Fluid dynamics of bubbly flows
vorgelegt von
Dipl.-Ing.
Thomas Ziegenhein
geb. in Jena
von der Fakultät III – Prozesswissenschaften
der Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktor der Ingenieurwissenschaften
- Dr.-Ing. -
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr.-Ing. Felix Ziegler
Gutachter: Prof. Dr.-Ing. Matthias Kraume
Gutachter: Prof. Dr.-Ing. Michael Schlüter
Gutachter: Dr. rer. nat. Dirk Lucas
Tag der wissenschaftlichen Aussprache: 8. Juli 2016
Berlin 2016
“Nur die Idee, die unbegründete Antizipation, der kühne Gedanke ist es, mit dem wir,
ihn immer wieder aufs Spiel setzend, die Natur einzufangen versuchen: Wer seine
Gedanken der Widerlegung nicht aussetzt, der spielt nicht mit in dem Spiel
Wissenschaft”
“Bold ideas, unjustified anticipations, and speculative thought, are our only means for
interpreting nature: our only organon, our only instrument, for grasping her. And we
must hazard them to win our prize. Those among us who are unwilling to expose their
ideas to the hazard of refutation do not take part in the scientific game.”
Karl Popper – Wien, 1934
ii
iii
Acknowledgement
Like probably all doctorate students at the beginning of their studies, I thought in chaotic
ways, changed my mind three times a minute and was eager to do everything in a
different way than others. Quickly, I learned that scientific progress is a slow progress
reasoned on undoubtedly arguments. Nevertheless, I was allowed to keep a bit of the
initial chaos, for this I would like to sincerely thank my scientific supervisor Dr. Dirk
Lucas. More than once, he was the only one who understands my unconventional ideas
and theories so that they could evolve over time. Those lead also to the purpose to build
an own experimental facility, which might be uncommon as a member of the theoretical
department. For this opportunity, I have to thank him again. In this context, I have to
thank the head of the experimental department Prof. Uwe Hampel to tolerate me in his
labs as well as the function as head of the project I worked in, the Helmholtz Energy
Alliance.
I sincerely thank Prof. Matthias Kraume for accepting me as an external doctorate
student as well as for the organization of my PhD at TU-Berlin. In addition, I want to
thank Prof. Michael Schlüter for accepting to be a referee for this thesis. I would
particularly thank my colleagues at HZDR for the many fruitful discussions, especially Dr.
Roland Rzehak, Dr. Sebastian Kriebitzsch, Dr. Eckhard Krepper and Tian Ma.
My gratitude and special thanks go to my friends; they tolerated me without grimness
the last three years in which I spend most of the time on my doctorate studies.
Finally yet importantly, I want to thank my father and my mother for supporting and
comfort me the last years.
This work was funded by the Helmholtz Association within the frame of the
Helmholtz Energy Alliance “Energy Efficient Chemical Multiphase Processes” (HA-E-
0004).
iv
v
Abstract
Bubbly flows can be found in many applications in chemical, biological and power
engineering. Reliable simulation tools of such flows that allow the design of new
processes and optimization of existing one are therefore highly desirable. CFD-
simulations applying the multi-fluid approach are very promising to provide such a
design tool for complete facilities. In the multi-fluid approach, however, closure models
have to be formulated to model the interaction between the continuous and dispersed
phase. Due to the complex nature of bubbly flows, different phenomena have to be taken
into account and for every phenomenon different closure models exist. Therefore,
reliable predictions of unknown bubbly flows are not yet possible with the multi-fluid
approach.
A strategy to overcome this problem is to define a baseline model in which the closure
models including the model constants are fixed so that the limitations of the modeling
can be evaluated by validating it on different experiments. Afterwards, the shortcomings
are identified so that the baseline model can be stepwise improved without losing the
validity for the already validated cases. This development of a baseline model is done in
the present work by validating the baseline model developed at the Helmholtz-Zentrum
Dresden-Rossendorf mainly basing on experimental data for bubbly pipe flows to bubble
columns, bubble plumes and airlift reactors that are relevant in chemical and biological
engineering applications.
In the present work, a large variety of such setups is used for validation. The
buoyancy driven bubbly flows showed thereby a transient behavior on the scale of the
facility. Since such large scales are characterized by the geometry of the facility,
turbulence models cannot describe them. Therefore, the transient simulation of bubbly
flows with two equation models based on the unsteady Reynolds-averaged Navier–
Stokes equations is investigated. In combination with the before mentioned baseline
model these transient simulations can reproduce many experimental setups without
fitting any model. Nevertheless, shortcomings are identified that need to be further
investigated to improve the baseline model.
For a validation of models, experiments that describe as far as possible all relevant
phenomena of bubbly flows are needed. Since such data are rare in the literature, CFD-
grade experiments in an airlift reactor were conducted in the present work. Concepts to
measure the bubble size distribution and liquid velocities are developed for this purpose.
In particular, the liquid velocity measurements are difficult; a sampling bias that was not
yet described in the literature is identified. To overcome this error, a hold processor is
developed.
The closure models are usually formulated based on single bubble experiments in
simplified conditions. In particular, the lift force was not yet measured in low Morton
number systems under turbulent conditions. A new experimental method is developed in
the present work to determine the lift force coefficient in such flow conditions without
the aid of moving parts so that the lift force can be measured in any chemical system
easily.
vi
vii
Zusammenfassung
Die Auslegung und Optimierung von Mehrphasen-Prozessen im Bereich der chemischen
und biologischen Verfahrenstechnik sowie der Energietechnik mithilfe von verlässlichen
Simulationswerkzeugen ist aufgrund ihrer Vielzahl wünschenswert. Im speziellen für
Blasenströmungen sind CFD-Simulationen die auf dem multi-fluid Ansatz basieren sehr
vielversprechend um mit Ihrer Hilfe komplette Reaktoren auszulegen. Dabei müssen
jedoch Schließungsmodelle formuliert werden, die die Wechselwirkungen zwischen der
dispersen und der kontinuierlichen Phase beschreiben. In Blasenströmungen müssen
verschieden Phänomene modelliert werden, für die es wiederum verschiedene
Schließungsmodelle gibt. Durch diese Komplexität ist es bis jetzt nicht möglich
Blasenströmungen die nicht vorher vermessen wurden verlässlich vorherzusagen. Eine
Möglichkeit hinzu verlässlichen Modellen ist die Definition eines baseline models in dem
alle Modelle und Modellkonstanten festgelegt sind. Durch die Validierung von
verschiedenen Experimenten mit solch einem Modellsatz können die Grenzen der
Modellierung ausgelotet werden, Defizite erkannt und Schrittweise verbessert werden
ohne die Gültigkeit für die bereits bestehenden Anwendungen zu verlieren.
In dieser Arbeit wird das baseline model welches am Helmholtz-Zentrum Dresden-
Rossendorf hauptsächlich für Blasenströmungen in Rohren validiert wurde
weiterentwickelt, indem es mit einer Vielzahl von Blasensäulen und Schlaufenreaktoren
validiert wird, die in der chemischen Industrie und Biotechnologie angewendet werden.
Solche dichtegetriebenen Strömungen zeigen charakteristische Strömungsmerkmale in
der Größe des Apparates. Solche großen Skalen können im Allgemeinen nicht durch ein
Turbulenzmodell abgebildet werden, wodurch die transiente Simulation von
Blasenströmungen mit Zweigleichungs-Turbulenzmodellen basierend auf den Reynolds
gemittelten Navier-Stokes Gleichungen untersucht wurde. In Kombination mit dem
baseline model konnten diese transienten Simulationen die Experimente reproduzieren
ohne Modellkonstanten anzupassen. Defizite existieren jedoch, die weiter untersucht
werden müssen um das baseline model weiter zu verbessern
Die Anforderungen an die experimentellen Daten bei einer Modellvalidierung sind sehr
hoch, so müssen diese soweit möglich jeden relevanten Aspekt von Blasenströmungen
beschreiben. Da solche umfassenden Daten in der Literatur selten sind, wurden eigene
Experimente für einen Schlaufenreaktor speziell zur CFD-Validierung durchgeführt. In
diesem Zusammenhang wurden Messkonzepte entwickelt, um die
Blasengrößenverteilung und Flüssiggeschwindigkeit bei hohen Gasgehalten zu
bestimmen. Bei der Messung der Flüssiggeschwindigkeit wurde eine
Stichprobenverzerrung identifiziert die bis jetzt noch nicht in der Literatur beschrieben
wurde. Um diesen Fehler zu beheben, wurde eine hold processor entwickelt.
Die erforderlichen Schließungsmodelle sind im Allgemeinen für
Einzelblasenexperimente in vereinfachten Modellsystemen formuliert. Im Besonderen
wurde die Lift-Kraft noch nicht in Systemen mit einer niedrigen Morton Zahl und unter
turbulenten Bedingungen vermessen. Deshalb ist in dieser Arbeit ein neue Methode
beschrieben die es erlaubt die Lift-Kraft in solchen Systemen zu bestimmen, dabei
werden, wie üblich, keine beweglichen Teile benutzt wodurch sich diese Methode für eine
Vielzahl von chemischen Stoffen eignet.
viii
ix
Contents
1 Introduction 1
1.1 Subject and motivation 1
1.2 Modelling bubbly flows 2
2 Simulation Methods 7
2.1 Computational fluid dynamics 7
2.1.1 Material balance equations – Reynolds transport theorem 7
2.1.2 General balance equation, Navier-Stokes equation and finite volumes 8
2.2 Eulerian averaging, two-fluid model and closure problem 11
2.3 Closure models and multiple bubble velocity classes 14
2.3.1 Drag force 15
2.3.2 Lift force 17
2.3.3 Multiple bubble classes 19
2.4 Turbulence modeling 20
2.4.1 Reynolds Averaged Navier Stokes (RANS) equations 21
2.4.2 Unsteady RANS equations 22
2.4.3 Large Eddy Simulations 23
2.4.4 Turbulence in Bubbly flows 24
2.5 Baseline concept for simulating bubbly flows 26
3 Experimental Methods 29
3.1 Experimental facility 29
3.2 Bubble size 32
3.2.1 Single bubbles 33
3.2.2 Systems with very low void fraction or narrow bubble size distributions 35
3.2.3 Systems with higher void fractions and wide bubble size distributions 36
3.3 Void fraction 38
3.3.1 Needle probe 38
3.3.2 2D-Videometry 38
3.4 Liquid velocity, turbulence & Sampling bias in bubbly flows 39
3.4.1 Particle Image velocimetry in bubbly flows 39
3.4.2 Particle tracking velocimetry with micro bubbles in bubbly flows 42
3.4.3 Sampling Bias in bubbly flows 47
x
3.4.4 Results 52
3.4.5 Conclusions 61
4 Eulerian bubbly flow simulations with the URANS equations 63
4.1 Modelling, setup and convergence criteria 63
4.2 Mesh and time step study 65
4.3 Influence of the virtual mass force 68
4.4 Influence of the bubble induced turbulence 72
4.5 Conclusions 74
5 Prediction of the large liquid structures of bubbly flows with the URANS equations
77
5.1 Simulation setup and experimental data 78
5.2 Results 79
5.2.1 Experiments of Becker et al. (1994) 79
5.2.2 Experiments of Pfleger et al. (1999) 81
5.2.3 Experiments of Julia et al. (2007) 83
5.3 Conclusions 90
6 Turbulence in bubbly flows using the URANS equations with separation in large
and small turbulence structures 93
6.1 Simulation Setup and experimental data 93
6.2 Results 95
6.3 Heterogeneous regime 95
6.4 Homogenous regime 97
6.5 Comparison with Large Eddy simulation 99
6.6 Conclusions 102
7 A complex validation case for CFD simulations: Airlift reactor 105
7.1 Setup 105
7.2 Results 106
7.2.1 Bubble size distribution 107
7.2.2 Liquid velocity and turbulence 109
7.2.3 Void fraction 114
7.3 CFD Simulations 115
7.3.1 Results 117
7.4 Conclusion 123
8 Lift force measurements in very low Morton number systems and high bubble
Reynolds number flows 125
xii
xiii
List of Figures
Figure 1-1 Grace Diagram (Grace, et al., 1976) for bubble shapes and bubble rise velocity. The Morton
number is denoted as ‘M’ in the diagram. ................................................................................................................... 3
Figure 1-2 Bubbles for different Eötvös Numbers in air/water except the right bubble is in air/water +
2ppm Triton X-100 at Morton number of 2.63E-11............................................................................................... 4
Figure 2-1 The difference between material and spatial perspective. Left a material control volume with
always the same bai; right a fixed control volume with the streaklines in blue. ...................................... 7
Figure 2-2 Flux around an infinite volume ............................................................................................................................... 10
Figure 2-3 Terminal velocity obtained with different drag laws in air/water systems ....................................... 17
Figure 2-4 Lift coefficient correlation of Tomiyama et al. (2002) for air-water systems. The dashed line
indicates the not measured region. ............................................................................................................................ 19
Figure 3-1 Sketch of the experimental setup and the ground plate with (right) (metrics in mm). ................. 29
Figure 3-2 Photograph of the experimental setup ................................................................................................................ 30
Figure 3-3 Size of the bubbles (top) and the standard deviation of the size (bottom) over the volume flow
rate for different needle sizes. ...................................................................................................................................... 31
Figure 3-4 Wobbling uncertainty .................................................................................................................................................. 31
Figure 3-5 Calculation of the solid of rotation of a bubble. ............................................................................................... 32
Figure 3-6 Main steps for bubble size determination of single bubbles, left to the right: Original image;
Edges with weak (grey) and strong edges (white); filled structures; result with major and minor
axis, area centroid and crossing points of the axis (white dots). .................................................................. 34
Figure 3-7 Bubble size measurement for a 0.6 mm inner diameter single needle sparger with 0.05 l/min
gas volume flow rate. ........................................................................................................................................................ 34
Figure 3-8: Automated detection of overlaid bubbles. a) Input with corrected contrast b) segmented
bubbles with positions (grey crosses) c) edge detection algorithm d) cutting out the overlaid
bubbles (dark grey). Only bubbles larger than 1.5 mm are treated. ........................................................... 35
Figure 3-9 Volume density function in the downcomer of an airlift reactor shown in Section 7, the shown
volume density function is obtained in the downcomer for Case 6 between y= 0.2 m and y=
0.3 m. ....................................................................................................................................................................................... 36
Figure 3-10 Bubbly flow for different volume flow rates, left, 3 l/min, right, 7 l/min. The used sparger
setup is discussed in Section 3.4. ................................................................................................................................. 36
Figure 3-11 Determination of the bubble sizes in bubble clusters by following the cluster over ten frames
with 200 frames per second recording speed. ...................................................................................................... 37
Figure 3-12 Bubble size distribution determined at two different heights and two different volume flow
rates; left, 2.2 l/min, right, 3.4 l/min. The sparger consists of four needles with 1.5 mm and four
needles with 0.6 mm inner diameter. ........................................................................................................................ 38
Figure 3-13 Algorithm for determining the bubbles in the PIV image for Case 3 (1): The obtained image
from the PIV camera. (2): Applying a median filter on (1). (3): Results obtained from a low
threshold on (2). (4): Results obtained from a high threshold on (2). (5): Hysteresis of the low
and high threshold value results (6): Closing the boundaries and filling the structures................... 41
Figure 3-14 Masking the bubbles and shadows. a) The original PIV image; b) the bubble masks (blue) and
shadow mask (black); c) obtained velocity vectors. ........................................................................................... 42
Figure 3-15 Determining the depth of field by filtering the edge strength. The distance in depth between
the test prints is 1 mm. a) Original picture, b) edge strength in grey shades and as graph along the
red-white dotted line. ....................................................................................................................................................... 43
Figure 3-16 Determination of the depth of field depending on the edge filter for three different
resolutions. ............................................................................................................................................................................ 44
Figure 3-17 Bubbles that are used for tracking in the field of maximum sharpness. ............................................ 44
Figure 3-18 Particle tracking using mini bubbles below 500 µm diameter in a 60 mm wide chanel. Bottom:
The original picture at t=t1; top: The selected particles at t=t1 and t=t1+Δt labeld with
different grey tones, Δt=1/1000 s. .......................................................................................................................... 45
Figure 3-19 Sampling bias in bubbly flows using BTV (top) and PIV (bottom). The tracked vertical velocity
(dashed blue line) and the count of the determined trajectories (continuous red line) are
smoothed with a moving average to represent the sampling bias clearly. ............................................... 49
xiv
Figure 3-20 The algorithm for the hold processor in space and time with an example on an Eulerian grid
with illustrated velocity vectors. ................................................................................................................................. 50
Figure 3-21 Comparison of the hold processor with the simple ensemble averaging used on the analytical
test function. ......................................................................................................................................................................... 51
Figure 3-22 Experimental setup used for the liquid velocity experiments. Left a sketch of the facility, the
measuring line is dotted red; right the ground plate of the bubble column with the holes for the
used needle sparger. ......................................................................................................................................................... 52
Figure 3-23 Left and the right measuring window of the left half of the column for case 13. ........................... 53
Figure 3-24 Count of the tracked micro bubbles for case 13 at the wall and towards the center; the center
is at x=0.125 m. ................................................................................................................................................................ 54
Figure 3-25 Comparing the results obtained by tracking different bubble sizes. a) The vertical liquid
velocity v for the different bubble sizes, b) the normal Reynolds stress tensor component v′v′, c)
the fluctuation probability function in 0.0375 m<x<0.05 m for two bubble groups, b) the
fluctuation probability function in 0.0875 m<x<0.1 m for two bubble groups. .............................. 55
Figure 3-26 Results of the BTV for all cases. a) Vertical liquid velocity v b) normal Reynolds stress
component v′v′ c) normal Reynolds stress u′u′ (u is the horizontal liquid velocity). .......................... 56
Figure 3-27 The influence of the sampling bias on the PIV results for different volume flow rates. ............. 57
Figure 3-28 The influence of the sampling bias on the BTV results compared to the PIV results for case 13.
