Encyclopaedia Index

### 1.1 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?

• 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 soilds with which it is in contact?

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

the answers to all these questions being of course different for each location and instant of time.

Methods of answering such questions are numerous and varied in nature; but it appears useful to separate them into four main classes, as will now be explained.

In later sections of the paper will be found discussions of the pros and cons of the four classes, with particular attention to their prospects for development, followed by more detailed attention to the class which the present author believes to have the brightest prospects, namely the multi-fluid class.

### 1.2 Definitions of the four classes

Turbulence models are here classified as:

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

These terms will now be explained.

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.

The so-called "k-epsilon" model is probably the best-known example nowadays.

MULTI-FLUID models express the fluctuations in a different way, 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.

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.

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.

### 1.3 Some members of the four classes

In order to explain to well-read readers more precisely what is meant by this classification, the following partial lists of members are provided.

CLASS I 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 [1]):

• zero-(differential)-equation models, such as those of Boussinesq [2] and Prandtl [3];

• one-equation models, such as those of Prandtl [4] and Bradshaw [5];

• two-equation models, such as those of Kolmogorov [6], Harlow and Nakayama [7], Saffmann [8], Spalding [9], Wilcox [10], Yakhot and Orszag [11] and many others;

• multi-equation models, such of those concerned with multiple levels of energy or scale [12, 13] or with further statistical quantities [14, 15, 16].

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

CLASS II is less densely populated. Nevertheless, a full listing would exceed available space.

The basic concept was already present in the writings of Reynolds [17] and Prandtl [3]; for they both conceived of the relative motion of fluid fragments; which fragments could be of significant size.

Scientists concerned with combustion [18, 19, 20, 21] 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 [22,23],

• the "eddy-dissipation concept" of Magnussen [24], and

• the Bray-Libby-Moss "flamelet" model [25, 26].

A more extensive exploration of the capabilities of two-fluid models was made later by the author and his co-workers [27 - 38]. 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 will be illustrated later in this paper, the multi-fluid idea has been extended to 4-, 14, and many-fluid models [39,40]. 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 [41, 42, 43, 44].

CLASS III appears to have originated in the chemical-engineering- science field with the publications by Curl [45] and Dopazo and O'Brien [46]. More recently Pope [47, 48, 49, 50], Kollmann [51] and Chen [52] have conducted a vigorous research campaign, which, to judge from the proceedings of a recent conference [53], is gathering momentum. Once again, it is the desire to be able to simulate combustion phenomena with quantitatuve ralism which provides much of the motivation.

CLASS IV, ie direct-numerical simulation, has been subjected to increasing attention as computers have become more and more powerful. Pioneers include Schumann [54] and Reynolds [55].

### 1.4 "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 pdf's are presumed rather than computed. Models of this kind may perhaps have started with the author's own paper of 1971 [14], in which a "two-spike" double-delta-function presumption was made. Subsequently, Lockwood and Naguib [56] made the "clipped-Gaussian" presumption; they were followed by others, including Kent and Bilger [57], Kolbe and Kollmann [58] and Rhodes et al [59].

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 ever emerged.

As will be shown below, especially in section 3.4, 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 will be shown to depend 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. Presumpton of the shape without such guidance is however likely to do as much harm as good.

### 2.1 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.

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.

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.

### 2.2 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 below 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 for the two-fluid model.

• Class-III methods work in a non-discretized, multi-dimensional, all-fluid-attribute space; "coarsening" for economy appears to require reductin if 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, discharge fewer bullets.

### 2.3 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 the necessary "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; but it is interesting to note that the generic Multi-Fluid Model is not so afflicted.

### 2.4 Prospects for the future

The present author is of the opinion that, although the simpler Class-I methods will (and should) long remain in use, some of the more complex ones, for example those which solve differential equations for the Reynolds stresses, will fall into disuse. The reason is that all single-fluid models represent a determined attempt to NEGLECT THE OBVIOUS.

Anyone who looks into the sky on most days sees clouds of various sizes, slowly forming or dissolving, coalescing or splitting, rising or falling, and sometimes with blue sky visible beween them. Smoke plumes from chimneys, the condensate streams from vehicle exhausts, and the discharges of rivers into the oceans, all tell the same story: turbulent fluids are fragmentary. They are to be understood as heterogeneous populations; and to suppose that all the "ethnic groups" within them will act in the same way is as unwise of the turbulence modeller as it would be of the politician.

Both Classes II and III recognise this, the first through its FPD (ie fluid population distribution) and the second thrugh its PDF (ie probability density function). In the author's view, each has good prospects of meeting the most pressing needs of combustion scientists within a few years, provided that research efforts are diverted to at least some extent from the less-promising Class-I endeavours.

