An Efficient Method for the Solution of the Incompressible Navier--Stokes Equations in Cylindrical Geometries

The article presents a fast pseudo-spectral Navier-Stokes solver for cylindrical geometries, which is shown to possess exponential rate of decay of the error. The formulation overcomes the issues related to the axis singularity, by employing in the radial direction a special set of collocation points together with standard Chebyshev polynomials. A multi-domain technique with patching interfaces yields significant improvements in the conditioning of the algebraic problems arising from the discretization procedure and allows for an enhanced near wall resolution of wall bounded shear flows. The elliptic kernel enjoys the efficiency of an analytic expansion of the harmonic extension, The method is tested by computing the formation of Taylor vortices in a rotating Couette flow for both axisymmetric and non-axisymmetric configurations, A direct numerical simulation of a turbulent pipe flow at moderate Reynolds number demonstrates the effectiveness of the method in as much as the axis singularity is concerned, Results compare well with reference experimental and numerical data,