Natural and forced convection compared

Differences

Mechanism on a heated vertical plate

Fluid next to the hot plate is heated, expands, becomes lighter and rises. Cooler fluid from the surroundings replaces it and a buoyancy-driven boundary layer forms.

      ^  ^  ^    rising warm
   |  |         turbulent
   |  |   -----------
   |  |    transition
   |  |   laminar layer
 hot plate --> grows with height

The layer starts laminar at the lower edge, grows in thickness, becomes turbulent when $Ra \gtrsim 10^9$, and $h$ then becomes nearly independent of height.

Dimensional analysis

Buoyancy force per unit mass is $g\beta\,\Delta T$, replacing the velocity $V$ of forced flow. Thus $h = f(L, \rho, \mu, k, c_p, g\beta\Delta T)$: 7 variables and 4 fundamental dimensions give 3 groups:

$$\frac{hL}{k}= f\left(\frac{\rho^2 g\beta\Delta T L^3}{\mu^2},\ \frac{\mu c_p}{k}\right)\;\Rightarrow\; Nu = f(Gr, Pr)$$

For a vertical plate, $Nu = 0.59\,(Gr\,Pr)^{1/4}$ for laminar flow and $Nu = 0.10\,(Gr\,Pr)^{1/3}$ for turbulent flow.