Three-dimensional analysis of the unidirectional oscillatory flow around a circular cylinder at low Keulegan-Carpenter and β numbers

In this paper, the oscillating flow around a circular cylinder is investigated numerically using both a three-dimensional and a two-dimensional model. Two important three-dimensional regimes of the Tatsuno & Bearman (J. Fluid Mech. vol. 211, 1990, p. 157) map are investigated: the asymmetric transverse-street regime D, and the double-pair diagonal regime F. The Stokes number is held constant (beta = 20) and the Keulegan-Carpenter number and the Reynolds number changed so that they match the conditions of these two regimes. The cross-sectional vortex streets (V-pattern in regime D and diagonal pattern in regime F) appear to be unstable, and switching from a pattern to its mirror-image mode occurs during the simulation. This switching is related to a two-dimensional instability in the flow field; this phenomenon can be reproduced by pure two-dimensional simulations. Three-dimensionality in the flow field always appears after the asymmetric vortex pattern has fully developed. The main effect of three-dimensionality in the flow field appears to be a rotation (circumferential effect) along the axial direction of the main sectional vortex patterns and a time delay along this axis of the switching from a two-dimensional mode to its mirror-image. These features contribute to generation of the three-dimensional sinuous S-mode as defined by Yang & Rockwell (J. Fluid Mech. vol. 460, 2002, p. 93). The three-dimensionality of the vorticity field affects the dynamic loads induced on the cylinder. The longitudinal component of the force acting on the cylinder appears to be weakly affected by three-dimensional effects, and so does its distribution in the axial direction. This finding explains why the results of two-dimensional simulations often agree fairly well with the data from laboratory experiments. Conversely, the transversal force appears to be significantly affected by the three-dimensional flow field, which suggests that an accurate numerical prediction requires the use of three-dimensional numerical models. Finally, a simplified conceptual model explains why the axial variation of the sectional transversal force always appears to be much larger than that of the corresponding longitudinal one. The model also explains why two-dimensional simulations tend to underpredict the r.m.s. value of the longitudinal force and to overpredict that of the transversal force, compared to the data from three-dimensional studies.