Finite element methods for viscous free-surface fluids including breaking and non-breaking waves

The objective of the present study is to formulate a more efficient numerical scheme based on finite element method to solve viscous free surface problems involving large free surface motion and distortion. The approach used in this study involves a new Arbitrary Lagrangian-Eulerian description of free surface problems. In this approach the rezoning grid is predetermined by explicit transformation and the Lagrangian calculation phase is no longer needed.;For the analysis, the flow is assumed to be viscous. It is governed by the two dimensional Navier-Stokes and continuity equations along with the full nonlinear kinematic free surface equation. A particular mapping technique is used to transfer the fluid region and its boundaries into a regular geometry in order to conveniently treat the moving free surface and an irregular bottom topography. This necessitates the transformation of the governing equation and the boundary conditions into more complicated equations. However, the transformed equations can be effectively handled by a suitable analytical and numerical procedure. The developed technique could be easily extended to analyze many other problems involving free surfaces.;A Galerkin finite element model is developed to model the transformed fluid region. The resulting discrete equations are solved iteratively by using Multi-Grid method. No artificial viscosity is introduced in the kinematics free surface equations to damp out the free surface oscillations in the region. The explicit finite difference method is employed to discretize the time variation in the nonlinear Navier-Stokes and free surface equations. A periodic or solitary wave is used to define the initial velocities and pressure of free surface profile as well as the distribution of these primitive values along the boundaries if needed.;Propagation of periodic and solitary waves in constant or variable depth medium are studied using this algorithm and wave run-up as well as wave deformation characteristic are obtained for various conditions. Results obtained by the numerical method are compared to the available numerical and experimental data obtained by others in order to demonstrate the workability of the proposed algorithm. The computed time-history of the breaking wave on constant depth and sloping bottom compares quite well with the available experimental and numerical data. For different topographies, some computed time histories of the wave deformation and breaking are also provided to evaluate the effects of wave height and sloping bottom. It is observed that the plunging breaker happens up to some extended.