OpenFOAM guide/The PISO algorithm in OpenFOAM

1 The PISO algorithm

The PISO (Pressure Implicit with Splitting of Operators) is an efficient method to solve the Navier-Stokes equations in unsteady problems. The main differences from the SIMPLE algorithm are the following:

• No under-relaxation is applied.
• The momentum corrector step is performed more than once.

2 Implementation of the PISO algorithm in OpenFOAM

The PISO algorithm is implemented in OpenFOAM as follows (Details can be found in the icoFoam standard solver provided with OpenFOAM):

• Define the equation for U

 
 fvVectorMatrix UEqn
 (
   	fvm::ddt(U)
      + fvm::div(phi, U)
      - fvm::laplacian(nu, U)
 );

• Solve the momentum predictor

 
 solve (UEqn == -fvc::grad(p));

• Calculate the $a_p$ coefficient and calculate U

 
 volScalarField AU = UEqn().A();
 U = UEqn().H()/AU;

• Calculate the flux

 
 phi = (fvc::interpolate(U) & mesh.Sf()) 
       + fvc::ddtPhiCorr(1/AU, U, phi);
 adjustPhi(phi, U, p);

• Define and solve the pressure equation and repeat for the prescribed number of non-orthogonal corrector steps

 
 fvScalarMatrix pEqn
 (
    fvm::laplacian(1.0/AU, p) == fvc::div(phi)
 );
 pEqn.setReference(pRefCell, pRefValue);
 pEqn.solve();

• Correct the flux

 
 phi -= pEqn.flux();

• Calculate continuity errors

 
# include "continuityErrs.H"

• Perform the momentum corrector step

 
 U -= fvc::grad(p)/AU;
 U.correctBoundaryConditions();

• Repeat from the calculation of $a_p$ for the prescribed number of PISO corrector steps.

3 References

J. H. Ferziger, M. Peric, Computational Methods for Fluid Dynamics, Springer, 3rd Ed., 2001.

H. Jasak, Error Analysis and Estimation for the Finite Volume Method with Applications to Fluid Flows, Ph.D. Thesis, Imperial College, London, 1996.