[Chapter 2 of the lecture CAD to SFT.
Click here for the start of the start of the lecture]
Contents

4.1 The requirements: realism; economy; balance

4.4 The IMMERSOL model for radiation

4.5 The MFM turbulence model, for turbomachines and combustors

Click here for a complete lecture on the Multi-Fluid Model of Turbulence

The just-described two-metal-block problem can also be used to exemplify the practical difficulties of simulating turbulence, and heat transfer by radiation and convection, in practical circumstances; and indeed, fortunately, means of surmounting them.

The difficulties result from the facts that:-

- numerous variously-shaped three-dimensional solid-plus-fluid elements, of the kind illustrated, are to be found in practical equipment; and all must be analysed simultaneously;
- even with the largest computers, very few grid intervals are available to represent the spaces between the solids;
- small inter-solid distances make Reynolds numbers low, so that turbulent and laminar transfers are comparable in magnitude;
- calculating distances between walls is necessary for both turbulence and radiation models; and, though easy in principle, it may be enormously time-consumimg in practice;
- the fluid often participates in the radiation process, by way of absorption, scattering and re-emission; and
- the radiative properties of materials vary significantly with both temperature and wavelength.

Thus, it is pointless to expend large resources on elaborate low- Reynolds number turbulence models if the grid fineness is hopelessly inadequate; or on complex geometrical view-factor calculations if fluid-participation and wave-length dependences are totally ignored.

Considerations of balance led to not-yet-conventional methods being used in the above example. They will now be discussed.

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Whatever turbulence model is employed, since the Reynolds number is likely to be low, the distance from the nearest wall must be known.

Computing this was not easy until the introduction of the LTLS method (Spalding, 1994), in which the wall-distance (and also the distance between walls) was computed by solving the equation:

div_grad L = - 1

This elliptic linear equation for a single variable is very easy to solve. In PHOENICS, the solution is carried out swiftly at the start of the computation, and the results are stored for subsequent use.

The following pictures show the resulting distributions of wall- distance and the distance-between-walls, for the two-block problem.

The results are exactly correct wherever the quantities in question have precise meanings; and elsewhere they are "plausible". [The quotes imply a need for discussion, for which there is no space here]

Fig 4.2-1 The distance from the wall

Fig 4.2-2 The gap between walls

As has been shown by Aganofer, Liao and Spalding (1996), conventional low-Reynolds-number models (eg Lam and Bremhorst, 1981) are computationally expensive and of doubtful realism.

However, there exists a simpler, more economical, and (in these circumstances) equally realistic model, which is described in that paper, and used here. This is the so-called LVEL model, which derives the local effective viscosity from the wall distance, the distance between the walls, and the local velocity.

The following picture shows the effective-viscosity distribution computed for the two-block problem.

Once again, it can be proved that the predicted distribution is precisely correct in simple circumstances, and plausible elsewhere.

Fig 4.3-1 The effective viscosity

Click here for the PHOENICS Encyclopaedia article on LVEL

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If turbulence modelling in domains which are crowded with immersed solids is difficult, no less so is the computation of the radiative heat exchange between the solids and the intervening medium.

In the results presented above, use was made of the IMMERSOL (ie IMMERsed SOLids) method (Spalding, 1996). This represents radiative transport of energy by way of a diffusion equation for radiosity, in which the diffusivity is proportional to:

1 / [ A + S + 1 / WGAP ]

where A and S are the absorption and scattering coefficients of the medium per unit length, and WGAP is the distance between solid walls.

Further formulae represent the radiosity jump at phase boundaries, and so enable the radiosity in the medium and the temperature in the solids to be computed from the solution of one (non-linear) equation.

In the following pictures, contour diagrams will be presented. These will display, in order:-

- TEM1, the temperature of the air and solids, the latter being what gave rise to the displacements and stresses discussed above;
- T3, the radiosity temperature, which equals TEM1 only within the solid, and
- QRY and QRX, the vertical- and horizontal-direction radiant fluxes.

The computations were completed within a few minutes, by the PHOENICS computer code mounted on a Pentium personal computer.

The results are plausible; but experimental verification is needed.

Fig 4.4-1 Gas and solid temperature

Fig 4.4-2 The radiosity temperature

Fig 4.4-3 y-direction radiation flux

Fig 4.4-4 x-direction radiation flux

The following points should be recognised in conclusion:

- no contributions to the sciences of fluid mechanics or heat transfer have been made by the models described;
- more exact embodiments of scientific knowledge could have been used, if there had been no resource limitation;
- however, the models recommended are
**affordable approximations**to that scientific knowledge; indeed they are possibly the only such ones which exist; - because they are affordable, easy
**extensions**to account for wave-length and temperature dependences can also be afforded; - users of commercial computer codes would be wise to ask whether those codes offer affordable approximations, or non-affordable exactnesses combined with tacit neglects.

Contents of section 4.5

- The fundamental ideas
- Why turbo-machinery designers need MFM
- A two-fluid-model prediction for turbo-machinery
- MFM for combustion processes
- The significance of the 2D population distribution
- An example; smoke production in a 3D gas-turbine-type combustor
- Discussion of the MFM smoke calculation
- Concluding remarks about MFM and its future

There are currently three approaches to the quantitative prediction of turbulent-flow phenomena, namely:-

- use of Kolmogorov-type models which solve equations for quantities such as energy and dissipation, ie k and epsilon;
- use of Monte Carlo methods, seeking to compute probability- density functions (ie PDFs) for important variables; and
- use of multi-fluid models (ie MFMs), which can be regarded computing DISCRETISED PDFs (Spalding, 1995).

Approach (2), of Dopazo/O'Brien type, is followed by some combustor specialists; but its expense deters all but the wealthiest.

