Heat transport enhancement and flow transitions in partitioned thermal convection

Partitions are an essential part of industrial reactors and thermal management devices whose primary purpose is to increase transport rates by obstructing flow in one direction and promoting in the other via secondary and small-scale motions. Inspired by such applications, we have investigated thermal convection in a two-dimensional square enclosure heated at the bottom and cooled at the top, with four additional thin vertical partitions arranged parallel to facilitate organized plume motions in the range of Rayleigh numbers ${10}^6$ to ${10}^9$. The large-scale classical circulation observed in thermal convection breaks down into many roll configurations based on the constriction gap ($\mathcal{S}$) between the partitions and the conduction walls. Due to their arrangement, we observed increased plume ejection, impact, and shear near the conduction walls when the partitions disturb the thermal boundary layers. The plume ejection and impact on either end of the constriction gap sets a pressure-driven forced convection on the conduction wall, thus increasing overall heat transport by at least an order of magnitude. We found the maximum heat transport when $0.2{\delta }_{\text{RB}}<\mathcal{S}<0.4{\delta }_{\text{RB}}$, where ${\delta }_{\text{RB}}$ is the time-averaged thermal boundary layer thickness in classical thermal convection. Using both the numerical simulations and a simple control volume-based analysis, we have estimated that the heat transport increases as ${\mathcal{S}}^3$ for small constriction gaps and as an inverse power of $\mathcal{S}$ for the large gap limit. With the help of energy dissipation, we have concluded that increasing plume intensity near the conduction walls leads to the observed high heat transport.

Figure 1

Schematic of a thermal convection in a square enclosure: (a) RB-convection without partitions, and (b) with four thin partitions. Enclosure is $H\times H$ in size and heated to $T_h$ at bottom and cooled to $T_c$ at top, and sidewalls thermally insulated. Gravity acts in negative $y$ direction. All four adiabatic partitions have a thickness $a$ and separated by a gap $2b$ from each other. Constriction gap between a partition and a conduction wall is $\mathcal{S}$. Instantaneous temperature (red for hot and blue for cold) and velocity vectors superimposed in background. (c) Typical numerical mesh with a size $394384$ used for $\mathcal{S}=0.07H$. Closeup views of meshes in boundary layers in red boxes.

Figure 2

Instantaneous temperature fields superimposed with velocity vectors: (a) classical square RB, and (b to p) for partitioned convection. Flow transition with a change in constriction gap, $\mathcal{S}=0.40H$ to $0.001H$. Color bar for dimensionless temperature $\theta =0.3$ (blue) and 0.7 (red). Partitions as black color strips. Simulations for $\text{Ra}={10}^8$ and $a=0.04H$.

Figure 3

Regime map based on the Fourier dominant mode on $\text{Ra}-\mathcal{S}$ plane for $a=0.04H$. Symbols are diamonds (elliptical roll), rightward triangles (square roll), leftward triangles (two roll), squares (three roll), stars (four roll), upward triangles (thermal columns), and circles (multicellular).

Figure 4

Time-averaged dimensionless temperature ($\overline{\theta }$) in vertical ($y$) direction: (a) $\mathcal{S}=0.40H$, (b) $3\times {10}^{-2}H$, (c) $7\times {10}^{-3}H$, and (d) ${10}^{-3}H$. The average profiles taken at five horizontal locations, $x=0.1H$ (red), $0.3H$ (blue), $0.5H$ (green), $0.7H$ (magenta), and $0.9H$ (black). Continuous lines for partition thickness $a=0.04H$ and dash lines for $a=0.08H$. Insets for $\overline{\theta }$ contours. Color ranges linearly from $\overline{\theta }=0.3$ (blue) to 0.7 (red).

Figure 5

Time-averaged (a) horizontal velocity $\overline{u}\left(x\right)$, and (b) velocity gradient $(\partial \overline{u}/\partial x)$, near hot conduction wall taken at midheight of constriction gap. Continuous lines for $a=0.04H$: RB (red), $\mathcal{S}={10}^{-1}H$ (blue), ${10}^{-2}H$ (green), and $7\times {10}^{-3}H$ (magenta). At a fixed constriction gap $\mathcal{S}=7\times {10}^{-3}H$: dash-dot lines for $a=0$, and dash lines for $a=0.08H$. Notation: $E$ for ejection, $I$ for impact, and light shaded rectangles for partitions. Simulations for $\text{Ra}={10}^8$.

