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.

An investigation is described for the nonlinear stability, at large Reynolds numbers R , of the Hagen-Poiseuille flow through a pipe of circular cross section when subjected to three-dimensional disturbances of typical relative size δ large enough to yield only a vanishingly small phase shift across the critical layer. A crucial size is δ = O ( R -⅓ ) since then this small phase shift is in tune with the small phase shift produced by the viscous wall layers. The critical layer, which is fully nonlinear and three-dimensional, and the wall layers, where the disturbance is greater than the basic flow and flow reversal occurs, are discussed in detail. Neutral solutions are then found to exist in the range c 01 < c 0 < 1 with N = 1, where c 0 is the non-dimensional wavespeed, c 01 = 0.284 and N is the azimuthal wavenumber; there is also evidence to suggest that no similar neutral solutions exist outside that range. The amplitude-dependence of the neutral modes follows and it is such that the cut-off value c 0 = c 01 + is approached as the amplitude shrinks, whereas centre modes with c 0 → 1 - are produced as the amplitude becomes relatively large.

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A numerical study of the generation of Tollmien-Schlichting (T–S) waves due to the interaction between a small free-stream disturbance and a small localized variation of the surface geometry has been carried out using both finite–difference and spectral methods. The nonlinear steady flow is of the viscous–inviscid interactive type while the unsteady disturbed flow is assumed to be governed by the Navier–Stokes equations linearized about this flow. Numerical solutions illustrate the growth or decay of the T–S waves generated by the interaction between the free-stream disturbance and the surface distortion, depending on the value of the scaled Strouhal number. An important result of this receptivity problem is the numerical determination of the amplitude of the T–S waves.

The similarity equations for rotationally symmetric flow above an infinite counter–rotating disk are investigated both numerically and analytically. Numerical solutions are found when α, the ratio of the disk's angular speed to that of the rigidly rotating fluid far from it, is greater than −0.68795. It is deduced that there exists a critical value αcr, of α above which finite solutions are possible. The value of α and the limiting structure as α → αcrare found using the method of matched asymptotic expansions. The flow structure is found to consist of a thin viscous wall region above which lies a thick inviscid layer and yet another viscous transition layer. Furthermore, this structure is not unique: there can be any number of thick inviscid layers, each separated from the next by a viscous transition layer, before the outer boundary conditions on the solution are satisfied. However, comparison with the numerical solutions indicates that a single inviscid layer is preferred.