..................................................................................................................................................................................................... 58
Figure 3-29 The probability density function of the upward liquid velocity fluctuations obtained with PIV
and BTV for case 13. a) Near the wall between 0.0375 m<x<0.05 m, b) towards the center
between 0.0875 m<x<0.1 m. .................................................................................................................................. 58
Figure 3-30 The vertical liquid velocity obtained with PIV with BTV for different gas volume flow rates. 59
Figure 3-31 Reynolds stress components obtained by using PIV and BTV. a) Normal Reynolds stress
components v′v′ and u′u′ (u is the horizontal liquid velocity), b) cross Reynolds stress
component u′v′. ................................................................................................................................................................... 60
Figure 4-1 Experimental setup....................................................................................................................................................... 64
Figure 4-2 Number density function of the bubble diameter in the experiment of Mohd Akbar et al. ........ 64
Figure 4-3 Mesh study for four different meshes. ................................................................................................................. 67
Figure 4-4 Time step study for different CFL numbers using the virtual mass force ............................................ 68
Figure 4-5 Time study for different RMS(CFL)-numbers without using the virtual mass force ...................... 69
Figure 4-6 Comparison between using the virtual mass force and not using the virtual mass force for a
superficial velocity of 13 mm/s and 3 mm/s. The curves for using the virtual mass force and not
using the virtual mass force for the 3 mm/s case are on the top of each other. ..................................... 70
Figure 4-7 Comparison of different bubble induced turbulence modeling approaches for 13 mm/s
superficial velocity. ............................................................................................................................................................ 72
Figure 4-8 Unresolved turbulent viscosity for different modeling approaches for 13 mm/s superficial
velocity. ................................................................................................................................................................................... 73
Figure 4-9 Comparison of the total upward turbulence intensity for different bubble induced turbulence
modeling approaches for 3 mm/s superficial velocity. ..................................................................................... 73
Figure 5-1 Setup of the experiments by Becker et al. (1994) (left), Pfleger et al. (1999) (middle and Julia et
al. (2007) (right). The measurement planes are dotted and points with measurements over time
are marked red. ................................................................................................................................................................... 79
Figure 5-2 Qualitatively results of the experiments (top) of Becker et al. (1994) and the simulations
(bottom). Pictures are taken every 5 seconds; the crosses in the simulation pictures mark the
measuring points Point A and Point B, respectively. .......................................................................................... 80
Figure 5-3 Liquid upward velocity at two different points obtained from experiments of Becker et al.
(1994) and simulations.................................................................................................................................................... 81
Figure 5-4 Comparison of the averaged results with the experiment at 0.75 m above the ground plate. ... 81
Figure 5-5 Liquid velocity in sideward direction at 0.25 m above the ground plate. ............................................ 82
Figure 5-6 Liquid upward velocity profiles at three different heights. ........................................................................ 82
Figure 5-7 Velocity vectors (left) and void fraction profiles at the center plane for three different cases.
The measuring lines at 0.15 m and 0.3 m height are marked black. ........................................................... 83
Figure 5-8 Velocity profiles at four different heights for pattern F8 and F11 with 29 mm/s superficial
velocity compared with the experiments. ............................................................................................................... 84
xv
Figure 5-9 Vertical velocity over time in the center of the column for pattern F11 at a superficial velocity
of 29 mm/s. Simulation results are shown at two heights. ............................................................................. 85
Figure 5-10 Velocity and gas void fraction profiles for pattern F11 for different superficial velocities at
two different heights. ........................................................................................................................................................ 86
Figure 5-11 Comparison of the bubble plume frequency obtained by experiments and simulations. Left
the Frequencies for the F11 pattern; right the frequencies for the F16 and F8 pattern. ................... 87
Figure 5-12 Averaged upward velocities and vector plots for pattern F11 at 36 mm/s superficial velocity
and F16 at 16 mm/s superficial velocity. ................................................................................................................. 88
Figure 5-13 Sensitivity regarding lift force for pattern F11, superficial velocity 29 mm/s. ............................... 88
Figure 5-14 Influence of the bubble size for two different flow patterns. .................................................................. 89
Figure 5-15 Influence of the turbulence model for flow pattern F16 and a superficial velocity of 16 mm/s.
..................................................................................................................................................................................................... 90
Figure 6-1 Sketch of the experimental setups. Right the setup of Julia et al. (2007); left the setup of Deen et
al. (2001). The measurement positions are shown as dotted lines. ............................................................ 94
Figure 6-2 Simulation results using different bubble sizes compared to experiments. Left the vertical
liquid velocity is shown; right the root mean square of the normal components of the Reynolds
shear stress tensor (𝑣 vertical, 𝑢 horizontal). ........................................................................................................ 95
Figure 6-3 Resolved and unresolved parts of RMS(v′v′) and RMS(u′u′) with and without BIT at 0.25 m
above sparger compared to experimental data by Deen et al. (2001). a) And b) using BIT; c) and
d) without a BIT model. ................................................................................................................................................... 97
Figure 6-4 Vertical liquid velocity and RMS values of the normal Reynolds stress components obtained
with the URANS modeling compared to experiment (Juliá, et al., 2007). a) And b) superficial
velocity of 29 mm/s. c) And d) superficial velocity of 43 mm/s. .................................................................. 98
Figure 6-5 Reynolds stresses in homogenous bubble columns for different superficial velocities. The four
superficial velocities from the present study (Juliá, et al., 2007) and the experiments by bin Mohd
Akbar et al. (2012) with the corresponding URANS simulations by Ziegenhein et al. (2015) are
shown. ...................................................................................................................................................................................... 99
Figure 6-6 LES compared with the URANS simulations for experiments of Deen et al. (2001). .................... 101
Figure 6-7 LES compared with the URANS simulations for the experiments of bin Mohd Akbar et al.
(2012) (cf. Section 4). a) And b) vertical liquid velocity and gas volume fraction for 3 mm/s and
13 mm/s superficial velocity, c) 𝑅𝑀𝑆(𝑤′𝑤′) for 13 mm/s superficial velocity (𝑤 is the upward
velocity), d) 𝑅𝑀𝑆(𝑤′𝑤′) for 3 mm/s superficial velocity. .............................................................................. 102
Figure 7-1 Experimental setup and the used ground plate setup. The red lines label the measuring
positions. .............................................................................................................................................................................. 106
Figure 7-2 Bubble size distributions in the riser. a) Number density for case 6 at two different heights b)
Number density for case 8 at two different heights c) averaged area density function at 0.2 m and
0.6 m d) averaged volume density function at 0.2 m and 0.6 m. ................................................................. 107
Figure 7-3 Pictures of the bubbly flow in the riser at a height of y=0.2 m. a) Case 4 b) case 6 c) case 8.108
Figure 7-4 Bubble sizes in the downcomer. a) Bubble sizes along the downcomer. b) Bubble sizes over the
width of the downcomer for case 8 averaged over height from y=0.3 m to y =0.4 m ................ 109
Figure 7-5 Situation in the downcomer, a) case 4 b) case 6 c) case 8. ....................................................................... 109
Figure 7-6 Sampling bias in the center of the riser for case 8 at y=0.2 m. ............................................................ 110
Figure 7-7 Liquid velocity profiles measured at two different heights. .................................................................... 110
Figure 7-8 Vertical velocity over time at two different positions for case 6, the time scale is arbitrary set to
zero for both and is not synchronized. Top: The vertical velocity over time in the left quarter of
the riser at x=0.095 m and y=0.2 m; every measuring point is moving averaged over 0.08 s.
Bottom: The vertical velocity over time in the center of the left downcomer at x=0.03 m
and y=0.6 m; every measuring point is moving averaged over 2 s. ....................................................... 111
Figure 7-9 Normal Reynolds stresses in the vertical (v’v’) and horizontal (u’u’) direction at two different
heights. .................................................................................................................................................................................. 113
Figure 7-10 Cross Reynolds stress u′v′ at two different heights. ................................................................................. 113
Figure 7-11 Void fraction in the riser at y=0.6 m (left) and along the downcomer (right). .......................... 114
Figure 7-12 Flow situation in the downcomer. a) Horizontal liquid velocity at two different heights, b)
void fraction profiles for case 6 at three different heights............................................................................. 115
Figure 7-13 The complete computational mesh and a magnification of the top region..................................... 116
xvi
Figure 7-14 Asymmetric flow behavior in the airlift reactor. From left to right: Case 4, case 6, case 8. ..... 117
Figure 7-15 Void fraction at y=0.6 m in the riser. ............................................................................................................ 118
Figure 7-16 Liquid velocity and normal components of the Reynolds stress tensor v′v′ and u′u′ (u is the
horizontal liquid velocity) at y=0.2 m (left) and y=0.6 m (right) with delineated internal walls.
................................................................................................................................................................................................... 119
Figure 7-17 Resolved and unresolved normal components of the Reynolds stress tensor v′v′ and u′u′ (u is
the horizontal liquid velocity) for case 6 at y=0.2 m (left) and y=0.6 m (right) with delineated
internal walls. ..................................................................................................................................................................... 120
Figure 7-18 Void fraction along the downcomer. ................................................................................................................ 121
Figure 7-19 Bubble size seperation along the downcomer. ............................................................................................ 121
Figure 7-20 Horizontal liquid velocity and gas void fraction in the downcomer compared to experiments
for case 6............................................................................................................................................................................... 122
Figure 7-21 Flow situation of the gas phase in the downcomer for case 6. From right to left: The gas void
fraction and the gas velocity in the center plane; the liquid velocity vectors from the aerial view in
the downcomer at 𝑦=0.415 m, the horizontal slip velocity in the downcomer at 𝑦=0.415 m. All
shown values are time averaged. .............................................................................................................................. 123
Figure 8-1 Setup to determine the lift force coefficient with the measuring area delineated form 𝑧=0.5 𝑚
to 𝑧=0.65 𝑚 (left) and the flow structure from PIV measurements (right). ....................................... 126
Figure 8-2 Back illumination (yellow) of the experimental setup. .............................................................................. 129
Figure 8-3 Side view (left), which is illuminated from left, and front view (right), which is illuminated
from the back. The red rectangles mark the same bubble and edge the pictures used in Figure 8-4.
................................................................................................................................................................................................... 130
Figure 8-4 Determining the position of the bubble in the side view (left) by adding half of the major axis
dM determined from the front view (right) to the left tip.............................................................................. 130
Figure 8-5 The void fraction function on a two dimensional cut in the center between z=0.4 m and z=
0.7 m for the drilled out needle and 800 ml/min driving volume flow rate. The color from red to
blue indicates the void fraction function from high to low values, respectively. The left picture
shows the determined local maxima and the right picture the spline that is taken as bubble trace.
................................................................................................................................................................................................... 131
Figure 8-6 Liquid velocity field determined with PIV for 800 ml/min. The left reactor wall is at x=0 m,
the driving flow at the right wall is between x=0.2 m and x=0.25 m. ................................................ 132
Figure 8-7 Normal components of the Reynolds stress tensor u′u′ and v′v′. Top: Two dimensional
distribution of u′u′ (left) and v′v′ (right) for 800 ml/min gas volume flow rate. Bottom: Profile at
y=0.6 m for all gas volume flow rates. ..................................................................................................................... 133
Figure 8-8 Vertical and horizontal liquid velocity along the x-axis for different heights generated by the
800 ml/min driving flow. Left the vertical velocity and right the horizontal velocity. ..................... 134
Figure 8-9 Velocities along two bubble traces over height for 800 ml/min driving flow. ................................ 135
Figure 8-10 Comparison of the empirical Wellek correlation with the results in air/water with turbulent
background flow. .............................................................................................................................................................. 136
Figure 8-11 Different bubble sizes in the lift force experiment (All pictures have the same scale). ............ 136
Figure 8-12 Number density function of the spherical equivalent diameter and major axis for 0.8 𝑙/𝑚𝑖𝑛
driving flow, except for the 1.5 mm needle 1.0 𝑙/𝑚𝑖𝑛 is plotted. ................................................................ 137
Figure 8-13 Determined terminal velocity. ............................................................................................................................ 138
Figure 8-14 Gas phase velocity field with vertical velocity as color for 800 ml/min driving flow. The left
reactor wall is at x=0 m, the driving flow at the right wall is between x=0.2 m and x=0.25 m.
Left: The velocity field obtained with the 0.3 mm inner diameter needle generating 2.27 mm
bubbles. Right: The velocity field obtained with the drilled out needle generating 4.13 mm
bubbles. ................................................................................................................................................................................. 138
Figure 8-15 Lift force coefficient along the averaged traces. ......................................................................................... 140
Figure 8-16 Averaged lift force coefficients along the averaged bubble traces for different shear rates. The
needle diameter is written at the points. ............................................................................................................... 141
Figure 8-17 Results of the present lift force measurements in turbulent air/water flow (Morton number of
around 2.63⋅10−11) compared to results from the literature for different Morton numbers.
The DNS by Dijkhuizen et al. (2010b) and the experiments of Tomiyama et al. (2002) are
conducted under laminar conditions....................................................................................................................... 142
xvii
List of Tables
Table 2-1 Different formulations of the time scale of the turbulence dissipation production .......................... 26
Table 2-2 Closure models in the baseline model. .................................................................................................................. 27
Table 3-1 Characteristic particle time scale for different measurement techniques in bubbly flows ........... 47
Table 3-2 The different gas volume flow rates used for the experiments, the values are refereed to
standard conditions. .......................................................................................................................................................... 52
Table 5-1 Setups of the experiments by Julia et al. (2007) used for simulations. ................................................... 79
Table 6-1 The superficial velocities and bubbles sizes. ...................................................................................................... 94
Table 7-1 Experimental parameters at standard conditions. ......................................................................................... 106
Table 7-2 Inlet conditions. ............................................................................................................................................................. 116
Table 8-1 Experimental conditions of the lift force experiments. ................................................................................ 128
xviii
xix
Nomenclature
1
Symbols
𝐴
Area
𝐿2
𝐶𝐿
Lift force coefficient
−
𝑑
Diameter
𝐿
𝐹
Differentiable function
𝐹
Force
𝑀𝐿𝑇−2
𝑔
Gravitational acceleration
𝐿 𝑇−2
𝑘
Turbulent kinetic energy
𝐿2 𝑇−2
𝑀
State function
−
𝒏
Normal boundary vector
−
𝑝,𝑃
Pressure
𝑀 𝐿−1𝑇−2
𝑃𝑘
Turbulent kinetic energy
production term
𝐿2𝑇−3
𝑆𝜙
Source term of 𝜙
𝑡
Time
𝑇
𝑢
Horizontal velocity
𝐿 𝑇−1
𝑣
Velocity
𝐿 𝑇−1
𝑣
Vertical velocity
𝐿 𝑇−1
𝑣′
Velocity fluctuation
𝐿 𝑇−1
𝑉
Volume
𝐿3
𝑉
Averaged velocity
𝐿 𝑇−1
𝒙
Position vector
−
𝑋
Lagrangian coordinates
−
Greek and other symbols
𝛼
Void fraction
−
𝒞𝑇
Count trajectories
−
𝔤
Body forces
𝐿 𝑇−2
𝛿
Interface thickness
𝐿
Δ
Difference of two values
−
𝜖
Turbulence dissipation rate
𝐿2𝑇−3
𝜇
Viscosity
𝑀 𝐿−1𝑇−1
𝜕𝑡
Partial derivative with
respect to the variable 𝑡
𝑇−1
𝜌
Density
𝑀 𝐿−3
ℑ
Efflux
∇
Nabla operator
−
ℜ
Non-convective flux
𝜎
Surface tension
𝑀 𝑇−2
𝜏
Characteristic time scale
𝑇
𝕋
Particle/micro bubble track
−
Φ
Field variable
−
1
Only symbols that are used more than once are given
xx
Indices
𝑏
Boundary
𝐵
Bubble; spherical equivalent bubble
diameter
𝐷
Drag
𝐵𝐼𝑇
Bubble induced turbulence
𝐹
Fluid flow
𝐺 or 𝑔
Gas phase
𝑀𝑎𝑗𝑜𝑟
Maximum axis of a bubble
𝑀𝑖𝑛𝑜𝑟
Maximum axis perpendicular to the major
axis
𝑃
Particle
𝐿,𝑙 or 𝐿𝑖𝑞.
Liquid phase
𝑟𝑒𝑙
Relative velocity
Dimensionless numbers
𝐶𝐷 – Drag force coefficient
2𝐹𝐷
𝜌𝑣𝑟𝑒𝑙
2𝐴
Eo – Eötvös number
Δ𝜌𝑔𝑑2
𝜎
Fr – Froude number
𝑣𝑟𝑒𝑙
𝑔𝑑𝐵
Mo – Morton number
𝑔𝜇4Δ𝜌
𝜌2𝜎3
Re – Reynolds number
𝜌𝑣𝑟𝑒𝑙𝑑𝐵
𝜇
St – Stokes number
𝜏𝑃𝑎𝑟𝑡𝑖𝑐𝑙𝑒
𝜏𝐹𝑙𝑜𝑤
We – Weber number
𝜌𝑣𝑟𝑒𝑙
2𝑑𝐵
𝜎
1
1 Introduction
1.1 Subject and motivation
Bubbly flows are well known and encounter in everyday life from refreshing drinks to
cooking. Beyond those trivial applications, aerated flows are a fundamental principle in
biological, chemical, energy and metallurgy engineering. In energy engineering, bubbles
are produced in heat exchangers, evaporators or cooling systems. In biological and
chemical engineering, in contrast, gas is supplied in a liquid for process intensification
due to a large interface and enhanced mixing. No matter how the bubbles originate, a
great interest exists to fully understand bubbly flows and predict their behavior with
suitable models and simulations.
Despite the high interest in modeling bubbly flows, a distinct lack of understanding
of such exists. This lack is due to the complex interface formed by the bubble. Even
fundamental hydrodynamic characteristics like the bubble size are very challenging to
measure and to model. Paired with the broad usage of bubbly flows, a better
understanding has a great potential to increase the efficiency and energy saving for
applications in nearly all fields of process engineering.
In the last years, new measuring techniques were developed for bubbly flows. In
particular, the measurement of the gas phase structure with impedance measuring
techniques (da Silva 2008) or computer tomography (Hampel et al. 2012) is available.
Moreover, liquid velocity measuring techniques that are used in single phase flows are
adopted and the capability to measure at higher gas load is increased, e.g. as shown by
Hosokawa & Tomiyama (2013) with the laser Doppler anemometry.
In addition to new measuring techniques, the ability to perform simulations of bubbly
flows with a complete resolved interface has been developed in the last years. From
simulating single bubbles at lower Reynolds numbers (Tryggvason et al. 2006) up to
several bubbles at higher Reynolds numbers (Roghair et al. 2013) the possibilities are
steadily increasing thanks to more powerful computer systems. Besides, innovative
methods of interface capturing like the combination of the level set and front tracking
method (Maric et al. 2015) increase the possibilities of resolved simulations.
