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# 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 ρ1 and ρ2 one needs only consider a single density ρ: the difference δρ = ρ1 - ρ2 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

$g' = g{\rho_1-\rho_2\over {\rho}}$.

(Note that the denominator may be either density without affecting the result because the change would be of order g(δρ / ρ)2). The most generally used dimensionless number would be the Richardson number.

The flow is therefore simpler because the density ratio (ρ1 / ρ2---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 δρ / ρ 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.

03-10-2013 05:06:04