Size invariance of the granular Rayleigh-Taylor instability

The size scaling behavior of the granular Rayleigh-Taylor instability [J. L. Vinningland et al., Phys. Rev. Lett. 99, 048001 (2007)] is investigated experimentally, numerically, and theoretically. An upper layer of grains displaces a lower gap of air by organizing into dense fingers of falling grains separated by rising bubbles of air. The dependence of these structures on the system and grain sizes is investigated. A spatial measurement of the finger structures is obtained by the Fourier power spectrum of the wave number $k$. As the size of the grains increases the wave number decreases accordingly which leaves the dimensionless product of wave number and grain diameter, $dk$, invariant. A theoretical interpretation of the invariance, based on the scaling properties of the model equations, suggests a gradual breakdown of the invariance for grains smaller than $\sim 70\mu \text{m}$ or greater than $\sim 570\mu \text{m}$ in diameter.

Figure 1

(Color online) (left) Side view of the experimental setup with the initial and final cell positions superimposed. (right) Typical snapshot from an experiment using grains of $230\mu \text{m}$ in diameter in a cell that is $91\times 141\times 2.3\text{mm}$.

Figure 2

Numerical snapshots. Each column represents different simulations using grains of diameter $d$, and each row represents snapshots at different times $t$.

Figure 3

Experimental snapshots. Each column represents a different experiment characterized by the diameter $d$ of the grains used, and each row represents a different time $t$.

Figure 4

(Color online) Plots of Fourier power spectra $S_j\left(k\right)$ (left) obtained at three different heights of the solid fraction field $\rho \left(x,y\right)$ (right) shown as a color coded data image [red (gray) is high $\rho$, blue (dark gray) is low $\rho$]. The stapled white lines indicate at which $y$ position the power spectra are obtained.

Figure 5

(Color online) Plots of $\overline{S}\left(dk\right)$ [$\overline{S}\left(k\right)$ in inset plots] obtained from numerical data. Each plot is a data collapse of $\overline{S}\left(dk\right)$ for all grain-diameters $d$ at three different times $t$ (the same times as in Fig. 2). The different colors represent different grain diameters ranging from $70$ to $490\mu \text{m}$ as indicated in the inset plots. The straight black lines are power laws obeyed by $\overline{S}\left(dk\right)$ at large and small scales: $\overline{S}\left(dk\right)\sim \text{constant}$ at large scales (growing white noise), and $\overline{S}\left(dk\right)\sim {\left(dk\right)}^{-2.5}$ at small scales.

Figure 6

(Color online) Plots of $\overline{S}\left(dk\right)$ [$\overline{S}\left(k\right)$ in inset plots] obtained from experimental data. Each plot is a data collapse of $\overline{S}\left(dk\right)$ for all grain-diameters $d$ at three different times $t$ (the same times as in Fig. 3). The different colors represent different grain diameters ranging from $80\mu \text{m}$ to $570\mu \text{m}$ as indicated in the inset plots. The straight black lines are power laws obeyed by $\overline{S}\left(dk\right)$ at large and small scales, as in the numerical case: $\overline{S}\left(dk\right)\sim \text{constant}$ at large scales, and $\overline{S}\left(dk\right)\sim {\left(dk\right)}^{-2.5}$ at small scales.

Figure 7

(Color online) (a) Plot of the mean wave number $⟨k⟩$ obtained from four different simulations with grains of diameter $d=140\mu \text{m}$ but with four different cell widths $w$. (b) Same plot for the standard deviation $\sigma$.