Viscous flow with large free surface motion

Abstract

An arbitrary Lagrangian-Eulerian (LE) Petrov-Galerkin finite elemen{technique is developed to study nonlinear viscous fluids under large free surface wave motion. A review of the kinematics and field equations from an arbitrary refe�ence is presented and since the major challenge of the ALE description lies in the mesh rezoning algorithm, various methods, including a new mixed formulation, are developed to update the mesh and lnap the moving domain in a more rational manner. Moreover, the streamline-upwind/Petrov-Galerkin formulation is implemented to accurately describe highly convective free surface flows. The effectiveness of the algorithm is demonstrated on a tsunami problem, the dam-break problem where the Reynolds number is taken as high as 3000, and a large-amplitude sloshing problem.

Comparison of wall boundary conditions for numerical viscous free surface flow simulation

“…The notation w denotes that the derivative is evaluated on the moving frame of reference of velocity w. This is the standard Arbitrary Lagrangian Eulerian formulation used by many investigators to simulate free surface flow in stationary tanks (Warburton & Karniadakis, 1997;Ramaswamy & Kawahara, 1987;Ramaswamy, 1989;Huerta & Wing Kam, 1988;Robertson, 2000). The conditions placed on the boundary of the fluid will be discussed in section 2.1.…”

“…The contact wall boundary problem has been overcome by many authors (Huerta & Wing Kam, 1988;Warburton & Karniadakis, 1997) by adopting a slip boundary condition on the free surface contact walls such that the non-permeability condition is maintained, whilst the tangential velocity is allowed to take a non-zero value by implementing a Neumann boundary condition at the wall. The resulting boundary conditions for the vertical side walls are,…”

“…Boundary element methods have been extensively used to computationally simulate such free surface flows (Celebi et al, 1998;Ferrant, 1996), though recent understanding of the computational efficiency of these algorithms has shown that the finite element method is a competitive numerical technique when a large number of degrees of freedom is used to discretise the fluid domain (Cai et al, 1998;Wu & Eatock Taylor, 1994). The finite element method has therefore become popular when simulating unsteady, inviscid free surface flows (Cai et al, 1998;Wu et al, 1998;Robertson & Sherwin, 1999) and in many cases has been extended to the solution of viscous free surface flows (Warburton & Karniadakis, 1997;Ramaswamy & Kawahara, 1987;Ramaswamy, 1989;Huerta & Wing Kam, 1988;Robertson, 2000).…”

“…The most widely used computational model is the no-shear force or friction free condition (Huerta & Wing Kam, 1988), which is referred to in this work as the slip condition. This boundary condition does not promote the generation of vorticity at free surface contact walls, where it is theorised that much of the dissipation of the energy of the fluid occurs (Keulegan, 1959;Miles, 1967).…”

“…This procedure alleviates the production of highly deformed elements which would affect the stability of the computations. The first application of the ALE formulation was used in conjunction with a finite difference scheme (Hirt et al, 1974) and has since been extensively utilised for free surface problems using the finite element method (Huerta & Wing Kam, 1988;Ramaswamy & Kawahara, 1987). Ho (Ho, 1995) developed an ALE approach to simulate free surface flow using a spectral element discretisation based on conforming quadrilaterals and this formulation has been extended by Warburton and Karniadakis (Warburton & Karniadakis, 1997) to hybrid discretisations incorporating triangular and tetrahedral elements (Sherwin & Karniadakis, 1995a,b), where the vorticity generation induced by a cylinder close to a free surface is investigated.…”