Nonlinear critical layers and their development in streaming-flow stability

Abstract

The effect of increasing disturbance size on the stability of a laminar streaming flow is considered theoretically at high Reynolds numbers Re. The theory has a rational basis that allows detailed understanding of the delicate physical balances controlling stability, and is presented with an accelerating boundary layer taken as the basic flow. The theory predicts that the scales and properties required to produce the Rayleigh situation (where the disturbances have wave speed and wavelength comparable to the typical speed and thickness respectively of the basic flow) in neutral stability are very different from those predicted by a classical approach, involving a relative disturbance size O(Re^−⅙) rather than the classical suggestion $O(Re^{-\frac{1}{3}})$. Before then, however, the disturbances undergo an abrupt alteration in scale and character as they pass through the just slightly smaller size $O(Re^{-\frac{7}{36}})$, with the stability structure changing from the relatively large-scale form of linear theory to the more condensed Rayleigh form by means of a nonlinear interaction within the critical layer. Strong higher harmonics of the fundamental disturbance are induced throughout the flow field by the velocity jump across the critical layer, but the phase jump remains the most significant property. Solutions for the nonlinear critical layer are recalculated and reanalysed. Also, the mean-flow correction produced by the nonlinear critical layer is shown to be smaller than the main part of the fundamental, owing to the regularity of the latter. As the Rayleigh stage is approached, the lateral variation of the induced pressure force through the critical layer begins to exert a considerable influence. Similar characteristics also arise in other fundamental streaming flows, and the implied Rayleigh stage is the subject of a subsequent investigation.