Which of these two classes will be the more useful cannot be foretold with certainty; so both should be developed further until their relative merits become clearer. Probably, as experience is gathered, each will borrow useful features from the other; they may even merge, as the finite-volume and finite-element methods of CFD have tended to do.

As to Class-IV methods, only a not-yet-in-sight increase in computer power can bring them out of the research laboratory into widespread use. They remain very valuable, however, as providers of insight, and even quantitatuve guidance, to the developers of methods of Classes II and III.

### 3.1 A 2-fluid model applied to the environment

In the atmosphere, and in natural waters, gravitational forces have a major effect on the fluid-dynamic phenomena, precisely because of the facte that fragments of denser fluid (colder air or more-saline water) are mingled with lighter ones. The latter tend to rise, and the former to fall, often with striking effects.

For example, hurricanes gather their destructive power from the fact that the upwardly moving moist-air fragments, as they rise to lower-pressure altitudes, shed much of their water content as rain. The latent heat of condensation has the effect of increasing still further the disparity of density between the upward- and downward-moving fluids, the relevant motion of which is therefore maintained, despite the friction between them.

Two-fluid models of turbulence are well able to simulate such phenomena. For example, the present author showed [57] how PHOENICS could be used to calculate the variation with time of the upward- and downward-moving members of the "atmospheric population" as a consequence of the heating and cooling of the surface of the earth.

temperature in upward-moving fluid
temperature in downward-moving fluid
Not surprisingly, the upward-moving fluid has a higher temperature than the downward-moving fluid.

volume fraction of upward-moving fluid
The upward-moving fluid occupies less than 50% of space near the hot surface.

The colour plots of the results of that study, which have just been seen, could not be easily produced for the printed paper. Instead therefore, simply to show what kinds of information can be provided by a two-fluid turbulence model, some line-printer output from the PHOENICS library case W975, which concerns flow between a a hot surface at rest and an upper cold moving surface. This is a Couette-flow idealization of an unstable atmosphere.

Each plot shows the variation with vertical distance of some properties of the two fluids, namely:

in Fig.1

• V1 and V2, the vertical velocity components of the fluids;
• W1 and W2, their horizontal velocity components;
in Fig.2
• H1 and H2, their temperatures;
• UP, the volume fraction of the rising fluid; and
and in Fig.3
• MDOT, the volumetric rate of mass transfer from one fluid to the other.

```
Fig.1       1 =   V1   2 =   V2   3 =   W1   4 =   W2
MINVAL= -1.871E-01 -1.871E-01  7.617E-13  7.617E-13
MAXVAL=  1.875E-01  1.875E-01  2.000E+00  2.000E+00
1.00 +....+....+....+...1+1..1+....+....+....+....+....4
+          1 1 1 1        1                       +
+   11 1 1                   1 1 1              443
+111                               1 1 1 1 1 444 3+
1                       4 4  4 4 4 4 4 4 4 4   331+
+  444 4 4 4 4 4 4 4 4  3 3  3 3 3 3 3 3 3 3 33   2
224    3 3 3 3 3 3 3                             2+
+4 333 2 2 2 2 2                              222 +
433              2 2                   2 2 2 2    +
+                    2         2 2 2 2            +
0.00 3....+....+....+....+...2+2..2+....+....+....+....+
0    non-dimensional vertical height            1.0

Fig.2       1 =   H1   2 =   H2
MINVAL=  0.000E+00  0.000E+00
MAXVAL=  1.000E+00  1.000E+00
1.00 1....+....+....+....+....+....+....+....+....+....+
+1                                                +
2 1                                               +
+2 111 1                                          +
+ 222    1 1 1 1 1 1 1  1 1  1 1 1 1 1            +
+    2 2 2 2 2 2 2 2 2  2 2  2 2 2     1 1 1 111  +
+                                  2 2 2 2 2 22 11+
+                                              22 +
+                                                21
+                                                 +
0.00 +....+....+....+....+....+....+....+....+....+....2
0    non-dimensional vertical height            1.0

Fig.3       1 = UP     2 = MDOT
MINVAL=  0.000E+00 -2.038E-02
MAXVAL=  1.000E+00  1.992E-02
1.00 2....+....+....+....+....+....+....+....+....+....+
+                                                 +
+                                                 +
+2                                                +
+ 2222 2 2 2 2 2 2 2 2       1 1 1 1 1 1 1 1 111111
+                       2 2                       +
111111 1 1 1 1 1 1 1 1       2 2 2 2 2 2 2 2 2222 +
+                                                2+
+                                                 +
+                                                 +
0.00 +....+....+....+....+....+....+....+....+....+....2
0    non-dimensional vertical height            1.0

```
Despite the crudeness of the graphical representation, information of physical interest can be derived. Thus, one may see from Fig.1 that the upward-rising fluid has the smaller horizontal velocity, for the understandable reason that it comes from a lower-velocity region.

