For liquid-gas bubbly flows using the Eulerian model Ansys Fluent can include the effect of wall lubrication forces on the secondary phases (bubbles). The wall lubrication force tends to push the secondary phases away from walls. For example, in a bubbly upward flow in a vertical pipe this force results in the dispersed phase concentrating in a region near, but not immediately adjacent to, the wall.
The wall lubrication force acting on a secondary phase, 
, in a primary phase, 
, has the general form:
 | (14–319) |
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where | |
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The wall lubrication models in Ansys Fluent differ in how they compute the wall lubrication
coefficient, 
in Equation 14–319.
Ansys Fluent offers the following models:
You can also specify a user-defined wall lubrication coefficient with a User Defined
Function using the DEFINE_EXCHANGE_PROPERTY macro. For details on using a UDF to define the wall
lubrication coefficient, see DEFINE_EXCHANGE_PROPERTY in the Fluent Customization Manual.
The model proposed by Antal et al. [20]
computes
the coefficient, 
, as:
 | (14–320) |
where 
and 
are non-dimensional coefficients, 
is the bubble/particle diameter, and 
is the distance to the nearest wall.
Note that 
is nonzero only within a thin layer adjacent to
the wall that satisfies:
 | (14–321) |
corresponding to 
with the default values of 
and 
. As a result, the Antal model will only be active on a sufficiently fine mesh
and grid independence can only be achieved with very fine meshes.
The Tomiyama model [656]
modifies the
wall lubrication force formulation of Antal [20]
based on the results of experiments with flow of air bubbles in
glycerin in a pipe. The expression for 
for the Tomiyama model is:
 | (14–322) |
In Equation 14–322, 
is the pipe diameter and 
depends on the Eotvos number, 
. 
is defined as
 | (14–323) |
where the Eotvos number, 
is defined as
and 
is the surface tensions coefficient. It has been noted by Frank et al. [190] that, though the Tomiyama model has been found to be superior to the
Antal model, it is restricted to flows in pipe geometries due to the dependence on pipe
diameter in Equation 14–322.
The model of Frank et al. [190], [191] removes
the dependence on pipe diameters in the Tomiyama model [656]. The Frank
model defines the coefficient 
as
 | (14–324) |
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where | |
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By default, 
, 
, and 
.
The model from Hosokawa et al. [258] is based on experimental
measurements that indicate that, for bubbles, the coefficient, 
(Equation 14–323) used in the Tomiyama and
Frank models depends not only on the Eotvos number, but on the phase relative Reynolds number.
The model proposes the following expression for 
.
 | (14–325) |
where the phase relative Reynolds number, 
, is defined
 | (14–326) |
By default, the Hosokawa model computes 
as in Equation 14–324. Note that Equation 14–325 is only valid for liquid-gas bubble flows.
The Lubchenko approach [395] departs from treating wall lubrication as an artificial force exerted on bubbles near the wall and uses a more fundamental treatment, which is based on a geometric correction arising from the spherical shape of the bubbles. From this, a functional dependence for the gas volume fraction is derived, which is used to construct a wall lubrication force that regularized turbulent dispersion in the near-wall region. The model accounts for the decreasing cross-sectional area of bubbles.
If lift is considered, you should use this model together with the Shaver-Podowski lift correction (see Shaver-Podowski Correction), which will damp the lift force near the wall, thus allowing the turbulent dispersion to remain as the only interfacial force acting perpendicular to the wall.
The Lubcheko's derivation for the gradient of volume fraction term takes the following form [395]:
 | (14–327) |
where
= gradient of the volume fraction of the dispersed
phase |
= volume fraction of the dispersed phase |
= distance from the wall |
= bubble diameter |
= wall normal vector |
Substituting the gradient of the volume fraction term in the specific turbulent dispersion approach by Equation 14–327 yields the following equations for the wall lubrication model when using it together with the Burns et al. or the Lopez de Bertodano turbulent dispersion models:
Burns et al. (Burns et al. Model)
(14–328)
Lopez de Bertodano (Lopez de Bertodano Model)
(14–329)
In the above equations:
= wall lubrication coefficient |
= turbulent dispersion coefficient |
= drag coefficient |
= slip velocity |
= turbulent viscosity |
= dispersion Prandtl number |
= density of the primary phase |
= turbulent dispersion of the primary phase |
Equation 14–328 and Equation 14–329 are valid as long as
</2. If ≥/2, the Lubchenko wall lubrication force will not be used, and will be equal to zero.
The Lubchenko wall lubrication model does not require any limiter or calibration coefficients in order to control its performance, thus significantly improving the generality, applicability, and ease of use of the approach.