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.

Introduction.-Turbulent flows feature a self-organized net transfer of kinetic energy from the large energetic length scales (L) to the small viscous and dissipative scales (η). The statistical properties of incompressible turbulent flows in the inertial range (η ≪ r ≪ L) have long been proposed to depend only on the rate of kinetic energy transfer (ε) and otherwise to follow universal relations [1]. Universality, thus, is expected to emerge as the turbulence intensity, i.e., the Reynolds number increases-the larger it is, the larger the expected universal inertial range should be. This prediction relies on dimensional arguments and is supported to some extent by experiments and by numerical simulations of the Navier-Stokes equation [2][3][4][5]. The central parameter is the Reynolds number given by (also known as the Taylor-scale Reynolds number), where u 0 is the rms velocity of the velocity fluctuations, λ 2 ¼ 15νu 02 =ε is the (square of the) Taylor scale, and ν is the kinematic viscosity.
In his seminal work, called K41, Kolmogorov [1] assumed that the statistics of a given flow are defined entirely by the rate of energy transfer to dissipation ε in the inertial range. Dimensional analysis then leads to the inertial-range scaling law S n ðrÞ ¼ C n ðεrÞ n=3 ; ð2Þ where the structure function of nth order is given by S n ðrÞ ¼ h½ðuðxÞ − uðx þ rÞ · r=jrjÞ n i. It is well known that the intermittent spatial distribution of ε leads to a dependence of the inertial-range scaling exponent of the longitudinal structure function S n on n that is a convex function of n and different from n=3 [6]. This does not hold for n ¼ 3, where the inertial-range scaling exponent is unchanged by the presence of intermittency. Scaling is expected to hold in the limit R λ → ∞ for statistically homogeneous and isotropic turbulence and there will always be corrections to these power laws in real flows and noninfinite Reynolds numbers. However, these corrections are expected to diminish with an increasingly extended inertial range, i.e., with increasing R λ . One way for power laws to be approached, for example, would be for the local slope of the structure functions to become constant and independent of r within an interval of spatial scales that increases with Reynolds number.
The way in which the limit of large R λ is approached appears to depend on the type of flow. For instance, it has been observed that structure functions in numerical simulations in a periodic box forced continuously at large scales may approach power laws faster with increasing R λ than in grid-generated turbulence decaying in wind tunnels (e.g., [7][8][9]). Atmospheric flows exhibit the largest R λ and the largest scale separation of any available data, but also exhibit anisotropy and nonstationarity that tend to obscure scaling [3,[10][11][12]. Low temperature liquid helium laboratory flows (e.g., [13]) reach atmospheric Reynolds numbers, but have until now been constrained by a lack of instrumentation with sufficient resolution [14][15][16].
In this Letter, we present experimental results from decaying turbulent flows in the Max Planck Variable Density Turbulence Tunnel over a wider range of R λ than in any previous experiment and find that ζ 2 ðrÞ-a measure for the local scaling exponent observed at a given r, see Eq. (3)-is independent of the Reynolds number, but dependent on the separation r. We discuss the possible physical origins of this unexpected result and consequences of the finding.
Experiment and results.-In the Max Planck Variable Density Turbulence Tunnel (VDTT) 88 m 3 of sulphur hexaflouride (SF 6 ) at pressures between 1 and 15 bar are circulated at up to 5.5 m=s. Turbulence is generated with an active grid with individually controllable paddles [17,18] at the inlet of a measurement section. The turbulence in the main measurement location on the tunnel center line 8.3 m downstream from the active grid is approximately statistically homogeneous and isotropic [18,19]. The developing turbulent boundary layers on the tunnel walls do not influence the central measurement region, and the turbulence decays downstream from the grid [20,21].
In the VDTT, the Reynolds number can be adjusted in three different ways by changing (i) the kinematic viscosity ν with pressure, (ii) the energy injection scale L with the active grid protocol, and (iii) the fluctuation velocity u 0 with the mean flow speed. The primary means, however, of effecting large changes in the Reynolds number was by the first mechanism (i), so that the small scales [η ¼ ðν 3 =εÞ 1=4 ] differed by up to 2 orders of magnitude between experiments, while the large scales differed by up to a factor of 4.