Fig. 2 reveals that it is hotter than the downward-moving fluid, no doubt because it has received heat from the warmer regions, closer to the ground.

Fig. 3 shows that, according to this model, the volume fractions of the two fluids are not uniform. Thus there is more upward-moving fluid in the upper half (ie the right-hand half on the diagram) of the layer than in the lower. It is the inter-fluid mass-transfer rate (MDOT) which maintains the balance.

Before leaving this topic, it is worth remarking, to those to whom the two-fluid model is new, that this Couette-flow study can be regarded as a refinement of the idea which Ludwig Prandtl presented seventy years ago as the "mixing-length model". Prandtl had to make assumptions, where the two-model permits calculations. Thus, Prandtl had to assume that the differences of the vertical velocity were equal to the differences of the horizontal velocity, and that the volume fractions of the two fluids were equal.

As mentioned above, the two-fluid concept was inherent in the thinking of both Reynolds and Prandtl. To develop it further is to continue a distinguished tradition.

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

The two-fluid model just discussed was invented in the early 1980's with combustion processes especially in mind. Some applications of this kind will be found in the same section of the PHOENICS library as the above Couette-flow example.

In the present section however, allusion will first be nade to an earlier (1970's) model of combustion process, which already contained the germ of the two-fluid idea. This is the "eddy-break- up" model [22], the essence of which is the supposition that a turbulent flame of pre-mixed fuel and oxidant can be most simply regarded as the intermingling of members of a two-component population of gas fragments: one component is completely burned, and therefore at high temperature; the other is cold and entirely unburned.

Thus, if the unburned-gas temperature is 0 degC and the burned-gas is 2000 degC, a mixture having a local-average temperature of 1000 degC is regarded as comprising 50% (by mass) of each component.

Such a population, while having the same energy content and specific volume as a homogeneous gas, differs from it profoundly in respect of chemical behaviour. For example, the rate of NOX production is much more than twice as great in the 2000 degC gas as the value it would have in twice as much 1000 degC gas.

Moreover, the rate of combustion in the 1000 degC gas would be much smaller than in the two-component mixture; for, at the surfaces between the hot and cold fragments, thin layers of gas having intermediate temperatures and concentration are formed by diffusion; and in these "flamelets" chemical reaction can be intense.

The quantitative part of the eddy-break-up model was the presumption that the local-average rate of this combustion reaction was controlled by the average size and the average relative velocity of the fragments, and that this rate could be deduced from hydrodynamic properties of the flow, for example the quantity "epsilon/k", ie the specific decay rate of turbulence energy. This rate term was multiplied by the product of the concentration of the two fluids in the mixture.

### (b) The four-fluid model

Although the two-fluid eddy-break-up model (EBU) has been widely used, especially in the "eddy-dissipation-concept" form (EDC) introduced by Magnussen [24], it appears to have taken twenty-five years for anyone to recognise that greater realism can probably be achieved by increasing the number of fluids in the population.

The modest step from two to four has however at last [40] been made and will now be described and illustrated.

The new supposition is that, in addition to the two "ethnicaly- pure" components of the EBU, which components will be here called A and D, two "mixed-race" components can also be distinguished, namely B (which can be formed by mixing two mass units of A with one of D) and C (for which the mixture ratio is reversed). B is supposed to be (like A) too cold to burn sufficiently; whereas the hotter C, which because of its origin contains unreacted fuel and oxidant, can burn vigorously at a rate controlled by chemical kinetics, being thus transformed into D.

The distinction from the EBU is that the latter's one-step mixed- is-burned presumption is replaced by the following set of parallel paths to combustion:

1. A + D --> C --> D
2. A + D --> B
B + D --> C --> D
3. A + C --> B
B + D --> C --> D

The mixing rates which produce B and C from A and D are supposed, as in the two-fluid EBU, to depend upon the products of the concentrations of the parent fluids, multiplied by a hydrodynamic term such as epsilon/k.

From the point of view of computation time, the increase from two fluids to four imposes no intolerable burden; for many more differential equations have usually to be solved in the hydrodynamic part of the simulation. However, the increase in realism is great; for the boundaries between the regimes of control of combustion rate by hydrodynamics on the one hand and by chemical kinetics on the other can be clearly delineated.

