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Simulating the detailed movement of a rising bubble can be challenging, especially when it comes to bubble path instabilities. A solution based on the Euler Lagrange (EL) approach is presented, where the bubbles show oscillating shape and/or instable paths while computational cost are at a far lower level than in DNS. The model calculates direction, shape and rotation of the bubbles. A lateral force based on rotation and direction is modeled to finally create typical instable path lines. This is embedded in an EL simulation, which can resolve bubble size distribution, mass transfer and chemical reactions. A parameter study was used to choose appropriate model constants for a mean bubble size of 3 mm. To ensure realistic solution, validation against experimental data of single rising bubbles and bubble swarms are presented. References with 2D and also 3D analysis are taken into account to compare simulative data in terms of typical geometrical parameters and average field values.

Understanding bubbles path instabilities during rise in bubble columns is a major challenge since the 1960s. Early measurements were performed by [

Simulating bubble hydrodynamics is an issue since many years ago. The most common approach to do this is the simplified Euler Euler (EE) method, where single bubbles are no longer resolved but a bubble number density is taken into account instead. Bubbles located in a computational cell are supposed to have identical velocity and bubble sizes are often reduced to characteristic mean values (d_{32}, d_{10}) for the calculation. Interaction of bubbles is no longer a problem, since simple source terms can be used to simulate changes in the bubble size distribution. This also leads to a much simpler way to calculate for an instable bubble path, namely a diffusion term is used. This gives a high loss of detail on the bubble scale while the overall spatial distribution of bubbles can be forecast with adequate precision.

The other extreme is a high detail direct numerical simulation (DNS) of a single bubble or rather a small bubble swarm. Bubbles are resolved in full detail including the interfacial area deformations and the hydrodynamics inside. Turbulence in- and outside the bubbles are also calculated in high detail, trailing eddies in the bubble wake flow are resolved. In doing so, path instabilities can be simulated consequently on the lowest level of scale but at a high computational load. The DNS is unsuitable for simulation of a whole bubble column reactor because of its sheer bubble number. Despite the detailed flow simulation of the single bubble, the DNS is not capable of simulating a reliable behavior when bubbles coalesce or breakage happens. Those processes take place on an even lower spatial scale at the bubble surface layer and need special modeling, e.g. sub-grid models [

Free rising ellipsoidal bubbles not only move in straight lines but can describe sinusoidal, zigzag or spiraling paths. The common Euler Euler (EE) simulation techniques can no longer resolve the actual movement patterns and Direct Numerical Simulations (DNS) tend to be very costly when simulating a larger number of bubbles. This work presents a solution to calculate the orientation and shape of bubbles using the Euler Lagrange (EL) approach. Advantages lie within the fast computation and the high level of detail. In comparison to DNS, the insides of the bubbles are not calculated in full detail but macroscopic models are employed. Every bubble is calculated individually, having its own size, direction and shape. The surrounding fluid will influence not only the bubble’s movement but also the rotation and shape. The actual calculation of the turbulent eddies behind the bubble will not be carried out, but an oscillation orientation model is used and model parameters are calibrated from experimental data. This enables the simulation of instable bubble paths, while the bubble number can easily exceed those of a DNS simulation. The EL approach is also capable of evaluating the bubble size distribution due to coalescence and break-up by stochastic modeling. The individual simulation of each bubble allows more detailed simulation of bubble dynamics, e.g. mass transfer or residence time characteristics, than it is possible in the EE approach. With the implemented shape factor estimation, physical models could be expanded to account for the deformed bubble surface area.

The general EL model describes bubbles as a point volume acting under Newtonian dynamics. Forces created by the surrounding fluid and neighboring bubbles accelerate these Lagrangian points through the domain. The continuous phase itself is calculated using Navier-Stokes equations and can be coupled to the interaction forces by a source term. All forces are calculated for each bubble individually, which produces an individual path for each bubble. Therein lies one of the advantage of the EL approach in comparison to the EE methods. Bubbles can coalesce and break, which gives a bubble size distribution, with even more detail than common method of moments approaches. Downside of the EL approach is the higher computational load, which is strongly dependent on the number of bubbles simulated. Nevertheless, the EL approach has been used frequently in several different simulations of bubbly flow [

The continuous phase is assumed to be incompressible, basis for calculation is a modified Navier-Stokes equation:

ρ ( ∂ u c ∂ t + ( u c ⋅ ∇ ) u c ) = − ∇ p + μ Δ u c + f (1)

Given is the continuous velocity u_{c}, the pressure p, the density ρ, the viscosity μ and the source term f. The source term f depicts the forces of the bubbles and will thereby serve as a coupling of the phases. Turbulence is computed using the standard RANS k-epsilon model [

S k = | ∑ F D | | u c − u b | (2)

S ε = C ε S k τ (3)

τ = d k ; C ε = 1.0 (4)

Here, S_{k} denotes the source term for the turbulent energy k, F_{D} stands for the drag force. S_{ε} is the turbulent dissipation with τ, the turbulent time scale.

