The paper presented at the 2nd Helsinki Colloquium on Process (Spalding,1995a) was the start of a series of publications on MFM, each of which has carried the idea a little further.
It showed how a four-fluid model of premixed-fuel combustion allowed phenomena to be predicted, and specifically those concerning the spread of a steady flame confined in a duct (Williams, Hottel and Scurlock, 1953) which were beyond the capability of the "two-fluid" eddy- break-up model. Much of this demonstration is reproduced in section 4.4 below.
A ten-fluid model of an axi-symmetrical turbulent jet was also presented.
This paper already introduced the notions of "promiscuous coupling" and "Mendelian splitting", which have proved to be central ingredients of later multi-fluid models; and it was there pointed out, but not illustrated, that two-, three- and more-dimensional population grids could be envisaged and used.
The essential idea was recognised as that of "discretization" of fluid-attribute space, which can be carried out in numerous ways, all of which should be equivalent if the "population grid" is made sufficiently fine.
The "fluid-population distributions" of multi-fluid models and the "probability-density functions" of probabilistic methods (Dopazo and O'Brien, 1974) were regarded as being equivalent, again in the fine-grid limit.
At the CTAC-95, Conference in Melbourne, a paper was presented (Spalding,1995b) which applied MFM to the turbulent Bunsen burner. This required the use of a two-dimensional fluid-population distribution, indeed the one which appears in section 1.6, above.
The results were plausible, but no attempt was made to ensure grid- independence, or to make comparisons with experiment.
A 100-fluid model was used (Spalding, 1995c) for the study of the shapes of the fluid-population distributions, both one- and two- dimensional, which appear (according to MFM) when combustible gases flow steadily into a reactor which is so "well-stirred" that macro-mixing is perfect.
It was shown there that the one-dimensional fdps could take a great variety of different shapes, far exceeding those which have been imagined by the practitioners of the "presumed-pdf" approach [see section 4.1f below].
The same conclusion about pdf shapes emerged from a later study (Spalding, 1996a), in which MFM modelling was applied to a one- dimensional steadily-propagating turbulent pre-mixed flame.
This study further demonstrated that the concept of grid refinement, which is familiar to all flow-simulation experts in connexion with the sub-division of time and of geometrical space, is equally applicable to the population grid. It was further shown that, for the case in question, results of adequate accuracy could be obtained with 10- or 20-fluid models; litlle further increase of accuracy resulted from using the 100-fluid model.
The four-fluid model has been employed in two transient explosion computations. One of these concerns a laboratory experiment in which flame propagates through a pre-mixed gas in a duct on the walls of which are a series of turbulence-promoting baffles. The results, reported by Freeman and Spalding (1995) are in good qualitative and semi-quantitative agreement with experiment.
The second application has been to the explosions recently carried out at Spadeadam, in Northern England, by the UK Steel Construction Institute. The results have not yet been published. It can however be said that the computations based upon the 4-fluid model were far from providing the worst agreement with experimental data.
Later research has shown that it would be wiser to employ a 10-fluid model, in order that the population-grid effects should not be troublesome.
However, the ways in which the presence of solid obstacles influence the intensity and length-scale of turbulence, and so of coupling- splitting rates, is at the present a matter of guess-work. No-one living in the world today, in the present author's belief, can quantify this influence reliably.
3.2 Unpublished work
More recently (Spalding,1996d) the following studies have been made, albeit not yet completed:-
The second, which probably breaks new ground for chemical engineers concerned with reactor-vessel design, reveals that it is much more important to simulate the micro-mixing than it is the macro-mixing.
The third, by introducing an idealised smoke-production reaction, shows how different are the amounts of this production predicted by single- and multi-fluid models.
The fourth study may be regarded as being of especial interest because, whereas all the others relied on a hydrodynamic model of turbulence, either mixing-length or k-epsilon, to provide the multiplier of the coupling-and-splitting-rate term, this one did not.
Instead, the rate was deduced from the fluid-population distribution of longitudinal velocity, and from a prescribed length scale. The first step towards a stand-alone MFM for turbulent flows has thus been taken. The next, namely the removal of the need for a length- scale prediction, is not far away.