One of the crucial issues of modeling chemical reaction in turbulent flows is the chemical closure problem. It is well known that in such flows, due to the highly non-linear dependence of chemical reactions on temperature, using the mean temperature and mean species concentrations for calculations of mean chemical reaction rates can cause significant errors.

The closure problem associated with non-linearities in the equations governing turbulent flow can be avoided by considering the joint probability density function (PDF) of the flow variables. The joint PDF of scalars, , provides a complete statistical description of the chemical and thermodynamic state. The use of a transport equation for is particularly attractive for reacting flows, since the effects of reactions appear in closed form, irrespective of the complexity and non-linearity of the reaction mechanism. However, reactive flows of practical interest usually involve many species. Consequently the dimensions of the PDF are large and finite-difference solutions of the PDF transport equation are impractical because of the large . Pope [80] developed a Monte Carlo algorithm that makes solving the PDF transport equation practical for general turbulent reactive flows. Rather than considering explicitly, the dependent variable in the simulation is represented by an N -member ensemble:

(10–2)

Here each of the members of the ensemble is referred to as a "particle". Although each particle is ascribed a unique number, , no ordering is implied. In fact, operations are performed either on all particles or particles selected at random. Thus, the numbering is a convenience that has no effect on the outcome. The ensemble average of any function is defined by

(10–3)

In the limit of large , Pope[80] showed that the ensemble average converges to the corresponding density-weighted average, that is,

(10–4)

For the general multiple reactive scalars, the transport equation for the joint PDF in the PaSR is derived by integrating the governing equation of the single-point joint scalar PDF over the reactor volume. The resulting PDF transport equation for the PaSR is

(10–5)

The first two terms on the right hand side of Equation 10–5 represent the effects of chemical reaction and the through-flow on the joint PDF, respectively. The last term represents the effect of micro-scale mixing on the PDF, which requires the use of a mixing model. The mixing model mimics the finite rate mixing of particles in the stochastic simulations. Two widely used mixing models are employed as options in the current PaSR model.

The simplicity of using a Monte Carlo method and a scalar PDF permits us to carry out simulations with detailed chemistry without significant computing costs.