Bubbles are modeled as point volumes acting under Newtonian dynamics. Their movement is calculated using a number of different forces.

m b d u b = d t ∑ F (5)

The sum of forces ∑ F consists of the buoyancy and weight force F_{B}, the drag force F_{D}, the lift force F_{L}, the virtual mass force F_{VM}, the wall lubrication force F_{W} and the bubble dispersion force F_{BD}. Here the subscripts b and c stand for the bubble and the continuous phase accordingly, the subscript rel identifies the relative differences between them. Furthermore, g denotes the gravitational acceleration, ρ stands for densities, u for velocity, V for the bubble’s volume, k for the turbulent kinetic energy and α for the phase fraction. Appropriate model parameters are denoted with C_{i}.

F B = m b g ( 1 − ρ C ρ b ) (6)

F L = m b ρ c ρ b C L u r e l × ∇ × u c (7)

F D = 3 4 C D m b ρ c d b ρ b | u r e l | u r e l (8)

F V M = − C V M ρ c V b ( D b u b D t − D c u c D t ) (9)

F W = − C W V b α b ρ c | u r e l | 2 n w a l l (10)

F B D = − C B D ρ c k c ∇ α b (11)

Di/Dt in Equation (9) denotes the material derivative, meaning that the derivative is made while following the bubble. Drag and lift force coefficients C_{D}, C_{L} are calculated using the models of [_{VM} = 0.5 according to [_{TD} = 0.1 [_{E} is evaluated, after which a new eddy is calculated.

t E = ( 3 2 ) 0.5 C μ 3 4 k ε ; C μ = 0.09 (12)

Then the movement direction of the eddy is uniformly chosen while its velocity follows a normal distribution with a variance dependent on the turbulent energy k.

| u T ¯ | = N ( 0 ; 2 3 k ) (13)

This turbulent velocity is added to the underlying continuous phase velocity for each bubble individually, which is then used to calculate the different bubble forces. Especially the drag force calculation is a crucial step to produce dispersion of the bubbles. In order to achieve the same amount of bubble dispersion like in the experiment, the turbulent velocity for drag force calculation had to be slightly increased.

Change in bubble volume due to the pressure drop while rising is calculated assuming an ideal gas. Mass transfer and/or chemical reactions are not considered in this simulation.

According to the state of the art of bubble simulations almost any model assumptions are based on bubbles having a spherical shape. Simplest example is the Sauter diameter d_{32}, which maps the mean volume/surface area ratio of a bubble population to one spherical bubble size. Collision frequency or rather probability, e.g. by [

In this work, the deformed bubble will be approximated with an oblate spheroid, an ellipsoid with two different axes a and c, where c > a. For simplicities sake, all following figures will contain a simplified 2D version of the spheroid in

The ratio of the axes is chosen to describe the bubble shape via the shape factor sf.

s f = 1 χ = a c (14)

A shape factor of sf = 1 describes a perfect spherical shape, while a lower value stands for a more deformed sphere and sf = 0 would describe an infinitely thinned spheroid. Some models are based on empirical measurements using dimensionless numbers for the calculation, others are physical models derived from the interfacial tension and pressure distribution. It turned out that a good result for the simulation problem was achieved by [

W e ( χ ) = 4 χ − 3 4 ( χ 3 + χ − 2 ) [ χ 2 a sec ( χ ) − ( χ 2 − 1 ) 1 2 ] 2 ( χ 2 − 1 ) − 3 (15)

Since the shape factor has to be computed as a function of the Weber number, an approximation has been used:

s f = 1 atanh ( W e W e c r i t ) + 1 (16)

Note that this equation is based on the dimensionless Weber number, which can represent changes in the shape induced by fluctuating relative velocities. The critical Weber Number We_{crit} = 3.745 describes the transition to irregular bubble shapes. If the current Weber number is higher than We_{crit} a shape factor of sf = 0.2 is chosen.

