ENGINEERING AND CFD

Project name: Eulerian and Lagrangian analysis of transport in thermal convection in thermal convection flows in large aspect ratio cells

The law of heat transfer remains one of the central questions in the research field of turbulent thermal convection with its numerous applications in atmospheric and astrophysical phenomena and technology. The Rayleigh-Bénard setup includes a fluid layer that is uniformly heated from below and cooled from above. It is the simplest turbulent thermal convection flow and serves as a paradigm for many real-life flows. Our present knowledge of the heat transport law in this setting, which connects the Nusselt number Nu, a dimensionless measure of turbulent heat transfer across the convection layer, with the Rayleigh number Ra, a dimensionless number that probes the vigor of convective turbulence, is still inconclusive for very high Rayleigh numbers. Laboratory experiments in closed cylindrical cells with diameters similar to the cell height for Ra ≥ 10¹² have reported different outcomes. Typical working fluids to obtain these large Rayleigh numbers are cryogenic helium or compressed sulfur hexafluoride. The resulting Prandtl numbers Pr are close to those of convection in air.

Our direct numerical simulations of turbulent Rayleigh-Bénard convection in closed cylindrical cells advanced to Rayleigh numbers as high as Ra = 10¹⁵ when the aspect ratio (which relates the diameter d to the height H) of the cell is chosen sufficiently small, d/H=1/10 for Pr = 1. Without a decrease of the aspect ratio to such a small value, direct numerical simulations which resolve all relevant scales in the flow cannot access such large Rayleigh numbers. We found that the classical scaling law of turbulent heat transfer, Nu~Ra^1/3, is satisfied. This scaling assumes that the thin boundary layers of the temperature and velocity fields at the top and bottom plates are marginally stable, a model that goes back to the seminal theoretical works by Malkus and Spiegel in the 50s and 60s of the past century. Even the prefactor of the detected transport law was very close to the theoretical predictions. For the highest Rayleigh number in our simulations and the chosen slender cell geometry, the ratio of the boundary layer thickness to the cell diameter is still about 1 to 1000. Figure 1 displays the boundary layer structure for three different Rayleigh numbers. It is seen how the turbulence fields become more and more filamented, developing ever finer thermal plumes structures and vortices. We also showed that the fluctuations, e.g. those of the wall shear stress components, grow steadily as Ra grows without an abrupt transition into a new state. Therefore, the Rayleigh numbers were varied over a range of 8 orders of magnitude.