Effect of velocity boundary conditions on the heat transfer and flow topology in two-dimensional Rayleigh-Bénard convection

The effect of various velocity boundary condition is studied in two-dimensional Rayleigh-Bénard convection. Combinations of no-slip, stress-free, and periodic boundary conditions are used on both the sidewalls and the horizontal plates. For the studied Rayleigh numbers $\text{Ra}$ between ${10}^8$ and ${10}^{11}$ the heat transport is lower for $\Gamma =0.33$ than for $\Gamma =1$ in case of no-slip sidewalls. This is, surprisingly, the opposite for stress-free sidewalls, where the heat transport increases for a lower aspect ratio. In wider cells the aspect-ratio dependence is observed to disappear for $\text{Ra}\ge {10}^{10}$. Two distinct flow types with very different dynamics can be seen, mostly dependent on the plate velocity boundary condition, namely roll-like flow and zonal flow, which have a substantial effect on the dynamics and heat transport in the system. The predominantly horizontal zonal flow suppresses heat flux and is observed for stress-free and asymmetric plates. Low aspect-ratio periodic sidewall simulations with a no-slip boundary condition on the plates also exhibit zonal flow. In all the other cases, the flow is roll like. In two-dimensional Rayleigh-Bénard convection, the velocity boundary conditions thus have large implications on both roll-like and zonal flow that have to be taken into consideration before the boundary conditions are imposed.

Figure 1

Temperature field snapshots for different simulations. Red and blue indicate hot and cold fluid, respectively. The color varies between $\theta =0.2$ and $\theta =0.8$. $\text{Ra}={10}^{11}$ and $\Gamma =1/2$ for lateral periodicity, showing zonal flow (left panel); $\text{Ra}={10}^{10}$ and $\Gamma =0.33$ for no-slip sidewalls, showing roll structures (center panel); and $\text{Ra}={10}^{10}$ and $\Gamma =0.33$ for stress-free sidewalls (right panel). The plates are no-slip in all cases. Movies can be found in the Supplemental Material [29].

Figure 2

Compensated $\text{Nu}/{\text{Ra}}^{2/7}$ as a function of Ra for stress-free and no-slip sidewalls and no-slip plates. The data are for aspect ratios $\Gamma =1$ and $\Gamma =0.33$. As expected, the stress-free Nu is consistently higher than for no-slip for the same value of $\Gamma$. For stress-free boundary conditions, the heat transport is higher for $\Gamma =0.33$ than for $\Gamma =1$ for all of the evaluated Ra, while for no-slip this is only observed at high Ra.

Figure 3

(a) Nu as a function of $\Gamma$ for stress-free sidewalls and no-slip plates. The data are for $\text{Ra}={10}^8$. At $\Gamma \approx 1$ there is a transition region between a roll state with one roll and a roll state with two horizontally aligned rolls, which is reflected by hysteretic jumps in Nu. (b) Compensated $(\text{Nu}-1)/{\text{Ra}}^{2/7}$ as a function of Ra for stress-free and periodic sidewalls and no-slip plates. Data are depicted for $\Gamma =1.5$ and $\Gamma =2$. The effect of the aspect ratio on Nu disappears for high Ra while the effect of the sidewall boundary condition remains.

Figure 4

(a) Nu vs $\Gamma$ for periodic sidewalls and no-slip plates for $\text{Ra}\in \{{10}^8,{10}^9,{10}^{10},{10}^{11}\}$; see the legend in (b). Jumps in Nu can be seen around $\Gamma \approx 1.25,\Gamma \approx 1.25,\Gamma \approx 1.10$, and $\Gamma \approx 1.05$ for $\text{Ra}={10}^8,\text{Ra}={10}^9,\text{Ra}={10}^{10}$, and $\text{Ra}={10}^{11}$, respectively. The Nusselt number statistics at lower $\Gamma$ than the jump are more difficult to converge than the other data points due to the bursting nature of $\text{Nu}\left(t\right)$ and have a larger error. They are included to indicate the change of flow state. In (b) Nu is compensated with $\text{Nu}(\Gamma =3)$.

Figure 5

Temperature field snapshots for $\text{Ra}=2.15\times {10}^8$ and $\Gamma =1.5$ with no-slip plates and stress-free sidewall boundary conditions of two different statistically steady states. The color-map ranges from $0\le \theta \le 1$. The area averaged mean temperature ${\langle \theta \rangle }_{A,t}$ is $0.44$ (left panel) and $0.56$ (right panel).

Figure 6

Temperature field snapshots with velocity vectors superimposed for stress-free plates and periodic sidewalls for $\text{Pr}=1$ and at $\text{Ra}={10}^6$ (left panel) and $\text{Ra}={10}^8$ (right panel) in a $\Gamma =2$ cell. The temperature ranges from $0\le \theta \le 1$. The size of the arrows is proportional to the absolute velocity. The flow is roll-like in (a) and zonal in (b). Corresponding movies can be found in the Supplemental Material [29].

Figure 7

(a) Nu as a function of dimensionless time in free-fall time units at $\Gamma =1/2$ for $\text{Ra}={10}^8$ and $\text{Ra}={10}^{11}$, NS plates, and PD sidewalls. At $\text{Ra}={10}^8$ we observe bursting behavior while at $\text{Ra}={10}^{11}$ we do not: The zonal flow is bursting for lower Ra and sustained for higher Ra. Note the large difference between the time scale of the bursts at $\text{Ra}={10}^8$ and the fluctuations at $\text{Ra}={10}^{11}$. (b) Probability density function of the time interval between the bursts $\Delta t$ for $\text{Ra}={10}^7$ and $\text{Ra}={10}^8$.

Figure 8

Mean horizontal velocity ${\langle u_x\rangle }_{x,t}$, normalized by its maximum, as a function of vertical position $z/L$ for different Ra (see legend) in (a) $\Gamma =1$ cells with lateral periodicity and no-slip plates, (b) $\Gamma =0.5$ cells with lateral periodicity and no-slip plates, and (c) $\Gamma =2$ cells with lateral periodicity and asymmetric plates.

Figure 9

(a) Temperature field snapshot for asymmetric plates BCs and periodic sidewalls at $\text{Ra}=4.64\times {10}^8$. The top boundary is stress-free and the bottom boundary is no-slip. Movies of roll-like and zonal flow with asymmetric BCs can be found in the Supplemental Material [29]. (b) Nu as a function of Ra for asymmetric plate BC's and periodic sidewalls. Simulations where the initial conditions is taken from the lower Ra neighboring data point are indicated with rightward-pointing triangles and where the initial condition is taken from higher Ra are indicated with leftward-pointing triangles.

Figure 10

(a) The area- and time-averaged temperature ${\langle \theta \rangle }_{A,t}$ as a function of Ra for asymmetric plate boundary conditions. Only the runs where the initial condition is taken from a lower Ra are included. (b) The horizontal- and time-averaged temperature ${\langle \theta \rangle }_{x,t}$ as a function of the vertical coordinate $z/L$ for $\text{Ra}={10}^7,\text{Ra}={10}^8$, and $\text{Ra}={10}^9$. The $\text{Ra}={10}^9$ data in this plot are taken from a simulation with zonal flow while the others are in a roll-like state. The dashed black line indicates ${\langle \theta \rangle }_{x,t}=0.5$. For both panels, the asymmetry is introduced by making bottom plate (i.e., $z/L=0$) no-slip, while the top plate (i.e., $z/L=1$) is stress free.