Chebyshev–Fourier collocation spectral method for the solution of swirling flow

Abstract The Chebyshev–Fourier collocation spectral method (CSM), in which the Chebyshev collocation method is applied in both radial and axial directions and the Fourier collocation method is applied in azimuthal direction, is developed to solve the Poisson equation with Neumann boundary condition in cylindrical coordinates. In order to avoid the axis at r = 0 , the Chebyshev–Gauss–Lobatto collocation point is adopted in radial direction and the number of points is set to be even. These direct solvers are developed to solve three-dimensional transient incompressible Navier-Stokes equations with projection method. The approach, in which the Neumann boundary condition is constructed for solving the Poisson equation based on the Chebyshev–Fourier CSM, is testified to be easily and efficiently implemented. Furthermore, the approach is applied for solving the swirling flows in an enclosed cylinder by the rotation of an end wall.