Dynamics of rising bubbles in initially quiescent liquids that are later on disturbed by falling drops

Abstract

Three-dimensional dynamics of rising bubbles in initially quiescent liquids that are later on disturbed by falling drops has been numerically investigated. The effects of various parameters such as Eotvos number (Eo), Morton number (M), size of the falling drop, single and multiple falling drops on the dynamics of three different cases represented as single bubble rising, inline bubbles rising and offset bubbles rising are investigated. From the results, it is observed that the final bubble rise velocity in disturbed liquid (with Eo = 38.8 and M = 9.71 × 10−4) due to the impact of falling drop with 0.01 m diameter is increased by 84.21% when compared to the rising bubble in undisturbed quiescent liquid. The same decreases with an increase in the diameter of falling drop up to 0.02 m and 0.03 m, respectively. For inline and offset bubbles, the disturbance caused by the falling drops reduces the final rise velocity by 110% and 113.4%. For Eo = 10.0 and M = 9.71 × 10−8, the low fluids internal resistance causes the bubble to rise faster and reach the liquid surface quickly. However, in the case of inline and offset rising bubbles, the bubbles reach the liquid surface even before the falling drop impacts.

Introduction

Understanding the dynamics of rising bubbles in initially quiescent liquids that are later on disturbed by falling drops is useful in many engineering and industrial applications such as boiling and condensation, fizzing, chemical reactors and pressurized water reactors. One of the quickest possible ways to understand this sort of bubble dynamics is with the use of numerical modeling. This acts as an initial inexpensive tool in the primary evaluation of the design and development of multiphase fluid systems. However, due to the pressure jump across the interface between the two phases as a result of surface tension, as well as due to the presence of discontinuities in the evaluation of fluid properties like density and viscosity, this numerical modeling is rather difficult as given by Larimi and Ramiar [1].

Dynamics of rising bubbles in quiescent liquids has been widely investigated by various researchers. However, most of such numerical investigations are only confined to two-dimensional (2D) analyses as they are less complex and computationally inexpensive. Very few limited studies presented conclusions by focusing on three-dimensional (3D) analyses. 3D numerical studies on bubble rising in viscous liquids are carried out by using a constructive phase-field lattice Boltzmann model and front tracking method as given in Zhang et al. [2] and Hua et al. [3]. Deformation of buoyancy driven air bubbles in shear thinning nanofluids is investigated in Rao et al. [4] by using level set method in COMSOL Multiphysics 4.3a. 3D volume of fluid (VOF) model was employed in Sun et al. [5] to understand the dynamic behavior of bubbles that are continuously rising in shear thinning fluid. Direct numerical simulations of rising bubbles with path instabilities at high Reynolds number have been carried out in Antepara et al. [6]. Dynamics of drop impact on surfaces are experimentally studied in Unnikrishnan et al. [7] at high Weber number. Coalescence dynamics of two unequal-sized drops of same liquid has been numerically investigated using coupled level set and volume of fluid method by Deka et al. [8]. 3D numerical simulation on collisions of two drops are presented in Nobari and Tryggvason [9] by solving full Navier–Stokes equations using a front tracking finite difference method.

Studying the dynamics of rising bubbles in flowing liquids is also an utmost important area of research because the movement of the surrounding liquid influences the motion and shape of the bubble. Quan [10] numerically investigated the effects of co-current flows on rising Taylor bubble. The observations conclude that the upward co-current tends to elongate the bubble, while the downward co-current makes the bubble flatter and shorter. A series of experimental and numerical studies were carried out in Kurimoto et al. [11], Fershtman et al. [12], Lei et al. [13], Nogueira et al. [14], Magnini et al. [15], Tomiyama et al. [16], Mirsandi et al. [17] and Abishek et al. [18] in order to understand the shape and dynamics of bubbles in flowing liquids. As observed in Abishek et al. [18], the dynamics of an air bubble rising in a vertical tube under steady and pulsatile co-current flow is studied by using OpenFOAM-2.1. The study concluded that a new type of ripple appears on the Taylor bubbles in pulsatile shear thinning flows.

Experimental investigation is carried out in Akers and Andrew [19] to study the dynamics of impact of solid sphere on the surface of viscoelastic wormlike micellar fluid. Rein [20] conducted experiments to understand the phenomena of liquid drop impacting with liquid surface. This phenomenon includes bouncing, coalescence and splashing on liquid surfaces. The phenomenon of liquid drop impact with the surface of a deep liquid pool in the context of bubble entrapment is experimentally studied in Raman et al. [21] using high-resolution digital photography. Numerical study is carried out in Deng et al. [22] to investigate the dynamic behavior of two simultaneous droplets impinging on stationary and moving liquid film. The study concluded that for larger separation gap between the droplets, a delay in the formation of central jet is observed. A combination of numerical, experimental and theoretical investigations on drop impact on to a liquid film of finite thickness is presented in Berberovic et al. [23]. In the numerical part of Berberovic et al. [23], advanced free-surface capturing model based on classical VOF model is used. Simultaneous and non-simultaneous impacts of single and multiple droplets on liquid film are numerically investigated in Liang et al. [24], Xu et al. [25] and Liang et al. [26]. The observations conclude that the droplet vertical spacing has negative effects on impact area and minimum residual film thickness.

