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Inlet boundary condition in SIMPLE

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Old   October 18, 2010, 07:28
Default Inlet boundary condition in SIMPLE
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Tobias Elmøe
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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 w-velocity component in a cubic control volume is (I'm using central difference scheme to evaluate the gradients):
\mu A \left[ \frac{w_E-w_P}{\delta}- \frac{w_P-w_W}{\delta}+ \frac{w_N-w_P}{\delta}- \frac{w_P-w_S}{\delta}+ \frac{w_T-w_P}{\delta}- \frac{\partial w}{\partial z}_b \right]= -S_w-S_p w_p
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 S_w=-\frac{\partial p}{\partial z}_b \Delta V?
My first thought is to use linear extrapolation to determine the pressure outside the domain. Using central differences, I then arrive to:
S_w = -\frac{\partial p}{\partial z}_b \Delta V \approx -\frac{p_T-p_P}{\delta}\Delta V
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 w_0 is known. I therefore approximate the remaining partial derivative as:
\frac{\partial w}{\partial z}_b \approx \frac{w_P-w_0}{0.5\delta}.

This approach would correspond to setting a_B=0, s_P = -\frac{\mu A}{0.5\delta} and S_w = -\frac{p_T-p_P}{\delta}\Delta V+\frac{\mu A}{0.5\delta}w_0.

Looking forward for any input

Regards,
Johnhelt
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Old   October 18, 2010, 10:32
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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

-\frac{\partial{p}}{\partial{z}}\Delta{V} = -\frac{p_t - p_P}{\Delta{z}}\Delta{V}?

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

\int_w^e\,\frac{\partial{u}}{\partial{x}} = \frac{u_e - u_w}{\Delta{x}}

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.
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Old   October 18, 2010, 11:29
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Tobias Elmøe
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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 \delta, so then the pressure gradient at P, following your notation, should then be:

-\frac{\partial p}{\partial z}\Delta V = -\frac{p_t-p_P}{0.5\delta}
However, the pressure at the 't' face is not known (I'm using collocated variables). You can find it by interpolation p_t = \frac{p_P+p_T}{2}, and therefore you end up with -\frac{\partial p}{\partial z}\Delta V = -\frac{p_T-p_P}{\delta}, 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:

\int_S \left( \frac{\partial w}{\partial z} \vec{k}\right)\cdot \vec{n} dS

where \vec{k} is the z-directional vector and \vec{n} is the surface normal vector. Carrying out the dot product and integrating you get:

A\left(\frac{\partial w}{\partial z}_t - \frac{\partial w}{\partial z}_b\right)

so I need to evaluate both gradients at the two surfaces, which can be done by central differences. However, at the bottom surface only w_b is known w_b = w_0, while the velocity at B is unknown (outside of domain). When using linear interpolation the gradient at b \frac{\partial w}{\partial z}_b of course ends up being identical to that between b and P.

Regards!
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Old   October 18, 2010, 15:53
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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.
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Old   October 19, 2010, 11:26
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Tobias Elmøe
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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 \left(S_w = 0\right), as well as velocity gradient at the boundary \left(\frac{\partial w}{\partial z}_b = 0\right) to 0 both, while specifying the velocity at the "bottom" face in the pressure-correction equation to the inlet velocity w_0, 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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