Biharmonic Viscosity with Smagorinsky model
Dear friends:
I'm working on a nearshore Boussinesq wave model. Currently I have a problem with numerical stability for some cases. In order to damp out small scale noises, I want to use biharmonic damping term with an artificial viscosity given in light of Smagorinsky (1963) and (1993). While I'm trying to implement it on a curvilinear coordinate, can anyone point me to a reference of biharmonic term calculation in general coordinates? Since it appears to me that the biharmonic operator \nabla^4 in nonorthogornal coordinates is quite complicated. Thanks, Wen 
Re: Biharmonic Viscosity with Smagorinsky model
If you introduce a biharmonic operator, you are changing the order of the PDEs. You would need additional boundary conditions.
If the biharmonic term is merely an artifact that the numerical equations need, please make sure that the term zeros out near the boundaries. The Smagorinsky viscosity (I guess you are refering to the LES model) does not (and does need to) go to zero at the boundaries. You may find the discrete biharmonic operator on general coordinates in papers involving transonic flow solvers. 
Re: Biharmonic Viscosity with Smagorinsky model
Dear Kalyan:
Thank you very much. I wasn't aware that I'll have to zero out the viscosity near boundaries. That could be hard if my domain has arbitrarily shaped islands in side. Yes my biharmonic damping will be only artificial part. Many people do Smagorinsky closure in LES and Large Scale Ocean Turbulence, but papers I got are too concise to gain much out of them. You mentioned transonic flow solvers, can you point it to me more clearly? I'll try to find sth in my lib. Thanks again, Wen 
Re: Biharmonic Viscosity with Smagorinsky model
coef *( Q_{i2}4 Q_{i1} +6 Q_i +4 Q_{i+1} +Q_{i+2}) for equal sized grid. But coef is quite empirical.

Re: Biharmonic Viscosity with Smagorinsky model
Sorry , there should be "" before "4".

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