However, from the above-mentioned literature, it is understood that only four types of experimental and numerical investigations were carried out till date. They are: (1) dynamics of rising bubbles in quiescent liquids as given in Zhang et al. [2], Rao et al. [4] and Antepara et al. [6]; (2) dynamics of drops coalescence as given in Unnikrishnan et al. [7], Nobari and Tryggvason [9]; (3) dynamics of rising bubbles in flowing liquids as given in Quan [10], Magnini et al. [15] and Abishek et al. [18]; and (4) dynamics of liquid drops impacting liquid film as given in Rein [20], Berberovic et al. [23] and Liang et al. [24]. As per the authors’ knowledge, no study has been reported till date on the 3D dynamics of rising bubbles in initially quiescent liquids that are later on disturbed by falling drops. This sort of study will rely on the combined physics of both literature areas mentioned in points 1 and 4 as stated above with coupling effect on one another. Thus, the present study focuses on these important aspects, as understanding of this underlying physics is very much essential in many applications such as closed loop power and refrigeration cycles as well as in the design and development of heat recovery steam generators. The present 3D numerical investigation is carried out by using volume of fluid (VOF) method in an open source C++ code called as OpenFOAM. The InterFOAM solver within OpenFOAM utilizes a model based on VOF method. This adopted numerical solver is initially validated against the experimental and numerical data available in the literature. Later on, this validated multiphase solver is used in the present investigation. The effects of various parameters such as Eotvos number, Morton number, size of the falling drop, single and multiple falling drops in different configurations for three different cases represented as single bubble rising, inline bubbles rising and offset bubbles rising are comprehensively investigated and presented. The present results are also useful in the design and development of multiphase fluid systems like chemical reactors and pressurized water reactors.

Problem description

Figure 1a represents the 3D geometrical domain adopted in the present numerical investigation. The length (L), height (H) and width (W) of the domain are chosen to be 0.08 m, 0.16 m and 0.08 m, respectively. As observed in Fig. 1b, at time t = 0 s, a pre-defined bubble of 0.01 m diameter (Db) is located exactly near the center of the domain at a distance of Db above the bottom face. Moreover, a pre-defined drop of 0.01 m diameter (Dd) is also located exactly near the center of the domain at a distance of Dd below the top face. It should be noted that for each test case as well as for each configuration, the properties of the falling drop are same as that of the liquid in which the bubble rises. Similarly, the properties of the medium in which the drop falls are also same as that of the gas comprised within the rising bubble.

Fig. 1
figure1

Representation of geometry: a Adopted domain in 3D and b location of bubble and drop in 2D

Single and multiple falling drops in a total of five different configurations as represented by Config-1 to Config-5 and as shown in Fig. 2 are used in the present study. Furthermore, two different types of pre-defined bubbles as observed in Fig. 3 are also adopted in the present investigation. Along with single pre-defined bubble rising as given in Config-1 of Fig. 2, two more types defined as inline bubbles rising and offset bubbles rising as seen in Fig. 3 are used. The configurations and types presented in Figs. 2 and 3 are selected in a way to understand how the bubble dynamics will vary when drops fall in different positions. This is because in real time systems, different drop and bubble configurations are possibly depending upon the working conditions. As the bubbles and drops are comprised of two different fluids with different density values, the pre-defined bubbles tend to rise from the starting instant (t = 0 s) as they are lighter in density than the surrounding quiescent liquid. At this same instance, the pre-defined drops also start falling, thereby causing them to impact with the surface of the quiescent liquid. This impact causes a disturbance in the initially quiescent liquid which in turn will adversely affect the dynamics of rising pre-defined bubbles.

Fig. 2
figure2

Types of pre-defined falling drops implemented: Config-1 (single falling drop), Config-2 (two adjacent falling drops separated by 1.5Dd), Config-3 (two adjacent falling drops separated by 3Dd), Config-4 (two offset falling drops separated by 0.5Dd) and Config-5 (two offset falling drops separated by 1.5Dd)

Fig. 3
figure3

Different types of pre-defined bubbles adopted: inline bubbles and offset bubbles

Mathematical representation

InterFOAM multiphase solver is used in the present 3D numerical investigation. It lies within OpenFOAM which is an open source C++ code. This solver utilizes a model based on VOF method which is a phase-fraction-based interface capturing approach and is used for two immiscible, isothermal and incompressible fluids. This implemented numerical solver is initially validated against the numerical and experimental data given in the literature. Subsequently, the present study is carried out with this validated multiphase solver. OpenFOAM version 5.0 is used in the present numerical investigation. The governing equations, boundary conditions and evaluation of the rise velocity of bubble used in the present investigation are explained in the below sections.

Governing equations

The Navier–Stokes equations for two immiscible, isothermal and incompressible fluids are solved in InterFOAM multiphase solver. Thus, apart from the interface between two fluids, the material properties remain constant in all other regions filled with any one of the two fluids. The equations for volume fraction and velocity as given in [27, 28] are as follows.

$$\frac{\partial \alpha }{\partial t} + {\mathbf{\nabla }}.(\alpha {\mathbf{U}} ) { + }{\mathbf{\nabla }}.(\alpha (1 - \alpha ){\mathbf{U}}_{\text{r}} ) { = 0}$$
(1)

Here, \(\alpha\) represents the volume fraction whose value in a cell will range between 0 and 1. In OpenFOAM, the necessary compression of the surface is achieved by introducing an extra artificial compression term \(\left( {{\mathbf{\nabla }}.(\alpha ( 1- \alpha ){\mathbf{U}}_{\text{r}} )} \right)\) [29, 30], where \({\mathbf{U}}_{\text{r}}\) is the relative velocity applied in a direction normal to the interface.

$${\mathbf{\nabla }}.{\mathbf{U}} = 0$$
(2)
$$\frac{{\partial \rho {\mathbf{U}}}}{\partial t} + {\mathbf{\nabla }}.(\rho {\mathbf{UU}} ) { = - }{\mathbf{\nabla }}p + {\mathbf{\nabla }}.({\varvec{\uptau}} + {\varvec{\uptau}}_{\text{t}} ) + \rho {\mathbf{g}} + {\mathbf{f}}_{\sigma }$$
(3)

