A numerical method for solving the equations for a vortex core

Abstract

A method is presented for calculating steady axially symmetric spiralling motions of an incompressible fluid at large Reynolds numbers. By making approximations of the boundary-layer type the Navier-Stokes equations are reduced essentially to a pair of non-linear parabolic equations. Initial conditions are specified on some upstream cross-section, and boundary conditions on the axis of symmetry and on some bounding surface of revolution. The method involves replacing the differential equations by sets of finite-difference equations, using first-order central differences in an implicit scheme. The calculation is by a marching technique, which proceeds step-by-step in the axial direction. For each step an iterative plan is followed. The finite-difference equations themselves are solved by straightforward matrix methods. A programme is developed for a digital computer of moderate size and examples of the application of the method are given.