Dynamics of three-dimensional turbulence from Navier-Stokes equations

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.

Figure 1

Comparison of theoretical predictions (44) for the anomalous exponents ${\zeta }_n$ with experimental and simulations data. Where the error bars are not visible in experimental and simulations data, they are smaller than the symbol size. The present theory is indicated by the solid line, and the She-Leveque theory by the dashed line. The inset shows the ratio ${\zeta }_n/n$ plotted against $1/n$. It is clear that the saturation tendency is visible in both the theory and experiment as well as simulations.

Figure 2

Left: Normalized moments of velocity gradients, $M_n$, for $n$ = 2, 3, 4, 5, and 6, from direct numerical simulations of the Navier-Stokes equations [41, 42]. For low Reynolds numbers, each of the moments is given by $(2n-1)!!$; i.e., they obey Gaussian statistics, as shown by horizontal lines. Each moment takes off, almost abruptly, to follow the fully developed anomalous scaling relations given by the theory, Eq. (23). The lowest moment undergoes a transition from a Gaussian to an anomalous state at a Reynolds number of about 9. Higher moments undergo this transition at lower Reynolds numbers $R_{\lambda }^{*}\left(n\right)$, for which sub-Gaussian fluctuations reflecting a more complex dynamics of forcing are responsible. These Reynolds numbers are shown on the right panel, where theory (dashed line through the circles) and simulations (the second dashed line) are compared. We do not deal with this particular aspect of the theory in this paper but refer the reader to Refs. [41, 42].

Figure 3

Comparison of theoretical predictions for the exponents $d_n$ through the transitional region with numerical simulations of Refs. [41, 42]. The numbers to the right are least-squares experimental fits to the power-law regions. Table 2 provides comparisons of the exponents.