On the Separating Flow Behind a Cylinder: Insights from the Principle of Minimum Pressure Gradient

Abstract

We study the separating flow over a circular cylinder with two objectives: (i) to demonstrate the validity of the condition of matching curvature, and (ii) to obtain a reasonable estimate of the separation angle in the subcritical regime (Re=10^4-10^5) without explicitly modeling the boundary layer. First, we study Roshko's free streamline model (1954); it is an ideal flow model with sheets of discontinuities that represent the separating shear layers in the near wake region. The model fails to predict the correct separation angle over a cylinder. Roshko attributed this discrepancy to the condition of matching curvature, which asserts that the curvature of the separating streamline at the separation point must match that of the cylinder. We show that such a condition is legitimate and is not the real culprit for the failure of Roshko's model in predicting separation. Second, we employ the principle of minimum pressure gradient (PMPG), which asserts that, an incompressible flow evolves by minimizing the total magnitude of the pressure gradient over the domain. Encouraged by the fact that the flow characteristics in the range Re=10^4-10^5 are fairly independent of Re, we aim to predict the separation angle in this regime without modeling the boundary layer -- a task that may seem impossible, though anticipated by Prandtl in his seminal paper (Prandtl 1904). Over the family of kinematically-admissible, equilibrium flows, we utilize the PMPG to single out the separating flow with the minimum pressure gradient cost. Interestingly, the obtained separation angles match experimental measurements over the regime Re=10^4-10^5.