Bénard-von Kármán vortex street in a Kelvin-Helmholtz–type confined geometry: Wavelength selection by shear flow instabilities

¹ Matière et Systèmes Complexes, CNRS and Université Paris Cité, UMR 7057 - Paris, France
² Laboratoire Interdisciplinaire des Sciences du Numérique, CNRS and Université Paris-Saclay, UMR 9015 Orsay, France

Received: 18 February 2025
Accepted: 20 November 2025

Abstract

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 stationary framework. In the present paper we present a stability analysis around the mean flow 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 the mean wavelength favored by shear flow instabilities is given by , where c is half the channel width, while the maximal wavelength predicted by marginal stability is equal to . Our available experimental data are in very good agreement with these results and with the resulting phase diagram.