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Connections between the multi-fluid and flamelet models of turbulent combustion


Brian Spalding, CHAM Ltd, London, England

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It is shown that the much-studied "flamelet model" (FLM) of turbulent combustion fits well within the framework of the more-recently developed Multi-Fluid Model (MFM).

MFM reproduces the results of FLM when:-

  1. the fuel-air ratio of the gases is uniform;
  2. the chemical reaction rate is fast compared with the rate of turbulent micro-mixing;
  3. the Reynolds number is high;
  4. appropriate formulations of the "offspring distribution" of the micro-mixing process are adopted.

However, whereas FLM requires limitations 1, 2 and 3, and is tied to a particular distribution formulation, MFM is free from these restrictions.

These points are explained and illustrated by way of computations for a "well-stirred reactor".



  1. The underlying physical concepts
    1. The flamelet model
    2. The multi-fluid model
    3. Connexions between FLM and MFM
  2. MFM applied to circumstances for which FLM may be valid
    1. The well-stirred reactor; description
    2. The PDFs predicted by MFM
    3. Comparison with the presumptions of FLM
  3. MFM in more general circumstances
    1. Non-uniform fuel ratio
    2. Low chemical reaction rate
    3. Low Reynolds numbers
  4. Conclusions
  5. References

1. The underlying physical concepts

1.1 The laminar-flamelet model

(a) The "eddy-break-up" model

A convenient way of introducing the laminar-flamelet model (FLM) of turbulent combustion [Refs 1, 2, 3, 4, 5, 6, 7] is as a refinement of the "eddy-break-up" model (EBU) [8]. The latter, first published in 1971, involved regarding a turbulent burning mixture as comprising inter-mingled fragments of just two gases, namely:

It was recognised that, at the interfaces between the two sets of fragments, thin layers of gas in various stages of incomplete combustion must exist; but the fraction of the total volume occupied by these gases was regarded as much less than unity; and the only important consequence of their existence was the primary chemical transformation (fuel + air --> products) which took place in them.

The rate of that transformation, per unit volume of the total space, was treated as being governed, however, by the rate of turbulent micro-mixing, for which the quantity:

epsilon / k
was a convenient measure.

Here, epsilon represents the volumetric rate of dissipation of the kinetic energy of turbulence, and k the kinetic energy itself.

Although the eddy-break-up model has proved rather successful in predicting overall combustion rates, its total neglect of the chemical kinetics has disbarred it from simulating secondary but important kinetically-controlled processes, such as NOX production in flames.

(b) The FLM

Whereas the EBU pays scant regard to the interface layer between the burned and unburned gases, the FLM concentrates attention upon it; and one of its major concerns is to compute the magnitude of the area of this interface per unit volume: sigma.

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The underlying notion is that, if sigma is known, then the volumetric rate of combustion omega should be capable of being computed from:

omega = sigma * Ulam
where Ulam stands for the laminar flame speed, which can be regarded as a property of the unburned-mixture composition and temperature, and of the pressure.

The attractiveness of this idea is that values of Ulam are often known from measurements on Bunsen burners, and similar laboratory devices; and their presence in the formula provides comfort to the combustion specialist for whom the uncertainties of turbulence are somewhat bewildering.

Numerous proposals have been made for the terms in a "sigma-transport equation". A review is to be found in Reference 7. Although they differ in detail, all contain, directly or indirectly, the epsilon / k term of the EBU; for they all have to conform to the experimental facts which that overall-rate term expresses.

The quantity Ulam is also to be found in them, in a secondary role; but recognition that the composition distributions in the interface are unlikely to be exactly the same as in laboratory-measurement circumstances has caused some authors to multiply Ulam by a (sometimes only-vaguely defined) "stretch factor".

The present purpose is not to enumerate the differences between the various proposals or to prefer one to another; instead it is to draw attention to the physical concept which unites them, both with the EBU and with the MFM.

