Fourier number in fluid mechanics
 

Hello everyone,

recently I noticed about the use of the Fourier number for heat transfer, which is

and it should be taken into account to set the proper time step when heat transfer is present. Have you ever used it? I have been told that it should be taken into account in natural convection, specially at low Prandtl number, it is, when your thermal diffusivity alpha increases.

This number appears only after a space-time discretization (in your example 1D) is performed. It makes sense for the analysis of the numerical stability constraint when explicit schemes are used. For example, a first order in time and second order in space (FTCS) scheme leads to fix a typical stability value <=1/2.

Yes it should be considered & it's super important for heat diffusion dominated problems.

In transient conjugate heat transfer problems for example, the Fourier number can limit your time-step much more than your convective Courant number (because alpha in solids is large).

Thanks for your answers. Most of the times, when I see this number, there is conjugate heat transfer. This number, should it be taken into account for single-phase heat transfer as well?

Yes! Again, this number appears only after a numerical discretization of the parabolic heat transfer is introduced. The physics dictates the value of alpha but dt and dx are of numerical character.
See Chap.2 http://mathcenter.hust.edu.cn/Upload...98740d59f8.pdf

Yes, it should be! However, usually the diffusion time-scale is small compared to convective time-scale in a fluid. For example, your cell Peclet number is usually more than 1 (and that is why you use upwind schemes). So usually your Courant number is usually limiting and not the Fourier number. However, I don't really have a feel for Fo vs Pe vs Co in natural convection. You should check to make sure. You are also right, for low Prandtl numbers the Fourier number becomes more important.