A Lagrangian Approach to Two-Dimensional Free Surface Flows

The two-dimensional, free surface motion of an inviscid isochoric fluid is investigated, through a lagrangian approach. Among the equations of motion, a pressure equation is also considered. It is deduced from the continuity constraint, in analogy with the classical eulerian formulation for isochoric fluids. The flow is numerically simulated by using standard finite differences, with staggered pressure and a predictor-corrector time integration scheme. The continuity constraint is directly enforced in the corrector substep. Moreover, a small perturbation analysis is performed, in order to obtain an approximated analytical solution of the lagrangian equations. The perturbation parameter is related to the amplitude of the initial velocity field. The expansion is carried out up to second order terms, accounting for metrical coefficients and the source term in the pressure equation. In presence of a gravitational field, the free surface elevation results a time periodic one in the first order solution, while the periodicity is lost at second order. The flow as a whole is not a time periodic one. Comparisons with the direct numerical simulation of the flow are discussed.