Flow past a cylinder: shear layer instability and drag crisis

Abstract

Flow past a circular cylinder for Re=100 to 10⁷ is studied numerically by solving the unsteady incompressible two-dimensional Navier-Stokes equations via a stabilized finite element formulation. It is well known that beyond Re ~ 200 the flow develops significant three-dimensional features. Therefore, two-dimensional computations are expected to fall well short of predicting the flow accurately at high Re. It is fairly well accepted that the shear layer instability is primarily a two-dimensional phenomenon. The frequency of the shear layer vortices, from the present computations, agree quite well with the Re^0.67 variation observed by other researchers from experimental measurements. The main objective of this paper is to investigate a possible relationship between the drag crisis (sudden loss of drag at Re ~ 2 × 10⁵) and the instability of the separated shear layer. As Re is increased the transition point of shear layer, beyond which it is unstable, moves upstream. At the critical Reynolds number the transition point is located very close to the point of flow separation. As a result, the shear layer eddies cause mixing of the flow in the boundary layer. This energizes the boundary layer and leads to its reattachment. The delay in flow separation is associated with narrowing of wake, increase in Reynolds shear stress near the shoulder of the cylinder and a significant reduction in the drag and base suction coefficients. The spatial and temporal power spectra for the kinetic energy of the Re=10⁶ flow are computed. As in two-dimensional isotropic turbulence, E(k) varies as k^-5/3for wavenumbers higher than energy injection scale and as k^-3 for lower wavenumbers. The present computations suggest that the shear layer vortices play a major role in the transition of boundary layer from laminar to turbulent state.