Backwards and forwards relative dispersion in turbulent flow: An experimental investigation

From particle tracking velocimetry we present an experimental measure of the ratio between backwards and forwards relative dispersion in an intermediate Reynolds number turbulent flow. Lack of time-reversal symmetry implies that their ratio may be different from 1. From a stochastic model, this has recently been studied by Sawford et al. [Phys. Fluids 17, 095109 (2005)] giving ratios between 5 and 20. We find a value of approximately 2 and discuss it in the context of the characteristics of the rate of strain tensor $s_{ij}$. An analysis of a direct numerical simulation by Biferale et al. [Phys. Rev. Lett. 93, 064502 (2004) and Phys. Fluids 17, 021701 (2004)] gives the same result.

Figure 1

Experimental setup.

Figure 2

(a) Standard deviations of the velocity components averaged over the $yz$ plane and the $xz$ plane. The dotted lines correspond to the radial components while the dashed lines correspond to the axisymmetric vertical component. (b) Same as in (a) but averaged over the $xy$ plane.

Figure 3

(a) $f\left(r\right)$ with the fitted von Kármán model on top (full line). (b) $⟨\delta a_{\mid \mid }\left(r\right)⟩$ with a linear fit.

Figure 4

(a) $⟨r^2\left(t\right)⟩∕r_0^2$ as a function of $t∕t_0$ in the forward case. The different dotted lines correspond to different initial separations $r_0$ from $4–8\eta$ to $24–28\eta$. A fit to data by Eq. (10) is plotted as the gray solid line. The dashed line is the backward case fit. (b) Backward case.

Figure 5

$(⟨r^{2∕3}⟩-r_0^{2∕3})∕(r_0^{2∕3}t∕t_0)$ vs $t∕t_0$. The different curves correspond to the different bins.

Figure 6

(a) Distance-neighbor function for forward dispersion for pairs starting at $8–12\eta$, (b) backward case. In both plots, the straight line is the prediction by Richardson.

Figure 7

(a) $⟨r\left(t\right){⟩}^2∕⟨r^2\left(t\right)⟩$ as a function of $t∕t_0$ for the forward case. The different lines correspond to different initial separations: $r_0$ increasing upward from $r_0=4\eta$ to $r_0=28\eta$ in bins of $4\eta$. The horizontal lines are the Richardson prediction at $0.67$ and the Gaussian prediction at $0.85$. (b) Backward case.

Figure 8

$⟨r^2\left(t\right)⟩∕r_0^2$ as a function of $t∕t_0$ for the two cases with $r_0=20\eta$. Fits by Eq. (10) are displayed on top by thin lines. The upper curve is the backward case.

Figure 9

$g_f$, $g_b$, and $g_b∕g_f$ (from bottom to top) as a function of $r_0∕\eta$.

Figure 10

(a) PDF of ${\Lambda }_i$. The PDFs are normalized with the rms of total strain $⟨s^2{⟩}^{1∕2}$. (b) Stretching rates as a function of time. From bottom to top: $\mathrm{var}\mathbf{\left(}{L}_f\right(t\left)\mathbf{\right)}$, $\mathrm{var}\mathbf{\left(}{L}_b\right(t\left)\mathbf{\right)}$, $⟨L_f\left(t\right)⟩$, and $⟨L_b\left(t\right)⟩$.