Turbulent Dispersion

The effect of turbulence in redistribution of non-uniformities in phase concentration is modeled by an additional turbulent dispersion force in the phase momentum equations. This term arises naturally when Reynolds averaging is applied to the instantaneous drag force.

For a simple derivation and further references, see Contribution of Drag to Turbulent Dispersion. With suitable closures, the term has the following form:

Figure 1. EQUATION_DISPLAY

\begin{matrix} {F_{i j}}^{T D} = A_{D} v_{T D} \\ v_{T D} = D_{T D} \cdot {\frac{\nabla α_{d}}{α_{d}} - \frac{\nabla α_{c}}{α_{c}}} \end{matrix}

(2022)

where:

${F_{i j}}^{T D}$ is the force per volume applied to the continuous phase $i$ momentum equation due to dispersed phase $j$ .
$v_{T D}$ is a relative drift velocity due to the use of volume-fraction weighted definitions of phase velocity.
Alternatively, $v_{T D}$ can be written:

Figure 2. EQUATION_DISPLAY

v_{T D} = D_{T D} \cdot {\nabla ln (α_{d}) - \nabla ln (α_{c})}

(2023)

The logarithmic form is easier to discretize on a collocated grid, so this form is used by default. This form is appropriate for simulating flows where separation or boiling/condensation processes can produce volume fractions differing by many orders of magnitude in adjacent cells.

The tensor diffusivity coefficient $D_{T D}$ is usually approximated isotropically from the continuous phase turbulent diffusivity:

Figure 3. EQUATION_DISPLAY

D_{T D} = C_{0} \frac{{ν_{c}}^{t}}{σ_{α}} I

(2024)

where:

${ν_{c}}^{t}$ is the continuous phase turbulent kinematic viscosity.
$σ_{α}$ is the turbulent Prandtl number for volume fraction. This is discussed in more detail in the following section.
The constant $C_{0}$ is not part of the model derivation, but has a default value 1 and you can vary it to test the sensitivity of the solution to this term.
$I$ is the identity matrix.

When a Reynolds stress tensor $R_{c}$ is modeled for the continuous phase, then an anisotropic approximation is also available:

Figure 4. EQUATION_DISPLAY

D_{T D} = C_{0} \frac{3}{2} \frac{C_{μ}}{σ_{α}} \frac{k_{c}}{ε_{c}} R_{c}

(2025)

Instead of choosing a model for the tensor diffusivity,

D_{T D}

, the essential difference between alternative dispersion coefficient models can be encapsulated as an effective Turbulent Prandtl Number,

σ_{α}

. This number signifies the ratio of momentum diffusivity over the volume fraction diffusivity due to continuous phase velocity fluctuations.

Note

For turbulent diffusion of energy (temperature) this ratio is known as Turbulent Prandtl number while for turbulent diffusion of concentration this ratio is known as Turbulent Schmidt number. Since there is no convention for general variables, the preference here is to name the quantity as Turbulent Prandtl number.

There are four modeling options for the Turbulent Prandtl number:

Constant, with default value 1 as for a passive scalar
User-specified expression
Tchen turbulent dispersion coefficient, recommended for small particles or bubbles
Inertial turbulent dispersion coefficient, recommended for larger particles or bubbles

For a multiple flow regime phase interaction, the turbulent dispersion force is calculated as:

Figure 5. EQUATION_DISPLAY

F^{T D} = (W_{f r} \frac{A_{D, f r} v_{p}^{t}}{σ_{p}} + W_{i r} \frac{A_{D, i r} v_{i r}^{t}}{σ_{i r}} + W_{s r} \frac{A_{D, s r} v_{s}^{t}}{σ_{s}}) (\frac{\nabla α_{s}}{α_{s}} - \frac{\nabla α_{p}}{α_{p}})

(2026)

where:

$\begin{matrix} v_{p}^{t}, & v_{i r}^{t}, & v_{s}^{t} \end{matrix}$ are the primary phase, intermediate regime, and secondary phase turbulent kinematic viscosity, respectively.
$\begin{matrix} σ_{p}, & σ_{i r}, & {\begin{matrix} σ \end{matrix}}_{s} \end{matrix}$ are the primary phase, intermediate regime, and secondary phase turbulent Prandtl number, respectively.

Tchen Turbulent Dispersion Coefficient

Thai-Van et al. [555] describe a model for effective turbulent diffusivity of particles. This is based on Simonin's extension of Tchen analysis of particle behavior in steady homogeneous turbulence.

Expressed as an effective Turbulent Prandtl number for the Turbulent Dispersion Force term, this model becomes:

Figure 6. EQUATION_DISPLAY

σ_{α} = σ_{0} \sqrt{1 + C_{β} ξ^{2}} \frac{1 + η}{b + η}

(2027)

where:

$σ_{0}$ is the unmodified Turbulent Prandtl number, assumed to be unity to represent basic passive diffusivity.
$C_{β}$ is a calibration coefficient for the crossing-trajectories effect.
$ξ$ is the particle slip velocity scaled by turbulent fluctuation velocity.
$η$ is the particle-eddy interaction time, scaled by particle relaxation time.
$b$ is the ratio of the coefficients of the continuous/disperse acceleration terms in the equation of motion for a particle.

For further details, see Tchen Closures.

The general effect of high crossing speed ( $ξ » 1$ ) in this model is to reduce correlation and dispersion through an increased effective Turbulent Prandtl number.

For bubbly flows (phase densities $ρ_{d} « ρ_{c}$ ) with suitable interaction times, the effect of virtual mass in this model could be to increase dispersion by up to a factor of 1.5.

Inertial Turbulent Dispersion Coefficient

If the particles are larger than the inner turbulent scale, they hold a substantial share of the fluctuating kinetic energy and particle-particle interaction is an important energy redistribution mechanism. Gas-particle flows with high particle concentrations are best modeled as granular flows, but for more dilute particle concentrations ( $α_{d}$ between about 1% to 10%), it is often possible to obtain useful results with an inertial correction to the turbulent dispersion.

The effective Turbulent Prandtl number for inertial dispersion of heavy particles is modeled as:

Figure 7. EQUATION_DISPLAY

σ_{α} = \frac{σ_{0}}{(1 + \frac{τ_{R}}{τ_{T}}) (1 + \frac{ρ_{c} C_{V M}}{ρ_{d} + ρ_{c} C_{V M}})}

(2028)

where:

$σ_{0}$ is the unmodified Turbulent Prandtl, assumed to be unity to represent basic passive diffusivity.
$τ_{R}$ is the particle relaxation time:

Figure 8. EQUATION_DISPLAY

τ_{R} = \frac{ρ_{d} α_{d}}{{A^{D}}_{c d}} (1 + \frac{ρ_{c}}{ρ_{d}} C_{V M})

(2029)

$τ_{T}$ is an eddy-turnover time:

Figure 9. EQUATION_DISPLAY

τ_{T} = C_{μ} \frac{k_{c}}{ε_{c}}

(2030)

$C_{V M}$ is a virtual mass coefficient for particle response, which is taken from the Virtual Mass Force model if that has been activated for simulation of an accelerating mean flow; otherwise it is assumed to have a constant value of 0.5.

The first factor in the denominator of Eqn. (2028) is accounting for the increase of the longer particle relaxation times ( $τ_{R} » τ_{T}$ ), while the second one originates from the virtual mass force (see Contribution of Virtual Mass to Turbulent Dispersion for details). The general effect of the model is to increase dispersion through a reduced effective Turbulent Prandtl number.