Without doubt, not every chemical reactor or heat exchanger can be examined with a
computer tomography technique and not every single bubble of a bubble column can be
simulated with a detailed resolved interface. Nevertheless, this new methods allow a deep
insight to hydrodynamics of bubbly flows that are used to develop and improve models
for simulation methods which are capable to simulate bubbly flows in complete vessels
or pipes. The simplest methods are zero dimensional based on an integral balance
equation of the complete system.
Since it is often needed to see local effects, methods are formulated in between a
resolved surface simulation and the integral balance formulation. One of these methods
is the Euler-Euler method that is based on the phase averaged Navier-Stokes equations
formulated for the continuous and dispersed phase (Ishii & Hibiki 2006). In this process,
the interface information is lost due to the averaging so that the interaction between the
continuous and dispersed phase has to be modeled. With this method, three-dimensional
simulations of large facilities are possible with locally resolved hydrodynamics. Using
1 Introduction
2
reliable models that are based on a deep understanding of bubbly flows, promising
results were obtained with this method in the recent years.
Despite the efforts being made in the last years to improve the models for the Euler-
Euler method by validating a great model variety, no consensus for modelling could be
found, especially in the area of chemical engineering (Tabib et al. 2008) (Masood &
Delgado 2014). However, based on long-term validation of pipe flows in the area of
nuclear safety engineering (Lucas et al. 2007a) (Frank et al. 2008) (Rzehak & Krepper
2013a) a baseline model has been suggested for bubbly flows (Rzehak & Krepper 2013b).
In such a baseline model, all well-known effects of bubbly flows are included and the
model parameters are fixed. The baseline model is improved by simulating a broad range
of experiments under different conditions with the fixed models. From such simulations,
shortcomings are identified and the models are strategically improved with specifically
dedicated experiments. Moreover, the baseline model is validated in a certain range of
usage so that a prediction of bubbly flows is possible within an assessable error. This
knowledge is essential for the process design of bubbly flows in all fields.
The superior aim of the present work is to improve the understanding of the
hydrodynamics in bubbly flows towards predictable simulations in all fields. In
particular, to adopt and extend the recently suggested baseline model for the Euler-Euler
method to the applications beyond nuclear safety. Besides widespread CFD simulations
of bubble columns, new measuring techniques and concepts were developed as well as
experiments were executed particular for CFD validation.
1.2 Modelling bubbly flows
To understand complex bubbly flows, the motion of single bubbles is investigated, which
is in general strongly affected by their shape. In simple quiescent fluids the shape can be
determined by knowing the Reynolds number, which is the ratio between inertial and
viscous forces
𝑅𝑒=𝜌𝑙𝑣𝑟𝑒𝑙𝑑𝐵
𝜇𝑙 ,
(1-1)
and the Eötvös/Bond number, which is the ratio between gravitational and surface
tension forces
𝐸𝑜=Δ𝜌𝑔𝑑𝐵
2
𝜎 .
(1-2)
Besides the material values density 𝜌, dynamic viscosity 𝜇 and surface tension 𝜎, the
bubble size and the slip/relative velocity 𝑣𝑟𝑒𝑙 between gas and liquid phase and the
bubble diameter 𝑑𝐵 is of importance. With these parameters the shape of bubbles can be
estimated as demonstrated by Grace et al. (1976) proposing the Grace Diagram shown in
Figure 1-1. Moreover, the rising velocity, which is equal to the relative velocity in
quiescent fluids, can be estimated by calculating the Morton number
𝑀𝑜=𝑔𝜇𝑙4Δ𝜌
𝜌𝑙2𝜎3 ,
(1-3)
which is denoted as 𝑀 in the Grace diagram.
1.2 Modelling bubbly flows
3
Figure 1-1 Grace Diagram (Grace et al. 1976) for bubble shapes and bubble rise velocity.
The Morton number is denoted as ‘M’ in the diagram.
Since the Morton number is only a function of the material properties of both phases,
the rising velocity is determined by knowing the bubble diameter and the materials. In
the present work, air in deionized water is used in general, which has a Morton number
of around 2.63⋅10−11. Thus, the wobbling regime is very large ranging from 𝐸𝑜=0.25
up to 𝐸𝑜=40 or from a bubble diameter of 1.4 mm up to 17 mm, respectively.
In Figure 1-2 bubbles in a range from 2.2 mm up to 9.8 mm with different Eötvös
numbers in air/water are shown, which is the size range investigated in the present work.
In addition, a bubble in air/water with 2 ppm of Triton X-100 is shown. In this case, the
bubble has a significantly different shape compared to the corresponding bubble in
water, but with an almost equal Morton number (Surface tension measured in steady
state conditions). The rising bubble has a slip velocity so that the impurity concentration
on the surface is not homogeneous distributed. This leads to the Marangoni convection,
which is not covered by the Grace Diagram. Such effects might be important in almost all
bubbly flows in which chemical reactions take place since a multi-material composite is
often present. The low integral concentration of 2 ppm illustrates that even small
impurity concentrations lead to large effects and the great complexity of bubbly flows in
technical use. Such complex impurity effects, however, are beyond the scope of this work
since the hydrodynamics of clean air/water systems are still not well understood.
Nevertheless, the findings are often compared to fully contaminated systems in which the
bubble behavior is similar for different impurities due to a very high concentration of
such.
1 Introduction
4
Figure 1-2 Bubbles for different Eötvös Numbers in air/water except the right bubble is
in air/(water + 2ppm Triton X-100) at Morton number of 2.63⋅10−11.
Although all bubbles in Figure 1-2 are in the wobbling regime distinct differences are
observed. For lower Eötvös numbers the surface is smooth and the shape is elliptical, with
increasing Eötvös number, the surface is disturbed and the shape becomes more random.
These differences lead to different interactions of the bubble with the surrounding fluid.
Usually, these interactions are modelled as forces that effect the bubble as well as the
surrounding fluid according to the third Newton’s law of motion.
The best-examined bubble force might be the drag force of a single bubble rising in a
stagnant liquid. For example, important work on this topic was done by Haberman &
Morton (1953) in the 50’s; however, still new findings are made to the drag force like the
dependency of the drag coefficient to the initial bubble shape (Tomiyama 2004). Without
doubt, the drag is not the only effect but it is often stated as the most important one. In
the present work, a variety of forces is used in order to model the complex bubble motion
in technical use.
In moving fluids, also the turbulence is of importance. The turbulence is in general
affected by the presence of a bubble, which has to be modeled as well. From single-phase
flow different approaches regarding turbulence modeling exists that are also used for
bubbly flows. In general, these approaches can be described by their capability to resolve
different scales of fluid motion. In technical use, large apparatuses have to be modeled so
that turbulence structures are almost completely modeled and barely resolved by the
approach. In contrast, fundamental flow behavior is studied with methods that fully
resolve the turbulence so that no modelling is needed. In the same way, approaches of
modeling bubbles can be described by their capability to resolve the interface of the gas
bubbles. Methods for technical use do not resolve the interface due to its complexity;
methods used for model development fully resolve the interface. The first one need the
just mentioned forces, the later one do not need any modeling of the bubble motion (for
very simple problems). Naturally, the turbulence model and interface handling have to
be consistent.
Simulations in which all time and length scales are resolved are called direct
numerical simulations (DNS). Indeed, DNS of bubbly flows are not fully resolving all
scales, for example, the coalescence of bubbles depends on a film rupture between the
bubbles that is in the range of nanometers. Such small scales are beyond the possibilities
and the present scope of DNS for bubbly flows, a discussion about this issue is for example
given by Tryggvason et al. (2013).
Nevertheless, DNS in the field of bubbly flows is a very active field; probably the most
investigated phenomenon is the drag of bubbles in an infinite fluid. The drag is correctly
reproduced by DNS up to high Reynolds numbers and large bubbles (Dijkhuizen et al.
2010a). Thus, the methods to capture the deformable interface are at a high level, which
1.2 Modelling bubbly flows
5
is also indicated by the simulations of Tripathi et al. (2015) proposing a new kind of break
up mechanism at low Morton numbers from DNS simulations. Moreover, the lift force of
bubbles in a shear field is a frequently topic of DNS. Whereas the calculation of the correct
drag is a challenging task, dealing with the lift force is a more complicated problem since
the bubble shape interacts with a shear field and the bubble wake. Therefore, a highly
unsteady problem over a wide range of time and length scales occurs so that only studies
were published containing just a small amount of bubbles and simulations (Rabha &
Buwa 2009) (Zhongchuna et al. 2014). However, Dijkhuizen et al. (2010b) performed in
a limited range of Reynolds numbers a large amount of DNS regarding the lift force and
compared it to own experimental results. Important effects could be reproduced with the
DNS but also deviations to their own experimental data arise, especially simulations in
fluids with impurities. Besides the problem of bubbles in an infinite fluid, also the
interaction of bubbles with a wall is recently investigated with DNS (Sugioka & Tsukada
2015).
DNS provide also room for simplifications so that DNS were realized for turbulent
bubble laden flows in channels with the assumption of rigid spheres (Santarelli &
Fröhlich 2012), small bubbles (Bolotnov et al. 2011) or in small deformation regimes
(Dabiri et al. 2013) (Murai et al. 2001). Summarizing, DNS can provide essential data for
single bubble phenomena up to turbulent flows with few bubbles. The recent methods
can be used for assistance in modelling of physical effects. These simulations are still
limited to low Reynolds numbers and/or only few simulations with few bubbles so that a
statistical reliable validation is difficult.
A next step towards simulations at higher Reynolds numbers and increasing amount
of bubbles is the Euler-Lagrange method. The bubbles are treated as point sources, thus
the interface information are no longer directly available so that the interaction of the
bubble and the fluid has to be modelled. The interactions are formulated as forces as
described above. Besides the drag force, effects like the lagging boundary development in
turbulent conditions (Basset force) up to bubble clustering effects are modeled. In
addition, since the bubbles are in general larger than the smallest turbulence length scale
a specific turbulence modelling is necessary. For this purpose, the most common methods
are the large eddy simulations (LES) (bin Mohd Akbar et al. 2012) (Jain et al. 2013) and
the unsteady Reynolds averaged Navier Stokes equations (URANS) (Muñoz-Cobo et al.
2012) (Besbes et al. 2015). Despite the simplification that bubbles are treated as points
without interface, every bubble is tracked separately in order to track bubble-bubble and
bubble-wall collisions as well. For some applications, bubbles can be merged to parcels
to reduce the numerical effort. The knowledge of the collisions is used for a detailed
modelling of the effects that arise from such, e.g. coalescence and break-up (Lau et al.
2014) (Gruber et al. 2013) (Jain et al. 2014).
In the Euler-Lagrange formulation, for every bubble a complete set of motion
equations has to be solved, which are expensive calculations. Furthermore, a mapping
from the Lagrange formulation of the bubbles to the Eulerian formulations of the fluid
and vice versa must be implemented (Kitagawa et al. 2001). Obviously, the maximum
possible bubble size is somehow restricted to the mesh size of the Eulerian fluid grid; else,
the assumption of a point source is no longer valid. The grid size, however, is connected
to the flow situation so that it is not freely selectable. Recently, methods that combine a
resolved interface for large bubbles and the point source formulation for smaller bubbles
1 Introduction
6
in the Lagrangian formulation are developed (Hua 2015). Nevertheless, bubbles exist that
are too small for a resolved interface treatment but are too large for the Lagrangian
formulation. To overcome this, Badreddine et al. (2015) proposed a new method to treat
bubbles in this transition, which gives reasonable results for simple test cases.
Overall, the Euler-Lagrange method is a powerful and often used tool for simulating
bubbly flows. Moreover, this field is a very active area of research proposing new ways of
simulations and methods. Every bubble has to be tracked separately, which is for
technical apparatuses usually far beyond the computational possibilities.
A higher level of modeling is the treatment of both phases as interpenetrating
continua in the Eulerian formulation. The dispersed gas phase is averaged in space
and/or time with the purpose to dissolve the interface of the bubbles so that a continuum
formulation is obtained. Without knowing anything about the interface structure, every
bubble-liquid, bubble-wall and bubble-bubble interaction has to be modeled.
Nevertheless, simulations in which every effect is seen locally, depending on the used
models, for very large apparatuses are possible. This approach is called Euler-Euler
formulation and is the objective of the present work. A detailed introduction is given in
the following sections.
In contrast to the Lagrangian approach, the Eulerian description has no limits
regarding the bubble size to cell size ratio. However, the modeling of very large bubbles,
which might be already treated as gas structures, in the Eulerian framework is difficult
especially looking at the turbulence modeling. For this problem, also promising attempts
were made to combine resolved and unresolved bubbly flow modeling, e.g. the
generalized two-phase flow (GENTOP) concept as described by Hänsch et al. (2012). Such
large bubbles that have to be resolved, however, are not in the focus of the present work.
The highest levels of modeling are the integral based methods that model parts of an
apparatus, a complete apparatus or a complete process module. The models are normally
designed for one existing process so that they are less applicable to others. Usually, the
hydrodynamics are given as an input and specific values or the yield is calculated in a
limited input range. Such methods are typically used in practice of process engineering.
The advantage of CFD methods compared to integral methods is obvious; the CFD
methods are general so that they can be used for every process configuration. In general,
CFD methods are used if local effects, e.g. complex geometrics, are important for the total
process. Such problems are called multiscale problems since small scales influence the
large scale and vice versa. Bubbly flows are such multiscale problems, even if the reactor
geometry is simple. Usually, the flow on every scale in bubbly flows is significantly
influenced by the bubbles. The local bubble concentration and size are in return
determined by the local flow parameters. Thus, the hydrodynamics in bubbly flows
cannot be modeled in general with simple zero or one-dimensional approaches.
Moreover, complex flow situations including heat and mass transport underline the need
of CFD for bubbly flows. Apart from interest in its own right, the hydrodynamic studies
from the present work are a good starting point for the investigation of such complex
situations including heat and mass transport.
7
2 Simulation Methods
2.1 Computational fluid dynamics
The formulation of conservative equations is the basic concept of fluid dynamics and so
the basic concept of computational fluid dynamics (CFD). In these fields, a separation
between material or Lagrangian coordinates and spatial or Eulerian coordinates is used.
At first, the Reynolds transport theorem used for the Lagrange perspective as a
generalization of the Leibniz’s rule for differentiation of integrals is derived. Afterwards,
the finite volume method is explained as an Eulerian perspective.
2.1.1 Material balance equations – Reynolds transport theorem
The Leibniz’s rule gives information how to derivate an integral with the limits of
integration as a function of the variable to derivate. Let be 𝑓(𝑡,𝑥) a sufficient smooth
function that the partial derivate of 𝑓(𝑡,𝑥), 𝜕𝑡𝑓(𝑡,𝑥) and 𝑓(𝑡,𝑥) exists and is continuous
in 𝑡 and 𝑥, then the Leibniz’s rule reads
𝑑
𝑑𝑡(∫ 𝑓(𝑡,𝑥)𝑑𝑥
𝑏(𝑡)
𝑎(𝑡))=∫ 𝜕𝑡𝑓(𝑡,𝑥)𝑑𝑥
𝑏(𝑡)
𝑎(𝑡)+𝑓(𝑡,𝑏(𝑡))𝑑𝑏(𝑡)
𝑑𝑡 −𝑓(𝑡,𝑎(𝑡))𝑑𝑎(𝑡)
𝑑𝑡 .
(2-1)
This rule can be also used for a three dimensional volume 𝑉(𝑡) that is changing over
time. This formulation is called the Reynolds transport theorem:
𝑑
𝑑𝑡(∫ Φ(𝑡,𝒙)𝑑𝑉
𝑉(𝑡) )=∫ 𝜕𝑡Φ(𝑡,𝒙)𝑑𝑉
𝑉(𝑡) +∮ Φ(𝑡,𝒙)𝒗𝒃 ⋅𝒏𝑑𝐴
𝜕𝑉(𝑡) .
(2-2)
The boundary of the volume 𝑉(𝑡) is written as 𝜕𝑉(𝑡), the velocity of the boundary is
written as 𝒗𝒃. As can be seen, the Reynolds transport theorem gives no information about
the surrounding flow field or matter of the surrounding volume. On the right hand side,
the first term is the change of the field variable Φ over time itself, whereas the second
term is the change of the field variable Φ caused by a movement of the boundary.
The material or Lagrangian coordinates are a way of defining the volume used in the
Reynolds transport theorem so that the volume consists always the same continuum-
matter, particles or other matters depending on the problem. In consequence, if the
volume is located in a velocity field the boundary velocity 𝒗𝒃 in Equation (2-2) is the same
velocity as the instantaneous velocity at the boundary of the volume when the volume is
defined as a material volume.
Figure 2-1 The difference between material and spatial perspective. Left a material
control volume with always the same material; right a fixed control volume with the
streaklines in blue.
2 Simulation Methods
8
In Figure 2-1 the material and spatial perspective is sketched, the blue material
volume in the left picture is shown at two different times, 𝑡0 and 𝑡1, and is moving with
the flow field. Exemplary, two points are highlighted in Lagrangian coordinates 𝑋.
Following them over time, the pathlines, which are in a steady flow equal to the
streamlines and streaklines, are obtained. Since a flow field is continuous, the position 𝑥
of every point in the material control volume can be calculated by knowing the flow field
itself, the starting point 𝑋(which are called Lagrangian coordinates) and the time 𝑡.
Consequently, a transformation to the spatial coordinates 𝑥, which are the coordinates
connected to the matter inside the volume, can be given as
𝑥=𝜃(𝑋,𝑡).
(2-3)
Defining the volume in the spatial perspective the volume is not depending on the
matter inside the volume. A volume in spatial coordinates for a fixed coordinate system
is illustrated In Figure 2-1 on the right hand side. If the matter that passes the spatial
volume is tracked, the streaklines are obtained. Obviously, if the considered volume is not
connected to matter inside the volume the boundary velocity 𝒗𝒃 in Equation (2-1) is not
the same as the velocity of the flow field. In the case that the volume is fixed in the used
reference coordinate system, the boundary velocity 𝒗𝒃 is zero and the second term in
Equation (2-1) vanishes.
Using the Stokes’ theorem, the surface integral on the right hand side of Equation
(2-2) can be written as the divergence of Φ(𝑡,𝒙)𝒗𝒃 inside the control volume so that a
conservative balance equation can be formulated:
𝑑
𝑑𝑡(∫ Φ(𝑡,𝒙)𝑑𝑉
𝑉(𝑡) )=∫ 𝜕𝑡Φ(𝑡,𝒙)+𝛁(Φ(𝑡,𝒙)𝒗𝒃) 𝑑𝑉
𝑉(𝑡) =𝑅𝐻𝑆 .
