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May 20, 2016, 04:29 
FVM for unsteady diffusion with convective BC.

#1 
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Abhi
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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.


May 20, 2016, 04:44 

#2  
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Arjun
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Quote:
You mean you have convective term in purely diffusion equation? Or do you have a transport equation and it also have convective part. 

May 20, 2016, 04:49 

#3 
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Abhi
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its a pure diffusion equation with a convective boundary condition.


May 20, 2016, 04:52 

#4 
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Filippo Maria Denaro
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you cannot use a convective BC.s (like d/dn = 0) in the pure diffusion problem!


May 20, 2016, 04:59 

#5  
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Abhi
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Quote:
k(T/n)=h(TsT_) 

May 20, 2016, 05:29 

#6 
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Filippo Maria Denaro
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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"


May 20, 2016, 05:32 

#7 
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Abhi
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I have mentioned what I meant by convective or mixed type BC. What you wrote is a a fixed flux (i.e neumann type BC) Can u plz elaborate on the application of the mixed type (mentioned in my above comment) BC for unsteady diffusion eqn using FVM.


May 20, 2016, 05:47 

#8 
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Filippo Maria Denaro
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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 

May 20, 2016, 05:56 

#9  
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Abhi
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Quote:
k(dt/dn)=h(TT) 

May 20, 2016, 06:09 

#10 
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Filippo Maria Denaro
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Have a look at
An introduction to computational fluid dynamics. The finite volume method. H. K. VERSTEEG and W. MALALASEKERA. 

May 20, 2016, 09:25 

#11 
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Filippo Maria Denaro
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in a 1D example, the equation is
dT/dt = d/dx(k*dT/dx) with the BC: dT/dn=h(TT)/k =q in a FV method, when you integrate the equation over each FV of measure h, you get dT_av/dt = (k*dT/dxe  k*dT/dxw)/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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