Wave Stripping

Depending on flow conditions and the geometry of the wall on which the film resides, the film can form droplets. This process is called film stripping.

Simcenter STAR-CCM+ can model film stripping due to:

Instabilities induced by adjacent fluid flow
Instabilities induced by body forces (for example, gravity and film acceleration)

Adjacent Flow and Body-Force Induced Instability

Due to external forces such as gravity or shearing at the interface between a fluid film and its surrounding fluid, the film surface can develop instabilities and start shedding droplets. Following Hervé Foucart [618], Simcenter STAR-CCM+ models film stripping as a three stage process:

The model assumes that for surface stripping:

Waves develop at the liquid-gas interface.
As these waves grow, they become unstable. When the film height becomes critical, ejection occurs.
The ejected fluid volume forms a cylinder, as a result of the Kelvin-Helmholtz instability.
The cylinder then breaks into droplets due to Rayleigh-Taylor instabilities.
The model calculates the radius of the child droplets as a function of the most unstable wavelength characterizing the surface instability.

The most unstable wavelength is the wavelength at which surface break-up is most likely to occur. This is calculated as the resonance wavelength from the dispersion equation:

Figure 1. EQUATION_DISPLAY

λ_{res} = \frac{2 π}{ρ_{f} f_{b} \cdot n} (\frac{1}{3} ρ {\tilde{v}}_{r}^{2} - \sqrt{{(\frac{1}{3} ρ {\tilde{v}}_{r}^{2})}^{2} - ρ_{f} f_{b} \cdot n σ})

(2749)

where ${\tilde{v}}_{r}^{2}$ is the square of relative velocity between the film and the surrounding fluid $(v_{f} - v) \cdot (v_{f} - v)$ , $σ$ is the surface tension, and $f_{b}$ is the body force acting on the film which also includes the inertial force. The minimum film height necessary for droplet ejection is then:

Figure 2. EQUATION_DISPLAY

h_{min} = c_{H} \frac{λ_{res}}{2 π}

(2750)

The film height to strip, that is the surface wave amplitude, is:

Figure 3. EQUATION_DISPLAY

h_{a} = {[{\frac{3}{4} (\frac{2}{3.78})}^{3}]}^{2} π λ_{res}

(2751)

The resulting droplet diameter is:

Figure 4. EQUATION_DISPLAY

D_{d} = c_{D} \sqrt{\frac{λ_{res} h_{a}}{π}}

(2752)

Note	For wave stripping into a dispersed phase, Eqn. (2752) is not used. The droplet diameter is specified manually rather than being computed.

The default setting for $c_{D}$ is 3.78. The number 3.78 in Eqn. (2751) arises from Rayleigh theory. It means that the diameter of each droplet that is generated from the cylinder is 3.78 times the diameter of that cylinder.

All droplets initially have the same velocity as the film and are placed at a point between the cell center and the boundary face center, depending on the radius of the ejected droplet.

Although no consideration of time scale is included in the original stripping model, the time scale can be defined in terms of a linear growth rate as:

Figure 5. EQUATION_DISPLAY

t_{b} = \frac{λ_{res} (ρ_{f} + ρ)}{| v_{r} | \sqrt{(1 - We) ρ_{f} ρ}}

(2753)

where $We$ is the Weber number. In Eqn. (2753), $We = 2 / 3$ , which is when the most unstable mode occurs.

Using this time scale, stripping is expected to occur $n_{b}$ times during a time-step $Δ t$ :

Figure 6. EQUATION_DISPLAY

n_{b} = c_{B} \frac{Δ t}{t_{b}}

(2754)

Free-Stream Velocity

The wave stripping model requires a measure of the free-stream velocity ( $v_{r}$ as used in Eqn. (2749)).

Simcenter STAR-CCM+ calculates two measures of the free-stream velocity and uses the highest.

One measure is the gas velocity in the nearest gas cell. The disadvantage with this measure is that the value depends on the mesh size, and the approximation becomes less valid as the mesh size decreases.
A second measure is a mesh-independent estimate of the free-stream velocity based on the shear velocity. This second measure is derived below.

At the free surface, on the gas side, the shear (or frictional) velocity is defined as:

Figure 7. EQUATION_DISPLAY

v_{*} = - \sqrt{\frac{τ}{ρ}} n_{τ}

(2755)

where $n_{τ}$ is the unit vector in the direction of the shear stress $τ$ .

It is assumed that the free-stream gas velocity can be approximated as:

Figure 8. EQUATION_DISPLAY

v_{r} \approx {(v)}_{int} + K_{*, \infty} v_{*} = v

(2756)

where ${(v)}_{int}$ is the velocity at the interface and $K_{*, \infty}$ is a scaling factor for the gas velocity.