In statistically homogeneous and isotropic turbulence, the average value of the velocity increment ${\delta }_{r}u=u(x+r)-u\left(x\right)$, where $x$ and $x+r$ are two positions in the flow and $u$ is the velocity in the direction of the separation distance $r$, is identically zero, and so to characterize the dynamics one often uses the Reynolds number based on $\sqrt{\langle {\left({\delta }_{r}u\right)}^2\rangle }$, which acts as the coupling constant for scale-to-scale interactions. This description can be generalized by introducing structure functions of order $n$, $S_n=\langle {\left({\delta }_{r}u\right)}^n\rangle$, which allow one to probe velocity increments including rare and extreme events, by considering ${\delta }_{r}u\left(n\right)=O\left(S_n^{1/n}\right)$ for large and small $n$. If $S_n\propto r^{{\zeta }_n}$, the theory for the anomalous exponents ${\zeta }_n$ in the entire allowable interval $-1<n<\infty$ is one of the long-standing challenges in turbulence (one takes absolute values of ${\delta }_{r}u$ for negative $n$), usually attacked by various qualitative cascade models. We accomplish two major tasks here. First, we show that the turbulent microscale Reynolds number $R_{\lambda }^T$ (based on a suitably defined turbulent viscosity) is 8.8 in the inertial range with anomalous scaling, when the standard microscale Reynolds number $R_{\lambda }$ (defined using normal viscosity) exceeds that same number (which in practice could be by a large factor). When the normal and turbulent microscale Reynolds numbers become equal to or fall below 8.8, the anomaly disappears in favor of Gaussian statistics. Conversely, if one starts with a Gaussian state and increases $R_{\lambda }$ beyond 8.8, one ushers in the anomalous scaling; the inference is that $R_{\lambda }^T\approx 8.8$ and remains constant at that value with further increase in $R_{\lambda }$. Second, we derive expressions for the anomalous scaling exponents of structure functions and moments of spatial derivatives, by analyzing the Navier-Stokes equations in the form developed by Hopf. We present a novel procedure to close the Hopf equation, resulting in expressions for ${\zeta }_n$ in the entire range of allowable moment order $n$, and demonstrate that accounting for the temporal dynamics changes the scaling from normal to anomalous. For large $n$, the theory predicts the saturation of ${\zeta }_n$ with $n$, leading to several inferences, two among which are (a) the smallest length scale ${\eta }_n=L{\text{Re}}^{-1}\ll L{\text{Re}}^{-3/4}$, where Re is the large-scale Reynolds number, and (b) that velocity excursions across even the smallest length scales can sometimes be as large as the large-scale velocity itself. Theoretical predictions for each of these aspects are shown to be in excellent quantitative agreement with available experimental and numerical data.

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• Turbulence
• Navier-Stokes equation