Mesh Update Techniques for Free-Surface Flow Solvers Using Spectral Element Method

This presentation will highlight the computational techniques we are developing for simulating incompressible free-surface flows using the Spectral Element Method (SEM) [9, 10]. These techniques include the arbitrary Lagrangian-Eulerian (ALE) formulation [2,3,5,11], mesh update and re-meshing methods [4, 7]. A brief introduction will be devoted to recalling the physical problem and its Navier-Stokes variational formulation in the framework of the ALE kinematic description coupled with the moving-grid governing equation [1]. Following that, the spectral element discretization based on a classical Galerkin formulation will be derived and associated with a discretization of the kinematic boundary condition for the free surface [1]. A decoupled approach has been considered for the moving-grid algorithm and for the ALE Navier-Stokes algorithm. Particular emphasis has been put on the moving-grid technique that relies on calculations of the mesh velocity inside the fluid domain. For consistency with our physical problem and with the numerical technique used (namely SEM), our choice of mesh update strategy is based on considering the mesh motion as a steady incompressible viscous fluid motion governed by the steady Stokes equations. The major advantage of this choice is to ensure the incompressibility of the mesh, leading to a valuable conservation of the volume of the spectral elements used in the computation of the ALE Navier-Stokes problem for the fluid (see Fig. 1). These volume conservation laws for the mesh both locally and globally will be discussed in details. The numerical calculations of the mesh velocity for cases in two dimensions have been carried out and results will be given for different choices of wall boundary conditions (see Fig. 2 and 3): free-slip, no-slip and mixed conditions. The issue of the existence of the contact points in presence of the free surface will be analyzed together with the choice of the value of the tangential mesh velocity at the free surface. The motion of the computational nodes has been obtained numerically by time integration of the mesh velocity, leading to an automatic mesh update procedure [7]. Subsequently the internal nodes and grid, at the spectral element level, are classically built using an iso-parametric mapping with a parent element [1] (see Fig. 1, mesh on the right). Convolution of the mesh and of the spectral elements, especially those located near the free surface, renders re-meshing necessary in order to maintain the convergence of the method. The re-meshing procedure uses the same methodology as the mesh update [7] but in addition requires the definition of a new mesh topology with a given non-overlapping element decomposition of the fluid domain. The application of mesh updating and re-meshing techniques at each time-step requires a procedure able to transfer the fields value (velocities and pressure) onto the newly created mesh. Several techniques including transfinite interpolation and projections have been tested. Finally the complete algorithm will be summarized and results for two two-dimensional test cases, unsteady flow of decaying vortex [6,8,12,13] (see Fig. 4) and monochromatic standing waves [6], will be presented. The presentation will end with conclusions and perspectives for this on-going research topic.