Circulation in High Reynolds Number Isotropic Turbulence is a Bifractal

The turbulence problem at the level of scaling exponents is hard in part because of the multifractal scaling of small scales, which demands that each moment order be treated and understood independently. This conclusion derives from studies of velocity structure functions, energy dissipation, enstrophy density (that is, square of vorticity), etc. However, it is likely that there exist other physically pertinent quantities with less complex statistical structure in the inertial range, potentially resulting in huge simplifications in the turbulence theory. We show that velocity circulation around closed loops is such a quantity. By using a large database of isotropic turbulence, generated from numerical simulations of the Navier-Stokes equations over a wide range of Reynolds numbers, we show that circulation exhibits, to excellent accuracy, a bifractal behavior at the highest Reynolds number considered: space filling for low-order moments, close but not identical to the 1941 paradigm of Kolmogorov, and a monofractal with a dimension of about 2.2 for higher orders. This change in character, occurring around the third moment for the highest Reynolds number considered here, is reminiscent of a “phase transition.” We explore the possibility that the transition point moves to higher-order moments as the Reynolds numbers increases—even though one may continue to regard the structure as bifractal for moments of sufficiently high order. We confirm that the circulation properties depend essentially on the area of the loop, not its shape, and that the relevant contour area in figure-eight loops is the scalar area and not the vector area. These results demonstrate an intrinsic simplicity in the statistical structure of turbulence when considering circulation around closed loops, thus motivating a paradigm shift in turbulence research.

Popular Summary

Fluid turbulence is made up of a vast number of swirling regions, known as eddies or vortices, that fluctuate wildly in both space and time. The wide ranges of both physical scales and amplitudes make it difficult for researchers to come up with manageable statistical descriptions of turbulent fluids, which are required for physical understanding and quantitative prediction. Here, we show that the statistics of certain small-scale structures in a turbulent fluid can be described quite simply, offering some hope for a great simplification to the statistical theory of turbulence.

Relying on numerical simulations, we show that velocity circulation around a closed loop assumes a statistically simple form over a wide range of Reynolds numbers—a dimensionless parameter that predicts flow patterns in different situations. We demonstrate that the circulation assumes a “bifractal nature,” which means that it requires only two exponents to fully describe its statistics, instead of the infinite number needed to measure how well small-scale fluctuations are correlated in space and time. We also show that the circulation distribution depends only on the area of the fluid loop area, as long as its size is not affected by the dissipation and large-scale stirring mechanisms of the flow.

This work points to a hidden structure in seemingly random isotropic turbulent flows and highlights the importance of “loop space” approaches in fluid dynamics, which hitherto have been popular in other areas of physics.

Figure 1

Typical circulation traces in the inertial range, for $R_{\lambda }=1300$ (computational domain $8192^3$) along an arbitrary edge of the cube of length $L_0$. The linear sizes $r$ of the loops are (a) $r/\eta =50$ and (b) $r/\eta =150$, corresponding, respectively, to the lower end and middle of the $4/5$ plateau in the normalized third-order structure function [24].

Figure 2

Probability density function $\mathcal{P}$ of normalized circulation around loops with fixed area, $A={\ell }_1{\ell }_2=4646{\eta }^2$, but with different edge lengths, calculated for the $4096^3$ data at $R_{\lambda }=650$. The dash-dotted lines correspond to contours with at least one side outside IR with the dimensions $({\ell }_1/\eta ,{\ell }_2/\eta )$ as follows: (6.6,704), (11.0,422.4), (16.5,281.6), (22.0,211.2), (26.4,176.0), and (33.0,140.8), in the direction of the arrows shown. The collapsed solid lines correspond to loops with both sides $({\ell }_1/\eta ,{\ell }_2/\eta )$ contained within IR, and correspond to (44.0,105.6), (52.8,88.0), and (66.0,70.4). The collapsed curves in IR verify the area rule to a good approximation, more broadly than Migdal’s anticipation for the tails (which drop off rapidly towards the extreme ends here because of finite sample sizes). The tails are approximately exponential in IR, especially for ${\mathrm{\Gamma }}_A>0$, but are fitted better by stretched exponential fits. The parameters of the stretched exponential vary only slightly from one loop to another in IR.

