Anomalous Exponents in Strong Turbulence in an Infinite Fluid Driven by a Gaussian Random Force

To characterize fluctuations in a turbulent flow, one usually studies different moments of velocity increments and/or 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 with $n\neq m$ cannot be simply related to each other which is the signature of anomalous scaling. 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 $Re^{tr}$ of the first emergence of anomalous scaling out of low-$Re$ Gaussian background. The obtained closed expressions for anomalous scaling exponents $\zeta_{n}$ and $d_{n}$ agree well with available in literature experimental and numerical data 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 shown that fluctuations of dissipation rate are responsible for both: deviation from Gaussian statistics and multiscaling of velocity field.