(2-4)
The right hand side (RHS) contains the efflux 𝔍 of the volume 𝑉 and a source term 𝑆Φ
inside the volume
𝑅𝐻𝑆=−∮ 𝔍 ⋅ 𝒏 𝑑𝐴
𝜕𝑉(𝑡)+∫ 𝑆Φ 𝑑𝑉
𝑉(𝑡)=∫ −𝛁𝔍+𝑆Φ 𝑑𝑉
𝑉(𝑡)
(2-5)
Additionally, in Equation (2-5) Stokes’ theorem is used to rewrite the surface integral.
Using the material perspective, the efflux is not containing any convective flux since the
volume 𝑉(𝑡) is defined in a way that all matter stay inside the volume. Therefore, the
efflux is containing in general transfer effects like the viscosity in the momentum
equation when the material perspective is used.
In the following of this work, the Lagrangian formulation/material perspective is
used for a dispersed phase like solid or gaseous particles (if they are resolved). Moreover,
the particles in such problems are simplified to points, so the efflux can be treated as
source terms and no surface deformation occur. The continuous phase (including an
averaged dispersed phase) is formulated in the Eulerian formulation/spatial perspective
treated by the finite volume method as discussed in the next section.
2.1.2 General balance equation, Navier-Stokes equation and finite volumes
A general balance equation can also be formulated from the Reynolds transport theorem;
however, it is common to formulate a general balance equation from a differential form
2.1 Computational fluid dynamics
9
as will be shortly shown in this section. The common way to formulate this form is from
the Eulerian perspective that is straightforward towards the finite volume method (FVM).
The later obtained results might not be exactly calculated with the methods shown here
since complicated solution methods for the FVM exists in commercial CFD codes which
to explain and review is beyond the scope of this work.
The general balance equation can be derived with an infinitesimal volume in
Cartesian coordinates as shown in Figure 2-2 without losing generality. in Figure 2-2 the
flux around an infinitesimal volume is shown, which summed up over all elements gives
(𝑒𝑥𝔍−𝑒𝑥𝔍+𝜕𝔍
𝜕𝑥𝑑𝑥)𝑑𝑦𝑑𝑧+(𝑒𝑦𝔍−𝑒𝑦𝔍+𝜕𝔍
𝜕𝑦𝑑𝑦)𝑑𝑥𝑑𝑧
+(𝑒𝑧𝔍−𝑒𝑧𝔍+𝜕𝔍
𝜕𝑧𝑑𝑧)𝑑𝑥𝑑𝑦=(𝜕𝔍
𝜕𝑥+𝜕𝔍
𝜕𝑦+𝜕𝔍
𝜕𝑧)𝑑𝑉
=(∇⋅𝔍)𝑑𝑉
(2-6)
The unit vectors are written as 𝑒𝑥,𝑦,𝑧, the product of the unit vector with the flux 𝔍 is the
flux in the direction of the unit vector. The flux 𝔍 is defined per area; therefore, the flux
is multiplied with the infinitesimal front surface of the volume. The expression can be
simplified by sum up the flux in every direction and take out the equal amount of the in
and out flow terms. It remains the effective flux ∇ ⋅ 𝔍 per volume.
In contrast to the material perspective, the efflux contains in the spatial perspective
also the convective part. Thus, the flux can be split up to ℑ=𝜌ϕ𝒗+𝔑 with 𝜌𝜙𝒗 the
convective flux and 𝔑 the non-convective flux. Besides the change of the efflux, the
properties inside the volume can change because of processes inside the volume itself,
this processes can be written as source terms 𝑆𝜙. Since no other processes than flux
through the boundaries or internal processes can change the properties inside the
volume, a general balance equation in arbitrary coordinates can be formulated
𝜕𝜌𝜙
𝜕𝑡 =−𝛁⋅(𝜌ϕ𝒗+𝔑)+𝑆𝜙 ⇔ 𝜕𝜌𝜙
𝜕𝑡 +𝛁⋅(𝜌ϕ𝒗)=−𝛁⋅(𝔑)+𝑆𝜙 .
(2-7)
The mass balance equation is obtained by setting 𝜙=1. Furthermore, there is no source
of mass and no non-convective transport of mass, thus 𝔑=0=𝑆𝜙. The well-known
continuity equation is obtained
𝜕𝜌𝜙
𝜕𝑡 +𝛁⋅(𝜌ϕ𝒗)=0 .
(2-8)
The conservation of momentum can be expressed by setting 𝜙=𝒗, 𝔑=−𝑝𝑰+𝑻
and 𝑆𝜙=𝜌𝖌. The transport term is the sum of the scalar pressure multiplied with the unit
tensor 𝑰 and the viscous stress tensor. The viscous stress tensor can be very complex
depending on the used fluid. In the present work, however, mainly water is used and the
viscous stress tensor can be simplified to 𝑻=1
2((∇𝒗)+(∇𝒗)𝑇). The displacement in a
fluid is the velocity so the distortion gradient is equal to the Jacobian matrix of the
velocity ∇𝒗. The viscous stress tensor is a linearization of the Green-Lagrange
deformation tensor 𝐸=1
2((∇𝒗+𝑰)(∇𝒗+𝑰)𝑇−𝑰)=1
2(∇𝒗+(∇𝒗)𝑻+(∇𝒗)𝑻∇𝒗), the
product of the Jacobian matrix with its transposed is taken as zero and the viscous stress
2 Simulation Methods
10
tensor is obtained 𝑻=𝑡𝑖𝑗=1
2(𝜕𝑣𝑖
𝑥𝑗+𝜕𝑣𝑗
𝑥𝑖). With 𝖌 called the body forces as momentum
source term the momentum conservation can be written as
𝜕𝜌𝒗
𝜕𝑡 +𝛁(𝜌𝒗𝒗)=−𝛁 ⋅ 𝑝+ 𝛁 ⋅ 𝑻 +𝜌𝖌 .
(2-9)
In the same way the energy conservation and species conservation equations can be
obtained, which is not shown at this point.
The Navier-Stokes equations are the summarization of the mass, momentum and
energy conservation equations. In general, no analytic solution exists on complex
computational domains and the problem has to be solved numerically. For this purpose
different numerical methods exist, the most widely-know methods are the finite element
method (FEM) and the finite volume method (FVM). Using the latter one, the
computational domain is split up in many well-defined finite volumes.
The finite volumes that are in sum the finite computational domain are generated in
a way that every surface of a finite volume has contact to another finite volume or the
boundary of the domain and that no gaps are present. Therefore, the volumes are
discretizing the computational domain and called mesh or grid in total and grid cell in
particular. Around the grid cell, a balance equation equal to the balance equation around
an infinite volume can be formulated. In general, the integral formulation of Equation
(2-7) is more advantageous for this purpose
∫𝜕𝜌𝜙
𝜕𝑡 𝑑𝑉
𝐺𝐶 + ∮ 𝜌ϕ𝒗 ⋅ 𝑛𝑑𝑨
𝜕𝐺𝐶 =−∮ 𝔑⋅𝑛𝑑𝑨
𝜕𝐺𝐶 +∫𝑆𝜙𝑑𝑉
𝐺𝐶 .
(2-10)
This equation is solved for every grid cell. Furthermore, since the sum of all grid cells
is equal to the domain the general integral balance equation is obtained by summation of
all grid cells. Therefore, the finite volume method is fulfilling the general balance equation
for the domain automatically, which is a benefit of this method.
Figure 2-2 Flux around an infinite volume
2.2 Eulerian averaging, two-fluid model and closure problem
11
The values of a problem during a simulation, however, are only known at discrete
points that are usually located in the centroid of the grid cells. Thus, the Integrals have to
be approximated somehow. The surface integrals are commonly approximated with the
Gaussian quadrature rule, the simplest one is to take the integral average multiplied with
one. The integral average is often taken as the value in the centroid of the grid cell surface,
which is correct if the problem is linear. This approximation is of second order,
nevertheless, the value in the centroid of the surface is not known since only the values
in the volume centroid are known. Consequently, the value in the surface centroid has to
be interpolated which can be done by just using the next known value in the direction of
the flow (upwind scheme – first order) or by interpolating linear between the
neighboring grid cell (central difference scheme – second order). The volume integral is
approximated similar, the value at the volume centroid is multiplied with the volume of
the grid cell, which is correct for linear problems. The approximation and interpolation
schemes can be more complex if the problem requires a complex treatment (Ferziger &
Peric 2002).
Summarizing, the material/Lagrangian and spatial/Eulerian perspective and the
corresponding balance equations for mass and momentum were introduced.
Furthermore, with the finite volume method a numerical method was shown, which is
able to solve these balance equations on a computational domain. The presented methods
can also be applied to multiphase problems; these methods, however, are only valid
whether one phase or only the interface itself is inside the considered volume. In the next
section, a way is shown how these methods are formulated for multiphase problems
without these limitations.
2.2 Eulerian averaging, two-fluid model and closure problem
In the following section the two fluid method for formulating a balance equation for a
volume with both phases inside is introduced. The main idea is to average the quantities
of both phases weighted with the volume fraction. The averaging can be done over time
at one point as it is shown in this section or by averaging over a volume; in both cases the
same equations are obtained. As a result, balance equations for every phase are obtained
that are coupled with interphase exchange terms. The cost of the averaging is the loss of
the interface information, which has to be modeled and put into the interphase exchange
terms. The advantage of the lost interface is that the simulation methods are simplified
and multiphase problems with a very large interface size can be simulated.
In the following, a simplification is done by assuming that the interface cannot store
mass or momentum. The consequence of the first assumption is that the interface is in an
equilibrium; the consequence of the second one is that the interface tension is not minded
in the balance equation. A more precise treatment of the interface can be obtained by
formulating the so-called jump conditions over the interface as discussed for example by
Aris (1962) or Delhaye (1974). In the present work, however, the above simplifications
are sufficient. In the following, a very short derivation of the two-fluid formulations is
given by using the averaging in time method based on the work of Ishii & Hibiki (2006).
The purpose is to give an idea of the two-fluid method and to introduce the notations used
in this work, a complete derivation can be found in literature.
2 Simulation Methods
12
At first, the state functions are defined which distinguish between the phases and the
interface. If a problem with two phases is given, two state functions and one interface
function is needed
𝑀𝑘(𝑥,𝑡)={1if the 𝑘𝑡ℎ− phase is present at point x at time t
0 else
𝑀𝑆(𝑥,𝑡)={1if the interface is present at point x at time t
0 else .
(2-11)
A general differentiable function inside the phases is written as 𝐹; the multiplication with
the state function is written as 𝐹𝑘. The thickness of the interface 𝛿 is assumed to vanish,
which is written as lim𝛿→0.
With the assumption of a vanishing interface thickness the Eulerian time average of
the function 𝐹 is defined as
𝐹(𝑥,𝑡)≡lim
𝛿→0 1
𝛥𝜏∫ 𝐹(𝑥,𝜏)𝑑𝜏
[𝛥𝜏]𝑇 .
(2-12)
With 𝜏 the averaging time, Δ𝜏 the average interval and [Δ𝜏]𝑇 the time interval without the
surface, which is the same as [Δ𝜏] with an assumed zero boundary thickness. Using this
averaging the local void fraction of the 𝑘𝑡ℎ-phase can be defined
𝛼𝑘≡𝑀𝑘
(𝑥,𝑡) .
(2-13)
Further, the mean value of the general function 𝐹 of the 𝑘𝑡ℎ-phase 𝐹𝑘
is calculated to
𝐹𝑘
=𝐹𝑀𝑘
.
(2-14)
Since the integral is a linear function, the averaged values are also linear
𝐹=∑𝐹𝑘
𝑘 ; 1=∑𝑀𝑘
𝑘=∑𝛼𝑘
𝑘 .
(2-15)
Therefore, the local void fraction can also be interpreted as a probability of seeing the
𝑘𝑡ℎ-phase at the place 𝑥 to the time 𝑡.
Besides the averaged values over the total time interval 𝐹 , the averaged values of only
the 𝑘𝑡ℎ-phase are also of interest which are called phase average
𝐹𝑘
≡𝑀𝑘𝐹𝑘
𝑀𝑘
=𝐹𝑘
𝛼𝑘 .
(2-16)
The phase-averaged values are obtained by weighting the Eulerian averaged values with
the state function. In particular, the phase average values are not depending on the
quantity of the phase since it is divided by this. Finally, the mass weighted averaged
values are needed since the conserved quantities are extensive values
𝜙𝑘
≡𝜌𝑘𝜙𝑘
𝜌𝑘
=𝜌𝑘𝜙𝑘
𝜌𝑘
.
(2-17)
Since the mass is conservative itself, the mass averaged quantities are additive and from
the definition of the mass weighted mean values follows
2.2 Eulerian averaging, two-fluid model and closure problem
13
𝜌𝜙
=∑𝜌𝑘𝜙𝑘
𝑘 .
(2-18)
Summarizing, a relation can be written between mixture properties and phase averaged
properties
𝜙=∑ 𝛼𝑘𝜌𝑘
𝜙𝑘
𝑘
∑ 𝛼𝑘𝜌𝑘
𝑘=∑ 𝜌𝑘
𝜙𝑘
𝑘
∑ 𝜌𝑘
𝑘 .
(2-19)
With the above shown averaging methods, the balance equations for multiphase
problems can be formulated. Nevertheless, a lot work has to be done to formulate the
differentiation and all fluxes for the averaged values correctly, this is referred to the
literature at this point, e.g. as is found in the book of Ishii & Hibiki (2006). The mass
conservation for the 𝑘𝑡ℎ-phase can be written after averaging as
𝜕𝛼𝑘𝜌𝑘
𝜕𝑡 +𝛁⋅(𝛼𝑘𝜌𝑘
𝒗𝒌
)=Γ𝑘 .
(2-20)
The mass exchange term between the phases is written as Γ𝑘; the sum of all mass
exchange terms is zero because it is assumed that the interface cannot store mass.
Moreover, the momentum balance equation can be formulated as
𝜕𝛼𝑘𝜌𝑘
𝒗𝒌
𝜕𝑡 +𝛁⋅(𝛼𝑘𝜌𝑘
𝒗𝒌
𝒗𝒌
)
=−𝛁⋅(𝛼𝑘𝑝𝑘
)+𝛁⋅[𝛼𝑘(𝑇𝑘
+ℭ𝑘
𝑇)]+𝛼𝑘𝜌𝑘
𝒈𝒌
+𝑆𝒌 .
(2-21)
The averaged viscous stress tensor of the 𝑘𝑡ℎ-phase 𝑇𝑘
can be obtained by phase
averaging the velocities. Additionally, an interfacial turbulent flux tensor ℭ𝑘
𝑇 is obtained
due to the averaging. Moreover, the interfacial momentum exchange term 𝑆𝒌 occurs with
the side condition
∑S𝑘
𝑘=𝑆𝑆=0 .
(2-22)
The sum of all interfacial exchange terms is zero because it is assumed that the
interface cannot store momentum. The interfacial momentum exchange term connects
the phases; all interface information that is lost due to the averaging process must be
covered in this term. If for example the interfacial exchange were zero, the phases would
barely ‘see’ each other and would just exists side by side.
The correct formulation of the interfacial exchange term is called closure problem
and is one topic of the present work. The exchange is in general formulated as force per
volume, which is called closure models, and contains for example the drag force between
the phases. The forces in the exchange term can describe complex phenomena like the
interaction of the wake structure behind bubbles and the bubble itself. In the next section,
the closure models are formulated and reviewed shortly.
Summarizing, the two-fluid model is introduced and the average balance equations
for the phases are given. Because of the averaging a probability function 𝛼𝑘, which is the
local void fraction, is obtained that quantifies the probability if the 𝑘𝑡ℎ-phase can be seen
2 Simulation Methods
14
at a point 𝑥 and time 𝑡. Therefore, the phases literally exists side by side, which is
reflected by formulating balance equations for every phase in the same space and time.
The connection of the phases is done by coupling terms, which have to be modelled
explicitly since the interface information is lost due to the averaging.
2.3 Closure models and multiple bubble velocity classes
In the following section, the above-described closure models without the turbulence
closures are described. In the present work, the drag, lift, turbulent dispersion, virtual
mass and wall force might be the most important. Noteworthy, a boundary layer that is
not in the steady state, which is modeled as so-called history force, affects all these forces.
Such effects are usually not taken into account because the other forces are measured in
more or less stagnant fluids under steady state conditions. The drag and lift force are
described in detail in the following sections, the others briefly in the following.
The turbulent dispersion force describes the effect of the turbulent fluctuations of the
liquid velocity on the bubbles. Burns et al. (2004) derived by Favre averaging the drag
force an explicit expression, namely
𝐹𝐷𝑖𝑠𝑝=34𝐶𝐷𝛼𝐺
𝑑𝐵|𝑣𝐺−𝑣𝐿|𝜇𝐿𝑡𝑢𝑟𝑏
𝜎𝑇𝐷 (1
𝛼𝐿+1
𝛼𝐺)∇𝛼𝐺 .
(2-23)
In analogy to molecular diffusion, 𝜎𝑇𝐷 is referred to as a Schmidt number. In principle, it
should be possible to obtain its value from single bubble experiments by evaluating the
statistics of bubble trajectories in well-characterized turbulent flows but to the authors
knowledge this has not been done yet. A value of 𝜎𝑇𝐷=0.9 is typically used.
A bubble translating next to a wall in an otherwise quiescent liquid experiences a lift
force. This wall lift force, often simply referred to as wall force, has the general form
𝐹𝑊𝑎𝑙𝑙=2
𝑑𝐵𝐶𝑊𝜌𝐿𝛼𝐺|𝑣𝐺−𝑣𝐿|2 𝑦 ,
(2-24)
in which 𝑦 is the unit normal perpendicular to the wall pointing into the fluid. The
dimensionless wall force coefficient 𝐶𝑊 depends on the distance to the wall y; the
coefficient is expected to be positive so that the bubble is driven away from the wall.
Based on the observation of single bubble trajectories in simple shear flow of glycerol
water solutions Tomiyama et al. (1995) and later Hosokawa et al. (2002) concluded the
functional dependence
𝐶𝑊(𝑦)=𝑓(𝐸𝑜)(𝑑𝐵
2𝑦)2 ,
(2-25)
where in the limit of small Morton number (Hosokawa et al. 2002)
𝑓(𝐸𝑜)=0.0217𝐸𝑜 .
