Encyclopaedia Index

PEA, an acronym for Partial Elimination Algorithm

  1. Its purpose
  2. Where it is used in PHOENICS
  3. Its nature

1. Its purpose

PHOENICS, by default, solves the equations for its variables in a 'segregated' manner; this means that, if it is to calculate the fields of phi1, phi2, phi3, etc, it does so by formulating algebraic equations for phi1 at all grid nodes and solving them immediately; then it does the same for phi2; then for phi3; and so on.

Since the not-yet-solved-for phi2 and phi3 values may appear in the coefficients or sources of the phi1 equations, another solution operation for phi1 must be conducted after phi2 and phi3 have been solved.

Indeed many repetitions of all the solutions operations are usually needed until the fields of phi1, phi2, phi3 etc are sufficiently consistent with each other, and with the equations governing them, for the solution to be regarded as having converged.

There is one set of circumstances in which 'many' becomes 'far too many': it is that in which:

  1. the phi1 equation contains a term such as:

    (phi2 - phi1) * linkco

  2. the phi2 equation contains a corresponding term:

    (phi1 - phi2) * linkco

  3. and the 'linking coefficient' linkco is so large that these terms greatly exceed all the others.

The purpose of the PEA is to prevent the slow-down of convergence which, without it, large linking coefficients inevitably produce.

2. Where it is used in PHOENICS

The Partial Elimination algorithm was first introduced for two-phase-flow problems, for the obvious reason that, when the two intermingled flows are very finely divided, the coefficient linking the first-phase temperature, say, with the second-phase temperature, is extremely large.

Later, when the IMMERSOL model of radiative heat transfer was introduced into PHOENICS, slow convergence was encountered whenever the emissivity of the radiating medium was very high. The PEA was therefore applied to this situation also, the two linked variables being TEM1, the first-phase temperature, and T3, the radiative temperature.

It is also permissible, as is sometimes preferred in combustion simulations, to use H1, the first-phase enthalpy, in place of TEM1.

The user needs to take no action in order to activate the PEA for these situations. Its action is automatic.

Internally, the PEA is coded within PHOENICS so as to be able to handle an arbitrary number of linked variables; users wishing to avail themselves of the facility are asked to contact CHAM for advice, giving details of the variables in question, and especially of the linking terms.

It should be remarked that the linkage does not have to be linear; indeed it is not for IMMERSOL, for which the linking term equals

constant * (TEM1**4 - T3**4)

3. Its nature

The PEA operates by substituting for phi2, in the ( phi2 - phi1 ) term above, a 'best-estimate' of what its value will be after solution.

This 'best estimate' is that which would be arrived at if phi2 were solved in a point-by-point manner, whereby its neighbour-point values are presumed (for the moment) not to change.

The equation for phi2P for a particular computational cell P can be expressed as:

sum over all neighbours of: ( const2N * phi2N )
+ source2P
- const2P * phi2P
+ linkco * (phi1P - phi2P) = 0

Therefore, in the equation (1) for phi1P, it is possible to replace the slow-convergence-causing term:
linkco * (phi2P - phi1P)
sum over all neighbours of: ( const2N * phi2N )
+ source2P
- const2P * phi2P

This substitution, together with the corresponding one for the phi2P equation when it is its turn to be solved, proves to be decisive: the linking coefficients may be arbitrarily large without slowing-down convergence.

There is a small disadvantage: the coefficients of both variables must be computed and stored before the solution of either variable can begin. However, this would become severe only if many variables were to be solved simultaneously. For two variables it is trivial.