The impacts of the ALE and hydrostatic-pressure approaches on the energy budget of unsteady free-surface flows G. Lipari a , E. Napoli b, * a School of Mechanical, Aerospace and Civil Engineering, University of Manchester, Manchester, UK b Dipartimento di Ingegneria Idraulica ed Applicazioni Ambientali, Universita ` degli Studi di Palermo, Palermo, Italy Received 21 February 2006; accepted 24 October 2007 Available online 1 November 2007 Abstract 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-pur- pose 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 g 0 /h > 0.01, where g 0 is the wave semi-amplitude and h is the undisturbed water depth. The hydro- static 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. Ó 2007 Elsevier Ltd. All rights reserved. 1. Introduction The moving interface between water and atmosphere is a key feature in environmental hydraulics. The numerical techniques representing the free-surface motion are influ- enced by the way the domain discretisation is performed, and a wealth of contributions has been dealing with this topic in the past decades. Based on the broad classification given by [1], three major groups can be identified. The first one (so-called fixed-grid Eulerian techniques) draws a computational domain that includes both interfacing fluids, cover them with a single mesh and then evaluate the emptiness or full- ness of the cells with mass-less tracers [2] or an appropriate scalar [3]; such approach is able to handle highly-distorted shapes, but the information on the interface position is not immediately available from grid points stored as geometric variables. The second one (so-called front-fixing tech- niques) solves the equations on a fixed domain for the aqueous domain by employing a time-dependent mapping into body-fitted curvilinear co-ordinates. The third one (so-called front-tracking techniques) creates a mesh for the aqueous domain only, let the boundary be displaced in compliance with the governing equations, and periodi- cally regenerates the whole grid so that the interface is always a gridline (references are listed later on); the inter- face position is thus directly computed within the time-inte- gration cycle but, on the downside, an excessive grid distortion can corrupt the computational accuracy in the interior domain. (So far, this approach has been tested with single-valued surfaces, although its amplification to over- turning waves has been envisioned and is an area of current research [4].) The formulation used in this paper belongs to the latter group. Numerical techniques blending the fea- tures of different approaches have also been devised – for example in [5], where a split-merge mechanism is applied to the near-surface computational nodes. 0045-7930/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compfluid.2007.10.005 * Corresponding author. E-mail address: napoli@idra.unipa.it (E. Napoli). www.elsevier.com/locate/compfluid Available online at www.sciencedirect.com Computers & Fluids 37 (2008) 656–673