MFM: The Governing Equations and the Underlying Assumptions

APPENDIX: THE GOVERNING EQUATIONS AND THE UNDERLYING ASSUMPTIONS
What governs the mass fractions of the fluids
(a) the processes

The task of numerical simulation of turbulent flow, heat transfer and chemical reaction, when a multi-fluid model is used, becomes that of computing, for selected locations in space and time, values of the mass fractions of the distinct fluids, together with the values of their associated continuously-varying attributes.

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

time-dependence,
convection,
diffusion (laminar and turbulent), &
sources and sinks.

These processes, and their interactions through the conservation laws of physics, are expressible by way of differential equations of well-known forms and properties. These are the so-called "transport equations", solution of which can be effected by many well- established algorithms and computer codes.

(b) The source terms in the fluid-mass-fraction equation

So well-known are these equations that attention will be paid only to the source terms, which are of two distinct kinds, according to whether the equation in question governs (1) the mass fraction of a fluid, or (2) one of its continuously-varying attributes.

The mass-fraction-of-fluid source expresses the rate transfer of mass, per unit mass of mixture (ie total population), from one fluid and to another.

This will be given the symbol MI>J, where I and J are indices denoting the fluids, and the > sign indicates the direction.

The total source of fluid J is, as a consequence, S{ MI>J }I, where S{ }I indicates the summation for all values of index I.

Further obvious consequences of the definition of MI>J are:

MI>J = - MJ>I , and MI>I = MJ>J = 0 .

(c) The PDA-change term

Let it be supposed that a physical process exists, tending to cause material to change its attribute, such as a heat source tending to increase enthalpy, or a chemical source tending to increase chemical-species concentration.

Let it further be supposed that the attribute in question is a PDA, ie one of those which distinguishes the population; and let the rate of change of attribute effected by the source be expressed by Pdot.

Then the relevant mass-fraction source of fluid J is:

M(J-1)>J = Pdot * M(j-1) / D(j-1) , if Pdot is positive, and

M(J+1)>J = - Pdot * M(j+1) / D(j+1) , if Pdot is negative, where:

M(j-1) & M(J+1) are respectively the mass fractions of the fluids having values just below and just above fluid J in the PDA dimension, and D(j-1) and D(j+1) are the corresponding PDA-interval widths.

(d) The fluid-interaction term

A quite distinct contribution to MI>J is that resulting from the interactions between the fluids resulting from diffusion, heat- conduction and viscous action as they move past, or collide with, each other in their turbulent motion.

It is reasonable to express this contribution, for fluid K, as:

S{ S{ Fk(i,j) * M(i) * M(j) * T(i,j) }J }I

wherein: the S{ }s have the same significance as before (and it is immaterial whether summation over I or J occurs first);

M(i) and M(j) are the mass fractions of fluids I and J;

T(i,j) measuring the turbulent motion, has dimensions 1/s;

Fk(i,j) is the fraction of mass lost in the IJ encounter which enters fluid K .

(e) The underlying physical mechanism

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:


                ______________________________
                xxxxxxxxxxxxxxx...............
                xxxxxxxxxxxxxxx...............
                xxxxxxxxxxxxxxx...............
                xxxxxxxxxxxxxxx...............
                xxxxxxxxxxxxxxx...............
                xxxxxxxxxxxxxxx...............
                ------------------------------

(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:


                ______________________________
                xxxxx.xxxxx.x.x.x..x.....x....
                xxxxxxxx.x.x.x.x.x..x.........
                xxxxxx.xx.x.x.x.x.x..x..x.....
                xxxx.xx.xxxx.x.x.x...x.x......
                xxxxxxxxx.xxx.x.x.x.x.........
                xxxxxxxx.x.x.x.x.x.x..x.......
                ------------------------------

(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...   ....
        ----   ----   ----   ---   ---   ----   ----   ----

(f) Coupling and splitting for 2D populations

If the population is a two-dimensional one, the promiscuous- Mendelian hypothesis would imply that the offspring would be shared among the cells enclosing the straight line joining F to M.


       |_______|_______|_______|_______|_______|_______|_______|
       |       |       |       |       |       |       |  M    |
       |       |       |       |       |       |       |   *   |
       |_______|_______|_______|_______|_______|_______*_______|
       |       |       |       |       |       |   *   |       |
       |       |       |       |       |       *       |       |
       |_______|_______|_______|_______|___*___|_______|_______|
       |       |       |       |       *       |       |       |
       |       |       |       |   *   |       |       |       |
       |_______|_______|_______*_______|_______|_______|_______|
       |       |       |   *   |       |       |       |       |
       |       |       *       |       |       |       |       |
       |_______|___* __|_______|_______|_______|_______|_______|
       |  F    *       |       |       |       |       |       |
       |   *   |       |       |       |       |       |       |
       |_______|_______|_______|_______|_______|_______|_______|

(g) 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:-

parental bias; and
differentially-diffusive.

(h) Hypotheses for Fk(i,j) and T(i,j)

The crux of multi-fluid modelling lies in the formulae chosen for the Fk(i,j) and T(i,j) functions. Physical intuition, mathematical analysis, guess-work and computational parsimony all play a part in their choices. The subject is therefore too large to treated here.

What will however be described are the simplest-of-all hypotheses, applicable to a one-dimensional uniformly-divided population, and used in the computations reported below. These are, with attention restricted to i < j , without loss of generality:

T(i,j) is independent of i and j ; and,


       Fk(i,j) = -0.5       for k=i or k=j  and j greater than i+1,
               =  0.0       for k less than i or k greater than j  
                            or  j=i+1,

               =  1/(j-i-1) for all other values of i, j and k.

These hypotheses are conveyed figuratively by the "Promiscuous- Mendelian diagram above

(i) The source term in the CVA equations

Let C(k) be the value of a continuously-varying attribute of fluid K. Then the source term in the C(k) transport equation will possess three terms, corresponding respectively to:

an attribute-changing source Cdot of the same nature as Pdot;
the mass sources of K resulting from the IJ encounters; and
diffusional (or even radiation-type) interactions between fluids present in the same location which are independent ot fluid-to-fluid contact.

APPENDIX: THE GOVERNING EQUATIONS AND THE UNDERLYING ASSUMPTIONS What governs the mass fractions of the fluids (a) the processes