Heat flux and k-e wall boundary conditions

Blob · June 5, 2009, 07:47

I have questions regarding boundary conditions (BCs) when the k-e model is used.

1. No-slip wall BC: What are the boundary conditions for the k-e equations here? I read somewhere k is zero at walls. But if either one is zero, then singularity occurs due to the $\frac{\epsilon}{k}$ and $\frac{k^2}{\epsilon}$ terms. But obviously the solver doesn't blow up or anything. So what is happening here?

2. Heat flux BC: When I specify a fixed heat flux at a wall boundary, the laminar flux equation is $q=-k \textunderscore thermal\frac{dT}{dx}$ . k_thermal is the conductivity. But since the k-e equation uses a turbulent thermal conductivity in the energy equation, what is k_thermal in this case? For analysis, I need to know $\frac{dT}{dx}=-\frac{q}{k \textunderscore thermal}$ at the boundary which is the reason I need the k_thermal.

Thanks a lot in advance for your input and thoughts.

ghorrocks · June 5, 2009, 09:08

Hi,

This is all discussed in the documentation. It is also discussed in any decent CFD textbook.

But in summary:
1) When you integrate the turbulence model to the wall then k is zero and e is undefined. That is why you cannot integrate k-e turbulence models to the wall without some special treatment of the e equation. k-w based models are good here as w id defined at the wall so this family of models can be integrated to the wall.

So rather than integrating k-e to the wall it is usual to apply wall functions. This means that you model the shear of the boundary layer outside all the complicated near-wall stuff rather than directly resolving it in your simulation. Now k and e have finite values and k-e now has a wall boundary condition.

2) The turbulence model assumes the conductivity enhancement is proportional to the viscosity enhancement. The proportionality constant is the Turbulent Prandtl Number. From this assumption wall function boundary conditions can be derived for temperature at the wall. Also for a basic introduction try http://en.wikipedia.org/wiki/Turbulent_Prandtl_number

In any CFX simulation with heat transfer you should have the wall heat flux. I think the default turbulent Prandtl Number is 0.7 so if you don't have the turbulent enhancement of conductivity in the results you can work it out from the turbulent viscosity and the Turbulent Prandtl Number.

Glenn Horrocks

Blob · June 5, 2009, 10:15

Thank you for the pointers, Glenn. I have been going through the manuals trying to piece together the k-e model/wall function/turbulent energy equation. Your comments clarify the missing links.

I'm interested in the BCs because I'm trying to do a model reduction of the governing PDEs (momentum, energy, k-e) by Galerkin projecting them onto subspaces created with the CFD data using proper orthogonal/singular value decomposition. If all works, I'm left with a few ODEs that dynamically approximate the transient flow. Incorporating the BC effects into the ODEs has been tricky with the k-e model as my laminar counterpart works beautifully while the turbulent one doesn't at all. If you are familiar with this model reduction method, please let me know. I'll do a separate post on this as well.

ghorrocks · June 6, 2009, 08:42

Hi,

In that case I think you need a basic textbook in turbulence modelling. "Turbulence modelling for CFD" by Wilcox is my standard text and comes highly recommended (although he does have a very strong bias to the k-w turbulence model but we can forgive him for that because he developed most of it)

Glenn Horrocks

Blob · June 8, 2009, 06:10

Great! Thanks a lot!

June 5, 2009, 07:47	Heat flux and k-e wall boundary conditions	#1
Blob New Member Join Date: Mar 2009 Posts: 8 Rep Power: 17	I have questions regarding boundary conditions (BCs) when the k-e model is used. 1. No-slip wall BC: What are the boundary conditions for the k-e equations here? I read somewhere k is zero at walls. But if either one is zero, then singularity occurs due to the $\frac{\epsilon}{k}$ and $\frac{k^2}{\epsilon}$ terms. But obviously the solver doesn't blow up or anything. So what is happening here? 2. Heat flux BC: When I specify a fixed heat flux at a wall boundary, the laminar flux equation is $q=-k \textunderscore thermal\frac{dT}{dx}$ . k_thermal is the conductivity. But since the k-e equation uses a turbulent thermal conductivity in the energy equation, what is k_thermal in this case? For analysis, I need to know $\frac{dT}{dx}=-\frac{q}{k \textunderscore thermal}$ at the boundary which is the reason I need the k_thermal. Thanks a lot in advance for your input and thoughts.

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June 5, 2009, 09:08		#2
ghorrocks Super Moderator Glenn Horrocks Join Date: Mar 2009 Location: Sydney, Australia Posts: 17,871 Rep Power: 144	Hi, This is all discussed in the documentation. It is also discussed in any decent CFD textbook. But in summary: 1) When you integrate the turbulence model to the wall then k is zero and e is undefined. That is why you cannot integrate k-e turbulence models to the wall without some special treatment of the e equation. k-w based models are good here as w id defined at the wall so this family of models can be integrated to the wall. So rather than integrating k-e to the wall it is usual to apply wall functions. This means that you model the shear of the boundary layer outside all the complicated near-wall stuff rather than directly resolving it in your simulation. Now k and e have finite values and k-e now has a wall boundary condition. 2) The turbulence model assumes the conductivity enhancement is proportional to the viscosity enhancement. The proportionality constant is the Turbulent Prandtl Number. From this assumption wall function boundary conditions can be derived for temperature at the wall. Also for a basic introduction try http://en.wikipedia.org/wiki/Turbulent_Prandtl_number In any CFX simulation with heat transfer you should have the wall heat flux. I think the default turbulent Prandtl Number is 0.7 so if you don't have the turbulent enhancement of conductivity in the results you can work it out from the turbulent viscosity and the Turbulent Prandtl Number. Glenn Horrocks

June 5, 2009, 10:15		#3
Blob New Member Join Date: Mar 2009 Posts: 8 Rep Power: 17	Thank you for the pointers, Glenn. I have been going through the manuals trying to piece together the k-e model/wall function/turbulent energy equation. Your comments clarify the missing links. I'm interested in the BCs because I'm trying to do a model reduction of the governing PDEs (momentum, energy, k-e) by Galerkin projecting them onto subspaces created with the CFD data using proper orthogonal/singular value decomposition. If all works, I'm left with a few ODEs that dynamically approximate the transient flow. Incorporating the BC effects into the ODEs has been tricky with the k-e model as my laminar counterpart works beautifully while the turbulent one doesn't at all. If you are familiar with this model reduction method, please let me know. I'll do a separate post on this as well.

June 6, 2009, 08:42		#4
ghorrocks Super Moderator Glenn Horrocks Join Date: Mar 2009 Location: Sydney, Australia Posts: 17,871 Rep Power: 144	Hi, In that case I think you need a basic textbook in turbulence modelling. "Turbulence modelling for CFD" by Wilcox is my standard text and comes highly recommended (although he does have a very strong bias to the k-w turbulence model but we can forgive him for that because he developed most of it) Glenn Horrocks

June 8, 2009, 06:10		#5
Blob New Member Join Date: Mar 2009 Posts: 8 Rep Power: 17	Great! Thanks a lot!