A 3D pseudospectral algorithm to simulate rotating flows in cylindrical cavities
Résumé
When simulating flows in cylindrical rotating cavities a difficulty arises from the singularities appearing on the axis. Its singularities are due to the presence of terms 1/r^n (n = 1, 2) in the Navier-Stokes equations, where r is the radial dis-tance. To avoid evaluating differential equation coefficients which are infinite at that point, the grid must exclude the origin or specific pole conditions must be imposed. An efficient and accurate pseudo-spectral method has been here developed using collocation Chebyshev polynomials in the radial and axial directions and Fourier approximation in the azimuthal direction. To avoid the difficulty on the axis without prescribing any pole conditions, a new approach based on the work of Heinrichs [W. Heinrichs J. Comp. Phys. 199 (2004) 66-86] has been developed. The calculation domain is defined as (r, θ , z) ∈ [−1, 1] × [0, 2π] × [−1, 1] using an even number N of collocation points in the radial direction. Thus, r = 0 is not a collocation point. The clustering of collocation points around the rotation axis is also avoided due to the utilization of a Gauss-Lobatto distribution. The flow being indeed laminar close to the axis in most of the rotating flows. In the azimuthal direction, the overlap in the discretization is avoided by introducing a shift equal to π/2K (K the number of mesh points in θ -direction) for θ > π in the Fourier transform. The accuracy of the method was checked on the exact steady and unsteady analytical solutions and the capability of the method to simulate complex flows is illustrated considering the well documented case of the vortex breakdown phenomenon.
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