An exact theory of three-dimensional fixed separation in unsteady flows

We develop a nonlinear theory for separation and attachment on no-slip boundaries of three-dimensional unsteady flows that have a steady mean component. In such flows, separation and attachment surfaces turn out to originate from fixed lines on the boundary, even though the surfaces themselves deform in time. The exact separation geometry is not captured by instantaneous Eulerian fields associated with the velocity field, but can be determined from a weighted average of the wall-shear and wall-density fields. To illustrate our results, we locate separation surfaces and attachment surfaces in an unsteady model flow and in direct numerical simulations of a time-periodic lid-driven cavity.