Multitime structure functions and the Lagrangian scaling of turbulence

We define and characterize multitime Lagrangian structure functions using data stemming from two swirling flows with mean flow and turbulent fluctuations: a Taylor-Green numerical flow and a von Kármán laboratory experiment. Data is obtained from numerical integration of tracers in the former case and from three-dimensional particle tracking velocimetry measurements in the latter. Multitime statistics are shown to decrease the contamination of large scales in the inertial range scaling. A timescale at which contamination from the mean flow becomes dominant is identified, with this scale separating two different Lagrangian scaling ranges. The results from the multitime structure functions also indicate that Lagrangian intermittency is not a result of large-scale flow effects. The multitime Lagrangian structure functions can be used without prior knowledge of the forcing mechanisms or boundary conditions, allowing their application in different flow geometries.

Figure 1

Illustration of the multitime method: (a) A synthetically generated signal $v$, composed of fast (small-scale) fluctuations $v_S$ superimposed on a slow (large-scale) trend $v_L$ (chosen in the example as a first degree polynomial), and normalized by the standard deviation of $v_S$, denoted by ${\sigma }_S$. The inset shows the power spectral density of $v_S$. (b) Normalized second-order structure function using 2 and 3 times.

Figure 2

Second-order tracers' velocity structure functions for the VK experiment (dots) and for the TG DNSs (solid curves) using 2, 3, and 4 times; for each of the Cartesian components $x$, $y$, and $z$ in panels (a), (b), and (c) respectively. A power law with exponent $-1/3\mathrm{m}$ which corresponds to sweeping by the mean flow given by $S_2\propto {\tau }^{2/3}$, is indicated by the dashed line. The timescale ${\tau }^{*}$ where this contribution is comparable to inertial range scaling is indicated by arrows (normalized by ${\tau }_L$) for both data sets. The insets show $S_2^{2\text{pt}}/S_2^{3\text{pt}}$ and $S_2^{2\text{pt}}/S_2^{4\text{pt}}$ for each Cartesian component. Labels are the same as in the main panels.

Figure 3

Multitime Lagrangian velocity structure functions $S_p^{n\text{pt}}$ of orders $p=1$ to 5, compensated by the nonintermittent Lagrangian inertial-range prediction $S_p\left(\tau \right)\propto {\tau }^{p/2}$ for tracers in the TG and VK data sets (represented by lines and individual markers, respectively). The top, middle and bottom row panels correspond to calculations using 2, 3, and 4 times, respectively. Each column gathers one individual Cartesian component of the particles' velocity. In all panels, the arrows indicate increasing order $p$.

Figure 4

Logarithmic derivative of the multitime structure functions of orders $p=1$ to 5; for tracers in the VK and TG data sets (represented by markers and lines, respectively). Panels at the top, middle, and bottom rows correspond to ${\xi }_p^{\text{npt}}$ calculated using 2, 3, and 4 times, respectively. Each column shows one individual Cartesian component of the particles' velocity ($x$, $y$, and $z$, respectively). The shaded areas correspond to the inertial range, for practical purposes identified as the range in $\tau /{\tau }_L$ for which $0.9\le {\xi }_2^{n\text{pt}}\le 1.1$.

Figure 5

Inertial-range local scaling exponents of order $p$, normalized by the exponent corresponding to $p=2$ and computed from multitime structure functions for the TG and VK data. The exponents are compared with other results from experiments and simulations with different flows from Refs. [41, 49, 50, 51]. From left to right: (a) ${\xi }_p/{\xi }_2$ for the $x$ component of the velocity, (b) for $y$, and (c) for $z$. In all panels, the dashed line indicates the nonintermittent (i.e., self-similar) Kolmogorov scaling

Figure 6

Absolute width of the inertial range, normalized by the Lagrangian correlation time, as a function of the number of times $n$ used to compute the structure functions for each Cartesian component of the velocity and for the VK experiment (triangles) and the TG DNS (squares).

Figure 7

Second-order structure functions using 2 and 3 times compensated by the inertial range prediction, for all values of the Reynolds number considered in this study. (a) shows the structure function for the TG simulations, labeled according to grid resolution. (b) shows the data corresponding to the VK experiment for different rotation frequencies of the propellers. Larger number of grid points in DNSs and higher rotation frequencies in experiments correspond to higher Reynolds numbers (see Table 1). A power law with exponent $-1/3$, which corresponds to sweeping by the mean flow given by $S_2\propto {\tau }^{2/3}$, is indicated by the dashed line.

Figure 8

Lagrangian velocity power spectra computed from the velocity correlation function ($E^{\text{2pt}}$, filled markers) and from the 3-times second-order structure function ($E^{\text{3pt}}$, empty markers), in log-log scale. Square and triangle markers denote results from simulations and experiments, respectively. The inset shows $E^{3\text{pt}}/E^{2\text{pt}}$ for each data set in log-lin scale.

Figure 9

Compensated second-order tracers' velocity structure functions in a ${1024}^3$ simulation of homogeneous and isotropic turbulence (HIT) with random forcing, using 2, 3, and 4 times, for each of the Cartesian components $x$, $y$, and $z$ of the velocity in (a), (b), and (c) respectively.