Effective Boundary Layer Length
x' is the effective length of the boundary layer.
For a laminar flat plate x' is equal to x. For boundary layers
with pressure gradients, the effective length is modified by the fact that
favorable pressure gradients are good for the boundary layer health and
so a boundary layer traveling in a favorable gradient is similar to one
that has traveled a shorter distance without the favorable gradient.
Thus, for laminar boundary layers: 
For turbulent boundary layers, Stratford derived his equations by
assuming a flat turbulent rooftop distribution before the start of pressure
recovery. In this case x' = x, the distance from the leading edge. When
a pressure gradient exists or the flow is laminar, Stratford's criterion
may still be used as long as the effective turbulent boundary layer length
is used. The effective distance, x', is the effective distance to the start
of recovery, xm, plus the actual distance traveled after the recovery section
begins.
For turbulent flow with a pressure gradient, note the difference between
this expression and that for the laminar boundary layer shown on a previous
card.
Another case of great interest involves the effective boundary layer length
with some laminar flow followed by transition to a turbulent boundary layer.
The same basic idea is used in this case:
x' = x'm + (x - xm)
and the problem is that of computing x'm in the presence of transition.
This is quite straightforward when Ue = constant. This special case is
discussed on the following card.
If we have a flat rooftop section with no pressure gradient we can estimate
the equivalent turbulent boundary layer length as follows:
If we assume that the momentum thickness is not changed in the transition
process, then we can compute the thickness of the laminar layer at transition
and length of the turbulent boundary layer which would lead to the same
momentum thickness.
The laminar flat plate momentum thickness is:
θ = 0.671 x Rex-.5
and for a turbulent flat plate (from Schlichting -- see also Kuethe and
Chow 408):
θ = 0.036 x Rex-.2
The effective turbulent length is then:
xm' = 38.7 xm Rexm-3/8