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15.2.2 Modeling of Turbulence-Chemistry Interaction

Equations  15.2-11 through 15.2-14 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 turbulence-chemistry interaction model. FLUENT applies the assumed-shape probability density function (PDF) approach as its closure model when the non-premixed 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 $p(f)$, can be thought of as the fraction of time that the fluid spends in the vicinity of the state $f$. Figure  15.2.3 plots the time trace of mixture fraction at a point in the flow (right-hand side) and the probability density function of $f$ (left-hand side). The fluctuating value of $f$, plotted on the right side of the figure, spends some fraction of time in the range denoted as $\Delta f$. $p(f)$, plotted on the left side of the figure, takes on values such that the area under its curve in the band denoted, $\Delta f$, is equal to the fraction of time that $f$ spends in this range. Written mathematically,

 p(f)\ \Delta f = \lim_{T \rightarrow \infty} \frac{1}{T} \sum_{i} \tau_i (15.2-15)

where $T$ is the time scale and $\tau_i$ is the amount of time that $f$ spends in the $\Delta f$ band. The shape of the function $p(f)$ depends on the nature of the turbulent fluctuations in $f$. In practice, $p(f)$ is unknown and is modeled as a mathematical function that approximates the actual PDF shapes that have been observed experimentally.

Figure 15.2.3: Graphical Description of the Probability Density Function, $p(f)$

Derivation of Mean Scalar Values from the Instantaneous Mixture Fraction

The probability density function $p(f)$, describing the temporal fluctuations of $f$ in the turbulent flow, can be used to compute averaged values of variables that depend on $f$. Density-weighted mean species mass fractions and temperature can be computed (in adiabatic systems) as

 \overline{\phi}_i = \int_{0}^1 p(f) \phi_i (f) d f (15.2-16)

for a single-mixture-fraction system. When a secondary stream exists, mean values are calculated as

 \overline{\phi}_i = \int_{0}^1 \int_{0}^1 p_1(f_{\rm fuel}) ... ...) \phi_i (f_{\rm fuel},p_{\rm sec}) df_{\rm fuel} dp_{\rm sec} (15.2-17)

where $p_1$ is the PDF of $f_{\rm fuel}$ and $p_2$ is the PDF of $p_{\rm sec}$. Here, statistical independence of $f_{\rm fuel}$ and $p_{\rm sec}$ is assumed, so that $p(f_{\rm fuel},p_{\rm sec}) = p_1(f_{\rm fuel})p_2(p_{\rm sec})$.

Similarly, the mean time-averaged fluid density, ${\overline{\rho}}$, can be computed as

 \frac{1}{\overline{\rho}} = \int_0^1 \frac{p(f)}{\rho(f)} df (15.2-18)

for a single-mixture-fraction system, and

 \frac{1}{\overline{\rho}} = \int_{0}^1 \int_0^1 \frac{p_1(f_... ... {\rho (f_{\rm fuel},p_{\rm sec}) } df_{\rm fuel} dp_{\rm sec} (15.2-19)

when a secondary stream exists. $\rho (f)$ or $\rho (f_{\rm fuel},p_{\rm sec})$ is the instantaneous density obtained using the instantaneous species mass fractions and temperature in the ideal gas law equation.

Using Equations  15.2-16 and  15.2-18 (or Equations  15.2-17 and  15.2-19), it remains only to specify the shape of the function $p(f)$ (or $p_1 (f_{\rm fuel})$ and $p_2(p_{\rm sec})$) in order to determine the local mean fluid state at all points in the flow field.

The Assumed-Shape PDF

The shape of the assumed PDF, $p(f)$, is described in FLUENT by one of two mathematical functions:

The double-delta function is the most easily computed, while the $\beta$-function most closely represents experimentally observed PDFs. The shape produced by this function depends solely on the mean mixture fraction, $\overline{f}$, and its variance, $\overline{f^{'2}}$. A detailed description of each function follows.

The Double Delta Function PDF

The double delta function is given by

 p(f) = \left \{ \begin{array}{lll} 0.5,& \qquad& f = \overl... ...'2}}} \\ 0, & \qquad & \mbox{elsewhere} \end{array} \right. (15.2-20)

with suitable bounding near $f$ = 1 and $f$ = 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 $\beta$-function PDF because it assumes that only two states occur in the turbulent flow. For this reason, it is available only for two-mixture-fraction simulations where the savings in computational cost is significant.

Figure 15.2.4: Example of the Double Delta Function PDF Shape

The $\beta$-Function PDF

The $\beta$-function PDF shape is given by the following function of $\overline{f}$ and $\overline{f^{'2}}$:

 p(f) = \frac{ f^{\alpha-1} (1 - f)^{\beta-1} } { \int f^{\alpha-1} (1 - f)^{\beta-1} df } (15.2-21)


 \alpha = \overline{f} \left[ \frac{ \overline{f}(1- \overline{f}) }{ \overline{f^{'2}} } - 1 \right] (15.2-22)


 \beta = (1 - \overline{f}) \left[ \frac{ \overline{f}(1- \overline{f}) } { \overline{f^{'2}} } - 1 \right] (15.2-23)

Importantly, the PDF shape $p(f)$ is a function of only its first two moments, namely the mean mixture fraction, $\overline{f}$, and the mixture fraction variance, $\overline{f^{'2}}$. Thus, given FLUENT's prediction of $\overline{f}$ and $\overline{f^{'2}}$ at each point in the flow field (Equations  15.2-4 and  15.2-5), 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  15.2-16 and 15.2-18 (or, for a system with a secondary stream, Equations  15.2-17 and 15.2-19).

This logical dependence is depicted visually in Figure  15.2.5 for a single mixture fraction.

Figure 15.2.5: Logical Dependence of Averaged Scalars $\overline{\phi_i}$ on $\overline{f}$, $\overline{f^{'2}}$, and the Chemistry Model (Adiabatic, Single-Mixture-Fraction Systems)

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