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Streamwise periodic heat trasnfer

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Old   May 10, 2011, 07:11
Default Streamwise periodic heat trasnfer
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Robert Sawko
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Dear All,

I am trying to implement heat transfer in interFoam but coupled with periodic boundary conditions in streamwise direction.

In this case the temperature field is not periodic in any obvious way. But if I focus on a 2D channel case and impose a constant wall temperature and a bulk inlet temperature a renormalised temperature field

\theta = (T - Tw)/(Tbin - Tw)

is self-similar in the sense that if I rescale the profile at the inlet it should coincide with the profile at the outlet.

My question is: how can I do it with interFoam on a cyclic patch? Can I apply anything else than a cyclic boundary condition on this patch? The rescaling would be fairly easy if I had a direct access to values. Is there any easy way of doing it?

Something like writing your new BC applicable on cyclic patch would be great.

I would be obliged for any comments or hints . So far I've been using this thread as a guidance:
Diverging result for Temperature field in interFoam
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Old   May 10, 2011, 09:34
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I cannot help you with this problem.

But why do you want to do it like this? Normally (in literature) temperature is made periodic using a source term (I performed 3D periodic channel and pipe flows using this source). References are Kawamura, Tiselj, Bergant (channel) and Saad (pipe).

Is there a specific reason why to avoid their approach? If you want I can send you one of the references.
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Old   May 10, 2011, 10:50
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Thank you for your comment!

I have to confess that ignorance occurs to me as the only reason. To which papers do you refer exactly?

I downloaded the papers by Kawamura 1998 and Tiselj 2001. The latter one entitled "Effect of wall boundary condition on scalar transfer in a fully developed turbulent flume" talks about constant wall temperature so I am looking at this one at the moment.

I can't see exactly why \theta is to be periodic after inclusion of the source term u^+/2u^+_b in eq (4). In constant heat flux scenario it's correct but for constant wall temperature... shouldn't it be different?
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Old   May 11, 2011, 07:09
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I thought I'll post a clarification after my yesterday evening reading.

stevenvanharen: You were right about the source term. This seems to be an expedient way of doing it. But I still can't see how these references can be applied to isothermal wall condition.

But I've found this:

S. V. Patankar, C. H. Liu, and E. M. Sparrow.
Fully Developed Flow and Heat Transfer in Ducts Having Streamwise-Periodic Variations of Cross-Sectional Area.
ASME J. of Heat Transfer, 99:180-186, 1977.

which was actually my initial reading on streamwise periodic heat flow. The problem of 'self-similar' profiles is well-addressed there. The difference is that the source term for normalised temperature has to be obtained by solving another equation (no 38 in the publication).

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Old   May 18, 2011, 11:20
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I had a major breakthrough in my problem and now I can gladly report that it actually works. Not everything is sorted out yet, but it seems to be progressing. Steven was right about the source term and my initial post and idea were probably misleading. It's much easier to incorporate a source term in the equations.

My only difficulty was that I didn't know the form of the source term and I couldn't agree with the form given in papers on constant heat flux. Fortunately Patankar paper nails it down. It's really just a bit of calculus

Instaed of solving for tempearture you solve for \theta which is

\theta = (T - Tw)/(Tbx - Tw)

where Tbx is the bulk temperature in the cross section. Moreover you need to define another field lambda which is

\lambda = dTbx/dx/(Tbx - Tw)

Now you can solve the equations for \theta and \lambda and these quantites happen to be periodic.

u d\theta/dx v d\theta/dy = \kappa (d2\theta/dx2 + d2theta/dy2 ) + \sigma

Source term
\sigma = [2 \kappa d \theta/dx - u \theta] \lambda + \kappa \theta [d lambda/dx + lambda^2]

It might look a bit long-winded but it actually makes sense. The solution requires you to iteratively solve for \theta, correct so that bulk \theta is one in each cross-section and than correct \lambda.

Currently I am working on the last bit i.e. correcting lambda. But I attach some results with this post. This is interFoam with heat transfer and periodic boundaries, laminar flow.
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File Type: jpg temperatuere_10x.jpg (29.4 KB, 113 views)
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Old   December 1, 2011, 12:50
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Mohsen KiaMansouri
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Thanks to both Robert Sawko & Steven van Haren for this useful topic.

There is also the Steven van Haren M.Sc. thesis that might be helpful:
"Testing DNS capability of OpenFOAM and STAR-CCM+"

That you can download in DELFT UNIVERSITY OF TECHNOLOGY:
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Old   May 3, 2012, 06:00
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Dear Robert,

I am really interested in this thread since it has been the subject of my M. Sc. Thesis. At present, I would like to implement in OpenFOAM the model proposed by Patankar for the constant wall temperature case (maybe in the modified form of Stalio and Piller, where only the average lambda over one periodic module is computed by using the integral form of the energy conservation equation). Did you succeed in correcting lambda? Can you give me any advice to perform this task avoiding numerical instability?

Thanks in advance,

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