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6.2 The "presumed-pdf" approach

(a) Origins

Recognising the need to account for the incompleteness of micro- mixing in turbulent reacting fluids, combustion modellers have sought to compute probability-density functons by:

  1. presuming their shapes, and
  2. computing their "moments" from differential transport equations.

Possibly the first example was that of Spalding (1971), who presumed a "two-spike" shape, and solved for the RMS fluctuations as well as for the mean value of the relevant fluid attribute.

Subsequently, Lockwood and Naguib (1975) made the "clipped-Gaussian" presumption; they were followed by others, including Kent and Bilger (1976), Kolbe and Kollmann (1980), Rhodes et al (1974) and Gonzalez and Borghi (1991).

"Clipped Gaussian" and "beta-function" presumptions are popular.

(b) Doubts

Despite their popularity, presumed-pdf approaches lack physical plausibility.

That SOME account for the non-uniformity of a turbulent mixture must be taken is sure; but to presume without evidence that one pdf shape is preferable to another is dangerous.

Use of the multi-fluid model has shown that a wide range of shapes can arise, even within the same flame.

MFM has also been shown that the differing shapes have significant effects on quantities of interest to engineers, for example the yields of desired nd undesired reaction products.

When the presumed shape is mathematically complex, some rather time- consuming computations can ensue. Therefore, if the presumed-pdf approach is to be used at all, simple shapes are to be preferred.

(c) Activation in PHOENICS

The only presumed-pdf model which is supplied with PHOENICS is the "two-spike" version, which forms part of the SCRS coding of the Advanced Chemistry Option.

It is activated by entering SOLVE(F,FSQ) which activates coding in gxfsqs.htm for the calculation of the source and sink terms of the variable FSQ.

This variable represents the root mean square fluctuations of the composition variable F.

The latter usually stands for "mixture fraction", ie the proportion of the local mixture which originated in the fuel stream, regardless of how much of it has been burned.

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