Directional change of tracer trajectories in rotating Rayleigh-Bénard convection

The angle of directional change of tracer trajectories in rotating Rayleigh-Bénard convection is studied as a function of the time increment $\tau$ between two instants of time along the trajectories, both experimentally and with direct numerical simulations. Our aim is to explore the geometrical characterization of flow structures in turbulent convection in a wide range of timescales and how it is affected by background rotation. We find that probability density functions (PDFs) of the angle of directional change $\theta (t,\tau )$ show similar behavior as found in homogeneous isotropic turbulence, up to the timescale of the large-scale coherent flow structures. The scaling of the averaged (over particles and time) angle of directional change $\mathrm{\Theta }\left(\tau \right)=\langle \left|\theta \right(t,\tau \left)\right|\rangle$ with $\tau$ shows a transition from the ballistic regime $[\mathrm{\Theta }\left(\tau \right)\sim {\tau }^c$ with $c=1]$ for $\tau \lesssim {\tau }_{\eta }$, with ${\tau }_{\eta }$ the Kolmogorov timescale, to a scaling with smaller exponent $c$ for ${\tau }_{\eta }\lesssim \tau \lesssim T_L$, with $T_L$ the Lagrangian integral timescale. This scaling exponent is approximately constant in the weakly rotating regime (Rossby number $\text{Ro}\gtrsim 2.5$) and is decreasing for increasing rotation rates when $\text{Ro}\lesssim 2.5$. We show that this trend in the scaling exponent is related with the large-scale coherent structures in the flow; the large-scale circulation for $\text{Ro}\gtrsim 2.5$ and vertically aligned vortices emerging from the boundary layers (BLs) near the top and bottom plates and penetrating into the bulk for $\text{Ro}\lesssim 2.5$. In the viscous BLs, the PDFs of $\theta (t,\tau )$ and scaling properties of $\mathrm{\Theta }\left(\tau \right)$ are in general different from those measured in the bulk and depend on the type of boundary layer, in particular whether the BL is of Prandtl-Blasius type ($\text{Ro}\gtrsim 2.5$) or of Ekman type ($\text{Ro}\lesssim 2.5$). When it is of Ekman type, a stronger dynamic coupling exists between the BL and the bulk of the flow, resulting in similar scaling exponents in BL and bulk.

Figure 1

(a) Sketch of the angle between subsequent time steps $\tau$ of the particle trajectory, where $\mathbf{x}\left(t\right)$ is the position of the particle at time $t$. (b) Sketch of the trajectory at a time increment shorter than the Lagrangian integral timescale, where the length scales $l_{\parallel }$ and $l_{\perp }$ follow form fitting a right triangle through the three subsequent points. (Reproduction of Fig. 1 in [17].)

Figure 2

Sketch of the measurement volumes. The gray cube in the center represents the measurement volume in the DNS of size $0.5H\times 0.5H\times 0.5H$, in the $x, y$, and $z$ directions, respectively, while the hatched part shows the measurement volume used in the experiments of size $0.4H\times 0.3H\times 0.25H$, with $H=200\mathrm{mm}$ being the cell height. The green hatched rectangular parallelepiped at the top represents the experimental measurement volume of size $0.4H\times 0.3H\times 0.25H$. In the DNS this volume is subdivided in the vertical direction into a nonboundary layer (non-BL) part of size $0.5H\times 0.5H\times (0.25H-{\delta }_{u}H)$ (dark green upper part), and a boundary layer (BL) part of size $0.5H\times 0.5H\times {\delta }_{u}H$ (light green lower part), with ${\delta }_u$ the dimensionless viscous boundary layer thickness, which varies between ${\delta }_u=0.0299H$ and $0.0058H$, depending on the rotation rate [11]. Note that this sketch is not in real scale, particularly the BL size is enlarged for visibility reasons.

Figure 3

Top: PDFs of the angle $\theta , P_{\theta }$, for different (nondimensional) time increments $\tau$ measured in (a) the center and (b) the nonboundary layer (non-BL) region near the top plate in both experiments (symbols) and direct numerical simulations (DNS, lines) of nonrotating Rayleigh-Bénard convection. Focusing on the right part of the figures, $\tau$ is increasing from bottom to top. Center: PDFs of $Q=1-cos\theta , P_Q$, for the DNS data now represented by lines with symbols in (c) the center and (d) the non-BL region. Bottom: $P_Q$, but now normalized by ${\gamma }_{2,3}\left(\tau \right)=2\left(\tau \right)/3$, with the average angle of directional change, for (e) the center and (f) the non-BL region. The black dashed line shows a Fisher distribution $F_{2,3}$. For clarity, symbols are plotted in each 5 data points.

