Encyclopaedia Index

### Contents

The multi-fluid model of turbulence, even though its essential ideas have been known for many years, has not yet been subjected to intense and prolonged research and development efforts such as have been lavished on the single-fluid models.

Now is therefore the time to consider what form such efforts might take, and to initiate the appropriate actions.

The author's current thoughts on these now follow.

### (a) Grid-refinement

It is necessary to establish, for a much wider range of problems than has been investigated so far, that refining the population grid (ie increasing the number of fluids) does indeed always lead to a unique solution, and then to establish how the accuracy of solution depends upon the grid fineness.

This needs to be done for two- and (at least a few) three- dimensional population grids as well as one-dimensional ones, before the soundness of the basic ideas and solution algorithms can be regarded as secure.

### (b) Consistency tests

When the micro-mixing constant becomes very large, the range of fluids present at a particular location reduces to 2 for a 1D population, to 4 for a 2D one and to 8 for a 3D one. It needs to be checked that the system of equations does indeed exhibit this behaviour, and that the right values are calculated for the fluid concentrations.

A prudent and insightful investigator will wish to devise other tests of a self-consistency or conservativeness character before he wholly trusts his mathematical apparatus.

### (c) Non-uniform, unstructured, and self-adaptive grids

So far, the author has used only uniform and structured grids, which have remained the same throughout the computation; yet there are obvious advantages to be gained by relaxing these restrictions.

Thus, it may be convenient to define the attribute to be discretised as the ratio of the current temperature to the maximum temperature in the field, which temperature may change throughout the period of the calculation. Such a grid would have to be self-adaptive.

Another kind of self-adaptive grid would change its uniformity as the calculation proceeded, so as to capture with maximum accuracy the shape of the fluid-population distribution.

Yet another would insert new subdivisions of the discretised attribute in regions of steep variation, and perhaps remove them from regions of less interest.

The best way of summarising the possibilities is to say that probably all the ingenious devices which specialists have invented for the better representation of variations in geometric space are likely to have their population-grid counterparts.

### (d) Computational improvements

Although the computer-time burden is not yet great, it may become considerable when fine-grid three-dimensional transient simulations have to be undertaken.

Fortunately, there is much which can be done to reduce the load. For example:-

• population grids may be made non-uniform, self-adaptive and unstructured, as just explained;

• the multi-fluid model can be confined to only those parts of the flow domain in which unmixedness is of physical significance;

• cell-wise simultaneous-variable-adjustment procedures can be used, with multi-grid (in both senses) acceleration devices;

• the employment of multi-block solution techniques.

It will also be necessary to work out what is the optimal discretiz- ation principle; for certainly it is not optimal to employ a grid which is extremely fine in geometric space but has too small a population grid (ie too few fluids) to represent adequately the fragmentariness of the flow.

### 6.2 Comparisons with experiment

In order to establish the predictive capabilities of MFM now and in the future, it is of course necessary to make comparisons with experimental data.

Three kinds of comparisons should be distinguished, namely:

1. those most desired by the potential end-users, for example between measured and predicted total smoke production for a particular gas-turbine combustor, or yields of main- and side-products for a paddle-stirred reactor;

2. those with well-known data on the velocity, temperature and concentration profiles in jets, wakes, boundary layers and simple turbulent flames; and

3. those which provide detailed measurements of probability-density functions of important fluid variables.

Were there no urgency about solving the practical problems of engineering and the environment, a rational research program would probably concern itself with kind (2) first of all, in order to establish a convincing prima facie case for MFM; then experiments would be conducted systematically so as to permit kind-(2) comparisons, which might lead to improvements to some components of the model.

Finally comparisons of kind (1) would be made, whereafter, if the comparisons were successful, industrial use of MFM would begin.

However, perhaps fortunately, there IS much urgency; for the models which are currently used for predicting turbulent chemical reaction in particular are far from satisfactory. It therefore seems reasonable that some comparisons of kind (1) should begin in the near future.

For example, it may be found that the Mendelian principle does not always apply; and it is certainly to be expected that, when the FPD's of two velocity components are in question, the coupling of two fragments with differing x-direction velocities will lead to offspring with differing y-direction velocities also. (See section 6.3 below.)

### (a) "Parental bias" in the "splitting" process

As has been mentioned above, the promiscuous-Mendelian hypothesis is unlikely to prove to be the best; and it is not hard to improve upon it, if the underlying mechanism of section 1.8b above is believed, as follows.

