Richardson number
Encyclopedia
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
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
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.
, 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.
relative to the forced convection
. The Richardson number in this context is defined as
where g is the gravitational acceleration,
is the thermal expansion coefficient, 
is the hot wall temperature, 
is the reference temperature, 
is the characteristic length, and 
is the characteristic velocity.
The Richardson number can also be expressed by using a combination of the Grashof number and Reynolds number,
Typically, the natural convection is negligible when
, forced convection is negligible when 
, and neither is negligible when 
. 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.
, 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.
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.
Lewis Fry Richardson
Lewis Fry Richardson, FRS was an English mathematician, physicist, meteorologist, psychologist and pacifist who pioneered modern mathematical techniques of weather forecasting, and the application of similar techniques to studying the causes of wars and how to prevent them...
(1881 – 1953). It is the dimensionless number that expresses the ratio of potential to kinetic energy
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
Boussinesq approximation
In fluid dynamics, the Boussinesq approximation is used in the field of buoyancy-driven flow . It states that density differences are sufficiently small to be neglected, except where they appear in terms multiplied by g, the acceleration due to gravity...
), it is common to use the reduced gravity
g' and the relevant parameter is the densimetric Richardson number
which is used frequently when considering atmospheric or oceanic flows.
If the Richardson number is much less than unity, buoyancy
Buoyancy
In physics, buoyancy is a force exerted by a fluid that opposes an object's weight. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus a column of fluid, or an object submerged in the fluid, experiences greater pressure at the bottom of the...
is unimportant
in the flow. If it is much greater than unity, buoyancy is dominant (in
the sense that there is insufficient kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
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
Potential energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...
in the system originally.
Aviation
In aviationAviation
Aviation is the design, development, production, operation, and use of aircraft, especially heavier-than-air aircraft. Aviation is derived from avis, the Latin word for bird.-History:...
, 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 convectionNatural convection
Natural convection is a mechanism, or type of heat transport, in which the fluid motion is not generated by any external source but only by density differences in the fluid occurring due to temperature gradients. In natural convection, fluid surrounding a heat source receives heat, becomes less...
relative to the forced convection
Forced convection
Forced convection is a mechanism, or type of heat transport in which fluid motion is generated by an external source...
. The Richardson number in this context is defined as
where g is the gravitational acceleration,
is the thermal expansion coefficient, is the hot wall temperature, is the reference temperature, is the characteristic length, and is the characteristic velocity.The Richardson number can also be expressed by using a combination of the Grashof number and Reynolds number,
Typically, the natural convection is negligible when
, forced convection is negligible when , and neither is negligible when . 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. Oceanography
In oceanographyOceanography
Oceanography , also called oceanology or marine science, is the branch of Earth science that studies the ocean...
, 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
Taylor–Goldstein equation
The Taylor–Goldstein equation is an ordinary differential equation used in the fields of geophysical fluid dynamics, and more generally in fluid dynamics, in presence of quasi-2D flows. It describes the dynamics of the Kelvin–Helmholtz instability, subject to buoyancy forces , for stably-stratified...
, used to model Kelvin–Helmholtz instability which is driven by sheared flows.
where N is the Brunt–Väisälä frequency.
The Richardson number defined above is always considered positive. An imaginary
Imaginary number
An imaginary number is any number whose square is a real number less than zero. When any real number is squared, the result is never negative, but the square of an imaginary number is always negative...
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.