Here, \({\mathbf{U}}\) represents velocity vector, \(\rho\) represents net density, p represents pressure, and turbulent and viscous stress tensors are represented by \({\varvec{\uptau}}_{\text{t}}\) and \({\varvec{\uptau}}\). Since no turbulence effect is considered in the present study, \({\varvec{\uptau}}_{\text{t}}\) remains zero. However, viscous stress represented by \({\varvec{\uptau}}\) in (3) will be evaluated as \(\mu ({\mathbf{\nabla U}}{ + }{\mathbf{\nabla U}}^{T} )\). The surface tension force is represented by \({\mathbf{f}}_{\sigma }\) which is modeled on the basis of continuum surface force (CSF) as given in the below equation.

$${\mathbf{f}}_{\sigma } = \sigma k{\mathbf{\nabla }}\alpha$$
(4)

The value of \(k\) is obtained as given in the below equation, and it represents the curvature.

$$k = - \nabla .\left( {\frac{\nabla \alpha }{{\left| {\nabla \alpha } \right|}}} \right)$$
(5)

The density (\(\rho\)) and viscosity (\(\mu\)) values in (2) and (3) rely on the volume fraction (\(\alpha\)) as well as on the individual densities and viscosities of two fluids as shown below.

$$\rho \, = \, \alpha \rho_{1} + \, (1 - \alpha )\rho_{2}$$
(6)
$$\mu \, = \, \alpha \mu_{1} + \, (1 - \alpha )\mu_{2}$$
(7)

The value of \(\alpha\) = 0 inside fluid 2 with density \(\rho_{2}\) and viscosity \(\mu_{2}\) and its value will be 1 inside fluid 1 with density \(\rho_{1}\) and viscosity \(\mu_{1}\).

The shape of bubbles moving in surrounding liquid is characterized in the present numerical investigation by using two very important dimensionless numbers called as Eotvos number (Eo) and Morton number (M). The importance of gravitational forces in comparison with surface tension forces is represented by Eo and is given as follows.

$${\text{Eo }} = \, \frac{{g(\rho_{\text{l}} - \rho_{\text{g}} )D_{\text{b}}^{2} }}{\sigma }$$
(8)

In the above equation, Db represents the diameter of the pre-defined bubble at time t = 0 s.

However, representing the bubble shape purely in terms of the physical properties of the surrounding medium is done through Morton number that is used along with Eo and is given as follows.

$$M \, = \, \frac{{g\mu_{\text{l}}^{4} (\rho_{\text{l}} - \rho_{\text{g}} )}}{{\rho_{\text{l}}^{2} \sigma^{3} }}$$
(9)

Here, \(\rho_{\text{l}}\) and \(\rho_{\text{g}}\) represent liquid and gas densities; g represents acceleration due to gravity; \(\sigma\) represents surface tension of the liquid; and \(\mu_{\text{l}}\) represents dynamic viscosity of the liquid.

The bubble terminal Reynolds number (ReT) is evaluated as shown below and is defined as the ratio of inertia force to viscous force.

$${\text{Re}}_{\text{T}} = \, \frac{{\rho_{\text{l}} U_{{z_{\text{T}} }} D_{\text{eq}} }}{{\mu_{\text{l}} }}$$
(10)

Here, \(U_{{z_{\text{T}} }}\) is the terminal velocity of the bubble in vertical direction (direction of the bubble rise) and \(D_{\text{eq}}\) is the equivalent diameter of the bubble at the last instant of time (just before the bubble merges up with the falling drop after its impact with the liquid surface/just before the bubble touches the liquid surface even before the drop impacts with it).

The physical properties of liquid and gas used in the present study for different Eo and M combinations are given in Table 1.

Table 1 Physical properties of liquid and gas for different Eo and M combinations

Boundary conditions

All the field variables within the specified 3D geometry are obtained by applying appropriate boundary conditions from the basis of validation. Slip boundary condition is implemented on all the four vertical faces of the domain, whereas for the bottom as well as top faces, a fixed velocity value of zero is implemented. For the case of volume fraction, all the six faces of the geometry are implemented with zero gradient boundary condition. Neumann boundary condition with a pressure gradient of zero is implemented on five faces of the geometry, whereas Dirichlet boundary condition with a fixed uniform pressure value of zero is implemented on the top face.

Evaluation of the rise velocity of bubble

The pre-defined bubbles tend to start rising from the starting instant (t = 0 s) as they are lighter in density than the surrounding quiescent liquid. At this same instance, the pre-defined drops also start falling, thereby causing them to impact with the surface of the quiescent liquid. This impact causes a disturbance in the initially quiescent liquid which in turn will adversely affect the dynamics of rising pre-defined bubbles. The rise velocity of the pre-defined bubble at respective time intervals is evaluated as given in the below equation.

$$U_{\text{b}} = \frac{{\sum \alpha U_{z} }}{\sum \alpha }$$
(11)

The value of \(\alpha\) = 0 within the liquid that surrounds the bubble whereas \(\alpha\) = 1 within the bubble. At the interface between the two fluids that is between the bubble and surrounding liquid, the value of \(\alpha\) lies between 0 and 1. The component of velocity in the direction of bubble rise that is in Z-direction is denoted by \(U_{z}\).

Validation

Experimental and numerical results from the literature are used to validate the InterFOAM multiphase solver that is used in the present numerical study. To do so, the geometrical domains with relevant boundary conditions under optimum grid sizes as given in the particular literature are implemented into the numerical solver.