(2-26)
The experimental conditions on which Eq. (2-26) is based are 2.2≤𝐸𝑜≤22 and −2.5≤
log10𝑀𝑜≤−6.0 which is still different from the water–air system with 𝑀𝑜=2.63⋅
10−11 . A recent comparison of this and other distance-dependencies that have been
proposed (Rzehak et al. 2012) has nonetheless shown that good predictions could be
obtained for a set of data on vertical pipe flow of air bubbles in water.
2.3 Closure models and multiple bubble velocity classes
15
The virtual mass is the inertia of the surrounding fluid that has to be taken into
account when a bubble or particle is accelerated relative to the surrounding continuous
phase
𝐹𝑉𝑀=𝐶𝑉𝑀𝛼𝐺𝜌𝐺(𝐷𝑢
𝐺
𝐷𝑡 −𝐷𝑢
𝐿
𝐷𝑡) ,
(2-27)
where 𝐷𝐷𝑡
⁄ denotes the substantial derivative. The coefficient 𝐶𝑉𝑀 is simply set to 0.5 as
suggested by Mougin and Magnaudet (2002).
2.3.1 Drag force
Bodies within a fluid that are moving with a relative velocity greater zero experience a
drag force counteracting to the motion in the opposite direction of the velocity. The drag
force 𝐹𝐷 is a function of the density 𝜌, viscosity 𝜇, the velocity 𝑣 and a length scale of the
object. Regarding to the Buckingham Π-Theorem (Buckingham 1914) two dimensional
groups can be formed from this 5 variables, which are taken as the Reynolds number and
the drag coefficient
𝐶𝐷≡𝐹𝐷
0.5𝜌𝑣2𝐴 .
(2-28)
The surface 𝐴 is a reference area, which is here defined as the projected area of the sphere
equivalent volume of the bubble (Haberman & Morton 1953).
Indeed, the drag coefficient is the sum of the frictional drag coefficient and the
pressure drag coefficient. The corresponding frictional force and pressure drag force are
obtained by integrating the shear stress over the surface and the pressure over the
surface, respectively. For completely laminar flow situations, the drag force is dominated
by the frictional force since the inertial forces are small compared to the viscous forces
as indicated by a particle Reynolds number far below one. Here, a theoretical solution of
the drag force is possible that is called the Stokes law; for rigid spheres, the drag
coefficient is calculated to 24 𝑅𝑒
⁄. For bubbles it can be shown that the drag coefficient is
16 𝑅𝑒
⁄ (Hadamard 1911) which is only the 2/3 of the rigid sphere since the bubble
surface is not rigid. However, through pollution and the resulting Marangoni convection
towards the bubble cap, the drag force is nearly the same as for the rigid sphere
(Lakshmanan & Ehrhard 2010).
Looking at bubbles, the drag force is equal to the buoyancy force if the bubble rises
with a constant velocity in a quiescent fluid. Thus, determining the drag coefficient is
reduced to determine the terminal velocity of the bubble. Based on a dimensional analysis
by Schmidt (1934) using the Π-Theorem and further work by Rosenberg (1950), the
bubble motion was described using the Reynolds number, the drag coefficient and the
Morton number. The Morton number is a combination of the Froude (Fr - ratio of inertia
to body force), Reynolds (ratio of inertia to viscous force) and Weber number (We - ratio
of inertia to surface forces). Based on this Haberman & Morton (1953) described based
on a large preceding experimental work of others the bubble motion for various liquids.
In the following years, several theoretical and experimental work was done regarding
the bubble motion, which is for example summarized by (Loth 2008). The drag
coefficient, however, is often formulated as a function of the Reynolds and Eötvös number
(ratio of surface tension to body forces) in which empirical and semi-empirical
2 Simulation Methods
16
formulations exists. Normally, three different conditions are distinguished, spherical
bubbles, distorted bubbles and cap bubbles with a blending function between them as for
example used by Ishii & Zuber (1979)
𝐶𝐷,𝑆𝑝ℎ𝑒𝑟𝑒=24
Re(1+0.1Re0.75) ,
(2-29)
𝐶𝐷,𝐷𝑖𝑠𝑡𝑜𝑟𝑡𝑒𝑑=23Eo0.5 ,
𝐶𝐷,𝐶𝑎𝑝=83 ,
𝐶𝐷=(max(𝐶𝐷,𝑆𝑝ℎ𝑒𝑟𝑒,min(𝐶𝐷,𝐷𝑖𝑠𝑡𝑜𝑟𝑑𝑡𝑒𝑑,𝐶𝐷,𝐶𝑎𝑝))) .
This drag law is among others measured for polluted air/water systems; differences
occur for pure systems as shown by Tomiyama et al. (1998).
Another approach of describing the drag coefficient is done by Bozzano and Dente
(2001) by the multiplication of a friction factor 𝑓 to a deformation factor (𝑎𝑟
⁄)2 – 𝑎 is the
major axis and 𝑟 the equivalent spherical radius – obtained from a minimization of the
total energy
𝑓=48
Re(1+12Mo13
⁄
1+36Mo13
⁄)+0.9 Eo32
⁄
1.4(1+30Mo16
⁄)+Eo3
2 ,
(2-30)
(𝑎𝑟)2≅10(1 + 1.3 Mo16
⁄)+3.1Eo
10(1 + 1.3 Mo16
⁄)+Eo ,
𝐶𝐷=𝑓⋅(𝑎𝑟)2 .
The discussed drag laws are compared in Figure 2-3; clearly, the differences between the
drag laws for pure systems and the contaminated systems are seen. For the CFD
simulations in the present work the drag law of Ishii and Zuber (1979) will be used. The
drag force formulate in frame of the two fluid averaging as explained in Section 2.2 reads
𝐹𝐷𝑟𝑎𝑔=3
4𝑑𝐵𝐶𝐷𝜌𝑙𝛼𝐺|𝑢
𝐺−𝑢
𝐿|(𝑢
𝐺−𝑢
𝐿) .
(2-31)
The gas velocity is denoted with 𝑢
𝐺 and the liquid velocity with 𝑢
𝐿.
For tracking mini bubbles in the sections below, the drag law of Bozzano and Dente
(2001) will be used since the experimental data for this drag law are validated also for
smaller bubbles.
2.3 Closure models and multiple bubble velocity classes
17
Figure 2-3 Terminal velocity obtained with different drag laws in air/water systems
2.3.2 Lift force
The lift force is usually a force acting mainly perpendicular to the main flow direction. In
contrast to the well-known aerodynamic lift force the lift force on particles and bubbles
is fundamental different. The lift on very small rigid particles due to the shear lift force
and particle rotation was described by Saffmann (1965) for very low particle and
vorticity Reynolds numbers. Nonetheless, larger bubbles also experience a lift force even
in complex flow situations as shown for example by Serizawa et al. (1975) by measuring
a wall peak in bubbly pipe flows. The lift force on bubbles is described by a shear induced
formulation (Zun 1980) based on the formulation for rigid spherical particles (Lawler
1971):
𝐹𝐿𝑖𝑓𝑡=−𝐶𝐿𝜌𝐿(𝑢
𝐺−𝑢
𝐿)×𝑟𝑜𝑡(𝑢
𝐿) .
(2-32)
The lift force coefficient is denoted with 𝐶𝐿, the liquid velocity with 𝑢
𝐿, the gas velocity
with 𝑢
𝐺, the liquid density with 𝜌𝐿. Various work was done calculating the lift force for
inviscid fluids at small shear rates (Auton 1987) and extending the work by Saffmann
(Mei & Klausner 1994) (Legendre & Magnaudet 1997) for viscid fluids ReP→0 and finite
shear rates. It was found that at very low particle and vorticity Reynolds numbers the lift
force is (23
⁄)2 of the lift force of a rigid sphere (Legendre & Magnaudet 1997). Based on
an extensive study on spherical non-deformable bubbles based on DNS calculations by
Legendre & Magnaudet (1998) two regimes can be identified, the viscid dominant regime
with ReP→0 and the inertia dominated ReP→∞ describing two different physical
effects. At low Reynolds numbers, the effect is dominated by the rapid diffusion of the
vorticity generated by the presence of the bubble and its asymmetrical transportation by
the far flow field. At high Reynolds numbers, the vorticity of the undisturbed flow is
asymmetrical distorted due to the presence of the bubble. In between, the effects get very
2 Simulation Methods
18
complicated and are depending on the shear rate (Legendre & Magnaudet 1998). From
the theoretical and numerical work, the lift coefficient was found to be 0.5 for higher
particle Reynolds number; however, some experimental work, for example by Zun (1980)
or Lance & de Bertodano (1994), suppose a lift force coefficient between 0.2 and 0.3.
Furthermore, the lift force is also influenced by the contamination of the water and even
a change of sign appear due to contamination was observed (Fukuta et al. 2008).
The complexity is increased when the bubbles get larger and the assumption of
spherical bubbles is not valid or the shear stress is that high that the bubbles are
deformed. For larger bubbles, the lift force is changed fundamentally as experimentally
shown by Kariyasaki (1987). Kariyasaki used a rotating belt in a channel and supplied gas
bubbles of different sizes; they observed that the lift force changes the sign for larger
bubbles. This change of sign could also be shown in DNS simulations (Ervin & Tryggvason
1997) (Bothe et al. 2006). Two fundamental effects describing the lift force for larger
bubbles might be identified. First, the non-spherical shape which is theoretically
examined using a fixed ellipsoid for low shear rates and inviscid fluids by Naciri (1992),
which is later studied by Adoua et al. (2009) using DNS calculations over a wide range of
Reynolds numbers and shear rates. The lift force for this fixed ellipsoid is significantly
influenced by the Reynolds number, shear rates and the aspect ratio of the ellipsoid.
Increasing the aspect ratio for specific shear rates and Reynolds numbers, the lift force
changes its sign caused by the vorticity generated at the ellipsoid surface. Besides the
change of sign, it could be shown experimentally by Tomiyama et al. (1999) (2002) for
high Morton numbers and low Reynolds numbers that the lift force coefficient might not
be influenced by the shear rate. Thus, the lift force might be dominated by the
deformation of the bubble surface by the shear flow and the produced vortexes at the
bubble surface itself. The empirical correlation for the lift coefficient by Tomiyama et al.
(2002) reads
𝐶𝐿={min[0.288tanh(0.121Re),𝑓(Eo⊥)], Eo⊥<4
𝑓(𝐸𝑜⊥), 4≤Eo⊥<10.7
−0.28 Eo⊥≥10.7 ,
(2-33)
with
𝑓(𝐸𝑜⊥)= 0.00105𝐸𝑜⊥
3−0.0159𝐸𝑜⊥
2−0.0204𝐸𝑜⊥+0.474 .
(2-34)
The modified Eötvös number 𝐸𝑜⊥ is calculated with the maximum horizontal diameter
𝑑⊥; usually the empirical formulation by Wellek et al. (1966) is used
𝑑⊥= 𝑑𝐵√1+0.163Eo0.757
3 .
(2-35)
The lift coefficient in air-water systems is drawn in Figure 2-4 to emphasis the change
of the sign, which is here at around 5.83 mm for the sphere equivalent diameter. The
measurements of Tomiyama et al. (2002) stopped at a modified Eötvös number 𝐸𝑜⊥ of
around 10.7 because larger bubbles were not stable, the constant value of -0.28 for larger
bubbles is usually assumed. Indeed, the lift coefficients obtained by Bothe et al. (2006)
2.3 Closure models and multiple bubble velocity classes
19
and Rabha & Buwa (2009) using DNS and VOF, respectively, indicate that the lift
coefficient decreases further with larger bubble size.
In the past, no experiments were published in which the lift force coefficient was
successfully measured in low Morton number systems like air/water. Kulkarni (2008),
however, determined indirect the lift force coefficient in an air/water bubble column. The
results are in the same area as the results measured by Tomiyama et al. (2002) but the
lift coefficients differ also over the column diameter and show a different behavior for
larger bubbles. Also, Lucas & Tomiyama (2011) investigated a large data obtained from
bubbly flows in pipes and could confirm that the sign of the lift force in water changes at
around 5.8 mm as the Tomiyama lift coefficient correlation predicts.
In the following CFD studies, the lift coefficient correlation of Tomiyama et al. (2002)
is used.
2.3.3 Multiple bubble classes
In practice, a distribution of different bubble sizes is present. The distribution has to be
covered since the behavior of the bubbles depends on their size. For example, small
bubbles have a positive lift force coefficient and large bubbles a negative. Thus, at least
two different velocity fields for the gas phase covering the bubbles below 5.83 mm (zero
of the Tomiyama lift force coefficient) and above are needed.
Nevertheless, for some cases this separation is not sufficient, for example in the
downcomer of an airlift reactor the bubble sizes are separated due to the different drag
force acting on them. This is discussed in detail in Section 7.3. Moreover, if coalescence
and break up effects are significant the bubble distribution must be discretized by bubble
classes (Krepper et al. 2008) or represented by a characteristic function (e.g. quadrature
method of moments) besides the modeling with different velocity fields. This topic is
Figure 2-4 Lift coefficient correlation of Tomiyama et al. (2002) for air-water systems. The
dashed line indicates the not measured region.
2 Simulation Methods
20
discussed e.g. by Jay Sanyal et al. (2005). In the present work the bubble sizes are
assumed fixed, the bubble size distribution is modeled with different gas phases having
their own velocity fields.
It should be noted, that for all forces the same bubble diameter is used, even when a
distinct distribution of the bubble size is present. Looking at the different non-linear
bubble force models this is not correct. In fact, every closure model needs its own bubble
size due to their non-linearity. Nevertheless, this is not a common practice, since the
bubble size distributions are normally not known, so that it will not used in the present
work also.
2.4 Turbulence modeling
Turbulence is a comprehensive phenomenon occurring in nearly all engineering
applications and, therefore, is an important effect in bubbly flows. Turbulence is the
apparently three-dimensional random fluid motion in space and time. In the following
section, the very basics of turbulence modelling are given which are needed to
understand the discussions in this work. The Einstein notation is used in this section for
this purpose. For further information, an introduction to turbulence can be found in the
books of Pope (2000) and Tennekes & Lumley (1972).
The momentum equation with constant density 𝜌 as given in Section 2.1.2 in
Cartesian coordinates using the Einstein notation reads
𝜌[𝜕𝑣𝑖
𝜕𝑡+𝑣𝑗𝜕𝑣𝑖
𝜕𝑥𝑗]=−𝜕𝑝
𝜕𝑥𝑖+𝜕𝑠𝑖𝑗
𝜕𝑥𝑗 .
(2-36)
With 𝑠𝑖𝑗 the viscous stress tensor as defined also in Section 2.1.2.
To investigate the velocity fluctuations, the velocity 𝑣 is decomposed into a mean
value and a fluctuation value, which is called Reynolds decomposition
𝑣𝑖=𝑉𝑖+𝑣𝑖′ .
(2-37)
The instantaneous velocity 𝑣𝑖 is the summation of the mean velocity 𝑉𝑖 and the
fluctuation 𝑣𝑖′. The mean value is determined by using the ensemble average of 𝑣𝑖, which
is denoted in the following with 〈⋅〉. This decomposition is done for all values in Equation
(2-36). Inserting the decomposition in Equation (2-36) the ensemble average of this
equation reads
𝜌[𝜕𝑉𝑖
𝜕𝑡+𝑉𝑗𝜕𝑉𝑖
𝜕𝑥𝑗]=−𝜕𝑝
𝜕𝑥𝑖+𝜕S𝑖𝑗
𝜕𝑥𝑗−𝜌〈𝑣𝑗′𝜕𝑣𝑗′
𝜕𝑥𝑗〉 .
(2-38)
Adding the divergence of the velocity on the right hand side of Equation (2-38) and
zero to the left hand side, which is possible since the divergence of the velocity is zero for
incompressible flows, results in
𝜌[𝜕𝑉𝑖
𝜕𝑡+𝑉𝑗𝜕𝑉𝑖
𝜕𝑥𝑗]=−𝜕𝑝
𝜕𝑥𝑖+𝜕
𝜕𝑥𝑗[𝑆𝑖𝑗−𝜌〈𝑣𝑖′𝑣𝑗′〉] .
(2-39)
The term 𝜌〈𝑣𝑖′𝑣𝑗′〉 is called the Reynolds stress tensor 𝜏𝑖𝑗, which arises out of the non-
linearity of the momentum conservation equation.
2.4 Turbulence modeling
21
Normally, by solving the momentum equation numerically always a discretization in
space and time is used that is larger than the smallest velocity fluctuation scales unless a
direct numerical simulation is performed. Therefore, the Reynolds stress tensor has to be
modelled using the values from the mean flow leading to the closure problem of
turbulence.
A frequently used concept is the Boussinesq approximation introducing an eddy
viscosity 𝜇𝑡
𝜌〈𝑣𝑖′𝑣𝑗′〉=2𝜇𝑇𝑆𝑖𝑗−𝜌〈𝑣𝑖′𝑣𝑖′〉 𝛿𝑖𝑗.
(2-40)
The Reynolds stress tensor is formulated by using the mean strain rate tensor 𝑆𝑖𝑗 as
defined above. The second term is needed since the trace of 𝑆𝑖𝑗 is zero. Thus, the problem
is reformulated by modelling the eddy viscosity and the trace of the Reynolds stress
tensor.
2.4.1 Reynolds Averaged Navier Stokes (RANS) equations
Different approaches were developed in the last century to model the Reynolds stress
tensor; the most often used in engineering problems are the two equations models. Two
equation models are based on the postulation of Prandtl (1961) using the square root of
the turbulent kinetic energy 𝑘 as the characteristic mixing velocity for the eddy viscosity
𝑘=12〈𝑣𝑖′𝑣𝑖′〉 .
(2-41)
A transport equation for 𝑘 can be formulated by formulating the fluctuation transport
equation. Afterwards, subtracting the momentum conservation formulated with the
Reynolds decomposing with the mean averaged transport equation, Equation (2-38), and
multiply it with the fluctuation itself (Wilcox 1994)
𝜌[𝜕𝑘
𝜕𝑡+𝑉𝑗𝜕𝑘
𝜕𝑥𝑗]= 𝜕
𝜕𝑥𝑖[−〈𝑝𝑣𝑖′〉𝛿𝑖𝑗−12〈𝑣𝑖′𝑣𝑖′𝑣𝑗′〉+𝜇𝜕𝑘
𝜕𝑥𝑗]−𝜏𝑖𝑗𝜕𝑉𝑖
𝜕𝑥𝑗−𝜖𝜌 .
