Pole condition in cylindrical coordinates

Roddy

I am trying to write a CFD code for incompressible, transient laminar flow of water insider a cylinder vessel subjected to microwave heating. The model is assuming that the thermophysical properties are temperature-dependent. The temperature is expected to rise from 300 K to around 400 K.

I am actually facing two problems:

1. Is the Boussinesq approximation applicable in this case? What I have understood from the literature (mostly from the Gray and Giorgini paper) that for adopting the Boussinesq approx. the temperature difference must not exceed 20 K for air and 2 K for water. Also, variable property effects in the governing equations must be neglected, except for the density where it appears in the gravitational body force terms in the momentum equations. Is it correct or not?

2. I have discretised the complete Navier-Stokes equations using finite difference method. It seems pretty fine. However, at the pole I am having a problem with discretising the Newtonian stress tensor, most precisely with the term \tau_{t\theta} which is singular at the pole. There is a formula for this term (at the pole) in the paper "Fully conservative Finite Difference Scheme in Cylindrical Coordinates for Incompressible Flow Simulations" by Morinishi, Vasilyev and Ogi (equation 37). The authors used L'Hopital's rule to remove the singularity. However, when I applied this formula to my code, the results are incorrect. I have checked the implementation of my codes several times and didn't see any programming error.

Could anyone helped me out please? I hope my explanation is clear. Thank you.