Initially, two different test cases of 2D benchmark numerical results from Hysing et al. [31] are used in the comparison as shown in Fig. 4 for pre-defined bubble that is rising in quiescent liquid. The non-dimensional radius of the pre-defined bubble is chosen to be 0.25 as given in Hysing et al. [31]. As observed in Fig. 4, the non-dimensional quantities such as bubble rise velocity and centroid are plotted with respect to non-dimensional time. The bubble rise velocity, centroid and time are non-dimensionalized by using scaling factors such as \(\sqrt {gD_{\text{b}} }\), \(D_{\text{b}}\), \(\frac{{D_{\text{b}} }}{{\sqrt {gD_{\text{b}} } }}\), respectively. It should be noted that the plot of the centroid refers to the position of the centroid as a function of the position in Z axis (vertical direction, i.e., direction of bubble rise). Figure 4a, b represents test case 1 in which the value for both density and dynamic viscosity ratios is chosen to be 10, whereas Fig. 4c, d represents test case 2 in which the values of density and dynamic viscosity ratios are chosen to be 1000 and 100, respectively. From Fig. 4, it is clearly understood that the results obtained with the present numerical solver are in reasonable agreement with the 2D benchmark numerical results in the literature.

Fig. 4
figure4

Validation with 2D benchmark numerical results in terms of non-dimensional quantities: a, c rise velocity of bubble with respect to time for test cases 1 and 2, b and d centroid of bubble with respect to time for test cases 1 and 2

Later on, 3D numerical results from Van Sint Annaland et al. [32] at different Eotvos and Morton numbers as given in Table 2 are also used in the validation of the present numerical solver. The radius of the initial spherical bubble is chosen to be 0.01 m as given in Van Sint Annaland et al. [32]. As observed from Table 2, the terminal bubble shapes obtained with the use of present numerical solver are having very good similarity with those from Van Sint Annaland et al. [32]. It is also found that the obtained terminal Reynolds number exhibits good agreement with the respective values from Van Sint Annaland et al. [32]. It is noticed that an acceptable maximum percentage deviation of 6.38% is obtained.

Table 2 Validation of present numerical solver using 3D numerical results of Van Sint Annaland et al. [32]

Finally, the experimental and numerical results of Brereton and Korotney [33] as well as Van Sint Annaland et al. [34] for the cases of offset and inline bubbles are used in validating the present numerical solver as given in Table 3. A 3D square column filled with quiescent liquid and containing two spherical bubbles with a radius of 0.005 m each is used here. As seen in Table 3, the position and shape of the bubbles obtained by using the present numerical solver at different times are compared. It is to be observed that, for both the cases of offset and inline bubbles, the two bubbles are initially separated by a distance of 0.015 m. However, for the case of offset bubbles, the offset distance between the two bubbles is initially chosen as 0.008 m. As an observation from Table 3, it is understood that the results obtained with the present numerical solver are in well agreement with the numerical and experimental results available in the literature.

Table 3 Validation with experimental and numerical results of Brereton and Korotney [33] as well as Van Sint Annaland et al. [34] for the cases of offset and inline bubbles

After validating the reliability of the present InterFOAM multiphase solver with the existing results available in the literature, the validated multiphase solver is then used in the present numerical investigation.

Domain independence test

The influence of domain boundary on the rising bubble is evaluated in the present numerical study by carrying out the domain independence test. This test will ensure us to obtain the relevant domain size that will have negligible effect on the rising bubble. As observed in Fig. 5, the rise velocities of the bubble at different times are plotted for four domain sizes (0.02 m × 0.02 m × 0.04 m, 0.04 m × 0.04 m × 0.08 m, 0.08 m × 0.08 m × 0.16 m and 0.16 m × 0.16 m × 0.32 m) in initially quiescent liquid under the absence of falling drop. To do this test, the values of Eo and M are chosen to be 38.8 and 9.71 × 10−4, respectively.

Fig. 5
figure5

Rise velocity of bubble with respect to time for different domain sizes under the absence of falling drop

For smaller domain size, as in case of 0.02 m × 0.02 m × 0.04 m, the obtained bubble rise velocity is lower when compared with others. This is because of the viscous effect imposed on the rising bubble by the boundary of the domain. Moreover, for any increment in the domain size beyond 0.08 m × 0.08 m × 0.16 m, no change in the rise velocity of the bubble is observed as seen in Fig. 5. Hence, this particular domain size of 0.08 m × 0.08 m × 0.16 m is adopted in the present study as the solution is domain independent. Furthermore, this adopted domain size also justifies the observation given in Hua and Lou [35] as well as Amaya-Bower and Lee [36]. This states that, for having negligible wall effects, the radial and axial domain sizes should be of 4Db and 12Db, respectively.

Grid independence test

The influence of grid size on the accuracy of solution is evaluated in the present numerical study by carrying out the grid independence test. This test will ensure us to obtain the optimum grid size beyond which the solution will not be altered. As observed in Fig. 6, the rise velocities of the bubble at different times are plotted for four grid sizes (40 × 40 × 80, 60 × 60 × 120, 90 × 90 × 180 and 120 × 120 × 240) in initially quiescent liquid under the absence of falling drop. To do this test, the values of Eo and M are chosen to be 38.8 and 9.71 × 10−4, respectively. It is clearly noticed that for any increment in the grid size beyond 60 × 60 × 120, no change in the rise velocity of the bubble is observed as seen in Fig. 6. Hence, this particular grid size of 60 × 60 × 120 is adopted in the present study as the solution is grid independent.

Fig. 6
figure6

Rise velocity of bubble with respect to time for different grid sizes under the absence of falling drop

Results and discussion

The present numerical investigation is carried out to investigate the 3D dynamics of rising bubbles in initially quiescent liquids that are later on disturbed by falling drops. The InterFOAM solver within OpenFOAM that utilizes a model based on VOF method is used to serve this purpose. The effects of various parameters such as Eotvos number, Morton number, size of the falling drop, single and multiple falling drops in different configurations for three different cases represented as single bubble rising, inline bubbles rising and offset bubbles rising are comprehensively investigated and presented.

Four test cases as given by Van Sint Annaland et al. [32] and as shown in Table 2 are used in the present investigation.

