Numerical Solutions to Free-Surface Axisymmetric Flows

Methods are developed for obtaining solutions by finite differences to free streamline axisymmetric, potential fluid flows. The formulation of the boundary value problem considers the velocity potential and Stokes stream function as the independent variables and the radial and axial coordinates as the dependent variables. The nonlinear partial differential equations are derived which describe axisymmetric flow in terms of the radial and axial coordinates. Because of the nonlinear nature of the equations, the solution is obtained by using a Newton-Raphson inner iteration, as well as the usual outer iteration customarily used for a finite difference solution of a partial differential boundary value problem of the elliptic type. Two axisymmetric problems have been investigated. The first is of a jet of inviscid incompressible fluid issuing from a nozzle into the free atmosphere. The second is the cavity flow resulting from a jet flowing past a body of revolution.