# Scaling in Rayleigh-Bénard convection

**Time: **
Thu 2022-12-01 10.30 - 11.30

**Location: **
Faxén, Teknikringen 8

**Participating: **
Erik Lindborg (KTH Engineering Mechanics)

The Nusselt-Rayleigh number problem is to predict how the heat transfer scales with the input parameters in a convection cell where the fluid is confined between two horizontal walls and the upper wall is kept at a lower temperature than the lower. The input is given by the Rayleigh and Prandtl numbers while the output, quantifying the heat transfer, is given by the Nusselt number.

In this talk I will first review theories and experiments addressing this classical problem. In particular I will consider the prediction of Kraichnan (1962) that there should exist an “ultimate regime”, where the Nusselt number scales with the Rayleigh number as Ra^(1/2), as opposed to other predictions of Ra^(1/3) or Ra^(2/7), which are closer to experimental results. The last twenty years there have been a number of experimental efforts by several groups to confirm the Kraichnan prediction and several claims that a transition to the ultimate regime has been observed at high Ra.

Then I will argue that Kraichnan’s prediction cannot be valid since it is based on the assumption that the boundary layers in a convection cell are similar to classical shear boundary layers. The width of the viscous layer of a classical shear boundary layer scales with Reynolds number as Re^(-1), apart from a logarithmic correction, while the width of a laminar boundary layer scales as Re^(-1/2). I will argue that the width of the boundary layers in a convection cell scales as Re^(-3/4) and show that the Nusselt number in this case will scale as Ra^(1/3), in close agreement with recent measurements and DNS. Finally, I will argue that this scaling is the ultimate high Ra regime.