Figure 6

Isocontours of time-averaged rms fields: (a)–(d) for horizontal velocity $u_{\text{rms}}$, (e)–(h) for vertical velocity $v_{\text{rms}}$, and (i)–(l) for temperature ${\theta }_{\text{rms}}$. Color bar differs in each row. Simulations for $\text{Ra}={10}^8$ and $a=0.04H$.

Figure 7

(a) $v_{\text{rms}}$ taken at midheight of the enclosure ($y=H/2$) for $\text{Ra}={10}^8$ and partition thickness $0.04H$. Lines are red (RB), blue ($\mathcal{S}={10}^{-1}H$), green ($\mathcal{S}=5\times {10}^{-2}H$), and magenta ($\mathcal{S}=7\times {10}^{-3}H$). $E$ for ejection, $I$ for impact, and light shaded rectangles for shear zones. (b) Enclosure area-averaged ${\text{Re}}_{\text{rms}}$ vs Ra. Symbols are $\times$ (RB), circle ($\mathcal{S}=0.30H$), upward triangle ($\mathcal{S}={10}^{-1}H$), asterisk ($\mathcal{S}=7\times {10}^{-2}H$), diamond ($\mathcal{S}=3\times {10}^{-2}H$), rightward triangle ($\mathcal{S}=9\times {10}^{-3}H$), and square ($\mathcal{S}=7\times {10}^{-3}H$). Dash lines are best fits in the form ${\text{Re}}_{\text{rms}}\propto {\text{Ra}}^{{\gamma }_{\text{Re}}}$.

Figure 8

(a) Time-averaged Nusselt number Nu vs Ra for $a=0.04H$. Symbols are $\times$ (RB), $+$ (Bao et al. [44] for 4 partitions), upward triangle ($\mathcal{S}={10}^{-1}H$), diamond ($\mathcal{S}=3\times {10}^{-2}H$), rightward triangle ($\mathcal{S}=9\times {10}^{-3}H$), and square ($\mathcal{S}=7\times {10}^{-3}H$). Dash lines are best fits in the form $\text{Nu}\propto {\text{Ra}}^{{\gamma }_{\text{Nu}}}$, computed for $\text{Ra}>{10}^7$. Additional symbols in parrot green for partition thickness $0.08H$. (b) Scaling exponents ${\gamma }_{\text{Nu}}$ and ${\gamma }_{\text{Re}}$ as a function of constriction gap $\mathcal{S}$.

Figure 9

(a) Nu vs $\mathcal{S}$ for $a=0.04H$. Shaded regions represent different flow states. Peak value Nusselt number ${\text{Nu}}_{\text{opt}}$ for a constriction gap ${\mathcal{S}}_{\text{opt}}$. (b) Normalized heat transport $\text{Nu}/{\text{Nu}}_{\text{opt}}$ vs $\mathcal{S}/{\delta }_{\text{RB}}$. Symbols are squares ($\text{Ra}=5\times {10}^7$), upward traingles ($\text{Ra}={10}^8$), diamonds ($\text{Ra}=5\times {10}^8$), and rightward traingles ($\text{Ra}={10}^9$). Dash lines for power-law fits on either side of ${\mathcal{S}}_{\text{opt}}$. Continuous lines to guide eye.

Figure 10

(a) Local Nusselt number $j\left(x\right)$ along conduction wall ($x$) for $\text{Ra}={10}^8$. Lines for RB (red), $\mathcal{S}={10}^{-1}H$ (blue), ${10}^{-2}H$ (green), and $7\times {10}^{-3}H$ (magenta). Dash magenta line for partition thickness $a=0.08H$, and dash-dot line for infinitesimal partition thickness $a=0$ at $\mathcal{S}=0.007H$. $E$ for ejection, $I$ for impact, and light shaded rectangles for shear zones. (b) Zone-based average local Nusselt number vs $\mathcal{S}$. Symbols are square (ejection), circle (impact) and diamond (shear). Filled symbols for $\text{Ra}={10}^8$ and open symbols for $\text{Ra}={10}^9$. Best fits in the form $j_{\text{zone}}\propto {\mathcal{S}}^{\zeta }$, where $\zeta =-0.21$ for ejection, $-0.17$ for impact, and $-0.3$ for shear zone. Inset for zone identification based on time-averaged temperature field. Error bars are of symbol size.

