Thermal convection in octagonal-shaped enclosures

Flow-reversal phenomena in a classical two-dimensional (2D) Rayleigh-Bénard convection, in a square enclosure, are usually explained through growth and merging of diagonally opposite counterrotating corner rolls. To probe further the corner-roll growth dynamics, we have altered the square enclosure edges by additional slanted conduction walls, so that the enclosure resembles an octagonal shape. We have performed a series of 2D numerical simulations by varying the slanted wall inclination angle ($\alpha$) from $0^{\circ }$ to ${45}^{\circ }$, to construct a detailed flow map in thermal convection in a range $5\times {10}^5\le \text{Ra}\le {10}^8$ and $0.8\le \text{Pr}\le 2.0$, where $\text{Ra}$ is the Rayleigh number and $\text{Pr}$ is the Prandtl number. Depending on $\text{Ra}$, $\text{Pr}$, and $\alpha$, flow features in the octagonal enclosure can exist in the form of a uniform circulation, a two-roll, a mixed, a periodic, a quasiperiodic, or multiple flow states superimposed on each other. The flow reversals in the octagonal enclosure take place in several ways, for example, by the ejection of mushroom-shaped plumes alternatively from the opposite slanted walls at low $\text{Ra}$ ($\approx {10}^5$) and high $\text{Pr}$ ($\approx 2$), by the corner-roll growth at high $\text{Ra}$ ($\approx {10}^8$) and low $\text{Pr}$ ($\approx 1.2$), and by the dipole at high $\text{Ra}$ ($\approx {10}^8$) and high $\text{Pr}$ ($\approx 2$). Strikingly, the dimensionless flow-reversal frequency scales linearly with an increase in $\alpha$, and the slope varies from 1.04 at low $\text{Ra}=5\times {10}^5$ to 0.328 at high $\text{Ra}={10}^8$. We have shown the flow reversals are a consequence of competition between the dipole (a two-roll state where a cold roll sits above a hot roll) and the quadrupole (the four corner rolls) modes with the monopole mode. A uniform circulation with flow-reversal results if the quadrupole mode wins, and a two-roll state with a reversal results, if the dipole wins. At high $\text{Ra}(\ge {10}^8)$, the dipole strengthens, and the core bulk region shows hydrodynamic instabilities in the form of turbulent-like engulfments. We have uncovered the mechanism responsible for the observed engulfments due to the increase in turbulence production in the core bulk region by the buoyancy. As a result, we have observed that total heat transport also increases up to $14\%$ when $\alpha$ is varied from $0^{\circ }$ to ${45}^{\circ }$.

Figure 1

Schematics of a square (a) and an octagonal enclosure (b). The square is $D$ in width and $H$ in height, with $D=H$. We have inscribed an octagon in a square as a geometry and studied thermal convection. The slanted (or inclined) walls are of length $l_1$, which make an angle $\alpha$ with the horizontal walls (of length $l_2$). Note, hot walls (in red) at bottom, cold walls (blue) at top, and adiabatic sidewalls (black). The walls of the octagonal enclosure satisfy $2l_{1}cos\alpha +l_2=D$. The geometry origin is taken at the bottom left corner, and directions are $x$ in the horizontal direction and $y$ in the vertical direction. The gravity acts along the negative $y$ direction. Three square subdomains (shown by dash lines) are chosen to extract the Fourier modes. The subdomain areas in dimensionless square units are 0.4624 (for green box), 0.36 (for blue box), and 0.2704 (for cyan box). The blue box region with $0.2D\le x\le 0.8D$ and $0.2H\le y\le 0.8H$ is considered as a bulk region. Small black square symbols represent numerical probe locations, and their $(x,y)$ coordinates are the probe $P1$ at $(0.1465,0.0852)$, the probe $P2$ at $(0.5,0.02)$, and the probe $P3$ at $(0.8535,0.0852)$. (c) A typical computational mesh used for a regular octagonal enclosure (with $\alpha ={45}^{\circ }$). Total number of computational meshes used is 102 400. Close-up views of computational mesh near the bottom wall and in the core bulk region are also shown. Slant represents additional conduction walls used.

