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### (a) The "discretized-PDF" approach

The state of a turbulent fluid may be in part characterised by probability-density functions, which represent the fluctuations of temperature, concentration, velocity, etc, at a given point in the flow.

Such a PDF may be represented as a histogram, which shows for what proportions of time the temperature, or other fluid attribute, lies within discrete arbitrarily-defined bands.

The following sketch illustrates such a discretized PDF.

### (b) Illustration of a discretized probability-density function

```
__
|  |
p =  proportion |  |                  ________
of     ^ |  |         __      |        |         __
time    | |  |        |  |__ __|        |        |  |
| |  |        |  |  |  |        |__      |  |
| |  |   _____|  |  |  |        |  |     |  |
|  |  |     |  |  |  |        |  |     |  |
|  |__|     |  |  |  |        |  |     |  |
|  |  |     |  |  |  |        |  |     |  |
|  |  |     |  |  |  |        |  |     |  |
|__|__|_____|__|__|__|________|__|_____|__|
----------> a = attribute
```
In this case the attribute intervals (bands) are of unequal width. Only equal-width intervals are used in the examples below, however.

### (c) The multi-fluid concept

The multi-fluid concept is based on regarding the material lying within a specific band of attribute values as a distinct fluid.

What in reality is a single turbulent fluid, in which temperatures vary continuously, is therefore treated as a population of fluids, distinguished by the temperature intervals which characterise them.

The number of fluids is chosen by the modeller; so, indeed, is the attribute which he uses to distinguish them.

The choice of attribute is akin to the choice of statistical variable (eg turbulence energy or RMS vorticity fluctuations) which every turbulence modeller must make; and

the choice of fluid number is like the choice of discrete time step and spatial-grid fineness with which all CFD users are familiar.

The multi-fluid turbulence model (MFM) is an equally artificial construct; what it discretizes is fluid-attribute "space".

### Graphical representation of the multi-fluid population

The next picture illustrates the concept in two ways. On the left is an FPD histogram. It derives from a study of a well-stirred chemical reactor.

The attribute in question is the reactedness, ie the extent to which the fluids in question are fully reacted.

The right-hand diagram is a qualitative illustration of the multi- fluid concept.

It shows intermingled fragments of various sizes and states.

The fragments are drawn as circles only for simplicity: circularity is NOT part of the MFM idea.

### (d) Calculating the fluid-population distribution (FPD)

MFM terminology desribes the turbulent fluid as a "population; and the "discretized PDF" is called a "fluid-population distribution", with the abbreviation: FPD.

A population is characterized by a set of values, fa(i), which represent the MASS-FRACTION OF THE I'TH FLUID, at any point in the domain of study.

The values of the fa(i) are computed by solving differential transport equations for each of them, these equations having the usual form:

D fa(i)/DT = div (gamma. grad (fa(i)) ) + net source of fluid i

It is these equations, together with the functions of fa(i) which provide the gamma and source terms, which constitute a multi-fluid model.

### (e) Quantities which can be derived from a PDF

It is possible, by integration of p.da, p*a.da, p*a**2.da, etc, to compute the time-average value of a, its root-mean-square value, and other "moments" representing the statistical state of the fluid.

The same is true of the fluid-population distribution, fa(i).

Thus, if the attribute were the instantaneous velocity (regardless of direction), the mean kinetic energy of turbulence for the population could be deduced.

In principle, if a sufficient number of moments are computed, the complete PDF (or FPD) can be reconstructed; but this is rarely attempted.

### (f) How MFMs differ from conventional turbulence models

The difference between an MFM and a conventional turbulence model is thus revealed:

An MFM solves differential equations for the mass fractions of members of the fluid population; it can easily deduce statistical quantities therefrom.

A conventional model solves diffferential equations for the statisical quantities; but these do not, in practice allow the population distribution to be deduced.

This makes the MFM superior whenever the FPD is needed, as it is for:-

• flows strongly influenced by fluctuating body forces;
• flows in which chemical reacion is important

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