Pressure-correction method is a class of methods used in computational fluid dynamics for numerically solving the Navier-Stokes equations normally for incompressible flows.
Contents
Common properties
The equations solved in this approach arise from the implicit time integration of the incompressible Navier–Stokes equations.
Due to the non-linearity of the convective term in the momentum equation that is written above, this problem is solved with a nested-loop approach. While so called global or inner iterations represent the real time-steps and are used to update the variables
The outer iterations comprise two steps:
- Solve the momentum equation for a provisional velocity based on the velocity and pressure of the previous outer loop.
- Plug the new newly obtained velocity into the continuity equation to obtain a correction.
The correction for the velocity that is obtained from the second equation one has with incompressible flow, the non-divergence criterion or continuity equation
is computed by first calculating a residual value
The name of this class of methods stems from the fact that the correction of the velocity field is computed through the pressure-field.
The discretization of this is typically done with either the finite element method or the finite volume method. With the latter, one might also encounter the dual mesh, i.e. the computation grid obtained from connecting the centers of the cells that the initial subdivision into finite elements of the computation domain yielded.
Implicit split-update procedures
Another approach which is typically used in FEM is the following.
The aim of the correction step is to ensure conservation of mass. In continuous form for compressible substances mass, conservation of mass is expressed by
where
The way of obtaining a velocity field satisfying the above, is to compute a pressure which when substituted into the momentum equation leads to the desired correction of a preliminary computed intermediate velocity.
Applying the divergence operator to the compressible momentum equation yields
The idea of pressure-correction also exists in the case of variable density and high Mach numbers, although in this case there is a real physical meaning behind the coupling of dynamic pressure and velocity as arising from the continuity equation