Anomalous exponents in strong turbulence
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2018 Elsevier B.V. To characterize fluctuations in a turbulent flow, one usually studies different moments of velocity increments and dissipation rate, [Formula presented] and [Formula presented], respectively. In high Reynolds number flows, the moments of different orders cannot be simply related to each other which is the signature of anomalous scaling, one of the most puzzling features of turbulent flows. High-order moments are related to extreme, rare events and our ability to quantitatively describe them is crucially important for meteorology, heat, mass transfer and other applications. In this work we present a solution to this problem in the particular case of the NavierStokes equations driven by a random force. A novel aspect of this work is that, unlike previous efforts which aimed at seeking solutions around the infinite Reynolds number limit, we concentrate on the vicinity of transitional Reynolds numbers [Formula presented] where the first emergence of anomalous scaling is observed out of a low-[Formula presented] Gaussian background. The obtained closed expressions for anomalous scaling exponents [Formula presented] and [Formula presented], which depend on the transition Reynolds number, agree well with experimental and numerical data in the literature and, when [Formula presented], [Formula presented]. The theory yields the energy spectrum [Formula presented] with [Formula presented], different from the outcome of Kolmogorov's theory. It is also argued that fluctuations of dissipation rate and those of the transition point itself are responsible for both, deviation from Gaussian statistics and multiscaling of velocity field.
