Some numerical solutions of unsteady free surface wave problems using the Lagrangian description of the flow

Abstract

Until very recently numerical solutions of unsteady, free surface flows invariably employed the Eulerian description of the motions. Perhaps the most widely used of these has been the marker and cell (MAC) technique developed by Fromm and Harlow (1963) and further refined by many others. In such a formulation the most difficult problem arises in attempting to reconcile the initially unknown shape and position of the free surface with a finite difference scheme and the necessity of determining derivatives at that surface (in a similar fashion few solutions exist with curved or irregular boundaries). But this difficulty can be surmounted by solving in a parametric plane in which the position and shape of the free surface are known in advance; such mappings have been successfully employed in steady flows (eg. Brennen (1969)). Whilst there are other possibilities (see John (1953), Brennen and Whitney (1970)) the Lagrangian description in its general form involves just such a parrametric plane. The present paper describes briefly a numerical method for the solution of the Lagrangian equations of motion for the inviscid, planar flow of a homogeneous or inhomogeneous fluid, taking full advantage of the flexibility of choice of the Lagrangian coordinates (a, b). More details and other results can be found in Brennen and Whitney (1970). Very recently Hirt, Cook and Butler (1970) published details of a method which solves the Eulerian equations of motion in a fashion similar to the MAC technique but uses a Lagrangian tagging space.