Boussinesq Approximation (buoyancy)

Boussinesq Approximation (buoyancy)

In fluid dynamics, the Boussinesq approximation (, 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 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 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.