Multiscaling in Strong Turbulence Driven by a Random Force

Turbulence problem is often considered as "the last unsolved problem of classical physics". It is due to strong interaction between velocity and/or velocity gradient fluctuations, a high Reynolds number flow is a fascinating mixture of purely random, close to Gaussian, fields and coherent structures where substantial fraction of kinetic energy is dissipated into heat. To evaluate intensity of fluctuations, one usually studies different moments of velocity increments and/or dissipation rate, characterized by scaling exponents $\zeta_{n}$ and $d_{n}$, respectively. In high Reynolds number flows, the moments of different orders with $n\neq m$ cannot be simply related to each other, which is the signature of anomalous scaling, making this problem "the last unsolvable". No perturbative treatment can lead to quantitative description of this feature. In this work the expressions for the moments of dissipation rate $e_{n}=\overline{{\cal E}^{n}}\propto Re^{d_{n}}$ and those of velocity derivatives $M_{2n}=\overline{(\partial_{x}u_{x})^{2n}}\propto \frac{v_{o}^{2n}}{L^{2n}}Re^{\rho_{2n}}$ are derived for an infinite fluid stirred by a white-in-time Gaussian random force supported in the vicinity of the wave number $k_{f}\approx \frac{2\pi}{L}=O(1)$, where $v_{0}$ and $L$ are characteristic velocity and integral scale, respectively. 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 $Re^{tr}$ of the first emergence of anomalous scaling out of Low-Re Gaussian background. The obtained closed expressions for anomalous scaling exponents $d_{n}$ and $\rho_{n}$ agree well with available in literature experimental and numerical data and, when $n\gg 1$, $d_{n}\approx 0.3n \ln(n)$.