Click here for Contents
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
Abstract of the March 1999 lecture
Contents of the present lecture
- 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
- 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:
where T is the absolute temperature.
- Other quantities which we need to evaluate are
time-averages of multiplication products, for
for shear stresses and
for chemical reactions,
where u and v are instantaneous velocity components,
and a and b are
- "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
- "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
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!]
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-unburned 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
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
- 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
- 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
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
The highlights of this paper can be seen by
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
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
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";