In this paper, we study the free streamline theory for the separating flow over a circular cylinder. The objective of this paper is twofold: (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 ($\text{Re}={10}^4–{10}^5$) without explicitly modeling the boundary layer, following Prandtl's conjecture in his seminal paper [L. Prandtl, in Proceedings of the Third International Mathematical Congress in Heidelberg (B. G. Teubner, Leipzig, 1904), pp. 484–491]. For the former goal, we study Roshko's free streamline model [Roshko, Tech. Rep. No. NACA-TN-3168 (NASA, Washington, D.C., 1954)]; it is an ideal flow model with sheets of discontinuities that represent the separating shear layers in the near-wake region. It is known that the model fails to predict the correct separation angle over a curved surface. Roshko attributed this discrepancy to the condition of matching curvature, deeming it invalid. This condition (known as the condition of smooth detachment) asserts that the curvature of the separating streamline at the separation point must match that of the cylinder. We show that the condition of matching curvature is legitimate and is not the real culprit for the failure of Roshko's model in predicting the correct separation angle. We also show that Roshko's model is nonunique. There are solutions in Roshko's family with matching curvature whose separation point agrees with experimental measurements. As for the second goal, we rely on the principle of minimum pressure gradient (PMPG), which asserts that an incompressible flow evolves from one instant to another in order to minimize the total magnitude of the pressure gradient over the domain. Encouraged by the fact that the flow characteristics in the range $\text{Re}={10}^4–{10}^5$ are fairly independent of $\text{Re}$, we aim to predict the separation angle in this regime without modeling the boundary layer—a task that may seem impossible, though anticipated in Prandtl's original paper that introduced the concept of boundary layer [Prandtl, Proc. Intl. Congress Math. (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 the experimental measurements over the $\text{Re}={10}^4–{10}^5$ regime.

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• Aerodynamics
• Boundary layer receptivity, stability & separation
• High-speed flow
• Vortex shedding
• Euler equation