Heat Transfer · Learn
Why a thin, sluggish layer of fluid clinging to a surface controls how fast it cools — and the three dimensionless numbers that tie speed, fluid, and heat together.
Convection is conduction into a fluid that then carries the heat away by bulk motion. We bundle all the messy fluid physics into one coefficient $h$:
The whole game is finding $h$, which depends on the fluid, the geometry, and — above all — the flow.
At the wall the fluid is stuck (the no-slip condition); far away it moves at the free-stream speed $U_\infty$. The thin sheared region between is the velocity boundary layer, and it grows along the surface as
Heat leaving the wall must first cross this layer by conduction before the flow can sweep it off. So the wall heat flux is set by the temperature gradient right at the surface, $q = -k\,(\partial T/\partial y)|_{y=0} = h(T_s-T_\infty)$. A thinner layer means a steeper gradient and a larger $h$ — which is exactly why blowing harder cools faster: it pins the layer thin.
| Number | Definition | Meaning |
|---|---|---|
| Reynolds | $\mathrm{Re}=\rho U L/\mu = UL/\nu$ | inertia vs. viscosity |
| Prandtl | $\mathrm{Pr}=\nu/\alpha$ | momentum vs. heat diffusion |
| Nusselt | $\mathrm{Nu}=hL/k$ | convection vs. conduction |
The Prandtl number decides which layer is thicker: the thermal layer relates to the velocity layer by $\delta/\delta_t \approx \mathrm{Pr}^{1/3}$. For air ($\mathrm{Pr}\approx0.7$) they're comparable; for water ($\approx6$) the thermal layer is thinner; for oil ($\sim10^3$) it's a sliver nested deep inside.
Solving the boundary-layer equations (Blasius) gives the local and average Nusselt numbers for laminar flow over a flat plate:
then $h = \mathrm{Nu}\,k/L$. Notice $h \propto x^{-1/2}$: it's highest at the leading edge, where the layer is thinnest, and tapers downstream.
Past about $\mathrm{Re}_x \approx 5\times10^5$ the orderly laminar layer trips into turbulence. It suddenly thickens, but the violent mixing scrubs heat off the wall far more effectively, so $h$ jumps up (and so does drag). Turbulent flow uses a different correlation, $\mathrm{Nu}_x = 0.0296\,\mathrm{Re}_x^{4/5}\mathrm{Pr}^{1/3}$.
Switch fluids and watch the red thermal layer slide relative to the blue velocity layer ($\mathrm{Pr}^{1/3}$). Crank the velocity and the yellow transition marker marches toward the leading edge. Open the lab →
EngineeringCandy · Learn · the theory behind the playground