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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. |

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! |

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. |

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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