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************************************************************* * A first draft of * * * * THE MULTI-FLUID MODEL OF TURBULENT FLOW, MIXING AND * * * * COMBUSTION * * * * by * * * * PROFESSOR BRIAN SPALDING * * * * of * * * * Concentration, Heat and Momentum Limited * * * * May 1996 * *************************************************************


An outline is given of the physical concepts and mathematical formulations of a multi-fluid model (MFM) of turbulence.

Similarities to, and differences from, earlier turbulence models are described.

References to recent works on MFM are provided, but with only brief summaries of their contents, the aim of the present paper being to collect and explain all the relevant ideas, without distraction by particular examples.

Consideration is given to the advantages likely to be derived from applying the MFM to practical problems of engineering and environmental science.

Although MFM is already capable of being applied to many practical problems, especially those involving combustion processes or other chemical reactions, research and development work can be foreseen as likely to improve its acceptability, its realism and its economy. A classified list of such tasks is provided.



1. An introduction to the multi-fluid-model concept 1.1 Some origins of the multi-fluid idea 1.2 Quantification of the multi-fluid concept (pdf's and fpd's) 1.3 An example of a multi-fluid gas mixture 1.4 The concept of dependent-variable discretization 1.5 Discretization possibilities 1.6 An example of a two-dimensional population 1.7 Interactions between fluids 1.8 Coupling and splitting 1.9 Influences of hydrodynamic factors 1.10 Why combustion modellers need MFM

2. Mathematical formulation 2.1 Differential equation for the conservation of a fluid 2.2 The source-sink term 2.3 The differential equation for a within-fluid property 2.4 Population-average quantities 2.5 Computational aspects

3. Recent work on MFM 3.1 Reports and publications 3.2 Unpublished work

4. Relations to other turbulence models 4.1 The four main classes of turbulence model 4.2 Advantages and disadvantages of the four classes 4.3 Similarities and differences between MFM and previous models 4.4 Generalising the eddy-break-up model

5. Application prospects 5.1 The possibility of immediate use 5.2 Applications to chemically-reacting flows 5.3 Applications to hydrodynamic and heat-transfer phenomena

6. Research and development tasks 6.1 Mathematical and conputational tasks 6.2 Comparisons with experiment 6.3 Conceptual developments

7. References


1. An introduction to the multi-fluid-model concept ---------------------------------------------------

1.1 Some origins of the multi-fluid idea ----------------------------------------

The fragmentary, intermittent and near-random nature of turbulent flows, as a qualitative concept, has been long been observed and described qualitatively; yet to express this observation adequately in useful quantitative terms is still very difficult.

In striving to do so, many authors have found it helpful to regard a turbulently-flowing system as consisting of an assemblage of fragments of many distinct fluids, which intermingle, exchange heat, mass and momentum, and possibly merge.

For example, Reynolds (1874) explained the transfer of heat to a surface in terms of the rate at which a hotter fluid moved towards that surface while a cooler fluid moved in the opposite direction; and Prandtl (1925) conceived of the transfer of momentum across (ie the shear stress within) a turbulent jet or boundary layer in much the same way.

Generalising Reynolds' concept, one might postulate a population of fluids distinguished by their temperatures.

To generalise Prandtl's concept, one would imagine a population of which the distinguishing attribute was the main-flow-direction component of velocity.

[More information on the origin and history of the multi-fluid idea can be found in section 4.1c below.]

1.2 Quantification of the multi-fluid concept (pdf's and fpd's) ---------------------------------------------------------------

A quantitative way of expressing the variability of a fluid attribute is by way of its probability-density function (abbreviated to pdf), the shape of which can be deduced from measurements of instantaneous local values by suitable processing of their variations with time.

The counterpart of the pdf, in the multi-fluid approach, is the "fluid-population distribution" (fpd)

The multi-fluid concept implies that the pdfs of any fluid attribute should be plotted as a series of spikes (ie delta- functions), corresponding to the fact that the population- defining space is discretized.

Fdp's are more commonly plotted as histograms, such as the following diagram, in which the amount of each fluid in the population element is represented by the area of a rectangle standing on a base centered on the value of the attribute which defines the element: __ | | __ amount | | __| | ^ | | __ | | |__ __ | | | __| |__ __| | | | | | | | | | | | | | | | |__ | | | | | __| | | | | | | | | | | | | | | | | | | | | | | | | | |__| | | | | | | | | |__ | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |__| | |__|__|__|__|__|__|__|__|__|__|__|__|__|__| ----------> attribute

The fpd of Reynolds' and Prandtl's two-fluid models would have to be represented as: __ | | | |__ | | | | | | | | | | | | | | | |__|__| or rather as: _____ | | | | | | | | | | | | | | | | | | | | | |__|__|

because neither author indicated that the two fluids might be present in unequal proportions.

1.3 An example of a multi-fluid gas mixture -------------------------------------------

If a turbulent flame results from the injection of a combustible gas into an air atmosphere, and the characterising attribute of the imaginary population is the fuel mass fraction, a multi-fluid model of the gas in the flame could be a population in which there intermingle fluid fragments consisting of, for example:

- pure air; - pure fuel; - gas having (say) 10% fuel and 90% air; - gas having (say) 20% fuel and 80% air; - gas having (say) 30% fuel and 70% air; and so on.

With the above numbers the model could be called an eleven-fluid one, with uniform sub-division of the fuel-fraction dimension.

In reality, concentrations can take any value between zero and unity; but the discretization principle entails that attention is focussed on a finite number of discrete concentration values. Mixtures which in fact have concentrations lying between the two nearest discrete values must then be represented by interpolation.

The material which carries one of the selected concentrations is regarded as a distinct fluid, and will be referred to as such below.

1.4 The concept of dependent-variable discretization ----------------------------------------------------

The example brings immediately into notice the arbitrariness of the multi-fluid concept; for three un-forced decisions have already been made, namely in respect of:-

(1) choice of distinguishing attributes, (here fuel fraction, rather than temperature or velocity);

(2) choice of number of fluids (here 11, whereas 110, 1100 or 11000 might have been chosen);

(3) choice of attribute difference between the fluids (here a uniform 10%, whereas one might have preferred to cluster all 9 of the "impure" fluids closely around the stoichiometric fuel- air value.

The arbitrariness of the discretization may be an obstacle to acceptance of the multi-fluid concept by purists, who may not at first recognise that the same objection can be raised to numerical simulations of flames in which GEOMETRICAL SPACE is discretized and TIME is also sub-divided into a finite number of intervals.

This analogy can be instructive as well as persuasive; for it reminds doubters that grid-independence tests can be conducted for fluid-attribute "grids" as well as for geometric ones.

1.5 Discretization possibilities --------------------------------

The analogy may also stimulate researchers to employ their skills in "adaptive gridding" so as to make the numerical computation of a fluid-population distribution both accurate and economical.

Further, just as grids which sub-divide geometrical space can be one, two- or three-dimensional, so can those which are employed to distinguish the fluids in an imaginary multi-fluid population.

Temperature, velocity and fuel-fraction have already been mentioned; and of course the number of usable distinguishing attributes is unlimited.

Turbulence energy, vorticity, fragment size, degree of reactedness and smoke content are a few of the other attributes which spring immediately to mind.

However, possibility and necessity are not the same: just as there is no need to use a multi-fluid model at all, so is there no necessity to discretize any more of the fluid-attribute variables than the modeller believes will improve flow-simulation capability.

This is an important observation; for it distinguishes multi-fluid models from probabilistic models (see, for example, Pope, 1982), which use Monte-Carlo methods for computing the pdf in a multi- dimensional space.

[More is written about the possible varieties of population grid in section 6.1 below.]

1.6 An example of a two-dimensional population ----------------------------------------------

Many turbulent combustion problems involve the separate entry of fuel- and oxidant-bearing streams. Gas-turbine combustors, for example, exhibit this feature.

Because fluctuations of both fuel-air-ratio and reactedness may be expected to be significant, consideration of a two-dimensional fluid population is appropriate.

The following diagram represents such a population schematically, as used for the numerical simulation of a turbulent Bunsen-burner flame (Spalding 1995b).

_________________________________________ |///////|///////|///////| | | f - mfu |///////|///////|///////| 16 | 20 | * mfu stands for ^ | inaccessible |_______|_______|_______| mass fraction | | fluid states | | | | of unburned | |///////|///////| 11 | 15 | 19 | fuel; | |_______|_______|_______|_______|_______| * fluid 1 is | |///////| | | | | fuel-free air; |///////| 6 | 10 | 14 | 18 | * fluids 13-16 |_______|_______|_______|_______|_______| are stoichio- | | | | | | metric; | 1 | 5 | 9 | 13 | 17 | * fluid 17 is the |_______|_______|_______|_______|_______| fuel-rich entry --------> mixture fraction, f stream.

The vertical dimension (f - mfu, ie the mass proportion of fuel- stream-derived material in the local mixture, less the mass proportion of unburned fuel) measures the extent of the reactedness of the gas, being at its maximum (namely f) when no unburned fuel remains, and at its minimum (namely 0) when all the fuel is unburned.

Other examples of two-dimensional populations might involve the attributes:-

* temperature and salinity (ie salt content), for use in simulations of sea-water flows in estuaries;

* along-wall velocity and normal-to-wall velocity in a boundary layer; and

* temperature and upward-direction velocity in a buoyancy-driven- flow simulation.

For example, a 2-dimensional concentration space, concerned with two mixture components A and B, might focus attention on 10 distinct values of the concentrations of each component. The whole set of relevant mixture-composition possibilities would then be represented by the 100 pairs of A and B concentration values.

Once again, real mixtures possessing intermediate compositions would be represented by interpolation between the nearest 4 of the 100 allowed possibilities.

Three-, four- and more-dimensional models can of course be envisaged. There is no limit.

1.7 Processes affecting the fluid-population distribution ---------------------------------------------------------

(a) The atmosphere as a suggestive example ------------------------------------------

Clouds in the sky may be regarded as easily recognisable manifestations of the multi-fluid character of the Earth's atmosphere, with water-vapour content as the distinguishing attribute.

Observation of such clouds reveals that:-

* they are convected across the sky by the wind, more or less at a uniform velocity;

* they move upwards and downwards at what may sometimes be differing velocities; and

* they may come into existence, grow in size, then diminish, and ultimately disappear.

Similar renarks can be made about smoke plumes from chimneys, and about the much larger plumes which are created by forest fires. The processes so revealed must be expected to take place in all multi- fluid populations.

(b) Convection and diffusion ----------------------------

About the first-mentioned (uniform-convection) process, there is little to be said, for it takes place whether the atmosphere is turbulent or not.

As to the second, two modes of relative motion between members of the fluid population can be distinguished, namely:-

- turbulent diffusion, as a consequence of the gradients of concentration; and

- "sifting convection", which is the relative motion according to which, by reason of gravity, lighter fluids move upwards more rapidly (or downwards more slowly) than heavier ones.

