Beta Probability Density Function (PDF)

The flamelet models provide the instantaneous value of any variable of interest in the reaction state-space, such as species mass fractions and temperature, if the instantaneous mixture fraction is provided. In addition, averaged values of reaction state-space variables are needed that account for turbulent fluctuations at any spatial location.

Four steps are necessary for obtaining these averaged values:

Perform a time average, as is done with the momentum equations. This yields the time-averaged value (the first moment) of $Z$ , $Z_{m e a n}$ , from the following equation:
Figure 1. EQUATION_DISPLAY

$\frac{\partial}{\partial t} (ρ χ Z_{m e a n}) + \nabla \cdot [ρ v Z_{m e a n} - (ρ D_{Z} + \frac{μ_{t}}{σ_{t, Z_{m e a n}}}) \nabla Z_{m e a n}] = 0$

(3498)

where $D_{Z}$ is the laminar mixture fraction diffusion coefficient, $μ_{t}$ is the turbulent viscosity, and $σ_{t, Z_{m e a n}}$ is the turbulent Schmidt number for the mixture fraction.
Derive an equation for the second moment $Z_{var}$ , also known as the variance, of the mixture fraction:
Figure 2. EQUATION_DISPLAY

$\frac{\partial}{\partial t} (ρ χ Z_{var}) + \nabla \cdot (ρ v Z_{var} - \frac{μ_{t}}{σ_{{t, Z}_{var}}} \nabla Z_{var}) = \frac{2 μ_{t}}{σ_{{t, Z}_{var}}} {(\nabla Z_{m e a n})}^{2} - C_{d} ρ \frac{ε}{k} Z_{var}$

(3499)

where:
Figure 3. EQUATION_DISPLAY

$Z_{var} = \bar{{(Z - Z_{m e a n})}^{2}}$

(3500)

$σ_{t, Z_{var}}$ is the turbulent Schmidt number for the variance and $C_{d}$ is a dissipation constant.
Assume a shape for the PDF that depends only on these two moments. This presumed PDF of the mixture fraction is a beta function of the form:
Figure 4. EQUATION_DISPLAY

$P (Z) = \frac{Z^{a - 1} {(1 - Z)}^{b - 1}}{\int_{0}^{1} Z^{a - 1} {(1 - Z)}^{b - 1} d Z}$

(3501)

where a and b are related to the mean and variance by:
Figure 5. EQUATION_DISPLAY

$a = \frac{Z_{m e a n}}{Z_{var}} [Z_{m e a n} (1 - Z_{m e a n}) - Z_{var}]$

(3502)

and:
Figure 6. EQUATION_DISPLAY

$b = \frac{(1 - Z_{m e a n})}{\overline{Z}} a$

(3503)
Obtain the averaged value of any quantity from the PDF as:
Figure 7. EQUATION_DISPLAY

$\tilde{ϕ} (Z) = \int_{}^{} ϕ (Z) P (Z) d Z$

(3504)

The mapping of the mean value of any scalar to the mean value of the mixture fraction and its variance is accomplished through a pre-computed table. A simple look-up and interpolation is all that is then necessary when these mean values are required.

Although a transport equation for variance of mixture fraction is usually solved in RANS (Eqn. (3499)), it is common practice to use an algebraic relationship in LES combustion. When assuming that production equals dissipation in the mixture fraction variance equation, the following relationship can be used:

Figure 8. EQUATION_DISPLAY

Z_{var} = C_{v} Δ^{2} {(\nabla Z_{m e a n})}^{2}

(3505)

where $C_{v}$ is a user–adjustable constant with a default value of 1/12. $Δ$ is the subgrid scale.

In order to fulfill the assumption that the production equals dissipation, the dissipation constant $C_{d}$ (Eqn. (3499)) can be computed internally based on the following equation:

Figure 9. EQUATION_DISPLAY

C_{d} = \frac{2 C_{s}^{2} C_{l}}{C_{v}}

(3506)

Here $C_{l}$ is the constant when computing the turbulent time scale (Eqn. (1495) for Smagorinsky Subgrid Scale Turbulence or Wale Subgrid Scale Turbulence). $C_{s}$ is the constant for computing the length scale ( $C_{s}$ in Eqn. (1388) for Smagorinsky Subgrid Scale Turbulence or $C_{w}$ in Eqn. (1398) for Wale Subgrid Scale Turbulence).

The beta PDF has been extended for multiple streams with the Flamelet Generated Manifold (FGM) model and the Chemical Equilibrium model.

