Turbulent combustion models for CFD in the year 2000

A summary and update of:
CFD as applied to combustion: past, present, and future; March 1999

Brian Spalding, of CHAM Ltd

Paper presented at I Mech E, London, December 14,1999
in "CFD - Technical Developments and future Trends"
Also available as: www.cham.co.uk/phoenics/d_polis/d_lecs/cmbstr5/cmbstr5.htm

Contents of the present lecture

1. Kolmogorov's "bright idea"
2. Presuming the PDFs; another "good idea at the time"
3. The direct route to the goal
4. Relation to "flamelet" and other models
5. Practical consequences
6. Concluding remarks
7. References

1. Kolmogorov's "bright idea"

• To attain the flow-prediction aims of CFD, we need to ascribe values to time-averages of non-linear functions of fluid variables, for example:
  <T**4>
  

  where T is the absolute temperature.

• Other quantities which we need to evaluate are time-averages of multiplication products, for example:
  <u*v>
  for shear stresses and
  <a*b>
  for chemical reactions,
  

  where u and v are instantaneous velocity components, and a and b are concentrations.

• "All" that we need, to enable us to do so, is knowledge of the relevant "probability-density functions", for example the left-hand figures in:
  • this for a single variable, and

  • this for a pair of variables.

  Note that the right-hand-side diagrams are reminders of the "inter-mingled fluid population" concept of a turbulent fluid, which underlines the "multi-fluid model"

• It is now possible to calculate the PDFs; but it was not in 1942, when A.N.Kolmogorov had a "bright idea", namely:
  • "Let's see if we can devise differential equations which have just one or two statistical quantities as the dependent variables.

  • "Then, if we can solve these equations, and

  • "if we can find sufficiently general empirical constants to insert in them, and

  • "if we can connect these quantities to the ones we want by empirical relationships,

  • "maybe we can do without the PDFs altogether".

• One of the empirical relationships (already proposed by Boussinesq) was that turbulent flows were sufficiently like laminar ones for shear stresses, for example <u*v> (say) to be related to gradients of time-mean velocity by way of an "effective viscosity"; then this could be computed from the "dreamed-up" equations for the statistical quantities.

• Other innovators, for example Ludwig Prandtl and Howard Emmons, had the same idea a little later; but it is fair to say that the whole of modern (sometimes ludicrously called "classical") turbulence modelling, springs from Kolmogorov's "bright idea".

• It was a good idea at the time; and it worked fairly well for the (rather undemanding) turbulent shear flows; but is no use at all for chemical reaction

  [or for flows in which body forces (gravity/swirl) act on fluids exhibiting density fluctuations; but that is not the subject of the present paper]

  .

2. Presuming the PDFs; another "good idea at the time"

That knowledge of the PDFs was needed for predicting reaction rates was obvious in the early 1970s; and the first idea was that it might suffice to presume their shape, and devise an additional differential equation so as to find out everything elsew hich was necessary.

This notion led to:

• the eddy-break-up model (EBU; Spalding, 1971)
• the concentration-fluctuations model (CFM; Spalding, 1971)
• the eddy-dissipation concept (EDC; Magnussen, 1976)
• the two-fluid model (2FM; Spalding, 1981)
• and innumerable variants on the same theme

All of these involved the supposition that any turbulent mixture could be treated as the inter-mingling of two fluids, the states and mixture fractions of which required to be computed from easy-to-formulate differential equations.

This represented an advance on Kolmogorov's "ignore-the-PDFs" approach; but it was not good enough.

[Somebody might have thought at the time: If two is not enough, what about four? or eight? or sixteen? etc? Refine the grid, dummy!

But that did not happen for another 24 years!]

1. fully-burned gas at the local time-average fuel-air ratio;
2. fully-unburned gas at the local time-average fuel-air ratio;
3. and a small amount of intermediate-state gas with a PDF which is the same as that prevailing in laminar steadily-propagating one-dimensional flames.

This enables CFD/chemistry specialists to perform expensive calculations; but, in the present author's view, has no other merit (if that is the right word) whatever.

3. The direct route to the goal

• Presumed-PDF methods are what are mainly used by "high-tech" engineering companies at the present time. Nevertheless direct methods of calculating PDFs have been available for many years.

