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July 14, 2005, 04:15 
Pole condition in cylindrical coordinates

#1 
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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 temperaturedependent. 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 NavierStokes 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. 

July 14, 2005, 06:17 
Re: Pole condition in cylindrical coordinates

#2 
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Try dicretezing using finite volume. One face lying in the pole is inifneitly thin and got no surface, so the terms drop out. Use the BC condition PHI(angle)=PHI(angle+PI), I think is written in the Mosihini paper as well. Then compute a velocity at the pole in x,y (singel valued) and then convert it to r,angle in each node/cell. This will introudce a small error of nonconservation which can be importnat if you use higher order accuarcy but it should work fine if you use second order even for LES and compressibel flow.


July 15, 2005, 03:17 
Re: Pole condition in cylindrical coordinates

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U will face the singularity problem on center only for radial momentum equation (provided if ur grid is staggerred). The simple way to deal with it is that u dont discretize this rmomentum eqn at the center...rather, obtain the radial velocity at the center by interpolation using the adjacent cells. This the way wat I implemented in my code n it is working fine... The way u r dealing with the singalurity problem should produce more accurate result as claimed my the paper u mentioned...but I dont know y r u getting wrong results...I again guess it is bcoz of some discretization/implementation error...
However, I am observing relatively high value of divergence at the boundary cells where the ghost cell velues are in use....if u have any idea to get rid of it, please let me know. 

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