Can Flow Separation from Curved Surfaces Be Predicted Without Boundary-Layer Calculations?

It is well known that flow separation from a curved surface is a purely viscous phenomenon that is governed by the flow behavior inside the boundary layer, and is typically characterized by the vanishing of the friction coefficient at the wall. However, in his seminal paper, where he introduced the boundary layer theory, Prandtl wrote a very interesting statement that is typically overlooked in the literature: “The most important applicable result of these investigations, is that in certain instances the fluid flow separates from the wall at a position fixed completely by external conditions." In this statement, Prandtl referred to some scenarios where the separation location is determined from the outer flow conditions, in contrast to the conventional wisdom. Prandtl did not specify what these scenarios are nor did he discuss the purported separation criterion based on outer-flow separation criterion. Exploring the different regimes of the flow over a circular cylinder, we pick the subcritical regime (10^4 < 𝑅𝑒 < 10^5) as an ideal candidate to test Prandtl’s hypothesis since the global characteristics (e.g., averaged force coefficients and separation angle) are fairly independent of Reynolds number over this regime. Moreover, there is a well-established theory for modeling the outer flow in this regime—the free streamline theory. We rely on the principle of minimum pressure gradient, 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. Hence, we pose the following problem: Over the family of kinematically-admissible, equilibrium flows provided by the free streamline theory, we single out the separating flow with the minimum pressure gradient cost. Interestingly, the obtained separation angles match the experimental measurements in the subcritical regime.