
Equations 15.211 through 15.214 describe the instantaneous relationships between mixture fraction and species fractions, density, and temperature under the assumption of chemical equilibrium. The FLUENT prediction of the turbulent reacting flow, however, is concerned with prediction of the averaged values of these fluctuating scalars. How these averaged values are related to the instantaneous values depends on the turbulencechemistry interaction model. FLUENT applies the assumedshape probability density function (PDF) approach as its closure model when the nonpremixed model is used. The assumed shape PDF closure model is described in this section.
Description of the Probability Density Function
The Probability Density Function, written as , can be thought of as the fraction of time that the fluid spends in the vicinity of the state . Figure 15.2.3 plots the time trace of mixture fraction at a point in the flow (righthand side) and the probability density function of (lefthand side). The fluctuating value of , plotted on the right side of the figure, spends some fraction of time in the range denoted as . , plotted on the left side of the figure, takes on values such that the area under its curve in the band denoted, , is equal to the fraction of time that spends in this range. Written mathematically,
(15.215) 
where is the time scale and is the amount of time that spends in the band. The shape of the function depends on the nature of the turbulent fluctuations in . In practice, is unknown and is modeled as a mathematical function that approximates the actual PDF shapes that have been observed experimentally.
Derivation of Mean Scalar Values from the Instantaneous Mixture Fraction
The probability density function , describing the temporal fluctuations of in the turbulent flow, can be used to compute averaged values of variables that depend on . Densityweighted mean species mass fractions and temperature can be computed (in adiabatic systems) as
for a singlemixturefraction system. When a secondary stream exists, mean values are calculated as
where is the PDF of and is the PDF of . Here, statistical independence of and is assumed, so that .
Similarly, the mean timeaveraged fluid density, , can be computed as
for a singlemixturefraction system, and
when a secondary stream exists. or is the instantaneous density obtained using the instantaneous species mass fractions and temperature in the ideal gas law equation.
Using Equations 15.216 and 15.218 (or Equations 15.217 and 15.219), it remains only to specify the shape of the function (or and ) in order to determine the local mean fluid state at all points in the flow field.
The AssumedShape PDF
The shape of the assumed PDF, , is described in FLUENT by one of two mathematical functions:
The doubledelta function is the most easily computed, while the function most closely represents experimentally observed PDFs. The shape produced by this function depends solely on the mean mixture fraction, , and its variance, . A detailed description of each function follows.
The Double Delta Function PDF
The double delta function is given by
with suitable bounding near = 1 and = 0. One example of the double delta function is illustrated in Figure 15.2.4. As noted above, the double delta function PDF is very easy to compute but is invariably less accurate than the alternate function PDF because it assumes that only two states occur in the turbulent flow. For this reason, it is available only for twomixturefraction simulations where the savings in computational cost is significant.
The Function PDF
The function PDF shape is given by the following function of and :
where
(15.222) 
and
(15.223) 
Importantly, the PDF shape is a function of only its first two moments, namely the mean mixture fraction, , and the mixture fraction variance, . Thus, given FLUENT's prediction of and at each point in the flow field (Equations 15.24 and 15.25), the assumed PDF shape can be computed and used as the weighting function to determine the mean values of species mass fractions, density, and temperature using, Equations 15.216 and 15.218 (or, for a system with a secondary stream, Equations 15.217 and 15.219).
This logical dependence is depicted visually in Figure 15.2.5 for a single mixture fraction.