Difference between revisions of "OpenFOAM guide/H operator"

Revision as of 15:37, 30 June 2010

The H operator is a form of shorthand notation that is not commonly used, but appears in OpenFOAM. It is comprised of a collection of terms from the momentum equation. A partially discretized form of the momentum equation is:

$A_p \mathbf U_p = \mathbf {S_m}_{,p} - \sum\limits_r A_r \mathbf{U}_r - \boldsymbol \nabla p^*$

Where:

subscript p is the cell index;
subscript r are related cells^[1];
A are the matrix coefficients;
U is the uncorrected velocity;
S is the discretization source term; and
p^* is the pressure from the previous timestep or initial guess.

The H operator is all terms on the right-hand side, excluding those involving pressure:

$\mathbf H_p = \mathbf {S_m}_{,p} - \sum\limits_r A_r \mathbf{U}_r$

Therefore the momentum equation becomes:

$A_p \mathbf U_p = \mathbf H_p - \boldsymbol \nabla p$

Due to its prevalence in solver algorithms, OpenFOAM implements the H operator directly in its matrix classes, including:

Notes

↑ What I'm calling related cells is conventionally called neighbours. But OpenFOAM has a different meaning for neighbours, so the term related cells is used.

[1] What I'm calling related cells is conventionally called neighbours. But OpenFOAM has a different meaning for neighbours, so the term related cells is used.

[1]

@@ Line 9: / Line 9: @@
 * '''U''' is the uncorrected velocity;
 * '''S''' is the discretization source term; and
-* ''p''<sup>*</sup> is the [[pressure]] from the previous timestep or initial guess.
+* ''p''<sup>*</sup> is the pressure from the previous timestep or initial guess.
-The '''H''' operator is all terms on the right-hand side, excluding those involving [[pressure]]:
+The '''H''' operator is all terms on the right-hand side, excluding those involving pressure:
 <math>\mathbf H_p = \mathbf {S_m}_{,p} - \sum\limits_r A_r \mathbf{U}_r</math>