(2-42)
The left hand side of Equation (2-42) describes the change of 𝑘 due to convection and
inner processes. The three terms in the brackets on the right hand side describe the
pressure diffusion, the turbulent transport and the molecular diffusion of 𝑘, respectively,
which are summarized by defining a quantity 𝜎𝑘 (Wilcox 1994)
−〈𝑝𝑣𝑖′〉−12〈𝑣𝑖′𝑣𝑖′𝑣𝑗′〉+𝜇𝜕𝑘
𝜕𝑥𝑗=(𝜇𝑇
𝜎𝑘+𝜇)𝜕𝑘
𝜕𝑥𝑗 .
(2-43)
The next term on the right hand side of Equation (2-42) is the production term of 𝑘 due
to the shear of the mean flow, which is therefore often named shear-induced
turbulence −𝑃𝑘. Finally, the last term is the sink term of the turbulent kinetic energy
named turbulence dissipation rate 𝜖
𝜖≡𝜈〈𝜕𝑣𝑖′
𝜕𝑥𝑗𝜕𝑣𝑖′
𝜕𝑥𝑗〉 ~𝑘3
2
𝑙 .
(2-44)
2 Simulation Methods
22
From a dimensional analysis, the turbulence dissipation might be described by the
turbulent kinetic energy and a length scale, which can be taken as the mixing length scale
(Wilcox 1994). Summarizing, the 𝑘-equation is written as
𝜌[𝜕𝑘
𝜕𝑡+𝑉𝑗𝜕𝑘
𝜕𝑥𝑗]= 𝜕
𝜕𝑥𝑖[−(𝜇𝑇
𝜎𝑘+𝜇)𝜕𝑘
𝜕𝑥𝑗]+𝑃𝑘−𝜖𝜌 .
(2-45)
Thus, an equation for the turbulent kinetic energy is obtained that is used for the
characteristic mixing velocity to estimate the eddy viscosity. Nonetheless, a mixing length
scale is needed to describe the eddy viscosity and, additionally, is needed to model the
turbulence dissipation rate 𝜖 since the dissipation rate is not accessible with mean values
by definition.
For the length scale modelling with a two-equation model, two approaches are
frequently used: Formulating a transport equation for the dissipation per unit turbulent
kinetic energy 𝜔 or formulating a transport equation for the dissipation rate 𝜖. The
transport equation for the latter one reads (Jones & Launder 1972)
𝜌[𝜕𝜌𝜖
𝜕𝑡 +𝑉𝑗𝜕𝜌𝜖
𝜕𝑥𝑗]= 𝜕
𝜕𝑥𝑖[−(𝜇𝑇
𝜎𝜖+𝜇)𝜕𝜖
𝜕𝑥𝑗]+𝐶1𝜖𝜖𝑘𝑃𝑘+𝐶2𝜖𝜌𝜖2
𝑘 .
(2-46)
For further studies a model distinguishing near wall and free turbulence by
blending between the 𝑘−𝜔 and 𝑘−𝜖 model called SST 𝑘−𝜔 model by Menter et al.
(2003) is used with
𝜌[𝜕𝜌𝜔
𝜕𝑡 +𝑉𝑗𝜕𝜌𝜔
𝜕𝑥𝑗]= 𝜕
𝜕𝑥𝑖[−(𝜇𝑇
𝜎𝜔+𝜇)𝜕𝜔
𝜕𝑥𝑗]+𝑃𝜔−𝑌𝜔+𝐷𝜔 .
(2-47)
The production term 𝑃𝜔 and dissipation term 𝑌𝜔 are more complicated than for the 𝑘 or 𝜖
equation as described by Menter et al. (2003). A derivation for these values is not shown
at this point. The term 𝐷𝜔 arises due to the blending between the model formulations.
2.4.2 Unsteady RANS equations
As described above, the length and time scales of turbulence are in general very small. By
the use of an extra eddy viscosity, these very small scales are modeled (or filtered as
discussed in the next section). Thus, just a coarse mesh is needed to resolve the remaining
scales. The RANS equations need only a relative coarse mesh compared to other methods,
which is a great advantage in technical use.
Nevertheless, turbulence is a complex phenomenon and depends on the local
geometry/local flow. Furthermore, the usage of a total kinetic turbulent kinetic energy
𝑘 in the transport equation implies the assumption of isotropic turbulence, which is often
not given. Anisotropic turbulence phenomena, however, occur often on the larger scales
whereas the assumption of isotropic turbulence phenomena is good for the smaller
scales. Thus, the small scales might be described with general models while the larger
scales should be resolved. Therefore, an improvement is reached by calculating the large-
scale fluctuations directly while the smaller scales are modeled. For this purpose, the
RANS equations are solved transient while the small scales are modeled by the same
equations as for the steady state case. This method is often called unsteady RANS
(URANS) or very large eddy simulation (VLES). Often, the relatively simple and fast
URANS calculations are even treated with stationary boundary conditions to study for
2.4 Turbulence modeling
23
example vortex shedding at bluff bodies, which gives reasonable predictions as discussed
by Spalart (2000).
With the URANS approach, the fluctuations of the velocity are decomposed in
resolved and unresolved parts. For comparison with experiments, both fluctuation parts
have to be considered to get the total fluctuation. In general, for transient simulations the
total time-averaged kinetic energy is simply equal to the sum of the squared averaged
velocity and the average of the squared fluctuations:
12𝜌 𝑣𝑣
=12𝜌(𝑣𝑣+𝑣′𝑣′
) .
(2-48)
Analog to above, 𝑣 is the average over time and 𝑣′ is the fluctuation around the average.
The modeled and resolved fluctuation that are obtained in the URANS approach can be
written as the sum of these two components
𝑣′=𝑣′
+𝑣′′ ,
(2-49)
where 𝑣′
denotes the resolved fluctuation and 𝑣′′ the modeled fluctuation. Using this
summation the turbulent kinetic energy for the velocity component 𝑣 can be written as
𝑣′𝑣′
=𝑣′
𝑣′
+𝑣′′𝑣′′
.
(2-50)
The modeled fluctuation 𝑣′′𝑣′′
is described by the above defined 𝑘𝑚𝑜𝑑 transport
equation. Since only the total modeled turbulent kinetic energy is known, 𝑣′′𝑣′′
is
calculated by
𝑣′′𝑣′′
=23𝑘𝑚𝑜𝑑 .
(2-51)
Whereas the resolved part is obtained from the transient simulation. The normal
component of the Reynolds stress tensor is therefore
𝑣′𝑣′
=𝑣′
𝑣′
+ 23𝑘𝑚𝑜𝑑
.
(2-52)
2.4.3 Large Eddy Simulations
Large eddy simulations (LES) are similar to URANS simulations, but the small scales are
specifically filtered using filter functions. The filtering of the small scales for LES is done
by a convolution of the value with a filter function. Different filter functions exist for
different purposes. The selection of the scales is regulated by varying the size of the filter
functions, which is often called filter length. The advantage of this selection is that the
small scales can be specifically modelled. Normally, the small scales are only needed for
the dissipation of the turbulent kinetic energy, so a simple model can be used.
By filtering the Navier-Stokes equations, the filtered Navier Stokes equations and an
unresolved part are obtained, similar to the URANS equations. Differences to the URANS
equations occur since the filters usually used in LES have not the same properties as the
RANS models. Nevertheless, also the filter functions have to fulfill specific requirements
(Lesieur et al. 2005) and cannot be chosen arbitrary. The filtered Navier Stokes equations
can be discretized using for example the finite volume (FV) method and can be solved
2 Simulation Methods
24
using the standard procedures with consideration of the properties of the filtering
process.
Usually, the filter is connected to the used discretization, e.g. on a rectangular mesh
using the FV method the Navier-Stokes equations are naturally filtered with the shape of
a box filter. Therefore, the models for the small scales are connected to the local grid size.
Consequently, with this procedure a mesh independent solution is usually not achieved.
The unresolved scale is usually called sub-grid-scale (SGS) and has to be modeled.
Often, the SGS are modeled using the Boussinesq-approximation so that a turbulent
viscosity is modeled. For this purpose, different model approaches exists, ranging from
algebraic models up to two equation or Reynolds-stress models. A common model is the
algebraic Smagorinsky model using the filter width Δ and the rate of strain tensor 𝑺
𝜈𝑡=(𝐶𝑆Δ)2|𝑺| .
(2-53)
The prefactor 𝐶𝑆 depends on the actual flow conditions; a constant value between 0.1 and
0.24 is usually taken.
Modelling the turbulence of bubbly flows is not trivial with the LES approach and an
active topic of research. Often, an additional turbulent viscosity is added to the modeled
turbulent viscosity obtained from the SGS modelling. Among others, the modelling of the
bubble-induced turbulence using an additional turbulent viscosity is discussed in the next
section.
2.4.4 Turbulence in Bubbly flows
The turbulence in bubbly flows is not very well investigated compared to the turbulence
in single-phase flows. The presence of bubbles can either increase or suppress the
turbulence (Serizawa & Kataoka 1990). The wake of the bubbles, for example, usually
leads to an increase in turbulence. In addition, the turbulence interaction of bubbles and
their wake structures with the surrounding turbulence might be very complicated, up to
now no reliable models exists. Moreover, the gas phase distribution in the reactor due to
the turbulent dispersion of the bubbles and the gas phase structure due to break up and
coalescence processes is influenced by the turbulence. The gas phase distribution and
bubble size in return influence the turbulence; the interaction of the turbulence with the
bubbles is in general depending on the bubble size/length as well as the turbulence length
scale, which are both more or less continuously distributed and both often not known.
Since the turbulence of such complex bubbly flows are not understood no straight
forward theory exists, as for single-phase flows. Thus, a discussion of this topic is beyond
the scope of this section and is referred to the literature (Kataoka & Serizawa 1989) (de
Bertodano et al. 1994) (Risso et al. 2008) (Riboux et al. 2010). Nevertheless, in the
following some frequently used modeling approaches are shown, which are also used for
CFD simulations.
In practice, the turbulence in bubbly flows is modeled by the superposition of the
single-phase flow (SP) with a bubbly flow (B). One approach is the assumption that the
eddy viscosity is superimposed
𝜇𝑇=𝜇𝑇,𝑆𝑃+𝜇𝑇,𝐵 .
(2-54)
2.4 Turbulence modeling
25
An early semi-empirical model is the well-known Sato model (Sato et al. 1981) that reads
in the formulation of the two fluid model
𝜇𝑇,𝐵=0.6𝜌𝛼𝑑𝑝|𝒗𝐺−𝒗𝐿| .
(2-55)
Originally, this model was developed for pipe flows with a zero equation turbulence
model. Nevertheless, using two equation turbulence models derived for single phase
flows and just adding the additionally viscosity afterwards good results are obtained in
pipe flows (Krepper et al. 2005) and bubble columns (Tabib et al. 2008). In addition, the
separation of the turbulent viscosities is also a useful method to investigate the
turbulence in bubbly flows (Hosokawa & Tomiyama 2004). These models are useful for
the sub grid scale modeling in large eddy simulations (Ma et al. 2015) and for Reynolds
stress models (Masood et al. 2014).
Despite the usefulness of such viscosity superposition, the turbulent kinetic energy is
usually underpredicted, especially in combination with one or more equation models.
The only contribution to the turbulent kinetic energy production is to the shear induced
production 𝑃𝑘 in Equation (2-45)
𝑃𝑘=𝜏𝑖𝑗𝜕𝑉𝑖
𝑥𝑗=𝜇𝑡(2𝑆𝑖𝑗𝑆𝑗𝑖)=(𝜇𝑇,𝑆𝑃+𝜇𝑇,𝐵)(2𝑆𝑖𝑗𝑆𝑗𝑖) ,
(2-56)
with 𝑆𝑖𝑗 the mean strain rate tensor. It is imaginable that this contribution is not enough
if wake effects significantly contribute to the turbulent kinetic energy. This
underprediction is significant if further effects in bubbly flows depending on the kinetic
energy are modeled, in particular the break up and coalescence effects.
Another approach is the superposition of the production terms 𝑃 in Equations (2-45)
to (2-47) with a bubble induced turbulence (BIT) term. The production of the turbulent
kinetic energy then reads
𝑃𝑘=𝜇𝑡(2𝑆𝑖𝑗𝑆𝑗𝑖)+𝑃𝑘,𝐵𝐼𝑇 .
(2-57)
The production term 𝑃𝑘,𝐵𝐼𝑇 is usually taken as the energy transferred to the surrounding
liquid due to the slip velocity
𝑃𝑘,𝐵𝐼𝑇=𝐶𝑘,𝐵𝐼𝑇𝐹𝐷𝑟𝑎𝑔|𝒗𝐺−𝒗𝐿| .
(2-58)
The prefactor 𝐶𝑘,𝐵𝐼𝑇 is normally taken to one, which is also used in the following.
Formulating the production terms of the energy dissipation and dissipation rate the
similar heuristic is used as for single-phase flows (Rzehak & Krepper 2013a), e.g.
𝑃𝜖,𝐵𝐼𝑇=𝐶𝜖,𝐵𝐼𝑇 𝑃𝑘,𝐵𝐼𝑇
𝜏 .
(2-59)
The production term is formulated by dividing the production of the turbulent kinetic
energy by a time scale 𝜏 with a prefactor 𝐶𝜖,𝐵𝐼𝑇. Using a dimensional analysis the time
scale can be formulated on four different ways as shown in Table 2-1.
2 Simulation Methods
26
Author
1/𝜏
𝐶𝜖,𝐵𝐼𝑇
Morel (1997)
(𝜖
𝑑𝐵
2)13
⁄
1
Troshko & Hassan (2001)
|𝒗𝐺−𝒗𝐿|
𝑑𝐵
0.45
Politano et al. (2003)
𝜖𝑘
1.93
Rzehak & Krepper (2012)
√𝑘
𝑑𝐵
1
Table 2-1 Different formulations of the time scale of the turbulence dissipation
production
All the models in Table 2-1 are formulated and/or validated for different pipe flows. The
latter one formulated by Rzehak & Krepper (2012) is taken for the simulations in bubble
columns.
2.5 Baseline concept for simulating bubbly flows
CFD simulations of dispersed bubbly flow on the scale of technical equipment are feasible
within the above-discussed Eulerian two-fluid framework of interpenetrating continua.
By the use of suitable closure models describing the physics on the scale of individual
bubbles or groups thereof. A large number of numerical work exists, in each of those
largely a different set of closure relations is compared to a different set of experimental
data. However, the used models are often explicitly picked and adjusted to fit the actual
experimental data. Therefore, a reasonable agreement for a specific case is reached but a
reliable predictive modeling is not given. Predictive simulation, however, requires a
model that works without any adjustments within the targeted domain of applicability.
As a step towards this goal, the HZDR CFD group attempted to collect the best
available description for all aspects known to be relevant for adiabatic bubbly flows to a
baseline model. This attempt was predominantly focused on pipe flows with external
pressure gradients. In the present work, the performance of this baseline in bubbly flows,
which are driven by the bubbles, in particular bubble columns, is investigated and
extended.
Aspects requiring closure for adiabatic bubbly flows that have to be fixed in the
baseline model are the exchange of momentum between liquid and gas phases, the effects
of the dispersed bubbles on the turbulence of the liquid carrier phase and processes of
bubble coalescence and breakup that determine the distribution of bubble sizes. Those
closure models require a reliable validation, preferably by the use of experiments, which
are dedicated to one specific problem over a large range of boundary conditions. Since
such experimental data not exist, also because the closure models are strongly coupled,
the complex validation problem has to be simplified. A reasonable simplification is to
confine on problems with a fixed bubble size distribution as discussed above. In this way
the sub-models for bubble forces and bubble-induced turbulence can be validated
independently of bubble coalescence and breakup processes. The validation of this
2.5 Baseline concept for simulating bubbly flows
27
partial problem is done by the use of experimental data from the literature as well as
experimental data that was produced in the framework of the present work.
Besides a set of closure models, which are summarized in Table 2-2, other aspects of
simulating bubbly flows have to be determined. Up to now, these aspects are not fixed in
the current baseline model version; in the present work, however, they are held constant.
In particular, the URANS formulation with the SST turbulence model (Menter et al. 2003)
is used to resolve the large-scale structures, which often occur in bubbly driven flow,
including a reasonable convergence criterion. Moreover, a slip condition for the gas phase
and a no-slip condition for the liquid phase are used at the walls. The surface that
normally occurs in bubbly driven flows is modeled with a degassing boundary with a
variable pressure on it (Ansys 2013). Here the pressure remains variable over the top of
the column, which might be interpreted as different surface heights at different positions
due to the flow, for other scalar quantities of the dispersed phase a constant gradient is
imposed. Thus, effects of a free surface such as waves or foaming is not considered. The
gas inlet is modeled by the use of mass flow through surfaces in the range of the generated
bubble sizes. Accordingly, the initial gas velocity is taken to the ratio of the gas volume
flow rate to the corresponding inlet area. Therefore, the initial velocity of the gas phase
in the experiments, which range from zero for single bubble formation to large velocities
for aerating with gas jets, is not directly modeled because the terminal bubble velocity is
usually reached after one or two cells (Sokolichin et al. 2004). Furthermore, no
experimental data with significant gas jets are used in the present work.
In addition, it is attended to use a simple mesh consisting of rectangular volume cells
of the same size, only for the simulations of an airlift reactor in Section 7.3 the size of the
rectangular volume cells differs over the calculation domain due to the complex
geometry. The size of the volume cells was determined for every validation case in a mesh
study separately containing at least three different meshes, which is shown once in
Section 4. A cells size of 5 mm was usually found to be sufficient. For the spatial
discretization a high-resolution scheme is used (Ansys 2013). For the transient
discretization, a second-order backward Euler scheme is used.
Drag force
Ishii and Zuber (1979)
Lift force
Tomiyama et al. (2002)
Wall force
Hosokawa et al. (2002)
Virtual mass
𝐶𝑉𝑀=0.5
Turbulent dispersion
Favre Averaging (Burns et al. 2004)
Table 2-2 Closure models in the baseline model.
Apart from interest in its own right, results obtained for the momentum exchange
restricted problem also provide a good starting point for the investigation of more
complex situations including heat and mass transport, phase change and chemical
reactions in bubbly flows.
