# 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. 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.

Boussinesq flows are common in nature (such as atmospheric fronts,
oceanic circulation, katabatic winds), 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

*g' * 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.

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.

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 air inside is lighter than that outside, so flows into the room and down towards
the floor. Now imagine it is cold inside and warmer outside: the air
flows in and up toward the ceiling. If the flow is Boussinesq (and
the room 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 the density enters the problem is via the
reduced gravity * g' * and this undergoes only a sign change when
changing from the warm room flow to the cold room flow.

An example of a non-Boussinesq flow would be bubbles rising in water. This flow is nothing like water falling in air: rising bubbles tend to form hemispherical shells, while falling water splits into raindrops. At small length scales surface energy enters the problem and confuses the issue.