By Brian Spalding of Concentration, Heat and Momentum, Ltd

Presentation at CODE Annual SEMINAR in Teraelahti, Finland, 3-4 October 2001

- It is widely recognised that realistic simulation of turbulent
chemically reacting gases requires attention to the
*inhomogeneity*of turbulent mixtures. - This inhomogeneity is expressible mathematically by way of PDFs, i.e.
*probability-density functions*. - Before any means of predicting PDFs were available (in the early 1970s),
it was necessary to
*guess their shape*, while computing their**amplitude**from a transport equation similar to those of the k-epsilon turbulence model. - When equations for predicting
**both**shapes and amplitudes of PDFs from physical principles were first formulated (early 1980s), the*Monte Carlo method*was adopted for solving them.This proved to be both computationally expensive and intellectually demanding.

The guessed-PDF practices therefore continued.

- Since 1995 a
*more economical and simpler*method of predicting PDF shape and size has become available. This is the multi-fluid model, and it has shown how rarely the guessed shape is correct. - It will be argued that continuing to use guesses rather than
computations can
*no longer be justified*. - Although further research-guided refinement is desirable, MFM is ready for use right now.

Notes:

- The under-lined "click here" items in the following text are intended for browser-using readers only.
- This document can be viewed as www.cham.co.uk\phoenics\d_polis\d_lecs\turb2001\mfm_comb.htm

- Historical notes
- The main features of MFM
- The relation between MFM and other models of turbulent combustion
- Applying and extending MFM
- Conclusions
- References

The advent of the gas-turbine and the development of rockets during
the 1940s and 1950s, stimulated much research on combustion; and the
simultaneous development of digital computers enabled quantitative
models for **laminar-flow** phenomena to be created.

For example, one-dimensional flame propagation though pre-mixed gases become completely understood already in the 1950s [Spalding,1955]; and, once the appropriate chemical-kinetic and transport-property data had been gathered, numerical predictions fitted experimental data rather well.

However, experiments on **turbulent** pre-mixed flames showed effects for
which there were no explanations. For example, Williams *et al*
[1949]
showed that the speed of propagation of a baffle-stabilized flame,
confined in a duct, **decreased** when the initial temperature
was raised; and it was very little dependent on the chemical
composition of the gases.

This behaviour was so unlike that of laminar flames that a new
hypothesis had to be devised for its explanation, namely the
**"Eddy-Break-Up Hypothesis" (EBU)** [Spalding, 1971a].

In modern terms, EBU can be regarded as a **"two-fluid" model**; for
it postulated:

- that the gas mixture consisted of inter-mingled fragments of fully-burned and fully-unburned gases; and
- that the rate of chemical reaction, i.e. of transfer of mass from the unburned to the burned state, depended only on local hydrodynamic properties of the turbulence (specifically epsilon/k)

Despite its simplicity, and its disregard of chemical-kinetic influences, EBU proved to be largely successful. It is still in widespread use.

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__1.2 Turbulent diffusion flames; the fluctuations-transport model__

**Laminar** diffusion flames, i.e. those in which the supplies of
fuel and oxygen are provided by separate streams, were well
understood in the early 1970s.

Attempts were made to fit turbulent diffusion flames into the same
theoretical framework by supposing, as did Boussinesq [1877] that
the turbulence enlarged the
** effective diffusion coefficient **of the gases; but these were not
completely successful.

The reason was clearly shown by the experiments of Hawthorne *et
al* [1949], which revealed what they called
**"unmixedness".**

This entailed that flames were **visibly** much longer than the
effective-diffusion-coefficient approach could explain.

To fit the experimental data, it proved necessary once again to invent a new hypothesis, namely that the gas at any point consisted of intermingling fragments having greater and smaller fuel-air ratios than the local mean value.

Then the root-mean-square value of fuel-air ratio differences was
computed
from a **"fluctuations-transport"** equation of the type used in the
then-popular hydrodynamic models of turbulence
[Spalding, 1971b].

In its original form, this model, which is referred to as the FTM below, can be seen as being simultaneously:

- another two-fluid model,
- and also the progenitor of the guessed-PDF school;

for it was supposed that, at any point, the fuel-air ratio could have one or other of only two values.

