Heat transfer mechanisms in bubbly Rayleigh-Bénard convection

The heat transfer mechanism in Rayleigh-Bénard convection in a liquid with a mean temperature close to its boiling point is studied through numerical simulations with pointlike vapor bubbles, which are allowed to grow or shrink through evaporation and condensation and which act back on the flow both thermally and mechanically. It is shown that the effect of the bubbles is strongly dependent on the ratio of the sensible heat to the latent heat as embodied in the Jakob number Ja. For very small Ja the bubbles stabilize the flow by absorbing heat in the warmer regions and releasing it in the colder regions. With an increase in Ja, the added buoyancy due to the bubble growth destabilizes the flow with respect to single-phase convection and considerably increases the Nusselt number.

Figure 1

(Color online) Vertical velocity in the plane of symmetry of the full cylinder in the absence of bubbles; $\text{Ra}=2\times {10}^5$, $\text{Pr}=1.75$, $\text{Nu}=4.75$. As throughout the paper, the velocity is made dimensionless by using the free-fall velocity ${\left[\beta gH\left(T_h-T_c\right)\right]}^{1/2}$.

Figure 2

(Color online) Interpolation (22) for the dependence of the single-bubble Nusselt number on the Péclet number.

Figure 3

(Color online) The line shows the numerical results for the left-hand side of Eq. (29) and the points those for the right-hand side. Equality of these two quantities is a stringent of the accuracy of the computation. $N_b=5000$.

Figure 4

(Color online) Averaged Nusselt number $\overline{\text{Nu}}$ vs Jakob number for three different numbers of bubbles.

Figure 5

(Color online) Ratio of the bubble source term (35) to the average Nusselt number (30). At small Ja, the additional bubble-induced buoyant forcing is the dominant effect, while at large Ja the bubbles act as direct carriers of heat from the bottom to the top.

Figure 6

(Color online) Comparison between the Nusselt numbers computed at the top and at the bottom boundaries for 5000 bubbles; the middle line is the average Nusselt number $\overline{\text{Nu}}=\frac{1}{2}\left({\text{Nu}}_h+{\text{Nu}}_c\right)$. Here, we took $N_b=5000$.

Figure 7

(Color online) Average void fraction in the up-flow and in the down-flow regions for $N_b=5000$ bubbles.

Figure 8

(Color online) Averaged radius of the bubble computed in the up-flow and the down-flow regions for $N_b=5000$ bubbles; $R_{b0}$ is the initial radius, $25\mu \text{m}$.

Figure 9

(Color online) Averaged bubble numbers in the up-flow and the down-flow regions for $N_b=5000$ bubbles.

Figure 10

(Color online) Vertical and horizontal cross sections (taken at $0.05H$, $0.5H$, and $0.95H$, respectively) of the vertical liquid velocity distribution in the cylinder for $\text{Ja}=0$ and $N_b=5000$ bubbles. The blue (dark) structure near the axis is the descending region of the toroidal vortex which prevails for small Jakob numbers. The absolute values of the velocities are two orders of magnitude smaller as compared to the two subsequent figures as convection is suppressed at $\text{Ja}=0$.

Figure 11

(Color online) Vertical and horizontal cross sections (taken at $0.05H$, $0.5H$, and $0.95H$, respectively) of the vertical liquid velocity distribution in the cylinder for $\text{Ja}=0.0935$ and 5000 bubbles. The blue (dark) structure near the axis is the descending region of the toroidal vortex which prevails for small Jakob numbers.

Figure 12

(Color online) Vertical and horizontal cross sections (taken at $0.05H$, $0.5H$, and $0.95H$, respectively) of the vertical liquid velocity distribution in the cylinder for $\text{Ja}=0.748$ and 5000 bubbles.

Figure 13

(Color online) Fourier modes of the kinetic energy in the angular direction for $N_b=5000$ bubbles. Mode 0 corresponds to a toroidal vortex and mode 1 corresponds to a circulatory motion in the cell with approximately horizontal axis.

Figure 14

(Color online) Fourier modes of the kinetic energy in the angular direction for $N_b=10000$ bubbles. Mode 0 corresponds to a toroidal vortex and mode 1 corresponds to a circulatory motion in the cell with approximately horizontal axis.