We derive an exact theory of three-dimensional steady separation and reattachment using nonlinear dynamical systems methods. Specifically, we obtain criteria for separation points and separation lines on fixed no-slip boundaries in compressible flows. These criteria imply that there are only four basic separation patterns with welldefined separation surfaces. We also derive a first-order prediction for the separation surface using wall-based quantities; we verify this prediction using flow models obtained from local expansions of the Navier-Stokes equations.

We develop a nonlinear theory for separation and attachment on no-slip boundaries of threedimensional 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 wallshear and wall-density fields. To illustrate our results, we locate separation surfaces and attachment surfaces in unsteady model flows and in direct numerical simulations of a time-periodic lid-driven cavity.