(c) Numerical-accuracy improvements during 1996

Contents

1. The problem
2. The solution in pre-1996 PHOENICS
3. The solution in version 2.2.1
4. Conjugate-heat-transfer problems
5. Momentum-transfer problems
6. Concluding remarks

The differences between the equations governing the conservation of mass, momentum and energy arise from the facts that the diffusion and convection fluxes across control-volume (ie cell) boundaries must be computed from more than one of the known resolutes.

      cartesian or polar          body-fitted            EH
                               non-orthogonal    EN .     .
                                                        * E .
              |/                               /      /   .    ES
              |  /                            |  /  /    EL
              |    /                          |    /
     P *-------- e   /-----* E         H      |/e    /
              |      |                 .     /|      |
                /    |            N.   .   /  |      |
                  /  |                 P *    |------|
                    /|                .    .
                                     L          S

By contrast, for a non-orthogonal body-fitted-coordinate grid, the temperature differences between P and N, S, H and L also have an influence, as do those between E and EN, ES, EH and EL.

The latter influenes arise indirectly, as follows:

• What are truly needed are magnitudes of the temperature-gradient vector and of its direction cosines at the centre of face e, for multiplication, in scalar-product manner, by the corresponding magnitudes of the face area itself.
• The simplest way to obtain these quantities is by interpolation between the cartesian-components of the temperature-gradient vector at P and those at E.
• The three components at P, for example, can be deduced straight- forwardly from three resolutes at that point, for examples PE, PH and PN.
• However, they can also be obtained from PE, PH and PS; or from PE, PS and PL; or from PE, PL and PN.
• In general, there is no reason to prefer any one of these four choices rather than another; therefore some average should be taken, in order that symmetry may be preserved.
• Since an equal number of choices can be made at point E, an unbiassed calculation of the cartesian components at e involves altogether eight resolute-to-cartesian-component calculations, followed by weighted-averaging.
• Moreover, first, a total of nine resolutes must be computed (five at P and five at E, but PE is shared between them); and this involves the attention to ten temperatures.

• The amount of computation involved in the above operations was regarded by the creators of the early PHOENICS as being too expensive to contemplate. They therefore chose to presume that the required cartesian components at e were the same as those at P; they thus ignored those at E, introducing thereby a potential source of asymmetry.
• A second source of asymmetry was their choice of PE, PN and PH as the only resolutes which would be used in the computation of the cartesian components at P.
• Provision was made for making other choices when one of the resolutes in question happened to pass through a blocked region, or to pass out of the domain, for example in a cell adjacent to a north boundary.
• However, whichever choices were made had this in common: the heat flux through a face depended on only four temperatures in 3D problems, and on only two in 2D ones.

During 1996, the BFC portion of PHOENICS has been substantially re- written, in order to remove the above-mentioned asymmetries. Specifically, the necessary calculation and weighted-averaging are now performed, whenever the PIL logical variable SYMBFC is set to TRUE.

Computer times and storage requirements are necessarily increased thereby; therefore the possibility of leaving SYMBFC as FALSE is allowed, for those users who do not need the extra accuracy.

During the course of this re-writing, many other problems have been resolved by careful attention to detail, none being more important than the removal of the influence of "unnatural" cells.

This problem, and its resolution, will now be described.

An example of unnatural cells

Suppose that it is desired to represent a bifurcating duct by taking and distorting a grid such as the following:

        ----------------------------------------------
     A  |  |  |  |  |  |  | B|  |  |  |  |  |  |  |  |  C
        |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
        ----------------------------------------------
     D  |  |  |  |  |  |  | E|  |  |  |  |  |  |  |  |  F
        ----------------------------------------------
        |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
     G  |  |  |  |  |  |  | H|  |  |  |  |  |  |  |  |  I
        ----------------------------------------------
 

If the BC and HI strips are then also twisted in manners which still leave the cells within them in perfectly satisfactory shape, the cells in the EF strip can take such very unrealistic shapes that the attempts of PHOENICS to compute areas, volumes and direction cosines can lead to highly unsatisfactory consequences.

To avoid these consequences, whether SYMBFC is TRUE or FALSE, means have now been provided for removing totally from account, in the BFC-geometry calculation, all cells which have been given the values VPOR = 0, PRPS = PORPRP, or PRPS = VACPRP.

While it is unsafe to guarantee that all grids which have reasonably-formed fluid-accessible cells will now perform well, no matter how "unnatural" are the shapes of the blocked cells, it can be claimed that all the difficulties to which CHAM's attention has been drawn by users have been solved by the new treatment.

(iv) Conjugate-heat-transfer problems

One problem to which no wholly satisfactory general solution has been found is that of predicting accurately the heat transfer across the wall of a highly-skewed cell when a large discontinuity of thermal conductivity exists at that wall.

    An example of this situation would be    /       /      /      /
    as sketched on the right, where the        /       / air  /      /
    components of temperature gradient in the    ----------------------
    horizontal direction are large               |      |copper|      |
                                                 |      |      |      |

(v) Momentum-transfer problems

The above discussion has been conducted in terms of temperature and heat transfer because the entities and processes involved are easier to grasp than those associated with velocity and momentum transfer.

Concerning the latter, only the following brief remarks will be made:-

• The PHOENICS practice of employing velocity resolutes as dependent variables, although it has many merits, does necessitate careful internal-to-EARTH treatment of the momentum-transport terms.
• The principle to be strictly adhered to in this treatment is that material transported across a cell face carries three momentum components, which are most usefully defined as those in the three cartesian directions.
• Therefore the resolute-to-cartesian-component and component-to- resolute calculation must be accurately handled at each cell face.

PHOENICS now has four distinct ways of handling fluid-flow problems in body-fitted-coordinate situations, namely:

1. the default option, ie staggered-grid velocity resolutes;
2. the CCV option, of Prakash-Rhie-Chow type (withdrawn early in 2006);
3. the CCM option, devised by Poliakov and Semin at CHAM in 1994; and
4. the GCV option, devised by Semin in 1996.

CHAM will welcome reports of users' experiences in order that such a comprehensive view can gradually be assembled; thereafter it will be communicated to users generally.

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