28
29
3 Experimental Methods
3.1 Experimental facility
The used experimental facility, which was build up during the present work, is a
rectangular Plexiglas® bubble column shown in Figure 3-1. The cross sectional area is
250×50 mm2 large. The total height is 1000 mm. The liquid level is set depending on the
application. The ground plate is made of 10 mm thick stainless steel and up to 18 needles
can be placed in the threaded holes shown in the right side of Figure 3-1. The holes are
displaced to each other. The bubble column is placed in a frame of profiles as shown in
Figure 3-2.
The sparger consists of single needles pressed in a bracket with an inner radius of 3
mm and are screwed in the ground plate. The deburred needles are cut flatly and various
sizes of them were used in the experiments. The gas volume flow rate is regulated by up
to two mass flow controllers from Omega Engineering.
For a proper design of the experiments, the bubble sizes generated by different
needles were determined with the methods described in Section 3.2. In Figure 3-3 the
spherical equivalent diameter of the projected area with its standard deviation over the
volume flow rate per needle is shown. In addition, the bubbles are wobbling during the
ascent. This wobbling is artificial because the bubble size is evaluated from the projected
area; the resulting artificial standard deviation is shown in Figure 3-4. The shown
wobbling standard deviation is the arithmetic average of the standard deviations
determined along the bubble tracks.
Figure 3-1 Sketch of the experimental setup and the ground plate with (right) (metrics
in mm).
3 Experimental Methods
30
Figure 3-2 Photograph of the experimental setup
3.1 Experimental facility
31
Figure 3-3 Size of the bubbles (top) and the standard deviation of the size (bottom) over
the volume flow rate for different needle sizes.
Figure 3-4 Wobbling uncertainty
3 Experimental Methods
32
3.2 Bubble size
Determining the bubble size distribution in bubbly flows is essential since all closure
models discussed above depend on them. Digital image analysis of bubbly flows is often
used in order to determine the bubble size distribution, which is described e.g. by Bröder
& Sommerfeld (2007) or Lau et al. (2013). Although the identification of objects from
photographs is a well-known technique, the reliable identification of bubbles from
images is still very challenging. The nature of bubbly flows, in particular bubble clusters
that consists of many overlapping bubbles, is very complex so that a reliable identification
of bubbles is problematic. Therefore, bubbles in complex bubbly flows are normally
identified by hand (e.g. by Mohd Akbar et al. (2012)).
In the following, the used and developed digital image analysis methods in the
present work are described, ranging from automatized single bubble identification to
methods of handling bubbles in dense clusters. In the first part, the determination of
single bubbles with an edge detector is described; further, the basic principles of bubble
detection are introduced. In the second part, an algorithm is developed to evaluate
whether a bubble is overlaid by another bubble or not. Using this algorithm, bubble sizes
can be automatically determined for very low void fractions and/or narrow bubble size
distributions. Finally, a method is developed to pick bubbles by hand in dense clusters
with the aid of the previous developed methods.
After the bubbles are identified, a representative bubble size has to be determined.
For this purpose, two reasonable assumptions exist, first the circular equivalent diameter
of the projected area and second the spherical equivalent diameter of the solid of rotation
of the projected area. Since the closure models are formulated by using the spherical
equivalent diameter of the bubble volume (Haberman & Morton 1953) (Tomiyama et al.
2002) the latter one might be a reasonable choice.
The solid of rotation is calculated by half rotating the left and the right half of the
bubble that is split by the rotation center as demonstrated in Figure 3-5. The red point
marks the rotation center of the bubble. The boundary (𝛺1+𝛺2) and the projected Area
(A1+A2) of the bubble is separated by the rotation symmetry (dashed line). The left and
right half of the bubble have an own centroid (blue and black point, respectively) with a
distance to the rotation symmetry of Δx1 and Δx2, respectively.
Figure 3-5 Calculation of the solid of rotation of a bubble.
Using the second Guldinus theorem and further assuming 𝐴1=𝐴2, the volume is
calculated by
3.2 Bubble size
33
𝑉𝑅𝑜𝑡𝑎𝑡𝑖𝑜𝑛=𝜋(A1Δ𝑥1+𝐴2Δ𝑥2)=𝜋𝐴𝑝𝑟𝑜𝑗𝑒𝑐𝑡𝑒𝑑
2(Δ𝑥1+Δ𝑥2) .
(3-1)
With the spherical equivalent diameter of this volume, the density functions (DF) are
calculated. For this purpose, the determined bubbles are separated in bubble size classes
𝑑𝑖−Δ𝑑𝑖2
⁄≤𝑑𝑏<𝑑𝑖+Δ𝑑𝑖2
⁄. The discretized density functions are then calculated by
𝐷𝐹𝑖=𝑓𝑖
∑𝑓𝑖𝑖 1
Δ𝑑𝑖 , 𝑓𝑖=∑ 𝑓(𝑑𝑏)
𝑑𝑖−Δ𝑑𝑖2
⁄≤𝑑𝑏<𝑑𝑖+Δ𝑑𝑖2
⁄ ,
(3-2)
With 𝑓 the characteristic function. The number density function is calculated by
using 𝑓(𝑑𝑏)=1, the area density function by 𝑓(𝑑𝑏)=𝜋𝑑𝑏2 and the volume density
function by 𝑓(𝑑𝑏)=𝜋 6
⁄ 𝑑𝑏3. The expected value (𝐸) is calculated by
𝐸𝑓=∑𝑓𝑖𝑑𝑖Δ𝑑𝑖
𝑖 .
(3-3)
For example, the Sauter diameter is the expected value of the area density function. This
diameter is in the following given as bubble diameter, if not explicitly said otherwise.
3.2.1 Single bubbles
In this section, the procedure to determine single bubbles is explained, as for example
shown in Figure 3-6. All algorithms are based on a modified Canny edge detector (Canny
1986). The basic algorithm is shortly introduced and results are shown. The
mathematically description of every step is not given at this point since it is standard
knowledge and can be found in the literature (Petrou & Petrou 2010) (Parker 2010). All
images were recorded with a backlight technique with diffusors between the light source
and the bubble column.
As a first step the images are generally preprocessed, particularly the images are cut,
rotated, the contrast is corrected and modified to the needs of the edge detecting
algorithms. The contrast is corrected by renormalizing the white and black pixels at the
end of the histogram and elongating the grey scale histogram to fit the complete grey
scale. For a higher edge detecting quality, a Median and a Gaussian Filter is used. Besides,
the pixel size and the reference point are determined with calibration grids on the
experimental facility to identify later the size and position of the bubbles.
The used edge-detecting algorithm can be separated in the steps: Two dimensional
derivation, edge thinning and edge selection. The two dimensional derivation is obtained
by a convolution using the Sobel operator. The edges are thinned to one pixel by taking
the pixel that is at the minimum of the absolute value function of the two dimensional
derivation along the normal direction of the bubble surface. The normal direction of the
surface is calculated from the arctangent function of the derivation in vertical and
horizontal direction. The normal direction is discretized to 45-degree angels
corresponding to the pixel discretization of the image. Afterwards, a hysteresis with two
thresholds is used for the edge selection so that three categories are obtained, below the
first (edge is deleted), in the middle of the first and second (weak edge) and above the
second threshold (strong edge). Afterwards, if a weak edge is direct or indirect over other
weak edges connected to a strong edge, the weak edge becomes a strong edge; else, the
weak edge is deleted. The output of this double threshold is shown in Figure 3-6.
3 Experimental Methods
34
Figure 3-6 Main steps for bubble size determination of single bubbles, left to the right:
Original image; Edges with weak (grey) and strong edges (white); filled structures; result
with major and minor axis and area centroid (white dots).
After the edges are detected, the bubbles have to be identified as such. At First, the
edges are closed by a simple dilatation and erosion step. Next, the inside and outside of
every bubble have to be defined. For this purpose, the detected edges are marked as white
and the rest as black. Afterwards, the obtained image is as often dilated until all the image
is covered by white. Then, the previous dilation step is considered and evaluated if a black
pixel exists that have in the original (preprocessed) image a grey value of 127 or larger.
If so, this pixel is stated as a point in the surrounding and the surrounding is obtained by
getting the connected black pixels from the image obtained from the edge detecting. If
there is no black pixel that has a value greater than 127 in the original (preprocessed)
image, the next previous dilatation step is considered and so on. This algorithm is very
stable and gives in almost all cases a very good result. The definition of the surrounding
of the bubbles is very important since the bubbles have also structures inside as can be
seen in Figure 3-6. Moreover, for complex problems like the particle tracking in bubbly
flows described below the definition of the surrounding is problematic since most of the
image is covered by very big bubbles or bubble clusters. Here, the described algorithm
also works very well. After the surrounding of the single bubbles is defined, the bubbles
are separated by nature and can be identified easily. Clearly, if bubbles are overlapping
an extra step for separation is necessary. Finally, the projected areas of the bubbles are
obtained and the bubble sizes are calculated as described above.
The limit of the shown algorithm is when two bubbles are touching. Nevertheless, the
algorithm already shows good results for a single needle sparger as shown in Figure 3-7.
Figure 3-7 Bubble size measurement for a 0.6 mm inner diameter single needle sparger
with 0.05 l/min gas volume flow rate.
3.2 Bubble size
35
3.2.2 Systems with very low void fraction or narrow bubble size distributions
In the following section, an algorithm is developed to determine whether a bubble is
overlaid by other bubbles or not. Identifying bubbles that are overlaid by other bubbles
is simple, but identifying whether a bubble is in front or not is more difficult as well as
the reconstruction of the bubble(s) in the back. Up to now, no reliable and/or efficient
methods exist for this purpose. Thus, only the non-overlaid bubbles are used to
determine the bubble size distribution, segmentation of the overlaid bubbles is not
conducted.
Determining the bubble size distribution from the non-overlaid bubbles only,
imposes the assumption that all bubbles have the same probability to be overlaid, else a
wrong result is obtained. This assumption is only appropriate if the bubble size
distribution is narrow.
The algorithm for the automated evaluation of the recordings is demonstrated in
Figure 3-8. At first, the raw picture is segmented by the use of an adaptive threshold to
divide the black surrounding of the bubble and the translucent inside. The adaptive
threshold is determined for every pixel by averaging over a 25×25 pixel area; usually
this value is subtracted by 5 grey values afterwards. After the adaptive threshold is
applied, the bubbles that have a translucent inside are identified by a simple divide and
conquer algorithm.
The centroid of the translucent inside is used as bubble position. Furthermore, if an
object appears completely black without a white inside, the use of the adaptive threshold
is repeated for this single object with a threshold determined with reduced averaging
area size. This leads to a better segmentation of the previously under-segmented
completely dark object.
Next, the result of the above-described edge-detecting algorithm is combined with
the result of the segmentation; if inside a closed edge more than two bubbles are found,
the area inside the edge is treated as overlaid bubbles. Finally, the bubble size distribution
is determined with the non-overlaid bubbles as shown in Figure 3-8.
Figure 3-8: Automated detection of overlaid bubbles. a) Input with corrected contrast b)
segmented bubbles with positions (grey crosses) c) edge detection algorithm d) cutting
out the overlaid bubbles (dark grey). Only bubbles larger than 1.5 mm are treated.
This algorithm is used to identify bubbles in single needle experiments at higher gas
volume flow rates. Moreover, the algorithm is used in the downcomer of the later
discussed airlift reactor; a determined volume density function is shown in Figure 3-9.
Among others, this determined volume density function was compared to a
3 Experimental Methods
36
comprehensive evaluating of the bubble size by hand using the algorithm described in
the next section; as a result, an underprediction of 5 % was found. In general, this
algorithm gives very good results in bubbly flows containing small bubbles with a narrow
bubble size distribution and low gas void fraction.
Figure 3-9 Volume density function in the downcomer of an airlift reactor shown in
Section 7, the shown volume density function is obtained in the downcomer for case 6
between y= 0.2 m and y= 0.3 m.
3.2.3 Systems with higher void fractions and wide bubble size distributions
In general, wide bubble size distributions as well as bubble clusters are found in bubbly
flows. No method or algorithm is published that can determine reliable bubble sizes in
such flows. The level reached are methods that are trying to reconstruct overlaid bubble
structures from single pictures (Bröder & Sommerfeld 2007) (Lau et al. 2013). Such
reconstruction methods are very limited and can only be used in flow situations with
simple shapes like spherical or ellipsoid. In real flow problems, larger bubbles have an
arbitrary shape as demonstrated in Figure 3-10. Even at the lowest flow rate of 3 l/min
with an integral hold up of around 2% bubble clusters occur and an automated
reconstruction of the overlaid bubbles might be not possible.
Figure 3-10 Bubbly flow for different volume flow rates, left, 3 l/min, right, 7 l/min. The
used sparger setup is discussed in Section 3.4.
3.2 Bubble size
37
In this context, a semi-automated algorithm is developed. The main idea is to follow
a bubble cluster over a certain time until all bubbles of the cluster were clearly seen, a
maximum time is exceeded or the cluster moves out of the observation area. The bubble
identification in general is again done by an edge-detecting algorithm based on the Canny
edge detector (Canny 1986). Despite a certain amount of automation, the used algorithms
need the input of a user for proper separation of complex clusters.
In Figure 3-11 the results for a bubble cluster recorded over ten frames are shown.
The projected area of the bubbles at frame zero, five and ten is determined refereed to
frame zero. The used edge-detecting algorithm suggests shapes of bubbles, the user has
to pick bubbles and/or separate them from others. In principal, the result is equal to a
handpicked bubble size distribution. Although the bubble clusters are followed over time
not all bubbles from the cluster could be evaluated; the bubble in the left corner of frame
zero is moving behind the larger bubbles the whole time. From experience, such bubbles
are relatively seldom and are neglected.
Depending on the distribution of the bubble sizes, 1000 to 4000 identified bubbles
were found to represent sufficiently narrow and broad bubble size distributions,
respectively. The repeatability of the method was tested for different flow conditions by
evaluating the same flow conditions three times, in all cases almost equal bubble size
distributions were obtained. The reliability of the method cannot be evaluated by
comparing the obtained bubble size distributions with other methods from the literature
since no valid methods exist. To get a feeling for the reliability, setups in which no
coalescence and break up is expected (relatively low gas hold but a broad bubble size
distribution and using salt as coalescence inhibitor) were evaluated at two different
heights. A narrow sparger setup was used in order to obtain a bubble plume at the lower
height and a more or less homogenized bubby flow at the higher. Almost the same results
at both heights were obtained as demonstrated in Figure 3-12. Thus, it is reasonable to
assume that the shown method is reliable and robust for complex bubbly flows.
Figure 3-11 Determination of the bubble sizes in bubble clusters by following the cluster
over ten frames with 200 frames per second recording speed.
3 Experimental Methods
38
Figure 3-12 Bubble size distribution determined at two different heights and two
different volume flow rates; left, 2.2 l/min, right, 3.4 l/min. The sparger consists of four
needles with 1.5 mm and four needles with 0.6 mm inner diameter.
3.3 Void fraction
3.3.1 Needle probe
The volume fraction is measured by the use of a conductivity needle probe. The
performance is discussed in several studies, e.g. by Le Corre et al. (2001) or by Manera et
al. (2009). In the present work, a single needle probe that is described by Da Silva et al.
(2007) and Schleicher et al. (2008) is used. The probe lance is assembled movable at the
top of the reactor.
A needle probe is an invasive method and not usable if the probe lance disturbs the
flow. Depending on the bubble size, the bubbles need a distinct relative velocity to the
probe to get reliable information. For example, in the downcomer of the later discussed
airlift reactor the bubbles are small and have a slow relative velocity to the needle so that
a needle probe is not useable.
3.3.2 2D-Videometry
For very small void fractions, the void fraction is determined with videometry. For
this purpose, the method for the bubble size measurement given in Section 3.2.2 and
shown in Figure 3-8 is used. The divide and conquer algorithm that identifies the bubbles
from the segmented pictures gives the count of all bubbles including the overlaid bubbles.
From the non-overlaid bubbles, the volume of the solid of revolution obtained by
revolving of the projected area is calculated. The average of these volumes is assumed
representative for the bubbly flow so that the total gas volume is calculated by
multiplying the total bubble count (including overlaid bubbles) with the averaged bubble
volume (excluding the overlaid bubbles). Performing this method in different areas of the
downcomer the two-dimensional void fraction distribution along the downcomer is
obtained. This void fraction is a volume-averaged value over the cross section of the
downcomer and a certain height.
Using the above-described manual bubble-picking algorithm, the results of the
automated void fraction algorithm are compared to an extensive evaluation by hand. As
a result, an underprediction of 15 % was found for the case with the highest void fraction
3.4 Liquid velocity, turbulence & Sampling bias in bubbly flows
39
and broadest bubble size distribution evaluated with the automated method. This error
is mainly caused by an underprediction of the large bubbles since they have a higher
probability to be overlaid by smaller bubbles, which leads to a smaller determined
averaged bubble volume. For lower void fractions and narrower bubble size
distributions, the error was found to be smaller.
3.4 Liquid velocity, turbulence & Sampling bias in bubbly flows
In the following section the particle tracking velocimetry (PTV) using naturally occurring
micro bubbles and the particle image velocimetry (PIV) using fluorescence particles are
described. The results of the determined velocity, turbulent kinetic energy and
fluctuation distribution with both methods are compared and discussed.
In single-phase flow, added tracer particles do not disturb the flow; in multiphase
flows, however, these particles often tend to accumulate at the interfaces and could affect
the flow. In addition, tracer particles contaminate the facility in general, which might
disturb chemical as well as biological processes. Besides, the seeding of particles can be
problematic, for example in oceanographic applications. To overcome such problems
naturally occurring micro bubbles can be used as tracer particles.
Tracking micro bubbles so that the velocity of the continuous phase can be
determined is not extensively investigated yet. Nevertheless, some applications can be
found in literature, e.g. in breaking waves (Ryu et al. 2005), behind propellers bubbles
(Graff et al. 2008), around dolphins (Fish et al. 2014) or in horizontal channels (Murai et
al. 2006). Noteworthy, micro bubbles can also be explicitly generated for particle tracking
as for example shown by Ishikawa et al. (2009).
In the present work, particle-tracking velocimetry with micro bubbles (BTV) in
bubbly flows is investigated. In a rectangular tabletop bubble column, the results
obtained by tracking micro bubbles in the range of 100-300 µm are compared to PIV
measurements. For BTV a volume illumination is used so that high gas void fractions can
be investigated. A two dimensional measuring plane is obtained by using a narrow depth
of field in combination with an edge filter technique.
