Rayleigh Number

The Rayleigh number is a product, $\mathrm{Ra} = \mathrm{Gr}\,\mathrm{Pr}$, and both halves of that sentence deserve saying. It is a product EXACTLY — nothing is measured to obtain it that was not already measured to obtain the Grashof and Prandtl numbers, and multiplying the two out gives $g\beta\Delta T L^{3}/(\nu\alpha)$, buoyancy divided by the product of the two diffusivities that damp it. So it is not an independent quantity. And yet it has its own name, its own thresholds and its own correlations, which would be strange if it were only shorthand.

The reason is a genuine experimental finding rather than a definition. When natural-convection data are plotted against Grashof number, fluids scatter: air, water and oil with the same Grashof number transfer heat at visibly different rates. Plot the same data against the PRODUCT and they collapse onto one curve. That collapse is a physical statement — it says buoyancy-driven transfer depends on the combination $\nu\alpha$ and not on the two diffusivities separately, and that a viscous fluid which also conducts well behaves like a thin fluid which conducts poorly. Once that was established there was no further reason to quote Grashof thresholds, and the literature stopped. Every natural-convection correlation you will meet — Churchill and Chu for plates and cylinders, the enclosure correlations, the horizontal-plate rules — is written in Rayleigh.

Two thresholds get quoted, and they are different kinds of thing. The first is $\mathrm{Ra} \approx 1708$, below which a fluid layer heated from below simply sits there and conducts, and above which it breaks into the hexagonal Rayleigh–Bénard cells you can watch in a pan of oil on a low burner. That figure is a genuine stability result, derived exactly for an idealised infinite layer between rigid plates, and it does not transfer to any other geometry without being re-derived. The second is $\mathrm{Ra} \approx 10^{9}$ for the laminar-to-turbulent transition on a vertical plate, and that one is a CONVENTION. The real transition wanders roughly between $10^{8}$ and $10^{10}$ depending on surface roughness, on how the flow was disturbed at the leading edge, and on how quiet the surrounding room is. Treat it as the centre of a range. When a calculation lands near it, the honest move is to run the correlation on both sides and see how much the answer actually moves — usually far less than the argument about which side you are on would suggest.

A working note on scale, because the numbers are unintuitive at first. A coffee cup gives a Rayleigh number around $10^{6}$, a room wall around $10^{9}$ to $10^{10}$, and the Earth's mantle something near $10^{20}$ — which is why the mantle convects despite being rock. And the Nusselt numbers that come out grow slowly, roughly as $\mathrm{Ra}^{1/4}$ in the laminar range and $\mathrm{Ra}^{1/3}$ in the turbulent one, so a hundredfold increase in Rayleigh buys only a threefold or fivefold increase in the coefficient. Natural convection is reliable and it is cheap; it is not, and cannot be made, powerful.