The Numerical Solution of the Navier- Stokes Equations for an Incompressible, Inviscid Fluid with a Free Surface

Abstract

This paper discusses a method of describing the time-dependent motion of an incompressible, inviscid (ideal) fluid in two dimensions. The fluid in this project has rigid boundaries on all sides except for a free surface over which there is assumed to be a uniform pressure. The effects of gravity and this free surface condition will be studied. To describe fluid's motion the Navier Stokes equations, in two forms – Lagrangian and Eulerian – will be utilizaed. Basically the method involves coupling the Eulerian and Lagrangian systems of equations. The Lagrangian equations follow the location of the free surface and all elements of the fluid, and also determines fluid particle velocities. Composed of a fixed coordinate system, the Eulerian system is used to compute the pressure at various points in the fluid. With the advent of the Computer Age, the non-linear terms of the Navier Stokes equations, which classical theory neglected, can be retained and solved by numerical methods. This project employs the IBM 360, a high speed computer with a large memory core, to solve the Navier Stokes equations by the finite difference method. Although much work on the numerical solution of these equations for different fluid conditions has been done in the last few years, there has been no previous solution for this particular problem.