Richardson number

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 [Modellers will be more familiar with the reciprocal of the square root of the Richardson number, known as the Froude number.]

: Ri = {ghover u^2}

where "g" is the acceleration due to gravity, "h" a representativevertical 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' hover u^2}

which is used frequently when considering atmospheric or oceanic flows.

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

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

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.

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 eta (T_ ext{hot} - T_ ext{ref})L}{V^2}

where "g" is the gravitational acceleration, eta is the thermal expansion coefficient, T_ ext{hot} is the hot wall temperature, T_ ext{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 Ri < 0.1, forced convection is negligible when Ri > 10, and neither is negligible when 0.1 < Ri < 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.

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.

:Ri = N^2/(du/dz)^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.

Notes


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