study their influence on the deformation of the bubble. This study will present four different shape regimes,

which are obtained by varying the Bo (Bond number) and Mo (Morton number) values within the

corresponding ranges of 1 < Bo < 103 and 5×10-8 < Mo < 102. In addition, simulations are performed using

large density and viscosity ratios of 1000 and 100, respectively. The results are comparable with great precision

to the numerical simulation and experimental data.

Key-Words: - Air bubble, VOF, terminal velocity, Fluent, Bond number, Morton number.

Received: December 27, 2022. Revised: October 30, 2023. Accepted: November 26, 2023. Published: December 29, 2023.

accommodate a wide range of residence times by

manipulating the gas and liquid flow rates, [1].

They are of particular interest for research. Despite

the widespread applications of bubble columns, the

interactions between hydrodynamics, mass transfer

mechanisms, chemical reactions, and yield and

product quality are to date poorly understood. Two-

phase flows are ubiquitous in nature and industrial

applications such as bioreactors, chemical industry,

petrochemical, biochemical, metallurgical

The problem of the rise of a bubble in a liquid at rest

of infinite extension is complex because it involves

very rich physics and coupled mechanisms.

buoyancy in an infinite liquid pool have been the

recirculation) is a limitation for bubble columns:

excessive back-mixing can limit the conversion

efficiency. The reactor may be equipped with

internals, baffles, or sieve plates, to overcome the

back-mixing problem with an inevitable

synthetic fuels via a gas conversion process

vertical column with a constant section. The study's

findings could help us comprehend the actual

physical phenomena and water column design.

Lastly, it suggests novel standards for selecting

mesh and geometry combinations that accurately

depict the actual phenomena.

The initial configuration for this numerical study

corresponds to the study of the rise of an air bubble

in the stagnant water. In this work, the single bubble

diameter of D = 10 mm centred at (x, y) =

(0.0056, 0) is studied. The dimensions of the

column are a height of 100 mm and a width of 40

mm, as presented in Figure 1. The wall boundary

can consider that the density and the viscosity of the

gas (air) contained in the bubble have negligible

effects compared to those of the surrounding liquid

(water). The bubble is subject only to gravitational

forces. The flow is controlled by two dimensionless

numbers: the Bond number (Bo =10), and the

Morton number (Mo = 5×10-6). The parameters of

the numerical simulation are chosen to obtain the

same Bond and Morton number values as in some of

the reference simulation, [8] and experiments, [9].

described on the à scale by the system of Navier-

Stokes equations:

3.1 The VOF Method

The CFD Fluent code has a variety of models

available to incorporate multiphase flow.

liquid-gas interface was monitored through the

uses a discrete function

Morton number:

Where,

(effective gravitational forces) and the surface

tension, but it could also be considered as a

dimensionless size value of the bubble. Mo

describes the properties of the surrounding fluid,

mainly focusing on viscosity and surface tension.

The Froud number is defined as:

And the Reynolds number is defined as:

property group, measures the relative importance of

viscosity and surface tension forces. Following a

similar definition, the Reynolds number signifies the

contribution between the inertia and viscous effects.

3.2 Numerical Method

Before being able to launch a numerical simulation,

it is necessary to carry out several steps, including

the construction of the geometry of the system and

its spatial discretization (mesh), the choice of

the axis (Ox) and walls on the horizontal and

vertical boundaries (Figure 1). The time step was set

to 10-4s. The initial position of the bubble is shown

in Figure 3.

carried out using Fluent. The geometries used are of

the two-dimensional asymmetric type. The

governing equations are solved using an

incompressible approach (pressure-based solver) in

the simulations for a water-air two-phase flow

experimental data, [9]. To be able to study the

temporal evolution of the bubble speed in the

vertical direction along the axis (Ox), it is necessary

to calculate for each time step (t), the average axial

speed (V (t): barycentric speed of the bubble)

defined as:

obtained from x = d/90.

Vectors may be used to illustrate how the velocity

field impacts the bubble, as seen in Figure 7. The

arrows represent the velocity field of the flow,

which is colored according to the vorticity of the

flow. By observing this figure, we see that the flow

low-viscosity liquid (

Mo=10, Bo=5×10-6). (a) Simulation,

(b) Experiments, [9].

Fig. 6: Evolution of bubble circularity (case: Bo=10,

Mo=5×10-6, =1.0)

respectively, and we choose four pairs of values

(Bo, Mo). Table 2 shows the combinations to

choose among the values of viscosity, surface

tension, and the corresponding time step for the

simulation. We note that the validation of our

numerical simulation is based on the shape map

developed by, [13]. These shape maps show the

shape of bubbles for most conditions of practical

interest and are based on dimensionless numbers

(Re, Mo, and Bo); Figure 9.

11, Figure 12, Figure 13 and Figure 14. These

figures show that the Reynolds number of the

bubble plays an important role, i.e., with an increase

in these values, the bubble will change from one

shape to another. The time course of the Reynolds

number of the four bubbles in cases 1 to 4 shown in

Figure 11, Figure 12, Figure 13 and Figure 14 is

quite different because the repulsive irrotational

effect acts in region t/(D/g)0.5 < 3.