  Thwaites (1949) developed an approximate method for determining the evolution
of laminar boundary layers. The approximation follows from an assumption that
the growth of a laminar boundary layer in the presence of pressure gradients
could be parameterized solely as a function of a flow parameter, $m =
\theta^2/\nu \frac{dU_e}{ds}$, thus reducing the von Karman momentum integral
to a first-order ordinary differential equation. This method is useful for the
analysis of laminar flows, and in computational potential flow solvers to
account for the viscous effects. However, for turbulent flows, a similar
approximation for turbulent boundary layers subjected to pressure gradients
does not yet exist. In this work, an approximate method for determining the
momentum thickness of a two-dimensional, turbulent boundary layer is proposed.
It is shown that the method provides good estimates of the momentum thickness,
when compared to available high-fidelity simulation data, for multiple boundary
layers including both favorable and adverse pressure gradient effects, up to
the point of separation. In the limit of high Reynolds numbers, it is possible
to derive a criterion for the onset of separation from the proposed model which
is shown to be in agreement with prior empirical observations (Alber,
\textit{$9^{th}$ Aerospace Sciences Meeting, 1971}). The sensitivity of the
separation location with respect to upstream perturbations is also analyzed
through this model for the NASA/Boeing speed bump and the transonic
Bachalo-Johnson bump
