Recent studies have brought into question the view that at sufficiently high Reynolds number turbulence is an asymptotic state. We present the first direct observation of the decay of turbulent states in Taylor-Couette flow with lifetimes spanning five orders of magnitude. We also show that there is a regime where Taylor-Couette flow shares many of the decay characteristics observed in other shear flows, including Poisson statistics and the coexistence of laminar and turbulent patches. Our data suggest that characteristic decay times increase super-exponentially with increasing Reynolds number but remain bounded in agreement with the most recent data from pipe flow and with a recent theoretical model. This suggests that, contrary to the prevailing view, turbulence in linearly stable shear flows may be generically transient.

We show that aqueous solutions of ammonium thiocyanate (NH 4 SCN) can be used to match the index of refraction of several transparent materials commonly used in experiments, while maintaining low viscosity and density compared to other common refractive index-matching liquids. We present empirical models for estimating the index of refraction, density, and kinematic viscosity of these solutions as a function of temperature and concentration. Finally, we summarize the chemical compatibility of ammonium thiocyanate with materials commonly used in apparatus.

Symmetry reduction by the method of slices quotients the continuous symmetries of chaotic flows by replacing the original state space by a set of charts, each covering a neighborhood of a dynamically important class of solutions, qualitatively captured by a 'template'. Together these charts provide an atlas of the symmetryreduced 'slice' of state space, charting the regions of the manifold explored by the trajectories of interest. Within the slice, relative equilibria reduce to equilibria and relative periodic orbits reduce to periodic orbits. Visualizations of these solutions and their unstable manifolds reveal their interrelations and the role they play in organizing turbulence/chaos. Today, it is possible to take a stroll through the high-dimensional state space of hydrodynamic turbulence and observe that turbulent trajectories are guided by close passes to invariant solutions of the Navier-Stokes equations. Charting how close these passes are is a geometer's task, but in order to place them on a map, one first has to deal with families of solutions equivalent under the symmetries of a given flow. Evolution in time decomposes the state space into a 'spaghetti' of time trajectories. Continuous spatial symmetries foliate it like the layers of an onion. In this visual tour of dynamics, we use a low-dimensional flow to illustrate how this tangle can be unraveled (symmetry reduction), and how to pick a single representative point for each trajectory (section it) and group orbit (slice it). Once the symmetry induced degeneracies are out of the way, one can identify and describe the prominent turbulent structures by a taxonomy of invariant building blocks (numerically exact solutions of the NavierStokes equations, finite sets of relative equilibria and infinite hierarchies of relative periodic orbits) and describe the dynamics in terms of near passes to their heteroclinic connections.

Dynamical systems with translational or rotational symmetry arise frequently in studies of spatially extended physical systems, such as Navier-Stokes flows on periodic domains. In these cases, it is natural to express the state of the fluid in terms of a Fourier series truncated to a finite number of modes. Here, we study a 4-dimensional model with chaotic dynamics and SO(2) symmetry similar to those that appear in fluid dynamics problems. A crucial step in the analysis of such a system is symmetry reduction. We use the model to illustrate different symmetry-reduction techniques. The system's relative equilibria are conveniently determined by rewriting the dynamics in terms of a symmetry-invariant polynomial basis. However, for the analysis of its chaotic dynamics, the "method of slices," which is applicable to very high-dimensional problems, is preferable. We show that a Poincaré section taken on the "slice" can be used to further reduce this flow to what is for all practical purposes a unimodal map. This enables us to systematically determine all relative periodic orbits and their symbolic dynamics up to any desired period. We then present cycle averaging formulas adequate for systems with continuous symmetry and use them to compute dynamical averages using relative periodic orbits. The convergence of such computations is discussed.