The speed of relative motion can be reasonably supposed to be the outcome of the balance between the buoyancy-force difference (which increases the speed) and friction between the different-velocity fluids (which decreases it).

Thus friction between fluids in relative motion is one of the interactions to be accounted for.

Heat transfer and mass transfer between the fluids, by reason of their differences in temperature and composition, are also certain to take place to some extent.

(c) Sources and sinks of individual fluids ------------------------------------------

Finally, the creation, growth, diminution and disappearance of clouds is a reminder that a multi-fluid model of turbulence must make provision, since mass, energy and concentration must obey their relevant conservation laws, for the transfer of mass from one element of the population into another.

Just as a human population which is decribed by its age-distribution (eg 50% under 30, and 25% over 60) or political allegiances (eg 40% left, 35% right, 20% centre and the rest don't-know) will change as its members grow older or change their opinions, so will a fluid-population-distribution change.

Heating will effect this if temperature is one of the distinguishing attributes; momentum sources will do so if the attributes are velocities; and chemical reaction will bring it about if one attribute is the mass concentration of one of the reactants.

Sources and sinks of the above kind can take place in a single fluid, on its own. However, multi-fluid turbulence modelling involves consideration also of sources and sinks which result from the interactions BETWEEN fluids. These are of such central importance as to need a section to themselves, which now follows.

1.8 Coupling and splitting -------------------------- (a) The general idea --------------------

The present author, (Spalding, 1995a) has used the somewhat anthropomorphic term "coupling and splitting" to describe the interactions between fluids.

This term suggests that two fluid fragments making contact act as parents; and that, after the contact, there exist (what is left of) the parents and a collection of offspring.

The coupling process is conceived as the approach of two fragments of unlike fluids, followed by their partial coalescence and mixing.

The splitting process is their subsequent break-up into new fragments which possess concentrations (or temperatures, velocities, etc), which are intermediate in value between those of the parent fluids.

Although more sophisticated hypotheses may prove to be more in accord with reality, it is the simplest which will be described here, namely that which has been called "promiscuous" and "Mendelian".

"Promiscuity" implies that any fluid will couple with any other, ie indiscriminately, at a rate depending only on its availability.

"Mendelian" implies that the offspring may possess the character- istics of either parent, in any proportion.

The following diagram may serve to convey this idea.

^ frequency in | population | father mother | ______ / / | | | promiscuous ****** | __|______|__ <------- coupling ---------> ******** | | .. | *| .. |* | |* *| *| -- |* | /-**--/ ____ Mendelian ______ /-----/ | / |/////| / | splitting | / |_____| / | |/////| v v / / | | | | _ _ _ _ _ _ /_________/ | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | ------------------------------------------------------------------- fluid attribute -------------------------->

(b) The underlying mechanisms envisaged ---------------------------------------

The way in which this redistribution of material between fluids is effected is imagined to be as follows:

(1) Two fragments of fluid are brought into temporary contact by the random turbulent motion, thus:

______________________________ xxxxxxxxxxxxxxx............... xxxxxxxxxxxxxxx............... xxxxxxxxxxxxxxx............... xxxxxxxxxxxxxxx............... xxxxxxxxxxxxxxx............... xxxxxxxxxxxxxxx............... ------------------------------

(2) Molecular and smaller-scale turbulent mixing proceses cause intermingling to occur, with the result that, after some time, the distribution of x (father) and . (mother) material within the coupling fragments appears as:

______________________________ xxxxx.xxxxx.x.x.x..x.....x.... xxxxxxxx.x.x.x.x.x..x......... xxxxxx.xx.x.x.x.x.x..x..x..... xxxx.xx.xxxx.x.x.x...x.x...... xxxxxxxxx.xxx.x.x.x.x......... xxxxxxxx.x.x.x.x.x.x..x....... ------------------------------

(3) Before the intermingling is complete, however, the larger-scale random motions cause the fragments to be plucked away again, with the result that the amounts of the material having the compositions of the parent fluids (pure x, and pure .) are diminished, while some fluid material of intermediate composition has been created, as shown below.

pure <------------ intermediate ---------> pure ____ ____ ____ ___ ___ ____ ____ ____ xxxx x.xx xxx. x.x .x. .x.. ...x .... xxxx xxxx .x.x .x. x.x ..x. .... .... xxxx xx.x x.x. x.x .x. x..x ..x. .... xxxx .xx. xxxx .x. x.x ...x .x.. .... xxxx xxxx x.xx x.x .x. x.x. .... .... xxxx xxxx .x.x .x. x.x .x.. x... .... ---- ---- ---- --- --- ---- ---- ----

(c) Coupling and splitting for 2D populations ---------------------------------------------

If the population is a two-dimensional one, as illustrated in the diagram of section 1.6 above, the promiscuous-Mendelian hypothesis would imply that the offspring of the coupling between parents such as F and M below would be shared among the cells enclosing some part of the straight line joining F to M, ie those containing asterisks in the diagram below.

|_______|_______|_______|_______|_______|_______|_______| | | | | | | | M | | | | | | | | * | |_______|_______|_______|_______|_______|_______*_______| | | | | | | * | | | | | | | * | | |_______|_______|_______|_______|___*___|_______|_______| | | | | * | | | | | | | * | | | | |_______|_______|_______*_______|_______|_______|_______| | | | * | | | | | | | * | | | | | |_______|___* __|_______|_______|_______|_______|_______| | F * | | | | | | | * | | | | | | | |_______|_______|_______|_______|_______|_______|_______|

(d) Alternative nomenclature and hypotheses -------------------------------------------

The fluid-interaction process can also be usefully called the "micro-mixing process", so as to distinguish it from the "macro- mixing" ones of convection and turbulent diffusion, which re- distribute fluids in space without however increasing the intimacy of their contact.

The corresponding term employed by the users of probabilistic models (eg Pope 1982) is simply "the mixing model".

Alternative fluid-interaction processes, which remain to be explored are:-

* parental bias; and

* differentially-diffusive.

These will be discussed below, in section 6.3, as will their probabilistic-model counterparts.

1.9 Influences of hydrodynamic factors --------------------------------------

Before the coupling-and-splitting idea can be used in a quantitative method of turbulent-flow prediction, some means has to be provided for determining its rate. This must be connected in some way with quantitative properties of the hydrodynamic flow, for example the sizes of the fluid fragments and their relative velocities.

One of the simplest hypotheses, which has its origin in the so- called "eddy-break-up" formula (Spalding,1971a) [see section 4.4 below], is that the rate is proportional to:

the mean fluid density, times the product of the mass fractions of the two colliding fluids, times one or other of: the mean velocity-gradient or the energy-dissipation-rate / turbulence-energy

The first of the two alternatives may be appropriate when the hydrodynamic calculations are based on the Prandtl (1925) mixing- length theory; and the second may be more suitable when some variant of the "k-epsilon" turbulence model (Harlow and Nakayama, 1968) is employed.

Such presumptions are however merely starting points, proposed because they provide links with conventional turbulence models.

The MFM allows insights which suggest more sophisticated formulae, especially those which recognise the influences on relative motion of the pressure gradient, which may be taken as being the same for all fluids in the mixture, and the density, which often differs greatly from fluid to fluid.

The Reynolds number is of course a further probable source of influence, as are the "Densitometric Froude Number" and other quantities expressing body-force action.

These matters will be further discussed in section 6 below.

1.10 Why combustion modellers need MFM --------------------------------------

The following section is inserted at the present point in order to provide motivation, at least for combustion specialists, to proceed further. It has been adaptted from an earlier publication (Spalding, 1995a).

(a) The practical background ----------------------------

Laminar flames are, in principle, easy to simulate by numerical calculations; for the equations governing them are known, and chemical-kineticists have provided data on the rates of the (admittedly very numerous) chemical reactions which take part. To simulate a laminar flame therefore, all that is needed is a very powerful computer, equipped with the appropriate software.

However, most flames of industrial and environmental importance, whether in furnaces, engines, oil-platform explosions, or forest conflagrations, are turbulent.

(b) The theoretical difficulty ------------------------------

Turbulent flames, by contrast with laminar ones, present difficulties of principle which have not so far been adequately resolved. They result from the facts that:

(1) turbulent fluctuations of temperature and concentration are of too small a scale (in both space and time) to be resolved by any currently available computer, so that local-average values of temperature and concentration must be used;

(2) the relation between the required local-average reaction rates and the local-average temperatures and concentrations is therefore not what it would be in the absence of fluctuations, ie in the circumstances for which chemical kineticists have reported their findings.

In order to understand this mathematically, suppose that a reaction between species A and B proceeds at a rate proportional to their instantaneous concentrations [A] and [B].

In the absence of fluctuations, the rate is thus:

constant * [A] * [B]

However, if (to take an extreme view) fluctuations cause [A] to be finite only when [B] is zero, and vice versa, the time-average reaction rate, which is proportional to the time-average product of [A] and [B], is zero.

The time-average reaction rate is even harder to predict when the effects of temperature and its fluctuations are brought into consideration; for reaction rates are non-linearly dependent on temperature, rising steeply with it. Thus, if the rate is proportional to:

T ** n,

where T stands for absolute temperature and n is an exponent between 6 and 10, it is easy to work out that

0.5 * (T1 ** n + T2 ** n)

has a very different magnitude from (0.5 * (T1 + T2) ) ** n.

The relation between reaction rate and temperature is in fact more more complex than a simple power law. The following diagram shows the typical shape. Its rise and fall result from the fact that the increase of temperature is accompanied by decreases in the amounts of available reactants; and, without them, reaction cannot proceed.

------------------------------------------------------------------------ Variation of reaction rate, oxygen content and unburned-fuel content for a sub-stoichiometric fuel-air mixture

^ |* oxygen---> x # # | * x # # | * x # # | * x # # rate | * x #<--reaction | unburned fuel---> * # x # rate | * # x # | # * x | # * x# | # * x | # * # | # * # ------------------------------------------------* 0 reactedness ( = (T - Tu)/(Tb - Tu) ) 1 ------------------------------------------------------------------------

(c) A homely illustration -------------------------

The point can also be explained and understood as follows:

when the man is a night-worker and his wife a day-worker, their contribution to the population explosion is hard to estimate.

Being present at the same place AND at the same time is as important to interactions between chemical species as it is to those between humans.

(d) Conclusion --------------

The multi-fluid model has been created, in part, to enable turbulent-flow phenomena involving chemical ractions to be understood, and then simulated numerically.

However, that is far from being its only purpose; for it can, potentially, provide simulations of a pure hydrodynamic or heat- transfer character which take better account of the physics than do conventional models.

