Numerical Simulation of 2D Tube Convection

We present results from numerical simulation of two-dimensional tube convection: natural convection in a vertical tube with an imposed unstable density gradient. Simulations are performed in OpenFOAM, an open-source Finite Volume-based toolbox. The flow is characterized by gradient Rayleigh number \({\text{Ra}}_{g}\) which is varied approximately from the onset of convection \({\text{Ra}}_{g} \sim 10^{3}\) across four decades to \({\text{Ra}}_{g} \sim 10^{7}\) , for a fixed Prandtl number \({\text{Pr}} = 1\) . We see that the flowfield is characterized by vertically stacked, counter-rotating eddies or rolls that scale with the diameter of the tube. We show that the average temperature profile in the tube is linear with modulations due to the rolls and that the mixing length model gives a good estimate for fluctuations of the velocities. Evidences for existence of at least three regimes were observed for the scaling relations for Nusselt number, and at high enough Rayleigh numbers, \({\text{Nu}}\) scaling in 2D tube convection shows trend similar to the \(1/2\) power scaling observed in 3D DNS and experiments.