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October 18, 2010, 07:28 
Inlet boundary condition in SIMPLE

#1 
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Tobias Elmøe
Join Date: Oct 2010
Location: Denmark
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Dear all. I'm currently writing my own FVM code, to be solved by the SIMPLE method. I've run into some issues in defining my boundary conditions that I hope you can help me clarify. Pressure and velocity are collocated.
Say the discretized momentum equation for the wvelocity component in a cubic control volume is (I'm using central difference scheme to evaluate the gradients): The partial derivative at the "bottom" face has on purpose not been approximated yet. I have two questions both with ideas that I hope can produce some comments. First, how do I approximate the pressuregradient source at the inlet ? My first thought is to use linear extrapolation to determine the pressure outside the domain. Using central differences, I then arrive to: Should I use higher order extrapolation, or will this suffice? Second question, how do I approximate the velocity gradient? I would specify my velocity there as the inlet velocity is known. I therefore approximate the remaining partial derivative as: . This approach would correspond to setting , and . Looking forward for any input Regards, Johnhelt 

October 18, 2010, 10:32 

#2 
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Join Date: Apr 2009
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Hi Johnhelt,
I'm not an expert on CFD or SIMPLElike algorithms, but for the past 9 months or so I've been modifying a code written for SIMPLER algorithm and have had to do some discretisations (for a 1D grid though). I'm assuming that P here refers to the boundary grid point on the bottom surface. Also the 0.5 in your velocity gradient comes from the fact that you're looking at a halfcell. So if you look at the control volume along the zdirection you'd have PtT where t is the face of the CV. So shouldn't the pressure gradient term be ? similarly for the other gradient terms because you integrate between the face of the CV. So for the grid along the xdirection WwPeE Now if you are looking at the boundary node P at the lefthandside (LHS) you do not have u_w and u_w can be taken as the value of u_P, which is what you've done when you took w_0 for w_B. 

October 18, 2010, 11:29 

#3 
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Tobias Elmøe
Join Date: Oct 2010
Location: Denmark
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Hi Lost Identity, and thanks for your fast reply. I think I forgot to mention some nomenclature..
The distance between two nodal points is , so then the pressure gradient at P, following your notation, should then be: However, the pressure at the 't' face is not known (I'm using collocated variables). You can find it by interpolation , and therefore you end up with , which is the same as I wrote. So a boost of confidence there :) I'm not sure I get your integration though. At the "bottom" face: BbPtT after replacing volume integrals with surface ones by the Gauss theorem, the finite volume discretization at the b and t surface gives: where is the zdirectional vector and is the surface normal vector. Carrying out the dot product and integrating you get: so I need to evaluate both gradients at the two surfaces, which can be done by central differences. However, at the bottom surface only is known , while the velocity at B is unknown (outside of domain). When using linear interpolation the gradient at b of course ends up being identical to that between b and P. Regards! 

October 18, 2010, 15:53 

#4 
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Hi,
Sorry I made a mistake in my integral, missed out the dx and also I was only thinking of 1D in it. I think that's how I would do it too if I were you. 

October 19, 2010, 11:26 

#5 
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Tobias Elmøe
Join Date: Oct 2010
Location: Denmark
Posts: 9
Rep Power: 8 
Ok a followup for those who care...
I'm solving the NavierStokes equation for incompressible (laminar) flow to a hole in a plate  and inside that hole as well, where a parabolic flow profile should form. Later I will apply a permeability term to look at the effect of deposition from the flow (filtration). I have implemented the solution using the SIMPLE method with on a structured grid. The whole thing is solved using the preconditioned conjugate gradient method. With regards to the boundary conditions: If I use the approach I wrote in the original post, I get oscillations in the velocity near the inlet in the converged solution. These oscillations, however, die out a few grid points away from the inlet. Instead, I found that specifying the pressuregradient , as well as velocity gradient at the boundary to 0 both, while specifying the velocity at the "bottom" face in the pressurecorrection equation to the inlet velocity , I get the correct velocity field  without oscillations near the inlet. I guess it corresponds to specifying the velocity far from the domain, however I could not find any good explanation to this in Versteeg's book.. perhaps someone here can explain? Regards, JH 

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