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FVM for unsteady diffusion with convective BC.

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Old   May 20, 2016, 04:29
Default FVM for unsteady diffusion with convective BC.
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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.
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Old   May 20, 2016, 04:44
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Quote:
Originally Posted by abhi084 View Post
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.

You mean you have convective term in purely diffusion equation? Or do you have a transport equation and it also have convective part.
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Old   May 20, 2016, 04:49
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its a pure diffusion equation with a convective boundary condition.
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Old   May 20, 2016, 04:52
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you cannot use a convective BC.s (like d/dn = 0) in the pure diffusion problem!
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Old   May 20, 2016, 04:59
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Originally Posted by FMDenaro View Post
you cannot use a convective BC.s (like d/dn = 0) in the pure diffusion problem!
Why we can't use convective BC's in Diffusion equation. If I am not wrong what you mentioned i.e d/dn=0 are the flux BC (Neumann type) which can very well be used with pure diffusion equation. What I am meaning by convective BC is more of a mixed type Boundary condition like
-k(\partialT/\partialn)=h(Ts-T_\infty)
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Old   May 20, 2016, 05:29
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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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Old   May 20, 2016, 05:32
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Originally Posted by FMDenaro View Post
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"
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.
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Old   May 20, 2016, 05:47
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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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Old   May 20, 2016, 05:56
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Originally Posted by FMDenaro View Post
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
Okay but can you suggest any literature/book about it's implementation. The direct implementation of fixed flux (ur q) is fairly straight forward whereas in case of the convective type BC (yes these type BC are called Convective type BC in heat transfer problem) I am getting problems. Your help would be highly appreciated. And can we move past the names, just can you help me in applying this BC to diffusion equation:

-k(dt/dn)=h(T_{s}-T_{\infty})
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Old   May 20, 2016, 06:09
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Have a look at
An introduction to computational fluid dynamics. The finite volume method. H. K. VERSTEEG and W. MALALASEKERA.
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Old   May 20, 2016, 09:25
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in a 1D example, the equation is

dT/dt = d/dx(k*dT/dx)
with the BC: dT/dn=-h(T-T)/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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