Figure 3

Circulation standard deviation $⟨{\mathrm{\Gamma }}_A^2{⟩}^{1/2}$ for two sets of contours with the same loop area and different aspect ratios (see insets to the upper left) as a function of the area, at $R_{\lambda }=1300$, computed on a $8192^3$ grid. The standard deviations seem to be independent of the loop perimeter, especially for inertial loops (roughly, $10^3\le A/{\eta }^2\le {10}^6$). Inset on the bottom right plots the ratio of the standard deviation of the two contours as a function of loop area. The ratio is quite close to unity, though smaller by a few percent. It is also not exactly a constant within IR. We also show the same ratio for a lower Reynolds number, $R_{\lambda }=240$. It is clear from the two Reynolds numbers that, within the accuracy just noted, the area rule holds increasingly well at higher Reynolds numbers. The dashed line indicates the K41 law, $⟨{\mathrm{\Gamma }}_A^2{⟩}^{1/2}\sim A^{2/3}$; it holds adequately for inertial loops though the actual least-squares fit gives a slope of 0.674 (which is about 1.2% higher).

Figure 4

Circulation standard deviation as a function of the area $A$ of the figure-eight loop (shown in the inset), calculated in a $4096^3$ grid at $R_{\lambda }=650$. The edge lengths of the figure-eight loop are chosen such that $L_1-L_2=\mathrm{\Delta }$, where $\mathrm{\Delta }$ is a fixed constant. Symbols (square), (circle), (diamond), (asterisk), and (triangle) correspond to $\mathrm{\Delta }/\eta =2$, 4, 8, and 16, respectively. The standard deviations collapse for different figure-eight loops with the same area and follow the scalar area rule in IR, as evidenced by the closeness to the $A^{2/3}$ scaling. The linear area dependence for small $A$ is shown, and the tendency to saturation for large areas is also apparent. Both these tendencies also occur in Fig. 3.

Figure 5

PDF of ${\mathrm{\Gamma }}_r^3/r^4⟨\epsilon ⟩$ at $R_{\lambda }=1300$ at different inertial range separations, $r/\eta =80$ (circle), $r/\eta =120$ (times), $r/\eta =171$ (plus), $r/\eta =246$ (asterisk), and $r/\eta =300$ (square). Insets (a) and (b) show portions of the PDF corresponding to negative and positive ${\mathrm{\Gamma }}_r$, respectively, on log-log scales. The PDFs nearly collapse across the inertial range separations. If they are examined on an expanded scale, some modest differences are seen around ${\mathrm{\Gamma }}_r^3/r^4⟨\epsilon ⟩=\pm 150$. Differences occur also towards the tails, as shown in the insets. The tails of each of the PDFs can be fitted by stretched exponentials with unequal stretching factors for the negative and positive tails. For the average curves, the negative part decays with a stretch factor of about 0.35, about twice as large as that for the positive ($\approx 0.17$).

Figure 6

Normalized circulation moments for various orders $p$ as a function of the linear dimension of the loop, for $R_{\lambda }=1300$. Inset shows corresponding local slopes ${\lambda }_p=d[\log ⟨{\mathrm{\Gamma }}_r^p⟩]/d[\log r]$, with vertical lines demarcating the inertial range. Error bars (which are often subsumed by the symbol thickness) indicate 95% confidence intervals. Horizontal straight lines drawn at $4p/3$ correspond to K41 exponents.

Figure 7

Scaling exponents as a function of moment order $p$; $R_{\lambda }=1300$. The data can be fitted by two separate lines, one below order 3 and one above it. For comparison purposes, the dashed line is the K41 result given by $4p/3$. The low-order data can be fitted best by the expression ${\lambda }_{|p|}=(1.367\pm 0.0095)p$ with no intercept as shown in inset (a) by the solid line. The high-order data can be fitted (solid line) by a monofractal model, ${\lambda }_{|p|}=1.1p+(3-D)$, with $D=2.2$. Both fits are determined by least-squares method. The error bar shown for the tenth moment is typical and is subsumed by the symbol thickness. Inset (b) compares the relative departures of circulation exponents at $R_{\lambda }=1300$ and longitudinal velocity increment exponents (filled triangle) at $R_{\lambda }=10340$ [26], from their respective K41 estimates, for integral orders. The dashed line at zero is the K41 scaling. The relative deviations from K41 for higher-order circulation exponents are about a third of those of the longitudinal structure function exponents (but the sign of these differences in both cases is the same).

Figure 8

Probability density function of circulation normalized by the $h\approx 1.4$ scaling for different inertial range separations, at $R_{\lambda }=1300$. The PDF cores collapse for $Y\in [-10,10]$ across the inertial range, while diverging for larger $|Y|$. Inset shows the integrand of $⟨Y^2⟩$ for a typical inertial separation, showing that the dominant contribution to lower-order circulation moments comes from the region where the PDFs collapse. This plot confirms that the lower-order circulation moments follow the 1.4 scaling with the space-filling dimension of three.

