A paper to be published in PHOENICS Journal, January, 1999

A turbulence model which predicts concentration fluctuations without using any conservation equations for statistical properties of the fluctuations is an application of Multi-Fluid concept of Brian Spalding.

The 1-, 3-, 5-, 9- and 17-fluid models are employed here to simulate the turbulent mixing resulting from the admission of two separate isothermal, coaxial jets of different composition into a concentric duct.

PLANT feature of PHOENICS 3.1 has been used to implement the multi-fluid model and transport equation for the root-mean-square of fluctuating concentration.

A comparison with experimental data and with transport equation predictions is presented.

- Introduction
- The problem considered
- Mathematical formulation
- Implementation
- Results
- Conclusion
- References
- Appendix: Q1 input file

D B Spalding (1995-1996) has analysed number of turbulent phenomena in terms of the concept: Multi-Fluid Model of Turbulence.

The present work extends Spalding's analysis to flow for which quantitative test data are available.

The emphasis of the present paper is, therefore, on finding out how fine a fluid-population-grid is needed for accuracy sufficient for comparison both with experiments and currently popular models which operate with the transport equations for statistical properties.

After the brief description of the problem in question given in Section 2, the main aspects of mathematical formulations are given in Section 3. In Section 4 the remarks are made on implementing the model. Results and discussion are contained in Section 5 followed by concluding remarks and the list of references.

The particular process considered is the mixing resulting from the admission of two separate, isothermal, coaxial jets of different composition into a concentric duct. The geometry of the duct and the system of coaxial jets are taken as those which have been used by Elgobashi, Pun and Spalding (1977).

The central idea of MFM is that of a "**population of fluids**", the
members of which are distinguished by one or more "population-
distinguishing attributes" (PDAs), which are then arbitrarily
discretised by the analyst.

It is supposed here that each of the fluids has a different mixture composition in terms of concentrations of central- and surrounding-jet components. The concentration dimension has thus been uniformly discretised in the range from unity (central fluid) to zero (surrounding fluid).

Up to 17 conservation equations are used to calculate each-fluid-frequency-in-population measure:- the mass fraction of the fluids. They contain the convection, diffusion and source terms and are solved in a conventional way (viz by PHOENICS, in elliptic mode) together with appropriate boundary conditions.

The statistical properties of the fluid population, e.g. mean concentration and root-mean-square fluctuating concentrations etc. are then deduced.

For comparative purpose the transport equation for the latter,**g**, is
also solved along with appropriate source terms and boundary conditions
as described by Elgobashi et al (1977).

These are as follows:

- Hydrodynamic turbulence is supposed to obey the standard K-epsilon transport equation.
- The effective viscosity is deduced from the above quantities as proportional to K**2 / epsilon, where K is the local energy of the turbulent motion, and epsilon is its rate of dissipation.
- The source/sink terms in the fluid-mass-fraction equations are:
first_fluid_mass fraction * second_fluid_mass fraction * 5.0 * the energy-dissipation-rate / turbulence-energy

shared according to a variant of coupling/splitting formula proposed by Spalding (1996).

The full account of above formulations for 5 fluid model is given in Appendix.

The source/sink constant (viz. 5.0) has not been fully optimised; but, as will be shown below, it fits some experimental observations and data obtained by the solution of the transport equation for the root-mean-square of fluctuating concentration component fairly well.

Specifically,it has been found to produce results which are in a good agreement with experimental data as will be described in Section 5.

Although there exists the special GROUND, MFMGR.FOR, which can handle many fluids, and both one- and two-dimensional populations, the results to be displayed have been created by use of the PLANT feature of PHOENICS as described by Zhubrin (1997).

The necessary formulae for calculations of source/sinks, physical properties, statistical operations, auxilliary computations and post-processing preparations have been set by way of appropriate statements in Q1 file.

Running SATELLITE then results in the generation of all relevant GROUND codings, compilation and re-linking.

Running private EARTH initiates the calculations and produces the field distributions of all relevant variables.

The relevant Q1 file for 5-fluid model is supplied as Appendix and the one for 17-fluid model is supplied in PLANT data-input library of PHOENICS 3.2.

The results of computations will now follow.