### (c) Examples of the use of the four-fluid model

The four-fluid model has been implemented in PHOENICS, and applied to transient turbulent-flame propagation through enclosures containing obstacles [58]. Colour plots of some of these calculations will be presented during the lecture.

A 2D transient flame modelled by the 4-fluid model The flame-arrival times in milli-seconds, after ignition at the left-hand end.

The duct is partly blocked by a series of baffles, and is open at the right-hand end.

The calculations were performed by Dr David Freeman, with the following results.

For the printed paper, the following line-printer plot was provided.

It shows the profiles of concentration of the four fluids in a steadily-propagating turbulent pre-mixed gas, the propagation direction being from the left (where the composition is 100% A) to the right (where the composition is 100% D). The flame is the thick region, in which the intermediate fluids B and C have finite concentrations.

Interpretation of Fig.5 is not difficult. Fluids A and D diffuse towards the flame, by reason of their concentration gradients, and there create B and C by mixing. It will be observed that, in the example shown, the maximum concentration of B is of the order of 0.4, whereas that of C is of the order of 0.0025, the explanation being that the chemical reactivity of the gas is large compared with the rate at which it is supplied by mixing.

```

Fig.5       1 = D      2 = C      3 = B      4 = A
MINVAL=  0.000E+00  6.127E-06  0.000E+00  0.000E+00
MAXVAL=  1.000E+00  2.482E-03  1.000E+00  1.000E+00
1.00 444444....+....+....+..22222..+....+....+..11111111
+     44444          22    222     11111111       +
+         444       22        221111              +
+            44    2         1122                 +
unburned          44 2       111    22            burned
gas               44     111       222            gas
+               23443333333         222           +
+             333   444    3333       222         +
+          3332   111 444      33333     2222     +
+     333332211111       44444     333333333222222+
0.00 33333322211111.+....+....+....444444444444444444444

0  distance normal to the flame ---- >          1.0
<<<<<----  direction of propagation  <<<<<<-----

```

### (d) Discussion of the four-fluid model

The new light thrown on turbulent flame proceses by the four-fluid model remains to be systematically exploited. In the author's view its main benefit will be educational; for it enables many types of flame to be explored at little expense, whereafter the understanding thus engendered can be put to quantitative use with fluid populations with more numerous components.

How many components should be used will of course depend upon the task which the modeller has set himself. If the overall rate of production of NOX is in question, one can reasonably surmise that breaking the temperature range into four intervals rather than two will certainly improve accuracy; but that will surely not be the end of the matter. The wise modeller will use as many fluids as are needed to give him the accuracy he requires for the purpose in hand, but no more.

It should thus be apparent that the multi-fluid approach, which involves esentially the selective discretization of fluid attributes, gives the modeller new freedoms; but they are freedoms of a kind with which he is familiar. He understands about refining grids of geometrical space and time; now he can do the same in composition space.

### (a) The dimensionality of the population

The analogy between familiar grid-refinement considerations and those governing the increase in the number of fluids has just been mentioned. Here is another aspect of it: dimensionality.

The four-fluid model increased the number of fluids by selecting an increasing number of fluid-defining values of a single attribute, namely the reactedness. It exemplified a one-dimensional fluid population.

However, multi-dimensional populations of fluids are generated when more than one attribute is used for fluid definition. This will now be illustrated, by way of the simplest example, namely a two- dimensional population.

Once again, an example relevant to combustion will be chosen; specifically, values of fuel/oxidant ratio will be selected, as well as those of reactedness, so as characterise the population.

### (b) The fourteen-fluid model

The following diagrammatic representation is copied from [40], where some of the main concepts of multi-fluid turbulence modelling are explained at greater length than has been possible here.

The 4-by-5 fluid-attribute grid, of which the ordinate can be regarded as reactedness and the abscissa as fuel-air ratio, represents a two-dimensional population.

```
************************************************************
Fig.6 The 20-fluid population, of which only 14 fluids can exist
_________________________________________
|///////|///////|///////|       |       |
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.

************************************************************
```
Fluids 6, 11, 16 and 20 are supposed to contain no unburned fuel; they thus represent completely-burned gases, of various fuel-air ratios, which can react no further.

Fluids 5, 9, 13, 14, 17 and 18 contain finite amounts of free fuel; but they are regarded as being too cold to burn, like fluid B above.

Fluids 10, 15 and 19 both contain fuel and are hot enough to burn; it is therefore they which, like fluid C in the four-fluid model, carry out the chemical-reaction process.

Fluid 10 thus becomes transformed into 11, fluid 15 into 16, and fluid 19 into 20.

