Boussinesq approximation
Encyclopedia
In fluid dynamics
, the Boussinesq approximation (businɛsk, named for Joseph Valentin Boussinesq) is used in the field of buoyancy-driven flow (also known as natural convection
). 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. The essence of the Boussinesq
approximation is that the difference in inertia
is negligible but gravity is sufficiently strong to make the specific weight
appreciably different between the two fluids. Sound waves
are impossible/neglected when the Boussinesq approximation is used since sound waves move via density variations.
Boussinesq flows are common in nature (such as atmospheric front
s, oceanic circulation, katabatic wind
s), industry (dense gas dispersion
, fume cupboard ventilation), and the built environment (natural ventilation, central heating
). The approximation is extremely accurate for many such flows, and makes the mathematics and physics simpler.
The approximation's advantage arises because when
considering a flow of, say, warm and cold water of density
and 
one needs only consider a
single density
: the difference
is negligible.
Dimensional analysis
shows that, under these circumstances, the only sensible
way that acceleration due to gravity g should enter into the equations of motion is in the reduced gravity
where
(Note that the denominator may be either density without affecting the result because the change would be of order
). The most generally used dimensionless number would be the Richardson number and Rayleigh number
.
The mathematics of the flow is therefore simpler because the density ratio (
, a dimensionless number) does not affect the flow; the Boussinesq approximation states that it may be assumed to be exactly one.
is of order unity.
For example, consider an open window in a warm room. The warm air inside is lighter than the cold air outside, which flows into the room and down towards the floor. Now imagine the opposite: a cold room exposed to warm outside air. Here the air flowing in moves up toward the ceiling. If the flow is Boussinesq (and the room is otherwise symmetrical), then viewing the cold room upside down is exactly the same as viewing the warm room right-way-round. This is because the only way density enters the problem is via the reduced gravity
which undergoes only a sign change when changing from the warm room flow to the cold room flow.
An example of a non-Boussinesq flow is bubbles rising in water. The behaviour of air bubbles rising in water is very different from the behaviour of water falling in air: in the former case rising bubbles tend to form hemispherical shells, while water falling in air splits into raindrops (at small length scales surface tension
enters the problem and confuses the issue).
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
, the Boussinesq approximation (businɛsk, named for Joseph Valentin Boussinesq) is used in the field of buoyancy-driven flow (also known as natural convection
Natural 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...
). 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. The essence of the Boussinesq
Valentin Joseph Boussinesq
Joseph Valentin Boussinesq was a French mathematician and physicist who made significant contributions to the theory of hydrodynamics, vibration, light, and heat....
approximation is that the difference in inertia
Inertia
Inertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to...
is negligible but gravity is sufficiently strong to make the specific weight
Weight
In science and engineering, the weight of an object is the force on the object due to gravity. Its magnitude , often denoted by an italic letter W, is the product of the mass m of the object and the magnitude of the local gravitational acceleration g; thus:...
appreciably different between the two fluids. Sound waves
Sound
Sound is a mechanical wave that is an oscillation of pressure transmitted through a solid, liquid, or gas, composed of frequencies within the range of hearing and of a level sufficiently strong to be heard, or the sensation stimulated in organs of hearing by such vibrations.-Propagation of...
are impossible/neglected when the Boussinesq approximation is used since sound waves move via density variations.
Boussinesq flows are common in nature (such as atmospheric front
Surface weather analysis
Surface weather analysis is a special type of weather map that provides a view of weather elements over a geographical area at a specified time based on information from ground-based weather stations...
s, oceanic circulation, katabatic wind
Katabatic wind
A katabatic wind, from the Greek word katabatikos meaning "going downhill", is the technical name for a drainage wind, a wind that carries high density air from a higher elevation down a slope under the force of gravity. Such winds are sometimes also called fall winds...
s), industry (dense gas dispersion
Air pollution dispersion terminology
Air pollution dispersion terminology includes the words and technical terms that have a special meaning to those who work in the field of air pollution dispersion modeling...
, fume cupboard ventilation), and the built environment (natural ventilation, central heating
Central heating
A central heating system provides warmth to the whole interior of a building from one point to multiple rooms. When combined with other systems in order to control the building climate, the whole system may be a HVAC system.Central heating differs from local heating in that the heat generation...
). The approximation is extremely accurate for many such flows, and makes the mathematics and physics simpler.
The approximation's advantage arises because when
considering a flow of, say, warm and cold water of density
and one needs only consider asingle density
: the difference is negligible.Dimensional analysis
Dimensional analysis
In physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions which describe it; for example, speed has the dimension length per...
shows that, under these circumstances, the only sensible
way that acceleration due to gravity g should enter into the equations of motion is in the reduced gravity
where(Note that the denominator may be either density without affecting the result because the change would be of order
). The most generally used dimensionless number would be the Richardson number and Rayleigh numberRayleigh number
In fluid mechanics, the Rayleigh number for a fluid is a dimensionless number associated with buoyancy driven flow...
.
The mathematics of the flow is therefore simpler because the density ratio (
, a dimensionless number) does not affect the flow; the Boussinesq approximation states that it may be assumed to be exactly one.Inversions
One feature of Boussinesq flows is that they look the same when viewed upside-down, provided that the identities of the fluids are reversed. The Boussinesq approximation is inaccurate when the nondimensionalised density difference is of order unity.For example, consider an open window in a warm room. The warm air inside is lighter than the cold air outside, which flows into the room and down towards the floor. Now imagine the opposite: a cold room exposed to warm outside air. Here the air flowing in moves up toward the ceiling. If the flow is Boussinesq (and the room is otherwise symmetrical), then viewing the cold room upside down is exactly the same as viewing the warm room right-way-round. This is because the only way density enters the problem is via the reduced gravity
which undergoes only a sign change when changing from the warm room flow to the cold room flow.An example of a non-Boussinesq flow is bubbles rising in water. The behaviour of air bubbles rising in water is very different from the behaviour of water falling in air: in the former case rising bubbles tend to form hemispherical shells, while water falling in air splits into raindrops (at small length scales surface tension
Surface tension
Surface tension is a property of the surface of a liquid that allows it to resist an external force. It is revealed, for example, in floating of some objects on the surface of water, even though they are denser than water, and in the ability of some insects to run on the water surface...
enters the problem and confuses the issue).