Richardson number

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The Richardson number is named after Lewis Fry Richardson (1881 – 1953). It is the dimensionless number that expresses the ratio of potential to kinetic energy ^[1]

$Ri = {gh\over u^2}$

where g is the acceleration due to gravity, h a representative vertical lengthscale, and u a representative speed.

When considering flows in which density differences are small (the Boussinesq approximation), it is common to use the reduced gravity g' and the relevant parameter is the densimetric Richardson number

$Ri={g' h\over u^2}$

which is used frequently when considering atmospheric or oceanic flows.

If the Richardson number is much less than unity, buoyancy is unimportant in the flow. If it is much greater than unity, buoyancy is dominant (in the sense that there is insufficient kinetic energy to homogenize the fluids).

If the Richardson number is of order unity, then the flow is likely to be buoyancy-driven: the energy of the flow derives from the potential energy in the system originally.

1 Aviation
2 Thermal convection
3 Oceanography
4 Notes

[edit] Aviation

In aviation, the Richardson number is used as a rough measure of expected air turbulence. A lower score indicates a higher degree of turbulence. Values in the range 10 to 0.1 are typical, with values below unity indicating significant turbulence.

[edit] Thermal convection

In thermal convection problems, Richardson number represents the importance of natural convection relative to the forced convection. The Richardson number in this context is defined as

$\mathit{Ri} = \frac{g \beta (T_\text{hot} - T_\text{ref})L}{V^2}$

where g is the gravitational acceleration, $β$ is the thermal expansion coefficient, $T hot$ is the hot wall temperature, $T ref$ is the reference temperature, $L$ is the characteristic length, and $V$ is the characteristic velocity.

The Richardson number can also be expressed by using a combination of the Grashof number and Reynolds number,

$\mathit{Ri} = \frac{Gr}{Re^2}.$

Typically, the natural convection is negligible when $R i < 0.1$ , forced convection is negligible when $R i > 10$ , and neither is negligible when $0.1 < R i < 10$ . It may be noted that usually the forced convection is large relative to natural convection except in the case of extremely low forced flow velocities.

[edit] Oceanography

In oceanography, the Richardson number has a more general form which takes stratification into account. It is a measure of relative importance of mechanical and density effects in the water column.

R i = N 2 / (d u / d z) 2

where N is the Brunt-Väisälä frequency.

The Richardson number defined above is always considered positive. An imaginary N indicates unstable density gradients with active convective overturning. Under such circumstances, N does not have an accepted physical meaning and the magnitude of negative Ri is not generally of interest. When Ri is small (typically considered below 1/4), then velocity shear is considered sufficient to overcome the tendency of a stratified fluid to remain stratified, and some mixing will generally occur. When Ri is large, turbulent mixing across the stratification is generally suppressed. A good reference on this subject is J.S. Turner, Buoyancy Effects in Fluids, Cambridge University Press, 1973.