Richardson number#

Named after: Lewis Fry Richardson (1881-1953).

$$\text{Ri} \stackrel{\text{def}}{=} \frac{g \beta (T_s - T_\infty) L}{U^{2}} \sim \frac{\text{buoyancy}}{\text{inertia}}$$

Measures buoyancy relative to inertial forcing. It indicates whether a flow behaves more like forced, mixed, or natural convection.

Name	Symbol	SI units	Dimension
gravitational acceleration	$g$	$\mathrm{m}\,\mathrm{s}^{-2}$	$\text L\,\text T^{-2}$
thermal expansion coefficient	$\beta$	$\mathrm{K}^{-1}$	$\Theta^{-1}$
surface-ambient temperature difference	$T_s - T_\infty$	$\mathrm{K}$	$\Theta$
characteristic length	$L$	$\mathrm{m}$	$\text L$
velocity	$U$	$\mathrm{m}\,\mathrm{s}^{-1}$	$\text L\,\text T^{-1}$

Mixed convection

Range	Regime	Description
0 – 0.25	forced convection	Inertial forcing dominates buoyancy. Temperature differences have a weak effect on the main flow.
0.25 – 1	mixed convection	Buoyancy and imposed motion both influence the flow and heat-transfer pattern.
1 – ∞	natural convection	Buoyancy dominates over imposed inertia. Flow structure is set mainly by density differences.

Name	Symbol	SI units	Dimension
gravitational acceleration	\(g\)	\(\mathrm{m}\,\mathrm{s}^{-2}\)	\(\text L\,\text T^{-2}\)
thermal expansion coefficient	\(\beta\)	\(\mathrm{K}^{-1}\)	\(\Theta^{-1}\)
surface-ambient temperature difference	\(T_s - T_\infty\)	\(\mathrm{K}\)	\(\Theta\)
characteristic length	\(L\)	\(\mathrm{m}\)	\(\text L\)
velocity	\(U\)	\(\mathrm{m}\,\mathrm{s}^{-1}\)	\(\text L\,\text T^{-1}\)