We recorded time series of the streamwise velocity using MEMS-fabricated 30 μm hot wires (nanoscale thermal anemometry probes, NSTAPs) [22][23][24] and conventional hot wires. The time series were converted into one-dimensional spatial velocity fields by assuming the advection of turbulent fluctuations with the local mean velocity U, i.e., x ¼ Ut (Taylor's hypothesis) [25]. We have verified that the turbulence intensities found in our experiments allow the application of Taylor's hypothesis by studying the effect of a random sweeping model [26,27] on model spectra [21,28] and by comparing the results of experiments with different turbulence intensities. In all experiments, the wire lengths were ≲4η. Details on the active grid as well as the measurement accuracy can be found in the Supplemental Material [29], which includes Refs. [17,22,24,[30][31][32][33][34][35][36].
Here we investigate structure functions (S n ) and their slopes as functions of the scale r=η (Fig. 1). These functions can typically be separated into three regimes: the viscous subrange [where S 2 approaches ðr=ηÞ 2 toward small scales], the inertial subrange [where it is hypothesized that S 2 ∼ ðr=ηÞ ζ n ], and the large scales (where S 2 approaches 2u 02 ¼ const toward large scales). Figure 1 (upper) shows S 2 and S 3 compensated by the K41 predictions [Eq. (2)], so that the inertial subrange presents as an approximate plateau, one that extends to larger r=η at larger R λ . We compare with atmospheric measurements (at R λ ≈ 17 000 [10]) and recent computer simulation of the Navier-Stokes equations in a periodic box (at R λ ¼ 2250 [5]). The data have been collapsed artificially at r=η ¼ 10 to offset errors in measurements of ε. The direct numerical simulation (DNS) data are in approximate agreement with our data up to about 10 3 η, where the experimental curves fall beneath the DNS. This difference occurs at scales that we show below are R λ -independent in the experiment. The difference between the experiments and DNS may be attributed to the forcing scheme applied in computer simulations at large scales [37].
We take special note of the third-order structure function at R λ ¼ 2680, since an exact result exists for the S 3 (and not for any other order) at R λ → ∞, which is the 4=5 law, S 3 ¼ −ð4=5Þεr [1,38]. Figure 1(top) shows that the data approach this scaling approximately in the vicinity of 200η.
To reveal power laws unambiguously, it is convenient to study the slopes of the structure functions or the local scaling exponents as functions of r, where ζ n ðrÞ is a constant independent of r wherever scaling laws prevail. We observe in Fig. 1 (bottom) that ζ 2 ðrÞ decreases monotonically. As expected due to intermittency, we do not observe the K41 prediction of ζ 2 ¼ 2=3. Models of intermittent turbulence predict ζ 2 ≳ 2=3 (e.g., [6]), consistent with a region of approximately constant ζ 2 ≈ 0.73 that we find between 100η and 200η.
In contrast to this expectation, note that in the interval of scales between about 300η and 5000η, ζ 2 is not constant and depends on r, decreasing from above 0.7 to below 0.6 (and below 2=3 given by K41). Within this range, it appears that ζ 2 ðrÞ, whatever its shape, is independent of R λ . To further interrogate the evolution of the structure functions with increases in the Reynolds number, we compare the local slopes [Eq. (3)] measured at different Reynolds numbers. Figure 2 shows the differences (β 2 ) between the local slopes measured at the highest R λ (5779) and those measured at lower R λ , such that For this calculation, the local slope at the highest R λ has been smoothed with splines to suppress noise (see red line in Fig. 2  βðr; R λ Þ ≈ 0 for all R λ , indicating the expected universality of the scaling exponent in the viscous subrange, where ζ 2 ¼ 2. Data for which R λ ≳ 2000 also exhibit a range of scales where βðrÞ ≈ 0 at intermediate scales r ≳ 100η, indicating that for these cases the local slopes (ζ 2 )whatever their particular values-are close to those in the highest Reynolds number dataset. For instance, the differences in ζ 2 ðrÞ between R λ ¼ 2033 and R λ ¼ 5779 are less than AE5% within about one decade in r=η. Above R λ ≈ 3000 this range further extends to over two decades in r=η. The implication is that, throughout these intermediate scales, ζ 2 ðrÞ does not change with increases in R λ . That is, while S 2 remains r dependent in a way not described by power laws, S 2 does not appear to depend on R λ at intermediate scales 10 2 < r=η < 10 4 . In summary, we observe two subranges in structure functions (indicated by the black arrows in the lower panel of Fig. 1) consistent with other laboratory experiments, DNS, and the experiments in the atmosphere: (1) a narrow interval centered near r ≈ 150η, whose width is approximately constant despite increases in R λ and within which a scaling exponent may be identified that is approximately the same for the experiments, DNS, and atmosphere (ζ 2 ≈ 0.73) and (2) an interval r > 300η that widens with increasing R λ as expected for the inertial range, but within which ζ 2 ðrÞ decreases with increasing r in a way independent of R λ , albeit with subtle differences between the experimental, DNS, and atmospheric data.