(c) The unifying concept: the "brief encounter"

The following diagram will serve to focus attention on what the models have in common; for it shows, in idealised manner (ie with neglect of all the actual geometrical convolutions), a volume in which two materials, A and B, are separated by an interface of small but finite thickness.

For EBU and FLM, A is unburned gas and B is burned gas (or vice versa). MFM allows more possibilities; but it reduces to the same when EBU- or FLM-favouring conditions prevail.

     Figure 1. Two fluid "blocks" in contact

     .                      / /           .
     .                      / /           .
     .                      / /           .
     .                      / /           .
     .         A            / /    B      .
     .                      / /           .
     .                      / /           .
     .                      / /           .
     .                      / /           .
       The interface layer __i

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The diagram represents conditions at a particular instant; and all model makers agree that:

The following diagram

(Figure 2.), extracted from a book published in 1955 (!) [9], shows how long such ideas have been prevalent.

Recognised either explicitly (in MFM) or implicitly (in FLM) by the various model makers is that contact between the A and B "gas blocks" is intermittent, which is to say that, as a consequence of the turbulence, two such blocks:

Such a process is nowadays easily subjected to numerical analysis, the results of which may be represented as in the following colour plot.

(Figure 3), which shows contours of reactedness on a plot having distance horizontal and time vertical.

What is noticeable is that the contours are symmetrical at early times, during which interdiffusion domiminates; and they become unsymmetrical later, when chemical reaction "catches up", just as was predicted by the old calculations of Figure 2.

The calculations were performed with the PHOENICS computer code, by the loading of library case 103 .

As will be explained in more detail below, it is the intermittency of these "brief encounters" which explains how volumetric reaction rate comes to be so closely connected with epsilon / k

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It will also be shown that the brevity of the contact is such that, at the high Reynolds numbers that are commonly encountered, the interface layer must always be much thinner than the typical block size.

(d) The secondary reactions

Why FLM is superior to EBU is not that it is able to predict better the volumetric rate of progress of the main combustion reaction; for it is doubtful whether it can do this more often than not in circumstances of practical interest.

Its superiority lies in the fact that, through being able to estimate the distributions of the temperature and of chemical species within the interface layers, it allows calculation of the effects of secondary reactions such as those giving rise to oxides of nitrogen.

As will be shown, MFM offers the same possibilities; and in more general circumstances; and, if care is taken to ensure that comparable chemical-kinetic and transport-property presumptions are made, there is no reason why MFM should not produced the same results as FLM, for conditions for which the latter is valid at all.

1.2 The multi-fluid model

(a) MFM fundamentals

The multi-fluid model of turbulence has been described in numerous recent publications [10 to 20]; and most of these, and other explanatory material, can be viewed on CHAM's web-site: www.cham.co.uk. MFM is also available for activation in the PHOENICS computer code.

Here, therefore, only the main relevant ideas will be reviewed, as follows:

  1. The concept of "blocks" of gas in contact, illustrated by Fig. 1. above, is central to the MFM idea; and the contacts are thought of as being of brief duration, long enough for some interdiffusion to occur, but not long enough to allow extensive equilibriation (except in low-Reynolds-number turbulence, as explained in section 3.3 below).

  2. Unlike FLM, MFM recognises that such encounters will occur between blocks of gas of whatever state happens to be present, not only fully-burned and fully unburned. It is therefore able to provide an explanation, which the FLM literature lacks, as to why the quantity epsilon / k is of such importance. (See section 1.2(c) below).

  3. It follows that the idealised instantaneous state of of the turbulent mixture envisaged by MFM would have to be represented by a large number of diagrams such as that of section 1.1(c) above, in each of which the materials A and B could be different, eg A1 and B1, A2 and B2, A3 and B3, etc.

  4. These materials might have the same fuel-air ratio, but differ in respect of their reactednesses; or they might differ in fuel-air ratio also.

  5. There is no requirement, in MFM, that the two adjacent material blocks should be different at all; so a diagram containing both Ax and Ax might appear.

