CFD as applied to combustion: past, present, and future; March 1999

in "CFD - Technical Developments and future Trends"

Also available as: www.cham.co.uk/phoenics/d_polis/d_lecs/cmbstr5/cmbstr5.htm

- Kolmogorov's "bright idea"
- Presuming the PDFs; another "good idea at the time"
- The direct route to the goal
- Relation to "flamelet" and other models
- Practical consequences
- Concluding remarks
- References

- 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:##### 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 <
> (say) to be related to gradients of time-mean velocity by way of an "effective viscosity"; then**u*v****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]

.

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**.

But that did not happen for another 24 years!]

So the next invention (by Bray, 1980) was the "flamelet" model, which involves the presumption that the turbulent mixture consist of:-
**fully-burned**gas at the local time-average fuel-air ratio; - fully-
**un**burned gas at the local time-average fuel-air ratio; - 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.

**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.

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.

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.

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?

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