Marginal turbulent separation due to an imposed adverse pressure gradient

The development of an asymptotic theory of turbulent separation has been hampered severely by the fact that a pressure increase of O(1) appears to be necessary to separate an initially firmly attached turbulent boundary-layer even in the limit Re --> infinity. A different situation arises if one considers boundary-layers subjected to adverse pressure gradients such that the wall shear stress vanishes eventually but immediately recovers. Similar to the case of laminar marginally separated pouts the pressure changes in the vicinity of the point of vanishing wall shear then are small. This allows for further analytical progress which among others suggests a self-consistent description of the separation process by means of how the classical logarithmic law of the wall is gradually transformed into the square-root law that holds at the point of zero skin friction.