-------------------------------------------------------------------- 2. MATHEMATICAL FORMULATION --------------------------- 2.1 Differential equation for the conservation of a fluid ---------------------------------------------------------

Let the mass fraction of a specific fluid in the local population of fluids be denoted by ma, where a represents the attribute, or set of attributes, which characterise this fluid.

Then the usual considerations of time-dependence, convection, diffusion and source-sink action, lead to the following differential equations for ma:

ma_t + V.grad ma = v.div (G.grad ma) + madot (1)

where: ma_t stands for the differential of ma with respect to time, V is the velocity vector of the mixture, v is its specified volume, G is the turbulent exchange coefficient and madot is the net rate of creation of ma per unit volume.

In this equation, V and G are taken as having the same values for all fluids of the population.

If however different values are appropriate for different fluids (as, for example, where one is heavier than the other and therefore moves at a different velocity), such fluid-specific features will be expressed by modification of the source-sink terms, madot.

2.2 The source-sink term ------------------------ (a) The various kinds of source and sink ----------------------------------------

Apart from the just-mentioned effects associated with velocity and diffusion-coefficient disparities, the phenomena to be expressed by madot are those which cause the fluid with attribute a to be created or destroyed. These may include the following:

(1) inter-fluid mixing; (2) chemical reaction; (3) within-fluid sources and sinks.

These will now be discussed in turn.

(b) (1) Exchange of mass between fluids ---------------------------------------

The individual fluid fragments which constitute the postulated population can be imagined to come into contact with each other in a random manner; and, while in contact, to interact by transfer of mass, momentum and energy.

Suppose, for example that the distinguishing attribute chosen for a population is temperature T, and that the discrete values of temperature which are chosen to describe the population are: Ta, Tb, Tc,....... Tz, where: Ta < Tb < Tc <.....< Tz.

Let the fluids be referred to by name as: A, B, C,.....Z.

Consider now the coupling of fragments of A with those of, say, E. It is probable on physical grounds that, after the contact is over, some fluid at temperature Ta and some at temperature Te will still exist; but fluids at the intermediate temperatures Tb, Tc and Td are also likely to have come into existence.

Thus, the contact will have reduced the values of ma and me, because the parents have given up some of their mass; but the values of mb , mc and md will have been increased, and in such amounts that the sum ma + mb + mc + md + me is unchanged.

In order that the source-sink terms ma, mb, mc etc can be computed, mathematical expressions must be devised to dictate:

- at what rate fragment-types a and e "couple" with each other; and

- how the offspring which they produce are distributed along the fluid attribute (ie temperature) scale.

(c) (2) Chemical-reaction sources --------------------------------- Let it now be supposed that the fluids in question can sustain an exothermic chemical reaction, so that fluid C, for example, would tend (if it were allowed), to increase its temperature at the rate Tc degrees per second.

It is of course NOT allowed, because the discretised nature of the multi-fluid model does not permit any fluid to exist with temperatures other than Ta, Tb, Tc, Td, etc.

How can this difficulty, which appears to be rather serious for a model which purports to be able to represent combustion processes, be resolved?

The answer is: by regarding the heating process as reducing the mass fraction of C and increasing that of D. This may be expressed quantitatively as follows. If the temperature rise at rate Tcdot continued for long enough, the temperature of the fluid would reach the next-allowed temperature Td.

How long is "long enough"? The answer is: (Td-Tc)/Tcdot.

The heating process can therefore be represented as causing a sink of C, at the rate:

mcdot = - mc . Tcdot/(Td-Tc) (2)

simultaneous with a source of D given by

mddot = mc . Tcdot/(Td-Tc) (3)

(d) (3) within-fluid sources and sinks -------------------------------------- The same technique can be used for any other source which tends to change the state of fluid in the discretized-attribute direction.

Such sources include:

- if the attribute is temperature, the emission or absorption of radiation;

- if the attribute is velocity, the resolutes of pressure gradient and body forces.

In all cases the mdot expressions are of the form:

source / attribute difference.

2.3 The differential equation for a within-fluid property ---------------------------------------------------------

The temperature-distinguished fluids just discussed can, so far as the multi-fluid concept is concerned, each have their own composition, and other fluid properties such as velocities, etc.

These too need to be computed; and a differential equation is available for each.

If Fb stands for one of the properties of fluid b, the equation is:

(mb.Fb)t + V.grad (mb.Fb) = v.div (G.grad (mb.Fb) + Sfb (4)

where b is used in place of a, so as to ease later discussion.

The terms on the left-hand side of the equation require little comment; for they amount to saying that the amount of property F per unit mass of fluid b, namely Fb, remains unchanged along a particle track in space and time.

The first term on the right-hand side can be best interpreted by decomposing the term inside the bracket as follows:

grad (mb.Fb) = mb . grad Fb + Fb . grad mb (5)

The first term on the right stands for the within-fluid-b diffusion of the property F, while the second term implies that any of fluid b which changes location as a result of random turbulent motion carries its F property with it.

The term SFb requires more discussion. It will ordinarily consist of two parts; the first corresponding to the within-fluid source (for example the rate of creation of oxides of nitrogen appropriate to the temperature of the fluid and to its chemical composition), and the second to the consequences of the coupling-and-splitting process.

In the latter regard, the simplest hypothesis is that the "offspring" possess the arithmetic - mean F values of their parents.

Thus, if the source of B (say) resulting from the couplings between A and D is mb(AD), the coupling-splitting contribution to Fb consists of terms such as:


Of course, since B can be formed from the coupling of A with any item in the alphabet from C to Z, the number of possible terms may be large.

There is something else to be said: whereas couplings between A and B can be left out of account when the production of offspring is concerned, because there are no allowable values of temperature between TA and TB, such couplings can produce sources and sinks of Fb.

Their magnitudes will be proportional to (TA-TB), and to the coupling rate.

2.4 Population-average quantities --------------------------------- The population-average value of a discretized variable, such as temperature in the above example, is given by:

Taz = Sum az , ma.Ta (6)

That of a non-discretized variable, such as F in the above example, is given by:

F az = Sum az , ma.Fa (7)

Of course, whereas it is possible to compute the population-average quantities from knowledge of the properties of all the individual fluids, the reverse is not true.

Many other population-characterizing quantities can be computed once the full fluid-population distribution (ie the set of ma's) and the properties of each fluid (ie the set of Fa's) are known. Of especial interest are the "moments", ie such quantities as:

T az(n) = Sum az (maTan) (8)

These are useful when it is desired to transfer information from one fluid-attribute grid to another, for example as part of a computer- time-economising adaptive-grid procedure.

2.5 Computational aspects -------------------------- Many computer programs are available which solve, by finite-volume or finite-element techniques, differential-equation sets such as (1) or (4).

In order to use these, it is necessary only to provide sufficient storage space for (what may become) an unusually large number of dependent variables (the ma's and Fb's), and also to provide coding for the source terms.

The amount of computation increases more than linearly with the number of variables, because the number of coupling-splitting events to be considered may increase as N*(N-1), where N is the number of fluids.

The word "may" is used, because it will often be possible to find physically- or numerically-based reasons for not considering all the theoretically possible events.

It is therefore probable that use of multi-fluid models for practical turbulent--flow computations will often cause computer time to become a matter of concern.

Lest it be thought, however, that it will be as burdensome as is that associated with the Monte-Carlo approach referred to above, the following considerations should be weighed:

(1) Experience has shown that even a very coarse discretization of fluid-attribute space can produce considerable increases in realism.

Thus the 25-year-old "eddy-break-up model" (Spalding,1972) which is a two-fluid model in the present nomenclature, is still in use because it is much more realistic than would be the use of a one- fluid model (ie complete neglect of the fluctuations), which is what it replaced.

Further, as has recently been demonstrated (Spalding, 1995a), the modest increase from two to four fluids brings significant increases in realism.

There appears to be no such coarse-grid possibility in the Monte- Carlo method, which requires a large number of particles in order to produce meaningful results at all.

(2) Serious attention to computations based on larger numbers of fluids (eg 100) has only just started (Spalding, 1995b).

Therefore no significant use of the many available economising techniques has been made. Doing so is only a matter of time, funding, and a sufficient drive from the practical users.


3. Recent work on MFM ---------------------

3.1 Reports and publications ---------------------------- (a) The first MFM report ------------------------

The paper presented at the 2nd Helsinki Colloquium on Process (Spalding,1995a) was the start of a series of publications on MFM, each of which has carried the idea a little further. It showed how a four-fluid model of premixed-fuel combustion allowed phenomena to be predicted, and specifically those concerning the spread of a steady flame confined in a duct (Williams, Hottel and Scurlock, 1953) which were beyond the capability of the "two-fluid" eddy- break-up model. Much of this demonstration is reproduced in section 4.4 below.

A ten-fluid model of an axi-symmetrical turbulent jet was also presented.

This paper already introduced the notions of "promiscuous coupling" and "Mendelian splitting", which have proved to be central ingredients of later multi-fluid models; and it was there pointed out, but not illustrated, that two-, three- and more-dimensional population grids could be envisaged and used.

The essential idea was recognised as that of "discretization" of fluid-attribute space, which can be carried out in numerous ways, all of which should be equivalent if the "population grid" is made sufficiently fine.

The "fluid-population distributions" of multi-fluid models and the "probability-density functions" of probabilistic methods (Dopazo and O'Brien, 1974) were regarded as being equivalent, again in the fine-grid limit.

(b) The first two-dimensional population ----------------------------------------

At the CTAC-95, Conference in Melbourne, a paper was presented (Spalding,1995b) which applied MFM to the turbulent Bunsen burner. This required the use of a two-dimensional fluid-population distribution, indeed the one which appears in section 1.6, above.

The results were plausible, but no attempt was made to ensure grid- independence, or to make comparisons with experiment.

(c) Detailed fpd studies for the "well-stirred reactor" -------------------------------------------------------

A 100-fluid model was used (Spalding, 1995c) for the study of the shapes of the fluid-population distributions, both one- and two- dimensional, which appear (according to MFM) when combustible gases flow steadily into a reactor which is so "well-stirred" that macro-mixing is perfect.

It was shown there that the one-dimensional fdps could take a great variety of different shapes, far exceeding those which have been imagined by the practitioners of the "presumed-pdf" approach [see section 4.1f below].

(d) Steady one-dimensional flame propagation --------------------------------------------

The same conclusion about pdf shapes emerged from a later study (Spalding, 1996a), in which MFM modelling was applied to a one- dimensional steadily-propagating turbulent pre-mixed flame.

This study further demonstrated that the concept of grid refinement, which is familiar to all flow-simulation experts in connexion with the sub-division of time and of geometrical space, is equally applicable to the population grid. It was further shown that, for the case in question, results of adequate accuracy could be obtained with 10- or 20-fluid models; litlle further increase of accuracy resulted from using the 100-fluid model.

