Experimental investigation of Lagrangian structure functions in turbulence

Lagrangian properties obtained from a particle tracking velocimetry experiment in a turbulent flow at intermediate Reynolds number are presented. Accurate sampling of particle trajectories is essential in order to obtain the Lagrangian structure functions and to measure intermittency at small temporal scales. The finiteness of the measurement volume can bias the results significantly. We present a robust way to overcome this obstacle. Despite no fully developed inertial range, we observe strong intermittency at the scale of dissipation. The multifractal model is only partially able to reproduce the results.

Figure 1

Experimental setup

Figure 2

Pdf $p\left(r\right)$ of the radial position of particles in the measuring volume ($★$ symbol). The symbols ◻ and ○ represent respectively particles starting and ending a trajectory. The black solid line has a slope equal to two and therefore represent the uniform distribution.

Figure 3

Top: PDF histogram of number of particles within randomly positioned balls of radius 10 mm (left) and 30 mm (right). The dots represent a Poisson distribution. Bottom: Variation of the ratio $\left(⟨N^2⟩-{⟨N⟩}^2\right)/⟨N⟩$ as function of ball radius. The horizontal line represent the Poisson value of unity.

Figure 4

Left: ${\sigma }_u(★)$ and ${\sigma }_a$ (●) as functions of filter width ${\sigma }_{filter}$ divided by Kolmogorov time scale ${\tau }_{\eta }$. Right: Flatness $F_u(★)$ and $F_a$ (●). The dotted vertical lines denote a filter of length $N=4$ corresponding to $0.23{\tau }_{\eta }$.

Figure 5

(Color online) Acceleration pdf $p\left(a_y\right)$ for the component along the axis of forcing in the tank. The pdf is normalized with the standard deviation of acceleration ${\sigma }_{a_y}$ for the same component. The different colors represent the different filter lengths $N$ ranging from 1 to 20 following the direction of the arrow. The two black curves are the Gaussian distribution and a stretched exponential fitted to the curve for $N=4$ (only partly visible as the thin black curve below the fitted stretched exponential).

Figure 6

Mean Greens function $G\left(r,\tau \right)$ as a function of $r$ at three different time lags $\tau$. Increasing towards the right, time lags are $\tau =\left\{5,13,35\right\}{\tau }_{\eta }$ with (solid lines) and without (dashed lines) compensation through $W\left(r\right)$.

Figure 7

$\text{log}{S}_p\left(\tau \mid r\right)/S_p\left(\tau \right)$ as a function of $\text{log}r/{\sigma }_x\left(tau\right)$ for $\tau ∊\left(15;25\right){\tau }_{\eta }$. The four panels represents $p=2,4,6,8$ with fits $r^p$ for large $r/{\sigma }_x\left(tau\right)$.

Figure 8

Relative error estimates for $S_p$ for $p=2$, 4, 6, and 8. Dashed lines are without compensation while solid lines are with compensation

Figure 9

$G\left(r,\tau \right)$ for $\tau =10{\tau }_{\eta }$. The full line is a Gaussian.

Figure 10

$C_0$: The circles represent the two horizontal directions while the squares represents the axial direction. The stars are the mean [Eq. (4)]. The arrow indicates the maximum $C_0=4.46$ at $\tau =2.96{\tau }_{\eta }$.

Figure 11

Lagrangian structure functions $S_p\left(\tau \right)$ as a function of time lag $\tau$. $p$ is increasing upwards with $p=2,4,6,8$. The different structure functions have been shifted vertically for clarity.

Figure 12

${\zeta }_p\left(\tau \right)$. Top: $p=4$, middle: $p=6$ and bottom: $p=8$ (solid curves). The shaded areas represent the multifractal predictions: the upper boundary is based on longitudinal Eulerian structure functions while the lower boundary is based on transverse Eulerian structure functions. The thick black vertical lines denote the critical time lags according to Sec. 3. We find the large time lag saturation levels to be 1.53, 1.76, and 1.93 for $p=4,6,8$. The error bars refer to statistical errors.

Figure 13

(Color online) Top: Sample particle. The color denotes the magnitude of the acceleration. Bottom: Particle acceleration in units of the standard deviation ${\sigma }_a$ for all three components.

Figure 14

(Color online) ${\zeta }_p\left(\tau \right)$. Top: $p=4$, middle: $p=6$ and bottom: $p=8$ (solid curves). The colors denote different filter $N$ [increasing upwards from $N=1$ (red)]. Arrows indicate minimum position for $N=1$ (left arrow) and $N=10$ (right arrow).