
The kinetic mechanisms of NOx formation and destruction described in the preceding sections have all been obtained from laboratory experiments using either a laminar premixed flame or shocktube studies where molecular diffusion conditions are well defined. In any practical combustion system, however, the flow is highly turbulent. The turbulent mixing process results in temporal fluctuations in temperature and species concentration that will influence the characteristics of the flame.
The relationships among NOx formation rate, temperature, and species concentration are highly nonlinear. Hence, if timeaveraged composition and temperature are employed in any model to predict the mean NOx formation rate, significant errors will result. Temperature and composition fluctuations must be taken into account by considering the probability density functions which describe the time variation.
The TurbulenceChemistry Interaction Model
In turbulent combustion calculations, FLUENT solves the densityweighted timeaveraged NavierStokes equations for temperature, velocity, and species concentrations or mean mixture fraction and variance. To calculate NO concentration, a timeaveraged NO formation rate must be computed at each point in the domain using the averaged flowfield information.
Methods of modeling the mean turbulent reaction rate can be based on either moment methods [ 404] or probability density function (PDF) techniques [ 163]. FLUENT uses the PDF approach.

The PDF method described here applies to the NOx transport equations only. The preceding combustion simulation can use either the generalized finiterate chemistry model by Magnussen and Hjertager or the nonpremixed combustion model. For details on these models, refer to Chapters
14 and
15.

The PDF Approach
The PDF method has proven very useful in the theoretical description of turbulent flow [ 164]. In the FLUENT NOx model, a single or jointvariable PDF in terms of a normalized temperature, species mass fraction, or the combination of both is used to predict the NOx emission. If the nonpremixed combustion model is used to model combustion, then a one or twovariable PDF in terms of mixture fraction(s) is also available. The mean values of the independent variables needed for the PDF construction are obtained from the solution of the transport equations.
The General Expression for the Mean Reaction Rate
The mean turbulent reaction rate can be described in terms of the instantaneous rate and a single or joint PDF of various variables. In general,
(20.1103) 
where ,... are temperature and/or the various species concentrations present. is the probability density function (PDF).
The Mean Reaction Rate Used in
FLUENT
The PDF is used for weighting against the instantaneous rates of production of NO (e.g., Equation 20.115) and subsequent integration over suitable ranges to obtain the mean turbulent reaction rate. Hence we have
or, for two variables
where is the mean turbulent rate of production of NO, S is the instantaneous rate of production given by, for example, Equation 20.115, and and are the PDFs of the variables and, if relevant, . The same treatment applies for the HCN or NH source terms.
Equation 20.1104 or 20.1105 must be integrated at every node and at every iteration. For a PDF in temperature, the limits of integration are determined from the minimum and maximum values of temperature in the combustion solution. For a PDF in mixture fraction, the limits of the integrations in Equation 20.1104 or 20.1105 are determined from the values stored in the lookup tables.
Statistical Independence
In the case of the twovariable PDF, it is further assumed that the variables and are statistically independent so that can be expressed as
(20.1106) 
The Beta PDF Assumption
is assumed to be a twomoment beta function as appropriate for combustion calculations [ 135, 245]. The equation for the beta function is
where is the Gamma function, and depend on , the mean value of the quantity in question, and its variance, :
(20.1108) 
The beta function requires that the independent variable assume values between 0 and 1. Thus, field variables such as temperature must be normalized. See Section 20.1.10 for information on using the beta PDF when using singlemixture fraction models and twomixture fraction models.
The Calculation Method for
The variance, , can be computed by solving a transport equation during the combustion calculation stage. This approach is computationally intensive and is not applicable for a postprocessing treatment of NOx prediction. Instead, calculation of is based on an approximate form of the variance transport equation.
A transport equation for the variance can be derived:
where the constants , and take the values 0.85, 2.86, and 2.0, respectively. Assuming equal production and dissipation of variance, one gets
The term in the brackets is the dissipation rate of the independent variable.
For a PDF in mixture fraction, the mixture fraction variance has already been solved as part of the basic combustion calculation, so no additional calculation for is required.