Figure 11

(a) Instantaneous pressure field at $\mathcal{S}={10}^{-2}H$ and $\text{Ra}={10}^8$. Zoom-up view for constriction zone. Color varies linearly from light brown $[p=-0.05\rho |\mathbf{g}\left|\beta (T_h-T_c)H\right]$ to light green $[p=0.05\rho |\mathbf{g}\left|\beta (T_h-T_c)H\right]$. Pressure field is negative in ejection zone and positive in impact zones. (b) Zone-averaged pressure gradient vs $\mathcal{S}$ in ejection (square), impact (circle), and shear (diamond) zones. For $a=0.04H$: filled symbols for $\text{Ra}={10}^8$, and open symbols for $\text{Ra}={10}^9$. Stars for $a=0$, and crosses for $a=0.08H$ both at $\text{Ra}={10}^8$.

Figure 12

Forces (integrated) as a function of $\mathcal{S}$: (a) in an ejection zone near hot plate and (b) in a constriction zone. Symbols are square (buoyancy), circle (pressure), diamond (viscous), and triangle (inertial). Filled symbols for partition thickness $a=0.04H$. In the thermal column, the force balance between pressure and buoyancy takes place. Stars for $a=0$ and crosses for $a=0.08H$. Simulations for $\text{Ra}={10}^8$.

Figure 13

Schematic of a steady two-dimensional partition convection for which a simple control-volume-based model is developed. Hot plate temperature $T_h$ and cold plate temperature $T_c$. Partitions (in gray color) have thickness $a$, and separated by a distance $2b$. Chosen control-volume is bounded in short dash lines and consists of a hot thermal column, two half cold columns, and four constriction gaps. Cold fluid (light blue) enters the constriction gap at a temperature $T_i$ with a velocity ${\mathcal{U}}_{\mathcal{S}}$, heated (light orange) to a temperature $T_e$, and rises through the core in between the partitions with a velocity ${\mathcal{U}}_{\mathcal{S}}$. Partition walls experience a shear stress ${\tau }_{\mathcal{S}}$ in the constriction, and ${\tau }_{\mathcal{C}}$ in the core. For one-dimensional model $\xi =0$ (midpoint in between partitions and on control-volume boundary) serves a reference point.

Figure 14

Shear Nusselt number (${\text{Nu}}_{\mathcal{S}}$) vs Constriction gap based on Eq. (13). Solutions are proportional to ${\mathcal{S}}^3$ for ${\text{Nu}}_{\mathcal{S}}<{\text{Nu}}_{\text{opt}}$, and to $1/{\mathcal{S}}^2$ for ${\text{Nu}}_{\mathcal{S}}>{\text{Nu}}_{\text{opt}}$. Symbols for $\sigma =0.5$ (circle), 1.0 (square), 2.0 (diamond), 2.43 (up triangle), and 3 (down triangle). Continuous line for ${\mathcal{S}}^3$ and dashline for $1/{\mathcal{S}}^2$. Inset for ${\text{Nu}}_{\text{opt}}$(=maximum value of ${\text{Nu}}_{\mathcal{S}}$) vs $\sigma$ in stars: green ($a={10}^{-3}H$), black ($a=0.04H$), and pink ($a=0.08H$). Parameters are $\text{Ra}={10}^8$ and $b=0.084H$.

Figure 15

Instantaneous energy dissipation for different $\mathcal{S}$: (a)–(e) for thermal energy dissipation (${\varepsilon }_T$), and (f)–(j) for kinetic energy dissipation (${\varepsilon }_u$). Note, color bars in 10-base logarithm. Dark red zones indicate high dissipation, and light white zones indicate low dissipation. Simulations for $\text{Ra}={10}^8$ and $a=0.04H$.

Figure 16

Time and area-averaged energy dissipation vs Ra: (a) $\langle {\overline{\varepsilon }}_u\rangle$ and (b) $\langle {\overline{\varepsilon }}_T\rangle$. Symbols are $\times$ (RB), upward triangle ($\mathcal{S}=0.10H$), diamond ($\mathcal{S}=0.07H$), rightward triangle ($\mathcal{S}=0.007H$), and $+$ for Ref. [62]. Best-fits in dash lines in the form ${\text{Ra}}^{{\gamma }_{\varepsilon }}$ for dissipation. (c) ${\gamma }_{\varepsilon }$ vs $\mathcal{S}$. Symbols are square (for kinetic energy dissipation) and triangles (thermal energy dissipation). Data of Ref. [62] in circles for comparison.

