A spectral solver for the three-dimensional Navier-Stokes equations in velocity-vorticity formulation

A numerical algorithm intended for the study of flows in a cylindrical container under laminar How conditions is proposed. High resolution of the flow field, governed by the Navier-Stokes equations in velocity-vorticity formulation relative to a cylindrical frame of reference, is achieved through spatial discretization by means of the spectral method. This method is based on a Fourier expansion in the azimuthal direction and an expansion in Chebyshev polynomials in the (nonperiodic) radial and axial directions. Several regularity constraints are used to take care of the coordinate singularity. These constraints are implemented, together with the boundary conditions at the top, bottom and mantle of the cylinder, via the tau method. The a priori unknown boundary values of the vorticity are evaluated by means of the influence matrix technique. The compatibility between the mathematical and numerical formulation of the Navier-Stokes equations is established through a tau correction procedure. The resolved flow field meets the incompressibility constraint and definition of the vorticity up to machine accuracy.