The Boussinesq Approximation
The Boussinesq approximation assumes that the density can be written on the form
(4-15)
where ρ0, α and T0 are constants and T is the temperature. All other material properties are assumed to be constant. Provided that αp and T−T0 are small enough, Equation 4-1 and Equation 4-2 reduce to
(4-16)
where K0 is the constant thermal conductivity.
The Boussinesq approximation is commonly used to simulate buoyancy-driven flows. In this case
(4-17)
where g is the gravity vector. A further useful simplification is possible by writing Equation 4-17 in terms of a potential, Φ:
The first part can be canceled out by splitting the true pressure, p, into a hydrodynamic component, P, and a hydrostatic component, −ρ0Φ such that P = p + ρ0Φ. This reduced Equation 4-16 to:
(4-18)
The gravity force is added on the form given by Equation 4-17 if Include gravity is selected in the fluid flow interface. The form in Equation 4-18 is obtained when selecting the Use reduced pressure option.
The main assumption in the Boussinesq approximation 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. It can also be observed the energy equation in Equation 4-18 retains both the viscous heating term and the pressure work term. These can, however, almost always be neglected in situations when the Boussinesq approximation is valid (see, for example, Ref. 7). But there are situations where they need to be retained (Ref. 8). In particular, the pressure work term can be of importance for liquids where Cp ≠ Cv (Ref. 9). An excellent discussion of the Boussinesq approximation and its limitations appears in Chapter 14 of Ref. 7.