Figure 17

Time and area-averaged dissipation vs $\mathcal{S}$: (a) $\langle {\overline{\varepsilon }}_u\rangle$ and (b) $\langle {\overline{\varepsilon }}_T\rangle$. Symbols are upward triangles ($\text{Ra}={10}^8$) and rightward triangles ($\text{Ra}={10}^9$). Best-fits show ${\mathcal{S}}^{-1/3}$ for $\mathcal{S}\gg {\mathcal{S}}_{\text{opt}}$. Data of Ref. [62] in circles for comparison. Stars for $a=0$ and crosses for $a=0.08H$ both at $\text{Ra}={10}^8$.

Figure 18

Ratio of time and area-averaged energy dissipation in core-bulk zone to total dissipation as a function of $\mathcal{S}$ for $a=0.04H$. Symbols are square (for kinetic energy dissipation), diamonds (for thermal energy dissipation). Filled symbols for $\text{Ra}={10}^8$ and unfilled for $\text{Ra}={10}^9$. For comparison, Zhang et al. [62] 2D square RB dissipation data for $\text{Ra}={10}^8$: circle (kinetic) and triangle (thermal). Cross symbols for partition thickness $a=0.08H$ and $\text{Ra}={10}^8$.

Figure 19

Grid independence test for mesh 2 (medium mesh with cells $N_2=1140624$) and mesh 3 (fine mesh with cells $N_3=2566404$) meshes used for $\mathcal{S}=0.07H$ at $\text{Ra}={10}^9$ and $\text{Pr}=1$. Closeup views for the computational meshes near the bottom wall and the sidewall. Panel (c) for instantaneous Nusselt number $\text{Nu}\left(t\right)$ for two different meshes. Red and green dashed lines for time-averaged Nu of medium and fine meshes, respectively.

Figure 20

Dominant Fourier mode amplitudes as function of time for constriction gaps $\mathcal{S}=$: (a) 0.30H, (b) $0.10H$, (c) $0.03H$, and (d) $0.01H$. In panels (e)–(f), time-averaged flow features with black color streamlines, and dominant modes. Color contours for time-averaged dimensionless temperature which linearly varies from 0.3 (blue) to 0.7 (red). Simulations for $\text{Ra}={10}^8$.

Figure 21

Time-averaged dimensionless temperature ($\overline{\theta }$) in vertical ($y$) direction for partition thickness $a=0$ (dash lines), $0.04H$ (continuous lines), and $0.08H$(dash-dot lines). Temperature profiles taken in cold (at $x=0.5H$) and hot (at $x=0.7H$) thermal columns. Time-averaged temperature contours on right side of main panel. Simulation for $\text{Ra}={10}^8$ and constriction gap $\mathcal{S}=7\times {10}^{-3}H$.

Figure 22

Isocontours of $u_{\text{rms}}, v_{\text{rms}}$, and ${\theta }_{\text{rms}}$ for partition thickness, $a=0, 0.04H$ and $0.08H$. Simulations for $\text{Ra}={10}^8$, and constriction gap $\mathcal{S}=7\times {10}^{-3}H$. Note, color bar range differs in each row.

Figure 23

(a) ${\text{Re}}_{\text{rms}}$ vs Ra and (b) Nu vs Ra. Partition thickness $a=0$ (circle), $0.04H$ (square), and $0.08H$ (triangle). Simulations for constriction gap $\mathcal{S}=7\times {10}^{-3}H$.

Figure 24

Normalized local Nusselt number along the hot conduction wall. Partition thickness $a=0$ (dash-dot line), $0.04H$ (continuous line), and $0.08H$ (dash line). Simulations for $\text{Ra}={10}^8$, and $\mathcal{S}=0.007H$.

Figure 25

Instantaneous thermal energy dissipation rate (${\varepsilon }_T$) in top row, and kinetic energy dissipation rate (${\varepsilon }_u$) in bottom row. Partition thickness $a=0, 0.04H$, and $0.08H$. Color bar for dissipation in logarithmic base 10, and it differs for the thermal energy and kinetic energy dissipation. Dark zones indicate high dissipation, and lighter zones indicate low dissipation. Simulations for $\text{Ra}={10}^8$, and constriction gap $\mathcal{S}=0.007H$.