W e = d ρ u r e l 2 σ (17)

Implementing the new bubble shape into the CFD framework necessitates further usage of the diameter definition of a spherical bubble. The volume of the sphere should be equal to the oblate spheroid volume, which leads to the following basic equations:

V sphere = V ellipsoid (18)

1 6 π d 3 = 1 6 π a c 2 (19)

c = d 3 s f 3 ; a = d 3 s f 2 3 (20)

Differently than with common rotation calculation, using the moment of inertia and torque action [

p ˙ = ω × p (21)

This rotation notation has been used by [

rigid ellipsoidal bodies in a Stokes flow. Based on these equations, a modified model for ellipsoidal bubbles was developed. In general, the bubble rotation is calculated using an explicit Euler algorithm:

d p = p ˙ d t (22)

d p ˙ = γ ¯ ( G − p ˙ ) d t + R p T d t (23)

Just as the bubble’s position and velocity, the orientation p changes due to the orientation change p ˙ . The bubble’s rotation relaxes against the outer rotation G with the factor γ ¯ . Here, γ ¯ denotes a simplified interaction of the bubble’s moment of inertia and torque. A low γ ¯ implies a high bubble mass/inertia and thus a slower rotation due to outer forces. The additional term p T resembles a random rotation due to turbulence, where R is used to scale the effect of turbulent randomness. It is generated similarly to the turbulent dispersion velocity u T . Since the effective moment of inertia of a gas bubble inside a fluid is unknown, γ ¯ has to be derived from experimental data.

G = J 1 1 2 ( r o t u C ) × p

+ J 2 λ ( S [ u C ] p − ( p T S [ u C ] p ) p ) (24)

+ J 3 ( − g − p )

λ = s f 2 − 1 s f 2 + 1 (25)

The outer rotation G is a summation of three main mechanics acting on the body Equation (24). The first line implies that the rotation of the surrounding fluid is transmitted to the bubble itself. The second line describes the rotation induced by shear stress in the surrounding fluid, S[u_{C}] is the symmetric part of the Jacobian matrix.

S [ u C ] = 1 2 ( ∇ ⋅ u C + ( ∇ ⋅ u C ) T ) (26)

The third line is an addition to the original model [_{i} to fit experimental data of (non-rigid) bubbles. This also compensates for the fact, that there is no Stokes flow around the bubble and its interface is mobile (slip condition).

Especially the third line of Equation (24) leads to an oscillatory system, which enables the bubble to describe sinusoidal or helical orientation characteristics. While the orientation of the bubble is changing, the vector of orientation change p ˙ will eventually point to a direction not perpendicular to p, thus rising in size and finally damping and stopping the oscillation. To prevent this, the direction change vector will be moved to the plane normal to the orientation vector by subtracting the part parallel to it:

p ˙ ′ = p ˙ − ( p ( p ⋅ p ˙ ) ) (27)

In total, this will lead to a slight damping of the oscillation, which can be eliminated by preserving the magnitude of the orientation change vector:

p ˙ ″ = | p | p ˙ ′ | p ˙ ′ | (28)

To preserve robust behavior a slight damping (δ = 0.2) is executed for the orientation change:

p ˙ ‴ = p ˙ ″ e − d t δ (29)

Finally, with an interaction force based on the orientation, the bubble experiences a drift perpendicular to the main movement direction. This results in a bubble trajectory describing helical and sinusoidal (zigzag) paths. This perpendicular force has the same direction as the change in direction [

p ˙ ~ F S (30)

This force is modeled as a modified drag force. Also, to imitate real bubble behavior, we limit the force to only occur on bubbles within a certain range of Reynolds number [

F S = { β | F D | p ˙ | p ˙ | | p ˙ | if R e = [ 500 , 1300 ] 0 else (31)

The additional side force due to bubble rotation is modeled with the magnitude of the current drag force F_{D} and the bubble rotation but will point in the direction of rotation. It turned out, that a scaling linear to the magnitude of the direction change | p ˙ | will lead to instabilities easily, which can be stabilized by using the square root | p ˙ | instead. The bubble path amplitude is calibrated with the parameter β, where a higher value implies a larger amplitude of the resulting oscillating path.

