Sharp transitions in rotating turbulent convection: Lagrangian acceleration statistics reveal a second critical Rossby number

In Rayleigh–Bénard convection (RBC) for fluids with Prandtl number $\text{Pr}\gtrsim 1$, rotation beyond a critical (small) rotation rate is known to cause a sudden enhancement of heat transfer, which can be explained by a change in the character of the boundary layer (BL) dynamics near the top and bottom plates of the convection cell. Namely, with increasing rotation rate, the BL signature suddenly changes from Prandtl–Blasius type to Ekman type. The transition from a constant heat transfer to an almost linearly increasing heat transfer with increasing rotation rate is known to be sharp and the critical Rossby number ${\text{Ro}}_c$ occurs typically in the range $2.3\lesssim {\text{Ro}}_c\lesssim 2.9$ (for Rayleigh number $\text{Ra}=1.3\times {10}^9$, $\text{Pr}=6.7$, and a convection cell with aspect ratio $\mathrm{\Gamma }=\frac{D}{H}=1$, with $D$ the diameter and $H$ the height of the cell). The explanation of the sharp transition in the heat transfer points to the change in the dominant flow structure. At $1/\text{Ro}\lesssim 1/{\text{Ro}}_c$ (slow rotation), the well-known large-scale circulation (LSC) is found: a single domain-filling convection roll made up of many individual thermal plumes. At $1/\text{Ro}\gtrsim 1/{\text{Ro}}_c$ (rapid rotation), the LSC vanishes and is replaced with a collection of swirling plumes that align with the rotation axis. In this paper, by numerically studying Lagrangian acceleration statistics, related to the small-scale properties of the flow structures, we reveal that this transition between these different dominant flow structures happens at a second critical Rossby number, ${\text{Ro}}_{c_2}\approx 2.25$ (different from ${\text{Ro}}_{c_1}\approx 2.7$ for the sharp transition in the Nusselt number Nu; both values for the parameter settings of our present numerical study). When statistical data of Lagrangian tracers near the top plate are collected, it is found that the root-mean-square values and the kurtosis of the horizontal acceleration of these tracers show a sudden increase at ${\text{Ro}}_{c_2}$. To better understand the nature of this transition we compute joint statistics of the Lagrangian velocity and acceleration of fluid particles and vertical vorticity near the top plate. It is found that for $\text{Ro}\gtrsim 2.25$ there is hardly any correlation between the vertical vorticity and extreme acceleration events of fluid particles. For $\text{Ro}\lesssim 2.25$, however, vortical regions are much more prominent and extreme horizontal acceleration events are now correlated to large values of positive (cyclonic) vorticity. This suggests that the observed sudden transition in the acceleration statistics is related to thermal plumes with cyclonic vorticity developing in the Ekman BL and subsequently becoming mature and entering the bulk of the flow for $\text{Ro}\lesssim 2.25$.

Figure 1

Sketch of the measurement volumes (not to scale). The gray cube in the center represents a measurement volume of size $0.25H\times 0.25H\times 0.25H$, in the $x$, $y$, and $z$ direction, respectively. The green rectangular parallelepiped at the top plate represents a measurement volume of size $0.25H\times 0.25H\times 0.05H$ that starts right under the top plate (spanning the range $0.95H<z<H$ vertically).

Figure 2

(a) The Nusselt number Nu as function of Ro, normalized by $\text{Nu}\left(\infty \right)$ for the nonrotating case. Squares show data from [11, 12] for $\text{Ra}=2.73\times {10}^8$ and $\text{Pr}=6.26$, circles show data for the current simulations at $\text{Ra}=1.3\times {10}^9$ and $\text{Pr}=6.7$, and triangles show experimental data by [15] (supplementary data) taken at $\text{Ra}=2.19\times {10}^9$ and $\text{Pr}=6.26$. Closed symbols are for DNS while the open symbols are for experiments. (b) The same datasets, but now showing a zoom of the data in the range $2\le \text{Ro}\le 4$. The vertical dotted line in both panels represents ${\text{Ro}}_{c_1}=2.7$, and the vertical solid line in the left panel represents ${\text{Ro}}_{c_2}=2.25$.

