Chapter 7: Optimal Prandtl Number for Heat Transfer Enhancement

In Rayleigh-Bénard convection, a layer of fluid is heated from below and cooled from above. The dynamics of the system depends on two control parameters. The Rayleigh number, which indicates the dimensionless temperature difference, and the Prandtl number.

In the following movie we look at the dynamics in the horizontal midplane. We see the temperature field at the horizontal midplane for a simulation with Prandtl 6.4, which is typical for water. The red regions indicate warm fluid that is flowing up, and the blue regions cold fluid that is flowing down. All fluid is moving up on one side of the cell and moving down on the other side. This indicates that there is one large-scale circulation in the system.

When the system is rotated about a vertical axis, as is indicated in the figure, the system is called rotating Rayleigh-Bénard convection. The rotation rate of the system is indicated by the Rossby number. Due to rotation, vertically aligned vortices are formed. Here we show a visualization of these vortices where red indicates warm fluid that is trapped in up-going vortices, and blue cold fluid that is trapped in down-going vortices. The behaviour of these vertically aligned vortices is seen in the following movie.

We see the temperature field at the horizontal midplane for a rotating Rayleigh-Bénard simulation with Prandtl 6.4. The red regions indicate vortices in which warm fluid is sufficiently transported upwards. The blue regions indicate vortices in which cold fluid is sufficiently transported downwards. One can see that these vertically aligned vortices are randomly moving around.

The following movie shows the dynamics close to the bottom plate as is indicated in the figure. We see that close to the bottom plate warm fluid is pulled into the vortices and is then efficiently transported in the vertical direction. This process is called Ekman pumping. Due to Ekman pumping, the heat transport with respect to the non-rotating case can increase significantly.

This figure shows the heat transport enhancement with respect to the non-rotating case for Rossby 0.3. The figure thus shows that the effect of Ekman pumping strongly depends on the Prandtl number. We find that the effect of Ekman pumping is highest at an intermediate Prandtl number, which is close to the Prandtl number of water. This suggests that two competing mechanisms control the effect of Ekman pumping.

For Prandtl of order one, hot fluid enters the vortices at the base and is then efficiently transported in a vertical direction. When Prandtl becomes lower, the thermal diffusivity increases. This means that for lower Prandtl, more heat spreads out in the bulk, which limits the heat transport enhancement. This is shown in the right picture for Prandtl 0.7.

In the right picture the temperature isosurface for a higher Prandtl number is indicated. The difference with respect to the water case is that the fluid that enters the vortices at the base is colder. This means that for higher Prandtl, less warm fluid is transported by the vertically aligned vortices than for lower Prandtl. This effect becomes stronger when Prandtl is increased, and this limits the heat transport enhancement due to Ekman pumping for higher Prandtl.