Macro Porosity Formulation

When the static pressure in a cell falls below the user-specified void pressure, a fraction of the liquid phase is converted to the gaseous void phase. The total fluid mass is kept constant.

This formulation applies to the Macro Porosity (fully coupled) model only. The Macro Porosity (pure thermal) model is a simple model that attempts to keep the mass of the primary phase constant for each liquid zone.

Macro Porosity (fully coupled) Model Criterion

At the same pressure, a certain amount of a light gas fills a larger volume than the same mass of a heavy liquid. Since the cell volume is constant, the pressure is increased. This mechanism is used to keep the pressure from decreasing significantly below the target void pressure.

The equation of state of an ideal gas, the void phase $void$ , is:

Figure 1. EQUATION_DISPLAY

p = ρ \frac{R_{u}}{M_{void}} T

(353)

with the universal gas constant $R_{u}$ = 8314.4621 J/kmol K, and $M_{void}$ is the molecular weight in kg/kmol.

Mass and cell volume are constant, and pressure increases during the phase conversion. Therefore, the following relation is fulfilled even for the minimum temperature in any cell during the simulation that is run at $p = p_{void}$ :

Figure 2. EQUATION_DISPLAY

p_{void} \leq ρ_{l} \frac{R_{u}}{M_{void}} T_{min}

(354)

Hence, the Macro Porosity (fully coupled) model can operate correctly when the molecular weight of the ideal gas void phase is:

Figure 3. EQUATION_DISPLAY

M_{void} \leq \frac{ρ_{l} R_{u} T_{min}}{p_{void}}

(355)

However, this criterion is easily met for common casting material properties:

$M_{air} = 28.9664 \leq 44898.1 = \frac{2700 \cdot 8314.4621 \cdot 100}{50000}$

Void Phase Source

The rate at which the liquid phase is converted to the void phase without pressure decreasing below $p_{void}$ must be determined.

The relative volume change of the cell that leads to a pressure increase up to $p_{void}$ is computed. The result is the fraction of the cell volume that must be filled with void gas at $p = p_{void}$ and $T = T_{cell}$ .

$Δ p$ is the difference between the user-specified void pressure $p_{void}$ and the static pressure $p$ in the cell:

Figure 4. EQUATION_DISPLAY

Δ p = p_{void} - p

(356)

This pressure difference causes a change in mixture density in the cell, which is approximated with the derivative of mixture density with respect to pressure $\partial ρ / \partial p$ . This approximation assumes that $\partial ρ / \partial p$ is constant for small pressure deviations:

Figure 5. EQUATION_DISPLAY

Δ ρ \approx Δ p \frac{\partial ρ}{\partial p}

(357)

The relation between a density change and the related volume change for constant fluid mass in a cell is:

Figure 6. EQUATION_DISPLAY

\begin{matrix} m & = & ρ V \\ ρ V & = & (ρ + Δ ρ) (V + Δ V) \\ \frac{ρ V}{ρ + Δ ρ} - V & = & Δ V \\ \frac{ρ}{ρ + Δ ρ} - 1 & = & \frac{Δ V}{V} \\ - \frac{Δ ρ}{ρ + Δ ρ} & = & \frac{Δ V}{V} \end{matrix}

(358)

Eqn. (358) gives the fraction of the cell volume that must be filled with the void phase to compensate (inverted sign) the volume change of the mixture when the pressure is increased again to void pressure:

Figure 7. EQUATION_DISPLAY

\frac{Δ V}{V} = \frac{Δ ρ}{ρ + Δ ρ}

(359)

A VOF-phase source is specified in Simcenter STAR-CCM+ in the units $\frac{Volume}{Volume-Time} = \frac{1}{s}$ . The relative volume change applied per unit time (second) and an under-relaxation factor gives the rate at which the void fluid is converted from the liquid phase to the gaseous phase.

Figure 8. EQUATION_DISPLAY

\begin{matrix} \frac{Δ V}{V} |_{void} & = & max [0, \frac{Δ ρ}{ρ + Δ ρ} s] \\ = & max [0, \frac{(p_{void} - p) \frac{\partial ρ}{\partial p}}{ρ + (p_{void} - p) \frac{\partial ρ}{\partial p}} s] \end{matrix}

(360)

Values without index are cell values of the mixture. The scaling factor $s$ is limited to values $s \geq 0$ . It allows an adaptation to the specific case where larger rates are required to increase the minimum pressure.

The rate is limited to positive values to prevent emitted gas being dissolved again.

Void Phase Source Linearization

The linearization of the void phase source term is the derivative of Eqn. (360) with respect to pressure (assuming $\partial ρ / \partial p$ is constant):

Figure 9. EQUATION_DISPLAY

\frac{\partial}{\partial p} (\frac{Δ V}{V} |_{void}) = \frac{\partial}{\partial p} [\frac{(p_{void} - p) \frac{\partial ρ}{\partial p}}{ρ + (p_{void} - p) \frac{\partial ρ}{\partial p}} s] = - \frac{p \frac{\partial ρ}{\partial p}}{{[ρ + (p_{void} - p) \frac{\partial ρ}{\partial p}]}^{2}} s

(361)

Macro Porosity Nomenclature

The terms that are used in the Macro Porosity (fully coupled) model formulation are defined below.


$ρ$	Density
$M_{void}$	Molecular weight of the void phase
$R_{u}$	Universal gas constant
$T$	Temperature
$V$	Volume
$p$	Pressure
$s$	Scaling factor