Encyclopaedia Index

### SOLVING the FINITE-VOLUME EQUATIONS

(See first: the finite-volume equations solved by PHOENICS)

### The nature of the problem

For each dependent variable F, there are as many algebraic equations as there are cells in the integration domain, ie NX*NY*NZ.

In addition to being numerous, these sets of equations are often non-linear and strongly coupled, thus necessitating solution in an iterative 'guess-and-correct' manner, the object of which is to reduce the imbalance between the left and right sides of every equation to a magnitude which is small enough to be negligible.

### The solution procedure

The iterative process is a complicated one, involving a multi-stage sequence of adjustments of values, repeated many times.

In the course of an iteration cycle, the coefficients and sources are regarded as temporarily constant, so that linear- equation solvers can be used to solve the equation sets. On the next cycle, the coefficients and sources are updated from the latest values of the auxiliary and dependent variables, and the linear equations re-assembled and solved.

Here only the salient features of the iteration cycle are described. The concepts to be explained are: 'slabwise solution', 'sweeps', 'whole-field solution' and 'parabolic'.

### Slab-wise solution

'Slabs' are arrays of cells having the same value of the low-to-high coordinate z. Many of the mathematical operations conducted by PHOENICS operate over a single slab. The fluid properties and the finite-volume-equation coefficients are always computed slab-by-slab; and velocities (always) and scalar variables (if the choice is so made by the SOLUTN command) may actually be solved at the same time.

Many cycles of adjustment can be performed for one slab before PHOENICS transfers its attention to the next slab. The value if LITHYD sets the maximum number.

A 'sweep' is a set of slabwise operations, conducted in sequence from the lowest-z slab to the highest-z slab.

If slabwise solutions of a variable is in operation, because the equations for values at one slab ordinarily make reference to values at the next-higher-z slab, later adjustments made at the higher slab will upset, to some extent, the balances which have just been struck at the lower one.

For this reason, many sweeps must be ordinarily be made, in succession; and the process should be continued, ideally, until all equations are in such perfect balance that further adjustments are unnecessary.

### Whole-field solution

'Whole-field solution' is a procedure which can be employed by PHOENICS for all variables except velocities. It reduces the number of sweeps which must be made in order to eliminate the imbalances in the equations; but it uses more computer storage.

Whole-field solution is most effective for phenomena such as 'pure' heat conduction, or irrotational (ie 'potential') fluid flow; for then a single sweep may suffice to give the solution.

### Solving parabolic problems

A 'parabolic' problem is one in which, although z-direction gradients in solved-for variables do exist, the higher-slab values do not appear in the lower-slab equations, because the coefficient aH is negligible for all points.

This situation often arises when there is a flow of fluid in the positive z-direction and the Reynolds number is high, as occurs in a duct, or a jet, or a boundary layer; for then the influence of downstream conditions on upstream ones is very small.

If the problem is parabolic, a single sweep through the integration domain therefore suffices; and great reductions in computer storage may result. In such circumstances, it is usually necessary to conduct several slabwise iteration cycles in order that imbalances in the equations are sufficiently reduced before the next slab is visited; for single-sweep parabolic calculations allow no 'second chance'.

### Some further points about solving the equations

• One-dimensional problems may be solved in the x (E-W), y (N-S) or z (H-L) directions.
• Two-dimensional problems may be solved in the x-y (NSEW), x-z (EWHL) or y-z (NSHL) planes.

In the first of these cases, ie the use of the x-y plane, there is only one slab; therefore sweeps and slab-wise iterations have identical significances; and there is no distinction between slab-wise and whole-field solution procedures.

In the second and third cases, 'slabs' reduce to 'strips' of adjoining cells, extending in the east-west or north-south direction respectively; but otherwise all the above statements about the solution procedure remain true.

• The parabolic option of PHOENICS operates only for steady-flow processes; for, since transient phenomena require the allocation offull computer storage in any case, no saving can be effected.
• Problems not characterised as parabolic are often called elliptic; in the present context this means merely that influences are exerted in both the positive and negative directions for all existent space dimensions.
• In time however, influences proceed in only one way, from 'early' to 'late'. This is why PHOENICS never needs four- dimensional storage of its dependent variables.

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