Encyclopaedia Index

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

Research workers in the field of turbulent combustion have been particularly aware of the difference between macro-mixing and micro-mixing, for which Hawthorne, Weddell and Hottel (1949), in a new-ground-breaking publication, coined the term "unmixedness".

They observed in their study of turbulent diffusion flames (i.e. those in which streams of pure fuel and pure air mix and burn), that at locations at which the time-average fuel-air ratio is stoichiometric, so that precisely enough air is present to enable the fuel to burn completely, much unburned fuel is still present.

This was because, although the macro-mixing was perfect, the micro-mixing was not.

The former process is that of bringing the materials together in the right time-average proportions; the latter is that which causes the instantaneous proportions also to be correct.

[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)

& 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:

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:-

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:-

Similar remarks 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:-

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:


  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:


  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:-

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.