The formulation and approximate solution of the boundary layer equations is historically the first entry in the array of the methods available in computational aerodynamics (Prandtl, 1904).
Back in the old days when calculations had to be done by hand, scientists (Prandlt, Blasius, Polhausen, Von Karman, Falkner and Skan) used their intelligence to design practical methods. These methods are usually of integral form and provide global quantities relative to the viscous layers, namely displacement and momentum thickness, shape factor, and higher order terms (if any).
By themselves the boundary layer equations are not a closed system from a physical point, since they require one boundary condition from the external inviscid flow (pressure or velocity distribution at the edge of the boundary layer). These conditions are easy to set on flat plates aligned with the free stream, but pose a challenging difficulty on more general cases, when flow separation is involved.
From a mathematical point of view, instead, the boundary layer equations are a classical singular-perturbation problem (Van Dyke, 1964). The singular perturbation theory is very useful to analyze the order of magnitude of the viscous layers and other important quantities. For basic boundary layer flows see the references reported below.
Like in the old days most computational methods today rely on integral equations. These equations are of the Von Karman type, but have additional terms to take into account at least second order terms, such as streamline curvature and wake effects (Le Balleur, 1978; discussion in Lock-Williams, 1987).
Methods based on the solution of the differential equations have become more rare, due to the difficulty of coupling strongly the viscous solution to the external inviscid field. Numerical methods for boundary layer solutions developed over the past 25 years have been almost exclusively designed for coupling with the external inviscid flow.
After the explosion of theoretical boundary layers and computational methods,
most aspects are today well known. The topics most open to understanding are still
the turbulent transition (especially at very low and very high Reynolds number),
three dimensional unsteady boundary layers (Cousteix, 1986; Le Balleur, 1993).