  6. Thus, MFM takes account of all gas-fragment encounters, not only those between the fully-burned and fully-unburned materials.

  7. In respect of those properties of the materials which are selected as "population-distinguishing attributes" (PDAs), only a finite number of values are allowed to appear.

    Lest this be regarded as a serious restriction, it should be remarked that it is far less severe than those which are fundamental to FLM; for:
    • in FLM, only two values, namely 0.0 and 1.0 are allowed of the "reaction-progress variable", ie that which as just been referred to as the reactedness, and which acts as the relevant PDA;
    • in such a case, MFM would typically have between 10 and 100 allowable values;
    • in FLM, only one PDA is allowed, typically (and perhaps always) the reaction-progress variable;
    • for simulating a combustion process, MFM will typically have two PDAs (eg the progress-variable and the fuel-air ratio), and may in principle have any number.

  8. Each of the allowable states of "block" material is treated as a distinct fluid, the mass fraction of which is computed by way of a conventional transport equation, having time-dependence, convection, diffusion and source-sink terms.

  9. Each such fluid may have any number of continuously-varying attributes (CVAs), the values of which are also computed from transport equations.

  10. The turbulent micro-mixing process is a consequence of "brief encounters" of the kind described in section 1.1(c) above, and results in the production of interface material at a rate per unit volume equal to:
    CONMIX * epsilon / k,
    where CONMIX is a constant.

  11. The rate of production of interface material from such encounters between fluid i and fluid j is:
    Mi * Mj * CONMIX * epsilon / k ,
    where Mi and Mj are the mass fractions of the two fluids.

  12. The intermediate-fluid-layer material (called "offspring" in MFM parlance), which is produced as a consequence of the encounter (called "coupling and splitting"), goes to increase the concentration of the appropriate population-member.

  13. If two PDAs are chosen (eg reactedness and fuel-air ratio), with N1 allowable values of the first, and N2 allowable values of the second, the population would contain N1 * N2 distinct fluids; the model would be called a N1 * N2-fluid model; and the number of diagrams illustrating their "coupling and splitting", referred to in 1. above, would be (N1 * N2) ** 2 .

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(b) Connexions with conventional turbulence-modelling concepts

The well-known epsilon and k quantities have already been referred to. It will be useful also to connect MFM concepts with three others, namely:

These quantities are inter-related by equally well-known relationships, namely:-

Since sigma has the dimensions of 1/length, being a measure of the average thickness of the gas fragments engaging in the coupling-and- splitting process, it can be expected to be proportional to the reciprocal of Lmix,

Therefore, to avoid having to introduce another constant, let it be taken as equal to this reciprocal, so that

sigma * Lmix = 1.0

Important note

It is necessary to distinguish between the sigma employed in the FLM from that which is employed in MFM; for FLM pays attention only to interfaces between fully-burned and fully-unburned gases; whereas MFM recognises that the coupling-and-splitting process is a consequence of the turbulence, which disregards the chemical compositions of the materials which are brought into contact.

Thus FLM's sigma has to be zero when there is no more unburned combustible; whereas MFM's sigma can reasonably be taken as inversely proportional to Lmix, regardless of the extent of the reaction.

It is now interesting to ask two questions, namely:

First, however, it is necessary to justify the linkage of the rate of offspring production to epsilon / k by a more detailed discussion of the energy-dissipation process than has been provided anywhere in the MFM literature until now.

(c) The mechanism of energy dissipation

If, as MFM supposes, the dominant behaviour pattern of the multi-fluid population is "coupling and splitting", it must be possible to use this description to explain how turbulent kinetic energy is dissipated. Such a use of the concept now follows.

It is interesting to observe that such comparisons of MFM predictions with experiment as have been made so far have favoured a value of between 5.0 and 10.0 for CONMIX

The value most favoured by users of the Eddy-Break-Up model, is much lower, namely around 0.5; but this is easily explained by the fact that EBU is a mere "two-fluid" model, i.e. one with a very coarse "population grid".

