Buoyancy-Driven Flows and Natural Convection
The importance of buoyancy forces in a mixed convection flow can be measured by the ratio of the Grashof and Reynolds numbers:
Equation 86
When this number approaches or exceeds unity, you should expect strong buoyancy contributions to the flow. Conversely, if it is very small, buoyancy forces may be ignored in your simulation. In pure natural convection, the strength of the buoyancy-induced flow is measured by the Rayleigh number:
Equation 87
where 
is the thermal expansion coefficient:
Equation 88
and 
is the thermal diffusivity:
Rayleigh numbers less than 108 indicate a buoyancy-induced laminar flow, with transition to turbulence occurring over the range of 108 < Ra <1010.
Icepak uses either the Boussinesq model or the ideal gas law in the calculation of natural-convection flows, as described in the following sections.
The Boussinesq Model
By default, Icepak uses the Boussinesq model for natural-convection flows. This model treats density as a constant value in all solved equations, except for the buoyancy term in the momentum equation:
Equation 89
where ρ0 is the (constant) density of the flow, T0 is the operating temperature, and β is the thermal
expansion coefficient. Equation 89 is obtained by using the Boussinesq approximation
to eliminate ρ from the buoyancy term. This approximation is accurate as long as
changes in actual density are small; specifically, the Boussinesq approximation is valid when β(T−T0) << 1.
Incompressible Ideal Gas Law
In Icepak, if you choose to define the density using the ideal gas law, Icepak will compute the density as
Equation 90
where R is the universal gas constant and pop is the Operating Pressure specified on the Design Settings dialog box Advanced tab (see Icepak Design Settings). In this form, the density depends only on the operating pressure and not on the local relative pressure field.
Definition of the Operating Density
When the Boussinesq approximation is not used, the operating density, ρ0, appears in the body-force term in the momentum equations as (ρ−ρ0)g.
This form of the body-force term follows from the redefinition of pressure in Icepak as
Equation 91
The hydrostatic balance in a fluid at rest is then
Equation 92
The definition of the operating density is thus important in all buoyancy-driven flows.