Figure 2

Left panels for instantaneous temperature and velocity fields. Panels (a) to (c) for uniform circulation (UC) at $\text{Pr}=0.8$, panels (d) to (f) for a stable two-roll structure (TRS) at $\text{Pr}=1$, and panels (g) to (i) for uniform circulation with reversals (UCR) at $\text{Pr}=1.8$. Time increases from left to right in each row; see animation 1 in the Supplemental Material [45]. Color bar for dimensionless temperature, $(T-T_c)/\mathrm{\Delta }T$, which ranges from 0.3 (blue) to 0.7 (red). Black small arrows for the velocity field. White dashed lines with arrows indicate large-scale (dominant) rolls with their direction of rotation and approximate size. In right panel (j), normalized angular momentum vs time: UC state at $\text{Pr}=0.8$ (red), TRS at $\text{Pr}=1$ (blue), and UCR at $\text{Pr}=1.8$ (black). Simulations for a square enclosure at $\text{Ra}={10}^7$.

Figure 3

(a) Instantaneous thermal and velocity fields in a square enclosure at $\text{Ra}=5\times {10}^5$ and $\text{Pr}=2$: (1) for two-roll state (TRS), (2 and 3) for a mixed state (MS), and (4) for a uniform circulation state (UC). White dashed lines with arrows indicate large- and small-scale rolls with their direction of rotation and approximate size. (b) Angular momentum vs time. Vertical dash lines in angular momentum plot indicate the time at which panel (a) snapshots are taken. (c) Flow map pattern on $\text{Ra}\text{−}\text{Pr}$ plane. Symbols are circles for UC, pentagrams for TRS, upward triangles for UCR, and leftward triangles for MSR. Black crosses correspond to flow reversals mentioned in Sugiyama et al. [15]. To guide the eye, we have shown the hand-drawn demarcation regions with dashed lines.

Figure 4

(a) Instantaneous temperature and velocity fields in octagonal enclosures at $\text{Ra}=5\times {10}^5$ and $\text{Pr}=2$. A quasiperiodic reversal state (QPR) for $\alpha ={24}^{\circ }$, and a periodic reversal state (PR) for $\alpha ={45}^{\circ }$. In the same panel, snapshots before and after the flow reversal are also shown. Magenta arrows indicate mushroom-shaped plume ejections. In panels (b) and (c), angular momentum vs time, spectral energy ($E)$ vs frequency ($f$), and phase plot of $L$ vs $dL/dt$ are shown. Note, variables are shown as dimensionless. (b) Periodic reversal (PR) with $\alpha ={45}^{\circ }$, with a blue line for $\text{Ra}=5\times {10}^5$, red line for $\text{Ra}=6\times {10}^5$ and $\text{Pr}=2$. (c) The same but a blue line for $\text{Ra}=5.55\times {10}^5$ and $\text{Pr}=1.8$, red line for $\text{Ra}={10}^6$ and $\text{Pr}=2$, and gray line for $\text{Ra}=7.143\times {10}^6$ and $\text{Pr}=1.4$ for QPR with $\alpha ={45}^{\circ }$.

Figure 5

Comparison of flow reversals in a square ($\alpha =0^{\circ }$) and octagonal ($\alpha ={45}^{\circ }$) enclosures at $\text{Ra}={10}^8$ and $\text{Pr}=1.8$. In top panels: (a) angular momentum vs time, (b) spectral energy ($E)$ vs frequency ($f$), and (c) phase plot of $L$ vs $dL/dt$ are shown. Red lines for the square enclosure and blue lines for octagon ($\alpha ={45}^{\circ }$). Note, variables are shown in as dimensionless. Time spent in a given state (in which $L$ is nearly constant) is measured between a circle followed by a square symbol. Inset of panel (b) indicates $E\sim 1/f$ at high frequencies. Bottom panels for snapshots of thermal and velocity fields (for before and after flow reversals).

Figure 6

Flow pattern map on $\text{Ra}\text{−}\text{Pr}$ plane for octagonl enclosures: (a) $\alpha ={12}^{\circ }$, (b) $\alpha ={24}^{\circ }$, and (c) $\alpha ={45}^{\circ }$. Seven distinct flow states such as uniform circulation (UC), uniform circulation with reversal (UCR), stable two-roll state (TRS), mixed state reversal (MSR), periodic reversal (PR), quasiperiodic reversal (QPR), and a combination of UCR+TRSR states are observed. Symbols are for UC (in circles and shaded red), UCR (upward triangles shaded in cyan), TRS (in pentagrams shaded green), MSR (in leftward triangles shaded blue), QPR (in squares shaded magenta), UCR+TRSR (in upward triangles shaded dark cyan) states, and PR (in diamonds shaded yellow). To guide the eye, hand-drawn curves are shown to demarcate the flow states.

