FVM for unsteady diffusion with convective BC.
I am working on a 3D cylindrical code for unsteady diffusion equation using FVM. When I am applying the Dirichlet BC and Neumann BC, I am getting grid independence however when I am applying the convective BC, the solution is changing with the refinement of grid? Can anybody explain this behavior and also how to apply the convective BC for FVM. I think I may be applying the convective BC incorrectly but then the solution should be erratic, what I am getting seems physically possible but then refinement of grid makes it spread more b/w the initial temp., To and ambient temp, Ta.
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You mean you have convective term in purely diffusion equation? Or do you have a transport equation and it also have convective part. |
its a pure diffusion equation with a convective boundary condition.
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you cannot use a convective BC.s (like d/dn = 0) in the pure diffusion problem!
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df/dn = q is a Neumann bc and can be used in the pure diffusion problem provided that q is the known physical flux. However, that is not denoted as "convective bc"
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q is the RHS (divided by k) of your bc and must be known.
This is a non homogeneous neumann bc, not a robin one |
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-k(dt/dn)=h(T-T) |
Have a look at
An introduction to computational fluid dynamics. The finite volume method. H. K. VERSTEEG and W. MALALASEKERA. |
in a 1D example, the equation is
dT/dt = d/dx(k*dT/dx) with the BC: dT/dn=-h(Thttp://www.cfd-online.com/Forums/vbL...99a1c854-1.gif-Thttp://www.cfd-online.com/Forums/vbL...b0bddec4-1.gif)/k =q in a FV method, when you integrate the equation over each FV of measure h, you get dT_av/dt = (k*dT/dx|e - k*dT/dx|w)/h where |e and |w are the fluxes location at the faces of the FV. Thus, when the equation has a face coincident to the boundary, you just substitute the known flux q. Note that to produce a finite solution for any time, the Neumann BC.s must fulfill a compatibility condition otherwise dT_av/dt -> +Inf |
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