Contact Angle

The contact angle $θ$ describes the influence of a solid wall on the surface tension force.

The contact angle is measured at the triple line, which is the line where the wall and both fluids are in contact. In Simcenter STAR-CCM+, for an interaction between phases, you set one fluid to be the primary phase and the other as the secondary phase. Values of contact angle smaller than $90 °$ mean that the primary phase is wetting the wall, as illustrated in part (a) of the following diagram:

When contact angle also depends on the triple line velocity, it is called dynamic contact angle. Some simple models are based on a step function that uses a single advancing and receding angle, while other models use a smoothed step function. These models work well for inertia-dominated flows because the dynamic contact angle does not depend strongly on the velocity.

Simcenter STAR-CCM+ implements the Kistler correlation, which works well for capillary-dominated flows.

The dimensionless capillary number $C a$ is defined as:

Figure 1. EQUATION_DISPLAY

C a = \frac{V μ}{σ}

(2614)

where:

$V$ is the triple line characteristic velocity
$μ$ is the dynamic viscosity of the primary phase (usually a liquid phase)
$σ$ is the surface tension force

The triple line velocity is defined as:

Figure 2. EQUATION_DISPLAY

V = - (V \cdot {\hat{n}}_{t})

(2615)

where:

$V$ is the relative velocity of the fluid and the corresponding wall at the triple line.
${\hat{n}}_{t}$ is the unit vector in the tangential direction pointing in the same direction as the volume fraction gradient of the primary phase ( $\nabla α$ )

The dynamic contact angle models attempt to account for the dependency of the advancing and receding contact angles on the triple line velocity.

Kistler Correlation

Simcenter STAR-CCM+ implements the Kistler correlation [624], which is an empirical dynamic contact angle correlation based on the capillary number $C a$ and utilizing the Hoffman function. The Kistler contact angle is defined as:

Figure 3. EQUATION_DISPLAY

θ_{k} = f_{H o f f} (C a + f_{H o f f}^{- 1} (θ_{s}))

(2616)

where $f_{H o f f}$ is the Hoffman function:

Figure 4. EQUATION_DISPLAY

f_{H o f f} (x) = \cos^{-1} (1 - 2 \tanh (5.16 {(\frac{x}{1 + 1.31 x^{0.99}})}^{0.706}))

(2617)

with $f_{H o f f}^{- 1}$ being its inverse. Simcenter STAR-CCM+ uses a slightly modified definition of the Hoffman function to achieve a closed form for its inverse function. In the modified Hoffman function, the $x^{0.99}$ term in the denominator is changed to a simple $x$ . The closed form of the inverse function is valid for contact angle values in the range 0–176 degrees. Advancing or receding contact angles that are larger than the upper bound are limited to 176 degrees.

The static contact angle $θ_{s}$ is inserted into Eqn. (2616) as either the static advancing or receding contact angle, depending on the sign of the capillary number. $θ_{s}$ is specified as the Advancing Contact Angle or Receding Contact Angle in the Advancing/Receding Properties.

Blended Kistler Correlation

This method is implemented to enhance the stability of the Kistler method. In Simcenter STAR-CCM+, a range for the equilibrium capillary number ( ${C a}_{e q}$ ) can be defined. Within the specified range $- {C a}_{e q} < C a < {C a}_{e q}$ , the resulting dynamic contact angle is blended with the equilibrium contact angle ( $θ_{e}$ ) as a weighted average:

Figure 5. EQUATION_DISPLAY

θ_{d} = f θ_{e} + (1 - f) θ_{k}

(2618)

While $θ_{e}$ is a user-specified value, the factor $f$ is determined within the ${C a}_{e q}$ range as:

Figure 6. EQUATION_DISPLAY

f = 0.5 + 0.5 \cos (\frac{C a}{{C a}_{e q}} π)

(2619)