Three-dimensional instability of axisymmetric flows: Solution of benchmark problems by a low-order finite volume method

Several problems on three-dimensional instability of axisymmetric steady flows driven by convection or rotation or both are studied by a second-order finite volume method combined with the Fourier decomposition in the periodic azimuthal direction. The study is focused on the convergence of the critical parameters with mesh refinement. The calculations are done on the uniform and stretched grids with variation of the stretching. Converged results are reported for all the problems considered and are compared with the previously published data. Some of the calculated critical parameters are reported for the first time. The convergence studies show that the three-dimensional instability of axisymmetric flows can be computed with a good accuracy only on fine enough grids having about 100 nodes in the shortest spatial direction. It is argued that a combination of fine uniform grids with the Richardson extrapolation can be a good replacement for a grid stretching. It is shown once more that the sparseness of the Jacobian matrices produced by the finite volume method allows one to enhance performance of the Newton and Arnoldi iteration procedures by combining them with a direct sparse linear solver instead of using the Krylov-subspace-based iteration methods.