### 5.5.7. Wall Lubrication Force

Under certain circumstances, for example, bubbly upflow in a vertical pipe, the dispersed phase is observed to concentrate in a region close to the wall, but not immediately adjacent to the wall. This effect may be modeled by the wall lubrication force, which tends to push the dispersed phase away from the wall.

Currently, ANSYS CFX has the following wall lubrication force models:

- The Antal Wall Lubrication Force Model

- The Tomiyama Wall Lubrication Force Model

- The Frank Wall Lubrication Force Model

#### 5.5.7.1. The Antal Wall Lubrication Force Model

The Antal model, given by Antal et al. (1991) [88], computes the wall lubrication force as:

| (5–82)

where

- The non-dimensional coefficients are defaulted to and . You can change these values.

- is the gas volume fraction.

- is the liquid density.

- is the unit normal pointing away from the wall.

- is the relative velocity difference between phases, in the plane of the nearby wall surface (that is, orthogonal to ).

- is the dispersed phase mean diameter.

- is the distance to the nearest wall.

Note that this force is only active in a thin layer adjacent to the wall; it is only active up to a cut-off distance of:

where with default values of and .

Hence, this force will only be activated on a sufficiently fine mesh, and grid convergence can be expected only on extremely fine meshes.

#### 5.5.7.2. The Tomiyama Wall Lubrication Force Model

Tomiyama (1998) [172] modified the wall lubrication force formulation of Antal, based on the results of experiments with the flow of air bubbles in glycerin in a pipe. The modification is as follows:

| (5–83)

Here, is the pipe diameter. Hence, although the model was found to be superior to Antal's [88] (Frank et al. 2004 [173]), it is restricted to flows in pipe geometries. As is the case for the Tomiyama lift force correlation, the coefficient is dependent on the Eotvos number, and hence on the surface tension between the two phases. Frank et al (2004) modified this correlation slightly to ensure continuous dependence of the wall lubrication coefficient on Eotvos number:

| (5–84)

#### 5.5.7.3. The Frank Wall Lubrication Force Model

Frank et al. ([177], [173]) generalized the Tomiyama model to produce the Frank Wall Lubrication Force model, which has no dependence on pipe diameter, and is given by:

| (5–85)

Note:

- preserves the same dependence on Eotvos number as the Tomiyama model.

- The cut-off coefficient, , determines the distance relative to the particle diameter over which the force is active. gives the same range as the Antal model with default constants.

- The damping coefficient, , determines the relative magnitude of the force. gives the same behavior as the Antal model with default constants. However, Frank et al. found that such high damping of the wall lubrication force was not able to sufficiently counterbalance the Tomiyama lift force in the near wall region.

- The power-law constant, , makes the force fall off with a variable potential law relationship: . It is recommended that be in the range: 1.5 to 2.

- In extensive validation exercises by Frank et al. [177], the following model constants were determined in order to produce the best possible agreement with experimental data for vertical bubbly flow in pipes: , , .

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