The profile of concentration (say) in the conjoined fragments, after some time will have some such "error-function-like" shape as indicated below on the left, to which coresponds a pdf of the shape shown on the right.

Clearly the pdf exhibits the largest frequencies near the extreme, ie the "parental" concentrations.

```
|******                                ^        |******
|      *****                           |        |*****
|           ****                       |        |****
|               ***                 conc-       |***
|                  **               entr-       |**
|                    *              ation       |*
|                     **               |        |**
|                       ***            |        |***
|                          ****                 |****
|                              *****            |*****
|                                   ******      |******
|-------------------------------------------    |---------------
----------- distance ------->               - frequency--->
```
Only if the profile of concentration were linear with distance would the Mendelian assumption be correct.

### (b) The effects of laminar Prandtl and Schmidt numbers

Let it now be supposed that the two fragments differ both in temperature and salinity. Then, while they are in contact, the profile of temperature will broaden much more rapidly than the profile of salinity. As a consequence, the offspring do NOT lie on the diagonal joining the two parental locations as indicated in section 1.8c above.

```
|_______|_______|_______|_______|_______|_______|_______|
|       |       |       |       |       |       |  M    |
^     |       |       |       |       |       |       |   *   |
|     |_______|_______|_______|_______|_______|_______|_______|
|     |       |       |       |       |       |       |  *    |
temp-   |       |       |       |       |       |       |       |
erat-   |_______|_______|_______|_______|_______|_______|_*_____|
ure     |       |       |       |       |       |       |       |
|     |      *|       |       |       |       |       |       |
|     |_______|_______|_______|_______|_______|_______|_______|
|     |     * |       |       |       |       |       |       |
|     |       |       |       |       |       |       |       |
|     |____*__|____ __|_______|_______|_______|______ |_______|
|  F    |       |       |       |       |       |       |
|   *   |       |       |       |       |       |       |
|_______|_______|_______|_______|_______|_______|_______|

----- salinity------------>
```
Instead, the offspring are more likely to lie along the lines of asterisks shown above, with much smaller salinity changes than temperature changes.

### (c) The coupling-splitting hypothesis for a two-velocity population

As a final example of how the coupling-splitting hypothesis can be modified in the direction of greater realism, let it be imagined that the attributes of a 2D population grid are the horizontal velocity U and the vertical velocity V.

Then let the collision be imagined of two fluid fragments which have the same values of V but differing values of U. So the father and mother lie on the same horizontal. Where do the offspring lie?

```
|_______|_______|_______|_______|_______|_______|_______|
|       |       |       |       |       |       |       |
^     |       |       |       |   *   |       |       |       |
|     |_______|_______|_______|_______|_______|_______|_______|
|     |       |       |       |       |       |       |       |
|       |   *   |   *   |   *   |   *   |   *   |       |
V     |_______|_______|_______|_______|_______|_______|_______|
| F     |       |       |       |       |       | M     |
|     |   *   |   *   |   *   |   *   |   *   |   *   |   *   |
|     |_______|_______|_______|_______|_______|_______|_______|
|     |       |       |       |       |       |       |       |
|     |       |   *   |   *   |   *   |   *   |   *   |       |
|     |_______|____ __|_______|_______|_______|______ |_______|
|       |       |       |       |       |       |       |
|       |       |       |   *   |       |       |       |
|_______|_______|_______|_______|_______|_______|_______|

-----     U   ------------>
```
The answer, it appears to the present author, must be "not only on the horizontal line FM"; for colliding fluid fragments are likely to generate motion in lateral directions as well as being checked or accelerated in the direction of their velocity difference.

Therefore some of the offspring must be deposited into boxes above and below the horizontal line, of course in such a way as to preserve momentum and to ensure that there is at least no gain of energy.

The above sketch illustrates this by its band of asterisks; but, before the idea can be expressed in a computer program, a precise offspring-distribution formula must be settled.

### (d) Other matters

Among the questions still to be addressed are:-

• How is the inter-fluid friction which opposes motion caused by differential body forces to be formulated?

• As the Reynolds number becomes lower, fluctuations die out. This can be ensured by increasing the MIXRATE quantity. But what dependence of MIXRATE on Reynolds number will fit the experimental data?

• In the mixing-layer model alluded to in section 3.2, the root- mean-square of the velocity fluctuations has been employed in a formula for the effective viscosity, in place of the usual k**0.5. However, MFM provides so much more information about the velocity pdf than is needed for the RMS value. In what way can that information be exploited so as to give more realistic simulations?

No answers are proposed at the present time.

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