Abstract
The instability of the Kármán vortex street is revisited under a spatio-temporal perspective that allows the taking into account of the advection of the vortices by the external flow. We analyse a simplified point vortex model and show through numerical simulations of its linear impulse response that the system becomes convectively unstable above a certain critical advection velocity. This critical velocity decreases as the aspect ratio approaches its specific value for temporal stability, and increases with the confinement induced by lateral walls. In the limiting unconfined case, direct application of the Briggs–Bers criterion to the dispersion relation gives results in excellent agreement with the numerical simulations. Finally, a direct numerical simulation of the $Re=100$ flow past a confined cylinder is performed, and the actual advection velocity of the resulting vortex street is found to be much larger than the critical advection velocity for convective instability given by our model. The Kármán vortex street is therefore strongly convectively unstable.
Highlights
The flow past a circular cylinder is one of the most famous and well-studied problems in fluid mechanics
The critical advection velocity va,0 given by our numerical technique remains to be compared with actual values of the advection velocity va of the Kármán vortex street that develops past an obstacle in order to determine the absolute or convective nature of the instability
We based our analysis on the point vortex model of Rosenhead (1929), accounting for the influence of confining walls, and we considered the vortices to be advected in the downstream direction at a certain unknown velocity
Summary
The flow past a circular cylinder is one of the most famous and well-studied problems in fluid mechanics. Spatio-temporal stability of the Kármán vortex street the secondary instability of spatially periodic flows resulting from the saturation of an unstable primary mode They showed that the single row of vortices induced by a mixing layer, while being subject to a secondary instability, becomes convectively unstable above a critical value of its advection velocity. In order to visualize the growth of the perturbation, the displacements from the two rows are first assembled together, resulting in vectors xm and ym for the horizontal and vertical displacements respectively, where even (odd) indices indicate lower (upper) row vortices The envelope of this composed signal is computed from its analytic representation, i.e. by setting all negative-wavenumber modes of its Fourier transforms to zero. It is the proportional decrease to zero of both ωi,m and k as p tends to p0, akin to a self-similar behaviour of the dispersion relation, that causes the impulse response wave packet to retain finite width while its height shrinks to zero when the system moves towards its stable configuration
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