
Turbulent combustion is governed by the reacting NavierStokes equations. While this equation set is accurate, its direct solution (where all turbulent scales are resolved) is far too expensive for practical turbulent flows. In Chapter 14, the species equations are Reynoldsaveraged, which leads to unknown terms for the turbulent scalar flux and the mean reaction rate. The turbulent scalar flux is modeled by gradient diffusion, treating turbulent convection as enhanced diffusion. The mean reaction rate is modeled by the finiterate, eddydissipation, or EDC models. Since the reaction rate is invariably highly nonlinear, modeling the mean reaction rate in a turbulent flow is difficult and prone to error.
An alternative to Reynoldsaveraging the species and energy equations is to derive a transport equation for their singlepoint, joint probability density function (PDF). This PDF, denoted by , can be considered to be proportional to the fraction of the time that the fluid spends at each species and temperature state. has dimensions for the species and temperature spaces. From the PDF, any thermochemical moment (e.g., mean or RMS temperature, mean reaction rate) can be calculated. The composition PDF transport equation is derived from the NavierStokes equations as [ 289]:
where
=  Favre joint PDF of composition  
=  mean fluid density  
=  Favre mean fluid velocity vector  
=  reaction rate for species  
=  composition space vector  
=  fluid velocity fluctuation vector  
=  molecular diffusion flux vector 
The notation of denotes expectations, and is the conditional probability of event , given the event occurs.
In Equation 18.21, the terms on the lefthand side are closed, while those on the righthand side are not and require modeling. The first term on the lefthand side is the unsteady rate of change of the PDF, the second term is the change of the PDF due to convection by the mean velocity field, and the third term is the change due to chemical reactions. The principal strength of the PDF transport approach is that the highlynonlinear reaction term is completely closed and requires no modeling. The two terms on the righthand side represent the PDF change due to scalar convection by turbulence (turbulent scalar flux), and molecular mixing/diffusion, respectively.
The turbulent scalar flux term is unclosed, and is modeled in FLUENT by the gradientdiffusion assumption
where is the turbulent viscosity and Sc is the turbulent Schmidt number. A turbulence model, as described in Chapter 12, is required for composition PDF transport simulations, and this determines .
Since singlepoint PDFs are described, information about neighboring points is missing and all gradient terms, such as molecular mixing, are unclosed and must be modeled. The mixing model is critical because combustion occurs at the smallest molecular scales when reactants and heat diffuse together. Modeling mixing in PDF methods is not straightforward, and is the weakest link in the PDF transport approach. See Section 18.2.3 for a description of the mixing models.