Pair dispersion in inhomogeneous turbulent thermal convection

Due to large-scale flow inhomogeneities and the effects of temperature, turbulence small-scale structure in thermal convection is still an active field of investigation, especially considering sophisticated Lagrangian statistics. Here we experimentally study Lagrangian pair dispersion (one of the canonical problems of Lagrangian turbulence) in a Rayleigh-Bénard convection cell. A sufficiently high temperature difference is imposed on a horizontal layer of fluid to observe a turbulent flow. We perform Lagrangian tracking of submillimeter-sized particles on a large measurement volume including part of the large-scale circulation (LSC), revealing some large inhomogeneities. Our study brings to light several insights regarding our understanding of turbulent thermal convection: (i) by decomposing particle Lagrangian dynamics into the LSC contribution and the turbulent fluctuations, we highlight the relative impact of both contributions on pair dispersion; (ii) using the same decomposition, we estimate the Eulerian second-order velocity structure functions from pair statistics and show that after removing the LSC contribution, the remaining statistics recover usual homogeneous and isotropic behaviors which are governed by a local energy dissipation rate to be distinguished from the global dissipation rate classically used to characterize turbulence in thermal convection; and (iii) we revisit the superdiffusive Richardson-Obukhov regime of particle dispersion and propose a refined estimate of the Richardson constant.

Figure 1

Convection cell and camera position viewed from above. The gray zone corresponds to the field of view observed by the three cameras. More details about dimensions and camera specifications are in the text.

Figure 2

Scheme of the convection roll and the measurement volume captured by the experimental acquisition setup, compared to the whole convection cell.

Figure 3

Three-dimensional visualizations of the mean velocity field $\overrightarrow{v^E}=v_x^E\vec{x}+v_y^E\vec{y}+v_z^E\vec{z}$. The two plots corresponds to two different viewing angles. The arrow size is proportional to velocity magnitude, going from 0 and 12 mm/s.

Figure 4

Sketch of the pair dispersion principle. At time $t_0$, two particles are separated with an initial distance $\left|\left|\overrightarrow{{\mathrm{\Delta }}_0}\right|\right|$, while at time $t>t_0$ they are separated by $\left|\left|\vec{\mathrm{\Delta }}\right|\right|$.

Figure 5

Pair dispersion for (a) small (1.9, 2.1, 2.3, 2.6, and 2.9 mm) and (b) larger initial separations ${\mathrm{\Delta }}_0$ (4.3, 5.4, 7.0, 8.9, 11.4, 14.6, 17.7, 23.7, 30.3, 38.7, 49.3, 62.9, and 80.3 mm).

Figure 6

Second-order Eulerian structure functions of each velocity component versus the total initial separation ${\mathrm{\Delta }}_0$. The black line represents the model from Eq. (14). The dashed part corresponds to the range where the model validity decreases. In the inset, the same axes and data are plotted except for $S_{v_y}^2$, for which the LSC term ${\beta }^2{\mathrm{\Delta }}_0^2/3$ from Eq. (14) is subtracted to isolate the small-scale turbulent contribution. Only the part corresponding to the solid black line in the main plot is shown in the inset.

Figure 7

Sketch of the mean flow in the {$\vec{y},\vec{z}$} plane. The red arrow represents the mean streamlines. The green arrows represent the velocity vectors of two particles separated by ${\mathrm{\Delta }}_0$.

Figure 8

Compensated plot of the pair dispersion $D_{{\mathrm{\Delta }}_0}^2/\left({\langle \epsilon \rangle }_{\mathrm{loc}}{t}^3\right)$ for short initial separations: $2.7\eta$, $3.0\eta$, $3.3\eta$, $3.7\eta$, and $4.0\eta$.

Figure 9

Pair separation ${\langle {\mathrm{\Delta }}^2\left(t\right)\rangle }_0$ for HIT computed from the ballistic-cascade model (adapted to account for the dissipative scaling of $S_2$ at small initial separations) [16], compensated by $\left(\epsilon t^3\right)$. The initial separations rise linearly in the range $[\eta ,10\eta ]$.