Anomalous Exponents in Strong Turbulence

Abstract: To characterize fluctuations in a turbulent flow, one usually studies different moments of velocity increments and dissipation rate, $\overline{(v(x+r)-v(x))^{n}}\propto r^{\zeta_{n}}$ and $\overline{{\cal E}^{n}}\propto Re^{d_{n}}$, 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 Navier-Stokes 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 $Re^{tr}$ where the first emergence of anomalous scaling is observed out of a low-$Re$ Gaussian background. The obtained closed expressions for anomalous scaling exponents $\zeta_{n}$ and $d_{n}$, which depend on the transition Reynolds number, agree well with experimental and numerical data in the literature and, when $n\gg 1$, $d_{n}\approx 0.19n \ln(n)$. The theory yields the energy spectrum $E(k)\propto k^{-\zeta_{2}-1}$ with $\zeta_{2}\approx 0.699$, 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.