- Concluding remarks about the nature of MFM
- Mathematical and computational tasks
- Tests of realism
- Conceptual developments

- Lack of novelty
- Opportunities for research and development
- Especially promising practical applications

- The multi-fluid concept has a long history. It is implicit in the Reynolds (1894) Analogy and the Prandtl (1925) mixing-length hypothesis, both of which contain the idea of two fluids, having different velocities, temperatures, etc, in relative motion.
- The idea that the rate of interaction between the fluids depends on the local turbulence, and is proportional to velocity gradient or to epsilon/k, appeared first in the eddy-break-up theory; and it is used also in the Monte-Carlo PDF-transport theory.
- The mathematics of interpenetrating fluids is well understood, having been developed for handling multi-phase (eg steam and water) flows. The two-fluid model of turbulence arose from this.
- There is therefore nothing essentially novel about what has been presented in this lecture, which simply carries well-established ideas a little farther than before.

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.

Such efforts as have been exerted (see ICOMP, 1994), apart from the author's own, have adopted the Monte Carlo method; this, however, has heavy computational requirements, and makes unusual intellectual demands. For these reasons, it is little used.

MFM is computationally less expensive than Monte Carlo; it employs computational methods which are familiar to all CFD practitioners; and it has advantages (such as grid-refinement and coarsening) of its own.

Now is therefore the time to consider what opportunities exist for its further development, and what research activities are needed so as to allow these opportunitires to be exploited.

In addition to those applications mentioned in this lecture, eg to turbo-machinery, paddle-stirred reactors and gas-turbine combustors, there are several application areas for which MFM is especially promising. They include:-

- cyclone separators;
- flame propagation in internal-combustion engines, both spark- and compression-ignition;
- power-station- and other industrial-furnace combustion;
- atmospheric air pollution, especially by dense gases, including hazardous-gas dispersion;
- micro- and meso-climatology;
- natural-waters pollution, and other oceanographic phenomena.

Already it has been applied (Svensson, 1996) to simulation of ground-water flow beneath Scandinavia since the last Ice Age!

- Population-grid refinement
- Consistency tests
- Non-uniform, unstructured, and self-adaptive grids
- Computational improvements
- Attention to the continuously-varying attributes (CVAs)

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 many one- two- and (at least a few) three- dimensional population grids, before the basic ideas and solution algorithms will be widely accepted as sound.

There is little doubt that they ARE sound; but it is also important to gather information about how the fineness necessary for (say) 5% accuracy depends upon the process in question, and about local conditions.

Such information will assist automatic-grid-adaptation studies.

When the micro-mixing constant becomes very large, the behaviour of the population of fluids must reduce to that of a single fluid, of which the PDA values vary from place to place.

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; and he will apply these tests in the largest possible variety of conditions.

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 PDA in regions of steep variation, and remove them from regions of less interest.

In summary, 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.

Although the computer-time burden is not yet great, it may become less tolerable when fine-grid three-dimensional transient simulations with complex chemical kinetics have to be undertaken.

Fortunately, numerous methods of load-reduction exist, including:

- 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.

Although CVAs have made their appearance in the examples presented above, namely as salt and smoke concentrations, they have played only a subordinate role.

There are however many practical applications in which the CVAs will be of greater importance. These include:-

- those combustion processes in which many chemical reactions take place at rates which are comparable with those of micro-mixing;
- multi-phase phenomena in which the liquid-fuel content of the (multi-) fluids distinguished by total (ie liquid and gas) fuel/air ratio has to be computed;
- purely-hydrodynamic flows in which some velocity components are PDAs while others are CVAs.

- Comparisons with experiment
- Comparison with the results of direct numerical simulation (DNS)
- The order of events

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:

- those most desired by the potential end-users, for example those 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;
- those with well-known data on the velocity, temperature and concentration profiles in jets, wakes, boundary layers and other simple turbulent flames; and
- those with experiments providing detailed measurements of probability-density functions of important fluid variables.

4. that with the results of direct numerical simulations.

DNS has become an increasingly popular tool of turbulence researchers; and indeed it is an excellent, albeit expensive, means of investigating what truly happens in a turbulent flow.

Until now, the outputs of DNS studies have been mainly used for comparison with the predictions of conventional (ie Kolmogorov-type) turbulence models; yet complete PDFs can equally-well be produced.

When serious research into MFM begins, one of its important elements will certainly involve the use of DNS for distinguishing between alternative micro-mixing concepts, and for establishing the appropriate constants.

Then experiments would be conducted systematically so as to permit kind-(3) comparisons, leading perhaps to improvements in particular model features.

Finally, comparisons of kind (1) would be made, whereafter, if the agreement proved satisfactory, 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), ie between MFM predictions and industrially-important phenomena, should begin in the near future.

- "Parental bias" in the "splitting" process
- The effects of laminar Prandtl and Schmidt numbers
- The coupling-splitting hypothesis for a two-velocity population
- Other matters
- Concluding remarks regarding the conceptual development of MFM

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 coupling-splitting mechanism is that of collision, limited-time contact and then separation.

The profile of concentration (say) in the temporarily-adjacent 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 of the diagram below.

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.

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 (e) of the Appendix.

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

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

As a final example of how the coupling-splitting hypothesis can be improved, let the attributes of a 2D population grid be the horizontal and vertical velocities, U and 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?

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 no gain of energy.

The sketch below 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.

|_______|_______|_______|_______|_______|_______|_______| | | | | | | | | ^ | | | | * | | | | | |_______|_______|_______|_______|_______|_______|_______| | | | | | | | | | | | * | * | * | * | * | | V |_______|_______|_______|_______|_______|_______|_______| | F | | | | | | M | | | * | * | * | * | * | * | * | | |_______|_______|_______|_______|_______|_______|_______| | | | | | | | | | | | | * | * | * | * | * | | | |_______|____ __|_______|_______|_______|______ |_______| | | | | | | | | | | | | * | | | | |_______|_______|_______|_______|_______|_______|_______| ----- U ------------>

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 mdot quantity. But what dependence of mdot on Reynolds number will fit the experimental data?
- Moreover, at lower Reynolds numbers, the structure of the turbulent mixture becomes more orderly. Probably therefore, the degree of allowable promiscuity will diminish. How should this be expressed.
- In the mixing-layer model alluded to in section 8.1, 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?

Evidently, even though MFM in its present form can already be of practical use for solving practical problems of engineering and science, the possibilities for extending and refining it are immense.

Conventional, ie Kolmogorov-type, turbulence modelling is widely regarded as having become subject the "law of diminishing returns". MFM, by contrast, is almost virgin territory, with no frontiers in sight.

The author commends it as a research area which is suitable for the young and adventurous, and for any who prefer exploring new intellectual territory to making miniscule improvements to well-worn and therefore unexciting ideas.