The Richardson number (Ri) is named after
Lewis Fry Richardson
Lewis Fry Richardson, FRS (11 October 1881 – 30 September 1953) was an English mathematician, physicist, meteorologist, psychologist, and pacifist who pioneered modern mathematical techniques of weather forecasting, and the application of s ...
(1881–1953). It is the
dimensionless number
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
that expresses the ratio of the
buoyancy
Buoyancy (), or upthrust, is an upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the pr ...
term to the
flow
Flow may refer to:
Science and technology
* Fluid flow, the motion of a gas or liquid
* Flow (geomorphology), a type of mass wasting or slope movement in geomorphology
* Flow (mathematics), a group action of the real numbers on a set
* Flow (psyc ...
shear term:
:
where
is
gravity,
is density,
is a representative flow speed, and
is depth.
The Richardson number, or one of several variants, is of practical importance in
weather forecasting and in investigating density and turbidity currents in oceans, lakes, and reservoirs.
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
Buoyancy (), or upthrust, is an upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the pr ...
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
Aviation includes the activities surrounding mechanical flight and the aircraft industry. ''Aircraft'' includes fixed-wing and rotary-wing types, morphable wings, wing-less lifting bodies, as well as lighter-than-air craft such as hot air ...
, 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
:
where ''g'' is the gravitational acceleration,
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
In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domin ...
,
:
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. However, buoyancy often plays a significant role in defining the laminar–turbulent transition of a
mixed convection
In fluid thermodynamics, combined forced convection and natural convection, or mixed convection, occurs when natural convection and forced convection mechanisms act together to transfer heat. This is also defined as situations where both pressur ...
flow.
In the design of water filled thermal energy storage tanks, the Richardson number can be useful.
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
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 instab ...
, used to model
Kelvin–Helmholtz instability which is driven by sheared flows.
:
where ''N'' is the
Brunt–Väisälä frequency
In atmospheric dynamics, oceanography, asteroseismology and geophysics, the Brunt–Väisälä frequency, or buoyancy frequency, is a measure of the stability of a fluid to vertical displacements such as those caused by convection. More precisely ...
.
The Richardson number defined above is always considered positive. A negative value of ''N²'' (i.e.
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
''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.
[A good reference on this subject is ]
References
{{Dimensionless numbers in fluid mechanics
Dimensionless numbers
Atmospheric dispersion modeling
Fluid dynamics
Buoyancy
Dimensionless numbers of fluid mechanics