Emergence of Multiscaling in a Random-Force Stirred Fluid

We consider the transition to strong turbulence in an infinite fluid stirred by a Gaussian random force. The transition is defined as a first appearance of anomalous scaling of normalized moments of velocity derivatives (dissipation rates) emerging from the low-Reynolds-number Gaussian background. It is shown that, due to multiscaling, strongly intermittent rare events can be quantitatively described in terms of an infinite number of different “Reynolds numbers” reflecting a multitude of anomalous scaling exponents. The theoretically predicted transition disappears at $R_{\lambda }\le 3$. The developed theory is in quantitative agreement with the outcome of large-scale numerical simulations.

Figure 1

Top panel: Normalized moments $M_n^{\mathcal{E}}=\overline{{\mathcal{E}}^n}/{\overline{\mathcal{E}}}^n$ as a function of the large-scale Reynolds number. Dashed lines: Theoretical predictions and numerical simulations of Refs. [5]. Squares are from Ref. [6] and asterisks from our large DNS database (see, e.g., Ref. [7]). Bottom panel: Transition region. Moments of velocity gradients in the low-$R_{\lambda }$ transitional range (present work). From bottom to top, $n=2$, 4, 6, 8, and 10. Horizontal dashed lines: Gaussian values at $(2n-1)!!$ for reference. The vertical dashed line is at 8.91 for reference.

Figure 2

Transitional Reynolds number $R_{\lambda ,1}(n)$ of the $n$th moment of velocity derivative. Blue: Numerical simulations of the present work. Red: Theoretical prediction (10) with $R_{\lambda ,n}^{\text{tr}}=8.91$.