Mixture Fraction

The mixture fraction is the elemental mass fraction that originated from the fuel stream. Mixture fraction is a conserved scalar that is 1 for the fuel stream and 0 for the oxidizer stream.

For the flamelet methods, Simcenter STAR-CCM+ solves a transport equation for mixture fraction and uses it to obtain species mass fractions.

Mixture Fraction

The mixture fraction

Z

is calculated using:

Figure 1. EQUATION_DISPLAY

Z = \frac{m_{f}}{m_{f} + m_{o x}}

(3489)

where

m_{f}

is the total mass of all elements that originate from the fuel stream at any spatial location and

m_{o x}

is the total mass of all elements that originate from the oxidizer stream.

For simulations with a third stream that participates in a reaction

m_{\sec}

the mixture fraction of the fuel stream is defined as:

Figure 2. EQUATION_DISPLAY

Z_{0} = \frac{m_{f}}{m_{f} + m_{o x} + m_{\sec}}

(3490)

and the mixture fraction of the secondary stream is defined as:

Figure 3. EQUATION_DISPLAY

Z_{1} = \frac{m_{\sec}}{m_{f} + m_{o x} + m_{\sec}}

(3491)

If an inert (unreactive) stream is also included, the mixture fraction of the inert stream

Z_{I}

is defined as:

Figure 4. EQUATION_DISPLAY

Z_{I} = \frac{m_{I}}{m_{f} + m_{o x} + m_{I} + m_{\sec}}

(3492)

Since the sum of all mixture fraction streams must be equal to 1.0, the elemental mass fraction that originates from the oxidizer stream

Y_{o x}

is determined by:

Figure 5. EQUATION_DISPLAY

Y_{o x} = 1 - (Z_{0} + Z_{1} + Z_{I})

(3493)

Z

is transported by convection and diffusion. Without phase change, there is no production since atomic elements are conserved in chemical reactions. For steady-state flows, there is no accumulation, so the transient term is zero. The Favre averaged mixture fraction equation is:

Figure 6. EQUATION_DISPLAY

\frac{\partial}{\partial t} (ρ Z_{m e a n}) + \nabla \cdot [ρ V - (ρ D_{f} + \frac{μ_{t}}{σ_{t, Z}}) \nabla Z_{m e a n}] = {\dot{ω}}_{s p r a y}

(3494)

Note	Overbars (for RANS averaging) and overtildes (for Favre averaging) are excluded for clarity.

A PDF shape as a function of mixture fraction mean

Z_{m e a n}

and mixture fraction variance

Z_{var}

is assumed to account for turbulent fluctuations in the mixture fraction (see Beta Probability Density Function (PDF)). The mixture fraction variance

Z_{var}

is either calculated from a transport equation or an algebraic expression.

Mixture Fraction Variance: The equation that is derived for the mixture fraction variance is:; Figure 7. EQUATION_DISPLAY

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

(3495)
where the mixture fraction variance $Z_{var}$ of each scaled mixture fraction is calculated as:
Figure 8. EQUATION_DISPLAY

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

(3496)
$σ_{t, Z_{var}}$ is the turbulent Schmidt number for the mixture fraction variance and $C_{d}$ is a scalar dissipation constant.

Inert Stream

The presence of an inert stream impacts the mixture fraction properties since the inert stream dilutes the overall mixture. When transport equations are solved for the mixture fraction in the presence of an inert stream, the Favre averaged inert fraction

Z_{I}

and the active fuel stream fraction

Z_{f}

are considered for each inlet. This transport equation is given by:

Figure 9. EQUATION_DISPLAY

\frac{\partial}{\partial t} (ρ Z_{I}) + \nabla \cdot [ρ V - (ρ D_{f} + \frac{μ_{t}}{σ_{t, Z_{var}}}) \nabla Z_{I}] = {\dot{ω}}_{s p r a y}

(3497)

For further information, see Flamelet Tables: Inert Streams.