Figure 7

Fourier mode amplitude vs time in the PR state for $\alpha ={45}^{\circ }$, $\text{Ra}=5\times {10}^5$, and $\text{Pr}=2$. Zoomed view of highlighted rectangular box in panel (a) is shown in panel (b). Instantaneous temperature and velocity fields at different times (i), (ii), (iii), and (iv) are shown in insets.

Figure 8

The Fourier mode amplitude vs time: (a) $\alpha =0^{\circ }$ in the UCR state and (b) $\alpha ={45}^{\circ }$ in combined UCR+TRSR state. Dashed rectangular boxes in panels highlight flow mode features. Zoomed views of panel (b) are shown in panels (c) and (d), along with corresponding flow fields. Simulations are for $\text{Ra}={10}^8$ and $\text{Pr}=1.8$.

Figure 9

Comparison of the Fourier mode amplitudes: (a) $\mathrm{RMS}\left[{\widehat{v}}_{(m,n)}\right]/\mathrm{RMS}[{\widehat{v}}_{(1,1)}]$ vs $\alpha$, and (b) cross-correlation coefficient (CC) between ${\widehat{v}}_{(m,n)}$ mode and ${\widehat{v}}_{(1,1)}$ mode. Symbols: circles for (2,2) mode, squares for (1,2) mode, triangles for (2,1) mode, diamonds for (1,3) mode, and crosses for (3,1). Simulations are for $\text{Ra}={10}^8$ and $\text{Pr}=1.8$.

Figure 10

Dimensionless flow-reversal frequency ($f{t}_E$) as a function of $\text{Ra}$ on a log-log plot. Open black circles for experiments in quasi-2D square cell with $\text{Pr}=5.4$ [39], which follow $f{t}_E\sim {\text{Ra}}^{-1.08}$ in the range $1.18\times {10}^8\le \text{Ra}\le 2\times {10}^8$, and follow $f{t}_E\sim {\text{Ra}}^{-2.83}$ in the range $2\times {10}^8<\text{Ra}\le 1.12\times {10}^9$. Filled black circles for experiments in quasi-2D cornerless cell with $\text{Pr}=5.4$ [40], which follow $f{t}_E\sim {\text{Ra}}^{-1.53}$ in the range $9.58\times {10}^7\le \text{Ra}\le 2\times {10}^8$. Yellow diamonds for numerical data of Sugiyama et al. [15] at $\text{Pr}=4.3$. Magenta asterisks for experimental data of Ni et al. [46] at $\text{Pr}=5.7$. Magenta dashed line approximately fits the magenta asterisks and is obtained from stochastic model proposed by Brown and Ahlers [50]. Our numerical data at $\text{Pr}=2$ are shown as upward triangles: Red for $\alpha =0^{\circ }$, blue for $\alpha ={12}^{\circ }$, green for $\alpha ={24}^{\circ }$, and cyan for $\alpha ={45}^{\circ }$. We have shown best power-law fits of our data with dashed lines. Red dashed line for $\alpha =0^{\circ }$ follows $f{t}_E\sim {\text{Ra}}^{-0.358}$, and cyan dashed line for $\alpha ={45}^{\circ }$ follows $f{t}_E\sim {\text{Ra}}^{-0.281}$. In the inset, we have shown $f{t}_E$ vs $\alpha /\pi$, where plus, pentagram, cross, and square symbols are for $\text{Ra}=5\times {10}^5$, $5\times {10}^6$, $5\times {10}^7$, and ${10}^8$, respectively. Inset shows the reversal frequency scales linearly with an increase in $\alpha$. The corresponding linear slopes are shown above the best fits. The eddy turnover time ($t_E$) decreases from $40\tau$ to $15\tau$ as $\text{Ra}$ increases from ${10}^5$ to ${10}^8$, respectively, which is consistent with the results of Ref. [55, 56].

