Multi-fluid models for simulating turbulent combustion

By Brian Spalding of Concentration, Heat and Momentum, Ltd

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

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  1. Historical notes
    1. Pre-mixed flames; the eddy-break-up model (EBU)
    2. Diffusion flames; the fluctuations-transport model (FTM)
    3. Eddy-dissipation concept (EDC)
    4. The full two-fluid model (F2FM)
    5. The four-fluid model (4FM)
    6. The fourteen-fluid model (14FM)
    7. The multi-fluid model (MFM)
  2. The main features of MFM
    1. Basic concepts
    2. One-, two- and more-dimensional PDFs
    3. The modeller's options
    4. Combustion-specific choices
  3. The relation between MFM and other models of turbulent combustion
    1. EBU, 4FM and 14FM
    2. FTM
    3. F2FM
    4. "PDF-transport"
    5. "presumed-PDF"
    6. Flamelet models
    7. Direct numerical simulation
  4. Applying and extending MFM
    1. MFM's readiness for practical use
    2. Combustion applications
    3. Chemical reactors
    4. Environmental applications
    5. Experimental verification
    6. Numerical-method improvements
    7. Conceptual developments
  5. Conclusions
  6. References

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1. Historical notes

1.1 Pre-mixed turbulent flames; the eddy-break-up model

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.

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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:

  1. that the gas mixture consisted of inter-mingled fragments of fully-burned and fully-unburned gases; and
  2. 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:

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.

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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:

  1. the mixture-average condition; and
  2. the "interstitial-fluids" condition;
and it was in the latter that the chemical reactions were supposed to take place.

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.

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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 :

  1. the confined pre-mixed flame of Williams et al [1949] and
  2. 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".

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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].

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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:

  1. the "mixture fraction", i.e. the mass fraction of material originating in the fuel-supply stream; and
  2. 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.)

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1.6 The multi-fluid model (MFM)

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.

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2. The main features of the multi-fluid model (MFM)

2.1 Basic concepts (MFM)

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

  1. 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.

  2. 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.

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  3. The source terms in these equations express:-

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  4. 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.

  5. 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).

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2.2 One-, two- and more-dimensional PDFs

One-dimensional PDFs (discretized) look like this or this or this. [Left-hand diagram only]

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

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

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

Examples of three-dimensional populations would be:

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2.3 The modeller's options

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

  1. which dependent variables to discretise,
  2. which to allow to vary continuously for each fluid; and
  3. how finely to discretise.

These choices can be made with the aid of:

  1. physical insight into what variables are of dominant importance; and
  2. 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

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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:-

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2.4 Combustion-specific choices

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.

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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.

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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.

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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..

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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.

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3. The relation between MFM and other models of turbulent combustion

3.1 EBU, 4FM and 14FM

Just as the 4-fluid model was an obvious (but 25 years delayed!) extension to EBU; and the 14-fluid model an obvious extension to 4FM, so is MFM a natural extension to, and generalization of, all of them.

It follows that:

can be re-created simply by giving MFM the appropriate settings.

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3.2 The fluctuations-transport model (FTM) and

The same is not quite true of FTM.

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.

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3.2 F2FM

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:

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3.4 "PDF-transport"

The notion of a "PDF-transport" equation was first presented by Dopazo and O'Brien (1974); and it can reasonably be argued that MFM, despite having evolved independently, by way of EBU, FTM and F2FM, is in essence a numerical implementation of the Dopazo-O'Brien idea.

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.

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The guess employed in the first FTM. namely that the PDF in a non-pre-mixed gas consisted of two spikes, inspired numerous authors to invent smoother, and therefore more plausible, profiles.

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.

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Figure CONMIX CONREA RB ave. R rms. R

10.0 100.0 0.0 0.577 0.448
7 10.0 50.0 0.0 0.472 0.427

100.0 100.0 0.0 0.937 0.197

100.0 50.0 0.0 0.922 0.202

100.0 25.0 0.0 0.897 0.206

100.0 10.0 0.0 0.815 0.199

10.0 10.01.0 0.739 0.354

100.0 50.0 1.0 0.963 0.145

100.0 10.0 1.0 0.927 0.151

100.0 5.0 1.0 0.884 0.148

100.0 1.0 1.0 0.541 0.114


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

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3.5 Flamelet models

Several authors, including Bray[1981] and Peters [1986] have focussed attention on the notion that probably, when burned-gas and unburned-gas fragments make contact in a turbulent mixture, their interfaces are occupied by relatively thin regions in which diffusion and chemical reaction dominate.

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.

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3.6 Direct Numerical simulation (DNS)

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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4. Applying and extending MFMs

4.1 MFM's readiness for practical use

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

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4.2 Combustion applications

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

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4.3 Chemical reactors

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.

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4.4 Environmental applications

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:

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4.5 Experimental verification

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:

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.

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4.5 Numerical-method improvements

It is not only the experimentalists of academia to whom MFM offers opportunities: the ingenuity of mathematicians is also needed.

The following thought may provide sufficient stimulus:

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4.6 Conceptual developments

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:

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5. Conclusions

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

  1. The now-six-years-old multi-fluid model of turbulent combustion is ready for practical use.

  2. 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.

