Dear all
I want to compute vorticity in axisymmetric flow. The components are as follows:
what to do to overcome singularity for axial vorticity at r=0 in FF++?
BR
what to do to overcome singularity for axial vorticity at r=0 in FF++?
Bonjour Mademoiselle,
In axisymmetric coordinates, the singularity at r=0 is not a true problem if you use the weak form (i.e variationnal if the operator is symmetric).
The tricks is the following :
- considere that the FF++ coordinates (x,y) are the axisymmetric coordinates (r,z)
- so, the differential element (Lebesgue’s measure) which is dV(r,q,z) = r dr dq fz (q = theta) becomes x dx dy dq
- now , integrate over q
- then, in these conditions, in the meridian plane, dV(r,z) = 2 pi dr dv, i.e., in FF++ coordinnates, dv(r,z) = x dx dy
Morality, you just must ponderate the integrands by x. Is the same to work in weighted Sobolev spaces.
For exemple, the weak form of the Laplace operator is :
int2d(Th) ( (dx(uh)*dx(vh) + dy(uh)*dy(vh))*x )
Please, note the presence of the weighting function (x,y) → x.
Very good and simple presentation of the axisymmetric formulation of FEM can be find in Krizek and Neittaanmaki, Finite Elements Approximation of Variationnal Problems and Applications, Pittman (1990). A more mathematically complex one in Bernardi, Dauge and Maday, Spectral Methods for Axisymmetric Domains, RMA 4, Gauthier-Villard (1999).
Je vous souhaite une bonne journée.