Within each fluid, the gases were regarded as being in chemical equilibrium. Once again, therefore, the influence of finite chemical reaction rates could not be accounted for.

The fluctuations-transport equation is still in widespread use, albeit in conjunction with more elaborate guesses about the shape of the PDF.

__1.3 The eddy-dissipation concept (EDC)__

Magnussen and Hjertager [1976] proposed a model which, in some respects, bridged the gap between EBU and FTM, and allowed chemical-kinetic limitations to have an effect.

It was again a two-fluid model, in that the state of the gas at any location was supposed to jump between two conditions; but these were:

- the mixture-average condition; and
- the "interstitial-fluids" condition;

Moreover, necessarily, the volume fraction of fluid **b** was supposed
to be much less than unity.

Further assumptions were made about the rates of heat and mass transfer between the two fluids, the details of which the present author will not presume to summarise.

For the purposes of the present lecture it suffices to emphasis
that EDC, and its later variants, allow no more than **two**
states of fluid to
co-exist at the same location.

__1.4 The full two-fluid model(F2FM)__

A more complete account was provided, much later [Spalding, 1983],
of how finite chemical reaction rates could be accounted for. This was
achieved by utilising the so-called
IPSA procedure that had
been developed for **two-phase** flows, such
as steam and water [Spalding, 1980].

This model was applied to both steady and unsteady flames, as illustrated by :

- the confined
pre-mixed flame of Williams
*et al*[1949] and - transition from deflagration to detonation.

Whereas the EBU and FTM models were adopted swiftly by CFD specialists, this was not the case with the full two-fluid model (F2FM).

Probably the reason was that F2FM introduced too many novelties at the same time; for example the two fluids were allowed to possess not only different fuel-air ratios and degrees of reactedness but also different velocity components.

Another was perhaps that not many specialists possessed the means, at that time, of solving more than one set of Navier-Stokes equations simultaneously.

Finally, EBU and FTM appeared to many to be "good enough" for practical purposes, a view which (strangely) can be encountered even today.

Nevertheless, phenomena could be predicted by the F2FM which are still outside the scope of all the popular turbulence models, for example "un-mixing".

__1.4 The four-fluid model (4FM)__

In 1995, a more modest step was proposed for improving on EBU (and
other 2-flid models: the number of fluids form two to four; and
differences of velocity between them were **not** allowed
[Spalding, 1995a].

This development enabled finite chemical reaction rates to be accounted for.

It was used successfully for simulating both steady and unsteady flames.
including:

the Williams turbulent
flame confined in a duct, and

an explosion in an off-shore oil platform
[Freeman and Spalding,1997].

__1.5 The fourteen-fluid model (14FM)__

Like EBU, 4FM handled pre-mixed gases only. When variations of both fuel-air ratio and reactedness were to be handled simultaneously, the minimum number of fluids needed to provide at least qualitative realism was 14.

This was used in order to simulate a
turbulent Bunsen-burner flame
[Spalding, 1995b] and so to compute:

the contours of concentration of individual fluids,
such as this,
and the PDFs at various locations such as
this.

It should be noted that a **two-dimensional PDF** was involved in this
model. The dimensions were:

- the "mixture fraction", i.e. the mass fraction of material originating in the fuel-supply stream; and
- the "reacted-fuel fraction", i.e. the mixture fraction minus the mass fraction of unburned fuel (which is akin to, but not quite the same as, the reactedness.)

The four- and fourteen-fluid models were first steps on the road towards the multi-fluid model which was first systematically presented in a conference paper [Spalding, 1996b] in Canada.

The "multi" in the name implies that a turbulent mixture can be regarded as a "population" having an arbitrary number of "ethnic" components.

These concepts will be expanded upon below.

