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.

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

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".

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.

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.

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.

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