(d) The average contact time

Since the interfacial layer grows as a result of molecular action, it is certain that its maximum thickness Llay (i.e. that which exists at the end of the "brief encounter" when "coupling" gives way to splitting) is of the order of (nulam * Tcont) ** 0.5 .

Also, the same analysis would show that the rate of production of interface material per unit volume is equal to:

It follows that:

Let now this contact time be compared with the "micro-mixing time", Tmix, defined by:

Tmix = 1.0 / Roff
If sigma is taken as the reciprocal of Lmix, and use is made of the above definitions of nuturb, Re, etcetera, it can be shown by algebraic substitution that:
Tcont/Tmix = constant / Re
where constant = CMU / (CONMIX * CD ** 2) which is of the order of unity.

It follows that, since Re is usually much greater than unity in a turbulent flame, the contact time is much smaller than the mixing time.

One implication is that the interfacial layer does not have enough time to grow so as to have a dimension comparable with those of the "blocks" of fluid which are enjoying their brief encounter. Indeed, it can be algebraically deduced from the above equations that:

Llay / Lmix = constant / Re also.

1.3 Connexions between FLM and MFM

(a) An important question

It has been remarked above, in connexion with

(Figure 4.) that conditions in the interfacial layer governing momentum exchange are symmetrical throughout the encounter, whereas those for the reactedness

(Figure 3.) are symmetrical only at the start.

Obviously the relevance of the epsilon/k quantity to the creation of interface material is greater if the encounter is broken off before the reactedness profile has become very unsymmetrical, i.e. before the laminar-flame propagation process has had time to dominate; so it is interesting to estimate whether this condition is likely to be satisfied.

The following argument allows this question to answered:-

The following summarising remarks about the connexions between FLM and MFM therefore appear to be appropriate:

  1. FLM and MFM both pay attention to the interfacial material which results from turbulence-occasioned encounters between fluid fragments of unlike composition.

  2. FLM considers only encounters between fully-burned and fully-unburned fragments, both having the same fuel-air ratio; wheareas MFM allows the fragments to have any pairs of values of reactedness and fuel-air ratio.

  3. FLM practitioners tend to think of the interfacial layers as being similar to steadily-propagating laminar flames (admittedly modified in some way by "stretching"); whereas the MFM presumptions suggest that they are mainly of the transient-inter-diffusion kind.

  4. Since a major aim of many users of FLM is to compute the production rates of secondary-reaction products, such as NOX, by postulating that profiles in the inter-facial layers are those of propagating flames, it must be suspected that those rates will be wrongly calculated.

    MFM also has a presumption about the profiles in the layer, as part of its coupling-and-splitting hypotheses. The only presumption which has been seriously explored so far, namely the "promiscuous Mendelian" one, implies symmetry within the inter-diffusion layer; but this can easily be remedied.

2. MFM applied to circumstances for which FLM may be valid

2.1 The well-stirred reactor; description

FLM requires, for validity:

These requirements are satisfied by PHOENICS Input-Library Case, L103, of which the descriptive part runs as follows.

  Stirred reactor with a 1D population distribution, and
  reactedness, ranging from zero to 1, as the population-
  distinguishing attribute.
  It is supposed that two streams of fluid         |           |
  enter a reactor which is sufficiently       A ====>          |
  well-stirred for space-wise differences          |  stirred  |
  of conditions to be negligible, but not          |  ///|/// C===>
  sufficiently for micro-mixing to be              |  reactor  |
  complete.                                   B ====>          |
  The two streams have the same elemental
  composition; but one may be more reacted than the other.

  The flow is steady, and the total mass flow rate per unit volume
  is 1 kg/s m**3 .

The flow is zero-dimensional in the geometrical sense; but, to give it significance for a three-dimensional combustor simulation, it might be regarded as representing a single computational cell in the 3D grid. Then the conditions in the inlet streams might represent those in the neighbouring cells, from which material enters by way of diffusion and convection.

