The impacts of the ALE and hydrostatic-pressure approaches on the energy budget of unsteady free-surface flows

This paper focuses on the energy budget in the calculation of unsteady free-surface flows on moving grids with and without using the ‘arbitrary Lagrangian–Eulerian’ (ALE) formulation or hydrostatic-pressure assumption. The numerical tool is an in-house general-purpose solver for the unsteady, incompressible and homogeneous Navier–Stokes equations in a Cartesian domain. An explicit fractional-step method and co-located finite-volume method are used for the second-order accurate integrations in time and space 
The test cases are nonlinear and linear irrotational standing waves, which allow to characterise the impacts of an ALE or Eulerian formulation with moving grids by comparison with the anticipated energy conservation. The study is also extended to viscous waves for varying wave-height-to-water-depth and basin aspect ratios. The Eulerian viewpoint produces marked overdamping as early as in the first wave period for the range of relative wave heights η0/h > 0.01, where η0 is the wave semi-amplitude and h is the undisturbed water depth. The hydrostatic calculations misrepresent the evolution of the potential and kinetic energies for h/L > 0.1, where L is the basin length, with spurious modes arising from different initial conditions.