A comprehensive study of the onset of boundary layer separation in the unbounded flow around a circular cylinder

Abstract

The onset of boundary layer separation in the unbounded flow around a circular cylinder is determined directly with a numerical experiment with accuracy to the second decimal point of the Reynolds number, Re. The governing equations are the steady state, two dimensional Navier-Stokes equations, which are solved with Galerkin finite elements. A new criterion, the entrance ratio E = D / L with D the diameter of the cylinder and L the entrance length of the domain, is introduced in addition to the traditional blockage ratio; the aim is to establish conditions for unbounded flow in both flow directions. A hypothesis is formulated and verified for the onset of separation leading to the conclusion that this flow undergoes a supercritical pitchfork bifurcation with respect to fixed points on the boundary. Results are presented for various blockage ratios, B, in the range 1.2238 · 10 - 8 ≤ B ≤ 0.005 and constant entrance ratio E = 2.4472 · 10 - 9 until asymptotic behavior is observed for the onset of separation in the Reynolds number, in the angle and the length of the smallest eddy. Onset of separation occurs in the region 6.36 ≤ R e < 6.37. The bifurcation diagram of the flow is calculated. Streamlines of the flow domain before and after the bifurcation point are shown, including images of the smallest possible eddies. In addition, coefficients of base suction and total drag are computed and shown in the regime 0.1 ≤ R e ≤ 40. The results are compared with available laboratory and numerical measurements.