The calculations to be reported have been conducted with a 25-fluid model, and with reactedness as the population-defining attribute.

The reaction-rate-versus-reactedness formula is of the well-known kind:

rate = CONREA1 * (1 - R) * R ** CONREA2

where R stands for reactedness.

The exponent, CONREA2, has been given the value 5.0, which represents sufficiently well for the present illustrative purposes the temperature dependence of combustion reactions.

Stream A has been chosen as being completely unburned (R=0.0) and stream B as fully reacted(R=1.0).

Various values have been chosen for the micro-mixing constant CONMIX and the "pre-exponential factor" of the reactivity, CONREA2

2.2 The PDFs predicted by MFM

In the following table, the second, third and fourth columns disclose the input parameters.

Clicking on the number in the first column will display:

The PDFs appear as histograms, with:

The references to "spikes" should be noted. Comparison of the values ascribed to the height of these spikes with the also-printed "maximum ordinates" shows that they are often much larger. This is the justification for the "two-fluids-mainly" idea which underlies FLM.

The average-reactedness and RMS-deviations appear in the fifth and sixth columns.

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 10.01.0 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

Some trends revealed by these figures will now be discussed, as follows:

2.3 Comparison with the presumptions of FLM

It is now possible to consider to what extent MFM confirms or contradicts the presumptions of the flamelet model of turbulent combustion.

The best confirmation of FLM presumptions is afforded by Figures

6 and

7; for both of these exhibit the large spikes of zero- and unity-reactedness gas, with low concentrations of gases of intermediate reactedness.

Whether the distributions of concentration of the intermediate gases is the same, for MFM and LFM, is another matter. As has already been indicated, LFM practitioners are inclined to believe (albeit without any closely-reasoned argument) that the distributions correspond to those of a steadily-propagating flame; whereas the foregoing analysis suggests that the transient-interdiffusion model is more appropriate.

When the sequence of Figures



10, and

11 is considered, support for the FLM concept becomes weaker, the last of these figure revealing a PDF which is quite unlike that which LFM presumes.

Even less supportive are Figures

15 and

16; for these again show distributions which are different from those to which FLM applies.

It must indeed be concluded that FLM can be expected to represent turbulent combustion in pre-mixed gases only when the chemical reaction rate (measured here by CONREA) is large compared with the micro-mixing rate (measured by CONMIX).

This condition is perhaps satisfied for a gasoline engine, where the fuel/air ratio may be close to stoichiometric throughout; but it is unlikely to prevail for combustors into which the fuel and air enter separately.

Separate introduction of fuel and air is the subject of the next section.

3. MFM in more general circumstances

3.1 Non-uniform fuel ratio

(a) The cases considered

A further series of calculations has been performed for the conditions in which the two streams of fluids entering the stirred reactor have differing fuel-air ratios. Such conditions lie beyond the capabilities of the flamelet model at the present time.

Specifically, stream A has been taken as pure air; and stream B has been taken as having a fuel/air ratio of twice the stoichiometric value and a reactedness of 50 %. The two streams are equal in flow rate.

A 100-fluid model has been employed for this study, with a two-dimensional "population grid", of which the PDAs (ie population-defining attributes) are:

  1. The so-called "mixture fraction", fmx, which represents the mass of material in the time-average local mixture which has emanated from the fuel-bearing stream, namely stream B in the present case; and

  2. the "burned-fuel fraction", fbu, which represents the mass fraction of material emanating from the fuel-bearing stream which has been oxidised.
    This equals:
    fmx - fub
    where fub = the mass fraction of unburned fuel.

  3. The last-named PDA is related to, but not precisely equal to, the reactedness; for this is not, as fbu is, strictly speaking a conserved property of the gas (because mixing a 50%-reacted mixture with a 100%-reacted mixture results in a 75%-reacted mixture only when 5veerthe fuel-air ratios are the same).
The input conditions have been selected, from the multiply-infinite range of possibilities, because they sufficiently illustrate what the multi-fluid model of turbulence has to say about chemical reactions in gases in which both fuel-air mixing and finite-rate chemistry are influential.

