BY : S.V.ZHUBRIN FOR : Technical-discussion meeting DATE : June, 1997 PHOENICS Version : 2.2.2
Spalding (1995-1996) has analysed number of turbulent phenomena in a view of his new invention : Multi-Fluid Models of Turbulence.
The present work extends Spalding's analysis to an application for which quantitative test has been made for MFTM ability to predict concentration fluctuations.
The 1-, 3-, 5-, 9- and 17-fluid models are employed to simulate the turbulent mixing resulting from the admission of two separate, isothermal, coaxial jets of different composition into a concentric duct as next pictures illustrate.
The hydrodynamic turbulence obeys a standard K-epsilon transport equations.
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 as follows:
The scheme hypotheses is that the coupling may only take place between those parent fluids which would produce the appropriate offsprings inheriting the ATTRIBUTES of either parent in EQUAL proportion.
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 2.2.
The purpose is to provide the newcomers of MFM approach with self-contained, transparent, easy-to-understand and ready-to-make-parametric-study example to let them start.
The relevant Q1 file for 17-fluid model is supplied in PLANT data-input library.
The picture 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 axial variation of concentartion fluctuations.
The radial variation of the averaged concentration fluctuations are shown in the next picture. 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.
The radial variation of concentartion fluctuations.
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.
The results of multi-fluid computations are compared with experimental measurements of concentration fluctuations extracted from the work of Elgobashi et all.
Next plot displays the predicted and measured axial variations of normalised averaged concentration fluctuations for Craya-Curtet number equals to 0.875.
Points are the experiments of Torrest and Ranz; green line - MFM prediction yellow line - transport equation for fluctuations.
The comparison of axial fluctuation variations.
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.
The number of tests performed have proved the self-consistency and conservativeness of the model formulation, the correct behavoir of the equation system under the influences of micro-mixing constant and of the "population grid".
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.