Bénard-von Kármán vortex street in a confined geometry: wavelength selection by Kelvin-Helmholtz instabilities
Résumé
We have reconsidered the formation and stability of a vortex street, induced in a rectangular container by a tape moving at high speed on its free surface. In a certain range of tape velocity and of geometrical aspect ratios, the liquid recirculates along the lateral sides of the pool, which induces two shear flows between the tape and these lateral sides, that undergo two coupled Kelvin-Helmholtz instabilities, giving rise to the vortex street. Contrary to the classical situation of a wake, behind an obstacle, the double row remains static which allows one to study its absolute stability in a static framework. In the present paper we have built a model of the two shear flow instabilities that clarifies the wavelength selection problem, inside the stability tongue predicted long ago by Rosenhead, and reduced by steric arguments that we found in a previous paper. In summary, in terms of fig. 1(b) notations, the mean-wavelength favored by Kelvin-Helmholtz instabilities is given by 2bmax ≈ πc, while the maximal wavelength predicted by marginal stability is equal to 2bc 1 ≈ 5.71 c. Our available experimental data are in very good agreement with these results and with the resulting phase diagram depicted on fig. 7 .
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