I have an incompressible flow that includes quite big temperature gradients. (Air flow , Re = 50- 500) -One possibility is to use the Boussinesq Approximation. Is there any case, that this approximation may intoduce a considerable error ?

-Is there any other Method that could be applied ?

-When may I ignore buoyancy ? Any good Literature?

Thank you in advance

If your temperature gradient is very large you may introduce considerable error into the flow field by using the Boussinesq approximation. What you will have to do is include the buoyancy force into the Navier-Stokes equations. If you do need to include the buoyancy force you have to not only add it as the body force term, but also retain the div(V) term in the N-S equations. This is the variable fluid property formulation for the N-S equations.

I don't have any listing of good books on this topic but try looking in books on heat transfer for a detailed description of these types of flows. A good problem to study to understand these buoyancy driven flows is the driven cavity problem. You should be able to find lots of these problems in the journals.

If big temperature difference in the sytem, the Boussinesq Approximation will give considerable error.

Then, if you use some commerical software, you can change the option of the density (change it to the temperature dependent density) and just put g=9.81 in the source term of momentum equation. The solver will treat the buoyancy force. If you use in-house code, you'd better modify the density by yourself . It may take time.

Considering the buoyancy force, you can use the dimensionless Richardson Number (Gr/Re**2) to determine your system, if this value lower lower than 1 , then the buoyancy force can be ignored. If not, you should take account it.

Good luck!

In the next paper you can find something about the limits of applicability of the Boussinesq approximation:

Gray DD. y A. Giorgini (1978), The validity of the Boussinesq approximation for liquida and gases, Int. J. Heat Mass Transfer, 19, pp 545 551

In the following PhD works you can find some comparisons of the results obtained using B. app. and considering varible physical properties:

Lankhorst A. (1991), Laminar and Turbulent natural convection in cavities, Delft University of Technology, Netherlands

Heiss A. (1987), Numerishe und Experimentelleuntersuchungen der laminaren und turbulent konvektion in einem geschlossenen behalter, Technischen Universuitat Munchen, Germany

I´ve used extended B. app. (means nonlinear density variation in the buoyancy term in the momentum eq. and variable dynamic viscosity) for water.

Alex