Effect of the size of falling drop

The effect of the size of falling drop on the volume fraction contours as well as on the rise velocity of the bubble is shown in Fig. 7 for Eo = 38.8 and M = 9.71 × 10−4. In the case of no falling drop as observed in Fig. 7d, the bubble rise velocity gradually increases with time and then becomes unchanged as the bubble attains its terminal velocity. This type of behavior is mostly common for the case of single bubble rising in quiescent liquids. However, in the case of falling drop with incremental size, the bubble rise velocity decreases when compared to the one with smaller size drop. This is because more thrust will be developed in the downward direction by the bigger falling drop after its impact with the liquid surface. This developed thrust will restrict the bubble to rise, thereby reducing its velocity as seen in Fig. 7d. From Fig. 7a–c, it can be noticed that the bigger sized falling drop will have high influence on the initially quiescent liquid. For a smaller drop diameter of 0.01 m, the impact starts at 0.44 s. From this point of impact, the bubble rise velocity decreases as observed in Fig. 7d until 0.5 s. This is because of the thrust developed in the downward direction by the falling drop that restricts the bubble to rise. Again, from 0.5 s, the rise velocity increases until 0.59 s even though the drop penetrates deep into the liquid. This is because the dispersed liquid as a result of the drop penetration will push the bubble upward from behind, thereby increasing its velocity. During the initial times after the drop impact, some amount of liquid will be dispersed toward the domain boundary because of the thrust developed as a result of the drop penetration into the liquid. This thrust is responsible for the initial decrement in the rise velocity of the bubble. Later on, the dispersed liquid will try to move behind the rising bubble, thereby uplifting it with incremental velocity even though the drop still penetrates into the liquid. This movement of the dispersed liquid behind the rising bubble can be observed through the velocity vectors in the liquid region plotted as shown in Fig. 8 for falling drop with diameter of 0.01 m. As observed in Fig. 8, from 0.5 to 0.59 s, more velocity vectors gradually point toward the region behind the bubble which shows that the gradual movement of liquid behind the rising bubble uplifts it with incremental velocity. The final rise velocity of the bubble in the case of Dd = 0.01 m is 84.21% higher than that of the one with no falling drop. The Deq values of the bubble in the case of Dd = 0.01 m, 0.02 m and 0.03 m are 0.0078 m, 0.00712 m and 0.00691 m, respectively, whereas the corresponding final Re values are found to be 27.5, 11.24 and 3.08.

Fig. 7
figure7

Effect of the size of falling drop on the volume fraction contours and rise velocity of the bubble for Eo = 38.8 and M = 9.71 × 10−4: a for Dd = 0.01 m, b for Dd = 0.02 m, c for Dd = 0.03 m and d bubble rise velocity with respect to time

Fig. 8
figure8

Velocity vectors in the liquid region under falling drop with Dd = 0.01 m for Eo = 38.8 and M = 9.71 × 10−4: a at time = 0.5 s, b at time = 0.55 s and c at time = 0.59 s

Effect of different configurations of falling drops as in Fig. 2

For single rising bubble

The effect of different configurations of falling drops on the volume fraction as well as on the rise velocity of single rising pre-defined bubble is shown in Fig. 9 for Eo = 38.8 and M = 9.71 × 10−4. The volume fraction for different falling drop configurations is given at final time that is the maximum time beyond which the rising pre-defined bubble will merge with the penetrating drop or it may reach the surface of the liquid. The bubble rise velocity is also plotted by focusing at different times after the impact of the falling drop with the liquid surface. From Fig. 9, it is observed that the rise velocity in the case of Config-3 is almost similar to the case in which falling drop is absent. This is because the two falling drops in Config-3 are separated by large distance (3Db) that has almost negligible effect on the rising bubble dynamics. This negligible effect is also responsible for the rising pre-defined bubble to attain sufficiently large time (0.7 s) before reaching the liquid surface when compared with other configurations as seen in Fig. 9. Moreover, from Fig. 9, it is also observed that the final Reynolds number in the case of Config-1 is 93.98% higher than that of the case in which falling drop is absent. This is because in the case of Config-1, a single drop falls exactly in line with the rising bubble. This in line falling drop will disperse some amount of liquid toward the domain boundary because of the thrust developed as a result of the drop penetration into the liquid after its impact with the liquid surface. This thrust is responsible for the initial decrement in the rise velocity of the bubble as explained in Fig. 7. Later on, the dispersed liquid will try to move behind the rising bubble, thereby uplifting it with incremental velocity even though the drop still penetrates into the liquid. Due to the inline nature of the falling drop with the rising bubble, this uplifting force will be higher for Config-1 that is responsible for higher value of Re when compared with other configurations. The Deq and final Re value of the bubble for Config-1 is 0.0078 m and 27.5. Furthermore, at 0.5 s, the bubble rise velocity is lower for the case of Config-4. This is because the distance of separation between the two offset falling drops is very small (0.5Db) in such a way that both the drops start merging as a single large unit starting from the first instance. This bigger falling drop will develop more downward thrust after its impact with the liquid surface. This developed thrust will impose higher restriction on the bubble to rise, thereby reducing its velocity in comparison with other configurations.

