Partial Fill Method

The partial fill method handles incompressible viscous liquids that only partially fill a volume, the rest of the volume being filled with air or another gas. The principal use is in modeling the filling stage of injection molding.

This method uses a fixed mesh and defines the interface by a constant value of the level set function. This is in contrast with the free surface model, where the mesh is not fixed.

The viscous flow solver uses a conservative level set technique to handle tracking of the fluid interface in contact with air. [242]. This method defines the interface between the phases by solving an equation for the level set function:

Figure 1. EQUATION_DISPLAY

\frac{\partial ϕ}{\partial t} + \nabla\cdot (v ϕ) = \frac{1}{μ} \nabla\cdot [- ϕ (1 - ϕ) n + ϵ (\nabla ϕ \cdot n) n]

(1084)

where:

$ϕ$ is a level set function that varies from 0 (dry region, no viscous fluid) to 1 (wet region, filled with viscous fluid) and $ϕ = 0.5$ is a sharp interface.
$n$ is the unit normal defined as $n = \frac{\nabla ϕ}{| \nabla ϕ |}$ .
$μ$ and $ϵ$ are model parameters. The parameter $ϵ$ must be chosen small enough to assure conservation of the area bounded by the sharp interface ( $ϕ = 0.5$ ), but there are numerical restrictions on how small $ϵ$ can be, since choosing values too small relative to the cell size causes inaccuracy in unit normal and curvature calculations. It also results in difficulties on reaching the full convergence in the reinitialization stage. Following Olsson et al. [273], Simcenter STAR-CCM+ uses:
Figure 2. EQUATION_DISPLAY

$ϵ = {C h}^{1 - d}$

(1085)

where:
- $C$ is an arbitrary constant, $C \approx O (1)$ .
- $h$ is cell size.
- $0 < d < 1$ is the diffusion exponent, default value 0.1, set in the Partial Fill model.

The method breaks Eqn. (1084) into two equations, one for advection and one for stabilization. The pure advection part at one time-step is:

Figure 3. EQUATION_DISPLAY

\frac{\partial ϕ}{\partial t} + \nabla\cdot (v ϕ) = 0

(1086)

A Streamline Upwind Petrov-Galerkin (SUPG) scheme stabilizes the pure advection equation. (See Streamlined Upwind Petrov-Galerkin.) The SUPG parameter, default value 1, is set in the Partial Fill model. The solution to this equation provides the initial guess for the stabilizing part, which is:

Figure 4. EQUATION_DISPLAY

\frac{\partial ϕ}{\partial t} + \nabla\cdot [ϕ (1 - ϕ) n - ϵ (\nabla ϕ \cdot n) n] = 0

(1087)

Eqn. (1087) is called the reinitialization stage.

To obtain a variational form for the reinitialization stage, we multiply the above equation by a weight function $\tilde{ϕ}$ and then integrate over the entire domain:

Figure 5. EQUATION_DISPLAY

\int_{Ω}^{} \tilde{ϕ} \frac{\partial ϕ}{\partial τ} d Ω + \int_{Ω}^{} \tilde{ϕ} \nabla\cdot [ϕ (1 - ϕ) n - ϵ (\nabla ϕ \cdot n) n] d Ω = 0, \forall \tilde{ϕ} \in H_{0}^{1}

(1088)

where $H_{0}^{1} (Ω)$ is the Sobolev space of square integrable functions for which the first derivatives are also square integrable, with the magnitude zero at the boundaries with essential (Dirichlet) boundary conditions.

Figure 6. EQUATION_DISPLAY

\int_{Ω}^{} \tilde{ϕ} \frac{\partial ϕ}{\partial τ} d Ω + \int_{Ω}^{} \tilde{ϕ} \nabla\cdot [- ϕ (1 - ϕ) n - ϵ (\nabla ϕ \cdot n) n] d Ω + \int_{\partial Ω}^{} \tilde{ϕ} [ϕ (1 - ϕ) n - ϵ (\nabla ϕ \cdot n) n] \cdot e_{n} d Ω = 0, \forall \tilde{ϕ} \in H_{0}^{1}

(1089)

where $e_{n}$ denotes the unit normal vector pointing outward to the boundary $\partial Ω$ enclosing the domain $Ω$ . The third (boundary) term in the above equation can be neglected on the wall, where there is zero flux of the field variable. However, a Dirichlet boundary condition ( $ϕ = 1$ ) for mass and velocity inlets can be imposed.

In summary, the method for solving the partial filling problem is:

Solve the pure advection (transport) equation, Eqn. (1086), for a single time-step.
Solve the reinitialization equation Eqn. (1087) to guarantee mass conservation.
Solve the flow equations only in the wet region (that is, where $ϕ > 0.5$ ) to determine pressure, velocity, and related values.
After the velocity field is determined, return to Step 1 and repeat the cycle until stopping criteria are met.

Surface Tension: To model the role of interfacial tension between the fluids, the surface tension can be added to the momentum equation as a body force as shown by Ruschak [281]. In this case, the surface tension body force $f_{s t}$ is related to the level set function according to:
$f_{s t} = - γ [\nabla\cdot (\frac{\nabla ϕ \nabla ϕ}{| \nabla ϕ |}) - \nabla | \nabla ϕ |]$; where $γ$ is the interfacial tension between the fluids.