(e) Transient flame propagation -------------------------------

The four-fluid model has been employed in two transient explosion computations. One of these concerns a laboratory experiment in which flame propagates through a pre-mixed gas in a duct on the walls of which are a series of turbulence-promoting baffles. The results, reported by Freeman and Spalding (1995) are in good qualitative and semi-quantitative agreement with experiment.

The second application has been to the explosions recently carried out at Spadeadam, in Northern England, by the UK Steel Construction Institute. The results have not yet been published. It can however be said that the computations based upon the 4-fluid model were far from providing the worst agreement with experimental data.

Later research has shown that it would be wiser to employ a 10-fluid model, in order that the population-grid effects should not be troublesome.

However, the ways in which the presence of solid obstacles influence the intensity and length-scale of turbulence, and so of coupling- splitting rates, is at the present a matter of guess-work. No-one living in the world today, in the present author's belief, can quantify this influence reliably.

3.2 Unpublished work --------------------

More recently (Spalding,1996d) the following studies have been made, albeit not yet completed:-

(1) grid-refinement studies for the well-stirred reactor, operating in the mode in which a single cold-gas stream enters and its flow rate is increased until sudden extinction occurs;

(2) simulations of transient macro- and micro-mixing in a paddle- stirred reactor containing initially-separated acidic and alkaline liquids.

(3) simulations of combustion of fuel and air in a gas-turbine-type chamber; and

(4) a study of the plane turbulent non-reacting mixing layer.

The first study has confirmed the validity, in different circumstances, of the above-mentioned findings about grid refinement.

The second, which probably breaks new ground for chemical engineers concerned with reactor-vessel design, reveals than it is much more important to simulate the micro-mixing than it is the macro-mixing.

The third, by introducing an idealised smoke-production reaction, shows how different are the amounts of this production predicted by single- and multi-fluid models.

The fourth study may be regarded as being of especial interest because, whereas all the others relied on a hydrodynamic model of turbulence, either mixing-length or k-epsilon, to provide the multiplier of the coupling-and-splitting-rate term, this one did not.

Instead, the rate was deduced from the fluid-population distribution of longitudinal velocity, and from a prescribed length scale. The first step towards a stand-alone MFM for turbulent flows has thus been taken. The next, namely the removal of the need for a length- scale prediction, is not far away.


4. Relations of MFM to other turbulence models ------------------------------------------------ 4.1 The four main classes of turbulence model --------------------------------------------- (a) Introduction ---------------- Turbulence models, in the sense used here, are collections of concepts about the nature of turbulent fluids which may be expressed in mathematical form in such a way as to constitute a method of prediction.

Questions to be answered by such a method include:

for a given set of flow-defining conditions (inlet geometry, outlet location and shape, internal baffles and sources of heat and momentum, fluid density, viscosity and heat capacity, number of thermodynamic phases, etc),

- will the flow be turbulent at all?

- if so what will be the intensity of the turbulence?

- what will be the consequent exchange rates of mass, momentum and energy between the fluid and the solids with which it is in contact?

- what will be the rates of chemical reaction within the fluid?

The answers to all these questions are of course different for each location in space and instant in time.

Methods of answering such questions are numerous and varied in nature; but it appears useful to separate them into four main classes, namely as:

I single-fluid models, II multi-fluid models, III probabilistic models, IV direct-numerical-simulation models.

These terms will now be explained.

(b) Single-fluid models; Class I --------------------------------

Single-fluid models are those in which the fluid condition is characterised by average values of velocity, temperature, etc, at each location and time instant, and by statistical representations of the fluctuations about those averages.

This class contains most of the models which are described in the text-books and are in use today. They include, to use the nomenclature of Launder and Spalding (1972):

- zero-(differential)-equation models, such as those of Boussinesq (1877) and Prandtl (1925);

- one-equation models, such as those of Prandtl (1945) and Bradshaw (1967);

- two-equation models, such as those of Kolmogorov (1942), Harlow and Nakayama (1968), Saffmann (1970), Spalding (1969), Wilcox (1993), Yakhot and Orszag (1986) and many others;

- multi-equation models, such of those concerned with multiple levels of energy or scale (Elhadidy, 1980; Kim and Chen, 1989) or with further statistical quantities (Daly and Harlow, 1970; Spalding, 1971b; Naot, Shavit and Wolfshtein, 1974).

The methods of the Reynolds-stress kind fall into the last-named grouping.

(c) Multi-fluid models; Class II --------------------------------

Multi-fluid models express the fluctuations by representing them as though there were many different fluids mingling within the same space, each with its own local and instantaneous velocities and temperatures.

The mingling of steam bubbles with water in a kettle is an extreme example; but the fluids are more usually of a single thermodynamic phase, as when tongues of flame rise above a garden bonfire.

The basic concept was already present in the writings of Reynolds (1874) and Prandtl (1925); for they both conceived of the relative motion of fluid fragments, which could be of significant size.

Scientists concerned with combustion (Shchelkin,1943; Wohlenberg, 1953; Howe and Shipman, 1965; Kuznetzov,1979) have found the concept much to their liking; for it is only by taking into account the fragmentariness of the burning gases that observed phnomena can be explained, even qualitatively.

The author would place in the two-fluid category:

- his own "eddy-break-up" model (Spalding, 1971a; Mason and Spalding,1973),

- the "eddy-dissipation concept" of Magnussen and Hjertager (1976), which was derived from it, and

- the Bray-Libby-Moss "flamelet" model (Bray and Libby,1981; Moss, 1980).

A more extensive exploration of the capabilities of two-fluid models was made later by the author and his co-workers (Spalding, 1982, 1983, 1984, 1987, 1993; Andrews, 1986; Fueyo, 1992; Malin, 1986; Xi, 1986; Wu, 1987; Ilegbusi and Spalding, 1987a,b; Pericleous and Markatos, 1991).

In these works, the two fluids were distinguished in a variety of ways, eg turbulent/non-turbulent, hotter/colder, upward/downward- moving; and two sets of Navier-Stokes equations were solved simultaneously.

Still more recently, as is discussed elsewhere in this paper, the multi-fluid idea has been extended to 4-, 14-, and many-fluid models (Spalding, 1995a,b,c). However the first publications on the population-of-fluids concept may have been those arising from study of the long-forgotten "ESCIMO" model of turbulent combustion (Spalding, 1979; Noseir, 1980; Tam, 1981; Sun, 1982).

(d) Probabilistic models; Class III -----------------------------------

Probabilistic models represent the near-randomness of turbulence by introducing some randomness of their own. Specifically, they employ Monte-Carlo methods to establish the probable distribution of fluid attributes within a multi-dimensional space of which the coordinates include the components of velocity, the temperature, etc.

This class appears to have originated in the chemical-engineering- science field with the publications by Curl (1963) and Dopazo and O'Brien (1974).

More recently Pope (1980, 1982, 1985), Chen and Kollmann (1988, 1990) and others have conducted a vigorous research campaign, which, to judge from the proceedings of a recent conference (ICOMP, 1994), is gathering momentum.

Once again, it is the desire to be able to simulate combustion phenomena with quantitatuve realism which provides much of the motivation.

(e) DNS models; Class IV ------------------------

Finally, direct-numerical-simulation models revert to the single- fluid approach, but without any built-in averaging. Their users solve the Navier-Stokes equations with extremely fine sub-divisions of space and time, deriving averages and statistical measures of fluctuations only after the solutions have been obtained.

Direct-numerical simulation, has been subjected to increasing attention as computers have become more and more powerful. Pioneers include Schumann (1973) and Reynolds (1975).

(f) "Presumed-pdf" models -------------------------

There is a class of model featuring "probability-density functions" which the present author would place in the single-fluid category, because the shapes of the pdfs are presumed rather than computed. Models of this kind may perhaps have started with the author's own paper (Spalding,1971b), in which a "two-spike" double-delta-function presumption was made.

Subsequently, Lockwood and Naguib (1975) made the "clipped-Gaussian" presumption; they were followed by others, including Kent and Bilger (1976), Kolbe and Kollmann (1980), Rhodes et al (1974) and Gonzalez and Borghi (1991).

Employment of these presumptions gave rise to additional computational expense, because all reaction-rate terms (for example) required the evaluation of integrals for each cell at each iteration; but no clear advantage in respect of generality of agreement with experiment (in the present author's opinion) ever emerged.

As is shown elsewhere in this paper, the multi-fluid model makes it possible to calculate the shape of the pdf (or FPD, to accord with MFM nomenclature) if the number of fluids is very large; and this shape depends greatly upon the dimensionless parameters which relate the coupling rates of the fluids on the one hand, and their chemical reaction rates on the other, to the mass flow rate into the cell.

A detailed parametric study of these effects by way of the multi- fluid model might perhaps lead to formulae with the aid of which one might know what shape to presume in particular circumstances. Presumption of the shape without such guidance is however likely to do as much harm as good.

4.2 Advantages and disadvantages of the four classes ---------------------------------------------------- (a) Ease of use ---------------

CLASS I must be regarded as the easiest to use, because many members of the class are embodied in widely-available software packages. For example, the Shareware (and therefore freely available) version of PHOENICS contains zero-, one-, and two-equation models; and, because of its open-source character, it allows other models of Class I to be built into it.

The latest versions of PHOENICS have more than a dozen identifiable Class-I turbulence models, and indeed many more if the possibility of sub-model interchanges betwen them are considered.

Most other general-purpose computer codes (eg FLUENT, FIDAP, FLOW-3D, CFX, TASC) now have a similarly wide range of models.

Ease of use is greatest for the 0-, 1-, and 2-equation models, and least for the Reynolds-stress models, for which convergence is not always guaranteed.

CLASS II models are less widely available in packages, although Shareware PHOENICS does possess a 2-fluid model which solves two sets of Navier-Stokes equations.

Multi-fluid models of this class are not yet widely available. Those which will be described below have been implemented by use of the open-source facilities of the latest PHOENICS; and it is probable that the implementation will be attached to the next or next-but-one release of the code.

There are no publicly available codes which embody the probabilistic models of CLASS III, but the US Government makes some available to its contractors. Although the methods may be simple for those who have become used to them, or to Monte-Carlo methods in other circumstances, getting started with Class-III methods is difficult.

Recently, however, a paper has been published which shows how the general-purpose code PHOENICS can be used as the basis for Monte- Carlo calculations of turbulent-mixing and -combustion processes (Fueyo, Larroya, Valino and Dopazo,1995)

Class IV methods would probably be regarded as the easiest to use, were it not for the fact that not even the largest computers in the world are large enough. On the surface, they appear to require nothing more than the ability to make a time-dependent non- turbulent flow simulation. But large-grid-size problems bring their own special difficulties, and means for resolving them; so this is also not an easy field of research to get into.

(b) Ease of understanding -------------------------

For the reasons just alluded to, Class-IV methods can be regarded as the easiest to understand, because the physical assumptions are those of laminar flow; and indeed there is no "modelling" (in the sense of substituting guesswork for ignorance) at all.

