Hi,

I have written a 2D steady-state and transient coupled NS solver using FEM. I am using 8-node velocity quadrilaterals elements coupled to 4-node quadrilateral elements, and Galkerin weighting.

However, the model fails to converge for highly- convective flows, so I am trying to implement SUPG.

I am uncertain where to implement SUPG weighting. I've read (Heinrich) that I only use SUPG weighting on the convective terms on the RHS, i.e. Md. + Kd = F, where F is the body force vector [fx fy 0], and the SUPG weighting is: wi = Ni + (alpha*H)/2*abs(V) [ udNi/dx + v dNi/dy ]

Does this sound correct? Any advice we be much appreciated before I go and try to code the algorithm.

-N

From the little that I have read about FEM, I understand that SUPG is obtained when only the convective terms are weighted in that manner. When all the terms are weighted in the same way then the scheme is called Galerkin Least Squares (GLS). For purely advective systems both are of course identical. For SUPG see the thesis, A HIGHER ORDER ACCURATE FINITE ELEMENT METHOD FOR VISCOUS COMPRESSIBLE FLOWS by Daryl Lawrence Bonhaus. It is available on the net (google search) A quote from above thesis:

"Since the fundamental source of instability in the Euler and Navier-Stokes equations is the dominance of the advective terms, it is unclear that the additional complexity of the GLS method over the SU/PG method has any real benefit."

For GLS, there is a an entire book by Bo-nan Jiang, "The Least-Squares Finite Element Method", Springer 1998.

Is there any thesis available online about optimization of Francis Turbine runner or designing method of francis turbine runner thanks