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### (a) Summary

A 40-fluid model is employed to simulate the plane turbulent mixing layer.

Each of the 40 fluids has a different main-flow-direction velocity.

The velocity dimension has thus been discretised.

The 40 conservation equations for the concentrations of the fluids contain the usual terms:

time-dependence, convection, diffusion and source,

solved in a conventional way (viz by PHOENICS, in parabolic mode).

### (b) The three physical assumptions

1. The length scale obeys a transport equation with a source, viz:

0.005 * root-mean-square_velocity_fluctuations.

2. The effective viscosity is taken as:

0.5 * length_scale * root-mean-square_velocity_fluctuations

3. The source/sink terms in the fluid-concentration equations are:

first_fluid_concentration * second_fluid_concentration * 5.0 * root-mean-square_velocity_fluctuations / length_scale

shared according to the coupling/splitting formula.

The above constants have not yet been optimised; but they fit the experimental data fairly well.

### (c) Some details of the computations

The last lines of the computer (200 MHz Pentium) output are:

RUN COMPLETED AT 12:53:57 ON SATURDAY, 01 JUNE 1996 MACHINE-CLOCK TIME OF RUN = 642 SECONDS.

So one does not have to wait long for results, some of which now follow.

First, some PHOTON plots will be shown.

These will be followed by some line-printer plots, and thereafter by some fluid-population distributions.

The grid is an expanding one, with 40 intervals across and 100 along the layer.

This is the distribution of the COMPUTED length scale

Here is the distribution of the average concentration (for all 40 fluids) of material emanating from the upper stream.

It is just as a conventional turbulence model (eg mixing-length or k-epsilon) would reveal;

but neither k nor epsilon are being computed.

But how much upper-stream fluid (fluid 40) remains in its pure state? Only this. Where has the rest gone?

What about contours of pure lower-stream fluid?

It also occupies very little space.

So what lies in between? 38 other fluids, variously intermingled. The contours for some of them will now be shown.

The contours for fluids: 5 ; 10 ; 15 ; 20 ; 25 ; 30 ; 35

### (d) Another means of representing the results

Here is a series of line-printer plots, showing the profiles of individual-fluid concentrations across the fully-developed layer.

Only a few are shown.

