NUMERICAL SOLUTION FOR UNSTEADY TWO-DIMENSIONAL FREE-SURFACE FLOWS

An ecient computational procedure is developed for solving fully nonlinear unsteady free-surface problems in two dimensions under gravity. Specically, we solve here a nonlinear Cauchy-Poisson problem for a ow commencing from rest with a given initial free surface. Our task is thus to solve Laplace's equation xx yy = 0 for the velocity potential (x; y; t), with vanishing velocity at innity for all nite time t, and two nonlinear boundary conditions on an (unknown) free surface, namely the requirement that it be a material surface, and that the pressure on it be equal to atmospheric. We use a semi-Lagrangian approach, in which the equation of the free surface is specied parametrically at any time t as x = X(s; t); y = Y (s; t) where s is a marker of a given free-surface point, notionally the initial arclength along the free surface. Then there are three equations determining the free surface evolution, namely (at xed s, i.e. for each separate free-surface point) dX dt = u dY dt = v d d