Decay of Turbulence at High Reynolds Numbers

Turbulent motions in a fluid decay at a certain rate once stirring has stopped. The role of the most basic parameter in fluid mechanics, the Reynolds number, in setting the decay rate is not generally known. This Letter concerns the high-Reynolds-number limit of the process. In a classical grid-turbulence wind-tunnel experiment that both reaches higher Reynolds numbers than ever before and covers a wide range of them ($10^4<\text{Re}=UM/\nu <5\times 10^6$), we measure the decay rate with the unprecedented precision of about 2%. Here $U$ is the mean speed of the flow, $M$ is the forcing scale, and $\nu$ is the kinematic viscosity of the fluid. We observe that the decay rate is Reynolds-number independent, which contradicts some models and supports others.

Figure 1

(a) The decay of the turbulent kinetic energy $u^2$ for seven representative Reynolds numbers as a function of time. The linear relationship between the data on the double-logarithmic scales of the figure indicates a power law dependence on time. The values of $\text{Re}\times 10^{-3}$ for each set of data were 26, 54, 140, 410, 820, 1700, 3200, and 4800 from bottom to top. Each set of data is shifted upward to separate it from the lowest curve, which is unshifted. The lowest curve, the master curve, is the mean of 99 sets of data acquired at different Reynolds numbers. We normalized the kinetic energy by that in the mean flow, $U^2$, and time by $M/U$. Time was offset by $t_0$, the virtual origin. (b) We found the virtual origin by fitting each decay curve to a power law as described in the text. Over the full range of Reynolds numbers, the virtual origin fluctuated with a standard deviation of less than 10% of its mean value of $t_{0}U/M=3.66$.

Figure 2

The exponents $n$ in Eq. (1) of power-law fits to the data quantify the energy decay rate. The line is a fit to our data ($\times$, $*$, $+$), for which the mean decay rate was $n=-1.18$ with a standard deviation of 0.02. The 95% confidence interval for each value of $n$ ($\pm 0.9\%$) is given approximately by the size of the symbols. Additional data drawn from the literature appear as follows. Stars: brown [23], blue [33], gray [42]. Diamonds: red [43], blue [44], black [45], green [40], yellow [46]. Squares: black [47], blue [48], brown [15], green [49], red [50], yellow [51]. Circles: black [3], red [52], blue [53], green [54], yellow [55]. Triangles: pointing down [56], black pointing left [57], red [38], blue [34], brown [17]. Closed symbols mark experiments that employed classical grids, as ours did, whereas open symbols mark those that employed modified grids such as fractal grids. Symbols with thin lines mark active-grid experiments. In some of these experiments, modifications to the grids were made deliberately to elicit changes in the decay [15, 17]. The inset shows on a linear scale of Re the relative decay exponents as described in the text. Our data reveal that the rate of decay is invariant with respect to the Reynolds number for $\text{Re}\gg 10^4$.

Figure 3

The correlation lengths $L$ were approximately a power law of the kinetic energy $u^2$ during the decay. Time proceeds from right to left in the graph, as indicated by the arrow. Here we combined all data without discrimination since there was no trend with respect to Re. To produce the smooth curve, we took the median values of $L/M$ and $u^2/U^2$ data falling within logarithmically spaced bins in $u^2/U^2$. The relationships posited by well-known theories are indicated by straight lines. Our data are consistent with Saffman’s theory, with slope $-1/3$.