Parameters γ ¯ , R and J_{i} (Equations (22) and (23)) are derived from a parameter study and with analysis of the oscillatory equations. The characteristic bubble path amplitudes, frequency and wavelength shown in [

p ¨ + ω 0 2 p = 0 (32)

with

p ¨ = γ ¯ J 3 p (33)

Without damping of the oscillation, the frequency f_{0} can be calculated with the parameters J_{3} and γ ¯ .

f 0 = ω 0 2 π = γ ¯ J 3 2 π (34)

It turned out, that this is the case for our oscillatory system, but only if the random rotation is set to zero. A case without random rotation was set up for this reason, the results are shown in

Most references are made on a basis of a stagnant liquid phase, such that the bubble rise velocity u_{b} can be easily estimated. This makes it easy to calculate an appropriate wavelength of the generated/observed bubble path using its characteristic

frequency.

λ = u b f 0 (35)

In case of a non-stagnant liquid, like in a bubble column, the bubble velocity has to be identified at first. It is therefore important to mention, that the wavelength of a bubble path in a dynamic system will differ from most reference experiments, which are made using a well-defined surrounding. Also, in some areas of the column a downward flow occurs, lowering the bubble’s rising velocity. To overcome this problem, the bubble velocity and/or wavelengths are averaged over a large number of bubble paths.

For the parameter study, simulation results were compared to experimental measurements from the Institut für Strömungstechnik und Thermodynamik, University of Magdeburg [

An automated analysis was used to estimate the characteristic path shapes,

given by the mean and standard deviation of the wavelength λ and amplitude A of all bubble paths. To ensure a correct analysis, only those trajectories were considered, that were of appropriate length (minimum of 30 measured positions). Unfortunately, a 3D reconstruction was not possible for the bubble paths, because the side view measurements were not made simultaneously with the front view.

In addition, the liquid velocity was measured using Particle Image Velocimetry (PIV). Small tracer particles were added to the fluid and illuminated by a laser light sheet. This laser projection grants insight to the cross-sectional liquid velocity inside the column.

The corresponding computational mesh was created using a rectangular grid of 28 × 28 × 146 cells with overlapping cells being removed and reshaped to fit the cylindrical shape. This results in a mesh with 90,000 cells, shown in ^{3} per cell (mean cell length of 0.5 cm), because the bubble size must be smaller than the cell size in our EL approach. Cells near the walls are not rectangular anymore, but are also solemnly populated by bubbles due to wall lubrication force. The main bubble flow therefore happens in the center part, where the cells are in perfect rectangular shape.

For boundary conditions (s.

Patch | Flow condition | ||
---|---|---|---|

liquid velocity | pressure | k/epsilon | |

walls | no-slip condition | zero gradient | zero gradient |

outlet | slip condition | ambient pressure | zero gradient |

internal field, starting value | (0 0 0) | ambient pressure | 0.001/0.0001 |

suitable for estimating hydrodynamics near the surface. Since our interest lies within the lower part of the column, this error is acceptable.

On the basis of the experimental measurements, the mean inlet diameter of the bubbles was set to d = 3 mm. With a throughput of 10 l/h this leads to a total number of 196.4 bubbles per second to inject from the four nozzles. Bubble breakup, coalescence and mass transfer was turned off in this case, since only the bubble movement is of concern and measurements showed only a minimum of bubble size variations during rise. The experimental measurement area is positioned in the lower 312 mm of the column only, but the whole height of the column was simulated to achieve the correct flow pattern.

Since there are 4 parameters in total to be calibrated, an automated parameter study software has been used (Dakota, Sandia National Laboratories) to simulate a variety of combinations. The target function was the characteristic bubble flow path parameters, which were compared by screenshots at a first sight. After choosing the most promising parameter interval, a detailed analysis of bubble path frequency and amplitude followed. Therefore, an automated analysis of the generated path lines was used to calculate for mean wavelength and amplitude (~600 trajectories per simulation). The same method has been used to characterize the experimental measurements. Since the experimental data was collected using a 2D visual acquisition, simulation results were also calculated using a 2D mapping. Additionally, a 3D analysis was made and compared to other reference measurements.

Simulated liquid velocity and its deviation are compared to the experimental measurements by the Institut für Strömungstechnik und Thermodynamik, University of Magdeburg in

column. The white line represents the zero value, making it easier to spot the upward and downward flow area. The scale depicted on the left side shows the detailed line plot positions (A)-(D) in the lower figures. The liquid velocity profile from the experimental measurements shows a slower flow than the simulation produced. Probable reason for this could lie within the modifications done to the drag force calculations in order to acquire the special bubble movement. It has not yet been optimized in order to sustain an appropriate liquid velocity. Also, the liquid phase resolution is quite low in order to maintain an appropriate aspect ratio of bubble length to cell length. Aim of this work lies within the simulation of correct bubble paths rather than optimal liquid phase hydrodynamics.

