On Lagrangian properties of turbulent Rayleigh–Bénard convection

We report on Lagrangian statistics of turbulent Rayleigh–Bénard convection under very different conditions. For this, we conducted particle tracking experiments in a $H=1.1$ -m-high cylinder of aspect ratio $\varGamma =1$ filled with air ( Pr = 0.7), as well as in two rectangular cells of heights $H=0.02$ m ( $\varGamma =16$ ) and $H=0.04$ m ( $\varGamma =8$ ) filled with water ( Pr = 7.0), covering Rayleigh numbers in the range $10^6\le {\textit {Ra}}\le 1.6\times 10^9$ . Using the Shake-The-Box algorithm, we have tracked up to 500 000 neutrally buoyant particles over several hundred free-fall times for each set of control parameters. We find the Reynolds number to scale at small Ra (large Pr ) as $ {\textit{Re}} \propto {\textit{Ra}}^{0.6}$ . Further, the averaged horizontal particle displacement is found to be universal and exhibits a ballistic regime at small times and a diffusive regime at larger times, for sufficiently large $\varGamma$ . The diffusive regime occurs for time lags larger than $\tau _{co}$ , which is the time scale related to the decay of the velocity autocorrelation. Compensated as $\tau _{co} {\textit {Pr}}^{-0.3}$ , this time scale is universal and rather independent of $ {\textit {Ra}}$ and $\varGamma$ . We have also investigated the Lagrangian velocity structure function $S^2_i(\tau )$ , which is dominated by viscous effects for times smaller than the Kolmogorov time $\tau _\eta$ and hence $S^2_i\propto \tau ^2$ . For larger times we find a novel scaling for the different components with exponents smaller than what is expected in the inertial range of homogeneous isotropic turbulence without buoyancy. Studying particle-pair dispersion, we find a Batchelor scaling ( ${\propto }\,t^2$ ) on small time scales, diffusive scaling ( ${\propto }\,t$ ) on large time scales and Richardson-like scaling ( ${\propto }\,t^3$ ) for intermediate time scales.