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MFM: the economical route to PDFs


Brian Spalding, CHAM Ltd, London, England

A lecture delivered at The Isaac Newton Institute, Cambridge, England, on April 30 1999

Abstract and contents

It is argued that:
  1. probability-density functions (PDFs) are more useful than statistical averages;
  2. the colliding-fluid-fragments model of Reynolds and Prandtl, when generalised, leads to the multi-fluid model (MFM) and thence to PDFs;
  3. the "eddy-break-up model" of 1971 was a small step in the right direction;
  4. the PDF-transport model of 1982 and the two-fluid model of 1987 were larger steps in two divergent nearly-right directions; now MFM can deliver the results sought by the former by using the mathematical techniques of the latter;
  5. MFM differs from Monte Carlo PDF transport in several respects: in particular, it allows "population-grid refinement and adaptation".
  6. So, Kolmogorov's introduction of transport equations for statistical averages was "a good idea at the time, but..."

  7. During the lecture, applications of MFM will be made to:
    1. the plane mixing layer,
    2. Some near-wall flows.
    3. a stirred reactor and
    4. a gas-turbine combustor.
    whereafter possible future developments will be discussed,

  8. References are provided

1. Probability-density functions are more useful than statistical averages

1.1 What PDFs are

1.2 Why we need PDFs

1.3 Why not get PDFs from "weighted averages"?

2. The colliding-fluid-fragments model of Reynolds and Prandtl

3. The "eddy-break-up" model; a small step in the right direction

4. The PDF-transport model of 1982 and the two-fluid model of 1987

5. The Multi-Fluid Model (MFM), and how it differs from Monte Carlo PDF-transport (MCPT)

  1. MFM focusses attention on discretized PDFs. It produces "battlement-shaped" histograms, whereas MCPT produces a cloud of points, through which one may be able to draw a curve.

  2. The fineness of MFM discretization is chosen by the analyst, who may test its adequacy by grid-refinement. Sometimes an extremely coarse population grid will suffice, as for example, in this reactor study.

  3. MFM does not need to have the same number of fluids at all points in the domain of study. In a combustor simulation, a single fluid will often suffice over a large proportion of the volume. An algorithm can be devised for dynamically determining the number of fluids needed to provide a given accuracy.

  4. Population grids can thus be "unstructured" and "self-adaptive", exploiting experience gained by CFD experts with space and time grids.

  5. There appear to be no economising counterparts to points b, c and d in MCPT.

  6. Because the local mass fraction of each fluid is a calculated and accessible variable, MFM allows "micro-mixing hypotheses" to be investigated which are more sophisticated than any formulated by MCPT practitioners.

  7. MFM distinguishes between (what the analyst chooses as) population-distinguishing attributes (PDAs) and continuously-varying attributes (CVAs), for example, in a combustor simulation: MCPT appears to enjoy no such freedom.

  8. MFM fits easily into conventional finite-volume-type solution algorithms, whereas MCPT requires, in addition, the Monte-Carlo apparatus and methodology.

  9. MFM concepts can be described rather easily in words, whereas (it appears) MCPT demands a daunting display of mathematical symbols.

  10. The computer expense associated with MFM is of the same order of magnitude as that associated with the hydrodynamics in a typical CFD application.

6. Kolmogorov's"bright idea"

Kolmogorov's 1942 paper said, in effect:
"Although we really want to know much more (eg the PDFs), perhaps we can get away with calculating a few statistical quantities"

The turbulence-modelling world has followed him.

Kolmogorov chose the energy, k, and the "frequency", k/epsilon, as his variables, as did Wilcox much later.

Particularly since the late 1960's, many other choices have been made, the most popular being k and epsilon; but all modellers have shared Kolmogorov's hope: that "a few statistical quantities" will suffice.

However, for reasons explained in section 1, they do not suffice, and never will. PDFs are what we must have; and MFM enables us to get them economically.

7. Applications of MFM

Extracts will now be presented from earlier lectures by the author.

7.1 The plane mixing layer,

This concerns the first-ever simulation of a much-studied turbulent flow which does not employ one the "classical" turbulence-model approaches.

7.2 The stirred reactor

This concerns a large three-dimensional transient flow simulation, to which introduction of the multi-fluid model added little compuational expense but much valuable insight.

In this case, the k-epsilon turbulence model is used for the hydrodynamical part of the calculation, showing that MFM easily co-exists with conventional models.

7.3 The gas-turbine combustor.

This recent lecture shows how the predicted smoke-generation rate in a three-dimensional steady-flow combustor differs considerable are according to whether the concentration fluctuations are or are not taken into account.

Also reported are computer times, and how they vary with the number of fluids employed; and a population-grid-independence study is also reported.

7.4 Future developments.

Clicking here leads to the final section of a 1998 lecture on MFM.

This sets out what is, in essence, a multi-man-year program of research. This, it is argued, could beneficially transform the capabilities of engineers and applied scientists to simulate turbulent-flow phenomena realistically.

However, it recognises that a formidable obstacle stands in the way of such an enterprise, namely the strong psychological hold which Kolmogorov's "bright idea" of 1942 still exerts.

Loosening that hold is one intent of the present lecture.

Will The Isaac Newton Institute assist?

Or must the world wait for The Einstein Institute to take an interest in turbulence?

The End

8. References

C Dopazo and EE O'Brien (1974)
Acta Astronautica vol 1, p1239
AN Kolmogorov (1942)
"Equations of motion of an incompressible turbulent fluid"; Izv Akad Nauk SSSR Ser Phys VI No 1-2, p56
SB Pope (1982)
Combustion Science and Technology vol 28, p131
O Reynolds (1874)
"On the extent and action of the heating surface of steam boilers"; Proc. Manchester Lit Phil Soc, vol 8, 1874
DB Spalding (1971)
"Mixing and chemical reaction in confined turbulent flames"; 13th International Symposium on Combustion, pp 649-657 The Combustion Institute
DB Spalding (1987)
"A turbulence model for buoyant and combusting flows"; Int. J. for Num. Meth. in Engg., vol 24, pp 1-23
Spalding DB (1995a)
"Models of turbulent combustion" Proc. 2nd Colloquium on Process Simulation, pp 1-15 Helsinki University of Technology, Espoo, Finland
DB Spalding (1999)
"Connexions between the Multi-Fluid and Flamelet models of turbulent combustion"; www.cham.co.uk; shortcuts; MFM
DC Wilcox (1993)
"Turbulence modelling for CFD", DCW Industries, La Canada, California