Beyond chaos: fluctuations, anomalies and spontaneous stochasticity in fluid turbulence

In this perspective, we consider the development of statistical hydrodynamics, focusing on the way in which the intrinsic stochasticity of turbulent phenomena was identified and is being explored. A major purpose of our discussion is to bring out the role of anomalies in turbulent phenomena, in ways that are not usually done, and to emphasize how the description of turbulent phenomena requires delicate considerations of asymptotic limits. The scope of our narrative includes selected historical aspects that are not usually emphasized, primarily owing to G.I. Taylor, as well as discussions of certain aspects of the laminar–turbulent transition, the behaviour of turbulent drag at intermediate Reynolds numbers (Re) and the statistics of fully developed turbulence that exhibit spontaneous stochasticity. This article is part of the theme issue ‘Frontiers of turbulence and statistical physics’.