Figure 11

(a) $\overline{\text{Nu}}$ vs $\text{Ra}$: blue circles for square ($\alpha =0^{\circ }$) and red diamonds for octagon enclosure ($\alpha ={45}^{\circ }$) at $\text{Pr}=1$. Green squares for Ref. [44] at $\text{Pr}=1$. Magenta stars for Ref. [57] at $\text{Pr}=5.3$. Dashed lines to guide the eye. (b) ${\overline{\text{Nu}}}_{\alpha }/{\overline{\text{Nu}}}_{sq}$ vs $\alpha /\pi$. Continuous lines are best least-square fits in the form $Y=a{X}^2+bX+c$, where $X=\alpha /\pi$, and $Y={\overline{\text{Nu}}}_{\alpha }/{\overline{\text{Nu}}}_{sq}$. The fit parameters range is $-0.65\le a\le -0.31$, $0.39\le b\le 0.63$, and $c=1$. Dashed lines represent a linear trend along with their corresponding slopes. (c) Effective increase in Nusselt number vs increase in wall conduction area of the octagon. (b) and (c) Simulations of $\text{Ra}={10}^8$ and different $\text{Pr}$. Symbols are red diamonds ($\text{Pr}=1$), magenta upward triangles ($\text{Pr}=1.4$), cyan rightward triangles ($\text{Pr}=1.6$), green leftward triangles ($\text{Pr}=1.8$), and blue circles ($\text{Pr}=2.0$).

Figure 12

(a) Convective heat flux ($j$) vs time for $\alpha =0^o$ (blue) and ${45}^{\circ }$ (red). In inset, zoomed view of $j$ with time during a flow reversal. Vertical dashed lines indicate times at which six snapshots of the velocity and temperature fields are taken. The snapshots are timed at (i) $t=6526\tau$, (ii) $6536\tau$, (iii) $6542\tau$, (iv) $6545\tau$, (v) $6550\tau$, and (vi) $6574\tau$. The flow-reversal process is described as (i) and (ii) for growth of corner rolls; (iii) for corner-roll reconnection; (iv) for flow reconfiguration; (v) for hot plumes falling and cold plumes rising; and (vi) for hot plumes rising and cold plumes falling. (b) PDFs of normalized convective heat flux $(j-\overline{j})/{\sigma }_j$. Blue line is for $\alpha =0^{\circ }$, green for ${12}^{\circ }$, magenta for ${24}^{\circ }$, and red for ${45}^{\circ }$. The normal Gaussian is represented by long dashed lines. The standard deviation (${\sigma }_j$) vs $\alpha$ is shown in inset of panel (b). Simulations for $\text{Ra}={10}^8$ and $\text{Pr}=2$.

Figure 13

$\overline{\text{Nu}}$ (time-averaged Nusselt number) dependence on $\mathrm{\Gamma }$ in 2D thermal convection. For $\text{Ra}={10}^8$, $\text{Pr}=1$ and $\mathrm{\Gamma }=1$, we find $\overline{\text{Nu}}=24.89$ from our simulations (see magenta circle), which agrees with the results of Ref. [44] (open black diamond). We have also collated data from other works to understand $\mathrm{\Gamma }$ dependence and show these data: cyan dashed line from Ref. [62]; cyan circles from Ref. [23] for $\text{Ra}={10}^8$ and $\text{Pr}=0.7$; black stars from Ref. [23] for $\text{Ra}={10}^8$ and $\text{Pr}=4.3$; black crosses from Ref. [24] for $\text{Ra}={10}^8$ and $\text{Pr}=1$; black leftward triangle from Ref. [61] for $\text{Ra}={10}^8$ and $\text{Pr}=4.38$; and black upward triangle from Ref. [54] for $\text{Ra}={10}^8$ and $\text{Pr}=0.71$. Our numerical data for the octagonal enclosures are shown with continuous lines: red for $\alpha ={12}^{\circ }$, green for $\alpha ={24}^{\circ }$, and blue for $\alpha ={45}^{\circ }$. Note, for a given $\alpha$, we have a $\overline{\text{Nu}}$ and a range of aspect ratios with a lower limit and an upper limit (which are connected by a horizontal line).

Figure 14

Long-time-averaged temperature fields for more than $7000\tau$: Top panels (a, b, and c) for $\text{Ra}={10}^7$, and bottom panels (d, e, and f) for $\text{Ra}={10}^8$. Panels (a) and (d) for $\alpha =0^{\circ }$, panels (b) and (e) for $\alpha ={24}^{\circ }$, and panels (c) and (f) for $\alpha ={45}^{\circ }$. Color bar for dimensionless temperature, ($\overline{T}-T_c)/\mathrm{\Delta }T$, in the range 0.3 (blue) to 0.7 (red). Time-averaged temperature profile along the vertical midline of the enclosure: panel (g) for $\text{Ra}={10}^7$ and panel (h) for $\text{Ra}={10}^8$. Thermal gradient at the walls increases with increase in $\alpha$. Simulations for $\text{Pr}=1$.