### (c) The coupling-and-splitting process

In the Bunsen-burner flame for which the model was devised, the "Adam and Eve" of the whole population are fluids 17 (fuel-rich, unburned) and 1 (pure air). The fluids of other kinds come into being as a result of "coupling", ie contact and intermingling), and "splitting", ie production of progeny having intermediate grid locations.

For example, fluid 5 can mix with fluid 19 so as to produce fluids 9, 10, 14 and 15, in proportions which the modeller is free to decide, on the basis of considerations which it would be distracting to discuss here.

A small stream of fluid 16 (stoichiometric combustion products) is supplied at the edge of the burner tube, to act as a lame holder.

### (d) The Bunsen-burner model

Fig.7, also copied from [40], illustrates the subject of the study and how its behaviour simulated by the use of PHOENICS. Especially noteworthy is the fact that the "parabolic option" is used. This is of great convenience; for it enabled the computations to be completed in about 5 minutes on a Pentium PC.

```
************************************************************
Fig. 7 Diagrammatic representation of the Bunsen-burner flame

. /////////
|  //flame/                | * PHOENICS is used to solve the
.  ///////                 |   differential equations for 14
|   /////     air at       |   fluid concentrations, as well as
.   ////      rest,        |   for two velocities and pressure;
|   ///       entrained    |
/   ///       into         | * The parabolic (ie marching-
|/  //        flame        |   integration) mode is used.
.// //                     |
| ////                     | * The Prandtl mixing-length model is
.  //                      |   used for the hydrodynamics, and to
^^^^^| /                   |   supply local coupling-and-splitting
burner: |  / ignition         |   rates.
fuel-rich    source           |
gas jet |                     | * The grid is 20 (horizontal) and
100 (vertical)

************************************************************
```

### (e) Results of the simulation

PHOENICS, when applied to this problem, computes the complete two-dimensional fields of concentrations of all fourteen fluids, by taking into account the flow, turbulent diffusion, chemical reaction and coupling/splitting processes.

The Prandtl mixing-length model is used for the hydrodynamics; and the parabolic mode of integration is used.

The 4 stoichiometric gases
f13 contours: unreacted
f14 contours: partly reacted
f15 contours: reacting
f16 contours: fully reacted

2 more fuel-lean gases
f5 contours: unreacted
f6 contours: fully reacted

Uncontaminated air
f1 contours:

The plots of the concentration profiles of the fourteen distinct fluids are interesting; but it is hard to appreciate all their implications. Therefore another mode of representation has been devised by the author's colleague Liao Gan-Li.

This shows the fluid population distributions directly, both as a conventional histogram and as a randomly distributed set of blobs. Otherwise the line-printer plots below may be inspected.

Colour pictures constitute a small selection of what can be deduced from the PHOENICS calculation.

The FPDs for the locations shown by IY (horizontal) and IZ (vertical):
FPD 1 , FPD 2 , FPD 3 , FPD 4 , FPD 5 , FPD 6 , FPD 7 , FPD 8 , FPD 9 , FPD 10 .

The following two line-printer plots, showing the profiles of concentrations of some of the fluids along the axis, provide some insight into how the fluids distribute themselves in the flame.

```
************************************************************
Fig.8 Axial profiles of the mass fractions of unburned fluids.
Fluid 17 is fuel-rich supply gas; fluid 1 is air

VARIABLE    1 = F17    2 = F13    3 = F9     4 = F5     5 = F1
MINVAL=  6.654E-04  5.363E-13  2.382E-13  1.963E-13  4.988E-13
MAXVAL=  9.988E-01  2.649E-02  4.516E-03  9.750E-04  2.688E-03

1.00 11111111111111.+....2223334444444..+....+....+55555
+             111  2  3344433    4444   555555    +
+               112  3 44   33     555554         +
+                2113 4 22    33555      4444444  +
Fn's +               2  344   22   5533              444
+              22 3441    2255   333              +
+              2344  11   552      333            +
+             244     1155  22        33333       +
+           4443     55511    222         333333333
+         4443 5555555   1111   222222            +
0.00 555555555555555+....+....+..11111111112222222222222
--------- vertical distance above burner -------->

************************************************************

Fig.9 Axial profiles of the mass fractions of burned fluids
Fluid 20 is fuel rich; fluid 16 is stoichiometric;
fluid 10 is fuel-lean; fluid 6 is even leaner

VARIABLE    1 = F20    2 = F16    3 = F10    4 = F6
MINVAL=  1.246E-03  2.333E-12  9.055E-16  2.237E-13
MAXVAL=  5.736E-02  3.175E-01  5.355E-04  1.271E-01

1.00 +....+....+....+....+...3333222222.+....+....+..444
+                      33  33    2222        444  +
+                     13  2 13       222  444     +
+                    13  2   133       44422      +
Fn's +                   1 3 2     133   4444   22222  +
+                  113 2       113444           222
+                 113 2         44333             +
+                113322      444  113333          +
+               113322    4444     111133333      +
+            1113332  44444           1111133333333
0.00 44444444444444444444444..+....+....+....+....111111
--------- vertical distance above burner -------->

************************************************************
```

### (f) The significance of the Bunsen-burner simulation

There are many important industrial flames, both desired as in furnaces and engines, and undesired as in oil-platform explosions, in which the fuel/oxidant ratio varies with position and time. Means have hitherto been lacking for representing such phenomena in any but a single-fluid manner, with consequent inadequacy of realism.

