Turbulence models for CFD in the 21st century


Brian Spalding, of CHAM Ltd

October, 2000

Invited lecture presented at ACFD 2000, Beijing

Click here for start of lecture


The two approaches to turbulence modelling

Both Osborne Reynolds (1884) and Ludwig Prandtl (1925) regarded turbulence as an expression of the near-random intermingling of sizeable fragments of unlike fluid, which, during a succession of brief encounters, tended to equilibrium.

However, their concept found no place in the family of turbulence models springing from Kolmogorov's (1942) proposal to attend only to statistical measures of the turbulent motion, such as energy and frequency.

The intermingling-fragments idea was nevertheless preserved in the models of Spalding (1971) and Magnussen (1976) ("eddy-break-up" and "eddy-dissipation", respectively) which are still used for combustion simulation.

It also featured in Spalding's (1987) "two-fluid" model of turbulence; and it is essential to the "multi-fluid" models of turbulence (Spalding, 1996) which are the subject of the present lecture.

Why the Kolmogorov approach has been popular until now

Limitations of computing power, and the seeming simplicity of the associated (1877) effective-viscosity concept of Boussinesq (1877), favoured adoption of the Kolmogorov rather than the Reynolds-Prandtl approach; and this road has now become so well-trodden that most CFD practitioners suppose, wrongly, that it is the only one which is open.

This would not matter if Kolmogorov-type models, for example k-epsilon (Harlow and Nakayama (1968)), allowed computation of the "probability density functions" needed for the simulation of non-linear processes such as radiation and chemical reaction; or if they could comprehend such real processes as "un-mixing"; but they do not.

Why the Reynolds-Prandtl approach is likely to be favoured from now on

"Intermingling-fragments" models of the kind conceived by Reynolds and Prandtl, do however permit these things; and the computing power needed for using them is easily available nowadays.

The lecture will explain how such "multi-fluid models" may be:

  1. formulated
  2. calibrated
  3. utilised for simulating engineering processes and equipment,
  4. subjected to numerical-accuracy tests, and
  5. further developed.

Similarities to, and differences from, the "pdf-transport" models of Dopazo and O'Brien (1974), and of Pope (1982), will be pointed out.


Click here for a historical overview

  1. Alternative concepts of turbulence
  2. An enlarged-viscosity model: LVEL
  3. Where enlarged-viscosity models fail
  4. Why intermingling-fragments models (IFMs, MFMs) can do better
  5. How multi-fluid models (MFMs) work
  6. Calibrating MFMs
  7. Extending MFMs
  8. Distinguishing MFMs from other models
  9. Conclusions
  10. References

1. Alternative concepts of turbulence

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2. An enlarged-viscosity model: LVEL

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3. Where enlarged-viscosity models fail

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4. Why intermingling-fragment models (IFMs) can simulate a wider range of phenomena than EVMs

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5. How multi-fluid models (MFMs) work

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6. Calibrating MFMs

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7. Extending MFMs

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8. Distinguishing MFMs from other models

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9. Conclusions

The argument presented in the foregoing lecture will now be summarised, as follows:

  1. Of the two main approaches to turbulence modelling, namely it is the second which is better able to represent physical reality.

  2. Reynolds-Stress models, although they do not use the Boussinesq enlarged-viscosity notion, are no better able to provide the needed PDFs.

  3. PDF-transport models based upon Monte Carlo methods, although directed at the right target, have built-in limitations which will continue to prevent their widespread use.

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  4. Because of:

    Reynolds' intermingling-fragments approach to turbulence has been almost entirely neglected.

  5. Now, however, computing power is more than adequate; and sufficient work has been done to demonstrate its practicability and promise.

  6. The author recommends that approach, as currently embodied in MFM, as the better basis for CFD in the Twenty-First Century.

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10. References

  1. J Boussinesq (1877) "Theorie de l'ecoulement tourbillant"; Mem. Pres. Acad. Sci. Paris, vol 23, 46
  2. C Dopazo and EE O'Brien (1974) Acta Astronautica vol 1, p1239
  3. FH Harlow & PI Nakayama (1968) "Transport of turbulence-energy decay rate"; Los Alamos Sci Lab U Calif report LA 3854
  4. AN Kolmogorov (1942) "Equations of motion of an incompressible turbulent fluid"; Izv Akad Nauk SSSR Ser Phys VI No 1-2, p56
  5. BE Launder, GJ Reece, W Rodi (1975) "Progress in the development of a Reynolds-Stress closure" JFM, vol 68, p 537
  6. BF Magnussen and BH Hjertager (1976) "On mathematical modelling of turbulent combustion with special emphasis on soot formation and combustion". 16th Int. Symposium on Combustion, pp 719-729 The Combustion Institute
  7. SB Pope (1982) Combustion Science and Technology vol 28, p131 Springer Verlag, New York, 1980, p115
  8. L Prandtl (1925) "Bericht ueber Untersuchungen zur ausgebildeten Turbulenz"; ZAMM vol 3, pp 136-139, 1925
  9. O Reynolds (1874) "On the extent and action of the heating surface of steam boilers"; Proc. Manchester Lit Phil Soc, vol 8, 1874
  10. DB Spalding (1971) "Mixing and chemical reaction in confined turbulent flames"; 13th International Symposium on Combustion, pp 649-657 The Combustion Institute
  11. DB Spalding (1987) "A turbulence model for buoyant and combusting flows"; International J. for Numerical Methods in Engineering vol 24, pp 1-23
  12. DB Spalding (1994) Poster session, International Heat Transfer Conference, Brighton, England
  13. DB Spalding (1996) "Multi-fluid models of Turbulence; Progress and Prospects; lecture CFD 96, the Fourth Annual Conference of the CFD Society of Canada, June 2 - 6, 1996, Ottawa, Ontario, Canada

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Turbulence-modelling high-lights through four half-centuries

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Links to explanations of the micro-mixing hypothesis of MFM