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February 23, 2016, 07:55 
Nondiffusive inflow boundary condition for transport equation

#1 
Senior Member
Join Date: Oct 2013
Posts: 381
Rep Power: 8 
I have implemented a boundary condition for the velocity field that acts as an inflow of a gas which I want to track with a transport equation. I don't use different thermophysical properties for it so MULES etc is not needed right now. The transport equation uses the mass fraction Y of the additional gas as primary variable which is solved for. It looks like this:
fvm::ddt(rho,Y)+fvm::div(rho*U,Y)fvm::laplacian(turb>muEff(), Y)=0 As boundary for Y I'm using fixedValue with uniform 1 as value because the inflow consists purely of this gas. However, this leads to an additional diffusive flux because q =  turb>muEff()*grad(Y) is nonzero. I need to find a way to have a diffusive flux which is zero at the boundary. Let's look at ways to solve this:
I would be happy to have a discussion about this and similar topics. Edit: I have done some tests with a fixedValue=1 boundary, solving only the transport equation without flow and found that it is best to use an additional explicit corrector term in which the diffusivity is zero everywhere except for the boundary. There is still a very small error but atleast in my case it is negligible (Mass fraction is ~1e7 at the end of the simulation). Alternatively one could set the diffusivity to nonzero also in the first cell layer but the results at the end of the test are very similar and this stops diffusion between the first and second cell layer. I found it somewhat surprising that it doesn't suffice to set the diffusivity to zero in the first two cell layers to prevent incoming diffusion. If there is any interest I can show a comparison of the results. Last edited by chriss85; February 24, 2016 at 04:59. 

May 17, 2016, 10:30 
nonorthogonal correction at boundary is ignored

#2 
New Member

Hi @chriss85,
According to Jasak's PhD thesis, the nonorthogonal correction at boundary is ignored (Error Analysis and Estimation for the Finite Volume Method with Applications to Fluid Flows, 3.3.3.1). So I think just set diffusivity to zero at inlet boundary is sufficient. On the other hand, I don't agree to set diffusivity to zero at inlet boundary. Since you are using convectiondiffusion equation to model the transportation and mixing of the specified gas, diffusion is expected everywhere. The only thing you have to take care of is that Y at inlet must greater than Y at the first cell layer (maybe greater than 1 cause unbounded schemes). How do you think? And, can you post the comparison of the results? Hope you still keep the results. 

May 23, 2016, 03:37 

#3 
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Join Date: Oct 2013
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Yes I still have the results luckily, although I actually don't use them now.
The target of this comparison was to find ways to suppress diffusion in a transport equation where the boundary is set to fixedValue 1. This problem arises when there is an inflow of the transported gas. The convection term of the transport equation requires the fixedValue 1 boundary condition. At the same time one must prevent a diffusion from the wall into the fluid. In my case I couldn't have incoming diffusion when there is no inflow because I was modelling an ablation process where the medium is solid on the other side of the boundary. The best way proved to be a fully implicit diffusion term plus an implicit diffusion term in which only the cells at the boundary (and possibly first cell layer) have a nonzero diffusivity. 

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