9.1 Concluding remarks about the nature of MFM
Contents of section 9.1
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.
(c) Especially promising practical applications
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:-
Already it has been applied (Svensson, 1996) to simulation of ground-water flow beneath Scandinavia since the last Ice Age!
9.2 Mathematical and computational tasks
Contents of section 9.2
(a) Population-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 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.
(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 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.
Fortunately, numerous methods of load-reduction exist, including:
There are however many practical applications in which the CVAs will
be of greater importance. These include:-
Three kinds of comparisons should be distinguished, namely:
(d) Computational improvements
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.
It will also be necessary to devise optimal overall discretization
strategies; for certainly it is not optimal to employ an extremely
fine geometric grid when the population grid is too coarse to
represent adequately the fragmentariness of the flow.
(e) Attention to the continuously-varying attributes (CVAs)
Although CVAs have made their appearance in the examples presented
above, namely as salt and smoke concentrations, they have played
only a subordinate role.
When it is recognised that the PDA/CVA distinction is arbitrary, and
that rational principles of selection are needed, it can be seen
that much experience needs first to be gained.
9.3 Tests of realism
Contents of section 9.3
(a) 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.
(b) Comparison with the results of direct numerical simulation
(DNS)
One more comparison type should be mentioned, namely:
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.
(c) The order of events
Were there no urgency about solving the practical problems of
engineering and the environment, a rational research program would
probably concern itself first of all with kinds (2) and (4), in
order to establish a convincing prima facie case for MFM.
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.
9.4 Conceptual developments
Contents of section 9.4
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.
Only if the profile of concentration were linear with distance would
the Mendelian assumption be correct.
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.
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.
(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 coupling-splitting mechanism is that of
collision, limited-time contact and then separation.
|****** ^ |******
| ***** | |*****
| **** | |****
| *** conc- |***
| ** entr- |**
| * ation |*
| ** | |**
| *** | |***
| **** |****
| ***** |*****
| ****** |******
|------------------------------------------- |---------------
----------- distance -------> - frequency--->
(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.
|_______|_______|_______|_______|_______|_______|_______|
| | | | | | | M |
^ | | | | | | | * |
| |_______|_______|_______|_______|_______|_______|_______|
| | | | | | | | * |
temp- | | | | | | | |
erat- |_______|_______|_______|_______|_______|_______|_*_____|
ure | | | | | | | |
| | *| | | | | | |
| |_______|_______|_______|_______|_______|_______|_______|
| | * | | | | | | |
| | | | | | | | |
| |____*__|____ __|_______|_______|_______|______ |_______|
| F | | | | | | |
| * | | | | | | |
|_______|_______|_______|_______|_______|_______|_______|
----- salinity------------>
(c) The coupling-splitting hypothesis for a two-velocity population
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.
|_______|_______|_______|_______|_______|_______|_______|
| | | | | | | |
^ | | | | * | | | |
| |_______|_______|_______|_______|_______|_______|_______|
| | | | | | | | |
| | * | * | * | * | * | |
V |_______|_______|_______|_______|_______|_______|_______|
| F | | | | | | M |
| | * | * | * | * | * | * | * |
| |_______|_______|_______|_______|_______|_______|_______|
| | | | | | | | |
| | | * | * | * | * | * | |
| |_______|____ __|_______|_______|_______|______ |_______|
| | | | | | | |
| | | | * | | | |
|_______|_______|_______|_______|_______|_______|_______|
----- U ------------>
(d) Other matters
Among the questions still to be addressed are:-
(e) Concluding remarks regarding the conceptual development of MFM
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.