Phase Interaction Topology

Continuous-Dispersed

The Continuous-Dispersed topology provides the means to model the interaction of a phase which is dispersed within another phase throughout the domain at all times.

In the following equations, the subscript $c$ denotes the continuous phase and the subscript $d$ stands for dispersed phase. The subscript $i j$ describes an interaction between phases $i$ and $j$ where $i$ and $j$ can be continuous or dispersed.

Multiple Flow Regimes

The Multiple Flow Regime topology is an extended topology definition where all types of multiphase flows, that is, dispersed (for example bubble, droplet, and particle flows) and mixed/stratified (for example free-surface flows) flows can be modelled. An example of such a multiple flow regime topology is shown in the diagram in the previous section.

To achieve this combined simulation, the flow topology must be classified into dispersed and intermediate two-phase flow. Simcenter STAR-CCM+ employs a volume-fraction based criterion to classify the flow regime into dispersed and intermediate two-phase flow.

As a first step, the co-existence of dispersed and intermediate two-phase flow entails that the pair of phases must be a set of primary ( $p$ ) and secondary ( $s$ ) phase rather than a pair of continuous and dispersed phase. Based on this definition, for a pair of primary and secondary phase, three flow regimes can be defined:


Flow Regime	Flow Topology
First Dispersed Regime	Secondary phase is dispersed in primary phase.
Intermediate Regime	Neither phase is dispersed in the other phase.
Second Dispersed Regime	Primary phase is dispersed in secondary phase.

In the intermediate regime of the multiple flow regime topology, neither phase is clearly dispersed in the other phase. By default, the intermediate regime models the mixed flow. To model separated flow additional modeling for Large Scale Interface (LSI) is used. The flow regime classification criterion is implicitly defined with the definition of the blending weight function, discussed in the following section.

Blending Weight Function

To understand this regime definition more clearly, consider a system of water as primary phase and air as secondary phase. In this scenario, the multiple flow regime model treats bubbly flow in the first dispersed regime, droplet flow in the second dispersed regime, and separated water-air flow in the large scale interface regime.

To enforce the classification of flow regime in each computational cell, the total linearized drag $A$ and the heat transfer coefficient $h$ are calculated by use of the weighted sum of the interaction of each flow topology regime as follows:

Figure 1. EQUATION_DISPLAY

A = \sum_{r = fr, ir, sr} W_{r} A_{r}

(1914)

Figure 2. EQUATION_DISPLAY

h = \sum_{r = fr, ir, sr} W_{r} h_{r}

(1915)

where $W_{r}$ is the weight function.

The following weight function methods are available:

Standard Blending Function
Gradient Based Blending Function
User-Specified Blending Function

Standard Blending Function

This blending function is suitable when modeling mixed flow in the intermediate regime.

Following the work of Höhne and others [476], the weight function for each flow topology regime is calculated as described below.

The first dispersed regime blending weight function, $W_{f r}$ :

Figure 3. EQUATION_DISPLAY

W_{f r} = \frac{1}{1 + e^{B (α_{s} - α_{f r t})}}

(1916)

where:

$α_{s}$ is the volume fraction of the secondary phase.
$α_{f r t}$ is the first dispersed regime terminus, which is the value of $α_{s}$ at which the first dispersed regime transits to the intermediate regime. The default value is taken as 0.3 based on literature [439].

The second dispersed regime blending weight function, $W_{s r}$ :

Figure 4. EQUATION_DISPLAY

W_{s r} = \frac{1}{1 + e^{B (α_{s r o} - α_{s})}}

(1917)

where:

$α_{s r o}$ is the onset of the second dispersed regime. It corresponds to the value of $α_{s}$ at which the interface regime transits to the second dispersed regime. The default value is taken as 0.7 based on literature [439].

The intermediate regime blending weight function, $W_{i r}$ , is calculated as:

Figure 5. EQUATION_DISPLAY

W_{i r} = 1 - (W_{f r} + W_{s r})

(1918)

The two thresholds, $α_{f r o}$ and $α_{s r t}$ give the flexibility to control the extent of the individual flow regimes. The width of the transition zone can be controlled by changing the exponent $B$ : increasing the value of $B$ decreases the width of the transition zone. The default value of $B = 70$ leads to the width of transition boundary as $α = 0.2$ .

The weight functions depending on the volume fraction of the secondary phase are shown in the following plot:

Gradient Based Blending Function

This blending function is suitable when modeling separated flow in the intermediate regime.

Following the work of P. Porombka and T. Höhne [476], the standard weight function is extended by incorporating a gradient-based correction. The gradient-based modification essentially leads to a smoother field of blending weight function.

The weight function for each flow topology regime is calculated as follows:

Figure 6. EQUATION_DISPLAY

W_{f r} = \frac{1}{1 + e^{B (α_{s} - α_{f r t})}} \cdot \frac{1}{(1 + c_{g} | \nabla α_{s} |)}

(1919)

Figure 7. EQUATION_DISPLAY

W_{s r} = \frac{1}{1 + e^{B (α_{s r o} - α_{s})}} \cdot \frac{1}{(1 + c_{g} | \nabla α_{s} |)}

(1920)

Figure 8. EQUATION_DISPLAY

W_{i r} = 1 - (W_{f r} + W_{s r})

(1921)

The factor $c_{g}$ in Eqn. (1919) and Eqn. (1920) controls the role of the gradient in the blending function. The default value is $c_{g} = 0.05$ . This gradient-based blending function cannot be plotted objectively on a XY plot since it involves volume fraction gradient calculation.

User-Specified Blending Function

The user specifies $W_{f r}$ and $W_{s r}$ using field functions. The intermediate regime weight function $W_{i r}$ is calculated internally as:

$W_{i r} = 1 - (W_{f r} + W_{s r})$

Simcenter STAR-CCM+ performs an internal check to ensure that $0 \leq W_{f r} \leq 1$ , $0 \leq W_{s r} \leq 1$ and $0 \leq (W_{f r} + W_{s r}) \leq 1$ .