# PISO vs Approximate vs Exact Projection methods for pressure-velocity coupling

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 August 8, 2017, 09:29 PISO vs Approximate vs Exact Projection methods for pressure-velocity coupling #1 Member   Katt Join Date: May 2017 Posts: 30 Rep Power: 7 Hi everyone; I have a pressure-velocity coupling question. Comparing several softwares, I came across two ways of iteratively solving the Navier-Stokes equations. 1. SIMPLE/ PISO/ PIMPLE (from OpenFOAM) 2. Approximate/ Exact Projection Method (Gerris/ Basilisk) Though I tried to understand the concept through Wiki, I dont see what really is the difference between all 3 methods. from my understanding (PS: all above mentioned codes use collocated scheme), Step1: From the available velocity at the previous time step stored at cell centers, they are interpolated to the cell faces (as Finite volumes compute cell face flux to be mass conservative) Step2: Apart from pressure gradient term, lump all the source/ sink terms along with the viscosity term and update the cell face momentum flux term Step3: Aiming to achieve a mass conserved solution at the new time, with the available momentum flux term and pressure gradient term the following relation is solved for iteratively: laplacian (pressure) = divergence (not mass conservative flux available from Step2) Step4: from the updated pressure, re-evaluate the mass conservative flux. Step5: Proceed to the next time step with the same process. I am aware that projection methods use Hodge-decomposition where: "Intermediate velocity = Mass conserved velocity + grad(pressure)" which ultimately leads to Step 3 above. Then where is the difference between these methods coming from? Can anyone help me, point to a suitable link where the difference is clearly explained. Thanks; Katt

 August 8, 2017, 10:52 #2 Senior Member   Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,246 Rep Power: 67 Exact or Approximate projection methods are different. The former satisfy the divergence-free constraint up to machine tolerance, the latter only up to the magnitude of the local truncation error. Often this difference is strictly related to the type of colocation of the velocities on the grid (staggered/non-staggered) as well as for the type of discretization of divergence and gradient operators.

 August 8, 2017, 11:18 #3 Member   Katt Join Date: May 2017 Posts: 30 Rep Power: 7 Hi, So please correct me if wrong. Exact projection: computationally expensive, mass conservative Approximate projection: computationally less expensive, approximately mass conservative. So, from a paper by Gerris developer (S. Popinet, 2003) he states as follows: "We will use both an exact projection for face-centred advection velocities and an approximate projection for the final projection of the cell-centred velocities. The detail of these two projections does not influence the general description of the Poisson solver". What I am wondering is why cant we do exact projection of cell faces and estimate the cell center velocity field from the available mass conservative cell face field computed? -- and What makes the projection methods of Chorin different from Issa's PISO method? Thanks and apologies if its a basic question. Too confused

 August 8, 2017, 11:39 #4 Senior Member   Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,246 Rep Power: 67 Not exactly, both EPM and APM are similar in terms of computational cost. I worked a lot on these methods and developed a formulation that mixes both approaches in a non-staggered but exact projection method exploting two set of velocities: faces and cell-centred. PISO is much more similar to SIMPLE procedure than a projection procedure wherein one perform one step for the intermediate velocity and then project it onto the subspace of divergence-free velocity fields.

August 8, 2017, 11:51
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Katt
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Quote:
 Originally Posted by FMDenaro PISO is much more similar to SIMPLE procedure than a projection procedure wherein one perform one step for the intermediate velocity and then project it onto the subspace of divergence-free velocity fields.
So, isnt it similar to projection method then. I see similar explanation in Wiki and the following paper on Gerris:
http://www.sciencedirect.com/science...985?via%3Dihub
From the paper (page - 6), I see they too compute an intermediate velocity field using the divergence condition and recompute the mass conservative flux.

 August 8, 2017, 11:56 #6 Senior Member   Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,246 Rep Power: 67 The projection method has only one projection step to compute the pressure gradients that ensure the continuity constraint. No iterations are performed further. PISO works differently: https://www.cfd-online.com/Wiki/PISO...Split_Operator https://en.wikipedia.org/wiki/PISO_algorithm

 August 16, 2017, 14:52 #7 Senior Member   Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,110 Rep Power: 60 One thing I noticed. In Chorin's projection method, the first? intermediate velocity field is computed completely ignoring the pressure gradient. The influence of the pressure gradient happens later in the projection step. In SIMPLE/PISO/PIMPLE, the pressure gradient is not neglected. One assumes a pressure field and then calculates the intermediate velocity.

August 16, 2017, 15:00
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Filippo Maria Denaro
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Quote:
 Originally Posted by LuckyTran One thing I noticed. In Chorin's projection method, the first? intermediate velocity field is computed completely ignoring the pressure gradient. The influence of the pressure gradient happens later in the projection step. In SIMPLE/PISO/PIMPLE, the pressure gradient is not neglected. One assumes a pressure field and then calculates the intermediate velocity.
When the pressure gradient is totally disregarded, one denotes such method as "pressure-free projection". Different formulations consider the presence of a previous pressure gradient, see for example Sec.2 here http://129.81.170.14/~cortez/Prints/accproj.pdf

 Tags navier stoke solver, piso, pressure velocity, projection method