Mixture Turbulence

Mixture Quantities

The mixture density $ρ_{m}$ is defined as a linear interpolation of the quantities of all the other $n$ phases weighted by the volume fraction $α_{i}$ :

Figure 1. EQUATION_DISPLAY

ρ_{m} = \sum_{i = 1}^{n} α_{i} ρ_{i}

(2445)

The mixture dynamic viscosity is also a linear interpolation:

Figure 2. EQUATION_DISPLAY

μ_{m} = \sum_{i = 1}^{n} α_{i} μ_{i}

(2446)

$k_{m}$ and $ε_{m}$ are calculated with the solution of the transport equations.

Turbulence kinetic energy is the energy per mass of the velocity fluctuations, so it is assumed that the mixture $k$ and phase $k$ are equal:

Figure 3. EQUATION_DISPLAY

k_{m} = k_{i}

(2447)

Likewise, $ε_{m}$ describes a dissipation rate density and is assumed equal for the mixture and phase:

Figure 4. EQUATION_DISPLAY

ε_{m} = ε_{i}

(2448)

The average mixture velocity is defined as:

Figure 5. EQUATION_DISPLAY

\bar{v_{m}} = \frac{\sum_{i = 1}^{n} α_{i} ρ_{i} v_{i}}{\sum_{i = 1}^{n} α_{i} ρ_{i}}

(2449)

Calculation of the mixture turbulent eddy viscosity assumes that the kinematic viscosities of the phases are equal:

Figure 6. EQUATION_DISPLAY

ν_{m}^{t} = ν_{i}^{t}

(2450)

Hence:

Figure 7. EQUATION_DISPLAY

μ_{m}^{t} = \frac{ρ_{m} μ_{i}^{t}}{ρ_{i}}

(2451)

And the phase turbulent viscosity is:

Figure 8. EQUATION_DISPLAY

μ_{i}^{t} = \frac{ρ_{i} μ_{m}^{t}}{ρ_{m}}

(2452)

K-Epsilon Mixture Turbulence

The transport equations for the mixture turbulent kinetic energy $k_{m}$ and the mixture turbulent dissipation rate $ε_{m}$ are given by:

Figure 9. EQUATION_DISPLAY

\frac{\partial}{\partial t} (ρ_{m} k_{m}) + \nabla\cdot (ρ_{m} k_{m} \bar{v_{m}}) = \nabla\cdot [(μ_{m} + \frac{μ_{m}^{t}}{σ_{k}}) \nabla k_{m}] + P_{m}^{k} - ρ_{m} (ε_{m} - ε_{0}) + S_{m}^{k}

(2453)

Figure 10. EQUATION_DISPLAY

\frac{\partial}{\partial t} (ρ_{m} ε_{m}) + \nabla\cdot (ρ_{m} ε_{m} \bar{v_{m}}) = \nabla\cdot [(μ_{m} + \frac{μ_{m}^{t}}{σ_{ε}}) \nabla ε_{m}] + [\frac{1}{t_{m}^{e}} C_{ε 1} P_{m}^{ε} - C_{ε 2} f_{2} ρ_{m} (\frac{ε_{m}}{t_{m}^{e}} - \frac{ε_{0}}{t_{m}^{0}}) + S_{m}^{ε}]

(2454)

where:

$t_{m}^{e} = \frac{k_{m}}{ε_{m}}$ is the mixture large eddy time scale
$C_{ε 1}$ and $C_{ε 2}$ are model coefficients
$f_{2}$ is a damping function.

K-Omega Mixture Turbulence Model

The transport equations for the mixture turbulent kinetic energy $k_{m}$ and the mixture specific dissipation $ω_{m}$ are given by:

Figure 11. EQUATION_DISPLAY

\frac{\partial}{\partial t} (ρ_{m} k_{m}) + \nabla\cdot (ρ_{m} k_{m} \bar{v_{m}}) = \nabla\cdot [(μ_{m} + {σ_{k} μ}_{m}^{t}) \nabla k_{m}] + P_{m}^{k} - ρ_{m} β^{*} f_{β^{*}} (ω_{m} k_{m} - ω_{0} k_{0}) + S_{m}^{k}

(2455)

Figure 12. EQUATION_DISPLAY

\frac{\partial}{\partial t} (ρ_{m} ω_{m}) + \nabla\cdot (ρ_{m} ω_{m} \bar{v_{m}}) = \nabla\cdot [(μ_{m} + {σ_{ω} μ}_{m}^{t}) \nabla ω_{m}] + P_{m}^{ω} - ρ_{m} β f_{β} (ω_{m}^{2} - ω_{0}^{2}) + S_{m}^{ω}

(2456)

Reynolds Stress Transport Mixture Turbulence Model

Mixture Dissipation Rate Equation

For the mixture phase $m$ , the dissipation rate equation is similar to the K-Epsilon Mixture Turbulence model:

Figure 13. EQUATION_DISPLAY

\frac{\partial}{\partial t} (ρ_{m} ε_{m}) + \nabla\cdot (ρ_{m} ε_{m} \bar{v_{m}}) = α_{m} \nabla\cdot [(μ_{m} + \frac{μ_{m}^{t}}{σ_{ε}}) \nabla ε_{m}] + \frac{ε_{m}}{k_{m}} [C_{ε 1} (\frac{1}{2} t r (P_{m}^{ε}) + \frac{1}{2} C_{ε 3} t r (G_{m})) - C_{ε 2} ρ_{m} ε_{m}] + \sum_{i \neq j} (m_{i j} ε_{j}^{(i j)} - m_{j i} ε_{i})

(2457)

where

$P_{m}^{ε}$ is the turbulent production of the mixture.
$G_{m}$ is the buoyancy production of the mixture.

See Reynolds Stress Transport Equation.