Pressure-correction method

This term refers to a class of methods used in computational fluid dynamics for solving the Navier-Stokes equations normally for an incompressible fluid.

Common Properties

The equations solved in this approach arise from the implicit time integration of the incompressible Navier-Stokes equations.

$\overbrace {\rho {\Big (}\underbrace {\frac {\partial \mathbf {v} }{\partial t}} _{\begin{smallmatrix}{\text{Unsteady}}\\{\text{acceleration}}\end{smallmatrix}}+\underbrace {\left(\mathbf {v} \cdot \nabla \right)\mathbf {v} } _{\begin{smallmatrix}{\text{Convective}}\\{\text{acceleration}}\end{smallmatrix}}{\Big )}} ^{\text{Inertia}}=\underbrace {-\nabla p} _{\begin{smallmatrix}{\text{Pressure}}\\{\text{gradient}}\end{smallmatrix}}+\underbrace {\mu \nabla ^{2}\mathbf {v} } _{\text{Viscosity}}+\underbrace {\mathbf {f} } _{\begin{smallmatrix}{\text{Other}}\\{\text{forces}}\end{smallmatrix}}$

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 $\mathbf {v}$ and $p$, based on a linearized system, and boundary conditions; there is also an outer loop for updating the coefficients of the linearized system.
The outer iterations comprise two steps:

1. solve the momentum equation for a provisional velocity based on the velocity and pressure of the previous outer loop.
2. 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

${\text{div}}\mathbf {v} =0$

is computed by first calculating a residual value ${\dot {m}}$, known as the mass flux, then using the mass flux get a new pressure value. The pressure value that is attempted to compute, is such that when plugged into momentum equations a divergence-free velocity field results. The mass flux is often also used for control of the outer loop.
The name of this class of methods stems from the fact that the correction of the velocity field is comptued 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 subdivison into finite elements of the computation domain yielded.

References

• M. Thomadakis, M. Leschinzer: A PRESSURE-CORRECTION METHOD FOR THE SOLUTION OF INCOMPRESSIBLE VISCOUS FLOWS ON UNSTRUCTURED GRIDS, Int. Journal for Numerical Meth. in Fluids, Vol. 22, 1996
• A. Meister, J. Struckmeier: Hyperbolic Partial Differential Equations, 1st Edition, Vieweg, 2002

External links

• ISNaS - incompressible flow solver