# Inlet boundary condition in SIMPLE

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 October 18, 2010, 07:28 Inlet boundary condition in SIMPLE #1 New Member   Tobias Elmøe Join Date: Oct 2010 Location: Denmark Posts: 9 Rep Power: 8 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 w-velocity 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 pressure-gradient 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 Senior Member   Join Date: Apr 2009 Posts: 118 Rep Power: 9 Hi Johnhelt, I'm not an expert on CFD or SIMPLE-like 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 1-D 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 half-cell. So if you look at the control volume along the z-direction you'd have P----t-----T 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 x-direction W---w---P---e---E Now if you are looking at the boundary node P at the left-hand-side (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 New Member   Tobias Elmøe Join Date: Oct 2010 Location: Denmark Posts: 9 Rep Power: 8 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: B----b----P----t----T 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 z-directional 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 Senior Member   Join Date: Apr 2009 Posts: 118 Rep Power: 9 Hi, Sorry I made a mistake in my integral, missed out the dx and also I was only thinking of 1-D in it. I think that's how I would do it too if I were you.

 October 19, 2010, 11:26 #5 New Member   Tobias Elmøe Join Date: Oct 2010 Location: Denmark Posts: 9 Rep Power: 8 Ok a follow-up for those who care... I'm solving the Navier-Stokes 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 pressure-gradient , as well as velocity gradient at the boundary to 0 both, while specifying the velocity at the "bottom" face in the pressure-correction 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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