Approach (3), of the same type, has been little publicised; but it is economical, easy to use, and contains the necessary physics.

Axial-flow compressors and turbines, as used in aircraft propulsion and in ground- (or sea-) level power production, are characterised by the rapid passing of one blade row behind another.

The slower-moving boundary-layer fluid from the upstream row becomes a "wake" of slower-moving fluid fragments, which are distributed across the entrance plane of the downstream row.

The turbulent mixture which passes from row to row through a turbo- machine is therefore best represented as a population of fluids, with (say) axial velocity as their distinguishing characteristic.

Approach (3), ie use of MFM, is a practicable means of calculating the population distribution and its influence on the mean flow.

Research on the exploitation of this possibility is only now starting; but its promise appears to be very great. Further research on Kolmogorov-type models is now hard to justify.

The lowest member of the MFM family is the two-fluid model (Spalding, 1987), with which some recent studies have been made.

There follow two pictures which show how the time-mean velocity distribution of a blade row differs according to whether a two- fluid or (as is customary) a single-fluid model is presumed.

The differences are qualitatively similar; but the small quantitative differences are what counts when blade-row losses are to be computed.

If two-fluid calculations can already provide meaningful guidance to turbo-machinery designers, much more can be expected from the full MFM treatment.

Unfortunately, most turbo-machinery researchers still follow each other down the approach-1 tunnel, with no light at the end!

Fig 4.5-1 Radial-velocity contours at outlet ; 1 fluid

Fig 4.5-2 Radial-velocity contours at outlet ; 2 fluids

Combustion-chamber designers need to be concerned that their designs not only burn their fuels efficiently but also reduce to the minimum the production of atmospheric pollutants such as smoke and oxides of nitrogen.

To try one design variant after another is hopelessly expensive of time and money; so computer simulation is their main recourse.

Computer simulation may, of course, be MISleading; and it is likely to be so if the models built into the computer code do not embody the best physical knowledge about the relevant processes.

A realistic MFM model of gas-turbine combustion would supposes that the gases at any location constitute a population distinguished at least two-dimensionally, the dimensions being:

- fuel/air ratio; and
- progress towards complete reaction.

Fig 4.5-3: a two-dimensional (reactedness/fuel-air ratio) population

The random coloured circles on the right illustrate the physical conception underlying MFM; each colour represents a different discrete-fluid state.

The proportions of fluid in each state are represented by the fullnesses of the 2D array of boxes on the left. They are what MFM calculates; and each of the 100 fluids considered here has its own temperature, smoke- and NOX-production rate, velocity, and so on.

A conventional single-fluid model would work out the average fuel- air ratio and the average degree of reactedness, and deduce the smoke- and NOX-production rates from those quantities; but it would be wrong. The reason is that the rate expressions are non-linear.

In mathematical terms:

the average of (A x B) is NOT equal to the average of (A) x the average of (B) .

In human terms, a day-worker wife and a night-worker husband may NEVER meet sufficiently to have offspring.

This subject is a large one, which cannot be sufficiently discussed in the present context. It must therefore suffice to state that predictions of smoke (or NOX) production are totally different for single- and for multi-fluid models. An example follows.

This concerns smoke production in an imaginary 3D combustor, into which is injected a fuel-rich gaseous mixture at one location and pure air at a succession of other locations.

The following picture shows the distributions of smoke concentrations at the outlet cross-section based on:-

- a conventional single-fluid model;
- a five-fluid model;
- a ten-fluid model; and
- a twenty-fluid model.

Fig 4.5-4 Smoke predicted by a conventional single-fluid model

Fig 4.5-5 Smoke predicted by 5-fluid model

Fig 4.5-6 Smoke predicted by 10-fluid model

Fig 4.5-7 Smoke predicted by 20-fluid model

Comparison between the diagrams shows that there is very little difference between the smoke predictions for 10 and for 20 fluids; so it will better to use the smaller number, to save computer time.

Computer times are, in any case, not very large, that for 20 fluids being only three times that for a single fluid.

However, when 100 fluids (say) do prove to be necessary on grounds of accuracy, there are many available means of reducing the computer times.

For example, there is no need to employ the same number in all parts of the field; instead, the number can be varied according to the local behaviour of the solution.

Once, indeed, that it is recognised that MFM entails nothing more than discretizing dependent variables in the same way as is routine for independent ones (space and time), the well-known techniques of grid-adaptation become available.

New turbulence models need to be tested, by comparison of predictions with experiments, before they can be relied upon as the basis for serious engineering designs.

Performing the tests may be expensive, in man-power at least; so the case for committing the expenditure must be closely argued.

The case for testing MFM rests on three considerations, namely:-

- evident need, demonstrated by the failure of Kolmogorov-type models for turbo-machines and combustors;
- the inherent plausibility of the basic idea, and its embodiment of sufficient physics to represent (in turbo-machines) the relative motion of the faster- and slower-moving fluids; and (in combustors) their differing reaction rates;
- recognition that it does what Monte-Carlo-based methods aim to do, but with much less expense.

Consideration (2) is easily understood by scientists, but less easily by those for whom novelty is a synonym for danger. It is, unfortunately (but for good reasons) the latter who are usually put in charge of decisions about money.

It is for them that consideration (3) has been cited; for it implies that MFM is not totally novel, and therefore not extremely dangerous; and it indicates that there is money (currently being spent on Monte-Carlo) which can be saved.

The author's view is that, within ten years, MFM will have become
accepted, fashionable and (probably too-credulously) widely used;
it certainly needs serious attention from aeronautical engineers
right now; and researchers into direct numerical simulation could
assist by casting their results in MFM form.