• The "how-to-do-it" idea was provided by Dopazo and O'Brien in 1974; however, those authors were not numerical analysts at the time, so provided no solutions.

• In 1982, Pope started to solve the relevant equations; but he used a "Monte Carlo" method, which proved to be expensive in terms of computer time. This may have given the "compute-the-PDF" approach a bad name. It is indeed little used in engineering practice.

• More recently, the present author made the even-more-direct approach of discretising the PDF, and solving for its ordinates. This so-called "Multi-Fluid-Model (MFM)" approach has proved to be simple in concept, economical in implementation, and realistic in its predictions.

  This is what "dummy" should and could have done many years before. Turbulence-modelling history is a catalogue of missed opportunities and false starts.]

• MFM can be regarded as what EBU should swiftly have developed into in the 1970s, having as many fluids, and as many PDF dimensions ( 2 will be quite enough for the time being), as the situation requires.

• MFM is "too new" (five-years-old!) to have been adopted in engineering practice.

• At some time in the next millennium it will be (the author believes); perhaps even in Year 2000.

4. Relation to "flamelet" and other models

Since the "laminar-flamelet model LFM" is the most "advanced" which is currently used by engineering companies, it is worth exploring the relations between it and MFM.

This has been done in a recent paper, which shows that MFM reduces to LFM in restricted circumstances; but it has a much wider range of validity.

The highlights of the just-mentioned paper can be seen by clicking here.

5. Practical consequences

MFM is not just a scientist's plaything: it can already be used to enable better designs to be distinguished from worse ones.

A recent paper illustrates this by showing how MFM enables the smoke-generating propensities of gas-turbine-combustor designs to be predicted.

The highlights of this paper can be seen by clicking here.

6. Concluding remarks

It is the author's view that all time spent on CFD calculations incorporating the "presumed-PDF" approach is wasted; and, if design decisions are based on their outcome, the desisions will be correct only by chance.

Those who have considered but do not use the alternative, namely calculating the PDFs, argue only:

• it is too expensive (which may be true of Monte Carlo, but is certainly not of MFM);

• what we have already is good enough (which is hard to prove);

• the superiority of MFM has not been proved (which is true of anything which one has not tried).

To these arguments it can only be answered that:

• Kolmogorov's idea was adopted only because of its inherent plausibility and practicability;

• the same was true of EBU, EDC, presumed-PDF, and all the rest;

• none of these were "proved", "validated", "generally accepted" before they were taken up; nor could they have been.

• How interesting it is that the conjectures of almost thirty years present such obstacles to the innovations of the 1990s!

• Will the new millennium allow us to be more adventurous?

7. References

1. DB Spalding (1971) "Mixing and chemical reaction in confined turbulent flames"; 13th International Symposium on Combustion, pp 649-657 The Combustion Institute
2. DB Spalding (1971) "Concentration fluctuations in a round turbulent free jet"; J Chem Eng Sci, vol 26, p 95
3. 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
4. Bray KNC in Topics in Applied Physics, PA Libby and FA Williams, Springer Verlag, New York, 1980, p115
5. SB Pope (1982) Combustion Science and Technology vol 28, p131
6. C Dopazo and EE O'Brien (1974)
   Acta Astronautica vol 1, p1239

   DB Spalding (1999) "The use of CFD in the design and development of gas-turbine combustors"; www.cham.co.uk; shortcuts; CFD

   DB Spalding (1995) "Models of turbulent combustion" Proc. 2nd Colloquium on Process Simulation, pp 1-15 Helsinki University of Technology, Espoo, Finland

   DB Spalding (1998) The simulation of smoke generation in a 3-D combustor, by means of the multi-fluid model of turbulent chemical reaction: Paper presented at the "Leading-Edge-Technologies Seminar" on "Turbulent combustion of Gases and Liquids", organised by the Energy-Transfer and Thermofluid-Mechanics Groups of the Institution of Mechanical Engineers at Lincoln, England, December 15-16, 1998

   Spalding DB (1999) "Connexions between the Multi-Fluid and Flamelet models of turbulent combustion"; www.cham.co.uk; shortcuts; MFM