# Mixing Models

In PDF methods, the effect of molecular diffusion on the composition is represented by a mixing model. Three different mixing models have been used in recent TNF calculations. These are:

• IEM or LMSE
• Modified Curl (MC)
• EMST

Each model causes the composition variance to decrease at the rate $C_{\phi}*\epsilon/k$  without affecting the mean composition. Here $C_{\phi}$ is the mixing-model constant, $k$ is the turbulent kinetic energy, and  $\epsilon$ is its dissipation rate. PDF methods are generally implemented as particle /mesh methods, in which the particle properties are advanced in time in fractional steps. For each computational cell (or grid node) there is an ensemble of N particles (in general of unequal numerical weight), and the mixing fractional step consists of applying a mixing model for the time step $\Delta t$.

The following sub-sections provide a brief description of the three mixing models and links to Fortran subroutines for performing the mixing fractional step to an ensemble of particles.

#### IEM/LMSE

The IEM model (Interaction by Exchange with the Mean, Villermaux & Devillon 1972) and the LMSE model (Linear Mean-Square Estimation, Dopazo & O’Brien 1974) stem from different physical concepts but result in the same model equation. According to the model, the composition of the nth particle $\phi^{(n)}(t)$ evolves by ordinary differential equation

$\frac {d\phi^{(n)}} {dt} =-\frac 1 2 C_{\phi}\frac \epsilon k (\phi^{(n)}-\tilde\phi)\\$

where $\tilde\phi$ is the Favre mean composition of the ensemble of particles.

#### Modified Curl

The Modified Curl mixing model (Janicka, Kolbe & Kollmann 1977) is a particle interaction model based on Curl’s model (Curl 1963). For the case of equal-weight particles, at a rate proportional to $C_{\phi}\epsilon/k$, pairs of particles (denoted by p and q) are randomly selected from the ensemble and their compositions are charged to

$\phi^{(p,new)}= \phi^{(p)}+\frac{1}{2}a(\phi^{(q)}- \phi^{(p})\\$

$\phi^{(q,new)}= \phi^{(q)}+\frac{1}{2}a(\phi^{(p)}- \phi^{(q})\\$

where a is a random number uniformly distributed in (0,1).

For the case of unequal weight particles, an implementation of modified Curl has been developed by Nooren et al. (1997) which conserves the mean and causes the correct variance decay. However, the effect on the PDF depends on the distribution of particle weights.

#### Test Program and Subroutines

The combustion group at TU Delft has provided a Fortran 90 test program for mixing models. Routines are included for IEM/LMSE, modified Curls, mapping closer models. An outline and download files are available below.

#### EMST

The Euclidean Minimum Spanning Tree mixing model (Subramaniam & Pope 1998) is a complicated particle-interation model, designed to overcome shortcomings of simpler models. In addition to the particle composition $\phi^{(n)}(t)$ and weight $w^{(n)}(t)$, the model involves a state variable $s^{(n)}(t)$.

A Fortran 90 subroutine implementing EMST is given at the URL below, where further explanation of the model is provided.

 Fortran subroutine for the EMST model tcg.mae.cornell.edu/emst

#### Comparison of Mixing Model Performance

The theoretical considerations concerning the relative merits of the different mixing models is discussed in Subramaniam & Pope (1998). In some simple test cases the models give substantially different results (see TNF6 proceedings, page 106). The relative performance of the models in the TNF target flames remains, to a large extent, an open question which should be answered at TNF7.

#### References

Curl, R.L. (1963). Dispersed phase mixing: I. Theory and effects of simple reactors. AIChE J. 9, 175-181.

Dopazo, C. and E.E. O’Brien (1974). An approach to the autoignition of a turbulent mixture. Acta Astronaut. 1, 1239-1266.

Janicka, J., W. Kolbe, and W. Kollmann (1977). Closure of the transport equation for the probability density function of turbulent scalar fields. J. Non-Equilib. Thermodyn. 4, 47-66.

Subramaniam, S. and S.B. Pope (1998). A mixing model for turbulent reactive flows based on Euclidean minimum spanning trees. Combust. Flame 115, 487-514.

Nooren, P.A., H.A. Wouters, T.W.J. Peeters, D. Roekaerts, U. Maas and D. Schmidt (1997). Monte Carlo PDF modeling of a turbulent natural-gas diffusion flame. Combust. Theory Modelling 1, 79-96.

Villermaux, J. and J.C. Devillon (1972). In Proc. Second Int. Symp. On Chemical Reaction Engineering, New York: Elsevier.