The interest in turbulent flow goes back many centuries, but progress has been very slow until recently, from the point of view of ab initio theory (starting from the Euler or Navier-Stokes equations). In the 20th century, innumerable applications, e.g., in aeronautics or atmospheric and oceanic circulation, have been a major driving force for progress. In the last eighty years, thanks (a) to the scaling predictions of Kolmogorov in 1941 (K41), (b) to Onsager's 1949 (Ons49) criterion for anomalous energy dissipation, and (c) to Kraichnan's discovery of the inverse energy cascade for 2D turbulence, we have had the beginnings of a theoretical understanding. Furthermore, from experiments and numerical simulations of the last forty years, it is now clear that simple K41 scale invariance is broken and that a turbulent fluid displays multifractal scaling, which has been modelled by using various ad hoc probabilistic models, so far not deduced from the hydrodynamical equations. Recently, th...
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