The present author would rank the other classes in the order I II III, in respect of understandability, with the qualification that some of the more complex single-fluid multi-equation models may be more difficult to grasp than the simpler two-fluid models.

It has to be admitted, however, that the two-fluid concept has not so far achieved much popularity; and this may in part result from reluctance to make the imaginative leap from steam/water mixtures on the one hand to hot-air/cold-air mixtures on the other.

Another misgiving has perhaps deterred more reflective persons, namely the thought: Why only TWO fluids? Perhaps the demonstration in the present paper that one can indeed handle any number of fluids will bring some reassurance.

The probabilistic models of Class III, are not, to this author's mind, easy to grasp at all. It is not so much the idea of multi- dimensional space that is difficult, but rather the esoteric mathematical language and symbols which, perhaps necessarily, the practitioners of these models employ.

To those who make the effort, Classes III and II can at first appear to be almost the same; so the next difficulty is that of understanding the differences. Of these, a crucial one is:-

* Class-II methods discretize some (but not all) fluid-attribute dimensions, in the manner familiar to all users of finite-volume or finite-element codes; and they can obtain useful results from very coarse discretizations, as use of the two-fluid model has shown.

* Class-III methods work in a non-discretized, multi-dimensional, all-fluid-attribute space; "coarsening" for economy appears to require reduction of the number of the attributes (eg species concentratons) which are considered.

It might be said that the Monte-Carlo approach of probabilistic methods is akin to the use of a shot-gun: many pellets are needed if the target is to be hit; whereas Class II methods work with rifles which, if carefully aimed, need discharge fewer bullets.

(c) Extent of validation ------------------------

Class I is by far the most extensively validated of the four; and although no widely-agreed answer can be given to the newcomer's reasonable question, "Which model is best?", advice can be given, based upon experimental knowledge, as to when it is permissible to use Prandtl's rather-simple mixing-length model, and when only a Reynolds-stress model will suffice.

However, it is just because of this extensive validation campaign that it is possible to assert that NONE of the Class-I models will suffice for the explanation of commonly-observed and practically- important phenomena. This has been suspected for many years; and the passage of time justifies the substitution of conviction for suspicion.

None of the other three models have been subjected to quantitative tests on the same scale. Probably the work on the two-fluid model by the author's former students at Imperial College represents the most systematic campaign; but this came to an end in 1988.

Comparison of Class-III predictions with experiments is being made in the USA for both simple and complex (eg gas-turbine-combustor) circumstances. The former comparisons appear to have drawn attention to a deficiency of all of the "mixing-model" assumptions which have been tried so far: they do not produce the right (Gaussian) probability-density-function shape in the late-time limit.

It is not clear whether this difficulty can be resolved while retaining the Monte-Carlo framework of calculation.

4.3 Similarities and differences between MFM and previous models ----------------------------------------------------------------

(a) Introduction ----------------

The brief review, in sections 4.1 and 4.2, of the whole turbulence- modelling scene, has been provided in order that the multi-fluid turbulence model can be understood in context.

MFM is not entirely new; but it has some significant points of novelty and advantage. The purpose of the present section is to bring these to the reader's attention, by pointing out similarities and differences.

(b) Comparison with single-fluid models ---------------------------------------

The basic mathematical structure of MFM is similar to that of single-fluid models of turbulence, in that differential equations are formulated and solved in order to represent the influences of:

- time-dependence, - convection, - diffusion and - sources and sinks.

MFM may also make use of single-fluid concepts such as "effective viscosity", "epsilon/k", mixing length, and root-mean-square fluctuations; however it also possesses other means of describing, calculating and responding to the influences of the fluctating nature of turbulent flows.

Like single-fluid turbulence models, MFM has its empirical "constants", which may turn out to be functions of the local Reynolds number of turbulence and of other dimensionless quantities; but, so far, little effort has been expended on determining what they are.

MFM has many more equations to solve than single-fluid models do; so computer times are bound to be greater, if the same computational grid is used. However, MFM uses the available computer power in a different way; for it devotes as much attention to differences in fluid properties prevailing at a single location as it does to differences from place to place. Therefore MFM may use coarser geometrical grids than have become usual for single-fluid models.

(c) Comparison with the 1982 two-fluid model --------------------------------------------

MFM differs from the two-fluid model worked on by the author and his colleagues in the early 1980s in two main ways, neither of which are essential. ________________________________ The first is that the | | states of the two fluids ^ | Fluid 2 | could roam freely over | | * | the "phase space" | | | defined by the fluid vert-| | attributes, whereas ical | | MFM restricts the attr-| | states of its fluids ibute| * | to discrete locations. | Fluid 1 | | | As soon as adaptive |______________________________| gridding is introduced horizontal attribute -----> however, MFM can enjoy the same freedom.

The second difference is that two complete sets of Navier-Skokes equations were solved for the two-fluid model, an achievement which may never be attainable for MFM, and has certainly not been attempted.

Nevertheless, the discrete fluids of MFM can indeed move relative to one another under the influences of body forces and fluid-to-fluid friction; and it appears that to allow each fluid its own Navier- Stokes equations would be needless indulgence. A "drift-flux" or "algebraic-slip" approximaton is likely to suffice.

It is worth noting that MFM has adopted one valuable feature of two- fluid thinking, namely the distinction between the fluid attributes which DISTINGUISH one fluid from the other and those which are merely POSSESSED by the fluid.

Thus, in the two-fluid model, the vertical velocity may be that which distinguishes the fluids; but each fluid still has its own distinct values of all three velocity components.

The difference between the models that IS essential, of course, is that between two and many; inevitably, MFM is the more capable of describing real flows.

(d) Comparison with the presumed-pdf models -------------------------------------------

MFM focusses attention on the probability-density functions of the fluid attributes for some of the same reasons as do users of the presumed-pdf approach, who nowadays form the majority of those who attempt to simulate turbulent combustion processes; for it is only knowledge of those functions which enables chemical-kinetic information to be rationally introduced.

However, MFM calculates the pdfs (ie its fluid-population distributions) and shows that the functions may take very varied shapes; whereas the basis of selection used by the shape-presumers is far from reliable.

Moreover, MFM finds it as easy to calculate two-dimensional pdfs as one-dimensional ones; presumed pdfs, by contrast, are perhaps always one-dimensional.

Of course, there can be no certainty, in advance of systematic comparison with experimental data, that the pdfs calculated by MFM in its present form are correct; and there is good reason to believe that the promiscuous-Mendelian coupling-splitting hypothesis is not always the most realistic (see section 6.3).

However MFM is at the beginning of its research life; whereas the presumed-pdf approaches may be near the end of theirs. It appears certain that MFM has the greater potential for development and improvement.

(e) Comparison with the probabilistic models --------------------------------------------

Here the abbreviation CPDFM (for Computed PDF model) is used in place of the longer "probabilistic model".

There is certainly a more than superficial relation between the PDF of the CPDFMs and the FPD of the MFM. In the limit of an infinite number of fluids (for the latter) and of particles (for the former), they may become quantitatively identical.

However, they ARE somewhat different concepts; they are computed by different mathematical techniques; they embody different physical hypotheses; and (possibly) they appeal differently to persons of differing educational backgrounds.

In respect of mathematics, CPDFMs employ Monte-Carlo methods, whereas MFMs use direct numerical solution. This means that CPDFMs focus on particles, whereas MFMs focus on cells. A consequence may prove to be that, for problems for which they can be reasonably compared, CPDFMs will prove to use more computer time, and MFMs to use more storage.

In respect of physics, CPDFMs employ something akin to "coupling", but not, it appears to "splitting". Nothing like the Mendelian concept appears so far to have been introduced; and perhaps indeed to do so would be impossible within the Monte-Carlo framework.

The 'mixing models' of the PDFMs are recognised, even by their strongest proponents, as being their major source of weakness; so it may be the constraints imposed by the mathematical technique which cause "relaxation to the local mean" to be used as the mixing model when a two-dimensional bluff-body flow is simulated (Correa and Pope, 1992), despite its known inadequacies.

There are also practical differences: MFMs can provide meaningful, and perhaps adequate, results with rather coarse population grids, eg no more than 20 fluids; but it is not clear that a corresponding saving is possible with CPDFMs.

Moreover, MFMs have always possessd, and need to retain, the notion that it is only few of the fluid properties which need to be discretised; whereas CPDFMs appear to adopt an all-or-nothing principle.

Finally, to bring to an end a comparison which deserves much more study and space, papers describing CPDFMs are written in language of a highly mathematical character, whereas descriptions of MFMs use more words and fewer symbols. Some users of turbulence models may base their choice between MFM and CPDFM on which is described in the language which is more congenial to them.

In summary, there appear to be sufficient differences between CPDFMs and MFMs, and (in the author's view) sufficient advantages on the side of the MFMs, to render the latter well worthy of the attention of those researchers who aim to provide engineers with the best predictive tools for turbulent combustion.

(f) Comparison with the eddy-break-up model -------------------------------------------

The eddy-break-up model (Spalding, 1971a) has had a surprisingly (in view of its crudeness) long-lasting influence: it inspired the invention of the eddy-dissipation concept (Magnussen,1976); it influenced the development of Pope's probabilistic model (1982); and it can usually be found to underly many models of the presumed-pdf kind.

It is now possible also to discern that the multi-fluid model which is described in the present paper is merely the long-delayed extension of the ideas of the original paper.

All persons with experience in computational fluid dynamics know that, in order to obtain improved accuracy, it is necessary to "refine the grid" sufficiently. MFM, it can be said, is merely a grid-refined eddy-break-up model.

In order to make clear the steps which led from EBU to MFM, the next section summarises part of the relevant publication (Spalding, 1995a).

4.4 Generalising the eddy-break-up model ----------------------------------------

(a) The two-fluid nature of the eddy-break-up model ---------------------------------------------------

The central idea of the EBU was that, where the time-average temperature of the gas was above that of the fully-unburned mixture, TU, and below that of the fully-burned mixture, TB, this was the consequence of its being made up of colder fragments inter-mingled with hotter fragments.

Moreover the cold fragments were as cold as they could be, namely of temperature TU; and the hot fragments were as hot as they could be, namely at temperature TB.

The mass fractions of cold and hot gas, MU and MB, were given by:

MU = 1 - MB

= (TB - TU) / (TB - TU).

Of course, combustion could not take place in the cold fragments; because they were too cold: nor in the hot fragments, because they were too hot.

Since combustion undoubtedly does take place, it was supposed that it did so at the inter-faces between the two types of fragments; but these were supposed to occupy only a small proportion of the volume.

