Abstract

The propagation and transformation of water waves over varying bathymetries is a subject of fundamental interest to ocean, coastal and harbor engineers. The specific bathymetry considered in this paper consists of one or two, naturally formed or man-made, trenches. The problem we focus on is the transformation of an incoming solitary wave by the trench(es), and the impact of the resulting wave system on a vertical wall located after the trench(es). The maximum run-up and the maximum force exerted on the wall are calculated for various lengths and heights of the trench(es), and are compared with the corresponding quantities in the absence of them. The calculations have been performed using the fully nonlinear water-wave equations, in the form of the Hamiltonian coupled-mode theory, recently developed by Papoutsellis et al. (Eur J Mech B/Fluids 72:199–224, 2018). Comparisons of the calculated free-surface elevation with existing experimental results indicate that the effect of the vortical flow, inevitably developed within and near the trench(es) but not captured by any potential theory, is not important concerning the frontal wave flow regime. This suggests that the predictions of the run-up and the force on the wall by means of nonlinear potential theory are expected to be nearly realistic. The main conclusion of our investigation is that the presence of two tandem trenches in front of the wall may reduce the run-up from (about) 20 to 45% and the force from 15 to 38%, depending on the trench dimensions and the wave amplitude. The percentage reduction is greater for higher waves. The presence of only one trench leads to reductions 1.4–1.7 times smaller.

Appendices

Appendix 1: Generation of the solitary wave

As mentioned in the main part of the present paper, to construct an exact, fully nonlinear, initial flow for a water-wave problem, we need to formulate and solve another, hopefully simpler, one. For solitary waves propagating over a horizontal bottom, the simplification comes from the fact that the problem can be reformulated as a steady (time-independent) one, in terms of a reference frame moving with the constant phase speed $c$ of the waves. In this appendix, we give a brief overview of the highly accurate, iterative solver of Clamond and Dutykh (2013) and Dutykh and Clamond (2014), which has been used for deriving all the initial conditions needed for the calculations of this work. The solution method goes as follows. The full 2D Euler equations in water of constant depth $d$, assuming that the free-surface elevation is localized near the origin and tends to zero at both infinities ($\pm \,\infty$), are reformulated as a variational principle with respect to the action functional:

$$\text{S}\,[\,\eta \,]\,\,\,\varvec{ = }\,\,\,\int\limits_{ - \,\,\infty \,\,\,\,\,\,\,\,}^{\,\,\,\,\,\,\, + \,\,\infty } {\left( {\frac{1}{2}\,c^{\,2} \,\eta \,\text{C}\,\{ \,\eta \,\} \,\, - \,\,\frac{1}{2}\,g\,\eta^{\,2} \,(\,1\,\, + \,\,\text{C}\,\{ \,\eta \,\} \,)} \right){\text{d}}\alpha } ,$$

(20)

where $\eta \,\, = \,\,\eta \,(\,\alpha \,)$ is the free-surface elevation (assumed time-independent in the appropriate reference frame), and $\text{C}$ is an appropriate nonlocal (pseudo-differential) operator, taking care of the substrate kinematics. Rendering the action functional $\text{S}$ stationary, yields the Euler–Lagrange equation:

$$\delta \text{S}\,\,\,\varvec{ = }\,\,\,c^{\,2} \text{C}\{ \,\eta \,\} \,\, - \,\,g\,\eta \,\, - \,\,\frac{1}{2}\,g\,\text{C}\,\{ \,\eta^{\,2} \,\} \,\, - \,\,g\,\eta \text{C}\,\{ \,\eta \,\} = 0,$$

