Inlet boundary condition in SIMPLE

johnhelt · October 18, 2010, 08:28

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):
$\mu A \left[ \frac{w_E-w_P}{\delta}- \frac{w_P-w_W}{\delta}+ \frac{w_N-w_P}{\delta}- \frac{w_P-w_S}{\delta}+ \frac{w_T-w_P}{\delta}- \frac{\partial w}{\partial z}_b \right]$

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

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

, $s_P = -\frac{\mu A}{0.5\delta}$ 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

lost.identity · October 18, 2010, 11:32

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.

johnhelt · October 18, 2010, 12:29

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

is known

, 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!

lost.identity · October 18, 2010, 16:53

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.

johnhelt · October 19, 2010, 12:26

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

, 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

October 18, 2010, 08:28	Inlet boundary condition in SIMPLE	#1
johnhelt New Member Tobias Elmøe Join Date: Oct 2010 Location: Denmark Posts: 9 Rep Power: 16	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): $\mu A \left[ \frac{w_E-w_P}{\delta}- \frac{w_P-w_W}{\delta}+ \frac{w_N-w_P}{\delta}- \frac{w_P-w_S}{\delta}+ \frac{w_T-w_P}{\delta}- \frac{\partial w}{\partial z}_b \right]$ 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 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 , $s_P = -\frac{\mu A}{0.5\delta}$ 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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October 18, 2010, 11:32		#2
lost.identity Senior Member Join Date: Apr 2009 Posts: 118 Rep Power: 17	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.

October 18, 2010, 12:29		#3
johnhelt New Member Tobias Elmøe Join Date: Oct 2010 Location: Denmark Posts: 9 Rep Power: 16	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 is known , 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!

October 18, 2010, 16:53		#4
lost.identity Senior Member Join Date: Apr 2009 Posts: 118 Rep Power: 17	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, 12:26		#5
johnhelt New Member Tobias Elmøe Join Date: Oct 2010 Location: Denmark Posts: 9 Rep Power: 16	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 , 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