# Solving Navier Stokes equations by projection method and predictor-corrector method.

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 February 9, 2017, 04:47 Solving Navier Stokes equations by projection method and predictor-corrector method. #1 Senior Member   Saideep Join Date: Apr 2015 Location: INDIA Posts: 203 Rep Power: 10 Hi guys; I am comparing two different CFD packages. OpenFoam and Gerris for studying two phase immiscible flow. According to OpenFOAM, the Navier Stokes equations are solved using a "Predictor-Corrector" step using PISO loop. In Gerris, the Navier Stokes equations are solved by a "time splitting Projection" method. Though from my understanding the summary of both the methods are the same: 1. Using an initial guess of pressure find an intermediate velocity (which considers the viscous and convective terms of present time step) field that is not mass conservative; 2. Using the non-divergent velocity field, update the pressure field by solving "Poissons equation"; 3. With known data on the correct pressure for present time step along with intermediate velocity field known update a mass conservative velocity field. However, I am unable to catch the difference between Chorin's projection method and Issa's PISO loop. Can anyone help me out? Thanks; SaiD

 February 9, 2017, 07:36 #2 Senior Member   Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,041 Rep Power: 64 Projection methods are based on the Hodge decomposition: in each time step, the intermediate velocity is projected onto the subspace of divergence-free velocity field by solving the Poisson equation. That is solved only once in each time step and is not iterated in "outer" iteration.

February 9, 2017, 08:44
#3
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Kaya Onur Dag
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Quote:
 Originally Posted by Saideep Hi guys; I am comparing two different CFD packages. OpenFoam and Gerris for studying two phase immiscible flow. According to OpenFOAM, the Navier Stokes equations are solved using a "Predictor-Corrector" step using PISO loop. In Gerris, the Navier Stokes equations are solved by a "time splitting Projection" method. Though from my understanding the summary of both the methods are the same: 1. Using an initial guess of pressure find an intermediate velocity (which considers the viscous and convective terms of present time step) field that is not mass conservative; 2. Using the non-divergent velocity field, update the pressure field by solving "Poissons equation"; 3. With known data on the correct pressure for present time step along with intermediate velocity field known update a mass conservative velocity field. However, I am unable to catch the difference between Chorin's projection method and Issa's PISO loop. Can anyone help me out? Thanks; SaiD

have a look at the red ferziger, this is covered in that book and explanations are very easy to get

 February 9, 2017, 09:14 #4 Senior Member   Saideep Join Date: Apr 2015 Location: INDIA Posts: 203 Rep Power: 10 Hi Denaro & Kaya; Thanks for your reply. PISO is an iterative approach (we specify number of pressure correction). The more corrections we make for Pressure the more accurate the divergence free flux we get. Unlike, Projection is a single step method with pressure fixed upon reaching suitable specified tolerance limit. (I am not well versed by Hodge decomposition but I understand the velocity is decomposed into a divergence free term which we are interested in and a pressure term. What I understood from Wiki) So, to conclude: Projection method = 1 PISO loop. Then, is PISO a more powerful approach to get a better conservative flux if both the methods have a fixed tolerance limit set by me. Thanks Kaya I will have a look right now. SaiD

 February 9, 2017, 11:39 #5 Senior Member   Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,041 Rep Power: 64 Well, more in details, we call Exact projection method when in a single step the divergence-free constraint is satisfied up to machine accuracy while the Approximate projection method is satisfied only up to the order of the local truncation error. However, since the Hodge decomposition is well posed setting only one boundary condition, any projection method suffers from an approximation in the setting of the tangential condition. On the other hand, PISO as well as other coupled method iterate continuity and momentum until the desired convergence is reached.

February 10, 2017, 17:31
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Bernd
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Quote:
 Originally Posted by FMDenaro while the Approximate projection method is satisfied only up to the order of the local truncation error.
As the OP mentioned the Gerris package it's worth noting that Gerris also uses an approximate pressure correction because approximate corrections can be stably used with co-located, i.e. non-staggered velocity grids which in turn allows for more efficient implementation of the adaptive, octree based grid structure.

February 10, 2017, 17:43
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Filippo Maria Denaro
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Quote:
 Originally Posted by zx-81 As the OP mentioned the Gerris package it's worth noting that Gerris also uses an approximate pressure correction because approximate corrections can be stably used with co-located, i.e. non-staggered velocity grids which in turn allows for more efficient implementation of the adaptive, octree based grid structure.
Yes, the original nature of the APM is for colocated grid with compact stencil for the pressure equation but it can be extended easily to general unstructured grids. However, an EPM can be implemented on non-staggered grids, too.

February 10, 2017, 18:10
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Bernd
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 Originally Posted by FMDenaro However, an EPM can be implemented on non-staggered grids, too.
That sounds interesting as it was my impression that any non-staggered grid scheme is supposed to be vulnerable to pressure instabilities/oscillations when using the exact, i.e. non-compact stencil. The only 'work-arounds' I heard of so far are the approximate correction APM and "Rhie Chow"-like interpolation schemes ...

In my personal experience I have found that the APM is stable and only shows small divergence errors in the highest frequencies (wave numbers) of the solution but over all the computed flow is practically indistinguishable from the solution obtained via exact pressure projection.

However, severe pressure oscillations did appear with the APM derived pressure when used with the itarativ pressure correction method (i.e. when one solves the poisson equation not for the pressure but for a pressure increment )

February 10, 2017, 18:19
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Filippo Maria Denaro
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Quote:
 Originally Posted by zx-81 That sounds interesting as it was my impression that any non-staggered grid scheme is supposed to be vulnerable to pressure instabilities/oscillations when using the exact, i.e. non-compact stencil. The only 'work-arounds' I heard of so far are the approximate correction APM and "Rhie Chow"-like interpolation schemes ... In my personal experience I have found that the APM is stable and only shows small divergence errors in the highest frequencies (wave numbers) of the solution but over all the computed flow is practically indistinguishable from the solution obtained via exact pressure projection. However, severe pressure oscillations did appear with the APM derived pressure when used with the itarativ pressure correction method (i.e. when one solves the poisson equation not for the pressure but for a pressure increment )

Yes, some years ago I worked a lot on this issue ... The APM produced some of these problems in my LES code and I wanted not to use the RC-interpolation to not introduce artificial dissipation. Thus, I developed an exact double-projection method that works on non-staggered grid. I had good and oscillation-free solutions without adding dissipation. The general idea can be implemented also on triangular (tetraedra in 3d) unstructured grids. If you are interested, you can find a couple of my published papers on IJNMF

February 13, 2017, 08:16
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Michael Prinkey
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Quote:
 Originally Posted by FMDenaro Yes, some years ago I worked a lot on this issue ... The APM produced some of these problems in my LES code and I wanted not to use the RC-interpolation to not introduce artificial dissipation. Thus, I developed an exact double-projection method that works on non-staggered grid. I had good and oscillation-free solutions without adding dissipation. The general idea can be implemented also on triangular (tetraedra in 3d) unstructured grids. If you are interested, you can find a couple of my published papers on IJNMF
I'm looking forward to reading those papers, Professor. I have never liked using RC for the reasons you cite.

February 13, 2017, 10:29
#11
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Filippo Maria Denaro
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Quote:
 Originally Posted by mprinkey I'm looking forward to reading those papers, Professor. I have never liked using RC for the reasons you cite.

we are discussing here:

double-projection method that works on non-staggered grid

 Tags navier stoke solver, piso, projection method