Dynamical and statistical properties of large scales in turbulence

We present a review of results concerning the properties of scales larger than the forcing scale in turbulent flows. In three-dimensional turbulence, when the power driving the flow is injected at a well-defined scale, the injection scale separates the small-scale range, where the Kolmogorov cascade takes place, from the large-scale range across which the mean energy flux is zero, suggesting that these modes are in equilibrium. We show that, in the case of spatially periodic forcing involving only a few modes, an equipartition energy spectrum is generated at large scales, and we discuss the deviations from equipartition observed with more complex forcings. In two-dimensional turbulence, the injection scale separates the direct enstrophy cascade towards small scales from the inverse energy cascade towards large scales. Because of the inverse cascade, one would not expect equilibrium statistical physics tools to be suitable for describing the large-scale structures. Yet this is precisely the situation in which such tools have been most widely used. We indeed show that the bifurcations observed between different large-scale regimes of two-dimensional turbulence, both in experiments and in direct numerical simulations (DNS), can be qualitatively described by the truncated Euler equations (TEE), whose solutions satisfy statistical equilibrium. This article is part of the theme issue 'Frontiers of turbulence and statistical physics'.