The working concepts of a multi-fluid model are **few**
and ** simple.**
They are as follows:-

- The fluid mixture is regarded as composed of an intermingling
**population of individual fluids**, each distinguished by the interval it occupies on the (discretised) PDF abscissa. - A
**differential equation**of the standard "conservation" type is solved for the**mass fraction**(i.e. PDF ordinate) of each member of the population; The solutions of these equations provide the PDF for every location and time.MFM therefore departs from the practice, introduced by Kolmogorov [1942], of solving equations for

**statistical properties**of the turbulent fluid, such as k, the turbulence energy. - The
**source terms**in these equations express:-- the postulated
**micro-mixing hypothesis**, which defines:- the frequency with which the different fluids
"collide"; and
- the re-distribution of material between population
members which ensues;

- the frequency with which the different fluids
"collide"; and
- the
**speed of movement**of material**in "population space"**, as when a heat source shifts material from low-temperature intervals into higher-temperature ones.

- the postulated
**Additional equations**, either differential or algebraic, are also solved for non-discretised, i.e.**continuously varying, dependent variables**, for example the velocity components of the distinct fluids, each of which will ordinarily have a different density and so be subject to different body forces.- Such operations of course increase computer times as compared with those required for Kolmogorov-type models; but the increases are not exorbitant (See below for an example).

In these pictures, the left-hand half gives the PDF; the right-hand half is merely a reminder of the "inter-mingling fluid" concept.

Populations of fluids may be **multi-dimensional**. Examples of
**two-dimensional** populations would be:-

- the use of:
- temperature and
- salinity

- the use of:
- fuel/air ratio and
- completeness of reaction

for simulating the flow and combustion of turbulent gases in a combustion chamber.

A discretised **two**-dimensional PDF looks like
this, or
this.

Examples of **three-dimensional** populations would be:

- the discretization of all three velocity components for the
detailed simulation of turbulent hydrodynamics; and
- the use of
**fragment size**as a third population dimension when temperature and salinity are the other two.

It is important to recognise that **the modeller can choose**
freely:-

**which**dependent variables to discretise,- which to
allow to
**vary continuously**for each fluid; and **how finely**to discretise.

These choices can be made with the aid of:

**physical insight**into what variables are of dominant importance; and**population-refinement**studies of essentially the same nature as are used to determine how finely it is necessary to sub-divide space and time.Example 3: how many fluids are needed for accuracy when predicting smoke generation

These choices may differ from place to place and from time to
time. MFM allows the possibility of using **"un-structured"** and
**"adaptive"** population grids.

It should also be understood that **MFM models can be combined with
enlarged-viscosity models**.

Thus it is common to use the k-epsilon model for the hydrodynamics when the phenomena of greater interest involve chemical reaction or radiation.

This was done in the examples shown here:-

- Example 4: Smoke generation in gas-turbine combustors; and
- Example 5: Chemical reaction in a paddle-stirred reactor;

**Choice (1): Mixture fraction as the only population-distinguishing attribute,**

Most practical combustion devices are of the "diffusion-flame" type, in the sense that the fuel and the oxidant enter the combustion space at different locations, and mix within that space.

Since the local fuel-air ratio has such a profound effect upon the combustion process, it is therefore obvious that the mixture fraction (MIXF) should be a PDA.

This is the choice which was made for the above-described simulation of the smoke-generating combustor.

Also made there was the 'mixed-is-burned' assumption, signifying that the composition of each component of the population (apart from its smoke content) depends only in MIXF. There was therefore no need to consider discretisation in the reacted-fuel-proportion dimension.

In the absence of heat losses, the temperature of each component is similarly dependent on MIXF alone. It is therefore possible to associate a smoke-generation rate with each population component.

The total smoke concentration of the mixture can then be calculated by adding together the contributions of the individual fluids.

In MFM parlance, the smoke concentration is a CVA, i.e. a continuously-varying attribute.

If heat losses, for example by radiation to cold walls, can
**not** be neglected, it is wise to treat the enthalpy also as a
CVA.

The same is true of NOX, if that is to be computed.

Indeed, if the validity of the mixed-is-burned assumption is doubtful, the reacted-fuel proportion can also be treated as a CVA.

**Choice (2): Reacted-fuel proportion as a second PDA**

If such an exploration of the effect of finite-rate main-reaction
chemistry demonstrated that strong departures from equilibrium were
possible, it would be wise to investigate their interaction with the
turbulence by using a **two**-dimensional population, with RFP (ie
reacted-fuel proportion) as the second population-distinguishing
attribute.