Further information about the definition of the model will be provided during the discussion of the results and of their displays, which may be inspected by clicking on the relevant number in the left-hand column of the following table.

Figure CONMIX CONREA average F average R
10.0 10.0 0.510 0.649
10.0 5.0 0.507 0.595
10.0 2.0 0.502 0.476
10.0 1.0 0.500 0.354
10.0 0.0 0.500 0.222
100.0 10.0 0.512 0.818
100.0 5.0 0.507 0.754
100.0 2.0 0.500 0.587
100.0 1.0 0.500 0.346
100.0 0.0 0.500 0.222

(b) Conclusions from inspection of the table

The runs conducted fall into two groups. The first five have the fixed CONMIX value of 10.0, and successively smaller CONREA values; and the next five have CONMIX = 100.0, and the same sequence of CONREA values.

It can be seen that:-

(c) Conclusions from inspection of the figures

All the figures have the same general form. They show a two-dimensional histogram on the left, and a graphical representation of the multi-fluid-model concept on the right.

The latter is similar to that of earlier one-dimensional population distributions; but the average-R value is printed, not the RMS deviation of F.

The left-hand diagram is however quite different from before; for it has to represent the proportions in which the total amount of material is distributed between ten fmx and ten fbu intervals.

"Spikes" no longer make any appearance.

The most-prevalent fluids are those whose "boxes" are most full of colour; the least-prevalent are those whose boxes are black.

The figures will now be discussed individually, in some detail.

Figure 17 shows that the total amount of material is fairly uniformly divided between the boxes lying in a rectangle of which the corners are:-

When the sequence of Figure 18 to 21 is considered, it is seen that the upper (higher-reactedness) boxes are less-and-less well filled, until, when CONREA has been reduced to zero, for

Figure 21, all the fluid is to be found in boxes close to the line joining the A-stream and B-stream states.

Indeed, were the "population grid" finer, the histogram would provide colour only along that line.

It will have been noticed that, in all the histograms, boxes lying to the left of the stream-A-to-stoichiometric-mixture line are absent.
The reason is that the states represented by such boxes are not physically attainable; for they correspond reactednesses which exceed unity.

Therefore, although it has been stated above that a 100-fluid model is in use, the statement is a slight exaggeration; for the concentrations of fluid in the "missing boxes" are necessarily zero, and so need not be computed.

(d) Comparison between FLM and MFM for the cases considered

The above heading is included for, completeness; but there is little to say, for the simple reason that FLM, like the eddy-break-up model (EBU) to which it is the successor, is silent about mixtures of differing fuel-air ratio.

It is however worth emphasising that the "brief-encounter" model, which some writers on FLM appear to grasp, does supply the means by which "flamelet-like" concepts can be applied to turbulent diffusion flames; and it is of course the Multi-Fluid Model which provides the unifying mathematical framework.

3.2 Low chemical reaction rate

As was mentioned at the start of the present paper, FLM is acknowledged by its proponents as being restricted in its validity to the circumstances in which the reactivity of the combustible mixtures is sufficiently high, in comparison with the turbulent mixing process, to ensure that the sum of the volume fractions of fully-burned and fully-unburned gases is close to unity.

In respect of pre-mixed (i.e. uniform-fuel/air-ratio) gases, section 2.3 has confirmed the correctness of this acknowledgement; and it has indicated that MFM permits prediction of combustion processes for which the high-reaction-rate condition is not satisfied.

When combustion of non-pre-mixed gases is in question, of the kind treated in section 3.1, the condition is hardly ever satisfied; for the "brief encounters" result in the creation of interfacial layers in which, perhaps, in some central region, the mixture ratio is near stoichiometric so that the gases are highly reactive. This must however surely be very rare; both in space and time.

