Abstract
The damped motion of driven water waves in a Hele-Shaw tank is investigated variationally and numerically. The equations governing the hydrodynamics of the problem are derived from a variational principle for shallow water. The variational principle includes the effects of surface tension, linear momentum damping due to the proximity of the tank walls and incoming volume flux through one of the boundaries representing the generation of waves by a wave pump. The model equations are solved numerically using (dis)continuous Galerkin finite element methods and are compared to exact linear wave sloshing and driven wave sloshing results. Numerical solutions of the nonlinear shallow water-wave equations are also validated against laboratory experiments of artificially driven waves in the Hele-Shaw tank.
Highlights
The study of water waves has been an important area of research for years; their significance becomes obvious when looking at ocean and offshore engineering or naval architecture
A variational principle for the Lagrangian density L describing the dynamics of shallow water waves with surface tension can be obtained by extending the well-known Luke’s variational principle [1] and is given by: 0 = δ
Despite the popularity of horizontal Hele-Shaw cells, the use of a vertical Hele-Shaw cell to study the hydrodynamics of water waves is fairly novel [18,27], and takes advantage of the fact that the flow becomes nearly two-dimensional and offers better visualisation of the wave phenomena
Summary
The study of water waves has been an important area of research for years; their significance becomes obvious when looking at ocean and offshore engineering or naval architecture. The variational principle for water waves can be extended to include forcing and/or dissipation, essentially by adding time-dependent internal or boundary conditions. This is necessary for most practical problems involving waves generated by a driven wavemaker or the removal of a sluice gate; for example, see Bokhove and Kalogirou [8] for a problem concerning. In contrast to the classical case, we retain inertial effects due to the proximity of the glass plates and we extend the variational principle to include an exponential time-dependent term representing linear damping effects. Starting from the one-dimensional shallow water equations, we add the effects of surface tension, linear damping and forcing through the boundary (representing the motion of the wave pump).
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