Wall Lubrication

Bubbles rising close to a wall experience a force from the fluid flowing asymmetrically around the bubble. This force, known as the Wall Lubrication Force, prevents the bubbles from touching the wall.

Coefficient Based Wall Lubrication

Antal and others [426] proposed the following model.

The force per unit volume experienced by a dispersed phase at a distance $y_{w}$ , from a wall is:

Figure 1. EQUATION_DISPLAY

F_{i j}^{W L} = {- C}_{W L} (y_{w}) α_{d} ρ_{c} \frac{{| v_{r, ∥} |}^{2}}{d_{p}} n

(2031)

where:

$α_{d}$ is the bubble volume fraction.
$d_{p}$ is the diameter of the bubble.
$C_{W L}$ is a function of inverse length. Its value decreases rapidly with increasing distance, $y_{w}$ .
$n$ is the outward facing unit normal at the nearest point on the wall, so that the force is pointing inwards to prevent the bubbles from contacting the wall.

The velocity scale in this model is based on the slip velocity component that is parallel to the wall:

Figure 2. EQUATION_DISPLAY

v_{r, ∥} = v_{r} - (v_{r} \cdot n) n

(2032)

The Antal model is implemented in Simcenter STAR-CCM+ using two calibration coefficients, $C_{w 1}$ and $C_{w 2}$ :

Figure 3. EQUATION_DISPLAY

C_{W L} (y_{w}) = \max {C_{w 1} + (\frac{C_{w 2}}{y_{w}}) d_{p}, 0}

(2033)

The coefficients are chosen such that $C_{w 1}$ becomes zero a few bubble diameters away from the wall. That is, when:

Figure 4. EQUATION_DISPLAY

\frac{y_{w}}{d_{p}} > \frac{{- C}_{w 2}}{C_{w 1}}

(2034 2035)

In practice, this means that the force is zero at five bubble diameters. The default value for $C_{w 1}$ is -0.01 and the default value for $C_{w 2}$ is 0.05.

Simcenter STAR-CCM+ also allows you to implement alternative models for the wall lubrication force by defining field functions for $C_{W L}$ .

Lubchenko Wall Lubrication

Lubchenko and others [511] used an approach in which the void fraction peak is resolved in the near-wall region. As turbulent force is the only interfacial force acting in the lateral direction perpendicular to the wall, the wall lubrication force is based on the regularization of turbulent dispersion.

In order to maintain a flat profile for the gas volume fraction, the wall lubrication force, $F_{W L T D}$ , must be in equilibrium with the turbulent dispersion force, $F_{T D}$ :

Figure 5. EQUATION_DISPLAY

F_{T D} + F_{W L T D} = 0

(2034 2035)

Considering the expression for turbulent dispersion force by Burns [437] yields the following expression for the wall lubrication force:

Figure 6. EQUATION_DISPLAY

F_{W L T D} = {\begin{matrix} \frac{3}{4} \frac{C_{D}}{d_{p}} v_{r} \frac{μ_{t}}{σ_{T D}} (1 + \frac{α}{1 - α}) \cdot α \cdot \frac{1}{y_{w}} \frac{d_{p} - 2 y_{w}}{d_{p} - y_{w}} n & y_{w} < \frac{d_{p}}{2} \\ 0 & y_{w} > \frac{d_{p}}{2} \end{matrix}

(2036)

where:

$C_{D}$ is the drag coefficient
$d_{p}$ is the diameter of the bubble
$v_{r}$ is the relative velocity between phases
$μ_{t}$ is the turbulent viscosity
$α$ is the bubble volume fraction
$y_{w}$ is the normal wall distance
$n$ is the outward facing unit normal at the nearest point on the wall, so that the force is pointing inwards to prevent the bubbles from contacting the wall.

Note that Eqn. (2036) assumes a quadratic dependence of volume fraction on wall-normal distance and, unlike other models, does not require the use of tuneable coefficients. This model can also be coupled to any turbulent dispersion model.