Steady Normal-Mode Vortices in Circular Cavities

This article establishes elementary families of steady two-dimensional (2D) vortices in circular enclosures filled with inviscid fluid. A normal-mode vortex is a continuous vortex that satisfies a Helmholtz equation for the streamfunction. An individual normal-mode vortex satisfies the steady 2D vorticity equation. A way to satisfy the vorticity equation with two superposed normal-mode vortices is to let one of them have circular streamlines, trivially satisfying the kinematic boundary condition at the circle boundary. The requirement for steady flow is that the wavenumber eigenvalues for the Helmholtz equations are identical. This eigenvalue has to be dictated by a normal-mode vortex that is fully 2D. In a circle sector with a designed angle, it is possible to add two fully 2D normal-mode vortices so that their superposed flow is steady. One such example is demonstrated, where the circle sector has an angle of $63.77^\circ$. While conventional stability analysis does not apply to these families of steady normal-mode vortices, there is a latent algebraic instability in terms of linearly growing offspring vortices evolving from an initial state where a small initial perturbation is added to the steady normal-mode vortex.