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

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Abstract
We formulate here a lowest order parabolic (LOP) theory for investigating the stability of two-dimensional spatially developing boundary layer flows. Adopting a transformation earlier proposed by the authors, and including terms of order R−2/3 where R is the local boundary-layer thickness Reynolds number, we derive a minimal composite equation that contains only those terms necessary to describe the dynamics of the disturbance velocity field in the bulk of the flow as well as in the critical and wall layers. This equation 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 theory). The LOP equation is shown to give results close to the full nonparallel theory, and is the highest-order stability theory that is justifiable with the lowest-order mean velocity profiles for the boundary layer.

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