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4. The multi-fluid model (MFM) of turbulence

(a) Introduction

It is here argued that there are three main approaches to the numerical simulation of turbulent flows, namely:-

                   > transport of statistical quantities, (1)
                  /      following Kolmogorov's idea
============>> /_________ > /
                   > multi-fluid, ie discretized PDF     (3)

It will be further argued that the third, a natural extension of the two-spike presumed-PDF approach, offers the best prospects.

(b) The local mixture as a population

The MFM approach to turbulence performs the numerical-simulation task by extending to "PDF space" the discretization (eg finite- volume) methods which are commonly applied only to geometric or time space.

The basic idea is that a fluid mixture, whether containing only one thermodynamic phase, or more than one phase (as in boiling, condensation or solid-fuel combustion), can be best treated as a "discretized population", as illustrated here.

      p =  proportion |  |                  ____
             of     ^ |  |         __      |    |___          __
            time    | |  |        |  |__ __|    |   |        |  |
      during which  | |  |        |  |  |  |    |   |__      |  |
      fluid has an  | |  |   _____|  |  |  |    |   |  |     |  |
      attribute value |  |  |     |  |  |  |    |   |  |     |  |
      within the      |  |__|     |  |  |  |    |   |  |     |  |
      abcsissa limits |__|__|_____|__|__|__|____|___|__|_____|__|
                                     ----------> a = attribute

(c) Method of description

The population can be described by:

(d) The dimensions of the population

If a single PDA is chosen, for example temperature, the population is 1-D.

A 2-D population is created if two PDAs are selected, for example temperature and vertical- direction velocity.

An example of the graphical representation of a two- dimensional population is shown below.

The coloured proportion of each box represent the mass fraction. A 2D population

(e) PDAs and CVAs

Three- and four-D populations can be envisaged; but they are harder to draw!

Material attributes which are not discretised (ie not selected to be PDAs) are called "continuously-varying attributes" (CVAs).

CVAs are treated as uniform in value (but different from each other) within each component of the population, and in each element of space-time.

PDAs are treated as uniform in value (but different from each other) only within each element of space-time.

The distinction between the two kinds of attribute is arbitrary; the analyst chooses as PDAs the attributes of which the fluctuation are most likely to be physically significant, for example the density when gravitation influences the flow.

(f) What is meant by "multi-fluid"

Each component of a population is regarded as being a distinct fluid. MFM models are called "2-fluid", "4-fluid", "100-fluid", etc, according to the number of distinct fluids which are in use.

If the selected PDA for a steam-water mixture is density and only two value bands are chosen, with the dividing value equal to the arithmetic-mean density of steam and water, the population is one- dimensional with two components.

It could be called a "2-fluid model"; and indeed the IPSA 2-phase- flow model, which has been in use for nearly two decades, is of this kind. An implication is that the steam and water temperatures, which are CVAs, have each only one value at a given point in the flow.

If it were then decided also to take account of the fluctuations of temperature, dividing the whole range into 10 equal intervals, temperature would have become the second PDA. The population would be 2-dimensional; and a "20-fluid" model would have resulted.

(g) What governs the mass fractions of distinct fluids

The task of numerical simulation of turbulence, with or without boiling, condensation, combustion, etc, thus becomes that of:

The said values are influenced by the physical processes of:-

These, and their interaction through the conservation laws of physics, are expressible by way of differential equations of well- known forms and properties.

(h) Solving the equations

So well-known are the equations that many widely-available computer programs are equipped to solve them. [ One such is the PHOENICS Shareware package.]

If, as usual, they are of the finite-volume kind, the programs compute the MFDFs and associated CVAs by solving a large number of inter-linked and non-linear balance equations by iterative techniques.

The most time- and attention-consuming part of the modelling exercise is then the formulation of the source-and-sink terms, which express (chiefly) how the distinct fluids interact with each other.

In the work to be reported below, the source-and-sink terms for the MFDFs have been derived from a physical hypothesis, based upon intuition rather that exact analysis.

(i) Coupling and splitting

The hypothesis has two parts, namely:

  1. that all the fluids may collide indiscriminately, in pairs, at rates which are proportional:-
  2. that, as a result of each collision, some material:
In what follows the colliding fluids are referred to as parents.

(j) The hypothesis: " Promiscuous Mendelian coupling and splitting"

The two "parent fluids", from different attribute-space locations, F and M, lose mass; but this is gained by their offspring at O1, O2, O3, etc.

The "promiscuous Mendelian Hypothesis" ^ frequency in | population father mother |/ ______ / / | | | promiscuous ****** | __|______|__ <------- coupling ---------> ******** | | .. | *| .. |* | |* *| *| -- |* | /-**--\ ____ Mendelian ______ /-----\ | /|//////| \ | splitting | / |_____| \ | |//////| v v / \ | | || | _ _ _ _ _ _ /_________\ | | || | | | | | | | | | | | | | | | | | |__||__| | | | offspring | | | | | |__|__| --------------F--------O1----O2---O3---O4---O5---O6------M--------- fluid attribute -------------------------->

(k) The measure of the turbulent motion, mdot

The above-mentioned quantity of dimension 1/time, ie mdot, is of course deducible directly from the breadth of the population, if the distinguishing attribute involves velocity, and if a length-scale measure is chosen.

It may be deduced indirectly if a different attribute is chosen.

Moreover, there is nothing to prevent more-conventional means being employed.

For example, if the modelling study is focussed on chemical reaction rather than on hydrodynamics, the latter may be handled by the k- epsilon model, and mdot can be taken as proportional to k divided by epsilon.

The quantity mdot can be regarded as the micro-mixing rate. Something similar is used in earlier models such as those which presume the shape of the PDF or use the Monte-Carlo approach.