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Discretization in PHOENICS

Dividing the computational domain to make arrays of finite-sized elements is known as discretization.

The locations in space and time at which pressure and other scalar dependent variables are computed are finite in number; and they are imagined to lie within a set of 'cells' which, added together, make up the whole space-time domain that is being considered. The following diagrams illustrate the conception, to some extent; but it must not be supposed that cells are always equal in size, or rectangular in shape.



                  N     H       The grid points shown are:-

                     |   /

                     |  /          P, the central point

                     | /           N, the north neighbour

                     |/            S, the south neighbour

           W---------P--------E    E, the east neighbour

                    /|`            W, the west neighbour

                   / |  `          H, the high neighbour

                  /  |    `T       L, the low neighbour

                 /   |             T, the neighbour in the

                /    S                earlier-time direction

              L

/------/------/------/------/ / / / / /| The grid points can /------/------/------/------/ / be thought of as / / / / /|/| being at the centres |-----|------|------|------| / / of a stack of boxes | | | | |/|/| (i.e. the volume |-----|------|------|------| / / elements) so shaped | | | | |/|/| as to fill entirely |-----|------|------|------| / / the space in which | | | | |/|/| the fluid flows. |-----|------|------|------| / / | | | | |/| | |-----|------|------|------|

Strictly speaking, the statement about the dependent-variable locations being within the cells requires modification: for PHOENICS computes values of velocities for locations on the walls of the cells. The following diagram illustrates this, a single cell with four of its neighbours being shown in plan view.



                    |       N         |

                    |                 |

                    |       ^         |

                    |       |n        |

              ------|-------|---------|-----

                    |       |         |

                    |                 |

                    |                 | e

         W       ------>    P       ----->            E

                    |                 |

                    |       ^         |

                    |       |         |

              ------|-------|---------|-----

                    |       |         |

                    |                 |

                    |       S         |

                    |                 |

Temperatures, pressures and concentrations are evaluated by PHOENICS for the locations like P, N, S, E, W which lie within cells; but west-to-east velocities are evaluated for the cell-wall locations like w and e; and south-to-north velocities are evaluated for locations like s and n.

A three-dimensional diagram would show in addition the low- and high-neighbour points for pressures, etc. viz. L and H; and the low-to-high velocities would then be represented by arrows at the cell walls, l and h. This convention is called the 'staggered-grid' arrangement.

Inspection of any printed output from PHOENICS will allow the foregoing description to be understood. Values will be seen to be printed for the x, y and z coordinates of the cell centres; and the values of the dependent variables will appear in tables, in which the locations are described by indices IX, IY and IZ for the three spatial coordinates, and by the index ISTEP for the time. In a steady-flow computation, ISTEP will appear as 1.


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