Encyclopaedia Index

(c) Numerical-accuracy improvements during 1996


  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

(i) The problem

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

Thus, for a cartesian, polar or other orthogonal grid, the flux of heat across the face e, above, can be rather accurately expressed in terms of the magnitude of the temperature difference between P and E.

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:

(ii) The solution in pre-1996 PHOENICS

(iii) The solution in version 2.2.1 and beyond

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

with the strip of cells from B to C "peeled off" upwards and bent back to the left, and that from H to I "peeled off" downwards and also bent to the left, while the cells in the strip from E to F are rendered uninfluential by being given zero volume porosity.

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|      |
                                                 |      |      |      |
Users are therefore advised to ensure that the cells on either side of such a discontinuity should be made as orthogonal as possible, if necessary by the introduction of one or more layers of such cells near the phase interface.

(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:-

(vi) Concluding remarks

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.

In principle, each of these methods, when applied to the same problem with a sufficiently fine grid, should produce the same solution. However, the number of possibilities is so vast that no comprehensive view can yet be formed as to which method is best in which circumstances.

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.