(b) The rate-determining process --------------------------------

The suppositions were then made that:-

(1) the rate of combustion per unit volume was proportional to

the rate of intermingling of the two types of fragments; and

(2) this rate was proportional to:


(3) the magnitude of the variable MIXRATE was proportional to either:

VELOCITY_GRADIENT * MIXING_LENGTH in the first model (Spalding, 1971b)

or to:

TURBULENCE_ENERGY_DISSIPATION_RATE / TURBULENCE_ENERGY in a later version (Mason and Spalding,1973).

(c) Successes and failures of EBU ---------------------------------

The model was successful in explaining certain otherwise inexplicable experimental findings, for example the fact that the angle subtended by the flame anchored in a plane-walled channel was nearly independent of approach-gas velocity.

However, it had no means for expressing the influence of chemical kinetics. Yet such an influence does exist, as witness the fact that, when the approach-gas velocity becomes very large, flame propagation abruptly ceases.

This latter shortcoming was addressed by the author already in the original reference, and later by Magnusson and Hjertager (1976), in rather similar ways; but, at least in the present author's opinions, nothing truly satisfactory emerged.

(d) The four-fluid model ------------------------

Nearly twenty-five years elapsed before (what now seems) an obvious next step was taken, namely to increase the realism of the model by increasing the number of fluids, ie, in the terms of the present paper, to refine the population grid.

The central idea was to suppose that the turbulent reacting mixture consisted of fragments of more than two kinds. Four being the minimum number which, it then appeared, would explain all the qualitatively observed facts, it was the number first used.

Specifically, the mixture was supposed to consist of four fragment classes, namely:

A, comprising fully-unburned gas; B, comprising a mixture of unburned gas and combustion products of too low a temperature for the chemical reaction rate to proceed with significant speed; C, comprising a mixture of unburned gas and combustion products of higher temperature, at which the chemical reaction rate is significant; and D, comprising fully-burned gas.

The eddy-break-up model had allowed for only A and D; the new 4-fluid model allowed for B and C in addition.

(e) The rates of production of A, B, C and D --------------------------------------------

By analogy with the formulation successfully used for the EBU, it is postulated for the 4-fluid model that, per unit mass of mixture:

* fluid B is produced from fluids A and D at the rate: 0.5 * MA * MD * MIXRATE

* fluid B is produced from fluids A and C at the rate: MA * MC * MIXRATE

* fluid C is produced from fluids A and D at the rate: 0.5 * MA * MD * MIXRATE

* fluid C is produced from fluids B and D at the rate: MB * MD * MIXRATE

In addition to these rates of production of one fluid (and corresponding diminution of the others):

* fluid D is produced from fluid C (only) at the rate: MC * CHEMRATE

CHEMRATE is, of course, the rate of chemical conversion of the unburned fuel in fluid C to combustion products. It is the element of the model which allows (as EBU did not) chemical-kinetic effects to be introduced rationally.

(f) The first application of the 4-fluid model ----------------------------------------------

The first application of the 4-fluid model was to the same problem as that for which the EBU was invented, namely that of steady turbulent flame spread in a plane-walled duct, through the upstream end of which flows both a stream of unburned combustible mixture, and a separate but thinner stream of fully-burned products to serve as an igniter, shown below. ------------------------------------------------------------------------

Steady turbulent flame spread in a plane-walled duct

Entrance ##################### Channel wall ############## Exit unburned gas-> :::::::::::: -> " ---> :::::::::: --> " ---> ::::::::: flame spreading to wall ---> " --->:::::: ----> burned gas -->:: - --- - --- - --Symmetry plane - --- - --- ----->


The first remarkable feature of the experimental results is that the rate of spread of the flame, as measured by its angle for a fixed inlet-stream velocity, is very little dependent on the fuel- air ratio of the incoming mixture; until, that is to say, this ratio becomes too rich or weak to sustain combustion at all.

The second remarkable feature is that the angle of the flame depends very little on the inlet-stream velocity. This implies that the rate of combustion somehow keeps pace with an increased supply of reactants.

(g) The numerical simulation ------------------------------

This process was simulated by activating the 4-fluid model within the PHOENICS computer program operating in "parabolic" (ie marching- integration) mode.

The magnitude of the MIXRATE quantity was computed in the same manner as for the eddy-break-up model, ie as proportional to the square root of the turbulence energy divided by the length scale; and

the quantity CHEMRATE was taken as equal respectively to

10000, 1000, 100 and 10 times

that which would suffice, if it were Fluid C which filled the duct, to consume all the fuel by the time that the duct exit was reached.

The following figures show, for the four different CHEMRATE values, the profiles of concentration of fluids A, B, C and D across the duct, at a distance half-way between the entrance and the exit.

------------------------------------------------------------------------ Profiles of A, B, C, and D for CHEMRATE = 10000 The symmetry plane is on the left, the wall on the right

VARIABLE D C B A MINVAL= 0.000E+00 0.000E+00 0.000E+00 0.000E+00 MAXVAL= 1.000E+00 1.000E+00 1.000E+00 1.000E+00 CELLAV= 3.992E-01 7.687E-04 1.871E-01 4.126E-01

1.00 +....+....+....+....+....+....+....+....+A.A.A.AA.A . A . 0.90 D DD + . D D A . 0.80 + DD + . D A . 0.70 + D A + . D . 0.60 + D A + . D . 0.50 + D A + . D . 0.40 + D B A + . BB B D AB B . 0.30 + B B A D B + . B B A B . 0.20 + B B A D B + . B B A D . 0.10 B BB B A A DD B + . A A A A D D B . 0.00 A.AA.A.A.AC.C.C+CC.C+CC.C+C.CC+C.CC+C.C.CB.B.B.BB.B 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 the abscissa is Y . min= 8.33E-04 max= 4.92E-02


Profiles of A, B, C, and D for CHEMRATE = 1000 The symmetry plane is on the left, the wall on the right VARIABLE D C B A MINVAL= 0.000E+00 0.000E+00 0.000E+00 0.000E+00 MAXVAL= 1.000E+00 1.000E+00 1.000E+00 1.000E+00 CELLAV= 3.722E-01 7.029E-03 1.883E-01 4.322E-01 1.00 +....+....+....+....+....+....+....+....+A.A.A.AA.A . A . 0.90 D D A + . D D . 0.80 + D D A + . D A . 0.70 + D + . D A . 0.60 + D + . D A . 0.50 + D + . D A . 0.40 + D B B A + . B BB D A BB . 0.30 + BB D B + . B B A D B . 0.20 + B A D B + . B B B A A D B . 0.10 B BB AA D B + . AA A A D D B . 0.00 A.AA.A.A.CC.C.C+CC.C+CC.C+C.CC+C.CC+C.C.CB.B.B.BB.B 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 the abscissa is Y . min= 8.33E-04 max= 4.92E-02


Profiles of A, B, C, and D for CHEMRATE = 100 The symmetry plane is on the left, the wall on the right VARIABLE D C B A MINVAL= 0.000E+00 0.000E+00 0.000E+00 0.000E+00 MAXVAL= 1.000E+00 1.000E+00 1.000E+00 1.000E+00 CELLAV= 2.234E-01 3.924E-02 1.919E-01 5.451E-01 1.00 +....+....+....+....+....+....+....+A.A.AA.A.A.AA.A . A . 0.90 + A + . A . 0.80 + + D D A . 0.70 + D A + . D . 0.60 + D A + . D . 0.50 + D A + . D A . 0.40 + B BB B BB + . B A B . 0.30 + BB D A B + . B A B . 0.20 + B B A D B + B B A DD B . 0.10 + C C C CA A C C D B + C CA A A A C C CC C C DD B . 0.00 A.A..+....+....+....+....+..CC+C.CC+B.B.BB.B.B.BB.B 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 the abscissa is Y . min= 8.33E-04 max= 4.92E-02


Profiles of A, B, C, and D for CHEMRATE = 10 The symmetry plane is on the left, the wall on the right

VARIABLE D C B A MINVAL= 0.000E+00 0.000E+00 0.000E+00 0.000E+00 MAXVAL= 1.000E+00 1.000E+00 1.000E+00 1.000E+00 CELLAV= 8.441E-02 4.799E-02 8.105E-02 7.864E-01 1.00 +....+....+....+...A+AA.A+A.AA+A.AA+A.A.AA.A.A.AA.A . A . 0.90 + A + . . 0.80 + + . A . 0.70 D + . . 0.60 + D A + . . 0.50 + A + . D . 0.40 + + . D B AB . 0.30 + B + . B C A B . 0.20 C C C B + . B A C . 0.10 B A C B + . A C C B . 0.00 A....+....+...D+CC.B+BB.B+B.BB+B.BB+B.B.BB.B.B.BB.B 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 the abscissa is Y . min= 8.33E-04 max= 4.92E-02


(h) Discussion of the results -----------------------------

Despite the crudeness of the line-printer plots, the following observations can be made:

(1) In the first three figures, the differences between the profiles of the unburned-gas (A) and the burned gas (D) concentrations are rather small. Evidently the change of CHEMRATE from 10000 down to 100 has had little effect of the angle of the flame, in qualitative agreement with experiment.

(2) However, the average values of the concentration of the reactive gas (C) (see CELLAV at the tops of the figures) are very small only when CHEMRATE is high (7.87E-04 for 10000, 7.03E-03 for 1000). For CHEMRATE = 100, the CELLAV of C has risen to 3.92E-02, signifying that this gas is not now so easily able to convert into products (D) all that is created by micro-mixing

(3) When, for the fourth figure, CHEMRATE drops to 10, the profiles become radically different: the burned-gas (D) concentration becomes small everywhere, signifying that the incoming hot-gas stream has simply been mixed with the colder unburned-gas stream without promoting any significant further reaction; and the concentration of C rises as a result of this mixing, without any of the diminution which would result from such reaction.

(i) Concluding remarks ----------------------

The above example has been discussed at some length because it shows how "refinement of the composition grid" enables new insights to be gained and improved realism to be added to a numerical simulation. It also perhaps has some historic importance.

Later studies have shown that, as any CFD expert would expect, grid refinement from 2 to 4 is not enough for grid-independent results to be attained; but (depending on the presumption made for the chemical kinetics) as few as 10 fluids will give acceptable accuracy.

-------------------------------------------------------------------- 5. Application prospects ----------------------------- 5.1 The possibility of immediate use ------------------------------------

(a) The implications of already-reported results ------------------------------------------------

In the author's opinion, the multi-fluid model is ready for use, right now, for all simulations in which chemical reactions in turbulent fluids are of prime importance. This opinion holds both for chemical-engineering equipment such as the paddle-stirred reactor, and for combustion chambers of engines and furnaces.

In so far as a turbulence model such as k-epsilon can be relied on for simulating the hydrodynamics, there is only one constant needed by the model, namely that which relates the "fluid-coupling" rate to epsilon/k; and this is known to be of the order of unity.

The results for the rate of reaction in the stirred vessel in (Spalding, 1996b) show very clearly the danger of using a single- fluid model. In particular, the common belief that single-fluid models are usable, when reaction is slow compared with turbulent mixing, is no longer tenable.

