A spectral algorithm in the vorticity-stream function formulation for two-dimensional flows with no-slip walls

An accurate and efficient spectral algorithm is presented for the solution of the incompressible Navier-Stokes equation in the vorticity-stream function formulation for two-dimensional flows which are bounded in one and periodic in the other direction, such as a periodic channel or an annulus. The functions are expanded in Chebyshev-Fourier series. The Navier-Stokes equation is solved implicitly for the viscous term, while the inertial term is computed in point space and the result fully dealiased with the 2/3 truncation scheme. The Poisson equation for the stream function and the Helmholtz equation originating from the implicit time integration of the viscosity term are both reduced via Chebyshev recursions to banded form and solved directly by LU decomposition. The number of operations for the solution of these equations is O(M N), where M and N are the number of Chebyshev and Fourier modes, respectively. No-slip conditions are expressed by integral solvability constraints on the vorticity field. These constraints are determined in a pre-processing step and add no additional CPU-time to the main computations. A numerical example of an unbalanced dipole colliding with a no-slip wall under the influence of Ekman damping is shown and compared with new experimental results from a rotating tank experiment. Furthermore, a numerical example of self-organization in an annular region of flute modes originating in a magnetized plasma and described by the Navier-Stokes equation is presented. (Authors)