Rayleigh's stability criterion describes the inviscid stability of rotating fluid flows. Despite the limitations of the criterion due to the assumptions used, it has been widely viewed as a general stability barrier in various rapidly rotating flows. However, contrary to previous belief, a linear instability is identified in Rayleigh-stable Taylor-Couette flow. The instability is found in cyclonic rapid rotation regime, for almost the entire range of the radius ratio of the cylinders.

Nonlinear equilibrium states characterized by strongly localized vortex pairs are calculated in the linearly stable parameter region of counterrotating Taylor-Couette flow. These subcritical states are rotating waves whose region of existence is consistent with the critical threshold for relaminarization observed in experiments. For sufficiently rapid outer cylinder rotation the solutions extend beyond the static inner cylinder case to corotation, thus exceeding, for the first time, the boundary defined by the inviscid Rayleigh's stability criterion.

We investigate the local self-sustained process underlying spiral turbulence in counter-rotating Taylor–Couette flow using a periodic annular domain, shaped as a parallelogram, two of whose sides are aligned with the cylindrical helix described by the spiral pattern. The primary focus of the study is placed on the emergence of drifting–rotating waves (DRW) that capture, in a relatively small domain, the main features of coherent structures typically observed in developed turbulence. The transitional dynamics of the subcritical region, far below the first instability of the laminar circular Couette flow, is determined by the upper and lower branches of DRW solutions originated at saddle-node bifurcations. The mechanism whereby these solutions self-sustain, and the chaotic dynamics they induce, are conspicuously reminiscent of other subcritical shear flows. Remarkably, the flow properties of DRW persist even as the Reynolds number is increased beyond the linear stability threshold of the base flow. Simulations in a narrow parallelogram domain stretched in the azimuthal direction to revolve around the apparatus a full turn confirm that self-sustained vortices eventually concentrate into a localised pattern. The resulting statistical steady state satisfactorily reproduces qualitatively, and to a certain degree also quantitatively, the topology and properties of spiral turbulence as calculated in a large periodic domain of sufficient aspect ratio that is representative of the real system.

Interactions between nearly bicritical modes in Taylor-Couette flow, which have been concerned with the framework of weakly nonlinear theory, are extended to fully nonlinear Navier-Stokes computation. For this purpose, a standard Newton solver for axially periodic flows is generalized to compute any mixed solutions having up to two phases, which typically arise from interactions of two spiral or Taylor vortex modes. Also, a simple theory is developed in order to classify the mixed solutions. With these methods, we elucidate pattern formation phenomena, which have been observed in a Taylor-Couette flow experiment. Focusing on the counter-rotating parameter range, all possible classes of interaction of various solutions with different azimuthal and axial wave numbers are considered within our computational restriction, and we observe numerous connection branches, e.g., footbridge solutions. Some of the mixed solutions result in a three-dimensional wavy spiral solution with axial relative periodicity or an axially doubly periodic toroidally closed vortex solution. The possible connection of the former solution family to spiral turbulence, which has been observed in highly counter-rotating Taylor-Couette flow, is discussed.

The present paper treats the large Reynolds number scaling of coherent structures in stably stratified flows sheared between two horizontally placed walls. Three-dimensional steady solutions are used to confirm the theoretical scaling. For small values of the Richardson number, the previously known scaling based on the vortex–wave interaction/self-sustaining process is found to give excellent predictions of the numerical results. When the Richardson number is increased, the maximum size of the vortices is limited by the Ozmidov scale. The largest possible Richardson number to sustain the vortices is predicted to be of order unity when the typical length scale of the vortices reaches the Kolmogorov scale. The minimum-scale vortices are governed by unit Reynolds number Navier–Stokes equations.

The effect of free stream coherent structures in the asymptotic suction boundary layer on the initiation of Görtler vortices is considered from both the ‘imperfect’ bifurcation and receptivity viewpoints. Firstly a weakly nonlinear and a full numerical approach are used to describe Görtler vortices in the asymptotic suction boundary layer in the absence of forcing from the free stream. It is found that interactions between different spanwise harmonics occur and lead to multiple secondary bifurcations in the fully nonlinear regime. Furthermore it is shown that centrifugal instabilities of the asymptotic suction boundary layer behave quite differently than their counterparts in either fully developed flows such as Couette flow or growing boundary layers. A significant result is that the most dangerous disturbance is found to bifurcate subcritically from the unperturbed state. Within the weakly nonlinear regime the receptivity of Görtler vortices to the free stream exact coherent structures discovered by Deguchi & Hall (J. Fluid Mech., vol. 752, 2014, pp. 602–625; J. Fluid Mech., vol. 778, 2015, pp. 451–484) is considered. The presence of free stream structures results in a resonant excitation of Görtler vortices in the main boundary layer. This leads to imperfect bifurcations reminiscent of those found by Daniels (Proc. R. Soc. Lond. A, vol. 358, 1977, pp. 173–197) and Hall & Walton (Proc. R. Soc. Lond. A, vol. 358, 1977, pp. 199–221; J. Fluid Mech., vol. 90, 1979, pp. 377–395) in the context of transition to finite amplitude Bénard convection in a bounded region. In order to understand the receptivity problem for the given flow the spatial initial value problem for this interaction is also considered when the free stream structure begins at a fixed position along the wall. Remarkably, it will be shown that free stream structures are incredibly efficient generators of Görtler vortices; indeed the induced vortices are found to be larger than the free stream structure which provokes them! The relationship between the imperfect bifurcation approach and receptivity theory is described.