PDF's would then arise of the kind which have already been seen above.

Another, with less colour but more content is shown here.

In this picture, the right-hand half is being use to show some information about a CVA.

Evidently, the mixed-is-burned presumption would NOT be justified in this case. If it had been, the PDF would have appeared like this, with most of the material in the uppermost population elements..

**Choice (3): Reacted-fuel proportion as the only PDA**

There are, of course, some practical circumstances in which the fuel-air ratio is almost uniform, whereas the major difference between the gas fragments is their degree of reactedness.

Combustion in a gasoline-engine cylinder is of this kind.

The PDF can therefore again be one-dimensional, with reacted-fuel fraction as the PDA.

The following picture shows an example of such a PDF,

The shapes depend greatly of the ratios of the micro-mixing (CONMIX) and the chemical-reaction rate (CONREA) to the local flow rate, as the following further cases illustrate:

case 2, case 3, case 4, and case 5,

To attempt to guess such shapes correctly would appear to be a hopeless enterprise; and to base engineering designs on the guesses an unwise one.

It follows that:

- the 4-fluid model, with its one-dimensional population and reactedness as its PDA, and
- the 14-fluid model with its two-dimensional population,

It is true that it is a two-fluid model; but the values of the population-distinguishing attributes are not fixed, as in the case of EBU, but vary with position within the flame, in a manner determined by the solutions of the equations for the mean and RMS-deviation values of MIXF.

However, MFM can do anything that FTM can do, and more, as is illustrated by the following figure extracted from a report by S.V.Zhubrin,

The figure shows that agreement is obtained between FTM and MFM when seventeen fluids are used; and of course MFM computes the PDF which FTM has to guess.

As compared with F2FM, MFM in its present form does both more and less.

It does **more** in that it can handle many fluids, not just two; but
it does **less** in that all its fluids share the same velocity
component. It can not therefore, as F2FM can, simulate the
differential acceleration of hotter and colder gases illustrated
above.

This deficiency will be removed by work currently in progress; but not as F2FM did, by allowing each fluid to have its own set of Navier-Stokes equations; for that would be needlessly expensive.

Instead, each fluid will have,
its own velocity **differences** from the mean; and these will
be calculated, as continuously varying attributes, by allowing for
only:

- inter-fluid friction; and
- differences of body force per unit volume.

However, the first such implementation was made by Pope [1982], who chose to adopt a Monte-Carlo method of solution; and prior to 1995, this was the only method which appears to have been employed by anyone.

The result has been that "pdf-transport" and "Monte-Carlo" have become so frequently associated that it seems best to treat "pdf-transport" and "multi-fluid" models as wholly distinct.

Because of the Monte Carlo method, the former appears to lack some conceptual and practical advantages which the "discretised-PDF" nature of MFM possesses.

However, given unlimited computer time, and care to employ precisely the same micro-mixing formulae, MFM and PDF-transport should produce the same answers.

Among the first were Lockwood and Naguib [1975].

"Clipped-Gaussian" and "beta-function" presumptions have both had their adherents; and large amounts of computer time have been consumed in exploring the implications of one or the other.

Unfortunately, none of the presumptions appear to have better claims than others to be preferred on theoretical or experimental grounds; and indeed the validity of the fluctuations-transport equation itself is little more than than a matter of faith.

MFM, even in its present rather primitive state, has shown that PDF shapes can be widely various. For example, to click on the links in the following table extracted from the 1998 lecture will reveal the variety.