  3. Where they are not valid, MFM will still produce results which are at least plausible, and probably more reliable.

  4. 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.

  5. The extent to which the foregoing assertions can be justified by example is, as always when new territory is being explored, rather small.

  6. It is therefore highly desirable that they should now be put to the practical test.

6. References


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

  1. MJ Andrews (1986) "Turbulent mixing by Rayleigh-Taylor instability"; PhD Thesis, London University
  2. J Boussinesq (1877) "Theorie de l'ecoulement tourbillant"; Mem. Pres. Acad. Sci. Paris, vol 23, 46
  3. 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
  4. Bray KNC in Topics in Applied Physics, PA Libby and FA Williams, Springer Verlag, New York, 1980, p115
  5. KNC Bray and PA Libby "Counter-gradient diffusion in pre-mixed turbulent flames"; AIAA J vol 19, p205, 1981
  6. Bray KNC Proc Roy Soc London A 431:315-355, 1990
  7. 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
  8. Cant RS, Pope SB, Bray KNC, Twenty-Third Symposium (International) on Combustion. The Combustion Institute, Pittsburgh, 1990, pp 809-815
  9. 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
  10. 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
  11. 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
  12. RL Curl (1963) AIChE J vol 9, p 175
  13. BJ Daly and FH Harlow (1970) "Transport equations in turbulence"; Phys Fluids, vol 13, p 2634
  14. C Dopazo and EE O'Brien (1974) Acta Astronautica vol 1, p1239
  15. 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
  16. 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)
  17. N Fueyo (1992) "Two-fluid models of turbulence for axi-symmetrical jets and sprays"; PhD Thesis, London University
  18. FH Harlow and PI Nakayama (1968) "Transport of turbulence-energy decay rate"; Los Alamos Sci Lab U Calif report LA 3854
  19. 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
  20. 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
  21. ICOMP-94-30; CMOTT-94-9; "Industry-wide workshop on computational modelling turbulence"; NASA Conference Publication 10165
  22. 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
  23. 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
  24. 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
  25. 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
  26. W Kolbe and W Kollmann (1980) "Prediction of turbulent diffusion flames with a four-equation turbulence model" Acta Astronautica, vol 71 p 91
  27. AN Kolmogorov (1942) "Equations of motion of an incompressible turbulent fluid"; Izv Akad Nauk SSSR Ser Phys VI No 1-2, p56
  28. VR Kuznetsov; USSR Fluid Dynamics vol 14, p328, 1979
  29. BE Launder, GJ Reece, W Rodi (1975) "Progress in the development of a Reynolds-Stress closure" JFM, vol 68, p 537
  30. BE Launder and DB Spalding (1972) "Mathematical Models of Turbulence", Academic Press
  31. FC Lockwood and AS Naguib (1975) "The prediction of the fluctuations in the properties of free, round-jet, turbulent, diffusion flames", Comb and Flame, vol 24 p 109
  32. JP Longwell (1954) "Selected combustion problems" Butterworths, p 508
  33. 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
  34. MR Malin (1986) "Turbulence modelling for flow and heat transfer in jets, wakes and plumes"; PhD Thesis, London University
  35. HB Mason and DB Spalding "Prediction of reaction rates in turbulent pre-mixed boundary-layer flows" Combustion Inst European Symposium, pp 601-606, 1973
  36. JB Moss "Simultaneous measurements of concentration and velocity in an open pre-mixed flame" Combustion Science and Technology, vol 22, pp115-129
  37. D Naot, A Shavit M Wolfshtein (1974);"Numerical calculation of Reynolds stresses in a square duct with secondary flow"; Waerme u Stoffuebertragung, vol 7, p151
  38. M Noseir (1980) "Application of the ESCIMO theory of turbulent combustion"; PhD Thesis, London University
  39. KA Pericleous and NC Markatos (1991) "A two-fluid approach to the modelling of three-dimensional turbulent flames", in Proc. Eurotherm Seminar 17, Springer Verlag
  40. Peters N, Twenty-First Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1986, p 1231
  41. SB Pope (1982) Combustion Science and Technology vol 28, p131 Springer Verlag, New York, 1980, p115
  42. SB Pope (1985) Progr Energy Combust Sci vol 11, pp119-192
  43. SB Pope (1990) "Computations of turbulent combustion; progress and challenges" Twenty-Third International Symposium on Combustion, The Combustion Institute, pp 591-612
  44. L Prandtl (1925) "Bericht ueber Untersuchungen zur ausgebildeten Turbulenz"; ZAMM vol 3, pp 136-139, 1925
  45. L Prandtl (1945) "Ueber ein neues Formelsystem fuer die ausgebildete Turbulenz", Nachr. Akad. Wiss. Goettingen
