
When heat is added to a fluid and the fluid density varies with temperature, a flow can be induced due to the force of gravity acting on the density variations. Such buoyancydriven flows are termed naturalconvection (or mixedconvection ) flows and can be modeled by FLUENT.
Theory
The importance of buoyancy forces in a mixed convection flow can be measured by the ratio of the Grashof and Reynolds numbers :
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 buoyancyinduced flow is measured by the Rayleigh number :
where is the thermal expansion coefficient:
(13.216) 
and is the thermal diffusivity:
(13.217) 
Rayleigh numbers less than 10 indicate a buoyancyinduced laminar flow, with transition to turbulence occurring over the range of 10 Ra 10 .
Modeling Natural Convection in a Closed Domain
When you model natural convection inside a closed domain, the solution will depend on the mass inside the domain. Since this mass will not be known unless the density is known, you must model the flow in one of the following ways:

For a closed domain, you can use the
incompressible ideal gas law only with a
fixed operating pressure. It
cannot be used with a floating operating pressure. You can use the
compressible ideal gas law with either
floating or
fixed operating pressure.

See Section 9.6.4 for more information about the floating operating pressure option.
The Boussinesq Model
For many naturalconvection flows, you can get faster convergence with the Boussinesq model than you can get by setting up the problem with fluid density as a function of temperature. This model treats density as a constant value in all solved equations, except for the buoyancy term in the momentum equation:
where is the (constant) density of the flow, is the operating temperature, and is the thermal expansion coefficient. Equation 13.218 is obtained by using the Boussinesq approximation to eliminate from the buoyancy term. This approximation is accurate as long as changes in actual density are small; specifically, the Boussinesq approximation is valid when .
Limitations of the Boussinesq Model
The Boussinesq model should not be used if the temperature differences in the domain are large. In addition, it cannot be used with species calculations, combustion, or reacting flows.
Steps in Solving BuoyancyDriven Flow Problems
The procedure for including buoyancy forces in the simulation of mixed or natural convection flows is described below.
Define Models Energy...
Define Operating Conditions
Define Materials...
Note that if your model involves multiple fluid materials you can choose whether or not to use the Boussinesq model for each material. As a result, you may have some materials using the Boussinesq model and others not. In such cases, you will need to set all the parameters described above in this step.
Define Boundary Conditions...
Solve Controls Solution...
You may also want to add cells near the walls to resolve boundary layers.
If you are using the pressurebased solver, selecting PRESTO! as the Discretization method for Pressure is another recommended approach.
See also Section 13.2.2 for information on setting up heat transfer calculations.
Operating Density
When the Boussinesq approximation is not used, the operating density appears in the bodyforce term in the momentum equations as .
This form of the bodyforce term follows from the redefinition of pressure in FLUENT as
The hydrostatic pressure in a fluid at rest is then
Setting the Operating Density
By default, FLUENT will compute the operating density by averaging over all cells. In some cases, you may obtain better results if you explicitly specify the operating density instead of having the solver compute it for you. For example, if you are solving a naturalconvection problem with a pressure boundary, it is important to understand that the pressure you are specifying is in Equation 13.219. Although you will know the actual pressure , you will need to know the operating density in order to determine from . Therefore, you should explicitly specify the operating density rather than use the computed average. The specified value should, however, be representative of the average value.
In some cases the specification of an operating density will improve convergence behavior, rather than the actual results. For such cases use the approximate bulk density value as the operating density and be sure that the value you choose is appropriate for the characteristic temperature in the domain.
Note that if you are using the Boussinesq approximation for all fluid materials, the operating density does not appear in the bodyforce term of the momentum equation. Consequently, you need not specify it.
Solution Strategies for BuoyancyDriven Flows
For highRayleighnumber flows you may want to consider the solution guidelines below. In addition, the guidelines presented in Section 13.2.3 for solving other heat transfer problems can also be applied to buoyancydriven flows. Note, however that no steadystate solution exists for some laminar, highRayleighnumber flows.
Guidelines for Solving HighRayleighNumber Flows
When you are solving a highRayleighnumber flow ( ) you should follow one of the procedures outlined below for best results.
The first procedure uses a steadystate approach:
The second procedure uses a timedependent approach to obtain a steadystate solution [ 139]:
where and are the length and velocity scales, respectively. Use a time step such that
(13.222) 
Using a larger time step may lead to divergence.
Postprocessing BuoyancyDriven Flows
The postprocessing reports of interest for buoyancydriven flows are the same as for other heat transfer calculations. See Section 13.2.4 for details.