Hele-Shaw beach creation by breaking waves: a mathematics-inspired experiment

Abstract

Fundamentals of nonlinear wave-particle interactions are studied experimentally in a Hele-Shaw configuration with wave breaking and a dynamic bed. To design this configuration, we determine, mathematically, the gap width which allows inertial flows to survive the viscous damping due to the side walls. Damped wave sloshing experiments compared with simulations confirm that width-averaged potential-flow models with linear momen- 2A n t h o n y T h o r n t o n e t a l .tum damping are adequately capturing the large scale nonlinear wave motion. Subsequently, we show that the four types of wave breaking observed at realworld beaches also emerge on Hele-Shaw laboratory beaches, albeit in idealized forms. Finally, an experimental parameter study is undertaken to quantify the formation of quasi-steady beach morphologies due to nonlinear, breaking waves: berm or dune, beach and bar formation are all classified. Our research reveals that the Hele-Shaw beach configuration allows a wealth of experimental and modelling extensions, including benchmarking of forecast models used in the coastal engineering practice, especially for shingle beaches.

On Wave-Driven “Shingle” Beach Dynamics in a Table-Top Hele-Shaw Cell

“…a) t = 3.25T , b) t = 3.5T , c) t = 3.75T , and d) t = 4.0T with period T = 1s. A similar yet different simulation is found in Thornton et al (2014).…”

“…Finally, we conclude in Section 6. The work in this article runs partially in parallel with our previous work (Thornton et al, 2014), but concerns different data sets, which shows that the experimental results are robust. In addition, a comparison between different realizations of the Hele-Shaw cell is made.…”

“…In Thornton et al (2014), a comparison between the potential flow model arising from (3) and a damped wave-sloshing experiment in the rectangular Hele-Shaw cell with fixed walls shows that experimental data and simulations compare reasonably well, for about one to one-and-a-half long-wave period. Hence, we have shown that for driven flows on scales much larger than the gap width, the linear momentum damping in (2) and (3) is a good leading-order model approximation because new waves keep coming in.…”

“…We performed various numerical simulations of these depth-averaged shallow water equations with linear momentum damping for β = 1 and different values of the gap width 2l, as well as different wave-maker amplitudes and frequencies around 1 Hz. A typical breaking wave in the form of a hydraulic bore was found to survive the damping and travel across this fixed sloping beach of circa 0.5m length in the corresponding numerical shallow water model for gap widths 2l > 1.5 mm and frequencies of about 0.5-1.3 Hz (Thornton et al, 2014). Sample profiles of h(x, t) andū(x, t) of one simulation are shown in Fig.…”

“…The final design step was to derive a depth-avaraged shallow-water model by averaging (2) over the depth and using the corresponding kinematic condition at the free surface (extending the classic derivation in, e.g., Pedlosky (1987)). The resulting shallow-water equations with linear momentum damping (Thornton et al, 2014) are as follows…”