Poloidal-toroidal decomposition in a finite cylinder. II. Discretization, regularization and validation

TL;DR

This paper develops a spectral method for solving Navier-Stokes equations in a finite cylinder using poloidal-toroidal decomposition, ensuring regularity and accuracy through smoothing and singularity elimination, validated against known solutions.

Contribution

It introduces a pseudo-spectral discretization approach with regularization techniques for Navier-Stokes in cylindrical geometry, validated by convergence and comparison with existing solutions.

Findings

Spectral convergence of the solution coefficients.

Agreement with exact polynomial solutions and previous studies.

Parallelization by azimuthal wavenumber is highly effective.

Abstract

The Navier-Stokes equations in a finite cylinder are written in terms of poloidal and toroidal potentials in order to impose incompressibility. Regularity of the solutions is ensured in several ways: First, the potentials are represented using a spectral basis which is analytic at the cylindrical axis. Second, the non-physical discontinuous boundary conditions at the cylindrical corners are smoothed using a polynomial approximation to a steep exponential profile. Third, the nonlinear term is evaluated in such a way as to eliminate singularities. The resulting pseudo-spectral code is tested using exact polynomial solutions and the spectral convergence of the coefficients is demonstrated. Our solutions are shown to agree with exact polynomial solutions and with previous axisymmetric calculations of vortex breakdown and of nonaxisymmetric calculations of onset of helical spirals. Parallelization by azimuthal wavenumber is shown to be highly effective.