A study of two-dimensional flow past regular polygons via conformal mapping

Abstract

In this paper we study two-dimensional flow around regular polygons with an arbitrary but even number of edges N and one apex pointing to the free stream, with comparison to circular-cylinder flow. Both inviscid flow and low-Reynolds-number viscous flow are addressed. For inviscid flow, we obtained the exact solution for pure potential flow through Schwarz–Christoffel transformation, with the emphasis on the role of edge number, N, on the flow details. We also studied the behaviour, stationary lines and stability of vortex pair and found new stationary lines compared to circular cylinder. For viscous flow we derived the equation of stream function in the mapped (circle) domain, based on which approximate expressions for the critical Reynolds numbers and Strouhal number, as functions of the edge number, are obtained. The Reynolds number is based on the diameter of the circumscribed circle. For the steady flow, the first critical Reynolds number is a monotonically decreasing function of N, while N → ∞ corresponds to that for circular cylinder. The bifurcation point is ahead of the bifurcation point for circular cylinder. For unsteady flow, the critical Reynolds number for vortex shedding and the Strouhal number are both monotonically decreasing functions of N.

On vortex shedding from a hexagonal cylinder

“…It is a well established fact for wakes behind circular cylinders (Fey et al [16]) that the Strouhal number increases with Re throughout the parameter range in which the vortex shedding remains two-dimensional. This trend was also found for square cylinder wakes (Sohankar et al [7]) and for wakes behind other regular polygons with N 6 by Tian and Wu [15]. At higher Reynolds numbers, however, no discernible Reynolds number effects on St were reported for square cylinder wakes by Vickery [5], Knauss et al [6], Dutta et al [11].…”

“…Sohankar et al [7], however, found that St varied from 0.148 (face orientation) to 0.169 (corner orientation) at a Reynolds number Re = 100, whereas the correlation (1) gives St = 0.165 for a circular cylinder at the same Reynolds number. The latter value is above the Strouhal number 0.1585 found behind the corneroriented hexagonal cylinder and is thus apparently in conflict with the anticipations of Tian and Wu [15]. Although not stated explicitly, the Strouhal numbers reported in Ref.…”

“…from face orientation to corner orientation, the Strouhal number is gradually increased such that St for a corner-oriented square cylinder is higher than for the sideoriented cylinder. For a corner-oriented regular polygon, Tian and Wu [15] showed that St decreased with an increasing number N of faces and approached the Strouhal number of a circular cylinder in the limit as N becomes infinitely large. It should therefore be expected that the Strouhal numbers found for the hexagonal cylinder are somewhat lower than those found for square cylinder wakes.…”

“…[14]. Two-dimensional flow past regular polygons at low Reynolds numbers has recently been subjected to an extensive analytical and computational investigation by Tian and Wu [15]. They considered infinitely long cylinders with polygonal cross-sections but only with an even number of faces and only with corner orientation.…”

Numerical investigation of flow rate measure using vortex counting at low Reynolds number

“…The basis of flow around circular cylinders have been reported in the standard reference by Zdravkovich [1] where Reynolds numbers range were given for different regimes of flow around smooth circular cylinder in steady flow. Tian and Wu [2] generalized a relationship between Strouhal and Reynolds numbers by an extensive analytical and computational investigation of two-dimensional flow around regular polygons at low Reynolds number. Lee [3]studied numerically the symmetrical wake flow developments around a tapered trapezoidal cylinder.…”