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Additional recommended knowledge
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 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 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 , resulting from spurious mass flux, then using this mass imbalance to 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 imbalance is often also used for control of the outer loop.
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.
Implicit Split-Update Procedures
Another approach which is typlically 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
Applying the divergence operator to the copmpressible 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
|This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Pressure-Correction_Methods". A list of authors is available in Wikipedia.|