Navier-Stokes Spectral Solver in a Finite Cylinder (Communications in Computational Physics, 2010)
A primitive variable spectral method for simulating incompressible viscousflows inside a finite cylinder is presented. One element of originality of the proposedmethod is that the radial discretization of the Fourier coefficients depends onthe Fourier mode, its dimension decreasing with the increase of the azimuthal modalnumber. This principle was introduced independently by Matsushima and Marcusand by Verkley in polar coordinates and is adopted here for the first time to formulatea 3D cylindrical Galerkin projection method. A second element of originality is the useof a special basis of Jacobi polynomials introduced recently for the radial dependencein the solution of Dirichlet problems. In this basis the radial operators are representedby matrices of minimal sparsity — diagonal stiffness and tridiagonal mass — providedhere in closed form for the first time, and lead to a Helmholtz operator characterizedby a favorable condition number. Finally, a new method is presented for eliminatingthe singular behaviour of the solution originated by the rotation of the lid with respectto the cylindrical wall. Thanks to these elements, the resulting Navier-Stokes spectralsolver guarantees the differentiability to any order of the solution in the entire computationaldomain and does not suffer from the time-step stability restriction occurringin spectral methods with a point clustering close to the axis. Several test examples areoffered that demonstrate the spectral accuracy of the solution method under differentrepresentative conditions.