modify laplacian: div(kgrad(au))

gunnar2 · August 5, 2011, 11:42

Hello,
I am trying to solve a type of heat equation:

$\partial u/ \partial t + \nabla\cdot\mathbf{v} u - \nabla\cdot k \nabla \alpha u =0$

The difference to the standard heat equation is the scalar field alpha.
The problem is that the last term cannot be expressed as fvm::laplacian(k,alpha*u), because apparently alpha*u is not the right class for this argument, and the compiler fails.

How can this PDE be solved?

Thanks
Gunnar

ata · August 6, 2011, 05:00

Hi
I think you can define a new field newfield=alpha*u and use fvc::laplacian(k,new)
good luck

gunnar2 · August 6, 2011, 06:11

But I need the implicit (fvm::laplacian), not the explicit (fvc::laplacian) term.

If there is no such term, it should be possible to use something like fvm::laplacian(k,u)*diag(alpha) for $\nabla\cdot k\nabla \alpha u$ .
diag(alpha) would be a sparse diagonal matrix with the entries of the field alpha on it's diagonal, but I don't think such matrix exists in Openfoam.
What do you think, does it make sense to implement it?

Thanks
Gunnar

ata · August 7, 2011, 05:10

Hi
I do not know

roth · August 7, 2011, 11:42

How about a little chain rule? i.e. split into:

Code:

\nabla \cdot ( k \alpha \nabla u ) + \nabla \cdot ( k u \nabla \alpha )

Then treat the first term implicit (by defining a new diffusivity of k\alpha) and second term explicit.

ata · August 7, 2011, 12:29

Hi
But you loose conservative form.

santiagomarquezd · August 8, 2011, 08:43

First of all the equation appears to be not consistent, the first and last terms are vectorial, meanwhile the second one is scalar.

Regards.

ziemowitzima · August 21, 2011, 14:53

Hi Gunnar,
If the only think you need is modification of the discretization matrix (by multiplying diag(alpha)), then you can change the matrix itself.
See the PISO algorithm from icoFoam:
http://openfoamwiki.net/index.php/IcoFoam
You can do similar and:
1. discretize alpha
2. UEqn.A() = UEqn.A()*alpha_discretized

maybe will work...
ZMM

kaifu · August 21, 2011, 16:44

$\nabla\cdot (k \nabla \alpha \mathbf{u}) =\nabla \cdot (k\alpha\nabla \mathbf{u})+\nabla\cdot (k \mathbf{u} \nabla \alpha)=\nabla \cdot (k\alpha\nabla \mathbf{u})+(\nabla \cdot k \mathbf{u})\nabla \alpha$

ziemowitzima · August 21, 2011, 17:36

Hi Kai,
Of course,
I would proceed using this transformation.
Currently I have some issues with laplacian operator in OFoam as well.
Could you see my post:
http://www.cfd-online.com/Forums/ope...n-problem.html

Maybe you will have some suggestions...
Thanks
ZMM

August 5, 2011, 11:42	*modify laplacian: div(kgrad(au))*	#1
gunnar2 New Member Join Date: Aug 2011 Posts: 6 Rep Power: 14	Hello, I am trying to solve a type of heat equation: $\partial u/ \partial t + \nabla\cdot\mathbf{v} u - \nabla\cdot k \nabla \alpha u =0$ The difference to the standard heat equation is the scalar field alpha. The problem is that the last term cannot be expressed as fvm::laplacian(k,alphau), because apparently alphau is not the right class for this argument, and the compiler fails. How can this PDE be solved? Thanks Gunnar Last edited by gunnar2; August 8, 2011 at 13:26. Reason: correction: replaced [math]\nabla\cdot\phi u[/math] with [math]\nabla\cdot\mathbf{v} u[/math]

August 7, 2011, 11:42		#5
roth Member Michael Roth Join Date: Mar 2009 Location: Guelph, Ontario, Canada Posts: 50 Rep Power: 17	How about a little chain rule? i.e. split into: Code: \nabla \cdot ( k \alpha \nabla u ) + \nabla \cdot ( k u \nabla \alpha ) Then treat the first term implicit (by defining a new diffusivity of k\alpha) and second term explicit.

August 8, 2011, 08:43		#7
santiagomarquezd Senior Member Santiago Marquez Damian Join Date: Aug 2009 Location: Santa Fe, Santa Fe, Argentina Posts: 452 Rep Power: 23	First of all the equation appears to be not consistent, the first and last terms are vectorial, meanwhile the second one is scalar. Regards. __________________ Santiago MÁRQUEZ DAMIÁN, Ph.D. Research Scientist Research Center for Computational Methods (CIMEC) - CONICET/UNL Tel: 54-342-4511594 Int. 7032 Colectora Ruta Nac. 168 / Paraje El Pozo (3000) Santa Fe - Argentina. http://www.cimec.org.ar

August 21, 2011, 16:44		#9
kaifu Member Kai Join Date: May 2010 Location: Shanghai Posts: 61 Blog Entries: 1 Rep Power: 15	$\nabla\cdot (k \nabla \alpha \mathbf{u}) =\nabla \cdot (k\alpha\nabla \mathbf{u})+\nabla\cdot (k \mathbf{u} \nabla \alpha)=\nabla \cdot (k\alpha\nabla \mathbf{u})+(\nabla \cdot k \mathbf{u})\nabla \alpha$ __________________ Kai

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August 6, 2011, 05:00		#2
ata Senior Member ata kamyabi Join Date: Aug 2009 Location: Kerman Posts: 323 Rep Power: 17	Hi I think you can define a new field newfield=alpha*u and use fvc::laplacian(k,new) good luck

August 6, 2011, 06:11		#3
gunnar2 New Member Join Date: Aug 2011 Posts: 6 Rep Power: 14	But I need the implicit (fvm::laplacian), not the explicit (fvc::laplacian) term. If there is no such term, it should be possible to use something like fvm::laplacian(k,u)*diag(alpha) for $\nabla\cdot k\nabla \alpha u$ . diag(alpha) would be a sparse diagonal matrix with the entries of the field alpha on it's diagonal, but I don't think such matrix exists in Openfoam. What do you think, does it make sense to implement it? Thanks Gunnar

August 7, 2011, 05:10		#4
ata Senior Member ata kamyabi Join Date: Aug 2009 Location: Kerman Posts: 323 Rep Power: 17	Hi I do not know

August 7, 2011, 12:29		#6
ata Senior Member ata kamyabi Join Date: Aug 2009 Location: Kerman Posts: 323 Rep Power: 17	Hi But you loose conservative form.

August 21, 2011, 14:53		#8
ziemowitzima Senior Member Mieszko Młody Join Date: Mar 2009 Location: POLAND, USA Posts: 145 Rep Power: 17	Hi Gunnar, If the only think you need is modification of the discretization matrix (by multiplying diag(alpha)), then you can change the matrix itself. See the PISO algorithm from icoFoam: http://openfoamwiki.net/index.php/IcoFoam You can do similar and: 1. discretize alpha 2. UEqn.A() = UEqn.A()*alpha_discretized maybe will work... ZMM

August 21, 2011, 17:36		#10
ziemowitzima Senior Member Mieszko Młody Join Date: Mar 2009 Location: POLAND, USA Posts: 145 Rep Power: 17	Hi Kai, Of course, I would proceed using this transformation. Currently I have some issues with laplacian operator in OFoam as well. Could you see my post: http://www.cfd-online.com/Forums/ope...n-problem.html Maybe you will have some suggestions... Thanks ZMM

modify laplacian: div(k*grad(a*u))

modify laplacian: div(kgrad(au))