The Boussinesq approximation assumes that variations in density have no effect on the flow field except that they give rise to a buoyancy force. The density is assigned a reference value, ρ0, everywhere except in the volume force term, which is set to

where g is the gravity vector. A further simplification is often possible. Because g can be written in terms of a potential, Φ, Equation 3-25 can be written as:

The first part can be canceled out by splitting the true pressure, p, into a hydrodynamic component, P, and a hydrostatic component, −ρ0Φ. Equation 3-17 and Equation 3-18 are expressed in terms of the hydrodynamic pressure P = p + ρ0Φ:

To obtain the Boussinesq approximation on this form, the flow must be defined as Incompressible with the Include gravity and Use reduced pressure options selected in the flow interface, and the Nonisothermal Flow multiphysics feature should be used to coupled the heat transfer and fluid flow interfaces.

In practice, the shift from p to P can be ignored except where the pressure appears in boundary conditions. The pressure that is specified at boundaries is the reduced pressure in this case. For example, at a vertical outflow or inflow boundary, the reduced pressure is typically a constant, whereas the true pressure is a function of the vertical coordinate.

The system formed by Equation 3-26 and Equation 3-27 has its limitations. The main assumption is that the density fluctuations must be small; that is, Δρ/ρ0 << 1. There are also some more subtle constraints that, for example, make the Boussinesq approximation unsuitable for systems of very large dimensions. An excellent discussion of the Boussinesq approximation and its limitations appears in Chapter 14 of Ref. 10.