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October 6, 2021, 16:28 
FEM for INS in Polar Coordinates on disk

#1 
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Jonas T. Holdeman, Jr.
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FEM references I have seen for INS in polar coordinates seem to be limited to an annulus which excludes the origin. Could anyone give some (accessible) FEM references for incompressible fluid flow in polar coordinates on a disk which includes the origin using the finite element method? The origin is a singular point of the coordinate system, but a regular point in physical coordinates. Problem examples might be uniform flow (as in the attachment), or driven (full) circular cavity. I have seen papers with circular cavity using spectral elements on the full disk but not finite elements. I am interested in how the origin is treated. The figures in the attachment were created using the FEM in polar coordinates with Dirichlet conditions on the outer boundary and one condition on the radial flow at the origin (necessitated by a problem with the radial basis function at the polar origin in my formulation). I used periodic BC on theta (joined along the negative xaxis) and all polar nodes at the origin mapped to a single node. Fig. (a) & (b) are vector plots on polar and Cartesian grids and Fig. (c) & (d) are stream function plots.


October 6, 2021, 16:44 

#2 
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Filippo Maria Denaro
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I can give my experience only in terms of FVM, considering that FVM is a special case of FEM.
I simply worked by building small cylinders with the axis at r=0 and N finite faces around it (r=dr, theta=0...2pi). Two closing surfaces are at z+dz/2 and zdz/2. This way I had no problem in the origin. I don't know if that can useful to be expressed in terms of general FEM. 

October 11, 2021, 14:01 

#3 
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Jonas T. Holdeman, Jr.
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No direct responses. perhaps this is the wrong forum to ask my request for references to handle coordinate singularities in this particular coordinate system.
Filippo: Thank you for your response. It seems that one approach is to circumvent the problem rather than attacking it directly. I understand that an approach in the global circulation (weather) problem is to use a spherical coordinate mesh but with a different mesh system on caps placed at the poles. I think your FVM approach might be a special case of this. Actually, I think a direct approach is fascinating. Any nonzero radial velocity component at the origin would seem to imply a point source or sink, i.e. a Dirac delta function. But the boundary condition is singularly discontinuous. For a continuous physical vector field, we have at the origin U_r(0,theta)= U_r(0,theta+pi) in polar coordinates. What looks like a source on one side of the origin looks like a sink on the other side. There is a cancellation of the sources over the support of the node to produce a continuous field. Is this some kind of 2D Gibbs phenomena? I use Hermitetype elements, with stream function and divfree velocity components as degreesoffreedom, which elements are the curl of a stream function element. The curl operator in PoCo introduces premultiplication of the derivatives by the matrix [r^1 , 0; 0, 1] (in Matlab notation) where r>0. My formulation requires post multiplication of the basis functions by the inverse of this evaluated at the basis node to preserve the divfree property under (affine) transformation. The The elements map smoothly to the origin r=0 (as can be seen from the figure) but the postmultiplication forces the radial shape function to zero. This results in a zero row and column in the assembled matrix. Hence I used an additional Dirichlet condition in the figure. Still open to suggestions as well as references. 

October 13, 2021, 13:57 

#4 
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I think your problem stems only from the fact that you do not think in general coordinates. Solving a transformed PDE is equivalent to solving the original PDE on an arbitrary (curved) mesh. So simply reinterpret your question in:
How to solve the incompressible NavierStokes equations on a polar like mesh? One thing that immediately gets my attention here is the fact that you have to use a different element type for the discretization at the singularity, namely triangles instead of quadriterals. Are you familiar with the collapsed coordinate transformation used in the spectral and finite element community? Regards 

Tags 
fem, incompressible flow, polar coordinates, singularity 
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