Lift

When the continuous phase flow field is non-uniform or swirling, particles experience a lift force perpendicular to the relative velocity.

This force, $F_{i j}^{L}$ , is derived by Auton et al. [430] as:

Figure 1. EQUATION_DISPLAY

F_{i j}^{L} = C_{L, effective} α_{d} ρ_{c} [v_{r} \times (\nabla \times v_{c})]

(2015)

where $C_{L, e f f e c t i v e}$ is calculated from the Lift Coefficient ( $C_{L}$ ) and the Lift-Correction ( $f_{l}$ ) as:

$f_{l} C_{L}$ for Spherical Particle Interaction Area Density model

$f_{l} α_{c} C_{L}$ for Symmetric Interaction Area Density model

You can set the lift coefficient, $C_{L}$ , but it is set to $C_{L} = 0.25$ by default, following Lance et al. [498]. Other models that take account of bubble size and distortion can be found in the literature. You can implement these models through field functions for the lift coefficient.

For a multiple flow regime phase interaction, the lift force is calculated only for the first dispersed regime and the second dispersed regime. The weight function of the intermediate regime is taken as zero.

The lift force is calculated as:

Figure 2. EQUATION_DISPLAY

F_{i j}^{L} = W_{f r} C_{L, f r} α_{s} ρ_{p} v_{r} \times (\nabla\times v_{p}) + W_{s r} C_{L, s r} α_{p} ρ_{s} v_{r} \times (\nabla\times v_{s})

(2016)

where:

$C_{L, f r}$ is the lift coefficient considering the secondary phase dispersed in the primary phase.
$C_{L, s r}$ is the lift coefficient considering the primary phase dispersed in the secondary phase.

Tomiyama Lift Coefficient

The Tomiyama lift coefficient has been evaluated based on experimental trajectories of single bubbles in a high-viscosity system. However, it is worth noting that Tomiyama et al. [558] found that the Tomiyama lift coefficient proposed also yields similar values to experimental data for a small bubble in a low viscosity, air-water system.

The lift coefficient that is proposed by Tomiyama et al. [558] is:

Figure 3. EQUATION_DISPLAY

C_{L} = {\begin{matrix} \min [0.288 \tanh (0.121 Re), f_{T}] & {Eo}_{d} < 4 \\ f_{T} & 4 \leq {Eo}_{d} \leq 10 \\ - 0.27 & 10 < {Eo}_{d} \end{matrix}

(2017)

where:

$f_{T} = 0.00105 {Eo}_{d}^{3} - 0.0159 {Eo}_{d}^{2} - 0.0204 {Eo}_{d} + 0.474$
$Re$ is the Reynolds number (see Eqn. (1971)).
${Eo}_{d}$ is the modified Eotvos number that is based on the maximum horizontal dimension of a bubble.

The empirical relations between ${Eo}_{d}$ and $Eo$ is defined as:

Figure 4. EQUATION_DISPLAY

{Eo}_{d} = Eo \times E^{- 2 / 3}

(2018)

where $E$ is the empirical correlation for the bubble aspect ratio of spheroidal bubbles in a fully contaminated system and is:

Figure 5. EQUATION_DISPLAY

E = \frac{1}{1 + 0.163 {Eo}^{0.757}}

(2019)

Note	The Tomiyama [557] drag coefficient was used by Tomiyama et al. [558] while evaluating their lift coefficients. Therefore, use the Tomiyama lift coefficient with the Tomiyama drag coefficient.

Sugrue Lift Coefficient

The Sugrue lift coefficient [550] takes into account drift phenomena, bubble interaction probability, and the maximum packing factor for dispersed bubbly flow. This model is expressed as the product of a Wobble function $f (W_{o})$ and a void fraction function $f (α)$ :

Figure 6. EQUATION_DISPLAY

C_{L} = f (W_{o}) f (α)

(2020)

where:

$f (W_{o}) = \min [0.03, 5.0404 - 5.0781 {W_{o}}^{0.0108}]$
The Wobble number is a simple dimensionless quantity that describes the unsteady behavior of bubbles in turbulent flow conditions. It is defined as $W_{o} = E_{o} \frac{k}{v_{r}^{2}}$ , where $k$ is the turbulent kinetic energy.
$f (α) = \max (1.0155 - 0.0154 e^{8.0506 α}, 0)$

Lift Correction

A strong correlation between the lift correction and drag correction has been observed and, in light of limited literature available for lift correction, a perfect correlation of lift correction with drag correction is approximated.

The following options are available:

Drag Correlated

This method uses the Drag Coefficient Correction as an approximation for the Lift Coefficient Correction.

This approximation is useful for cases when swarming is modeled.

Podowski Near Wall Adjustment

The Podowski near wall adjustment [544] is a simplified correction that neglects wall lubrication. This method brings the lift coefficient to zero near the wall, and sets it to the nominal lift coefficient in the bulk of the flow. Only the effects of turbulent dispersion are modeled, which leads to a flat volume fraction profile in the near-wall region. In order to recover the gas fraction peak near the wall, it is recommended to activate the Wall Lubrication model.

The lift coefficient $C_{L}$ is adjusted as:

Figure 7. EQUATION_DISPLAY

C_{L} = {\begin{matrix} 0 & \frac{y}{D_{b}} < 0.5 \\ C_{L 0} (3 {(\frac{2 y}{D_{b}} - 1)}^{2} - 2 {(\frac{2 y}{D_{b}} - 1)}^{3}) & 0.5 \leq \frac{y}{D_{b}} \leq 1.0 \\ C_{L 0} & 1.0 < \frac{y}{D_{b}} \end{matrix}

(2021)

where:

$C_{L 0}$ is the nominal lift coefficient.
$D_{b}$ is the average bubble diameter.
$y$ is the lateral distance from the wall.

Field Function