Fig. 9
figure9

Effect of different configurations of falling drops on the volume fraction as well as on the rise velocity of single rising pre-defined bubble for Eo = 38.8 and M = 9.71 × 10−4

For inline rising bubbles

The effect of different configurations of falling drops on the volume fraction as well as on the rise velocity of inline rising pre-defined bubbles is shown in Fig. 10 for Eo = 38.8 and M = 9.71 × 10−4. As observed in Fig. 3, the distance between the bubbles is chosen to be 1.5Db and the latter bubble is present at a distance of Db above the bottom face of the domain. The volume fraction for different falling drop configurations is given at final time that is the maximum time beyond which the rising pre-defined bubble will merge with the penetrating drop or it may reach the surface of the liquid. The bubble rise velocity is also plotted by focusing at different times after the impact of the falling drop with the liquid surface. By the time the drop impacts with the liquid surface, the two rising bubbles already merge with each other in all the configurations. Therefore, the rise velocities are plotted by considering merged bubbles. In the rise velocity plot, the velocities at respective times before merging up of the bubbles are also considered by taking the velocities of both the bubbles into account. The time at which the falling drops will impact the liquid surface is 0.44 s for Config-1, Config-2 and Config-3, whereas this time will be 0.42 s for Config-4 and 0.4 s for Config-5, respectively. The rising front bubble will create a wake right behind it. This wake will completely influence the back inline bubble causing it to rise faster due to less resistance as a result of less drag force from the surrounding liquid. As a result of this, the back inline bubble will merge with the rising front bubble and the time at which both the bubbles start merging is 0.33 s. After merging of the two inline bubbles into a single large bubble, the rise velocity decreases (from 0.33 to 0.44 s). This is because the tendency of the single large bubble to rise will be lower as a result of greater drag force offered by the surrounding liquid. Beyond 0.44 s, the bubble rise velocity reduces further due to the developed downward thrust that restricts the bubble to rise, thereby reducing its velocity as observed in Fig. 10. Furthermore, from Fig. 10, it is also observed that the final bubble rise velocity for Config-4 is lower than that of all other configurations. This is because as explained in Fig. 9, the distance of separation between the two offset falling drops is very small (0.5Db) in such a way that both the drops start merging as a single large unit starting from the first instance. This bigger falling drop will develop more downward thrust after its impact with the liquid surface. This developed thrust will impose higher restriction on the bubble to rise, thereby reducing its velocity in comparison with other configurations. The change in the sign of the rise velocity of the bubble represents that the trajectory of the bubble is displaced toward the bottom of the domain due to the influence of the drops. It is also observed that the value of Deq for Config-4 is found to be 0.01043 m and its final Re is 127.48% lower than that of one obtained for the case of no falling drop.

Fig. 10
figure10

Effect of different configurations of falling drops on the volume fraction as well as on the rise velocity of inline rising pre-defined bubbles for Eo = 38.8 and M = 9.71 × 10−4

For offset rising bubbles

The effect of different configurations of falling drops on the volume fraction as well as on the rise velocity of offset rising pre-defined bubbles is shown in Fig. 11 for Eo = 38.8 and M = 9.71 × 10−4. The volume fraction for different falling drop configurations is given at final time that is the maximum time beyond which the rising pre-defined bubble will merge with the penetrating drop or it may reach the surface of the liquid. The bubble rise velocity is also plotted by focusing at different times after the impact of the falling drop with the liquid surface. From Fig. 11, it is observed that the trends in the bubble rise velocities are similar to those obtained in Fig. 10 but the quantification values are different. The time at which the falling drops will impact the liquid surface is 0.44 s for Config-1, Config-2 and Config-3, whereas this time will be 0.42 s for Config-4 and 0.4 s for Config-5, respectively. The rising front bubble will create a wake right behind it. This wake will influence the back offset bubble causing it to rise faster due to less resistance as a result of less drag force from the surrounding liquid. As a result of this, the back offset bubble will merge with the rising front bubble and the time at which both the bubbles start merging is 0.36 s. After completely merging of the two offset bubbles into a single large bubble, the rise velocity decreases (from 0.4 to 0.44 s). This is because the tendency of the single large bubble to rise will be lower as a result of greater drag force offered by the surrounding liquid. Beyond 0.44 s, the bubble rise velocity reduces further due to the developed downward thrust that restricts the bubble to rise, thereby reducing its velocity as observed in Fig. 11. Furthermore, from Fig. 11, it is also observed that the final bubble rise velocity for Config-4 is lower than that of all other configurations. This is because as explained in Fig. 9, the distance of separation between the two offset falling drops is very small (0.5Db) in such a way that both the drops start merging as a single large unit starting from the first instance. This bigger falling drop will develop more downward thrust after its impact with the liquid surface. This developed thrust will impose higher restriction on the bubble to rise, thereby reducing its velocity in comparison with other configurations. It is also observed that the value of Deq for Config-4 is found to be 0.00997 m and its final Re is 142.24% lower than that of one obtained for the case of no falling drop.

Fig. 11
figure11

Effect of different configurations of falling drops on the volume fraction as well as on the rise velocity of offset rising pre-defined bubbles for Eo = 38.8 and M = 9.71 × 10−4