The Bunsen-burner simulation just presented shows that a two- dimensional multi-fluid models can be created which will allow the fragmentariness of such flames to be taken more properly into account.

It seems certain that, when systematically exploited, this possibility will lead to better understanding and predictive capability. For example, smoke and NOX production in gas-turbines should become easier to control.

### (a) The problem considered

The stirred reactor is important both as a theoretical concept and a practical device.

As a concept, it disregards variations of fluid attributes with geometrical position; this simplifies the task of mathematical analysis. Often, to emphasise this feature, it is called the "WELL-stirred reactor".

As a practical device, of which the so-called "Longwell bomb" [62] is an example, it is used for the determination of chemical reaction rates under high-temperature conditions; the accuracy of this determination depends of course depends on just how "well- stirred" the reactor actually is.

The multi-fluid model of turbulence allows the "how well" question to be explored quantitatively, and with insight, in a manner that is quite beyond any single-fluid model; and it may be of assistance in engineering analysis and design also, because there exist types of equipment in which the fluid-to-fluid differences associated with the unmixedness may be far more imoprtant than the variations with position.

What will therefore be considered here is the behaviour of a reactor into which are injected two streams of fluid; these may differ in reactant ratio (eg fuel/air ratio), in degree of reactedness, or both.

If these streams have the SAME reactant ratio, or if they have different ratios but do not react chemically, one-dimensional populations result. Otherwise they are two-dimensional.

The motive of the study is educational at the present stage, the interesting questions being:-

• can we get results at all?
• are they qualitatively reasonable?
• do the results depend in understandable ways on input data (stream flow rates, inlet conditions, micro-mixing-rate presumptions, chemical-reaction-rate presumptions)?

Once positive answers have been obtained to these questions, it will be timely to institute a more detailed research program.

### (b) The controlling parameters

It is useful to relate the behaviour of the reactor to the ratios of the rates of:

1. micro-mixing, in terms of (say) epsilon/k;

2. chemical reaction, in terms of (say) the mass rate of consumption of the fastest-reacting fluid in the population, divided by the mass in the reactor;

to:

• the rate of flow, in terms of the mass-inflow rates of each fluid, divided by the mass in reactor.

Also important, of course, is the reaction-rate function, which defines how the rate depends on the composition and degree of reactedness.

### (c) The calculations performed

In the results to be presented, the micro-mixing ratio is represented by the constant CHSOA, and the chemical reaction ratio by the constant CHSOB.

Values from 0.1 to 1000 have been employed for the first, and from 0.0 to 1.E6 for the latter. Both one-dimensionaal and 2-dimensional populations have been employed.

The reaction-rate-versus-reactedness relationship has been supposed to be of the well-known form illustrated by the sketch of Fig.9; the magnitude of the maximum value is supposed to depend upon the unburned reaction ratio, being greatest for the stoichiometric ratio and diminishing rapidly to zero for larger and smaller ratios.

```
************************************************************

Fig.9  Typical dependence of reaction rate upon reactedness
for a gas of fixed reactant ratio (in the unburned state)
^
|* reactant 1----->x               # #
|    *                x           #   #
|        *               x       #     #
|           *               x   #       #
rate |               *              x         #<--reaction
| reactant 2------> *        #    x       #  rate
|                       *   #        x     #
|                         # *           x
|                       #       *          x#
|                     #             *         x
|                  #                    *     #
|           #                               *  #
#-----------------------------------------------#
0    reactedness ---------------------->        1

************************************************************
```
One hundred fluids are considered, evenly distributed in respect of both reactedness and reactant ratio.

Both one- and two-dimensional populations are considered; the first comprises fluids of uniform (stoichiometric) fuel-air ratio, in which the fluids differ in respect of reactedness alone; in the second, variations of both fuel-air ratio and reactedness are present, because the one of the two entering streams is weak in fuel and the other is rich.

