  In many engineering systems operating with a working fluid, the best
efficiency is reached close to a condition of flow separation, which makes its
prediction very crucial in industry. Providing that wall-based quantities can
be measured, we know today how to obtain good predictions for two and
three-dimensional steady and periodic flows. In these flows, the separation is
defined on a fixed line attached to a material surface. The last case to
elucidate is the one where this line is no longer attached to the wall but on
the contrary is contained within the flow. This moving separation is probably,
however, the most common case of separation in natural flows and industrial
applications. Since this case has received less attention during the past few
years, we propose in this study to examine some properties of moving separation
in two-dimensional, unsteady flows where the separation does not leave a
signature on the wall. Since in this framework separation can be extracted by
using a Lagrangian frame where the separation profile can be viewed as a
hyperbolic unstable manifold, we propose a method to extract the separation
point defined by the Lagrangian saddle point that belongs to this unstable
manifold. In practice, the separation point and profile are initially extracted
by detecting the most attracting Lagrangian coherent structure near the wall,
and can then be advected in time for following instants. It is found that
saddle points, which initially act as separation points in the viscous wall
flow region, remarkably preserve their hyperbolicity even if they are ejected
from the wall toward the inviscid region. Two test cases are studied, the
creeping flow of a rotating and translating cylinder close to a wall, and the
unsteady separation in the boundary layer generated by a planar jet impinging
onto a plane wall.
