how to apply a symmetry boundary?
 

hey guys
i am working on coding a simulation by SIMPLE method, the flow is laminar and axissymmetric, all the momentum and pressure correction equations are written in cylindrical form.
however i found it conquers error especially near the axis.the velocity derivative du/dr is not zero.

here is how i applied the symmetry boundary condition;
for all u-velocity(parallel to the symmetry axis) cells near the axis, the discrete momentum equation is:
ap*u(i,j) = ae*u(i,j+1) + aw*u(i,j-1) + an*u(i+1,j) + as*u(i-1,j) +dp/dz*r*dz*dr
then i set u(i-1,j) = u(i,j) and repeat

i found that the problem maybe caused by that the "as" coefficient is very close to zero near the axis because radius is near to zero.
any ideas about it?

or just i have any thing other wrong?

plz tell me your advice,thx

The FVM (as well as the FEM) does not need a BC at r=0 for an axisymmetric problems as it is a natural BC. You may notice that the face area there is 0, so the relevant coefficient would be 0. Using u(i-1)=u(i) [I assume i-1 is a ghost cell] will violate the true BC. Actually you do not need ghost cells either in the FVM.