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The '''Richardson number''' ('''Ri''') is named after [[Lewis Fry Richardson]] (1881 &ndash; 1953). It is the [[dimensionless number]] that expresses the ratio of [[Potential Energy|potential]] to [[kinetic energy]]. In certain fields the [[Froude number]] is more commonly used and is the reciprocal of the square root of the Richardson number or Ri = 1/Fr<sup>2</sub>.
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:<math>
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\mathrm{Ri} = \frac{\text{potential energy}}{\text{kinetic energy}} = \frac{gh}{u^2}
</math>
 
where ''g'' is the [[standard gravity|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 (buoyancy)|Boussinesq approximation]]), it is common to use the reduced gravity
''g' '' and the relevant parameter is the densimetric Richardson number
 
:<math>  
\mathrm{Ri} = {g' h\over u^2}
</math>
 
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 ==
[[File:Fernwärmespricher Theiss.JPG|thumb|District heating accumulation tower of the power plant Theiss, with volume of 50000 cubic meters]]
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
 
: <math>
\mathrm{Ri} = \frac{g \beta (T_\text{hot} - T_\text{ref})L}{V^2}
</math>
 
where ''g'' is the gravitational acceleration, <math>\beta</math> is the [[thermal expansion coefficient]], ''T''<sub>hot</sub> is the hot wall temperature, ''T''<sub>ref</sub> 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]],
 
: <math>
\mathrm{Ri} = \frac{\mathrm{Gr}}{\mathrm{Re}^2}.
</math>
 
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. <ref>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)</ref>
 
== 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.
 
:<math>\mathrm{Ri}  = N^2/(\mathrm{d}u/\mathrm{d}z)^2 </math>
 
where ''N'' is the [[Brunt–Väisälä frequency]].
 
The Richardson number defined above is always considered positive. A negative value of ''N²'' (i.e. [[complex number|complex]] ''N'') indicates unstable density gradients with active convective overturning. Under such circumstances the magnitude of negative Ri is not generally of interest.  It can be shown that Ri < 1/4 is a necessary condition for velocity shear to overcome the tendency of a stratified fluid to remain stratified, and some mixing (turbulence) will generally occur. When Ri is large, turbulent mixing across the stratification is generally suppressed.<ref>A good reference on this subject is {{cite book |first=J. S. |last=Turner |title=Buoyancy Effects in Fluids |location= |publisher=Cambridge University Press |year=1973 |isbn=0-521-08623-X }}</ref>
 
==Notes==
{{reflist}}
 
{{NonDimFluMech}}
 
[[Category:Dimensionless numbers]]
[[Category:Atmospheric dispersion modeling]]
[[Category:Fluid dynamics]]
[[Category:Buoyancy]]

Latest revision as of 19:51, 10 September 2014

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