A Study On Flow Past Regular Polygons

Flow past polygons has found wide application. It is rare to consider that about apolygon with an arbitrary edge number. In this paper we focused on the features of two-dimensional flow around regular polygons, and both potential flow and low-Reynoldsnumber viscous flows are addressed. We studied the flow features varying with the edgenumber, with comparison to flow past a circular cylinder.With the aid of Schwarz-Christoffel mapping, we obtained the general equation forflow past polygons, e.g. exact solution for pure potential, the equation of stream functionin the mapped (circle) domain, and the Eigen functions for disturbance evolution. Fromthe view of these equations, we studied the general differences of flow features betweenpolygons and circles.For potential flow, using the exact solution for pure potential, we studied the purepotential flow details and the singularity at apices varying with the edge number, N, atdifferent flow directions. We also studied the behavior, stationary lines and stability, ofvortex-pair. We found some new stationary lines for vortex pairs compared to circularcylinder, and different stabilities corresponding to the edge number and direction.Forviscousflow,weobtaincriticalReynoldsnumbersandStrouhalnumbersvaryingwith the edge number, with the aid of the equation of stream function in the mapped(circle) domain, and the eigen functions for disturbance evolution and CFD methods.For the steady flow when separation first occurs, we obtained the 1st criticalReynolds numbers (the value when flow separation firstly occurs) varying with N, anddifferent streamlines topologies after separation occurs, compared with the circular situa-tion. Withone apex locating on the rear stagnation point, the1stcriticalReynoldsnumberis larger than that for a circular cylinder, and is a monotonically decreasing function ofN. While N tends to infinity, the 1st critical Reynolds number corresponds to that for acircular cylinder. The bifurcation point (position where separation firstly occurs) is aheadof the bifurcation point for circular cylinder (at the rear stagnation point). With one mid-point of the edge locating on the rear stagnation point, the 1st critical Reynolds numberis smaller than that for a circular cylinder, and is a monotonically increasing function ofN. While N tends to infinity, the 1st critical Reynolds number corresponds to that for a circular cylinder. The bifurcation point is just at the rear stagnation point.For unsteady flow, we studied the 2nd critical Reynolds number (the value whenunsteady flow firstly formed) and the Strounal number varying with the edge number. Wefoundthe2ndcriticalReynoldsnumberandtheStrouhalnumberarebothlargerthanthosefor a circular cylinder, and are monotonically decreasing functions of N. While N tendsto infinity, the Reynolds number and Strouhal number correspond to those for a circularcylinder.