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22.2.8 Surface Tension and Wall Adhesion

The VOF model can also include the effects of surface tension along the interface between each pair of phases. The model can be augmented by the additional specification of the contact angles between the phases and the walls. You can specify a surface tension coefficient as a constant, as a function of temperature, or through a UDF. The solver will include the additional tangential stress terms (causing what is termed as Marangoni convection) that arise due to the variation in surface tension coefficient. Variable surface tension coefficient effects are usually important only in zero/near-zero gravity conditions.

Surface Tension

Surface tension arises as a result of attractive forces between molecules in a fluid. Consider an air bubble in water, for example. Within the bubble, the net force on a molecule due to its neighbors is zero. At the surface, however, the net force is radially inward, and the combined effect of the radial components of force across the entire spherical surface is to make the surface contract, thereby increasing the pressure on the concave side of the surface. The surface tension is a force, acting only at the surface, that is required to maintain equilibrium in such instances. It acts to balance the radially inward inter-molecular attractive force with the radially outward pressure gradient force across the surface. In regions where two fluids are separated, but one of them is not in the form of spherical bubbles, the surface tension acts to minimize free energy by decreasing the area of the interface.

The surface tension model in FLUENT is the continuum surface force (CSF) model proposed by Brackbill et al. [ 27]. With this model, the addition of surface tension to the VOF calculation results in a source term in the momentum equation. To understand the origin of the source term, consider the special case where the surface tension is constant along the surface, and where only the forces normal to the interface are considered. It can be shown that the pressure drop across the surface depends upon the surface tension coefficient, $\sigma$, and the surface curvature as measured by two radii in orthogonal directions, $R_1$ and $R_2$:

\begin{displaymath} p_2 - p_1 = \sigma \left( \frac{1}{R_1} + \frac{1}{R_2} \right) \end{displaymath} (22.2-10)

where $p_1$ and $p_2$ are the pressures in the two fluids on either side of the interface.

In FLUENT, a formulation of the CSF model is used, where the surface curvature is computed from local gradients in the surface normal at the interface. Let $n$ be the surface normal, defined as the gradient of $\alpha_q$, the volume fraction of the $q$th phase.

\begin{displaymath} n = \nabla \alpha_q \end{displaymath} (22.2-11)

The curvature, $\kappa$, is defined in terms of the divergence of the unit normal, $\hat{n}$ [ 27]:

\begin{displaymath} \kappa = \nabla \cdot \hat{n} \end{displaymath} (22.2-12)


\begin{displaymath} \hat{n} = \frac{n}{\vert n\vert} \end{displaymath} (22.2-13)

The surface tension can be written in terms of the pressure jump across the surface. The force at the surface can be expressed as a volume force using the divergence theorem. It is this volume force that is the source term which is added to the momentum equation. It has the following form:

\begin{displaymath} F_{\rm vol} = \sum_{{\rm pairs}\; ij,\; i < j} \sigma_{ij} \... ... \nabla \alpha_i} {\frac{1}{2} \left( \rho_i + \rho_j \right)} \end{displaymath} (22.2-14)

This expression allows for a smooth superposition of forces near cells where more than two phases are present. If only two phases are present in a cell, then $\kappa_i=-\kappa_j$ and $\nabla \alpha_i = - \nabla \alpha_j$, and Equation  22.2-14 simplifies to

\begin{displaymath} F_{\rm vol} = \sigma_{ij} \frac{\rho \kappa_i \nabla \alpha_i} {\frac{1}{2} \left( \rho_i + \rho_j \right)} \end{displaymath} (22.2-15)

where $\rho$ is the volume-averaged density computed using Equation  22.2-4. Equation  22.2-15 shows that the surface tension source term for a cell is proportional to the average density in the cell.

Note that the calculation of surface tension effects on triangular and tetrahedral meshes is not as accurate as on quadrilateral and hexahedral meshes. The region where surface tension effects are most important should therefore be meshed with quadrilaterals or hexahedra.

When Surface Tension Effects are Important

The importance of surface tension effects is determined based on the value of two dimensionless quantities: the Reynolds number, Re, and the capillary number, Ca; or the Reynolds number, Re, and the Weber number, We. For Re  $\ll 1$, the quantity of interest is the capillary number:

\begin{displaymath} {\rm Ca}=\frac{\mu U}{\sigma} \end{displaymath} (22.2-16)

and for Re  $\gg 1$, the quantity of interest is the Weber number:

\begin{displaymath} {\rm We}=\frac{\rho L U^2}{\sigma} \end{displaymath} (22.2-17)

where $U$ is the free-stream velocity. Surface tension effects can be neglected if Ca  $\gg 1$ or We  $\gg 1$.

Wall Adhesion

An option to specify a wall adhesion angle in conjunction with the surface tension model is also available in the VOF model. The model is taken from work done by Brackbill et al. [ 27]. Rather than impose this boundary condition at the wall itself, the contact angle that the fluid is assumed to make with the wall is used to adjust the surface normal in cells near the wall. This so-called dynamic boundary condition results in the adjustment of the curvature of the surface near the wall.

If $\theta_w$ is the contact angle at the wall, then the surface normal at the live cell next to the wall is

\begin{displaymath} \hat{n} = \hat{n}_w \cos \theta_w + \hat{t}_w \sin \theta_w \end{displaymath} (22.2-18)

where $\hat{n}_w$ and $\hat{t}_w$ are the unit vectors normal and tangential to the wall, respectively. The combination of this contact angle with the normally calculated surface normal one cell away from the wall determine the local curvature of the surface, and this curvature is used to adjust the body force term in the surface tension calculation.

The contact angle $\theta_w$ is the angle between the wall and the tangent to the interface at the wall, measured inside the first phase of the pair listed in the Wall panel, as shown in Figure  22.2.2.

Figure 22.2.2: Measuring the Contact Angle
\begin{figure} \psfig{file=figures/fig-vof-thetaw.ps,width=4in} \end{figure}

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