Response Models

Where there is a clearly-defined dispersed phase, then turbulent response provides a well-defined model for dispersed phase effective turbulent diffusivity and standard wall functions.

The following equations are again presented in terms of the $k - ε$ model. The turbulent stress is modeled by using the eddy-viscosity concept as given by Eqn. (1147) with the single-phase turbulent viscosity given by:

Figure 1. EQUATION_DISPLAY

μ^{t} = ρ C_{μ} \frac{k^{2}}{ε}

(2459)

where:

C_{μ}

is a model coefficient in the

k - ε

model. The modified

k - ε

equations are solved for the continuous phase and the turbulence of the dispersed phase is correlated to that of the continuous phase. The correlation is provided by the response function

C_{t}

Figure 2. EQUATION_DISPLAY

C_{t} = \frac{| v_{d}' |}{| v_{c}' |}

(2460)

where $v_{d}'$ is the velocity fluctuation of the dispersed phase and $v_{c}'$ is the velocity fluctuation of the continuous phase.

If the continuous phase uses the $k - ε$ model, then the dispersed-phase turbulent kinetic energy $k_{d}$ is:

Figure 3. EQUATION_DISPLAY

k_{d} = C_{t}^{2} k_{c}

(2461)

With these definitions, the dispersed phase turbulent eddy viscosity is:

Figure 4. EQUATION_DISPLAY

μ_{d}^{t} = \frac{ρ_{d}}{ρ_{c}} C_{t}^{2} μ_{c}^{t}

(2462)

where:

$ρ_{d}$ is the dispersed phase density
$ρ_{c}$ is the continuous phase density

Issa

The Issa model provides $C_{t}$ for a bubbly flow. It estimates $C_{t}$ to be up to 3 for very small volume fractions of dispersed phase, but rapidly decreasing to unity for dispersed phase volume fractions greater than about 5%.

The Issa turbulence response model is defined as a correlation for the turbulence response coefficient $C_{t}$ with a volume fraction correction [537]:

Figure 5. EQUATION_DISPLAY

C_{t} (α_{d}) = 1 + {({C_{t}}^{*} - 1)}^{- f (α_{d})}

(2463)

where:

Figure 6. EQUATION_DISPLAY

f (α_{d}) = 180 α_{d} - 4.71 \times 10^{3} α_{d}^{2} + 4.26 \times 10^{4} α_{d}^{3}

(2464)

${C_{t}}^{*}$ is computed from the Issa model [567]:

Figure 7. EQUATION_DISPLAY

{C_{t}}^{*} = \frac{3 + β}{1 + β + 2 (ρ_{d} / ρ_{c})}

(2465)

$β$ is defined as:

Figure 8. EQUATION_DISPLAY

β = \frac{2 A_{i j}^{D} l_{e}^{2}}{α_{d} μ_{c} R e_{t}}

(2466)

and:

Figure 9. EQUATION_DISPLAY

l_{e} = C_{μ} \frac{k_{c}^{3 / 2}}{ε_{c}}

(2467)

Figure 10. EQUATION_DISPLAY

R e_{t} = \frac{| v_{c}' | l_{e}}{v_{c}}

(2468)

Figure 11. EQUATION_DISPLAY

| v_{c}' | = \sqrt{\frac{2}{3} k_{c}}

(2469)

Tchen

The Tchen model provides a dispersed phase turbulent diffusivity for gas flows laden with heavy particles, as specified by Thai-Van et al. [555]. However, this reference remarks that this model can also be sufficient for bubbly flows because of the weak influence of the eddy diffusivity of the bubbly phase.

From Thai-Van et al. [555], the turbulence diffusivity for turbulent gas flows laden with heavy particles can be modeled as:

Figure 12. EQUATION_DISPLAY

{ν_{d}}^{t} = τ_{I} \frac{1}{3} q_{c d} + \frac{1}{2} τ_{R} \frac{2}{3} {q_{d}}^{2}

(2470)

where:

$q_{c d}$ is the dispersed-phase average of the product of dispersed and continuous phase velocity fluctuations
$q_{d}$ is half the dispersed-phase average of the product of dispersed and dispersed phase velocity fluctuations

For more details on the closure of these correlations, see Tchen Closures.