Figure | CONMIX | CONREA | RB | ave. R | rms. R |

6 | 10.0 | 100.0 | 0.0 | 0.577 | 0.448 |

7 | 10.0 | 50.0 | 0.0 | 0.472 | 0.427 |

8 | 100.0 | 100.0 | 0.0 | 0.937 | 0.197 |

9 | 100.0 | 50.0 | 0.0 | 0.922 | 0.202 |

10 | 100.0 | 25.0 | 0.0 | 0.897 | 0.206 |

11 | 100.0 | 10.0 | 0.0 | 0.815 | 0.199 |

12 | 10.0 | 10.0 | 1.0 | 0.739 | 0.354 |

13 | 100.0 | 50.0 | 1.0 | 0.963 | 0.145 |

14 | 100.0 | 10.0 | 1.0 | 0.927 | 0.151 |

15 | 100.0 | 5.0 | 1.0 | 0.884 | 0.148 |

16 | 100.0 | 1.0 | 1.0 | 0.541 | 0.114 |

Moreover:

- all the above have been derived from on a
**single**version of the MFM micro-mixing hypothesis (there are several); and - in reality
**two-dimensional**PDF's are needed, which no FTM user has until now dared to "presume".

While the notion is not implausible, a body of theory and computation has been built upon it which, in the author's opinion, is disproportionate.

The MFM theory, conceptually, also recognises that there may be such regions; but it allows also for their non-appearance and for the influences of such non-dimensional quantities as Reynolds number and Peclet number based on laminar flame speed.

The relation of MFM to flamelet theory has been discussed at length in a lecture devoted to the subject [Spalding, 1998]

Flamelet theory has nothing to say about combustion in non-pre-mixed gases.

Finally, **direct numerical simulation** [Schumann, 1973] should
be mentioned;
not because DNS is a turbulence model but in order to lead to
the following remark:

Whereas DNS has sometimes been used as a means of deriving the constants and functions of Kolmogorov-type models, such as k-epsilon, it could now perhaps be better be used for testing and augmenting the micro-mixing hypotheses of MFM.

Since all that is involved is the appropriate post-processing of the results of DNS computations, this should not be difficult to contrive.

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It is here argued that MFM is ready for practical use **now**,
for the following reasons:

- Although it has been necessary to introduce
**new words**and acronyms (e.g. population, PDA, CVA etc) to describe the new features of MFM, the model itself is**not**new; it is merely an natural extension of models which have been used for many years. - The eddy-break-up model, for example, is to be found, in some form
or other, in almost every CFD code which simulates turbulent
combustion.
As has been explained above, MFM is merely an obvious extension of EBU ideas.

Everyone knows that CFD predictions improve when the grid is refined in geometrical space-time: MFM is simply EBU with

**grid refinement in "population space".** - The importance of PDFs for describing the local non-uniformity of
turbulent gas mixtures is widely recognised; which is why many CFD
codes embody PDF
**presumptions**.But why should one guess when one can calculate? This is what MFM allows

Moreover, even though it is to be expected that the predictive accuracy of MFM will improve as a consequence of easily conducted research, it has already been shown that it predicts root-mean-square fluctuations just as well as the fluctuations-transport model which "PDF-presumers"

**must**employ.So why not use it, and get the PDFs as well?

Moreover, MFM can produce two-dimensional PDFs as easily as 1D ones; and these are certainly needed for combustion processes.

- The PDF-transport model (PDFTM) of Pope and his followers, has been
studied, and to some extent validated, for many years. It
receives significant research funding; and it is recognised by
many specialists as the one which they would use if it were not so
expensive.
MFM can now be regarded as "the poor man's PDFTM"; and it is not only cheaper to use: it can do much more.

- In summary, it is hard to make a reasonable case for
**not**using MFM whenever turbulent chemical reaction is of importance.

Among the turbulent-combustion applications for which MFM is suitable in its present state are:

- Flame propagation in gasoline engines, for which no other
physically-well-founded model exists;
- Explosions in off-shore oil platforms, and industrial equipment,
in which the presence of obstacles significantly modifies the
turbulence.
- Gas-turbine combustors, which must be designed so as to emit as
little smoke and NOX as possible.
- Gas-fired furnaces for space-heating or power production.
- Oil-, coal-, wood- and peat-fired furnaces, in which the fuel
particles will need to be computed as "continuously-varying
attributes".
- Diesel engines, which are similar, with the complications of
time-dependence and large pressure variations.
- Flames resulting from spills of gasoline or liquefied petroleum
gas.