Effect of different types of fluids under Config-1

For single rising bubble

Figure 12 shows the velocity of single rising bubble with respect to time for different fluids under Config-1 of Fig. 2. Different Eo and M values as seen in Fig. 12 represents different fluid types. From Fig. 12, it is observed that the rise velocity of bubble increases gradually during the initial times and then becomes unchanged as it reaches terminal velocity. However, in the lateral times (after the falling drop impacts with the liquid surface), there seems to be fluctuations in the rise velocity because of the disturbance created in the initially quiescent liquid due to the impact of the falling drop with liquid surface. In the case of no falling drop, no disturbance in the rise velocity is observed as the bubble rises up in a quiescent liquid till the end. But in the case of falling drop, as seen for Eo = 38.8 and M = 9.71 × 10−4, the rise velocity decreases from 0.44 s and then again increases from 0.5 s. This is because as explained in Figs. 7 and 9, the falling drop will disperse some amount of liquid toward the domain boundary because of the thrust developed as a result of the drop penetration into the liquid after its impact with the liquid surface. This thrust is responsible for the initial decrement in the rise velocity of the bubble. Later on, the dispersed liquid will try to move behind the rising bubble, thereby uplifting it with incremental velocity even though the drop still penetrates into the liquid. Among all the Eo and M values given in Fig. 12, Eo = 1.0 is the lowest value and its corresponding M = 1.26 × 10−3 is the highest value. For the same density difference of two fluids, the lower value of Eo represents incremental value of surface tension and higher value of M represents incremental value of liquid dynamic viscosity. The values of dynamic viscosity and surface tension for Eo = 1.0 and M = 1.26 × 10−3 are 0.5249 kg/m sec and 0.3881 kg/sec2, respectively. These higher values are responsible for higher fluids internal resistance which in turn is responsible for lower bubble rise velocity as seen in Fig. 12. Moreover, from Fig. 12, it is also observed that the rise velocity increases after 0.6 s for Eo = 1.0 and M = 1.26 × 10−3. This is because of the impact of the falling drop with the liquid surface that is responsible for the uplifting of rising bubble with incremental velocity. Furthermore, for the case of Eo = 10.0 and M = 9.71 × 10−8, the obtained dynamic viscosity value is 90.14% lower than the one obtained for Eo = 10.0 and M = 9.71 × 10−4. This lower dynamic viscosity value is responsible for low fluids internal resistance which in turn is responsible for higher bubble rise velocity for Eo = 10.0 and M = 9.71 × 10−8. However, beyond 0.4 s, the bubble rise velocity decreases drastically. This is because the low fluids internal resistance causes the bubble to rise faster, thereby reaching the liquid surface quickly. During this reach, the bubble comes very close to the penetrating drop, thereby undergoing the thrust that is being developed by the drop. This thrust will drastically restrict the bubble to rise causing its velocity to decrease all of a sudden as observed in Fig. 12. From Fig. 12, it is observed that the final rise velocity for Eo = 10.0, M = 9.71 × 10−8 is 104.76% lower than the value for Eo = 38.8, M = 9.71 × 10−4 under Config-1.

Fig. 12
figure12

Velocity of single rising bubble with respect to time for different fluids under Config-1

For inline rising bubbles

The velocity of inline rising bubbles with respect to time for different fluids under Config-1 is shown in Fig. 13. From Fig. 13, it is observed that the bubble rise velocity increases with time. This is because the rising front bubble will create a wake exactly behind it and this wake will completely influence the inline back bubble causing it to rise faster due to less resistance as a result of less drag force from the surrounding liquid. However, in the lateral times as also observed in Fig. 12, there seems to be fluctuations in the rise velocity because of the disturbance created in the initially quiescent liquid due to the impact of the falling drop with liquid surface. In the case of no falling drop, no disturbance in the rise velocity is observed as the bubbles rise in a quiescent liquid till the end. But, the rise velocity in the case of no falling drop increases initially and then starts decreasing after 0.33 s. This is because at 0.33 s, both the inline bubbles start merging to form a single large bubble whose tendency to rise will be lower because it will be under the influence of more drag offered by the surrounding liquid. For both the fluids with Eo = 1.0, M = 1.26 × 10−3 and Eo = 10.0, M = 9.71 × 10−8, no fluctuation in the rise velocity is observed. This is because the low fluids internal resistance in the case of Eo = 10.0, M = 9.71 × 10−8 is responsible for the rising inline bubbles to reach the liquid surface before the impact of the falling drop with the liquid, whereas the high surface tension in the case of Eo = 1.0, M = 1.26 × 10−3 is responsible for the inline bubbles to rise very slowly in such a way that the disturbance caused by the impact of falling drop will be far and has no effect on the rising inline bubbles.

Fig. 13
figure13

Velocity of inline rising bubbles with respect to time for different fluids under Config-1

In the case of Eo = 10.0, M = 9.71 × 10−4, the falling drop impacts with the liquid surface at 0.46 s. Due to this impact, a downward thrust will be developed that will impose restriction on the bubbles to rise, thereby reducing its velocity. As explained in Fig. 7, during the initial times after the drop impact, some amount of liquid will be dispersed toward the domain boundary because of the thrust developed as a result of the drop penetration into the liquid. This thrust is responsible for the initial decrement in the rise velocity of the bubbles. Later on, the dispersed liquid will try to move behind the rising bubbles, thereby uplifting them with incremental velocity even though the drop still penetrates into the liquid. However, for Eo = 38.8 and M = 9.71 × 10−4 under Config-1, the falling drop impacts with the liquid surface at 0.44 s. By this time, the rising inline bubbles merged together and reach close to the liquid surface, thereby coming very close to the penetrating drop undergoing the thrust that is being developed by the drop. This thrust will drastically restrict the bubble to rise causing its velocity to decrease all of a sudden as observed in Fig. 13. From Fig. 13, it is observed that the final rise velocity for Eo = 38.8, M = 9.71 × 10−4 under Config-1 is 110% lower than the value of same fluid under the absence of falling drop.

