Grashof Number

Stand near a radiator on a cold morning and you can feel air rising off it. Nobody is blowing that air; the radiator warms the layer touching it, warm air is less dense than cool air, and the difference in density is enough to lift it. That is natural convection, and it is the form of heat transfer most of us meet first — a hot mug, a window in the sun, a person in a still room — and have the least vocabulary for. The Grashof number is the vocabulary. It asks whether the buoyancy a temperature difference creates is strong enough to overcome the fluid's own stickiness, $\mathrm{Gr} = g\beta\,\Delta T\,L^{3}/\nu^{2}$, and it plays exactly the role in natural convection that the Reynolds number plays in forced convection. Reynolds compares the inertia a pump supplies against viscosity; Grashof compares the buoyancy a temperature difference supplies against the same viscosity. Where Reynolds has a fan, Grashof has gravity and a density difference doing the work for free.

Read the pieces and the physics is legible. On top, $g\beta\Delta T$ is the buoyant acceleration: how much lighter the heated fluid has become, times gravity. The length appears CUBED because buoyancy acts on a volume while the viscous drag opposing it acts on a face — so a tall wall is not a little more convective than a short one, it is dramatically more. Underneath, $\nu^{2}$ is the viscosity resisting twice over, once in setting up the motion and once in damping it. Note the presence of $g$, which no forced-convection group contains: turn gravity off and a hot plate in a spacecraft simply sits in a growing pocket of its own warm air, which is why cooling in orbit needs fans that a laboratory bench does not.

The characteristic length is a convention, not a measurement. This is the point at which most Grashof calculations go wrong, and the reason to be blunt about it. For a vertical plate, $L$ is the HEIGHT. For a horizontal plate it is the area divided by the perimeter. For a cylinder or a sphere it is the DIAMETER. These are not different measurements of the same thing; they are different agreements, adopted because the correlations that use them were fitted that way. The same wall in the same air has different Grashof numbers under different conventions, and because $L$ is cubed the difference is orders of magnitude — a 0.5 m wall taken on its height gives roughly a thousand times the Grashof number of the same wall taken on a 0.05 m thickness. A Grashof number without its length convention stated is not a number, it is a rumour. Before feeding one into a correlation, find the sentence in the source that says what $L$ is.

β is 1/T for an ideal gas and measured for everything else, and the T in that substitution is ABSOLUTE. For air at a film temperature of 320 K, $\beta = 1/320 = 0.003125$ per kelvin, exactly — no table needed. Use Celsius there and the answer is nonsense. For liquids there is no shortcut at all, and water is the reason it is worth belabouring: its expansion coefficient swings by more than a factor of ten between 0 and 100 °C, and it passes through ZERO at about 4 °C. Above that temperature water expands as it warms, as everything else does; below it, water CONTRACTS as it warms. So the densest water in a lake sits at 4 °C at the bottom, colder water floats above it, and ice forms at the surface rather than at the bed. That is why a lake freezes from the top down, why fish survive the winter, and — closer to this page — why a natural-convection calculation in cold water can return a Grashof number of nearly nothing and be perfectly correct.

Two practical habits. Evaluate all the properties at the FILM temperature, the average of the surface and the bulk fluid, because the boundary layer is where everything in this problem happens and it is neither at wall temperature nor at room temperature. And do not look for a Grashof threshold: unlike Reynolds, this group does not decide the flow regime on its own. The laminar-to-turbulent judgement in natural convection is made on the Rayleigh number, $\mathrm{Gr}\cdot\mathrm{Pr}$, which is where this calculation should go next.