**1. Introduction**

Multi-phase flow simulation, such as vapour bubbles is complicated. Various numerical methods, such as Front-tracking, Marker and Cell, Volume of Fluid, Level Set, Lattice-Boltzmann, were invented to mimic such phenomena – see **Figure 1**. These methods, however, have their own characteristics strengths and weaknesses (e.g. see summary in [1, 2]). Hybrid methods, as a result, have emerged to harness the merits of their parent methods to improve accuracy and to tackle complex multi-phase flow applications. Combining the strength of the VOF method and the Front-Tracking method, Aulisa et al. [3] Three-Dimensional (3D) method tracks the interface as a Lagrangian but finds the intersection of the surface mesh with control volume faces and locally remeshes the surface contour while preserving the tracked volume. Aulisa's method, however, requires permanent markers which cannot be seeded or removed after the simulation is executed, and leads to resolution issues for spherical bubble expansion problem. To improve the method prescribed in Aulisa et al. [3], InterSection Marker (ISM) method [4] was devised. The ISM method eliminated the need for permanent markers and addressed the

#### **Figure 1.**

*Taxonomy of selective numerical methods for multi-phase flows, and the introduction to the ISM method.*

local surface resolution issue in volume inflationary type problems. The ISM method was previously successfully applied to air bubble rise simulations which were adiabatic in nature [5]. However, ISM method's ability to calculate interfacial area more accurately (uncertainty in the order of 1–2%) than conventional VOF methods proved it an ideal candidate for multi-phase simulations involving heat and mass transfers across the interface, such as rising vapour bubbles in superheated or sub-cooled water. During the simulation, the predicted vapour bubble properties such as size, shape and velocity were compared against the past works and found to be in good agreement.

## **2. A brief introduction to the InterSection marker (ISM) method**

In the search for higher surface tracking fidelity, the InterSection Marker (ISM) method [4, 5] was developed where the proper determination of the interfacial area is critical, such as for the heat and mass transfers process across the interface separating two-phase fluids. An in-depth description of the ISM method is out of the scope of this chapter, and the reader should consult [4, 5] for details. Below, however, highlight the key features of the ISM method to provide the reader with a basic understanding.

The ISM method uses a Lagrangian surface mesh co-located within a uniform Eulerian mesh where upon flow-field qualities such as pressure, velocity and temperature are calculated. The total surface is modelled as a connected series of discrete interfaces (planar polygons), each located within their own *cubic control volume* (CCV). Each planar polygon intersects the edges of the control volume, and the combination of cell-edge intersections uniquely identifies the type of polygon a control volume holds.

of these polygons are carried out to maintain planar surface during translation/ deformation – see **Figure 3**. A triangular tessellation pattern is the preferred option because three points randomly translated will always form a plane. Additional intersection-marker combinations of non-planar-type interfaces were also identified (details in [4, 5]), which are necessary to prevent the modelled interface from

After identification of the planar polygons and their sub-divisions, the next step in the ISM method is to identify the component points of the interface – as shown in **Figure 4**: (i) the intersection markers where the interface crosses the control volume cell edges, (ii) the cell face conservation points which allow composite curves to be modelled, and (iii) the raised centroid whose position is calculated to satisfy volumetric conservation. The Volume-of-fluid (VOF) is then calculated by

collapsing and folding onto itself.

*Planar surfaces co-located within cubic cells can be of 3 to 6 sides.*

*Numerical Investigation of Rising Vapour Bubble in Convective Boiling Using an…*

*DOI: http://dx.doi.org/10.5772/intechopen.96303*

**Figure 2.**

**Figure 3.**

**123**

*Subdivision of planar polygons.*

The ISM method identifies the type of interface residing in a cell by the combination of cell-edge intersections that interface makes. Total of 51 combinations of basic set of planar-type interfaces had been identified: 8 intersection marker combinations for 3 sided interfaces, 15 for 4 sided, 24 for 5 sided, and 4 for 6 sided – see **Figure 2**. Given the combination of cell edge intersections is unique, a *look-up* table can be used to identify the type of interface located within each cell [4, 5] in a manner similar to that used in the marching cube method [6]. Further subdivisions *Numerical Investigation of Rising Vapour Bubble in Convective Boiling Using an… DOI: http://dx.doi.org/10.5772/intechopen.96303*

#### **Figure 2.**

local surface resolution issue in volume inflationary type problems. The ISM method was previously successfully applied to air bubble rise simulations which were adiabatic in nature [5]. However, ISM method's ability to calculate interfacial area more accurately (uncertainty in the order of 1–2%) than conventional VOF methods proved it an ideal candidate for multi-phase simulations involving heat and mass transfers across the interface, such as rising vapour bubbles in superheated or sub-cooled water. During the simulation, the predicted vapour bubble properties such as size, shape and velocity were compared against the past works

*Taxonomy of selective numerical methods for multi-phase flows, and the introduction to the ISM method.*

**2. A brief introduction to the InterSection marker (ISM) method**

In the search for higher surface tracking fidelity, the InterSection Marker (ISM) method [4, 5] was developed where the proper determination of the interfacial area is critical, such as for the heat and mass transfers process across the interface separating two-phase fluids. An in-depth description of the ISM method is out of the scope of this chapter, and the reader should consult [4, 5] for details. Below, however, highlight the key features of the ISM method to provide the reader with a

The ISM method uses a Lagrangian surface mesh co-located within a uniform Eulerian mesh where upon flow-field qualities such as pressure, velocity and temperature are calculated. The total surface is modelled as a connected series of discrete interfaces (planar polygons), each located within their own *cubic control volume* (CCV). Each planar polygon intersects the edges of the control volume, and the combination of cell-edge intersections uniquely identifies the type of polygon a

The ISM method identifies the type of interface residing in a cell by the combination of cell-edge intersections that interface makes. Total of 51 combinations of basic set of planar-type interfaces had been identified: 8 intersection marker combinations for 3 sided interfaces, 15 for 4 sided, 24 for 5 sided, and 4 for 6 sided – see **Figure 2**. Given the combination of cell edge intersections is unique, a *look-up* table can be used to identify the type of interface located within each cell [4, 5] in a manner similar to that used in the marching cube method [6]. Further subdivisions

and found to be in good agreement.

*Heat Transfer - Design, Experimentation and Applications*

basic understanding.

**Figure 1.**

control volume holds.

**122**

*Planar surfaces co-located within cubic cells can be of 3 to 6 sides.*

**Figure 3.** *Subdivision of planar polygons.*

of these polygons are carried out to maintain planar surface during translation/ deformation – see **Figure 3**. A triangular tessellation pattern is the preferred option because three points randomly translated will always form a plane. Additional intersection-marker combinations of non-planar-type interfaces were also identified (details in [4, 5]), which are necessary to prevent the modelled interface from collapsing and folding onto itself.

After identification of the planar polygons and their sub-divisions, the next step in the ISM method is to identify the component points of the interface – as shown in **Figure 4**: (i) the intersection markers where the interface crosses the control volume cell edges, (ii) the cell face conservation points which allow composite curves to be modelled, and (iii) the raised centroid whose position is calculated to satisfy volumetric conservation. The Volume-of-fluid (VOF) is then calculated by
