A spectral method for Taylor vortex flow and Taylor-Couette flow

A spectral method has been developed which simulates Taylor vortex flow. Axisymmetry was assumed which reduced the incompressible Navier-Stokes equations to two dimensions. The code was used to undertake an investigation into the non-uniqueness of the Taylor vortex flow state. The spectral method was then extended to three dimensions to solve for Taylor-Couette flow in general. In particular the latter method was used to simulate wavy vortex flow. A new feature of the numerical method is that the Poisson and Helmholtz solvers are based on a spectral Tau approach. It was developed as an alternative to a spectral collocation approach, where all computations are done in real space. In the Tau method the equations are firstly expressed in matrix form in spectral space. Then a matrix inversion gives the solution in spectral space. Finally, an inverse transform gives the solution in real space. As the equations were in cylindrical coordinates there was a complexity which was overcome, this being the representation in spectral space of factors proportional to 1/r and 1/r² multiplied with the derivative terms. The equations were firstly rewritten so that the factors were proportional to r and r². The spectral form of the matrix representing these factors were easily determined analytically. Another feature of the numerical method is that higher order Neumann boundary conditions for the pressure were applied in the pressure step of the operator splitting method. This was done to ensure that the velocity field is at least second order time accurate.