Fluid-population-grid-refinement studies requires detailed attention. We must find out how fine a grid is needed for the accuracy to be reasonable enough for comparison purposes.

There will now be presented a series of graphs, all for the same boundary conditions and the same value of mixing constant.

Specifically, the models will have **3**, **5**, **9** and **17 fluids** with uniformly
distributed fluid attribute. It was found that further refinement of
fluid population grid has little influence on the computational results.

Fig.1 shows the variations with axial distance the averaged concentration fluctuations at the axis. In each case, it is normalised by the local concentration of the central-jet fluid and axial distance is normalised by duct radius.

The full curves represent multi-fluid predictions and the points are the computations of the transport equations for the square of the concentration fluctuations.

The radial variation of the averaged concentration fluctuations are shown in Fig.2. The full lines are multi-fluid predictions and the points are the computations of the transport equations for the square of the concentration fluctuations.

The ordinates represent values of the averaged concentration fluctuations normalised by its value at the axis of symmetry, and the abscissae is radial distance normalised by the duct radius.

Fair agreement is seen between multi-fluid predictions and transport equation for turbulence quantity for finest ( not too much ) fluid-population grid used for all above calculations.

In Fig.3 and Fig.4 the results of numerical computations are compared with experimental measurements of concentration fluctuations extracted from the work of Elgobashi et all (1977): green lines - MFM predictions, yellow lines - transport equation for the root-mean-square fluctuation. The points are either the experiments of Torrest and Ranz for Craya-Curtet number 0.875, Fig.3, or of Becker, Hottel and Williams for Craya-Curtet number 0.345, Fig. 4.

These comparisons show that single constant of Multi-Fluid Model provides fair agreement between calculations and experiment for the cases when a region of recirculation is appeared downstream of the inlet plane.

Fig.3 displays the predicted and measured axial variations of normalised averaged concentration fluctuations for a Craya-Curtet number of 0.875.

Fig.4 shows the predicted and measured axial variations of normalised averaged concentration fluctuations for a Craya-Curtet number of 0.345.

Overall, it can be said that fair agreement between MFM predictions and experiments is observed.

The axial location of maximum concentration fluctuations is well reproduced.

MFM is slightly over-predicted the actual values in comparison with transport equation calculations for a Craya-Curtet number of 0.875.

For the flow with Craya-Curtet number of 0.345, it seems that the inlet concentration fluctuations in experiment were not negligible. It results in under-prediction of both transport equation model and MFM which treat the flow to be well stirred at the inlet plane with no concentration fluctuations whatsoever.

The use of multi-fluid approach together with K-epsilon turbulence model for hydrodynamics has been found to give satisfactory prediction for the concentration fluctuations in confined jet flow.

It has been established, that refining the population grid (ie increasing the number of fluids) does indeed lead to a unique solution, and though grid refinement from 3 to 5 is not enough for grid-independent results to be attained; but as few as 17 fluids give acceptable accuracy.

Concentration fluctuation results have provided further validity of the analysis put forward by Spalding. For the present case, where k-epsilon model is used 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 the best value is found to be 5.0.

The ability of the model to reproduce the results of the solution of transport equations for turbulence quantities, as well as experimental observations is especially gratifying.

More reliable experimental data are still needed for a more extensive validation of the multy-fluid model. However, it seems that sufficient confidence has already been generated.

S E Elgobashi, W M Pun and D B Spalding (1977) " Concentration fluctuation in isothermal turbulent confined coaxial jets". Chemical Engineering Science, v. 32, pp. 161-166

D B Spalding (1995) "Models of turbulent combustion" Proc. 2nd Colloquium on Process Simulation, pp 1-15 Helsinki University of Technology, Espoo, Finland

D B Spalding (1995) "Multi-fluid models of Turbulence", European PHOENICS User Conference, Trento, Italy

D B Spalding (1996) "Older and newer approaches to the numerical modelling of turbulent combustion". Keynote address at 3rd International Conference on COMPUTERS IN RECIPROCATING ENGINES AND GAS TURBINES, 9-10 January, 1996, IMechE, London

S V Zhubrin (1997) " PLANT - the method of implanting physical-model formulae into the executable sover module", VIIIth International PHOENICS User Conference, Seville, Spain.