Figure 15

(a) Wall thermal gradients measured normal to the bottom conduction walls ($\mathbf{\nabla }\overline{T}·\mathbf{n}=\frac{\partial T}{\partial n}$) for $\alpha =0^{\circ }$, ${12}^{\circ }$, and ${45}^{\circ }$. (b) Wall-length-averaged thermal gradient ($\overline{\xi }=-\frac{1}{l_i}\int \mathbf{\nabla }\overline{T}·d{l}_i$) for the left, right, and the middle (horizontal) conduction walls. Here $l_i$ is the length of a particular conduction wall. The plus symbols in panel (b) show $\overline{\text{Nu}}$ as a function of $\alpha$. Simulations for $\text{Ra}={10}^8$ and $\text{Pr}=2$. We have considered time averages from $500\tau$ to $15000\tau$ with a time gap of $50\tau$ to compute wall thermal gradients. Within this time average, we did not observe the expected left-right symmetry in wall-thermal gradient on the conduction wall.

Figure 16

Panels (a) and (b) for zone-area-averaged instantaneous kinetic energy per unit mass ($E_{\mathrm{kin}}$) vs time. We have compared Zone I and Zone II information of the octagonal enclosures with the square enclosures. In panel (c), time-averaged kinetic energy (${\overline{E}}_{\mathrm{kin}}$) as a function of $\alpha$. Note, Zone I for core bulk region and Zone II for near wall shear layers. Symbols are circles (for Zone I) and squares (for Zone II). Note, the kinetic energy is normalized with ${\kappa }^2\text{Ra}\text{Pr}/H^2$ to make it dimensionless. Simulations for $\text{Ra}={10}^8$ and $\text{Pr}=1$.

Figure 17

Time series of velocity and temperature at probe $P1$, probe $P2$, and probe $P3$ locations. Probe locations can be noted from Fig. 1. Top row for horizontal velocity, middle row for vertical velocity, and bottom row for temperature. Dark blue lines for $\alpha =0^{\circ }$ and light gray lines for $\alpha ={24}^{\circ }$. Simulations for $\text{Ra}={10}^8$ and $\text{Pr}=1$.

Figure 18

Zone-area-averaged instantaneous work done by buoyancy ${\mathcal{W}}_B$. Panels are for (a) $\alpha ={12}^{\circ }$ and (b) $\alpha ={45}^{\circ }$ in comparison to a square RB convection ($\alpha =0^{\circ }$). Panel (c) is for time-averaged work done by buoyancy (${\overline{\mathcal{W}}}_B/{\mathcal{W}}_C$). Here ${\mathcal{W}}_C={\rho }_m\beta \mathrm{\Delta }T\left|g\right|U$. Symbols are circles (for Zone I and $\text{Pr}=1$), squares (for Zone II and $\text{Pr}=1$), triangles (for Zone I and $\text{Pr}=2$), and stars (for Zone II and $\text{Pr}=2$). Simulations are for $\text{Ra}={10}^8$.

Figure 19

In each panel: first column for instantaneous thermal field ($T$), second column for thermal anomaly ($\theta$), third column for vorticity ($\mathrm{\Omega }$), and fourth column for local Richardson number (${\text{Ri}}_{\mathrm{RBC}}$). Panels are for $\alpha =$ (a) $0^{\circ }$, (b) ${12}^{\circ }$, (c) ${24}^{\circ }$, and (d) ${45}^{\circ }$. Simulations are for $\text{Ra}={10}^8$ and $\text{Pr}=1$ at time $t=2400\tau$. See animation 5(a–d) in the Supplemental Material [45] for more details.

Figure 20

Area-averaged vorticity transport terms such as the buoyancy ($\left|\mathbf{g}\right|\beta \frac{\partial T}{\partial x}$), the viscous ($\nu {\nabla }^2\mathrm{\Omega }$), and the advection ($\mathbf{u}·\mathbf{\nabla }\mathrm{\Omega }$) terms as functions of time. Panels are for (a) $\alpha =0^{\circ }$, (b) $6^{\circ }$, (c) ${12}^{\circ }$, and (d) ${24}^{\circ }$. Note, the terms are area averaged and normalized by $U^2/H^2$ to make them dimensionless. Simulations are for $\text{Ra}={10}^8$, $\text{Pr}=1$.

