A numerical method for the study of nonlinear stability of axisymmetric flows based on the vector potential

We develop in this paper a numerical method to simulate three-dimensional incompressible flows based on a decomposition of the flow into an axisymmetric part, in terms of the stream function and the circulation, and a non-axisymmetric part in terms of a potential vector function. The method is specially suited for the study of nonlinear stability of axially symmetric flows because one may follow neatly the raising of the different non-axisymmetric modes, their nonlinear development, and their nonlinear interaction. The numerical technique combines finite differences on a non-uniform grid in the axial direction, a Chebyshev spectral collocation technique in the radial direction, and a Fourier spectral method in the azimuthal direction for the non-axisymmetric vector potential. As an example to check the efficiency and accuracy of the method we apply it to the flow inside a rotating circular pipe, and compare the resulting travelling waves with previous stability results for this problem, for different values of the Reynolds and the swirl numbers.