Figure 4

(a) PDFs of the angle $\theta , P_{\theta }$, for different (nondimensional) time increments $\tau$, measured in the viscous BL at the top plate for direct numerical simulations (DNS) of nonrotating Rayleigh-Bénard convection. (b) PDFs of $Q=1-cos\theta , P_Q$, for DNS. (c) $P_Q$, but now normalized by ${\gamma }_{1,2}\left(\tau \right)={\mathrm{\Theta }}^2\left(\tau \right)/4$, with $\mathrm{\Theta }$ the average angle of directional change, together with a Fisher distribution $F_{1,2}$ (black dashed line). For clarity, symbols are added to the lines each 5 data points in panels (a) and (c) and each data point in panel (b).

Figure 5

(a) The average angle of directional change $\mathrm{\Theta }\left(\tau \right)=\langle \left|\theta \right(t,\tau \left)\right|\rangle$ as a function of the nondimensional time increment $\tau$, for both DNS (lines with closed symbols) and experiments (open symbols), measured in the center (red squares) and near the top plate (blue circles) of the Rayleigh-Bénard cell. In the DNS the top measurement volume is subdivided in a nonboundary layer (non-BL) and boundary layer (BL) region, indicated by the green lines with triangles. (b) $\mathrm{\Theta }\left(\tau \right)$, normalized by ${\tau }^{1/2}$, measured using DNS in the center (red lines with squares) and the non-BL region (blue lines with circles). For the DNS symbols are added to the lines for visibility with an interval depending on $\tau$.

Figure 6

Top: PDFs of $Q=1-cos\theta , P_Q$, for different Rossby numbers $\text{Ro}$, measured using direct numerical simulations (DNS) in (a) the center and (b) the nonboundary layer (non-BL) region near the top plate of the Rayleigh-Bénard cell. PDFs are normalized by ${\gamma }_{2,3}\left(\tau \right)={\mathrm{\Theta }}^2\left(\tau \right)/2$, with $\mathrm{\Theta }\left(\tau \right)$ the average angle of directional change. The black dashed line represents the Fisher distribution $F_{2,3}$. For each $\text{Ro}$ time increments $\tau =0.3$, 1, and 5 are shown. Center: (c) $P_{\theta }$ and (d) $P_Q$, measured in the BL for $\text{Ro}=0.1$. Bottom: (e) $P_Q$, normalized by ${\mathrm{\Theta }}^2\left(\tau \right)$, measured in the BL for $\text{Ro}\ge 2.5$ (blue lines with crosses) and $\text{Ro}<2.5$ (red lines with circles). For each $\text{Ro}, \tau =0.1$, 1 and 5 are shown. The black solid and black dashed lines represent $3F_{2,3}$ and $4F_{1,2}$, respectively. Symbols are added to lines each data point in panel (d) and each 5 data points in all other panels for visibility.

Figure 7

The average angle $\mathrm{\Theta }\left(\tau \right)=\langle \left|\theta \right(t,\tau \left)\right|\rangle$ as a function of $\tau$ (in dimensionless time units), measured using DNS in (a) the center, (b) the nonboundary layer (non-BL) region, and (c) the boundary layer (BL) region near the top plate of the Rayleigh-Bénard cell. Symbols are added to the lines for visibility with an interval depending on $\tau$. (d) The scaling exponents $c$, given by $\mathrm{\Theta }\sim {\tau }^c$, for the curves of panels (a)–(c) in an intermediate time range ${\tau }_{\eta }<\tau \lesssim T_L$, with ${\tau }_{\eta }$ the Kolmogorov time scale and $T_L$ the Lagrangian integral timescale. The vertical black dashed line corresponds to $\text{Ro}=2.5$ and the solid horizontal black line is the HIT prediction $c=0.5$.

Figure 8

(a) Scaling exponents without and with shear correction ($c$ and $c_s$, open and closed symbols, respectively), measured in the center of the Rayleigh-Bénard cell for ${\tau }_{\eta }<\tau \lesssim T_L$, with ${\tau }_{\eta }$ the Kolmogorov timescale and $T_L$ the Lagrangian integral timescale. Red circles correspond to a relative volume of size $V/V_{\mathrm{tot}}=0.16$ and blue squares correspond to $V/V_{\mathrm{tot}}=0.65$, where $V=\mathrm{\Delta }x\times \mathrm{\Delta }y\times \mathrm{\Delta }z$ with ${\mathrm{\Delta }}_i$ the size of the measurement volume in the $i\mathrm{th}$ direction. The vertical black dashed line corresponds to $\text{Ro}=2.5$ and the solid horizontal black line is the HIT prediction $c=0.5$. (b) $c$ and $c_s$ for $\text{Ro}=\infty$ as a function of $V/V_{\mathrm{tot}}$. The measurement volume is always centered in the middle of the RBC cell and has the same size in all directions, such that $\mathrm{\Delta }x=\mathrm{\Delta }y=\mathrm{\Delta }z$.

Figure 9

PDFs of the angle of directional change $\theta , P_{\theta }$, for different Rossby numbers and for a time increment of (a) $\tau =0.1$ and (b) $\tau =10$, measured numerically in the center of the Rayleigh-Bénard cell. For clarity, symbols are plotted each 5 data points.