Universal Velocity Statistics in Decaying Turbulence

In turbulent flows, kinetic energy is transferred from large spatial scales to small ones, where it is converted to heat by viscosity. For strong turbulence, i.e., high Reynolds numbers, Kolmogorov conjectured in 1941 that this energy transfer is dominated by inertial forces at intermediate spatial scales. Since Kolmogorov’s conjecture, the velocity difference statistics in this so-called inertial range have been expected to follow universal power laws for which theoretical predictions have been refined over the years. Here we present experimental results over an unprecedented range of Reynolds numbers in a well-controlled wind tunnel flow produced in the Max Planck Variable Density Turbulence Tunnel. We find that the measured second-order velocity difference statistics become independent of the Reynolds number, suggesting a universal behavior of decaying turbulence. However, we do not observe power laws even at the highest Reynolds number, i.e., at turbulence levels otherwise only attainable in atmospheric flows. Our results point to a Reynolds number-independent logarithmic correction to the classical power law for decaying turbulence that calls for theoretical understanding.

Figure 1

Top: $S_2$ and $S_3$ ($S_3$ trusted for $r>50\eta$) compensated by the K41 predictions at three Reynolds numbers and compared with numerical simulations at $R_{\lambda }=2250$ [5] (gray triangles). The arrows indicate $r=L$. $S_3$ approaches the $4/5$ law between about $150\eta$ and $300\eta$, and the tilt of $S_2/(ϵr{)}^{2/3}$ toward a positive slope in this interval is a signature of intermittency, which generates ${\zeta }_2>2/3$. Bottom: the local slopes ${\zeta }_2(r)$ [Eq. (3)] reveal additional details in the shapes of the structure functions, including a monotonic decrease over a range of scales up to and beyond $L\gtrsim {10}^4\eta$. In addition to ${\zeta }_2(r)$ of the datasets above, we show atmospheric data at $R_{\lambda }\approx 17000$ [10] (gray squares). The orange line is for reference and is proportional to $1-b\log (r/\eta )$. The black arrows indicate the subranges we identify within the inertial range. Inset: local slope of $S_3$ for the three VDTT datasets.

Figure 2

Bottom: the local scaling exponent differences ${\beta }_2(r,R_{\lambda })$ [Eq. (4)] from their values at $R_{\lambda }=5779$ for $413\le R_{\lambda }\le 4998$. The local slopes for $R_{\lambda }=5779$ are reproduced from Fig. 1. Red shading indicates $\pm 5\%$ intervals. The data have been shifted vertically in proportion to $\log (R_{\lambda })$, and the colors indicate $\log (R_{\lambda })$. A representative sample of 29 datasets is shown. Top: reference case ${\zeta }_2(r)$ measured at $R_{\lambda }=5779$. The red line indicates the smoothed data used to calculate $\beta (r,R_{\lambda })$.

Figure 3

Difference in the local scaling exponent ${\zeta }_2$ at two different values of $r$ within the inertial range: $200\eta$, where the statistics can be assumed to be the most isotropic, and $1200\eta$, which tends to be more anisotropic due to its relative proximity to the large scales. When $R_{\lambda }>2000$, the inertial range extends from $200\eta$ to $1200\eta$ and beyond. To the right of this point, changes in the difference of ${\zeta }_2$ are smaller than to the left. The squares indicate data from the VDTT with a passive grid of rigid bars installed [20]. The light blue lines are a model of decaying turbulence in a confined domain [9] for different choices of the parameter $-A_k$ [0.6 (light), 0.7, 0.9 (dark)]. The black horizontal line indicates power law scaling in $S_2$. The measured values of $L/\eta$ differ slightly from those read off from this graph due to a different ways of estimating $ϵ\sim u^3/L$.