In the author's opinion, the multi-fluid model is ready for use, right now, for all simulations in which chemical reactions in turbulent fluids are of prime importance. This opinion holds both for chemical-engineering equipment such as the paddle-stirred reactor, and for combustion chambers of engines and furnaces.
In so far as a turbulence model such as k-epsilon can be relied on for simulating the hydrodynamics, there is only one constant needed by the model, namely that which relates the "fluid-coupling" rate to epsilon/k; and this is known to be of the order of unity.
The results for the rate of reaction in the stirred vessel in (Spalding, 1996b) show very clearly the danger of using a single- fluid model. In particular, the common belief that single-fluid models are usable, when reaction is slow compared with turbulent mixing, is no longer tenable.
It has also been shown that the computer-time penalty associated with use of MFM is not a serious one.
Therefore, it can be plausibly argued, some form of MFM is the ONLY turbulence model which can be justified for use with homogeneous-reaction simulation from now on.
A counter-argument may be raised by users (who are numerous) of the "presumed-pdf" approach, to the effect that they are already taking sufficient account of fluctuations.
To rebut this counter-argument, one has only to inspect the computed fluid-population distributions which have appeared in the recent publicationss cited in the present paper.
These FDPs have the same significance as the PDFs; and they are of such varied shapes that to suppose that anyone could safely "presume" them would be extremely rash.
How about the use of a probabilistic method instead? If the computer times are low enough, and the computer programs easy enough to use, the case for using such methods is as strong as for MFM. Certainly, although the underlying micro-mixing models are different, there is no quantitative evidence which yet favours one rather than the other.
5.2 Applications to chemically-reacting flows
Certainly the reacting-flow applications of MFM are of high importance. They include:-
Let item (1) be considered first; and let it be supposed that the temperature is so high that, at the prevailing pressure (usually atmospheric), all energy-producing reactions proceed very rapidly towards completion.
If a single-fluid model is employed, the reaction will be supposed to be completed at the downstream section at which the time-average fuel/air mixture fraction is stoichiometric or less.
A multi-fluid model will reveal that some unburned fuel is present at locations downstream of that section; and the equipment designer seeking 100% combustion efficiency will want to know how much larger he must make his combustion chamber.
The answer will of course depend on the details of the geometry of the system; which is why a detailed three-dimensional analysis is needed.
MFM is already able to provide this, with some margin of error, of course.
MFM, in order to provide reliable predictions for most circumstances, will need to take account of the fact that light-gas fragments are accelerated (or decelerated) by pressure gradients more easily than dense-gas fragments, so giving rise to what is sometimes called "counter-gradient diffusion". The phenomenon is perhaps most important (and least taken into account) in simulations of reciprocating-engine (ie gasoline or diesel) combustion.
In gas turbines, perhaps, a one-dimensional fluid population will suffice, the distinguishing attribute being the fuel-air ratio.
Gasoline engines may also be simulated adequately by a one- dimensional population, with however reactedness (or mixture fraction minus unburned fuel) as the distinguishing attribute.
Diesel engines, on the other hand, because the mixing and the ignition processes are not separated in time, will probably require two-dimensional populations of the kind discussed in section 1.6.
There are innumerable papers which describe how, once the pdf is known, the rate of production of oxides of nitrogen, smoke, and other undesirable side-products, can be computed. To this author's surprise, many of these claim to have demonstrated sufficient agreement between predictions and experimental data to justify the assumptions.
The methods of calculation are ingenious, and frequently very time-consuming; yet they almost invariably rest on precarious foundations. One of these is usually some variant of the two-fluid eddy-break-up model; and the other will be some presumption about the probability-density of fuel/air ratio or reactedness, but not usually of both.
Sometimes the pdf is taken to be that which corresponds to an assemblage of flamelets, ie one-dimensional steadily-propagating pre-mixed flames.
The author's current view is that the PDFs computed by the MFM as it stands today, ie without any refinements whatsoever, are likely to be as least as realistic as any that are currently in use.
His recommendation is therefore to combine the best chemical-kinetic models of the current literature with the pdfs, one- or two-dimensional according to circumstance, computed by the MFM.
As already mentioned, the four-fluid model has been used with some success; and it can be be expected that further insight will be gained when:
[Further applications to reacting-flow situations remain to be discussed]
5.3 Applications to hydrodynamic and heat-transfer phenomena
The "MFM revolution" will not be complete until the reliance on an underlying hydrodynamic model, for example the k-epsilon model which has been used in several of the examples referred to above, has been totally dispensed with.
Here it needs to be noted the k is nothing but a root-mean-square kinetic-energy fluctuation; and MFM is well able to compute RMS values, as has been seen above.
All that is needed therefore is for velocity or energy to be made the discretized variable, and for something similar to be done about the length scale; then the underlying hydrodynamic model can be dropped.
Work along these lines has already enabled simple shear flows to be simulated by MFM in a stand-alone manner (Spalding, 1996b). It will be reported elsewhere.
It is flows with strong body-force effects which are likely to provide the greatest rewards; for single-fluid models simulate them least well. Therefore atmospheric and oceanographic flows, where gravitational effects are significant, and turbo-machinery flows, where centrifugal forces are large, may be the first to be subjected to MFM analysis.
In order that MFM should contribute significantly to heat-transfer engineering, it will have to throw quantitative light on behaviour close to walls, where viscous effects become dominant. Nothing has been done to explore these effects, in the MFM context, so far.