Effect of shear on coherent structures in turbulent convection

We study the effect of shear on the coherent structures near the hot surface, namely, line plumes, in turbulent Rayleigh-Bénard convection (RBC) and turbulent mixed convection (MC) for the range of near-surface Rayleigh numbers $5.75\times {10}^7\le {\text{Ra}}_w\le 2.17\times {10}^9$ and shear Reynolds numbers $8.02\times {10}^2\le \text{Re}\le 15\times {10}^3$ for a Prandtl number range of $10.1\ge \text{Pr}\ge 0.7$ in water and air. Plumes are visualized by particle scattering in MC in air while they are extracted from the particle image velocimetry fields in RBC in water. We also use the planforms of plume structure obtained by Gilpin et al. [J. Heat Transfer 100, 71 (1978)] in MC in water using electrochemical visualization, as well as those obtained by Pirozzoli et al. [J. Fluid Mech. 821, 482 (2017)] in simulations. The planforms of plume structure show that shear aligns the line plumes and increases their mean spacing $\lambda$. An increase in ${\text{Ra}}_w$ decreases $\lambda$, while the resulting increase in Re in RBC, due to the increase of larger large-scale flow strength, counteracts this effect. Further, plumes are seen more spaced and smeared in air, compared to that in water, due to the lower Pr. We show that these complex dependences of $\lambda$ on ${\text{Ra}}_w$, Re, and Pr in RBC and MC can be described by a common scaling law ${\lambda }^{*}=\lambda -{\lambda }_0=S{Z}_{\mathrm{sh}}/D$, where ${\lambda }_0$ is the mean plume spacing in the absence of shear, $S={\text{Re}}^3/{\text{Ra}}_w$ is a shear parameter, $Z_{\mathrm{sh}}=\nu /U_{\mathrm{sh}}$ is the viscous-shear length, and $D$ a function of Pr.

Figure 1

(a) Schematic of the setup for steady MC experiments in air at $\text{Pr}=0.7$; (b) planform view of the plume structure obtained at ${\text{Ra}}_w=1.55\times {10}^8$ and external flow rate $Q=1736$ lpm with the dark line plumes marked with short, white linear segments for measurement of length. The image is of size $98.5\mathrm{cm}\times 47.7\mathrm{cm}$.

Figure 2

Schematic of the setup for steady RBC experiments in water.

Figure 3

Dimensionless horizontal divergence fields $[{\mathbf{\nabla }}_H·u/(V_{LS}/{\lambda }_0)]$, overlaid over the horizontal velocity vector fields, in a horizontal plane at a height $h_m$ from the hot surface in RBC in water. (a) ${\text{Ra}}_w=1.09\times {10}^8,\text{Re}=832$, and $h_m=1.5\mathrm{mm}$; (b) ${\text{Ra}}_w=6.03\times {10}^8,\text{Re}=2490$, and $h_m=1.3\mathrm{mm}$. Plumes are the colored regions. Shear dominant areas are shown by the red polygon. $L_p$ is measured by adding up the length of the black lines in the plume regions in the shear-dominant regions. The large-scale flow velocity $V_{\mathrm{LS}}$ is estimated from the Reynolds number based on $V_{\mathrm{LS}}$ given by ${\text{Re}}_{\mathrm{LS}}=V_{\mathrm{LS}}H/\nu =0.55{\text{Ra}}_w^{4/9}{\text{Pr}}^{-2/3}$ [16].

Figure 4

Planforms of plume structure in MC in air at ${\text{Ra}}_w=1.55\times {10}^8$. (a) $\text{Re}=13.28\times {10}^3$; (b) $\text{Re}=14.48\times {10}^3$; (c) $\text{Re}=15\times {10}^3$. Flow is from top to bottom. The images are of size $98.5\mathrm{cm}\times 47.7\mathrm{cm}$.

Figure 5

Planforms of plume structure at approximately the same Re at two different ${\text{Ra}}_w$ in MC in air. (a) $\text{Re}=9.58\times {10}^3$ at ${\text{Ra}}_w=5.75\times {10}^7$; (b) $\text{Re}=9.6\times {10}^3$ at ${\text{Ra}}_w=1.55\times {10}^8$. Flow is from top to bottom. The images are of size $98.5\mathrm{cm}\times 47.7\mathrm{cm}$.

Figure 6

Variation of the mean plume spacing with the shear velocity. Hollow symbols indicate MC experiments in air at $\text{Pr}=0.7$ for the following ${\text{Ra}}_w$: $○,{\text{Ra}}_w=5.75\times {10}^7$; $\square ,{\text{Ra}}_w=1.01\times {10}^8$; $\diamond ,{\text{Ra}}_w=1.24\times {10}^8$; $▵,{\text{Ra}}_w=1.55\times {10}^8$. Filled symbols indicate RBC experiments in water for the following ${\text{Ra}}_w$ and Pr: $•,{\text{Ra}}_w=1.09\times {10}^8,\text{Pr}=5.24$; $\blacksquare ,{\text{Ra}}_w=2.79\times {10}^8,\text{Pr}=5.18$; $♦,{\text{Ra}}_w=6.03\times {10}^8,\text{Pr}=5.09$. $\times$, MC experiments by Gilpin et al. [17] in water at $\text{Pr}=10.1$ and ${\text{Ra}}_w=2.17\times {10}^9$; $✳$, MC simulations by Pirozzoli et al. [15] at $\text{Pr}=1$ and ${\text{Ra}}_w=5\times {10}^7$; $\text{----},{\lambda }_0$ given by (1) for ${\text{Ra}}_w=6.03\times {10}^8$ and $\text{Pr}=5.09$; $—,{\lambda }_0$ given by (1) for ${\text{Ra}}_w=1.55\times {10}^8$ and $\text{Pr}=0.7$. The inset shows the variation of the dimensionless plume spacing with Reynolds number; $—,{\text{Ra}}_{{\lambda }_0}^{1/3}{\text{Pr}}^{-n_1}=47.5$ (3).

Figure 7

Variation of ${\lambda }^{*}$, the difference between the mean plume spacing with shear $\left(\lambda \right)$ and without shear $\left({\lambda }_0\right)$ normalized by the viscous-shear length $Z_{\mathrm{sh}}$ (8), with the dimensionless shear parameter $S$ (7); The symbols are as per Fig. 6 and Table 1; $—$, (19). The inset shows the variation of the prefactor $D$ in (19) with Pr; $—$, (20); $––$, (21).

Figure 8

Variation of the ratio of the mean plume spacing with shear $\left(\lambda \right)$ and that without shear $\left({\lambda }_0\right)$ with the dimensionless shear parameter $S$ (7). The symbols are as per Fig. 6 and Table 1; $—$, (24).