It therefore appears proper to conclude that the restriction of FLM to high-reactivity conditions is a very severe restriction indeed.

3.3 Low Reynolds numbers

The third of the limitations to which FLM is subject is that the Reynolds number must be high. This is true of MFM also, as it has been developed so far; but it is interesting to consider which of the models has the better prospect of breaking free from the limitation.

Proponents of FLM have not, to the author's knowledge, pronounced explicitly on this matter; and it is not for him to speak for them. But the following can be said from the MFM side:-

4. Conclusions

The foregoing discussion may be held to justify the following conclusions:-

  1. The conceptual framework of the Multi-Fluid Model of turbulent combustion is large enough to accommodate the hypotheses of the Flamelet Model.
  2. It also enables the limitations of FLM to be perceived and understood.
  3. MFM is not itself subject to those limitations, being able to handle non-uniformites of fuel-air ratio, low reaction rates, and (although this is still speculative) low Reynolds numbers.
  4. Although both MFM and FLM focus attention on the interfacial layers between bodies of unlike gases in contact, MFM places its emphasis on the transient inter-diffusion process, while FLM pays more attention to the propagation of flame.
  5. In so far as a significant expenditure of computer time is spent by FLM practitioners, when calculating the rates of secondary chemical reactions, on the basis of the presumption that the PDFs are those pertaining to laminar propagating flames, it may well be that the time is wasted; for their PDF presumption is not well borne out by the present study.

5. References

  1. Bray KNC in Topics in Applied Physics, PA Libby and FA Williams, Springer Verlag, New York, 1980, p115
  2. Peters N, Twenty-First Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1986, p 1231
  3. Williams FA, Combustion Theory, Menlo Park CA, 1985
  4. Cant RS, Pope SB, Bray KNC, Twenty-Third Symposium (International) on Combustion. The Combustion Institute, Pittsburgh, 1990, pp 809-815
  5. Candel S, Veynante D, Lacas F, Maistret E, Darabiha N & Poinsot T, in Recent Advances in Combustion Modelling Lattoutourou B (Ed). World Scientific, Singapore, 1990
  6. Bray KNC Proc Roy Soc London A 431:315-355, 1990
  7. Duclos VM, Veynante D and Poinsot T Combustion and Flame 95: 101-117 (1993))
  8. Spalding DB, (1971) 13th International Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, pp 649-657
  9. Spalding DB (1995a) "Models of turbulent combustion" Proc. 2nd Colloquium on Process Simulation, pp 1-15 Helsinki University of Technology, Espoo, Finland
  10. Spalding DB (1995b) "Multi-fluid models of turbulent combustion"; CTAC95 Conference, Melbourne, Australia
  11. Spalding DB (1995c) "Multi-fluid models of Turbulence", European PHOENICS User Conference, Trento, Italy
  12. Spalding DB (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
  13. Spalding DB (1996b) "Multi-fluid models of Turbulence; Progress and Prospects" lecture presented at CFD 96, the Fourth Annual Conference of the CFD Society of Canada, June 2 - 6, 1996, Ottawa, Ontario, Canada
  14. Spalding DB (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
  15. Spalding DB (1997) "Boiling, condensation, multi-phase flow, chemical reaction and turbulence; the multi-fluid approach" Lecture at International Symposium on The Physics of Heat Transfer in Boiling and Condensation 21-24 May, 1997, Moscow
  16. Spalding DB (1998a) "CAD to SFT, with Aeronautical Applications" Proceedings of the 38th Israel Annual Conference on Aerospace Systems, Israel, Feb 25-26; pp s7 1-32
  17. Spalding DB (1998b) "Turbulent mixing and chemical reaction; the multi-fluid approach ". Lecture at the University of Delft, Netherlands
  18. Spalding DB (1998c) "The modelling of turbulent combusting systems via MFM". Invited Lecture at Eurotherm 56, Heat Transfer in Radiating and Combusting Systems, 1-3 April 1998, Delphi, Greece
  19. Spalding DB (1998d)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