Discussion.-We assembled a dataset of active-gridgenerated turbulence over a wider range of R λ than ever before measured in a single setup and found that the second-order velocity increment statistics quantified by ζ 2 ðrÞ is independent of R λ , but remains dependent on r within a broad range of intermediate scales consistent with the inertial range. The short range of approximately constant ζ 2 near r=η ≈ 150 is consistent with the oscillation in structure described in Refs. [39,40] and is not necessarily indicative of a nascent power law. In other words, the expected r independence of ζ 2 for R λ → ∞ is not observed in this dataset.
Our results agree with those from computer simulations and atmospheric flows up to ≈400η, noting that our Reynolds numbers are higher than in simulations and our experiments are better controlled than the atmosphere. In the following, we discuss how statistical anisotropy and nonstationarity may impact our results.
It is well known that real flows do not exhibit the ideal conditions of isotropy and statistical stationarity under which the theoretical limits are constructed, which can cause deviations from theoretical predictions [3,7,[41][42][43]. Anisotropic contributions can be separated from isotropic ones by expanding S n in spherical harmonics, provided one has directional information [43,44]. Scales must be considered relative to the scale of the source of anisotropy, which is on the order of the energy injection scale L.
Anisotropic contributions decrease relative to isotropic ones toward small scales. We choose ζ 2 ð200ηÞ as a reference value within the region exhibiting the smallest variation of ζ 2 with respect to changes in r. If anisotropy were the primary contributor to the observed variation in ζ 2 toward larger spatial scales, one would expect that an increase in scale separation L=η ∼ R 3=2 λ would tend to cause the anisotropic contributions to diminish. Figure 3 shows that this is not likely the case for R λ > 2000: ζ 2 at a fixed scale r 0 < L (we chose 1200η) does not approach ζ 2 ðr ¼ 200ηÞ with increasing L=η. Thus, anisotropy is unlikely to fully explain the scale dependence of ζ 2 depicted in Fig. 1.
Wind tunnel flows are not statistically stationary due to the downstream decay of turbulent kinetic energy, which is in contrast to computer simulations in periodic boxes that are forced and do not decay. The influence of decay on onedimensional turbulence statistics can be estimated using closure models for the statistically averaged Navier-Stokes equations (e.g., the Karman-Howarth equation) [38]. We follow Yang et al. [9], which accounts for a constant integral length scale, as this condition prevailed approximately in our experiments. We altered the model to include an intermittent inertial-range exponent (see Supplemental Material [29], which includes Refs. [9,45,46]). Adapting the remaining free parameter of the model to fit our data at low R λ , we estimate that R λ ≈ 10 6 would be necessary for a decade of inertial-range scaling to appear when the model is applied. These Reynolds numbers are close to those found FIG. 3. Difference in the local scaling exponent ζ 2 at two different values of r within the inertial range: 200η, where the statistics can be assumed to be the most isotropic, and 1200η, which tends to be more anisotropic due to its relative proximity to the large scales. When R λ > 2000, the inertial range extends from 200η to 1200η and beyond. To the right of this point, changes in the difference of ζ 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=η differ slightly from those read off from this graph due to a different ways of estimating ε ∼ u 3 =L.
on Jupiter [47]. The model, however, does not capture the detailed shape of the observed structure functions. Furthermore, our data suggest an even slower approach (if any) to a power law than the model, which tends to follow the data up to R λ ≈ 2000, but not beyond (see Fig. 3).
The data do not exclude an R λ dependence of the oddorder local slopes leading to power-law scaling (including the 4=5 law) at finite R λ . Our data suggest that R λ ≈ 10 5 would be required for the 4=5 law to appear. Differences in scaling behavior between odd and even orders have recently been attributed to gradual increases in the velocity time series (ramp) followed by sudden drops (cliff) [48]. In general, third-order statistics are sensitive to the transfer of energy, whereas second-order statistics measure the distribution of energy.
This Letter demonstrates a range of scales outside the viscous range whose shape is R λ -independent to within experimental error. Our findings suggest that the inertial range is more complex than anticipated and an increase in R λ might not necessarily yield an approach to power laws. The observed behavior questions the appropriateness of power laws for the description of real-world flows at large Reynolds numbers, in general. They therefore ultimately put into question the practical relevance of a separation of turbulence into viscous, inertial, and large scales, at least at a quantitative level.