It has become fashionable to appply CFD to the design of the large paddle-stirred chemical reactors which are used in chemical industry; and most commercial CFD codes possess some such capability.

However, the be-all and end-all of such reactors is to effect a
controlled **reaction**; and designing for this requires the
ability to predict the micro-mixing process.

Only MFM provides this at present.

Clicking here will lead to an example of such an application.

Oil- and LPG-spills have already been mentioned; but there are other environmental hazards to the simulation of which MFM can make a contribution. One example may suffice:

- Spilled chemicals may damage animal and plant life, whether on
land or under water.
- Living beings may be able to withstand small doses of the
chemical but may succumb to larger ones.
- Therefore, in order to predict the effect of a spill in
turbulent atmospheres or waters, it is necessary to be able to
predict the proportion of the most-toxic fluid in the population.
- predicting the mixture-average or RMS-fluctuation values is not
enough.
- Only MFM can do this.

The assertion that MFM is ready for use now by no means implies
that further research is not desirable. It **is**; and most
desirable of all would be the experimental measurement of PDFs which
would permit confirmation, or would lead to refinement, of the underlying
physical hypotheses.

The latter, and the uncertainties attending them, have not been emphasised in the present lecture; but full accounts can be accessed by clicking:

here, for
an account of "coupling and splitting"; or

here, or
here, or
here,
for
an account of "the brief encounter".

Preferably such experiments would be carried out on simple and easily controlled flows such as:

- the well-known plane turbulent mixing layer; or
- the recently-devised "puff-jet";

and there now exist easy-to-use procedures for systematically adjusting constants to fit CFD data.

It is therefore to be hoped that the academic-research community will soon see the opportunities which the un-tilled field of MFM presents to them.

The following thought may provide sufficient stimulus:

- One of the sectors of CFD which has received much inventive
attention is that of "grid-generation"; by which has always until
now been understood the creation of grids
**in geometrical space**. - MFM has introduced the idea of
**"population grids"**, the generation of which, in the current implementation of MFM, is in its infancy. - All the grids shown so far have been uniform, structured,
orthogonal and fixed in time.
- However, most descriptors of geometrical grids, such as:
**"unstructured", "non-orthogonal", "self-adaptive",**etc, could find advantageous application to population grids. - Surely
**some**young mathematicians will rise to the challenge?

Experimental and mathematical researchers will be very welcome; but even more so those imaginative scientists who can perceive which limitations of the current MFM are most disadvantageous, and then remove them.

For example:

- MFM can be applied to hydrodynamic phenomena as well as mixing
and chemical reaction, as shown
above.
However, momentum transfer in a "brief encounter" is more complex than heat and mass transfer; for two fluid fragments which collide "head-on" will scatter material into the lateral directions.

Research and thinking on this topic has only just begun.

- Transport of heat and mass in "brief encounters" is also not
without its complexities. Thus, when the Schmidt number exceeds the
Prandtl, a brief encounter imparts more heat than matter to the
"offspring", as is evident from study of
the "un-mixing example".
Introducing the effect into MFM requires physical intuition expressed in mathematical terms.

- Finally, the current MFM has a primitive
length-scale-modifying formula which applies to the whole
population.
Perhaps however the length scale should be a new PDA?

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

- The now-six-years-old multi-fluid model of turbulent combustion is
ready for practical use.
- In those limited circumstances in which more primitive models
(namely EBU, EDC, FTM, "flamelet")
are truly valid, MFM will probably produce the same limited results,
but much more besides.
- Where they are
**not**valid, MFM will still produce results which are at least plausible, and probably more reliable. - Where Monte-Carlo-based "PDF transport" has been found to produce
satisfactory results, the same results can probably be produced via
MFM, but with much smaller computational expense and greater ease of
understanding.
- The extent to which the foregoing assertions can be justified by
example is, as always when new territory is being explored, rather
small.
- It is therefore highly desirable that they should now be put to the practical test.