Extension of the Beta PDF to Multiple Streams

To obtain the averaged value of any quantity from the beta PDF for multiple streams, the following equation is used:

Figure 10. EQUATION_DISPLAY

ϕ = \int_{0}^{1} P (Z_{1}) \int_{0}^{(1 - Z_{i})} P (Z_{2}) \dots \int_{0}^{(1 - Σ_{i = 1}^{n - 2} Z_{i})} P (Z_{n - 1}) ϕ (Z_{1}, Z_{2}, \dots, Z_{n - 1}) d Z_{1} d Z_{2} \dots d Z_{n - 1}

(3507)

The complexity of computing the multiple integral comes from the limits that are not constant. To simplify this computation, the Scaled Mixture Fraction space is introduced:

Figure 11. EQUATION_DISPLAY

\begin{matrix} s_{1} = Z_{1} \\ s_{2} = \frac{Z_{2}}{1 - Z_{1}} \\ \begin{matrix} . \\ : \\ s_{n - 1} = \frac{Z_{n - 1}}{1 - Σ_{i = 1}^{n - 2} Z_{i}} \end{matrix} \end{matrix}

(3508)

Now Eqn. (3507) can be reformulated as:

Figure 12. EQUATION_DISPLAY

ϕ = \int_{0}^{1} P (s_{1}) \int_{0}^{1} P (s_{2}) \dots \int_{0}^{1} P (s_{n - 1}) ϕ (s_{1}, s_{2}, \dots, s_{n - 1}) d s_{1} d s_{2} \dots d s_{n - 1}

(3509)

Four steps that are described in Eqn. (3498) to Eqn. (3503) can be reformulated as follows to encapsulate the multiple stream beta PDF approach:

Since the scaled mixture fraction is not a conserved value, it is still necessary to perform a time average of the transport equations for the original mixture fraction space for each mixture fraction $i$ :
Figure 13. EQUATION_DISPLAY

$\frac{\partial}{\partial t} (ρ χ Z_{m e a n, i}) + \nabla \cdot [ρ v Z_{m e a n, i} - (ρ D_{Z} + \frac{μ_{t}}{σ_{t, Z_{m e a n}}}) \nabla Z_{m e a n, i}] = 0$

(3510)

where $D_{Z}$ is the laminar mixture fraction diffusion coefficient, $μ_{t}$ is the turbulent viscosity, and $σ_{t, Z_{m e a n}}$ is the turbulent Schmidt number for the mixture fraction.
Compute the scaled mixture fraction space.
Derive an equation for the second moment of every scaled mixture fraction $s_{i}, Z_{var, s i}$ , also known as scaled mixture fraction variance:
Figure 14. EQUATION_DISPLAY

$\frac{\partial}{\partial t} (ρ χ Z_{var, i}) + \nabla \cdot (ρ v Z_{var, i} - \frac{μ_{t}}{σ_{t, Z_{m e a n}}} \nabla Z_{var, i}) = \frac{2 μ_{t}}{σ_{t, Z_{m e a n}}} {(\nabla Z_{m e a n, i})}^{2} - C_{d} ρ \frac{ε}{k} Z_{var, i}$

(3511)

where:
Figure 15. EQUATION_DISPLAY

$Z_{var, i} = \bar{{(Z_{i} - Z_{m e a n, i})}^{2}}$

(3512)

$σ_{t, Z_{m e a n}}$ is the turbulent Schmidt number for the mixture fraction and $C_{d}$ is a dissipation constant.
Assume a shape for the PDF that depends only on these two moments. This presumed PDF of the scaled mixture fraction is a beta function of the form:
Figure 16. EQUATION_DISPLAY

$P (s_{i}) = \frac{s_{1}^{a_{1} - 1} {(1 - s_{i})}^{b_{1} - 1}}{\int_{0}^{1} s_{1}^{a_{1} - 1} {(1 - s_{i})}^{b_{1} - 1} d s_{i}}$

(3513)

where $a$ and $b$ are related to the mean and variance by:
Figure 17. EQUATION_DISPLAY

$a_{1} = \frac{s_{i, m e a n}}{Z_{var, s i}} [s_{i, m e a n} (1 - s_{i, m e a n}) - Z_{var, s i}]$

(3514)

and:
Figure 18. EQUATION_DISPLAY

$b_{i} = \frac{(1 - s_{i, m e a n}) a_{i}}{s_{i, m e a n}}$

(3515)