(21)

which is known as the Babenko equation for gravity solitary waves (Babenko 1987). The latter, separating the linear from the nonlinear part, is rewritten in the form

$$\text{L}\,\{ \,\eta \,\} = \text{N}\{ \,\eta \,\} ,$$

(22)

where

$$\begin{aligned}&\text{L}\{ \,\eta \,\} \,\,\, \equiv \,\,\,c^{\,2} \,\eta \,\, - \,\,g\,\text{C}^{\, - \,\,1} \{ \,\eta \,\} \; \,\,\,\,\,\,\,\,\,\ {\text{and}} \\ &\text{N}\,\{ \,\eta \,\} \,\,\, \equiv \,\,\,g\,\text{C}^{\, - \,\,1} \{ \,\eta \,\text{C}\,\{ \,\eta \,\} \,\} \,\, + \,\,\frac{1}{2}\,g\,\eta^{\,2} .\end{aligned}$$

(23)

Equation (22) is solved using Petviashvili’s iterations [see, e.g., Petviashvili (1976) and other references in the works of Clamond and Dutykh]:

$$\eta_{\,n\,\, + \,\,1} = S_{\,n}^{\,2} \,\text{L}^{\,\, - \,\,1} \{ \,\text{N}\,\{ \,\eta_{\,n} \,\} \,\} ,\,\quad S_{\,n} = \frac{{\int_{ - \,\,\infty }^{ + \,\,\infty } {\eta_{\,n} \,\text{L}\{ \,\eta_{\,n} \,\} \,{\text{d}}{\kern 1pt} \alpha } }}{{\int_{ - \,\,\infty }^{ + \,\,\infty } {\eta_{\,n} \,\text{N}\,\{ \,\eta_{\,n} \,\} \,{\text{d}}{\kern 1pt} \alpha } }},$$

(24)

where $S_{{\,{\text{n}}}}$ is a stabilization factor. The calculations are efficiently performed in the Fourier domain, and communicated to the physical domain by means of the fast Fourier transform (FFT). The iteration process uses as initial guess the KdV solution, in the form:

$$\eta_{\,0} \,(\,\alpha \,) = d\,{\kern 1pt} (\,F^{\,2} \,\, - \,\,1\,)\,{\kern 1pt} \text{sech}^{\,2} \,(\,\kappa \,\alpha /2\,),$$

(25)

where $\kappa$ is calculated from the equation $F^{\,2} = \kappa \,d\,{\kern 1pt} \tan \,(\,\kappa \,d\,)$, given the Froude number $F\,\, \equiv \,\,c/\sqrt {g\,d}$ of the desired solitary wave. The iterations stop when the following criteria:

$$\begin{aligned}&\left\| {\,\eta_{\,n\,\, + \,\,1} \,\, - \,\,\eta_{\,n} {\kern 1pt} } \right\|_{\,\infty } \,\, < \,\,\varepsilon_{\,1} , \\ &\left\| {\,\text{L}\,\{ \,\eta_{\,n} \,\} \,\, - \,\,\text{N}\,\{ \,\eta_{\,n} \,\} {\kern 1pt} } \right\|_{\,\infty } \,\, < \,\,\varepsilon_{\,2} , \end{aligned}$$

(26)

are met, where the tolerance parameters $\varepsilon_{\,1}$, $\varepsilon_{\,2}$ can be as small as the machine accuracy. Let it be noted that this solver has been tested for solitary waves with amplitude-to-depth ratio up to 0.796.

Appendix 2: Deviation of numerical from experimental results, for the cases considered in Sects. 4.1 and 4.2

In Table 3, the deviations of numerical predictions, $R_{{\,{\text{num}}}}$, from the corresponding experimental values, $R_{\,\exp }$, concerning the two peaks of the transformed solitary wave over a shelf and over a step, are summarized in percentages, calculated as:

$$\frac{{R_{{\,{\text{num}}}} - R_{\,\exp } }}{{R_{\,\exp } }}\,\,100\,\,\% .$$

Table 3 Deviation of various numerical methods from the corresponding experimental results, for the transformation of a solitary wave over a shelf and over a step considered in Sect. 4

The considered numerical methods are those appearing in Figs. 2 and 4 (Sect. 4), and the percentage discrepancies refer to the peak overestimation and time advancement, for each of the two peaks of the transformed solitary wave, at the gauges $g_{\,3}$ and $g_{\,4}$.