Asymmetric vortex pair in the wake of a circular cylinder

Abstract

Stationary configurations of two asymmetric point vortices in the wake of an infinite circular cylinder, spinning or not about its axis, are analytically investigated using an ideal fluid approximation. Four different vortex configurations (patterns) in the wake of a spinning cylinder are found in the case when vortex asymmetry is weak; each configuration is associated with a certain direction of the Magnus force. The qualitative relation between a pattern and a direction of the Magnus force is in agreement with experimental data. Also obtained are asymmetrical vortex configurations in the wake of a nonspinning cylinder. HE flowfield developing around an infinitely long circular cylinder in a crossflow was extensively studied over the last century both analytically and experimentally. It is currently a wellknown fact that several characteristic flow patterns in the wake of the cylinder are associated with the crossflow Reynolds number Re.1 In particular, when 6 <Re < 30 (the Reynolds number is based on the cylinder's diameter, and the ranges cited ar approximate since they can be affected by the roughness of the cylinder's surface), a pair of concentrated vortices is observed in the wake. The vortices are symmetric relative to the direction of the crossflow and equidistant from the cylinder; both the strength and the distance of the vortices from the cylinder increase with the Reynolds number. When Re increases above 30, the vortices may become asymmetric. In this case, a side force (or a Magnus force; hereafter both names will be used synonymously) is exerted upon the cylinder. With a further increase in the Reynolds number, additional pairs of asymmetric vortices are shed from the cylinder. If the cylinder spins about its axis, then the respective wake vortices are inherently asymmetric; both the loci of the vortices and the side force acting on the cylinder depend on the cylinder's spin rate and on the crossflow Reynolds number.2 The majority of the analytical studies of the problem were conducted in the framework of the ideal fluid approximation.36 The latter may be summarized as follows. The fluid is assumed inviscid and incompressible; it is postulated that there exists a wake behind the cylinder; it is further postulated that the wake is comprised of several stationary point vortices, the number of the wake vortices deriving from the respective physical flow pattern (cf. the previous paragraph). Each of the cited studies addressed different equilibrium configurations of the vortices. Thus, stationary configurations were found for several pairs of symmetric vortices near a circular cylinder,3'4 for several pairs of symmetric vortices behind an oscillating circular cylinder confined between walls,5 and for several pairs of symmetric vortices behind a noncircular cylinder.6 All of these studies dealt with symmetric vortices behind a nonspinning cylinder. To the best of our knowledge, asymmetric vortices behind either spinning or nonspinning cylinders were never investigated (although the case of a spinning cylinder without wake vortices was solved by Magnus 7 in 1853). This problem is addressed in the present publication.