Numerical Solutions of the Time-Dependent Incompressible Viscous Flow over a Disk or a Sphere

摘要

The Navier-Stokes equations for incompressible axisymmetric flow over a disk or a sphere are solved by means of finite difference approximation. The Reynolds number range covered is 1-1000. Time-dependent Stream function-Vorticity formulation in cylindrical and spherical polar coordinate systems is adopted. The surface of the disk and the sphere is conveniently represented by coordinate lines. Explicit central Dufort-Frankel differencing in time is employed. Second order accuracy conservative differencing is used for the spatial variables. For the disk case, vorticity and stream function are defined on mesh points, while the velocity components are defined at the midpoints of the mesh cells. The latter leads to difficulties with the implementation of boundary conditions. Therefore, in the sphere case all the dependent variables are defined on mesh points. Separate linearized criterions for diffusion and convective terms were used to guide the selection of an empirical relation between time and space increments of the entire Navier-Stokes equations. It was found that in the initial phase of the solution, especially at low Reynolds numbers, time steps considerably smaller than those suggested by the stability criterion must be used. The condition to be observed for 'stable' solutions appears to be that the fractional change of the vorticity at any point over a time step is small. (Author)