Momentum and heat transport scalings in laminar vertical convection

We derive the dependence of the Reynolds number $\text{Re}$ and the Nusselt number $\text{Nu}$ on the Rayleigh number $\text{Ra}$ and the Prandtl number $\text{Pr}$ in laminar vertical convection (VC), where a fluid is confined between two differently heated isothermal vertical walls. The boundary layer equations in laminar VC yield two limiting scaling regimes: $\text{Nu}\sim {\text{Pr}}^{1/4}{\text{Ra}}^{1/4}, \text{Re}\sim {\text{Pr}}^{-1/2}{\text{Ra}}^{1/2}$ for $\text{Pr}\ll 1$ and $\text{Nu}\sim {\text{Pr}}^0{\text{Ra}}^{1/4}, \text{Re}\sim {\text{Pr}}^{-1}{\text{Ra}}^{1/2}$ for $\text{Pr}\gg 1$. These theoretical results are in excellent agreement with direct numerical simulations for $\text{Ra}$ from ${10}^5$ to ${10}^{10}$ and $\text{Pr}$ from ${10}^{-2}$ to 30. The transition between the regimes takes place for $\text{Pr}$ around ${10}^{-1}$.

Figure 1

(a) Sketch of the conducted DNS of VC in a cylindrical cell filled with a fluid of $\text{Pr}=0.1$ (dotted circle), $\text{Pr}=1$ (upward triangle), $\text{Pr}=10$ (dotted square), and $\text{Ra}={10}^6$ (dotted diamond), $\text{Ra}={10}^7$ (downward triangle), in a $(\text{Ra},\text{Pr})$ plane. (b) Eight isosufaces of the instantaneous temperature $T$, as obtained in the DNS for $\text{Pr}=1$ and $\text{Ra}={10}^8$. The left vertical wall of the container is heated ($T=T_{+}$), while the right vertical wall is cooled ($T=T_{-}$); the other walls are adiabatic.

Figure 2

(a), (c) $\text{Ra}$ dependences and (b), (d) $\text{Pr}$ dependences of (a), (b) the mean thermal dissipation rate ${\epsilon }_{\theta }$ and (c), (d) the mean kinetic dissipation rate ${\epsilon }_u$, as obtained in the DNS of VC for (a), (c) $\text{Pr}=0.1$ (dotted circle), $\text{Pr}=1$ (upward triangle), $\text{Pr}=10$ (dotted square), and for (b), (d) $\text{Ra}={10}^6$ (dotted diamond) and $\text{Ra}={10}^7$ (downward triangle). Dashed lines in (c) show the corresponding exact values for RBC.

Figure 3

(a), (c) $\text{Ra}$ dependences and (b), (d) $\text{Pr}$ dependences of (a), (b) the Nusselt number and (c), (d) the Reynolds number, as obtained in the DNS of VC for (a), (c) $\text{Pr}=0.1$ (dotted circle), $\text{Pr}=1$ (upward triangle), $\text{Pr}=10$ (dotted square), and for (b), (d) $\text{Ra}={10}^6$ (dotted diamond) and $\text{Ra}={10}^7$ (downward triangle). For laminar vertical convection the simulations support the scalings $\text{Nu}\sim {\text{Pr}}^{1/4}{\text{Ra}}^{1/4}, \text{Re}\sim {\text{Pr}}^{-1/2}{\text{Ra}}^{1/2}$ for small $\text{Pr}$ [Eq. (22)], and $\text{Nu}\sim {\text{Pr}}^0{\text{Ra}}^{1/4}, \text{Re}\sim {\text{Pr}}^{-1}{\text{Ra}}^{1/2}$ for large $\text{Pr}$ [Eq. (20)]. There is a transition from one scaling regime to another at $\text{Pr}\approx 0.1$.