Calculation of the simulated fluctuating velocity is made using the k-epsilon turbulence model. Values described here (s.

| u ′ | = 2 3 k (36)

The upper figure shows the fluctuating velocity deviation of the experiment by the Institut für Strömungstechnik und Thermodynamik, University of Magdeburg (left) and simulation (right) on the middle plane of the column. Again, the scale on the left side shows the line plot positions A-D. In the lower sections, the fluctuation velocity is similar to the measured values from the experiments. In higher sections, the simulation underestimates the level of turbulence, leading to low fluctuating velocities in comparison to experimental data. Here again, a standard parameter set was used for the BIT model [

According to literature [_{0} = 5 s^{−1}. With its mean free rising velocity of u_{b} = 35 cm/s the bubble path wavelength is λ = 7 cm. According to Equation (34) this would recommend setting the parameter γ ¯ J 3 ≈ 1000 .

Anyhow, analysis of the experimental data from the University of Magdeburg yielded a mean bubble path wavelength of λ = 4.14 cm and an amplitude of A = 1.87 mm. Mean bubble rise velocity is u_{b} = 29 cm/s in this case which gives a frequency of f_{0} = 7 s^{−}^{1} and a product of parameters of γ ¯ J 3 ≈ 2000 . This was only a first estimation, since the random rotation has not been set to zero in the final simulation.

Experimental data from 4 different measurements on the same properties were used to reconstruct and analyze bubble trajectories (s.

frames implies a measuring time of 1, 2.5 or 5 seconds accordingly. In order to estimate characteristic trajectory wavelengths and amplitudes, the points of inflexion of each trajectory were determined. Since main movement direction is upwards, the inflexion point distance in vertical direction was used to identify the wavelength while the horizontal distance was used for the amplitude. Averaging over all trajectories yields the characteristic wavelength and amplitude with their deviations. This method was used for both, the experimental and the

Case name | Mean wavelength λ | Std. dev. wavelength | Mean amplitude A | Std. dev. amplitude |
---|---|---|---|---|

Air V = 10 l/h 0.1 khz 1-100 | 40.6024 | 12.5520 | 1.8762 | 1.2405 |

Air V = 10 l/h 0.1 khz 1-250 | 41.2272 | 13.1363 | 1.8810 | 1.2360 |

Air V = 10 l/h 0.1 khz 1-500 | 41.7505 | 13.2859 | 1.8777 | 1.1981 |

Air V = 10 l/h 0.1 khz 251-500 | 42.0848 | 13.4051 | 1.8642 | 1.1510 |

Experiments mean value | 41.4162 | 13.0948 | 1.8747 | 1.2064 |

Simulation 1 | 41.3764 | 14.6372 | 1.8069 | 1.1588 |

Simulation 2 | 41.0091 | 14.6410 | 1.7593 | 1.1135 |

Simulation 3 | 41.0273 | 14.6988 | 1.7472 | 1.1651 |

simulative data. Deviation of wavelength and especially of amplitude is quite high, which is partly owed to the 2D analysis of the bubble paths. Since a flat zigzag path can only be seen correctly from one perspective, the calculated 2D amplitude will hardly correspond to the true 3D amplitude, it will generally underestimate the true value. This is why the standard deviation of the analyzed amplitude show rather high values in all experimental and simulation cases.

With the appropriate set of parameters, the simulated bubble path characteristics properly match the experimental values (_{1}) and shear (γJ_{2}) are very low in contrast to the oscillation (γJ_{3}), while the random factor (R) is of the same magnitude. Thus, the impact of rotational/shear flow around bubbles plays a minor role on the rotation while it is dictated mostly by turbulent eddies hitting the bubbles.

A visual comparison of the three most promising bubble paths in

Case name | γ ¯ J 3 | R | β | γ ¯ J 1 , γ ¯ J 2 |
---|---|---|---|---|

Simulation 1 | 1300 | 1950 | 0.15 | 32.50 |

Simulation 2 | 1350 | 1620 | 0.15 | 33.75 |

Simulation 3 | 1250 | 1875 | 0.15 | 31.25 |

simulation, a parameter for random rotation is used, but the analysis does not concern the overall distribution of bubbles. The spread/diffusion of bubbles in the upper area however (height > 150 mm) yields no (qualitatively) visible difference between experiment and simulation.