  46. O Reynolds (1874) "On the extent and action of the heating surface of steam boilers"; Proc. Manchester Lit Phil Soc, vol 8, 1874
  47. RP Rhodes, PT Harsha and CE Peters (1974) "Turbulent-kinetic- energy analyses of hydrogen-air diffusion flame" Acta Astronautica vol 1 p 443
  48. PG Saffmann (1970) "A model for inhomogeneous turbulent flow"; Proc Roy Soc London vol A317 pp 417-433
  49. U Schumann (1973) "Ein Verfahren zur direkten numerischen Simulation ..." Thesis 1854 U Karlsruhe
  50. S Shaanen, JH Ferziger, WC Reynolds (1975) "Numerical Simulation of turbulence in the presence of shear"; Dept Mech Engg Stanford U, Rep TF-6
  51. KI Shchelkin USSR J Tech Phys vol 13, p 520, 1943
  52. DB Spalding (1955) "Some fundamentals of combustion"; Butterworth's, London
  53. DB Spalding (1969) "The prediction of two-dimensional steady turbulent elliptic flows" ICHMT Seminar, Herceg Novi, Yugoslavia
  54. DB Spalding (1971a) "Mixing and chemical reaction in confined turbulent flames"; 13th International Symposium on Combustion, pp 649-657 The Combustion Institute
  55. DB Spalding (1971b) "Concentration fluctuations in a round turbulent free jet"; J Chem Eng Sci, vol 26, p 95
  56. D B Spalding (1980), 'Numerical Computation of Multiphase Flow and Heat-transfer', Contribution to 'Recent Advances in Numerical Methods in Fluids', pp 139-167 Eds. C Taylor andamp; K Morgan, Pineridge Press, Swansea.
  57. DB Spalding (1982) "Chemical reaction in turbulent fluids" IC CFDU Report 82/8, June 1982
  58. DB Spalding 1979 "The influences of laminar transport and chemical kinetics on the time-mean reaction rate in a turbulent flame"; 17th International Symposium on Combustion, pp431-440, The Combustion Institute.
  59. DB Spalding (1983) "Towards a two-fluid model of turbulent combustion in gases, with special reference to the spark- ignition engine" Inst Mech Eng, London, paper c53/83, 1983
  60. DB Spalding (1984)"The two-fluid model of turbulence applied to combustion phenomena" 22nd AIAA Meeting, Reno, Nevada
  61. DB Spalding (1987) "A turbulence model for buoyant and combusting flows"; International J. for Numerical Methods in Engineering vol 24, pp 1-23
  62. DB Spalding (1993) Lecture delivered at "First International Conference on Air Pollution", Monterrey, Mexico. The text is included in "Lecture on the two-fluid turbulence model" supplied with the PHOENICS software package
  63. DB Spalding (1995a) "Models of turbulent combustion" Proc. 2nd Colloquium on Process Simulation, pp 1-15 Helsinki University of Technology, Espoo, Finland
  64. DB Spalding (1995b) "Multi-fluid models of turbulent combustion"; CTAC95 Conference, Melbourne, Australia
  65. DB Spalding (1995c) "Multi-fluid models of Turbulence", European PHOENICS User Conference, Trento, Italy
  66. DB Spalding (1996a) "Older and newer approaches to the numerical modelling of turbulent combustion". Keynote address at 3rd International Conference on COMPUTERS IN RECIPROCATING ENGINES AND GAS TURBINES, 9-10 January, 1996, IMechE, London
  67. DB Spalding (1996b) "Multi-fluid models of Turbulence; Progress and Prospects; lecture to be presented at CFD 96, the Fourth Annual Conference of the CFD Society of Canada, June 2 - 6, 1996, Ottawa, Ontario, Canada
  68. DB Spalding (1996c) "Progress report on the development of a multi- fluid model of turbulence and its application to the paddle- stirred mixer/reactor", invited lecture at 3rd Colloquium on Process Simulation, Espoo, Finland, June 12-14
  69. DB Spalding (1987) "A turbulence model for buoyant and combusting flows"; International J. for Numerical Methods in Engineering vol 24, pp 1-23
  70. DB Spalding (1994) Poster session, International Heat Transfer Conference, Brighton, England
  71. 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
  72. RLT Sun (1982) "Application of the ESCIMO theory to turbulent diffusion flames"; PhD Thesis, London University
  73. LT Tam (1981) "The theory of turbulent flow with complex chemical kinetics"; PhD Thesis, London University
  74. D.C.Wilcox, 'Reassessment of the scale determiing equation for advanced turbulence models', AIAA J., Vol.26, No.11, p1299, (1988).
  75. Williams FA, Combustion Theory, Menlo Park CA, 1985
  76. GC Williams, HC Hottel and AC Scurlock,(1949), 3rd Symposium on Combustion, p21, Williams and Wilkins
  77. WJ Wohlenberg "Minimum depth for flame front for stable combustion ... in a gaseous system at constant pressure" 4th International Symposium on Combustion p 796 Williams and Wilkins, Baltimore, 1953
  78. JZ Wu (1987) "The application of the two-fluid model of turbulence to ducted flames" IC CFDU Report, June 1987
  79. ST Xi (1986) "Transient turbulent jets of miscible and immiscible fluids"; PhD Thesis, London University
  80. V Yakhot and S Orszag (1986) "Renormalization group analysis of Turbulence. I. Basic theory" J. Sci Comp, vol 1, no 1, pp 1-51