Figure 21

Root-mean-square (RMS) values of vorticity terms vs $\alpha$. Symbols shown for buoyancy (circles), viscous (squares), and advection (triangles) terms. The advection terms increase significantly for $\alpha >{12}^{\circ }$. Dashed lines are best linear fits. Simulations for $\text{Ra}={10}^8$ and $\text{Pr}=1$.

Figure 22

Isocontour plots of time-averaged RMS fluctuations: (a–c) for horizontal velocity, (d–f) for vertical velocity, and (g–i) for temperature. Left panels for $\alpha =0^{\circ }$, middle panels for $\alpha ={12}^{\circ }$, and right panels for $\alpha ={24}^{\circ }$. Note, color bars range differently in each panel. In each row, $\alpha =0^{\circ }$, ${12}^{\circ }$, and ${24}^{\circ }$ are shown. Simulations are for $\text{Ra}={10}^8$ and $\text{Pr}=1$.

Figure 23

Instantaneous isocontours of turbulent production terms: left panels for shear production ($P_K$), and right panels for buoyancy production ($B_K$). In each panel, results for $\alpha =0^{\circ }$ and $\alpha ={24}^{\circ }$ are shown. Note, color bar ranges differently in each panel. Simulations are for $\text{Ra}={10}^8$ and $\text{Pr}=1$.

Figure 24

Time- and area-averaged mechanical energy budget terms in the turbulent core bulk region (Zone I) as functions of $\alpha$. Symbols are crosses ($B_K$), circles (${\epsilon }_K$), squares ($P_K$), diamonds ($D_K$), and downward triangles (${\mathrm{\Pi }}_K$). Simulations for $\text{Ra}={10}^8$ and $\text{Pr}=1$.

Figure 25

Fourier mode decomposition is carried out for the vertical velocity in the three subdomains. Lines are green (for the green square domain), blue (for the blue square domain), and cyan (for the cyan square domain). In each panel, a particular Fourier flow mode amplitude as a function of dimensionless time is shown: (1,1) mode (a), (2,2) mode (b), (1,2) mode (c), (2,1) mode (d), (1,3) mode (e), and (3,1) mode (f). The mode amplitudes from the complete octagon domain (without separating it into square domains) are shown as a red line. Simulations for $\text{Ra}={10}^8$, $\text{Pr}=1.8$, and $\alpha ={45}^{\circ }$ enclosure.

Figure 26

Time evolution of the dominant Fourier modes for square enclosure at $\text{Ra}=5\times {10}^5$ and $\text{Pr}=2$. At 1, 2, 3, and 4 we have shown dashed lines for which four snapshots are shown in Fig. 3.

Figure 27

Normalized angular momentum vs time for mixed state with reversal (MSR) in a square enclosure for the parameters $\text{Ra}=5\times {10}^5$ and $\text{Pr}=2$. Zoom of rectangular box shown in Fig. 3.

Figure 28

The ratio of $|{\widehat{v}}_{2,2}|/|{\widehat{v}}_{1,1}|$ for $\text{Ra}={10}^8$ and $\text{Pr}=1.8$ for $\alpha =0^{\circ }$ (green) and $\alpha ={45}^{\circ }$ (red).

Figure 29

Interaction between different Fourier modes during reversal. Upward arrow indicates significant influence of the specific modes.

Figure 30

Schematics of different flow states observed in the given range of $\text{Ra}\text{−}\text{Pr}$. Note, hot (H) and cold (C) rolls and large-scale circulation (LSC). In QPR and PR, mushroom-shaped giant plumes eject from the incline walls.

Figure 31

Normalized mean reversal frequency vs slanted angle for $\text{Ra}=5\times {10}^7$ and $\text{Pr}=2$. For a square enclosure ($\alpha =0^{\circ }$) flow reversals take place by UCR, and for $\alpha ={12}^{\circ }$ to ${45}^{\circ }$ case flow reversals take place by both UCR and TRSR. UCR is shown by a blue square, and TRSR is shown by black triangle symbols.

Figure 32

Steps involved in reversal of two-roll state (i.e., TRSR): Snapshots of the velocity and temperature fields during TRSR at $t/\tau =10,897$ (i), 10 918 (ii), 10 930 (iii), 10 936 (iv), 10 942 (v), and 10 945 (vi). The dimensionless temperature, $(T-T_c)/\mathrm{\Delta }T$, which ranges from 0.3 (blue) to 0.7 (red). Black small arrows for the velocity field. Simulation for $\text{Ra}={10}^8$ and $\text{Pr}=1.8$.