Arbitrary Lagrangian–Eulerian finite element analysis of free surface flow using a velocity–vorticity formulation

This paper describes the application of velocity–vorticity formulation of the Navier–Stokes equations for two-dimensional free surface flow using an arbitrary Lagrangian–Eulerian method. The velocity Poisson equations and the vorticity transport equations are solved using a finite element method to obtain the velocity and the vorticity fields in the interior region of the computational domain. The boundary-fitted coordinates system is adopted to solve the boundary equations for kinematic and dynamic conditions at the free surface using a finite difference method. The numerical model for the velocity–vorticity formulation is validated for a square cavity flow at Re=400 and 1000. The solitary wave reflected from a vertical wall is chosen as a test case for comparison and validation of the free surface flow model. Then the proposed numerical model is used to obtain flow results for the following free surface flow cases: (i) interaction between two opposite solitary waves, (ii) seiche phenomenon in a rectangular reservoir, and (iii) solitary wave through a submerged rectangular structure in a viscous fluid. The efficiency of the present numerical model for numerical treatment of free surface flows is discussed. Furthermore the advantage of this formulation with respect to primitive variables formulation is addressed from the computational point of view.