Stability of Disturbance Waves in Developing Shear Flows: A Review of ‘Ad-hoc’ Methods

This paper attempts to induce some formalism in the study of stability of developing shear flows, by use of so called ‘ad-hoc’ methods wherein all terms upto and inclusive of a particular order are lumped together in the same equation. However, whether the ‘ordering’ is perturbation or asympotic is somewhat vague. The classical example of stability of a developing shear flow is the stability problem of boundary-layer flow over a flat-plate including the non-parallel effects. Other examples are free shear flows; also, problems involving flow over alternate rigid and porous panels with suction, or suction from one wall and injection from the opposite wall. The quick method for stability analysis is to use the local Orr-Sommerfeld solution, using the quasi-parallel approximation. This is sometimes augmented by inclusion of some non-parallel terms, perhaps in an ad-hoc manner, and so called ‘improved solutions’ are obtained. The present paper attempts to establish a formal framework for these methods so that these methods are no more termed ‘ad-hoc’.