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 [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.
Aviation
In aviation, the Richardson number is used as a rough measure of expected air turbulence. A lower value 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
\[\mathit{Ri} = \frac{g \beta (T_\text{hot} - T_\text{ref})L}{V^2}\]
where g is the gravitational acceleration, \(\beta\) is the thermal expansion coefficient, \(T_\text{hot}\) is the hot wall temperature, \(T_\text{ref}\) 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. In the design of water filled thermal energy storage tanks, the Richardson number can be useful. [2]
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, as described by the Taylor–Goldstein equation, used to model Kelvin–Helmholtz instability which is driven by sheared flows.
\[\mathit Ri = N^2/(du/dz)^2 \]
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 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.[3]
Notes
- ↑ Modellers will be more familiar with the reciprocal of the square root of the Richardson number, known as the Froude number.
- ↑ Robert Huhn Beitrag zur thermodynamischen Analyse und Bewertung von Wasserwärmespeichern in Energieumwandlungsketten, ISBN 978-3-940046-32-1, Andreas Oberhammer Europas größter Fernwärmespeicher in Kombination mit dem optimalen Ladebetrieb eines Gas- und Dampfturbinenkraftwerkes (Vortrag 2007)
- ↑ A good reference on this subject is Script error
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