It has also been shown that the computer-time penalty associated with use of MFM is not a serious one.

Therefore, it can be plausibly argued, some form of MFM is the ONLY turbulence model which can be justified for use with homogeneous-reaction simulation from now on.

(b) The available alternatives ------------------------------

A counter-argument may be raised by users (who are numerous) of the "presumed-pdf" approach, to the effect that they are already taking sufficient account of fluctuations.

To rebut this counter-argument, one has only to inspect the computed fluid-population distributions which have appeared in the recent publicationss cited in the present paper.

These FDPs have the same significance as the PDFs; and they are of such varied shapes that to suppose that anyone could safely "presume" them would be extremely rash.

How about the use of a probabilistic method instead? If the computer times are low enough, and the computer programs easy enough to use, the case for using such methods is as strong as for MFM. Certainly, although the underlying micro-mixing models are different, there is no quantitative evidence which yet favours one rather than the other.

5.2 Applications to chemically-reacting flows ---------------------------------------------

(a) A list of the main application areas ----------------------------------------

Certainly the reacting-flow applications of MFM are of high importance. They include:-

(1) the calculation of the efficiency of combustion of: gaseous, liquid and/or solid fuels in furnaces, whether for: - production of electricity via steam-raising; - the production of materials such as steel, glass, bricks, pottery or petroleum products; - the destruction of waste; or - the heating of buildings or cities;

(2) the calculation of the efficiency of combustion of: gaseous and/or liquid fuels in engines, whether: - gas turbines for aircraft propulsion; - gas turbines for power production at ground level; - gasoline-fired spark-ignition engines; or - oil-fired diesel engines;

(3) the calculation of the rates of production of (usually) undesirable side-products such as: - oxides of nitrogen; - smoke;

in any of the above types of equipment;

(4) the simulation of reactions other than combustion in reactors of the kinds employed in the chemical, pharmaceutical and food-manufacturing industries;

(5) the simulation of flame propagation in hazardous circumstances such as those arising in off-shore oil-platform explosions, where the information sought is the pressure-versus-time profile on bounding walls.

(b) The relevance of MFM to the computation of combustion efficiency --------------------------------------------------------------------

Let item (1) be considered first; and let it be supposed that the temperature is so high that, at the prevailing pressure (usually atmospheric), all energy-producing reactions proceed very rapidly towards completion.

If a single-fluid model is employed, the reaction will be supposed to be completed at the downstream section at which the time-average fuel/air mixture fraction is stoichiometric or less.

A multi-fluid model will reveal that some unburned fuel is present at locations downstream of that section; and the equipment designer seeking 100% combustion efficiency will want to know how much larger he must make his combustion chamber.

The answer will of course depend on the details of the geometry of the system; which is why a detailed three-dimensional analysis is needed.

MFM is already able to provide this, with some margin of error, of course.

MFM, in order to provide reliable predictions for most circumstances, will need to take account of the fact that light-gas fragments are accelerated (or decelerated) by pressure gradients more easily than dense-gas fragments, so giving rise to what is sometimes called "counter-gradient diffusion". The phenomenon is perhaps most important (and least taken into account) in simulations of reciprocating-engine (ie gasoline or diesel) combustion.

In gas turbines, perhaps, a one-dimensional fluid population will suffice, the distinguishing attribute being the fuel-air ratio.

Gasoline engines may also be simulated adequately by a one- dimensional population, with however reactedness (or mixture fraction minus unburned fuel) as the distinguishing attribute.

Diesel engines, on the other hand, because the mixing and the ignition processes are not separated in time, will probably require two-dimensional populations of the kind discussed in section 1.6.

(c) The computation of pollutant production -------------------------------------------

There are innumerable papers which describe how, once the pdf is known, the rate of production of oxides of nitrogen, smoke, and other undesirable side-products, can be computed. To this author's surprise, many of these claim to have demonstrated sufficient agreement between predictions and experimental data to justify the assumptions.

The methods of calculation are ingenious, and frequently very time-consuming; yet they almost invariably rest on precarious foundations. One of these is usually some variant of the two-fluid eddy-break-up model; and the other will be some presumption about the probability-density of fuel/air ratio or reactedness, but not usually of both.

Sometimes the pdf is taken to be that which corresponds to an assemblage of flamelets, ie one-dimensional steadily-propagating pre-mixed flames.

The author's current view is that the PDFs computed by the MFM as it stands today, ie without any refinements whatsoever, are likely to be as least as realistic as any that are currently in use.

His recommendation is therefore to combine the best chemical-kinetic models of the current literature with the pdfs, one- or two-dimensional according to circumstance, computed by the MFM.

(d) Flame propagation in explosion circumstances ------------------------------------------------

As already mentioned, the four-fluid model has been used with some success; and it can be be expected that further insight will be gained when:

(1) 10 or more fluids are employed, so that the chemical-kinetic data can be more accurately embodied, and

(2) the fluids are allowed to move relative to one another under the influence of pressure gradients.

It is of course the second of these features which will allow the acceleration of the flame propagation resulting from its semi-confinement to be properly represented.

[Further applications to reacting-flow situations remain to be discussed]

5.3 Applications to hydrodynamic and heat-transfer phenomena ------------------------------------------------------------

The "MFM revolution" will not be complete until the reliance on an underlying hydrodynamic model, for example the k-epsilon model which has been used in several of the examples referred to above, has been totally dispensed with.

Here it needs to be noted the k is nothing but a root-mean-square kinetic-energy fluctuation; and MFM is well able to compute RMS values, as has been seen above.

All that is needed therefore is for velocity or energy to be made the discretized variable, and for something similar to be done about the length scale; then the underlying hydrodynamic model can be dropped.

Work along these lines has already enabled simple shear flows to be simulated by MFM in a stand-alone manner (Spalding, 1996b). It will be reported elsewhere.

It is flows with strong body-force effects which are likely to provide the greatest rewards; for single-fluid models simulate them least well. Therefore atmospheric and oceanographic flows, where gravitational effects are significant, and turbo-machinery flows, where centrifugal forces are large, may be the first to be subjected to MFM analysis.

In order that MFM should contribute significantly to heat-transfer engineering, it will have to throw quantitative light on behaviour close to walls, where viscous effects become dominant. Nothing has been done to explore these effects, in the MFM context, so far.

-------------------------------------------------------------------- 6. Research and development tasks ---------------------------------

The multi-fluid model of turbulence, even though its essential ideas have been known for many years, has not yet been subjected to intense and prolonged research and development efforts such as have been lavished on the single-fluid models.

Now is therefore the time to consider what form such efforts might take, and to initiate the appropriate actions.

The author's current thoughts on these now follow.

6.1 Mathematical and conputational tasks ----------------------------------------

(a) Grid-refinement -------------------

It is necessary to establish, for a much wider range of problems than has been investigated so far, that refining the population grid (ie increasing the number of fluids) does indeed always lead to a unique solution, and then to establish how the accuracy of solution depends upon the grid fineness.

This needs to be done for two- and (at least a few) three- dimensional population grids as well as one-dimensional ones, before the soundness of the basic ideas and solution algorithms can be regarded as secure.

(b) Consistency tests ---------------------

When the micro-mixing constant becomes very large, the range of fluids present at a particular location reduces to 2 for a 1D population, to 4 for a 2D one and to 8 for a 3D one. It needs to be checked that the system of equations does indeed exhibit this behaviour, and that the right values are calculated for the fluid concentrations.

A prudent and insightful investigator will wish to devise other tests of a self-consistency or conservativeness character before he wholly trusts his mathematical apparatus.

(c) Non-uniform, unstructured, and self-adaptive grids ------------------------------------------------------

So far, the author has used only uniform and structured grids, which have remained the same throughout the computation; yet there are obvious advantages to be gained by relaxing these restrictions.

Thus, it may be convenient to define the attribute to be discretised as the ratio of the current temperature to the maximum temperature in the field, which temperature may change throughout the period of the calculation. Such a grid would have to be self-adaptive.

Another kind of self-adaptive grid would change its uniformity as the calculation proceeded, so as to capture with maximum accuracy the shape of the fluid-population distribution.

Yet another would insert new subdivisions of the discretised attribute in regions of steep variation, and perhaps remove them from regions of less interest.

The best way of summarising the possibilities is to say that probably all the ingenious devices which specialists have invented for the better representation of variations in geometric space are likely to have their population-grid counterparts.

(d) Computational improvements ------------------------------

Although the computer-time burden is not yet great, it may become considerable when fine-grid three-dimensional transient simulations have to be undertaken.

Fortunately, there is much which can be done to reduce the load. For example:-

* population grids may be made non-uniform, self-adaptive and unstructured, as just explained;

* the multi-fluid model can be confined to only those parts of the flow domain in which unmixedness is of physical significance;

* cell-wise simultaneous-variable-adjustment procedures can be used, with multi-grid (in both senses) acceleration devices;

* the employment of multi-block solution techniques.

It will also be necessary to work out what is the optimal discretiz- ation principle; for certainly it is not optimal to employ a grid which is extremely fine in geometric space but has too small a population grid (ie too few fluids) to represent adequately the fragmentariness of the flow.

6.2 Comparisons with experiment -------------------------------

In order to establish the predictive capabilities of MFM now and in the future, it is of course necessary to make comparisons with experimental data.

Three kinds of comparisons should be distinguished, namely:

(1) those most desired by the potential end-users, for example between measured and predicted total smoke production for a particular gas-turbine combustor, or yields of main- and side-products for a paddle-stirred reactor;

(2) those with well-known data on the velocity, temperature and concentration profiles in jets, wakes, boundary layers and simple turbulent flames; and

(3) those which provide detailed measurements of probability-density functions of important fluid variables.

Were there no urgency about solving the practical problems of engineering and the environment, a rational research program would probably concern itself with kind (2) first of all, in order to establish a convincing prima facie case for MFM; then experiments would be conducted systematically so as to permit kind-(2) comparisons, which might lead to improvements to some components of the model.

Finally comparisons of kind (1) would be made, whereafter, if the comparisons were successful, industrial use of MFM would begin.

However, perhaps fortunately, there IS much urgency; for the models which are currently used for predicting turbulent chemical reaction in particular are far from satisfactory. It therefore seems reasonable that some comparisons of kind (1) should begin in the near future.

For example, it may be found that the Mendelian principle does not always apply; and it is certainly to be expected that, when the FPD's of two velocity components are in question, the coupling of two fragments with differing x-direction velocities will lead to offspring with differing y-direction velocities also. (See section 6.3 below.)

6.3 Conceptual developments ---------------------------

(a) "Parental bias" in the "splitting" process ----------------------------------------------

As has been mentioned above, the promiscuous-Mendelian hypothesis is unlikely to prove to be the best; and it is not hard to improve upon it, if the underlying mechanism of section 1.8b above is believed, as follows.

