Wall-Bound Droplets

Wall-bound droplets are a type of Lagrangian phase that is bound to the wall. This phase type is used to model droplets that slide along solid surfaces. The component of the particle equation of motion that is tangential to the surface is used to force the motion of the droplet along the surface.

Wall-bound droplets are injected onto the wall boundaries and are kept on the wall during their tracking. Particle tracking is performed on the triangulated boundary faces.

Wall-bound droplets also interact with the continuous phase to exchange momentum (drag force) and heat. The momentum source due to the drag force and the heat source due to convection are applied to the continuous phase in the fluid region.

Wall-bound droplets can be absorbed by fluid film. In that case, mass, momentum, and heat sources are added to the fluid film phase in the shell region.

Spherical Cap

Simcenter STAR-CCM+ employs the Spherical Cap model [660] to compute the shape of wall-bound droplets. The equilibrium contact angle $θ_{e}$ defines the spherical cap that the droplet forms on the surface.

The volume of the spherical cap is defined as:

Figure 1. EQUATION_DISPLAY

V = \frac{π}{3} r^{3} (2 - 3 \cos (θ_{e}) + (\cos (θ_{e}))^{3})

(3004)

where $r$ is the radius of the spherical cap.

The volume of an equivalent sphere is given by:

Figure 2. EQUATION_DISPLAY

V = V_{sphere} = \frac{4}{3} π r_{sphere}^{3}

(3005)

where $r_{s p h e r e}$ is the radius of the equivalent sphere.

The spherical cap radius can be expressed as:

Figure 3. EQUATION_DISPLAY

r = r_{sphere} (\frac{4}{2 - 3 \cos θ_{e} + (\cos θ_{e})^{3}})^{\frac{1}{3}}

(3006)

The following geometrical information describes completely the shape of the spherical cap:

Height: $h = r \cdot (1 - \cos θ_{e})$
Contact area between the droplet and the wall: $A_{contact} = π r^{2} \cdot \sin^{2} (θ_{e})$
Upwind projected area in the surrounding flow: $A_{d} = \frac{2 θ_{e} - \sin (2 θ_{e})}{2} r^{2}$
Surface area: $A_{s} = 2 π r h = 2 π r^{2} (1 - \cos θ_{e})$

Drag Force

For all forces that act on the wall-bound droplet parcels, only the force component that is tangential to the wall is retained. For the drag force evaluation, the particle Reynolds number is based on the height of the spherical cap. The velocity of the continuous phase at the droplet position is evaluated by using a blended wall function that defines the dependency of the wall-tangential velocity component $u^{+} = v / v_{*}$ on the wall distance $y^{+} = y ρ v_{*} / μ$ . The continuous phase velocity is evaluated at half of the height of the spherical cap.

Heat Transfer

The rate of heat transfer from the continuous phase to the wall-bound droplet is defined as:

Figure 4. EQUATION_DISPLAY

Q_{c} = h_{g} A_{s} (T_{g} - T_{p})

(3007)

where:

$h_{g} = \frac{N u k_{g}}{h}$ is the heat transfer coefficient, defined with the height of the spherical cap.
$k_{g}$ is the conductivity of the continuous gas phase.
$A_{s}$ is the surface area of the spherical cap.
$T_{g}$ is the temperature of the continuous gas phase.
$T_{p}$ is the wall-bound particle temperature.

Pohlhausen: The Nusselt number is calculated by the Pohlhausen correlation [651]:
Figure 5. EQUATION_DISPLAY

$N u = 0.664 {Re}_{p}^{\frac{1}{2}} \Pr^{\frac{1}{3}}$

(3008)
where ${Re}_{p}$ is the particle Reynolds number and $\Pr$ is the Prandtl number. ${Re}_{p}$ is always based on the height of the spherical cap.

Adhesion Force

For a wall-bound droplet to move on a surface, external forces need to act on the droplet. These forces need to overcome the adhesion force, which retains the droplet in its position on the surface. While the droplet slides, the adhesion force reduces the impact of the other external forces. The Adhesion Force model calculates the maximum of the retaining force, always acting in the direction that is opposite to the motion of the droplet.

The contact angle hysteresis is defined as the difference between the advancing and receding contact angles. The advancing contact angle corresponds to the largest contact angle at the point in time when the droplet starts sliding on the surface due to external forces. The receding contact angle is the smallest contact angle at that moment. While the droplet is moving on the surface, the retaining force acts on the droplet in the reverse direction. The amount of retaining force is related to the contact angle hysteresis [658].

The adhesion force for circular droplets [658] is given by:

Figure 6. EQUATION_DISPLAY

F = \frac{2}{π} σ D_{c o n t a c t} (\cos θ_{r} - \cos θ_{a})

(3009)

with:

D_{contact} = 2 r \cdot \sin θ_{e}

where:

$θ_{r}$ is the receding contact angle.
$θ_{a}$ is the advancing contact angle.
$σ$ is the surface tension of the droplet.

When $F$ is scaled with a characteristic length such as its contact-area diameter, a size-independent parameter is obtained. The angles $θ_{r}$ and $θ_{a}$ are independent of droplet size. The contact angle hysteresis depends on the nature of the surface and the contacting liquid.