Routes to turbulence in Taylor-Couette flow.

Fluid flows between rotating concentric cylinders exhibit two distinct routes to turbulence. In flows dominated by inner-cylinder rotation, a sequence of linear instabilities leads to temporally chaotic dynamics as the rotation speed is increased. The resulting flow patterns occupy the whole system and sequentially lose spatial symmetry and coherence in the transition process. In flows dominated by outer-cylinder rotation, the transition is abrupt and leads directly to turbulent flow regions that compete with laminar ones. We here review the main features of these two routes to turbulence. Bifurcation theory rationalizes the origin of temporal chaos in both cases. However, the catastrophic transition of flows dominated by outer-cylinder rotation can only be understood by accounting for the spatial proliferation of turbulent regions with a statistical approach. We stress the role of the rotation number (the ratio of Coriolis to inertial forces) and show that it determines the lower border for the existence of intermittent laminar-turbulent patterns. 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)'.

Data-driven modeling and prediction of non-linearizable dynamics via spectral submanifolds.

We develop a methodology to construct low-dimensional predictive models from data sets representing essentially nonlinear (or non-linearizable) dynamical systems with a hyperbolic linear part that are subject to external forcing with finitely many frequencies. Our data-driven, sparse, nonlinear models are obtained as extended normal forms of the reduced dynamics on low-dimensional, attracting spectral submanifolds (SSMs) of the dynamical system. We illustrate the power of data-driven SSM reduction on high-dimensional numerical data sets and experimental measurements involving beam oscillations, vortex shedding and sloshing in a water tank. We find that SSM reduction trained on unforced data also predicts nonlinear response accurately under additional external forcing.

Phase Transition to Turbulence in Spatially Extended Shear Flows.

Directed percolation (DP) has recently emerged as a possible solution to the century old puzzle surrounding the transition to turbulence. Multiple model studies reported DP exponents, however, experimental evidence is limited since the largest possible observation times are orders of magnitude shorter than the flows' characteristic timescales. An exception is cylindrical Couette flow where the limit is not temporal, but rather the realizable system size. We present experiments in a Couette setup of unprecedented azimuthal and axial aspect ratios. Approaching the critical point to within less than 0.1% we determine five critical exponents, all of which are in excellent agreement with the 2+1D DP universality class. The complex dynamics encountered at the onset of turbulence can hence be fully rationalized within the framework of statistical mechanics.

Second-Order Phase Transition in Counter-Rotating Taylor-Couette Flow Experiment.

In many basic shear flows, such as pipe, Couette, and channel flow, turbulence does not arise from an instability of the laminar state, and both dynamical states co-exist. With decreasing flow speed (i.e., decreasing Reynolds number) the fraction of fluid in laminar motion increases while turbulence recedes and eventually the entire flow relaminarizes. The first step towards understanding the nature of this transition is to determine if the phase change is of either first or second order. In the former case, the turbulent fraction would drop discontinuously to zero as the Reynolds number decreases while in the latter the process would be continuous. For Couette flow, the flow between two parallel plates, earlier studies suggest a discontinuous scenario. In the present study we realize a Couette flow between two concentric cylinders which allows studies to be carried out in large aspect ratios and for extensive observation times. The presented measurements show that the transition in this circular Couette geometry is continuous suggesting that former studies were limited by finite size effects. A further characterization of this transition, in particular its relation to the directed percolation universality class, requires even larger system sizes than presently available.

Liquidlike sloshing dynamics of monodisperse granulate.

Analogies between fluid flows and granular flows are useful because they pave the way for continuum treatments of granular media. However, in practice it is impossible to predict under what experimental conditions the dynamics of fluids and granulates are qualitatively similar. In the case of unsteadily driven systems no such analogy is known. For example, in a partially filled container subject to horizontal oscillations liquids slosh, whereas granular media of complex particles exhibit large-scale convection rolls. We here show that smooth monodisperse steel spheres exhibit liquidlike sloshing dynamics. Our findings highlight the role of particle material and geometry for the dynamics and phase transitions of the system.

High-precision Taylor-Couette experiment to study subcritical transitions and the role of boundary conditions and size effects.

A novel Taylor-Couette system has been constructed for investigations of transitional as well as high Reynolds number turbulent flows in very large aspect ratios. The flexibility of the setup enables studies of a variety of problems regarding hydrodynamic instabilities and turbulence in rotating flows. The inner and outer cylinders and the top and bottom endplates can be rotated independently with rotation rates of up to 30 Hz, thereby covering five orders of magnitude in Reynolds numbers (Re = 10(1)-10(6)). The radius ratio can be easily changed, the highest realized one is η = 0.98 corresponding to an aspect ratio of 260 gap width in the vertical and 300 in the azimuthal direction. For η < 0.98 the aspect ratio can be dynamically changed during measurements and complete transparency in the radial direction over the full length of the cylinders is provided by the usage of a precision glass inner cylinder. The temperatures of both cylinders are controlled independently. Overall this apparatus combines an unmatched variety in geometry, rotation rates, and temperatures, which is provided by a sophisticated high-precision bearing system. Possible applications are accurate studies of the onset of turbulence and spatio-temporal intermittent flow patterns in very large domains, transport processes of turbulence at high Re, the stability of Keplerian flows for different boundary conditions, and studies of baroclinic instabilities.

The onset of turbulence in pipe flow.

Shear flows undergo a sudden transition from laminar to turbulent motion as the velocity increases, and the onset of turbulence radically changes transport efficiency and mixing properties. Even for the well-studied case of pipe flow, it has not been possible to determine at what Reynolds number the motion will be either persistently turbulent or ultimately laminar. We show that in pipes, turbulence that is transient at low Reynolds numbers becomes sustained at a distinct critical point. Through extensive experiments and computer simulations, we were able to identify and characterize the processes ultimately responsible for sustaining turbulence. In contrast to the classical Landau-Ruelle-Takens view that turbulence arises from an increase in the temporal complexity of fluid motion, here, spatial proliferation of chaotic domains is the decisive process and intrinsic to the nature of fluid turbulence.