The profile of concentration (say) in the conjoined fragments, after some time will have some such "error-function-like" shape as indicated below on the left, to which coresponds a pdf of the shape shown on the right.

Clearly the pdf exhibits the largest frequencies near the extreme, ie the "parental" concentrations.

|****** ^ |****** | ***** | |***** | **** | |**** | *** conc- |*** | ** entr- |** | * ation |* | ** | |** | *** | |*** | **** |**** | ***** |***** | ****** |****** |------------------------------------------- |--------------- ----------- distance -------> - frequency--->

Only if the profile of concentration were linear with distance would the Mendelian assumption be correct.

(b) The effects of laminar Prandtl and Schmidt numbers ------------------------------------------------------

Let it now be supposed that the two fragments differ both in temperature and salinity. Then, while they are in contact, the profile of temperature will broaden much more rapidly than the profile of salinity. As a consequence, the offspring do NOT lie on the diagonal joining the two parental locations as indicated in section 1.8c above.

|_______|_______|_______|_______|_______|_______|_______| | | | | | | | M | ^ | | | | | | | * | | |_______|_______|_______|_______|_______|_______|_______| | | | | | | | | * | temp- | | | | | | | | erat- |_______|_______|_______|_______|_______|_______|_*_____| ure | | | | | | | | | | *| | | | | | | | |_______|_______|_______|_______|_______|_______|_______| | | * | | | | | | | | | | | | | | | | | |____*__|____ __|_______|_______|_______|______ |_______| | F | | | | | | | | * | | | | | | | |_______|_______|_______|_______|_______|_______|_______|

----- salinity------------>

Instead, the offspring are more likely to lie along the lines of asterisks shown above, with much smaller salinity changes than temperature changes.

(c) The coupling-splitting hypothesis for a two-velocity population -------------------------------------------------------------------

As a final example of how the coupling-splitting hypothesis can be modified in the direction of greater realism, let it be imagined that the attributes of a 2D population grid are the horizontal velocity U and the vertical velocity V.

Then let the collision be imagined of two fluid fragments which have the same values of V but differing values of U. So the father and mother lie on the same horizontal. Where do the offspring lie?

|_______|_______|_______|_______|_______|_______|_______| | | | | | | | | ^ | | | | * | | | | | |_______|_______|_______|_______|_______|_______|_______| | | | | | | | | | | | * | * | * | * | * | | V |_______|_______|_______|_______|_______|_______|_______| | F | | | | | | M | | | * | * | * | * | * | * | * | | |_______|_______|_______|_______|_______|_______|_______| | | | | | | | | | | | | * | * | * | * | * | | | |_______|____ __|_______|_______|_______|______ |_______| | | | | | | | | | | | | * | | | | |_______|_______|_______|_______|_______|_______|_______|

----- U ------------>

The answer, it appears to the present author, must be "not only on the horizontal line FM"; for colliding fluid fragments are likely to generate motion in lateral directions as well as being checked or accelerated in the direction of their velocity difference.

Therefore some of the offspring must be deposited into boxes above and below the horizontal line, of course in such a way as to preserve momentum and to ensure that there is at least no gain of energy.

The above sketch illustrates this by its band of asterisks; but, before the idea can be expressed in a computer program, a precise offspring-distribution formula must be settled.

(d) Other matters -----------------

Among the questions still to be addressed are:-

* How is the inter-fluid friction which opposes motion caused by differential body forces to be formulated?

* As the Reynolds number becomes lower, fluctuations die out. This can be ensured by increasing the MIXRATE quantity. But what dependence of MIXRATE on Reynolds number will fit the experimental data?

* In the mixing-layer model alluded to in section 3.2, the root- mean-square of the velocity fluctuations has been employed in a formula for the effective viscosity, in place of the usual k**0.5. However, MFM provides so much more information about the velocity pdf than is needed for the RMS value. In what way can that information be exploited so as to give more realistic simulations?

No answers are proposed at the present time.

-------------------------------------------------------------------- 7. References ---------------

MJ Andrews (1986) "Turbulent mixing by Rayleigh-Tay lor instability"; PhD Thesis, London University WH Avery & RW Hart (1953) Ind Engg Chem vol 45 p 16 34 J Boussinesq (1877) "Theorie de l'ecoulement tourbillant"; Mem, Pre. par. div. Sav. Paris, vol 23 P Bradshaw, DR Ferriss & NP Atwell (1967) "Calculation of boundary-layer development using the turbulent energy equation"; J Fluid Mech, vol 28, p 593 KNC Bray and PA Libby "Counter-gradient diffusion in pre-mixed turbulent flames"; AIAA J vol 19, p205, 1981 JY Chen & 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 & 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 & 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 MA 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 DJ 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 N Fueyo, JC Larroya, L Valino, C Dopazo (1995) "A combined CFD-Montecarlo method for the solution of the scalar PDF equation in turbulent reaction" European PHOENICS User Conference, Trento, Italy M Gonzalez & R Borghi (1991) "A Lagrangian intermittent model for turbulent combustion; theoretical basis and comparison with experiments", Turbulent Shear Flows 7, Springer Verlag FH Harlow & PI Nakayama (1968) "Transport of turbulence-energy decay rate"; Los Alamos Sci Lab U Calif report LA 3854 NM Howe and CW Shipman (1965) "A tentative model for rates of combustion in confined turbulent flames" 10th International Symposium on Combustion, p 1139 The Combustion Institute, 1965 ICOMP (1994) 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 & RW Bilger (1976) "The prediction of turbulent diffusion flame fields and nitric oxide formation" 16th International Symposium on Combustion, The Combustion Institute p 1643 SW Kim and CP 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 & W Kollmann (1980) "Prediction of turbulent diffusion flames with a four-equation turbulence model" Acta Astronautica, vol 71 p 91 AN Kolmogorov (1942) "Equations of motion of an incompressible turbulent fluid"; Izv Akad Nauk SSSR Ser Phys VI No 1-2, p56 VR Kuznetsov (1979); USSR Fluid Dynamics vol 14, p328 BE Launder and DB Spalding (1972) "Mathematical Models of Turbulence", Academic Press FC Lockwood & AS Naguib (1975) "The prediction of the fluctuations in the properties of free, round-jet, turbulent, diffusion flames", Comb & Flame, vol 24 p 109 JP Longwell (1954) "Selected combustion problems" Butterworths, p 508 BF Magnussen and BH Hjertager (1976) "On mathematical modelling of turbulent combustion with special emphasis on soot formation and combustion" 16th International Symposium on Combustion, pp 719-729 The Combustion Institute MR Malin (1986) "Turbulence modelling for flow and heat transfer in jets, wakes and plumes"; PhD Thesis, London University HB Mason and DB Spalding (1973) "Prediction of reaction rates in turbulent pre-mixed boundary-layer flows" Combustion Inst European Symposium, pp 601-606 JB Moss (1980) "Simultaneous measurements of concentration and velocity in an open pre-mixed flame", Combustion Science and Technology, vol 22, pp 115-129 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 M Noseir (1980) "Application of the ESCIMO theory of turbulent combustion"; PhD Thesis, London University 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 SB Pope (1982) Combustion Science and Technology vol 28, p131 SB Pope (1985) Progr Energy Combust Sci vol 11, pp119-192 SB Pope (1990) "Computations of turbulent combustion; progress and challenges" Twenty-Third International Symposium on Combustion, The Combustion Institute, pp 591-612 L Prandtl (1925) "Bericht ueber Untersuchungen zur ausgebileten Turbulenz"; ZAMM vol 3, pp 136-139, 1925 L Prandtl (1945) "Ueber ein neues Formelsystem fuer die ausgebildete Turbulenz", Nachr. Akad. Wiss. Goettingen O Reynolds (1874) "On the extent and action of the heating surface of steam boilers"; Proc. Manchester Lit Phil Soc, vol 8, 1874 RP Rhodes, PT Harsha & CE Peters (1974) "Turbulent-kinetic- energy analyses of hydrogen-air diffusion flame" Acta Astronautica vol 1 p 443 R Sala and DB Spalding (1973) "A mathematical model for an axi-symmetrical diffusion flame in a furnace". La Rivista di Combustibili, vol 27, pp 180-186 PG Saffmann (1970) "A model for inhomogeneous turbulent flow"; Proc Roy Soc London vol A317 pp 417-433 S Shaanen, JH Ferziger, WC Reynolds (1975) "Numerical Simulation of turbulence in the presence of shear"; Dept Mech Engg Stanford U, Rep TF-6 KI Shchelkin USSR J Tech Phys vol 13, p 520, 1943 U Schumann (1973) "Ein Verfahren zur direkten numerischen Simulation ..." Thesis 1854 U Karlsruhe DB Spalding (1969) "The prediction of two-dimensional steady turbulent elliptic flows" ICHMT Seminar, Herceg Novi, Yugoslavia DB Spalding (1971a) "Mixing and chemical reaction in confined turbulent flames" 13th International Symposium on Combustion, pp 649-657 The Combustion Institute DB Spalding (1971b) "Concentration fluctuations in a round turbulent free jet"; J Chem Eng Sci, vol 26, p 95 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. DB Spalding (1982) "Chemical reaction in turbulent fluids" IC CFDU Report 82/8, June 1982 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 DB Spalding (1984)"The two-fluid model of turbulence applied to combustion phenomena" 22nd AIAA Meeting, Reno, Nevada DB Spalding (1987) "A turbulence model for buoyant and combusting flows"; International J. for Numerical Methods in Engineering vol 24, pp 1-23 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 DB Spalding (1995a) "Models of turbulent combustion" Proc. 2nd Colloquium on Process Simulation, pp 1-15 Helsinki University of Technology, Espoo, Finland DB Spalding (1995b) "Multi-fluid models of turbulent combustion"; CTAC95 Conference, Melbourne, Australia DB Spalding (1995c) "Multi-fluid models of Turbulence", European PHOENICS User Conference, Trento, Italy 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 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 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 DB Spalding (1966d) Unpublished work at CHAM Ltd RLT Sun (1982) "Application of the ESCIMO theory to turbulent diffusion flames"; PhD Thesis, London University LT Tam (1981) "The theory of turbulent flow with complex chemical kinetics"; PhD Thesis, London University ST Xi (1986) "Transient turbulent jets of miscible and immiscible fluids"; PhD Thesis, London University V Yakhot & S Orszag (1986) "Renormalization group analysis of Turbulence. I. Basic theory" J. Sci Comp, vol 1, no 1, pp 1-51 DC Wilcox (1993) "Turbulence modelling for CFD", DCW Industries, La Canada, California GC Williams, HC Hottel and AC Scurlock,(1949), 3rd Symposium on Combustion, p21, Williams and Wilkins 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 JZ Wu (1987) "The application of the two-fluid model of turbulence to ducted flames" IC CFDU Report, June 1987