Figure 3

(a) Root-mean-square (rms) values of the horizontal and vertical acceleration components, $a_{xy}^{\text{rms}}$ and $a_z^{\text{rms}}$, respectively, as a function of Ro. Statistics are collected in the center (open symbols) and near the top plate (closed symbols). (b) The ratio between the rms values of the horizontal and vertical acceleration components, $R_{a^{\text{rms}}}=a_{xy}^{\text{rms}}/a_z^{\text{rms}}$, for the center (blue crosses) and the top (red asterisks). In both panels the inset shows a zoom of the data in the range $2\le \text{Ro}\le 2.75$ and the vertical solid line represents ${\text{Ro}}_{c_2}=2.25$, the second critical Rossby number, clearly different from ${\text{Ro}}_{c_1}=2.7$ (shown as the dotted line). Some of the error bars have the same size as the symbols, or are even smaller, and therefore not visible (the error bars are based on the differences between the results from the Lagrangian data obtained in the top and bottom half of the convection cell).

Figure 4

(a) Kurtosis and (b) skewness of the horizontal (squares) and vertical (circles) acceleration statistics as a function of Ro. Statistics are collected in the center (open symbols) and near the top plate (closed symbols). In both panels the inset shows a zoom of the data in the range $2\le \text{Ro}\le 2.75$. The vertical solid line represents ${\text{Ro}}_{c_2}=2.25$, and the vertical dotted line ${\text{Ro}}_{c_1}=2.7$. Some of the error bars have the same size as the symbols, or are even smaller, and therefore not very well visible (the error bars are based on the differences between the results from the Lagrangian data obtained in the top and bottom half of the convection cell).

Figure 5

The viscous BL thickness ${\delta }_{\nu }/H$ for cylindrical Rayleigh–Bénard convection with $\text{Ra}=1.3\times {10}^9$, $\text{Pr}=6.7$ and $\mathrm{\Gamma }=1$. The BL thickness is measured from the DNS as the position of the maximum of the horizontal rms velocity. The horizontal dashed line shows the kinetic BL thickness ${\delta }_{\nu }/H=0.032$ and the sloping solid black line shows the theoretical prediction based on the linear Ekman BL theory, where ${\delta }_{Ek}=\sqrt{\nu /\mathrm{\Omega }}$; see Ref. [18]. The black dashed-dotted vertical line indicates the Rossby number where the viscous and Ekman BLs intersect ($\text{Ro}\approx 1.4$). The vertical solid and dotted lines indicate ${\text{Ro}}_{c_2}$ and ${\text{Ro}}_{c_1}$, respectively.

Figure 6

Rms values of the (a) horizontal and (b) vertical acceleration components, both normalized by the respective rms acceleration values of the nonrotating case, $a_{xy}^{\text{rms}}\left(\text{Ro}\right)/a_{xy}^{\text{rms}}\left(\infty \right)$ and $a_z^{\text{rms}}\left(\text{Ro}\right)/a_z^{\text{rms}}\left(\infty \right)$, respectively. Rms values are computed in horizontal slabs of thickness ${\mathrm{\Delta }}_{z_i}=0.001H$ and central position $z_i$ as a function of Ro. The legend in panel (b) is the same as in panel (a) and in both panels the inset shows a zoom of the data in the range $2\le \text{Ro}\le 2.75$. The vertical solid and dotted lines represent ${\text{Ro}}_{c_2}=2.25$ and ${\text{Ro}}_{c_1}=2.7$, respectively.

Figure 7

Joint PDFs of the horizontal Lagrangian velocity of passive tracers, $u_y$, and the vertical vorticity, ${\omega }_z$, measured in the top measurement volume for six different Rossby numbers.

Figure 8

Joint PDFs of the vertical Lagrangian velocity of passive tracers, $u_z$, and the vertical vorticity, ${\omega }_z$, measured in the top measurement volume for six different Rossby numbers.

Figure 9

Joint PDFs of the horizontal Lagrangian acceleration of passive tracers, $a_{xy}$, and the vertical vorticity, ${\omega }_z$, measured in the top measurement volume for six different Rossby numbers.