For offset rising bubbles

The velocity of offset rising bubbles with respect to time for different fluids under Config-1 is shown in Fig. 14. All the fluids in Fig. 14 exhibit similar resemblance to their corresponding trends in Fig. 13 but with different quantification values. The difference in the quantification values is because the two rising pre-defined bubbles in Fig. 14 are offset from each other. From Fig. 14, it is observed that the bubble rise velocity increases with time. This is because the rising front bubble will create a wake exactly behind it and this wake will influence the offset back bubble causing it to rise faster due to less resistance as a result of less drag force from the surrounding liquid. However, in the lateral times as also observed in Fig. 12, there seems to be fluctuations in the rise velocity because of the disturbance created in the initially quiescent liquid due to the impact of the falling drop with liquid surface. In the case of no falling drop, no disturbance in the rise velocity is observed as the bubbles rise in a quiescent liquid till the end. But, the rise velocity in the case of no falling drop increases initially and then starts decreasing after 0.4 s. This is because at 0.4 s, both the offset bubbles completely merge together to form a single large bubble whose tendency to rise will be lower because it will be under the influence of more drag offered by the surrounding liquid. For both the fluids with Eo = 1.0, M = 1.26 × 10−3 and Eo = 10.0, M = 9.71 × 10−8, no fluctuation in the rise velocity is observed. This is because the low fluids internal resistance in the case of Eo = 10.0, M = 9.71 × 10−8 is responsible for the rising offset bubbles to reach the liquid surface before the impact of the falling drop with the liquid, whereas the high surface tension in the case of Eo = 1.0, M = 1.26 × 10−3 is responsible for the offset bubbles to rise very slowly in such a way that the disturbance caused by the impact of falling drop will be far and has no effect on the rising offset bubbles. In the case of Eo = 10.0, M = 9.71 × 10−4, the falling drop impacts with the liquid surface at 0.46 s. Due to this impact, a downward thrust will be developed that will impose restriction on the bubbles to rise, thereby reducing its velocity. As explained in Fig. 7, during the initial times after the drop impact, some amount of liquid will be dispersed toward the domain boundary because of the thrust developed as a result of the drop penetration into the liquid. This thrust is responsible for the initial decrement in the rise velocity of the bubbles. Later on, the dispersed liquid will try to move behind the rising bubbles, thereby uplifting them with incremental velocity even though the drop still penetrates into the liquid. However, for Eo = 38.8 and M = 9.71 × 10−4 under Config-1, the falling drop impacts with the liquid surface at 0.44 s. By this time, the rising inline bubbles merged together and reach close to the liquid surface, thereby coming very close to the penetrating drop undergoing the thrust that is being developed by the drop. This thrust will drastically restrict the bubble to rise causing its velocity to decrease all of a sudden as observed in Fig. 14. From Fig. 14, it is observed that the final rise velocity for Eo = 38.8, M = 9.71 × 10−4 under Config-1 is 113.4% lower than the value of same fluid under the absence of falling drop.

Fig. 14
figure14

Velocity of offset rising bubbles with respect to time for different fluids under Config-1

Conclusions

The present numerical investigation is carried out to investigate the 3D dynamics of rising bubbles in initially quiescent liquids that are later on disturbed by falling drops. The InterFOAM solver within OpenFOAM that utilizes a model based on VOF method is used to serve this purpose. The effects of various parameters such as Eotvos number, Morton number, size of the falling drop, single and multiple falling drops in different configurations for three different cases represented as single bubble rising, inline bubbles rising and offset bubbles rising are comprehensively investigated and presented. The following conclusions are made from the present investigation.

  • The final bubble rise velocity in disturbed liquid (with Eo = 38.8 and M = 9.71 × 10−4) due to the impact of falling drop with Dd = 0.01 m is increased by 84.21% when compared to the rising bubble in undisturbed quiescent liquid. The same decreases with an increase in the diameter of falling drop up to 0.02 m and 0.03 m, respectively.

  • Config-1 results in 93.98% higher value of final Reynolds number when compared with that of no falling drop case. This is because in the case of single falling drop, from 0.5 to 0.59 s, more velocity vectors gradually point toward the region behind the bubble which shows that the gradual movement of liquid behind the rising bubble uplifts it with incremental velocity.

  • For inline and offset rising bubble cases, the disturbance caused by the falling drops reduced the final rise velocity. The corresponding final Reynolds number values for Config-4 are 127.48% and 142.24% lower than that of the one obtained for the case of no falling drop. This is because of the merging of the two offset falling drops into a single large unit that develops more downward thrust after its impact.

  • For Eo = 10.0 and M = 9.71 × 10−8, the dynamic viscosity value is 90.14% lower than the one obtained for Eo = 10.0 and M = 9.71 × 10−4. This is responsible for low fluids internal resistance that causes the bubble to rise faster and reach the liquid surface quickly. In the case of single rising bubble, the bubble comes very close to the penetrating drop, thereby undergoing the thrust that is being developed by the drop, whereas in the case of inline and offset rising bubbles, the bubbles reach the liquid surface even before the falling drop impacts with the liquid.

  • For the cases of inline and offset rising pre-defined bubbles in fluid with Eo = 38.8 and M = 9.71 × 10−4, the final rise velocities are 110% and 113.4% lower than the values of same fluid under the absence of falling drop.

The present results will be very much helpful in closed loop power and refrigeration cycles as well as in the design and development of heat recovery steam generators.

Abbreviations

D :

Diameter (m)

Eo:

Eotvos number

f :

Force vector (kg m s−2)

g :

Acceleration due to gravity (m s−2)

H :

Height (m)

k :

Curvature (m−1)

L :

Length (m)

M :

Morton number

p :

Pressure (kg m−1 s−2)

Re:

Reynolds number

t :

Time (s)

U :

Velocity vector (m s−1)

W :

Width (m)

α :

Volume fraction

µ :

Dynamic viscosity (kg m−1 s−1)

ρ :

Density (kg m−3)

σ :

Surface tension (kg s−2)

\({\varvec{\uptau}}\) :

Viscous stress tensor (N m−2)

\({\varvec{\uptau}}_{\text{t}}\) :

Turbulent stress tensor (N m−2)

b:

Bubble

d:

Drop

eq:

Equivalent

g:

Gas

l:

Liquid

r:

Relative

T:

Terminal

z :

Vertical direction (direction of bubble rise)

1:

Lighter fluid (gas)

2:

Heavier fluid (liquid)

2D:

Two-dimensional

3D:

Three-dimensional

VOF:

Volume of fluid

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Correspondence to Rajesh Nimmagadda.

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Nimmagadda, R. Dynamics of rising bubbles in initially quiescent liquids that are later on disturbed by falling drops. J Braz. Soc. Mech. Sci. Eng. 42, 526 (2020). https://doi.org/10.1007/s40430-020-02612-y

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Keywords

  • Dynamics
  • Bubbles
  • Drops
  • Quiescent
  • Eotvos
  • Morton