Figure 9

Compensated probability density function of the normalized circulation $Y$, with scaling exponent $h=1.1$ and dimension $D=2.2$, for different inertial range separations, at $R_{\lambda }=1300$. The upticks at the tails of the PDFs are due to finite sampling. Inset shows the integrand of $⟨Y^8⟩$ for a typical inertial separation, demonstrating that the dominant contribution to higher-order circulation moments comes from the collapsed region. This plot confirms that the higher-order circulation moments follow the scaling ${\lambda }_p=hp+(3-D)$, with the Hölder exponent $h=1.1$ and fractal dimension $D=2.2$.

Figure 10

Circulation flatness $F(r)$ versus spatial separation $r$ from DNS at $R_{\lambda }=240$, $512^3$ (triangle), $R_{\lambda }=650$, $4096^3$ (square), and $R_{\lambda }=1300$, $8192^3$ (circle). The flatness from the DNS at $R_{\lambda }=1300$ computed on a $16384^3$ box is also shown (filled diamond) and agrees well with $F(r)$ from the $8192^3$ box, for $r/\eta >10$. Arrows on the ordinate show the limit $F(r\to 0)=⟨{\omega }^4⟩/⟨{\omega }^2{⟩}^2$, which is the flatness of the vorticity component normal to the circulation plane. At the largest scales, $F(r)$ is close to the Gaussian flatness of three (see dashed line). Inset compares the circulation flatness with those of the longitudinal (times) and transverse (plus) velocity increments at $R_{\lambda }=1300$ in the inertial range within the dashed vertical lines. Unlike circulation with a tendency towards constancy in the inertial range, the velocity increments show no tendency towards constancy in IR.

Figure 11

The peak magnitude of logarithmic local slope of circulation flatness in the inertial range, plotted against $R_{\lambda }$. The solid line is a least-squares fit. If extrapolated, the minimum of the logarithmic derivative will reach zero, corresponding to constant flatness in Fig. 10, at $R_{\lambda }\approx 1900$. The data of Ref. [13] at $R_{\lambda }=216$ (filled square) are shown for comparison. Inset shows the logarithmic local slopes of circulation flatness as a function of $r/\eta$, up to $r\sim O(L)$ for $R_{\lambda }=1300$ (upside down triangle) and $R_{\lambda }=240$ (circle). Error bars indicate 95% confidence intervals.

Figure 12

The relative difference of the circulation scaling exponent ${\lambda }_p$ from the K41 exponent $4p/3$ as a function of the Taylor-scale Reynolds numbers $R_{\lambda }$, for orders $p=4$, 6, 8, and 10. Data from Ref. [13] (pentagon) for orders $p=4$, 6, 8, and 10, at $R_{\lambda }=216$, and the highly resolved DNS for orders $p=4$, 6 and 8, at $R_{\lambda }=1300,16384^3$ (filled diamond), are provided for comparison. Exponential least-squares fits through the data are shown by the solid lines. Also shown are error bars.

Figure 13

The function $\psi (r)$ [left-hand side of inequality (13)] as a function of normalized spatial separation $r/\eta$ for different Reynolds numbers. The curves are seen to collapse, for $r\ll L$ ($L$ is the integral scale), at the different Reynolds numbers shown, reasonably well. The limit $\psi (r\to 0)=1/3$ is marked by the horizontal dashed line. Dash-dotted line indicates the corresponding K41 slope.

Figure 14

Probability density function of circulation, normalized by its standard deviation, on a $4096^3$ grid at $R_{\lambda }=650$, calculated using [Eq. (a1)] line integral of velocity (dashed lines) and [Eq. (a2)] surface integral of vorticity (solid lines), around square loops of size $r$. Different sets of curves correspond to, $r/\eta =1.1$, 6.6, 41.8, and 125.4, increasing in the direction shown. Insets (a) and (b) compare the circulation moments from the two quadratures for orders 2 and 8, respectively.

Figure 15

Peak values of PDF of normalized circulation, ${\mathrm{\Gamma }}_A/u^{\prime }L$, at $4096^3$, $R_{\lambda }=650$, around closed rectangular contours with fixed area, but different perimeters, $A={\ell }_1{\ell }_2=4646.4{\eta }^2$, as a function of side, ${\ell }_1$. The plot is symmetric around ${\ell }_1=\sqrt{A}$ due to statistical isotropy of the underlying velocity field (which holds by construction). Solid symbols correspond to contour dimensions $({\ell }_1,{\ell }_2)$ with both legs in IR, while open symbols correspond to contours with at least one leg outside IR. The PDF peaks tend to an approximately constant value for inertial loops.