### (d) Results for the one-dimensional population

Colour plots of typical results are displayed in the following pictures,

s5 , s4 , s3 , s2 , s1

The following discussion refers to the corresponding line- printer output in Fig. 10 (a), (b), (c) and (d).

Fig.10 (a) shows something akin to the "clipped-Gaussian" shape presumed by Lockwood and Naguib, as does also Fig.10 (b). The chemical-reaction-rate constant is low in both cases, so that the FPD is nearly symmetrical.

Figs 10 (c) and (d) show that, as the chemical-reaction-rate constant increases, the FPD becomes more and more unsymmetrical. The reason is, of course, that the higher-temperature high-reactedness gas disappears more rapidly than the cooler low-reactdness gas.

```
************************************************************
Fig. 10(a): chsoa   = 1.000E+01 chsob   = 1.000E+00
MINVAL=  4.496E-03     MAXVAL=  9.114E-02
1....+....+....+....+....+....+....+....+....+....1  max
+                                                 +
+                                                 +
+                                                 + fluid
+                                                 + con-
+                                                 + centr-
+                                                 + ation
+                                                 +
+                                                 +
+             111111111111111111111               +
+1111111111111.+....+....+....+....111111111111111+ zero
< fluid 1 ............................. fluid 100 >

************************************************************
Fig. 10(b): chsoa   = 2.000E+01 chsob   = 1.000E+00
MINVAL=  2.365E-03     MAXVAL=  4.775E-02
1....+....+....+....+....+....+....+....+....+....1  max
+                                                 +
+                                                 +
+                                                 + fluid
+                                                 + con-
+                      1111                       + centr-
+                    111  111                     + ation
+                 111        11                   +
+              1111            1111               +
+        1111111                  111111          +
+111111111+....+....+....+....+....+....1111111111+ zero
< fluid 1 ............................. fluid 100 >

************************************************************
Fig. 10(c): chsoa   = 2.000E+01 chsob   = 1.000E+01
MINVAL=  1.860E-03     MAXVAL=  5.240E-02
1....+....+....+....+....+....+....+....+....+....+  max
+                                                 1
+                                                 +
+                                                 + fluid
+                                                 + con-
+                                                 + centr-
+                                                 + ation
+      111111111111                               +
+ 11111            111111                         +
+1                       1111111111               +
+....+....+....+....+....+....+...1111111111111111+ zero
< fluid 1 ............................. fluid 100 >

************************************************************
Fig. 10(d): chsoa   = 2.000E+01 chsob   = 2.000E+01
MINVAL=  1.491E-03     MAXVAL=  1.167E-01
1....+....+....+....+....+....+....+....+....+....+  max
+                                                 +
+                                                 +
+                                                 + fluid
+1                                                + con-
+1                                                + centr-
+ 1                                               1 ation
+ 11                                              +
+  1111                                           +
+     11111111111                                 +
+....+....+....+1111111111111111111111111111111111+ zero
< fluid 1 ............................. fluid 100 >

************************************************************
```

### (e) Results for the two-dimensional population

Colour plots of typical two-dimensional FPDs now follow, each consisting of two halves.

The left-hand half is the 2D histogram, in which the horizontal axis represents mixture ratio and the vertical axis reactedness. The extent to which any box is filled with colour indicates how much of the fluid in question is present.

The right-hand half is simply a qualitative reminder of the multi-fluid concept.

In the first picture, the micro-mixing constant is large, while the chemical-reaction-rate constant is zero. CHSOA=200.0; CHSOB=0.0
The fluids having the largest representation in the population are those near the middle of the diagonal joining the two inlet states (bottom left and top right)

In the second, the micro-mixing constant is 20.0, while the chemical-reaction-rate constant is still zero. CHSOA=20.0; CHSOB=0.0
The reduced mixing constant entails that the fluids are more evenly spread along the diagonal.

In the third, the micro-mixing constant is 20.0, while the chemical-reaction-rate constant is finite. CHSOA=20.0; CHSOB=1.0
Now the fluids are spreading upwards into the "reacted" part of the population space.

In the fourth, the micro-mixing constant is large, while the chemical-reaction-rate constant is increased 20-fold. CHSOA=20.0; CHSOB=20.0
The increased reaction constant has caused most of the fluid, of whatever mixture ratio, to be nearly fully reacted.

The following discussion concerns the corresponding line-printer output in Fig. 11 (a),(b) and (c),

Fig.11 (b) is the easiest to understand; for it shows the population to be almost confined to the diagonal connectinq the points representing the two entering streams, viz the fuel-lean unreacted gas at the bottom left and the fuel-rich fully reacted gas at the top right. Confinement to the diagonal is a consequence of the low value of the reaction-rate constant, CHSOB.

