FINITE-VOLUME EQUATIONS solved by PHOENICS

CONTENTS

• Integration and interpolation
• The default interpolation assumptions
• The diffusion fluxes

Integration and interpolation

Within EARTH, PHOENICS solves sets of algebraic equations which represent the consequences of:

• integrating the differential equations over the finite volume of a computational cell and (for transient problems) over a finite time; and
• approximating the resulting volume, area and time averages by way of interpolation assumptions.

The default interpolation assumptions

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:

1. in time-dependent terms, all fluid properties are presumed to be independent of position within a cell, so that the integral over the cell volume of d(ri*rhoi*φ_i)/dt is replaced by:

   [(ri*rhoi*f_i*V),new - (ri*rhoi*f_i*V),old]/[t,new - t,old]

   for each cell, where V is the cell volume.

2. in convection terms, all fluid properties are uniform over cell faces; further, the "new" (i.e. end-of-time-interval) values are supposed to prevail throughout the time interval; and, except in respect of the velocities for which the face-centre values are stored, the values prevailing at the cell face are those at the nearest grid node on the "upwind" side of the face.

   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

   but by:

   ri,E * rhoi,E * Ui,e * Ae if Ui,e <0

   Likewise, the flux of variable f_i across the east cell face is taken as the product of the mass flux and the value of f_i at the upwind node, ie by

   f_i,P * ri,P * rhoi,P * Ui,e * Ae if Ui,e > 0

   but by:

   f_i,E * ri,E * rhoi,E * Ui,e * Ae if Ui,e < 0

3. In terms representing diffusion (and heat conduction and viscous action), the property gradients and the transport properties which they multiply are uniform over cell faces; further, the "new-time" values in a time-dependent calculation are supposed to prevail throughout the time interval.
4. In terms representing sources, the nodal values are supposed to prevail over the whole of the cell volume, and the "new-time" values (ie the late-time ones) are supposed to prevail over the whole of the time interval.

The diffusion fluxes

The diffusion fluxes are taken to be the product of the cell-to-cell difference in the f_i 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) / (exchange 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 + aP 

• F stands for the dependent variable currently in question;
• the letters P, E, W, N, H and L denote the locations at which this variable is computed, in accordance with the north-south-east-west-high-low convention described above; and
• T denotes the earlier-time value.

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.

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