Evidence for internal structures of spiral turbulence

Mean structure of the supercritical turbulent spiral in Taylor-Couette flow

The large-scale laminar/turbulent spiral patterns that appear in the linearly unstable regime of counter-rotating Taylor-Couette flow are investigated from a statistical perspective by means of direct numerical simulation. Unlike the vast majority of previous numerical studies, we analyse the flow in periodic parallelogram-annular domains, following a coordinate change that aligns one of the parallelogram sides with the spiral pattern. The domain size, shape and spatial resolution have been varied and the results compared with those in a sufficiently large computational orthogonal domain with natural axial and azimuthal periodicity. We find that a minimal parallelogram of the right tilt significantly reduces the computational cost without notably compromising the statistical properties of the supercritical turbulent spiral. Its mean structure, obtained from extremely long time integrations in a co-rotating reference frame using the method of slices, bears remarkable similarity with the turbulent stripes observed in plane Couette flow, the centrifugal instability playing only a secondary role. This article is part of the theme issue 'Taylor-Couette and related flows on the centennial of Taylor's seminal Philosophical transactions paper (Part 2)'.

Generalized quasi-linear approximation and non-normality in Taylor-Couette spiral turbulence

Taylor-Couette flow is well known to admit a spiral turbulence state in which laminar and turbulent patches coexist around the cylinder. This flow state is quite complex, with delicate internal structure, and it can be traced into certain regimes of linear stability. This behaviour is believed to be connected to the non-normality of the linear operator, which is itself a function of the control parameters. Using spiral turbulence in both linearly stable and unstable regimes, we investigate the effectiveness of the generalized quasi-linear approximation (GQL), an extension of quasi-linear theory designed to capture the essential aspects of turbulent flows. We find that GQL performs much better in the supercritical regime than the subcritical. By including only a small number of modes in the nonlinear interactions, GQL simulations maintain a turbulent-like state when in the supercritical regime. However, a much larger number is required to avoid returning to the laminar state when in the subcritical regime. This article is part of the theme issue 'Taylor-Couette and related flows on the centennial of Taylor's seminal Philosophical Transactions paper (part 1)'.

Exact invariant solution reveals the origin of self-organized oblique turbulent-laminar stripes

Wall-bounded shear flows transitioning to turbulence may self-organize into alternating turbulent and laminar regions forming a stripe pattern with non-trivial oblique orientation. Different experiments and flow simulations identify oblique stripe patterns as the preferred solution of the well-known Navier-Stokes equations, but the origin of stripes and their oblique orientation remains unexplained. In concluding his lectures, Feynman highlights the unexplained stripe pattern hidden in the solution space of the Navier-Stokes equations as an example demonstrating the need for improved theoretical tools to analyze the fluid flow equations. Here we exploit dynamical systems methods and demonstrate the existence of an exact equilibrium solution of the fully nonlinear 3D Navier-Stokes equations that resembles oblique stripe patterns in plane Couette flow. The stripe equilibrium emerges from the well-studied Nagata equilibrium and exists only for a limited range of pattern angles. This suggests a mechanism selecting the non-trivial oblique orientation angle of turbulent-laminar stripes.

It is known from experiments and simulations that stripes of alternating laminar and turbulent regions can form in plane Couette flow. Here the authors find an exact invariant solution which captures the detail of the spatial structures of these patterns and identifies their origin.

Oblique laminar-turbulent interfaces in plane shear flows

Localized structures such as turbulent stripes and turbulent spots are typical features of transitional wall-bounded flows in the subcritical regime. Based on an assumption for scale separation between large and small scales, we show analytically that the corresponding laminar-turbulent interfaces are always oblique with respect to the mean direction of the flow. In the case of plane Couette flow, the mismatch between the streamwise flow rates near the boundaries of the turbulence patch generates a large-scale flow with a nonzero spanwise component. Advection of the small-scale turbulent fluctuations (streaks) by the corresponding large-scale flow distorts the shape of the turbulence patch and is responsible for its oblique growth. This mechanism can be easily extended to other subcritical flows such as plane Poiseuille flow or Taylor-Couette flow.

Symmetry-breaking Hopf bifurcations to 1-, 2-, and 3-tori in small-aspect-ratio counterrotating Taylor-Couette flow

The nonlinear dynamics of Taylor-Couette flow in a small-aspect-ratio wide-gap annulus in the counterrotating regime is investigated by solving the full three-dimensional Navier-Stokes equations. The system is invariant under arbitrary rotations about the axis, reflection about the annulus midplane, and time translations. A systematic investigation is presented both in terms of the flow physics elucidated from the numerical simulations and from a dynamical system perspective provided by equivariant normal form theory. The dynamics are primarily associated with the behavior of the jet of angular momentum that emerges from the inner cylinder boundary layer at about the midplane. The sequence of bifurcations as the differential rotation is increased consists of an axisymmetric Hopf bifurcation breaking the reflection symmetry of the basic state leading to an axisymmetric limit cycle with a half-period-flip spatiotemporal symmetry. This undergoes a Hopf bifurcation breaking axisymmetry, leading to quasiperiodic solutions evolving on a 2-torus that is setwise symmetric. These undergo a further Hopf bifurcation, introducing a third incommensurate frequency leading to a 3-torus that is also setwise symmetric. On the 3-torus, as the differential rotation is further increased, a saddle-node-invariant-circle bifurcation takes place, destroying the 3-torus and leaving a pair of symmetrically related 2-tori states on which all symmetries of the system have been broken.

From temporal to spatiotemporal dynamics in transitional plane Couette flow

Laboratory experiments point out the existence of patterns made of alternately laminar and turbulent oblique bands in plane Couette flow (PCF) on its way to or from turbulence as the Reynolds number R is varied. Many previous theoretical and numerical works on the problem have considered small-aspect-ratio systems subjected to periodic boundary conditions, while experiments correspond to the opposite limit of large aspect ratio. Here, by means of fully resolved direct numerical simulations of the Navier-Stokes equations at decreasing R, we scrutinize the transition from temporal to spatiotemporal behavior in systems of intermediate sizes. We show that there exists a streamwise crossover size of order L{x}~70-80 h (where 2 h is the gap between the plates driving the flow) beyond which the transition to or from turbulence in PCF is undoubtedly a spatiotemporal process, with typical scenario turbulent flow→riddled regime→oblique pattern→ laminar flow, whereas below that size it is more a temporal process describable in terms of finite-dimensional dynamical systems with the scenario chaotic flow→laminar flow (via chaotic transients). In the crossover region, the oblique pattern stage is skipped, which leads us to suggest that an appropriate rendering of the patterns observed in experiments needs a faithful account of streamwise correlations at scales at least of the order of that crossover size.