Eigenmodes of a Counter-Rotating Vortex Dipole

Highly resolved solutions of the two-dimensional incompressible Navier-Stokes and continuity equations describing the evolution of a counter-rotating pair of vortices have been obtained accurately and efficiently by spectral-collocation methods and an eigenvalue decomposition algorithm. Such solutions have formed the basic state for subsequent three-dimensional biglobal-eigenvalue-problem linear-instability analyses, which monitor the modal response of these vortical systems to small-amplitude perturbations, periodic along the homogeneous axial spatial direction, without the need to invoke an assumption of azimuthal spatial homogeneity. A finite element methodology (Gonzalez, L. M., Theofilis, V., and Gomez-Blanco, R., "Finite Element Numerical Methods for Viscous Incompressible Biglobal Linear Instability Analysis on Unstructured Meshes," AIAA Journal, Vol. 45, No. 4, 2007, pp. 840-855) has been adapted to study the instability of vortical flows and has been validated on the Batchelor vortex (Mayer, E. W., and Powell, K. G., "Viscous and Inviscid Instabilities of a Trailing Vortex," Journal of Fluid Mechanics, Vol. 245, 1992, pp. 91-114). Subsequently, the instability of the counter-rotating dipole has been analyzed; aspects monitored have been the dependence of the results on the Reynolds number, the value of the (nonzero) axial velocity considered, and the time at which the quasi-steady basic How has been monitored. Essential to the success of the analysis has been the appropriate design of a calculation mesh, as well as exploitation of the symmetries of the basic state. The spatial structure of the amplitude functions of all unstable eigenmodes reflects the inhomogeneity of the basic state in the azimuthal spatial direction, thus providing a posteriori justification for the use of the biglobal-eigenvalue-problem concept.