[ Note: This list contains not only papers directly referred to above, but also some which appear in other documents regarding MFM ]

- MJ Andrews (1986) "Turbulent mixing by Rayleigh-Taylor instability"; PhD Thesis, London University
- J Boussinesq (1877)
*"Theorie de l'ecoulement tourbillant"*; Mem. Pres. Acad. Sci. Paris, vol 23, 46 - P Bradshaw, DR Ferriss and NP Atwell (1967) "Calculation of boundary-layer development using the turbulent energy equation"; J Fluid Mech, vol 28, p 593
- Bray KNC in
*Topics in Applied Physics*, PA Libby and FA Williams, Springer Verlag, New York, 1980, p115 - KNC Bray and PA Libby "Counter-gradient diffusion in pre-mixed turbulent flames"; AIAA J vol 19, p205, 1981
- Bray KNC
*Proc Roy Soc London A*431:315-355, 1990 - Candel S, Veynante D, Lacas F, Maistret E, Darabiha N and Poinsot T,
in
*Recent Advances in Combustion Modelling*Lattoutourou B (Ed). World Scientific, Singapore, 1990 - Cant RS, Pope SB, Bray KNC,
*Twenty-Third Symposium (International) on Combustion*. The Combustion Institute, Pittsburgh, 1990, pp 809-815 - JY Chen and W Kollmann (1988) "PDF modelling of non-equilibrium effects in turbulent non-premixed hydrocarbon flames"; 22nd Int. Symp. on Combustion, Combustion Inst pp 645-653
- JY Chen and W Kollmann (1990) "Chemical models for PDF modelling of hydrogen-air non-premixed turbulent flames"; Combustion and Flame, vol 79, pp 75-99
- SM Correa and SB Pope (1992) "Comparison of a Monte Carlo PDF/ finite-volume model with bluff-body Raman data" Twenty-Fourth International Combustion Symposium The Combustion Institute, pp279-285
- RL Curl (1963) AIChE J vol 9, p 175
- BJ Daly and FH Harlow (1970) "Transport equations in turbulence"; Phys Fluids, vol 13, p 2634
- C Dopazo and EE O'Brien (1974) Acta Astronautica vol 1, p1239
- M.A.Elhadidy (1980),'Applications of a low-Reynolds-number turbulence model and wall functions for steady and unsteady heat-transfer computations', PhD Thesis, University of London
- D Freeman and DB Spalding (1995) "The multi-fluid turbulent combustion model and its application to the simulation of gas explosions"; The PHOENICS Journal (to be published)
- N Fueyo (1992) "Two-fluid models of turbulence for axi-symmetrical jets and sprays"; PhD Thesis, London University
- FH Harlow and PI Nakayama (1968)
*"Transport of turbulence-energy decay rate"*; Los Alamos Sci Lab U Calif report LA 3854 - WR Hawthorne, DE Weddell and HC Hottel (1949) "Mixing and combustion in turbulent jets" Third Symposium on Combustion, published by Williams and Wilkins pp 266-288
- NM Howe and CW Shipman "A tentative model for rates of combustion in confined turbulent flames" 10th International Symposium on Combustion, p 1139 The Combustion Institute, 1965
- ICOMP-94-30; CMOTT-94-9; "Industry-wide workshop on computational modelling turbulence"; NASA Conference Publication 10165
- JO Ilegbusi and DB Spalding (1987) "A two-fluid model of turbulence and its application to near-wall flows" IJ PhysicoChemical Hydrodynamics , vol 9, pp 127-160
- JO Ilegbusi and DB Spalding (1987) "Application of a two-fluid model of turbulence to turbulent flows in conduits and shear layers" I J PhysicoChemical Hydrodynamics, vol 9, pp 161-181
- JH Kent and RW Bilger (1976) "The prediction of turbulent diffusion flame fields and nitric oxide formation" 16th International Symposium on Combustion, The Combustion Institute p 1643
- S.W.Kim and C.P.Chen (1989), 'A multi-time-scale turbulence model based on variable partitioning of the turbulent kinetic energy spectrum', Numerical Heat Transfer, Part B Vol 16 pp193
- W Kolbe and W Kollmann (1980) "Prediction of turbulent diffusion flames with a four-equation turbulence model" Acta Astronautica, vol 71 p 91
- AN Kolmogorov (1942)
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