Navier-Stokes equilibrium solutions of a viscous fluid confined between two infinite parallel plates that can independently stretch or shrink in orthogonal directions are studied. It is assumed that the admissible solutions satisfy spatial self-similarity in the stretching or shrinking perpendicular coordinates. The nonlinear steady boundary-value problem is discretized using a spectral Legendre method, and equilibrium solutions are found and tracked in the two-dimensional parameter space by means of pseudoarclength continuation Newton-Krylov schemes. Different families of solutions have been identified, some of which are two-dimensional and correspond to the classical Wang & Wu self-similar flows arising in a plane channel with one stretching-shrinking wall; C. A. Wang and T. C. Wu, Comput. Math. Applic., 30, 1-16 (1995). However, a large variety of three-dimensional solutions have also been found, even for low stretching or shrinking rates. When slightly increasing those rates, some of these solutions disappear at saddle-node bifurcations. By contrast, when both plates are simultaneously stretching or shrinking at higher rates, a wide variety of new families of equilibria are created-annihilated in the neighbourhood of cuspidal codimension-2 bifurcation points. This behaviour has similarities with the one observed in other planar and cylindrical self-similar flows.

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.

We investigate two distinct scenarios of spatial modulation that are candidate mechanisms for streamwise localisation of waves in two-dimensional plane Poiseuille flow. The first one stems from a symmetry-breaking bifurcation that disrupts the half-shift & reflect equivariance of Tollmien-Schlichting waves (tsw). A new state, an asymmetric tsw (atsw), emerges from unstable lowerbranch tsws at subcritical Reynolds number and undergoes subharmonic Hopf bifurcations that lead to branches of asymmetric time-periodic space-modulated waves (matsw). Streamwise modulation does not evolve into localisation within the range of parameters explored. In breaking the last standing remnants of the reflectional symmetry about the channel midplane, atsw and matsw admit a bias towards either one of the channel walls, thus bearing a potential for explaining near-wall structures that are typical of developed turbulence. The second scenario follows the fate of a branch of time-periodic space-modulated tsws (mtsw) initially discovered by Mellibovsky & Meseguer [J. Fluid Mech. 779, R1 (2015)]. We find that these waves can lead to localisation but the mechanism is not new, as they do so through their connection, by means of a codimension-2 bifurcation point, with other known localising mtsws. The codimension-2 point is however responsible for the appearance of mtsws that exclusively bridge upper-branch tsw-trains of different number of replicas. In this respect, these mtsws possess all required properties that single them out as possible constituents of the strange saddle that governs domain-filling turbulent dynamics at high Reynolds numbers.

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In this paper we explore the stability and dynamical relevance of a wide variety of steady,time-periodic, quasiperiodic and chaotic flows arising between orthogonally stretching par-allel plates. We first explore the stability of all the steady flow solution families formerlyidentified by Ayats et al. [Ayats, R., Marques, F., Meseguer, A. and Weidman, P., Flowsbetween orthogonally stretching parallel plates, Phys. Fluids, 33, 024103 (2021)], con-cluding that only the one that originates from the Stokesian approximation is actually sta-ble. When both plates are shrinking at identical or nearly the same deceleration rates,this Stokesian flow exhibits a Hopf bifurcation that leads to stable time-periodic regimes.The resulting time-periodic orbits or flows are tracked for different Reynolds numbers andstretching rates, whilst monitoring their Floquet exponents in order to identify secondaryinstabilities. It is found that these time-periodic flows also exhibit Neimark-Sacker bifur-cations, generating stable quasiperiodic flows (tori) that may sometimes give rise to chaoticdynamics through a Ruelle-Takens-Newhouse scenario. However, chaotic dynamics is un-usually observed, as the quasiperiodic flows generally become phase-locked through a res-onance mechanism before a strange attractor may arise, thus restoring the time-periodicityof the flow. In this work we have identified and tracked four different resonance regions,also known as Arnold tongues or horns. In particular, the 1 : 4 strong resonance regionis explored in great detail, where the identified scenarios are in very good agreement withnormal form theory.