Within EARTH, PHOENICS solves sets of algebraic equations which represent the consequences of:
A wide variety of interpolation assumptions may be selected by the user; but, if he makes no specific selection, the "fully-implicit- upwind" set is used by default, because of its reliability. Its use implies that:
[(ri*rhoi*Fi*V),new - (ri*rhoi*Fi*V),old]/[t,new - t,old]
for each cell, where V is the cell volume.
For example, if Ui,e denotes the x-directed velocity resolute of phase i stored at the east face of a cell, and if Ae denotes the area of the face, the mass flux of phase i across the east face of cell P (the neighbour of which is cell E) is given by:
ri,P * rhoi,P * Ui,e * Ae if Ui,e > 0
ri,E * rhoi,E * Ui,e * Ae if Ui,e <0
Likewise, the flux of variable Fi across the east cell face is taken as the product of the mass flux and the value of Fi at the upwind node, ie by
Fi,P * ri,P * rhoi,P * Ui,e * Ae if Ui,e > 0
Fi,E * ri,E * rhoi,E * Ui,e * Ae if Ui,e < 0
The diffusion fluxes are taken to be the product of the cell-to-cell difference in the Fi values and the cell-face area, divided by the resistance to diffusion represented by the integral over the distance between the cell centres of:
(distance increment) / (exchamge coefficient)
Thus, for the simplest case of brick-shaped cells, and single- phase flow, the diffusion flux of variable Fi from cell P to cell E is computed, if the default harmonic-mean option is chosen by the last argument of the SOLUTN command:
(FE - FP) * Ae / (Pe/GP + eE/GE) where Ae is the cell-face area, Pe and eE are the distances frpm the cell centres to the cell face, and GE and GP are the values of the relevant exchange coefficient appropriate to cells P and E.
(See TR 99 for full information about the above and alternative formulations)
Several variants from these default options are available to the user, simply by the setting of available data-input "switches".
If new variants are invented by the user, he may introduce them via the access facilities provided, eg by use of FORTRAN sequences introduced into GROUND subroutines.
Whatever the user's options, the result is likely to be a set of equations having the form:
aE*FE + aW*FW + aN*FN + aS*FS + aH*FH + aL*FL + aT*FT + S FP = --------------------------------------------------------- aE + aW + aN + aS + aH + aL + aT + aPHere:
The a's are coefficients, temporarily treated as though they were constants.
Those with subscripts N, S, E, W, H and L express the interactions between neighbouring cells by way of diffusion and bulk motion (ie convection), while aT expresses the time-dependence effect.
The a's have the dimensions of mass per unit time. They therefore increase with the geometric size of the cells, and have easy-to-determine physical significances.
S and aP express the influence of a source of the entity F. The total contribution of the source term to the balance of F for the cell is S - aP*FP; this is known as the linearised- source formulation, which helpfully promotes rapidity of convergence, and so reduces the cost of computation.
See the Lecture-notes-on PHOENICS entry of POLIS for further information.