Low-order parabolic theory for 2D boundary-layer stability

We formulate here a lowest order parabolic (LOP) theory for investigating t he stability of two-dimensional spatially developing boundary layer flows. Adopting a transformation earlier proposed by the authors, and including te rms of order R-2/3 where R is the local boundary-layer thickness Reynolds n umber, we derive a minimal composite equation that contains only those term s necessary to describe the dynamics of the disturbance velocity field in t he bulk of the flow as well as in the critical and wall layers. This equati on completes a hierarchy of three equations, with an ordinary differential equation correct to R-1/2 (similar to but different from the Orr-Sommerfeld ) at one end, and a "full" nonparallel equation nominally correct to R-1 at the other (although the latter can legitimately claim higher accuracy only when the mean flow in the boundary layer is computed using higher order th eory). The LOP equation is shown to give results close to the full nonparal lel theory, and is the highest-order stability theory that is justifiable w ith the lowest-order mean velocity profiles for the boundary layer. (C) 199 9 American Institute of Physics. [S1070-6631(99)01006-5].