Each is normalised, so as to have the same maximum ordinate.

```

Fluid 1: maximum value in the cross-section =  1.000E+00
1.00 11.11+....+....+....+....+....+....+....+....+....+
0.90 +    1                                            +
0.80 +                                                 +
0.70 +                                                 +
0.60 +                                                 +
0.50 +     1                                           +
0.40 +                                                 +
0.30 +       1                                         +
0.20 +                                                 +
0.10 +        11                                       +
0.00 +....+....+.1111.111+1111+1111+111.1111.111.1111.11
0   .1   .2   .3   .4   .5   .6   .7   .8   .9  1.0
the abscissa is       Y  .  min= 1.25E-03 max= 9.87E-02

Fluid 5: maximum value in the cross-section =  7.608E-02
1.00 +....+...1+....+....+....+....+....+....+....+....+
0.90 +       1 1                                       +
0.80 +                                                 +
0.70 +           1                                     +
0.60 +                                                 +
0.50 +     1      1                                    +
0.40 +             1                                   +
0.30 +              1                                  +
0.20 +                1                                +
0.10 +    1            11 1                            +
0.00 11.11+....+....+....+.111+1111+111.1111.111.1111.11
0   .1   .2   .3   .4   .5   .6   .7   .8   .9  1.0
the abscissa is       Y  .  min= 1.25E-03 max= 9.87E-02

Fluid 10: maximum value in the cross-section =  5.576E-02
1.00 +....+....+.1..+....+....+....+....+....+....+....+
0.90 +         1  1                                    +
0.80 +             1                                   +
0.70 +        1     1                                  +
0.60 +                                                 +
0.50 +                1                                +
0.40 +       1         1                               +
0.30 +                  1                              +
0.20 +     1              11                           +
0.10 +                      11 1                       +
0.00 11.111....+....+....+....+.111+111.1111.111.1111.11
0   .1   .2   .3   .4   .5   .6   .7   .8   .9  1.0
the abscissa is       Y  .  min= 1.25E-03 max= 9.87E-02

Fluid 15: maximum value in the cross-section =  4.988E-02
1.00 +....+....+...11....+....+....+....+....+....+....+
0.90 +            1   1                                +
0.80 +           1     1                               +
0.70 +                                                 +
0.60 +         1        1                              +
0.50 +                    1                            +
0.40 +        1            1                           +
0.30 +                      11                         +
0.20 +       1                 1                       +
0.10 +     1                    111 1                  +
0.00 11.111....+....+....+....+....+.11.1111.111.1111.11
0   .1   .2   .3   .4   .5   .6   .7   .8   .9  1.0
the abscissa is       Y  .  min= 1.25E-03 max= 9.87E-02

Fluid 20: maximum value in the cross-section =  4.911E-02
1.00 +....+....+....+.111+....+....+....+....+....+....+
0.90 +              1     1                            +
0.80 +             1       1                           +
0.70 +                      1                          +
0.60 +            1          1                         +
0.50 +                         1                       +
0.40 +           1              1                      +
0.30 +         1                 1                     +
0.20 +                            1 1                  +
0.10 +       11                      11 11             +
0.00 11.1111...+....+....+....+....+....+.11.111.1111.11
0   .1   .2   .3   .4   .5   .6   .7   .8   .9  1.0
the abscissa is       Y  .  min= 1.25E-03 max= 9.87E-02

Fluid 25: maximum value in the cross-section =  5.186E-02
CELLAV=  1.609E-02
1.00 +....+....+....+....+111.+....+....+....+....+....+
0.90 +                  1    1                         +
0.80 +                 1       1                       +
0.70 +                1         1                      +
0.60 +              1            1                     +
0.50 +                            1                    +
0.40 +             1                1                  +
0.30 +            1                  11                +
0.20 +           1                      11             +
0.10 +         1                          11 1         +
0.00 11.1111.11+....+....+....+....+....+....+11.1111.11
0   .1   .2   .3   .4   .5   .6   .7   .8   .9  1.0
the abscissa is       Y  .  min= 1.25E-03 max= 9.87E-02

Fluid 30: maximum value in the cross-section =  5.932E-02
1.00 +....+....+....+....+...1+111.+....+....+....+....+
0.90 +                      1     1                    +
0.80 +                     1        1                  +
0.70 +                    1          1                 +
0.60 +                  1             1                +
0.50 +                                  1              +
0.40 +                 1                 1             +
0.30 +                1                   11           +
0.20 +              1                        1         +
0.10 +           111                          11 1     +
0.00 11.1111.111....+....+....+....+....+....+....111.11
0   .1   .2   .3   .4   .5   .6   .7   .8   .9  1.0
the abscissa is       Y  .  min= 1.25E-03 max= 9.87E-02

Fluid 35: maximum value in the cross-section =  8.043E-02
1.00 +....+....+....+....+....+....+111.1....+....+....+
0.90 +                            1      1             +
0.80 +                           1        1            +
0.70 +                          1          1           +
0.60 +                         1             1         +
0.50 +                       1                         +
0.40 +                      1                 1        +
0.30 +                     1                   1       +
0.20 +                  1 1                      1     +
0.10 +              1 11                          1    +
0.00 11.1111.111.111+....+....+....+....+....+....+11.11
0   .1   .2   .3   .4   .5   .6   .7   .8   .9  1.0
the abscissa is       Y  .  min= 1.25E-03 max= 9.87E-02

Fluid 40: maximum value in the cross-section =  1.000E+00
1.00 +....+....+....+....+....+....+....+....+....+.1.11
0.90 +                                             1   +
0.80 +                                            1    +
0.70 +                                                 +
0.60 +                                           1     +
0.50 +                                         1       +
0.40 +                                        1        +
0.30 +                                       1         +
0.20 +                                    11           +
0.10 +                               11 11             +
0.00 11.1111.111.1111.111+1111+1111+1...+....+....+....+
0   .1   .2   .3   .4   .5   .6   .7   .8   .9  1.0
the abscissa is       Y  .  min= 1.25E-03 max= 9.87E-02
```

### (e) Yet another means of representing the results

The results can be processed and displayed in many different ways.

Particularly interesting are plots of:
fluid-population distributions (FPDs) ,
which (in the infinite-fluid-number limit) are akin to:
probability-density functions (PDFs) .

The FPDs are on the left in the next pictures.

The displays on the right are reminders of how the population of fluids might be distributed in a single computational cell, namely at random.

The colours in the FPDs do not match those used in the contour plots.

That FPD was not very interesting because the population consisted almost entirely of fluid 1, ie lower-stream fluid.

A series of FPDs will now be shown, for a succession of locations across the layer, at grid nodes with indices IY = :-

10; 15; 20; 25; 30; 35;

### (f) The RMS velocity fluctuations

There is much information to be extracted from the FPDs.

All that has been used so far is the root-mean-square velocity fluctuation, as an input to the formulae for:-

1. length-scale growth,
2. effective viscosity,
3. micro-mixing between fluids (ie coupling and splitting).

This MNSQ quantity is of course obtainable point-wise. Its contour plot is self-similar.

The value along the "spine" of the layer is approxinately 20 %, which is of the same order of magnitude as is found for the square root of the kinetic energy of turbulence, to which MNSQ is closely analogous.

### (g) Conclusions

• Much more has been learned from generalising the 2-fluid idea to multi-fluid than from any indirect approach via k, epsilon, etc.;
• The concept is simple and the computations EASY.
• The model provides:
• new insights,
• new opportunities, and
• new freedom from long-suffered constraints.
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