When the micro-mixing constant, CHSOA, is made larger, the central part of the accessible domain is more heavily populated; and this makes it possible for more chemical reaction to occur. As a consequence, the average reactedness of the fluids increases, which accounts for the "thickened-wedge" shape of the triangle.

When CHSOB is increased to 10.0, even with the smaller value of CHSOA, the movement to the high-reactedness regions is even more pronounced. This is shown in Fig.11 (c).

```
************************************************************

Fig. 10(a): chsoa=10, chsob=1

TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT   0
L0000000000000000000000000000000000000000     R   |
L0000000000000000000000000000000000000 2222   R   |
L000000000000000000000000000000000    2222  00R   |
L0000000000000000000000000000000 22  22   0000R   //
L000000000000000000000000000 22222222  0000000R
L00000000000000000000000   22   222   00000000R  f - mfu
L000000000000000000000 22      22  00000000000R   |
L00000000000000000  222     22   0000000000000R   |
L00000000000000   22     222  0000000000000000R   |
L00000000000  2222    222    00000000000000000R   |
L0000000 22      222222   00000000000000000000R   //
L00  22222   2222222     000000000000000000000R
L       222222222      00000000000000000000000R
L 864    0000        0000000000000000000000000R
---------------->  f ----------------------->

************************************************************
Fig. 10(b): chsoa=5, chsob=1

TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT   0
L0000000000000000000000000000000000000000     R   |
L0000000000000000000000000000000000000  2222  R   |
L0000000000000000000000000000000000    222  00R   |
L0000000000000000000000000000000  2222  000000R   //
L000000000000000000000000000  22      00000000R
L000000000000000000000000     2    00000000000R  f - mfu
L0000000000000000000000  22    000000000000000R   |
L000000000000000000         000000000000000000R   |
L000000000000000         000000000000000000000R   |
L000000000000         000000000000000000000000R   |
L0000000  2222     000000000000000000000000000R   //
L000  222        00000000000000000000000000000R
L  22222   00000000000000000000000000000000000R
L 86420000000000000000000000000000000000000000R

---------------->  f ----------------------->

************************************************************
Fig. 10(c): chsoa   = 5.0 chsob   = 10.0

TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT   0
L0000000000000000000000000000000000000000     R   |
L00000000000000000000000000000000000000   22  R   |
L0000000000000000000000000000000000000      00R   |
L000000000000000000000000000000000       00000R   //
L000000000000000000000000000000000    00000000R
L000000000000000000000000000000000000000000000R  f - mfu
L000000000000000000000000000000000000000000000R   |
L000000000000000000000000000000000000000000000R   |
L000000000000000000000000000000000000000000000R   |
L000000000000000000000000000000000000000000000R   |
L000000000000000000000000000000000000000000000R   //
L00000      0000000000000000000000000000000000R
L 2222                       00000000000000000R
L08  42222222  4444444     2222  0000000000000R

---------------->  f ----------------------->
```

### (f) Conclusions from the stirred-reactor studies

With reference to the questions posed in section 3.4.(a) above, it can be stated:-

• the relevant equations proved easy to solve, no convergence difficulties being encountered;
• all results were reasonable and understandable;
• unprecedented insight was provided by contriving that PHOENICS accepted input, and displayed the results graphically, in a swift and interactive manner.

Limitations of space prevent the full presentation of the results which were obtained in this preliminary study; and indeed one of the major findings has proved to be this: that the MFM provides so much more information about what takes place in a turbulent flame (if the hypotheses are realistic) that new means will have to be developed for first assessing and then conveying its significance.

Further publications are therefore in preparation.

### 4. Concluding remarks

The author hopes to have persuaded some of his readers and hearers that:

• exploration of the implications of multi-fluid models of turbulence is lkely to be worth while; and also that

• it is not difficult to engage in.

He does not put forward "The Multi-Fluid Model" as a rival to "The K-Epsilon Model", "The Reynolds-Stress Model", or "The RNG Model"; for the The's in all these names suggest a definiteness of concept which is at variance with reality; and the multi-fluid family of models is especially extensive.

Further, multi-fluid models can come in varieties of which nothing has been said in this paper. Thus:

• the fluid-attribute grids may be non-uniform, self-adaptive, and unstructured;
• the number of fluids considered can vary in number or kind from one part of the geometric domain to another;
• the fluid-defining attributes may include velocities, as well as scalars; and
• the coupling-and-splitting hypotheses can (and should) be formulateed so as to account for Prandtl- and Schmidt-number effects.

Conclusion:

• There is interesting work here for a whole army of research workers;

• so the author hopes to have inspired someone to raise at least a platoon.

### 5. References

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wbs