New facts on the transition to three-dimensionality in a cylinder wake T. AKBAR , G. BOUCHET , J. DU ˇ SEK Institut de M´ ecanique des Fluides et des Solides 2, rue Boussingault, 67000 Strasbourg R´ esum´ e: L’´ etude concerne l’apparition de la tri-dimensionnalit´ e dans le sillage d’un cylindre. Nous effectuons une analyse lin´ eaire et des simulations pleinement non-lin´ eaires et examinons les structures tridimensionnelles. L’analyse lin´ eaire reproduit les r´ esultats connus. L’´ ecoulement bi-dimensionnel devient instable pour des perturbations tri-dimensionnelles via une bifurcation sous-critique qui donne naissance au mode A avec une longueur d’onde dans l’envergure de 4.02 diam` etres du cylindre au nombre de Reynolds de 189,5. L’´ ecoulement de base bi-dimensionnel subit une bifurcation suivante au nombre de Reynolds de 262 faisant apparaˆ ıtre un mode B avec une longueur d’onde de 0.83 diam` etres. Les nouveaux r´ esultats indiquent une stabilit´ e de la branche fortement non-lin´ eaire de la bifurcation sous-critique loin au dessous du seuil de l’instabilit´ e lin´ eaire. L’´ etude du comportement sous-critique n´ ecessite une prise en compte de grandes ´ echelles sous-harmoniques. L’´ etude du mode B dans un contexte pleinement non-lin´ eaire montre qu’aucune nouvelle bifurcation ne semble donner naissance au mode B. Nous montrons la pr´ esence du mode B d` es le nombre de Reynolds de 190 dans une simulation avec une p´ eriodicit´ e en envegure de 4d. Progressivement, le mode B domine le mode A dans le sillage proche. Abstract : This study deals with onset of three dimensionality in the flow past a circular cylinder. Both linear analysis and non-linear simulations have been performed and three dimensional patterns have been investigated. The linear analysis confirms earlier results. The two dimensional flow behind a circular cylinder becomes unstable to three dimensional perturbations through a subcritical bifurcation to an instability mode A with a spanwise wavelength of 4.02 cylinder diameter at a Reynolds number 189.5. The two dimensional base flow undergoes a next bifurcation at a Reynolds number 262 with a spanwise wavelength of 0.83 cylinder diameter giving rise to the B mode. The new results show the strongly non-linear branch of the subcritical bifurcation to be stable well below the linear instability threshold. A reliable investigation of the sub-critical behaviour necessitates to account for large spanwise sub- harmonic scales. No new bifurcation seems to explain the presence of the B-mode. We show that the B-mode is already present at in a simulation with spanwise periodicity of . It progressively dominates the A mode in the near wake. Mots clefs : cylinder wake, onset of three-dimensionality, non-linear effects 1 Introduction Since many decades, the flow behind a circular cylinder has been a subject of great interest for researchers and engineers because of its applications in many fields. The works by Williamson [1], [2], [3], [4], Thompson et al. [5], [6], [7], [8] and Barkley & Henderson [9] are those who provided, experimentally and numerically, worthy results shedding light on the three dimensional instabilities in the wake of a circular cylinder. The two modes of three dimensional instabilities were first observed experimentally by Williamson [1], [2], Miller & Williamson [10] and Leweke & Williamson[11] and are named mode A and mode B instabilities. The mode A vortex shedding appears at a Reynolds number of approximately and has a spanwise wavelength of approximately , where is the cylinder diameter. The spanwise wavelength of mode A decreases from to as the Reynolds number increases. The mode B is found to appear at a Reynolds number , having a spanwise wavelength equal to one cylinder diameter. At Reynolds number the mode A is strong and mode B is weak. With the increase of Reynolds number beyond the redistribution of energy takes place between the modes and the mode B becomes dominant at the Reynolds number of . Zhang et al. [12] found another mode (different from modes A and B) in the flow behind an infinite bar. It is named as mode C instability. To produce mode C, authors perturbed the flow on one side of the wake of a circular cylinder by placing a tiny wire close to a circular cylinder which in turn leads to a subharmonic mode with a spanwise wavelength in 1