Exact theory of unsteady separation for two-dimensional flows

We use a dynamical systems approach to extend Prandtl's steady separation criterion to two-dimensional unsteady flows with no-slip boundaries. Viewing separation profiles as non-hyperbolic unstable manifolds in the Lagrangian frame, we obtain explicit Eulerian formulae for the location of flow separation and reattachment on fixed and moving boundaries. We also derive high-order approximations for the unsteady separation profile in the vicinity of the boundary. Our criteria and formulae only use the derivatives of the velocity field along the boundary, and hence are of use in monitoring and controlling separation. In particular, we predict unsteady flow separation points and separation angles from distributed pressure and skin-friction measurements along the wall. As an example, we predict and verify separation points and separation profiles in variants of a two-dimensional oscillating separation-bubble flow.