Another point of concern is the behavior of the simulated orientation in comparison with the velocity of the bubble. In order to describe this relations, different deduced angles are used; the movement angle lies between the vertical axis (gravitational direction) and movement direction, the orientation angle is between the short axis direction and vertical axis and the drift angle lies between orientation and movement vectors. With 2D analysis of the simulation data, the orientation angles show a normal distribution with mean value near to zero (

Bubble velocity and orientation vectors roughly point to the same direction, this has been shown in several references [

right) with a mean value of 19.2˚. Experimental data shows, that the largest drift angles occur at the most outer points of the oscillation, especially when a planar zigzag path is fulfilled [

When taking a closer look at single trajectories of one of the bubbles rising in the swarm it will reveal characteristic parts of the bubble’s movement. As shown in [

We can observe influence of the flow in terms of a strong drift in the bubble paths (s.

calculate for a “dynamic radius” of the path. A flattened spiral would look like an ellipsoid in the X-Y view (s. _{b} = 0.264 m/s). Oldest related work [_{eq}, maximum inclination angles from 20˚ to 30˚ and characteristic frequencies in the range of 0.09 < St < 0.15 (St = fL/U = L/λ). In our case, the simulated amplitudes are far lower (~1.5 d_{eq}) but inclination angles and frequency are similar (St = 0.073 in our case). Only severe difference is the stated maximum drift angle of only 2˚, which is far higher in our simulation case.

Most deviations to experimental and other simulative work venture from the drift angle characteristics. Since the side force F_{s} (Equation (31)) is responsible for the horizontal movement, other models should achieve more suitable drift angles. Reference yields only few insights on modeling of this particular force [

The presented EL simulation is capable of simulating unstable sinusoidal/spiraling bubble paths using macroscopic models. Bubble orientation, rotation and shape are calculated to achieve characteristic movement. Due to the assumption of bubbles describing rotational spheroids, the additional parameters that have to be calculated reduce to a shape factor, rotation and orientation vectors. A force induced by the bubbles rotation produces the lateral force leading to an oscillation movement.

The macroscopic orientation and rotation model uses a simple vector-based approach. The usage of a spherical coordinate system was intentionally rejected to keep the model as simple as possible. The model equations are easy to compute and allow simulating of a large number of bubbles at a time. Unfortunately, only few comparable models exist [

A parameter study was used to fit the model constants to experimental data for mean bubble size of d = 3 mm. Parameter estimation was made using the amplitude and wavelength of the typical spiral movement. It turned out that most 2D measurements cannot reflect characteristic path parameters entirely. Especially, orientation angles are problematic in 2D analysis, because the perspective view only permits seeing angles perpendicular to the line of sight. In order to still achieve a good fit, the simulation results were mapped to a 2D point of view and compared to the experimental data. After parameter fitting, comparison to reference bubble path data was made using 2D and 3D analysis and could prove correct reproduction of unstable bubble paths. Evaluation of the 3D bubble path in a bubble swarm is difficult, since most references only supply 2D camera setups or single bubble trajectories in 3D analysis systems. Comparison to DNS simulation and single rising bubble path data could also show good agreement. Amplitude and wavelength of the simulated bubble path are in unison with the measurements. Detailed comparison of DNS results of the drift angle reveals slight disagreement.

For further improvement of the model, a predictive parameter approach should be used to also cover different bubble sizes. Interaction of deformed bubbles is not considered in the EL model shown here, and this could include collision, break-up, mass transfer and other shape dependent processes.

Funding by the Deutsche Forschungsgemeinschaft (DFG) within the RTG GrK 1932 “Stochastic Models for Innovations in the Engineering Sciences”, project area P1, is gratefully acknowledged. We want to thank the Group of K. Zähringer, Laboratory of Fluid Dynamics and Technical Flows, University of Magdeburg, for supplying experimental data on the bubble paths and liquid velocity PIV measurements.

Weber, A., Bart, H.-J. and Klar, A. (2017) Simulating Spiraling Bubble Movement in the EL Approach. Open Journal of Fluid Dynamics, 7, 288-309. https://doi.org/10.4236/ojfd.2017.73019