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May 13, 2012, 18:08 
Pressure correction in fractional step method

#1 
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Peter
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Can someone explain to me how the pressure correction step works in cylindrical coordinates? I have been working on a code for a while and my pressure correction step is not producing a divergence free field.


May 14, 2012, 03:44 

#2 
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Filippo Maria Denaro
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did you check if your method is divergencefree in Cartesian coordinates? fractional step is a method to split pressure and velocity in the time integration, the system of coordinate is not involved


May 14, 2012, 10:18 

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How does the divergence change before and after the correction step?
Is the divergence localized in certain regions after the correction step? How accurately is your pressure equation being solved? Are you using an axisymmetric cylindrical grid, is weird stuff happening at the axis of symmetry? 

May 14, 2012, 11:11 

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Quote:
Quote:
I initially start with a velocity profile of 1 everywhere in the field. When integrated it produces a value of 1. I expect that at any time, if I integrate the velocity profile, I should also get a value of 1. However as my flow progresses to steady state, I get values up to 1.2 

May 14, 2012, 11:19 

#5 
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Filippo Maria Denaro
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Are you using an exact or approximate projection method?
You must be careful in the innerouter product of discretized divergencegradient operators 

May 14, 2012, 11:38 

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May 14, 2012, 11:43 

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Filippo Maria Denaro
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The pressure equation must be discretized from the Div Grad (*) operators, you should not use the Laplace operator. This allows you to substitute the correct BC in terms of the normal component of the pressure gradient


May 14, 2012, 12:01 

#8 
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Peter
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So instead of solving for the pressure in one step, could I split it into two steps? the first step would be multiplying the divergence of the intermediate velocity (not divergent free) by the inverse of the discretized divergence matrix. Then multiply that result by the inverse of the discretized gradient matrix.


May 14, 2012, 13:24 

#9 
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Peter
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Also are there any books or papers that I can look up that talk about how to do this?


May 16, 2012, 12:18 

#10 
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Peter
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for those interested, I found this paper which essentially talks about my problem. Unfortunately I still have not found the bug in my code. I expect my code to produce a poiseuielle like profile that peaks at 1.5 however my code peaks at ~1.8 and is not divergence free.
If anyone has seen similar problems, please let me know. thanks http://www.sciencedirect.com/science...45793004001148 

May 16, 2012, 17:09 

#11  
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Arjun
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How do you arrive at this equation?? If it is a pressure correction equation then it should have a 'deltat' term somewhere!!! 

May 16, 2012, 17:16 

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Filippo Maria Denaro
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May 16, 2012, 17:24 

#13  
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Peter
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I didn't put in the time because it cancels once applied in the navier stokes equations. I have checked and I don't think that is the problem. 

May 16, 2012, 17:29 

#14 
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Filippo Maria Denaro
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is not a problem if you then compute
Vn+1= V*  Grad P In any case, I am quite sure you have problems in setting pressure BCs ... you must write Div (Grad P) and substitute n.Grad P at boundaries 

May 16, 2012, 17:39 

#15 
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Peter
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I think I have accounted for that. Since my code is on a staggered grid, I'm only calculating pressure at the interior points. At the edges, I assume that the ghost point outside the boundary is the same as the first interior point. For example the second derivative in the R direction should be . Substituting for neumann bcs I get at the axis and at the wall.


May 16, 2012, 18:32 

#16  
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Arjun
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I do not have time right now to figure this out. But I will try to imbibe what you wrote. I am assuming the P in your equation is pressure correction and not the pressure itself. I asked that because I never remember deriving pressure correction in fractional step method that is independent of delta T. 

May 16, 2012, 19:21 

#17 
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You can not apply zero gradient at the axis by ghost points. Axis has pressure gradients. Yes this is the case, he is calculating for ( deltT * Pressurecorrection). 

May 17, 2012, 10:57 

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There normal component of the gradient of the pressure should be zero at the axis, if we are assuming axisymmetric flow and the fact that Ur must be zero at the axis.
What are you using for the inlet/oulet conditions? 

May 18, 2012, 11:56 

#19  
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If anyone is interested or if anyone thinks seeing the code might help, then i can post my matlab code. I'm pretty sure that this poisson equation solver is the last bug in my code. 

May 22, 2012, 00:04 

#20 
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Peter
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I think I figured out where the error is. When I formulate the finite difference matrix for the laplacian, I essentially solve Ax = b. However A is nearly singular and so my solution ends up being innacurate. How can I increase the accuracy?


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