Using PTV and PIV the view on the measuring plane is hindered in general because of
passing bubbles. This leads to a sampling bias that has a significant effect on the
measurements. This effect is described in the present work; further, a method is
developed to overcome the bias.
The sampling bias in general is a well-known phenomenon for many applications. For
example, in single-phase flow it is described for the usage of LDA (Hoesel & Rodi 1977)
(Edwards 1987). Based on that, the sampling bias for bubbly flows by the use of LDA was
described recently by Hosokawa & Tomiyama (2013). Nonetheless, a sampling bias in
multiphase flow is not restricted to LDA measurements, it might occurs for all measuring
techniques that are affected by the phases.
3.4.1 Particle Image velocimetry in bubbly flows
Particle image velocimetry in bubbly flows is a well-known method and is frequently
used since the late nineties of the last century (Oakley et al. 1997). The fundamental
principle is to illuminate tracer particles with a laser once or multiple times; the velocity
is determined by the particle displacement.
In multiphase flow, the problem arises to distinguish between the phases. The
bubbles are usually masked from the images (Brücker 2000) (Deen et al. 2002) for this
3 Experimental Methods
40
purpose. An early method that is described by Jakobsen et al. (1996) uses edge detecting
algorithm to identify the brighter illuminated bubbles, which is also used recently by
Pang & Wei (2013). Later, the use of fluorescence particles imprinted with Rhodamine is
used (Lindken et al. 1999). Rhodamine is fluorescenting in the yellow to red spectra
whereas the lasers used for PIV often emitting green light. A color filter for green light is
used in order to reduce reflections coming from the bubbles. The fluorescence particles,
however, tend to accumulated at the bubble surfaces and the bubbles are still present in
the recording.
Noteworthy, besides cutting the bubbles out, the phases can be distinguished by
different displacement peaks (Delnoij et al. 2000). Another method is the shadowgraph
technique which is described e.g. by Lindken & Merzkirch (2002) and is frequently used
in bubbly flows (Fujiwara et al. 2004) (Bröder & Sommerfeld 2007) (Sathe et al. 2010)
(Deen et al. 2001). The bubbly flow is illuminated with an additional light source so that
only bubbles by a second camera or in an extra pulse are recorded. The resulting complex
experimental setup is a disadvantage of those techniques. For that reason, fluorescence
particles are used in combination with digital image analysis in order to identify the
bubbles in the same recording with the PIV particles.
The flow is seeded with 20-50 µm PMMA particles imprinted with Rhodamine from
microParticles GmbH in Berlin. The particles are illuminated by a two dimensional laser
light sheet from the side. A double-pulsed laser is used with a difference of 1/2500 s
between the pulses, the double pulse is generated every 1/10 s. Every pulse is recorded
separately by one high-speed camera.
The recorded pictures are separated in rectangular interrogation areas, which are 2
mm large and are overlapping 50%. The commercial software Davis 8.2.1 is used for this
purpose. Hence, the PIV methods are not explicitly discussed at this point, a detailed
explanation of those can be found in various books, e.g. in the book of Raffel (2007). For
the phase discrimination, the complete interrogation areas that touch a bubble or a
shadow are excluded. Thus, the bubbles and shadows have to be identified from the
recordings.
A median filter is used in order to eliminate the tracer particles from the image
(Lindken et al. 1999), which is demonstrated in Figure 3-13. After the median filter is
applied (2), two thresholds are used. First, a low threshold value is applied to get more
or less closed bubble boundaries (3); many structures remain that do not belong to
bubbles. Second, a high threshold value is used to get only the bubbles (4). The obtained
bubbles are not complete, but almost all structures belong reliable to bubbles. The
pictures obtained from both thresholds are segmented and combined. If an object from
(4) touches an object from (3), this object (from (3)) is taken as a bubble. The bubble
structures are colored black in (5). Finally, the identified bubbles are dilated and eroded
to close possible disrupted boundaries (6). The artificial coalescence that is seen in (6) is
not adverse because possibly overexposed areas are identified as well.
3.4 Liquid velocity, turbulence & Sampling bias in bubbly flows
41
(1)
(2)
(3)
(4)
(5)
(6)
Figure 3-13 Algorithm for determining the bubbles in the PIV image for case 3 (1): The
obtained image from the PIV camera. (2): Applying a median filter on (1). (3): Results
obtained from a low threshold on (2). (4): Results obtained from a high threshold on
(2). (5): Hysteresis of the low and high threshold value results (6): Closing the
boundaries and filling the structures.
The laser that illuminates the particles is coming from the left in Figure 3-13 (1) so
that the bubble located in the sheet produce a shadow in which no particles are seen.
Therefore, such shadows have to be cut out as well. The same procedure as described for
the bubbles is used for this purpose. The result of the bubble and shadow detecting is
shown in Figure 3-14. In the top right corner the shadow is not detected (marked red in
b)) because the reflections of the other bubbles re-illuminate it, which is representative
for a distinct problem using PIV in multiphase flow in general; the light is reflected
arbitrary at the bubble surface. Such reflections lead to a scattering of the accurate
produced laser sheet. Consequently, particles might be seen that are not in the quasi two-
dimensional measuring area originally generated by the laser sheet.
3 Experimental Methods
42
a)
b)
c)
Figure 3-14 Masking the bubbles and shadows. a) The original PIV image; b) the bubble
masks (blue) and shadow mask (black); c) obtained velocity vectors.
Despite fluorescenting particles are used, nearly all bubbles in the bubble column are
seen. Bubbles that are not in the laser sheet are illuminated by the scattered light from
the bubbles in it and other reflections. These bubbles are seen bright in the recorded
pictures and are cut out by the post processing. The out of plane bubbles can be identified
in Figure 3-14 by the missing shadow behind them (light comes from left).
Therefore, the velocity information in general behind or before a bubble (related to
the view of the recording camera) in the bubble column is not measureable, besides, the
bubbles between the laser sheet and the camera naturally block the view in general. This
behavior leads to a sampling bias discussed below.
3.4.2 Particle tracking velocimetry with micro bubbles in bubbly flows
In contrast to the frequent use of PIV, PTV is rarely used in bubbly flows and only few
examples are found in literature. However, PTV is an often-used technique in single-
phase flows (Dimotakis et al. 1981). In the present work, micro bubbles as tracking
particles are used (Bubble tracking velocimetry – BTV).
In contrast to PIV, a volume illumination is used so that the experimental set-up is
simplified and higher gas volume fractions can be realized. A camera setup with a narrow
depth of field in combination with an edge filter is used in order to obtain a two
dimensional measuring plane. The technique is demonstrated in Figure 3-15. The field of
maximum sharpness is situated between the fourth and the fifth object from the left. The
blurring is increasing with increasing distance to the field of maximum sharpness. The
edge-detecting algorithm (Canny 1986) is applied on the a); the normalized edge strength
is shown along the red-white dotted line in picture b). Here, only the edges of the squares
are of interest, so the borders of the objects are excluded from the diagram in order to
3.4 Liquid velocity, turbulence & Sampling bias in bubbly flows
43
constitute the method clearly. A hysteresis on the edge strength is applied so that the
blurred edges are excluded. For the shown example, the hysteresis would cut out all edges
with an edge strength below 0.8.
With a similar test set-up, the depth of field was calibrated to 2 mm; the results of this
calibration are shown in Figure 3-16. The lowest measurable depth of field with the
present setup is 2 mm because the test object displacement in depth is 1 mm similar to
that shown in Figure 3-15. In practice, a resolution of 21 Px/mm with a normalized edge
filter of 0.75 gives good results so that a sufficient large area of view is still seen. These
settings will be used for the measurements in this work. The method applied to bubbly
flows is demonstrated in Figure 3-17. The marked bubbles, which are in the maximum
field of sharpness, are used for tracking.
a)
b)
Figure 3-15 Determining the depth of field by filtering the edge strength. The distance
in depth between the test prints is 1 mm. a) Original picture, b) edge strength in grey
shades and as graph along the red-white dotted line.
3 Experimental Methods
44
Figure 3-16 Determination of the depth of field depending on the edge filter for three
different resolutions.
Figure 3-17 Bubbles that are used for tracking in the field of maximum sharpness.
After the micro bubbles are identified, the weighted averaged position of the pixels
that belongs to the micro bubble is calculated in order to calculate the bubble position
𝒙𝑃=1
∑𝜔𝑗
𝑗 ∑𝜔𝑗𝒙𝑗
𝑗.
(3-4)
With 𝒙𝑃 the resulting positon vector, 𝒙𝑗 the position vector of the 𝑗𝑡ℎ-pixel, 𝜔𝑗 the scalar
weight function of the 𝑗𝑡ℎ-pixel and 𝑗 the pixel index over all pixels assigned to the micro
bubble. In general, the mass weighted centroid is the correct position for particle
tracking. Since this information is often not accessible, other methods are required as
discussed by Feng et al. (2007) and by Saunter (2010). Here, the position is determined
by calculating the centroid of the projected area (𝜔𝑗=1 ∀𝑗), which is reliable since the
micro bubbles are spherical and have a sharp boundary.
3.4 Liquid velocity, turbulence & Sampling bias in bubbly flows
45
After the positions are determined in all pictures that belong to a burst, the particle
tracks have to be determined. In principal, this problem can be formulated to define
similarity conditions that group positions to one track. For problems with a very high
particle concentration, this can be a challenging task and is still an active discussion in
literature. For the present setup, however, the concentration of the micro bubbles is low;
hence, the nearest particle in the next image is used to connect the particles to a track:
𝕋i={𝐱P,k(t),𝐱P,j(t+Δt)∈ith Burst|‖𝐱P,k(t)−𝐱P,j(t+Δt)‖<d
∩∀𝐱p(t+Δt): ‖𝐱P,k(t)−𝐱P(t+Δt)‖≤‖𝐱P,k(t)−𝐱P,j(t+Δt)‖}.
(3-5)
The micro bubble positions 𝐱P,k(t),𝐱P,j(𝑡+𝛥𝑡) in a burst of pictures are in the set 𝕋𝑖 if
the distance between 𝐱P,k(t) and 𝐱P,j(𝑡+𝛥𝑡)is smaller than 𝑑 and 𝐱P,j(𝑡+𝛥𝑡) is the
nearest position at time 𝑡+ 𝛥𝑡 compared to 𝐱P,k(t); the set 𝕋𝑖 is called the 𝑖𝑡ℎ- track.
From experience, the distance 𝑑 is chosen to be equivalent to a velocity of 1.5 m/s divided
by the measuring frequency for the discussed bubble column experiments. The set
formulated in Equation (3-5) is only meaningful if two or more micro bubbles are not
crossing each other during the recording. This does almost not exist because the
recording frequency is high so that the micro bubble displacement is low.
In Figure 3-18 an example of the BTV method in a 60 mm wide channel is shown. Two
pictures are taken consecutively with a recording frequency of 1000 Hz. In general, it is
assumed that the micro bubbles move linear, hence the velocity of the 𝑖𝑡ℎ-track is
calculated by the displacement multiplied with the recording frequency:
𝒗𝑷,𝒊(𝑡𝑖,𝒙
𝒊)=1
Δ𝑡(𝒙𝑷,𝒊(𝑡𝑖+Δ𝑡)−𝒙𝑷,𝒊(𝑡𝑖)), 𝒙𝑃,𝑖∈𝕋𝑖 .
(3-6)
With 𝒗𝑷,𝒊 the velocity vector of the 𝑖𝑡ℎ-track, 𝑡𝑖 and 𝒙
𝒊 the time and position of the
calculated velocity vector, respectively. Since the movement is assumed linear, the
position 𝒙
𝑖 of the velocity vector is calculated by
𝒙
𝑖=𝒙𝑷,𝒊(𝑡𝑖)+12 (𝒙𝑷,𝒊(𝑡𝑖+Δ𝑡)−𝒙𝑷,𝒊(𝑡𝑖)) ; 𝑡𝑖=𝑡𝑖+12 (Δ𝑡), 𝒙𝑃,𝑖∈𝕋𝑖 .
(3-7)
The time 𝑡𝑖 of the velocity vector 𝒗𝑷,𝒊 is taken as the time in between the frames.
Figure 3-18 Particle tracking using mini bubbles below 500 µm diameter in a 60 mm wide
chanel. Bottom: The original picture at t=t1; top: The selected particles at t=t1 and t=
t1+Δt labeld with different grey tones, Δt=1/1000 s.
3 Experimental Methods
46
The demand of linear movement of the micro bubbles leads to a high recording
frequency. On the contrary, a high recording frequency leads to very small displacements
if the particles have a small velocity. The pictures are recorded in discrete pixels so that
the pixel-discretization error becomes significant for very small displacements. In order
to obtain sufficient displacements, the particles are usually tracked at least four times by
a pulsed camera every 1/1600 s; if the micro bubble does not move its own radius from
one picture to another, the next picture will be taken. The first and the last picture are
used if a bubble does not move its own radius during the four-recorded frames.
It should be noted that the time interval between two pictures becomes important if
the particles are strongly accelerated in this time. This problem is also reduced by
recording several images in one burst of several pictures because the recording
frequency is increased; this topic is discussed for example by Feng et al. (2011).
In general, the naturally occurring micro bubbles are used as tracking particles so
that the size of them cannot be controlled. Micro bubbles in the range of 100 – 300 µm
exist in large numbers in the investigated bubbly flows. Such are large compared to tracer
particles usually used for velocity measurements, but the micro bubbles are distinctly
lighter than such tracers are. The micro bubbles, however, have a terminal velocity that
has to be taken into account.
The Stokes number (St) gives information about if particles are capable to follow all
turbulent scales of the flow and, therefore, if the determined particle velocities give
reliable information about the flow. The Stokes number is equal to the ratio between the
characteristic time scale of the particle and the flow
𝑆𝑡≡𝜏𝑃
𝜏𝐹 .
(3-8)
For velocity measurements very small, buoyancy neutral tracer particles are used that
fulfill 𝑆𝑡≪1. If the Stokes number is around one or even larger, important deviations
occur from the perfect flow following behavior. For particles with a greater density than
the continuous phase and a Stokes number around one the particles are accumulating in
region with high strain (Maxey 1987), which can be quantified with the second invariant
of the strain tensor 𝜕𝑢𝑖
𝜕𝑥𝑗𝜕𝑢𝑗
𝜕𝑥𝑖 (Squires 1990). It could be shown that larger particles than the
Kolmogorov length with a density similar to the flow show a similar behavior as heavier
particles smaller than the Kolmogorov length (Xu & Bodenschatz 2008) (Bourgoin et al.
2011). The wake effects of the larger particles and the Faxen correction of the added mass
becomes important, as dicussed for example by (Calzavarini et al. 2011). More recently,
the behavior of micro bubbles is studied in turbulence with Stokes numbers below one
(Mercado et al. 2012), for 75 µm bubbles with a Stokes number around one (Volk et al.
2008) and above (Prakash 2013). In comparison to heavy particles, bubbles show a
different behaviour; however, compared to tracer particles also a different behavior and
clustering effects depending on the turbulence structure occur. Therefore, it can be
assumed that such clustering effects also occur in bubbly flows when the micro bubbles
are used for particle tracking. This is important because if the micro bubbles, for example,
tend to accumulate in the wake region of the larger bubbles, this regions might be
overrated in the averaging process, hypothetically. This topic, however, is not yet
investigated and the effect of the wake and turbulence structure in bubbly flows on the
3.4 Liquid velocity, turbulence & Sampling bias in bubbly flows
47
accumulation of micro bubbles are speculative, hence these possible effects are neglected
at this point but should be keept in mind.
The characteristic time scale of complex bubbly flows is not known so that the Stokes
number cannot be used at this point. To get an idea, however, the results obtained with
different micro bubble sizes are compared below. Moreover, the characteristic time scale
is used in order to compare the micro bubbles with tracers used in similar experiments,
which is shown in Table 3-1. The characteristic time scale, which is the time constant of
the exponential decay of the particle velocity due to drag, is calculated by taking the
virtual mass into account (Calzavarini et al. 2008) to
𝜏𝑃=1
12𝑑𝑃
2
𝜈2𝜌𝑃+𝜌𝐹
3𝜌𝑓 .
(3-9)
In general, the time scale of the used micro bubbles is comparable to other methods.
Their time scale is in the range of the PIV particles used by Deen et al. (2001). LDA
particles, however, have a distinctly smaller time scale whereas the polystyrene particles
used for Computer-Automated Radioactive Particle Tracking (Luo & Al-Dahhan 2008)
have a significantly larger time scale.
Reference
Time scale 𝝉𝑷
Particles
Method
Present work
100 µm: 0.3 ms
200 µm: 1.1 ms
300 µm: 2.5 ms
Bubbles
𝑑𝑃: 100-300 µm
Particle tracking
Deen et al. (2001)
0.21 ms
PMMA particles
𝑑𝑃: 50 µm
PIV
Julia et al. (2007)
0.03 ms
Hole glass particles
𝑑𝑃: 20 µm
LDA
Luo & Al-Dahhan
(2008)
53 ms
PS particles
𝑑𝑃: 800 µm
Computer-
Automated
Radioactive Particle
Tracking (CARPT)
Table 3-1 Characteristic particle time scale for different measurement techniques in
bubbly flows
In contrast to buoyancy neutral tracer particles, the density of the micro bubbles is
smaller than the density of the liquid. The rising velocity has to be subtracted from the
measured velocity. The rising velocity is calculated by using the drag law by Bozzano &
Dente (2001).
3.4.3 Sampling Bias in bubbly flows
A sampling bias occurs if a not representative sample, in which some values are less likely
included than others, is picked. If the liquid velocity is measured with BTV or PIV in
bubbly flows, such a not representative sample is picked. Bubbles that are passing the
field of view hinder the view on the measuring plane. However, these large bubbles drive
3 Experimental Methods
48
the flow so that higher velocities occur just when many of these bubbles are in the field
of view. Since these velocities are less likely measured because the large bubbles hinder
the view, a sampling bias occurs. It should be noted, that the sampling bias is not caused
by the bubbles inside the measuring plane, but by the bubbles out of it.
The sampling bias is demonstrated in Figure 3-19. Clearly, the count of the velocity
information is low when the vertical velocity is high and vice versa. The above-described
behavior leads to this negative correlation. Other mechanisms might be identified that
cause a sampling bias, for example the re-illumination of the shadows by the other
bubbles for the PIV measurements or the micro bubble generation by the system for the
BTV measurements. The hindered view on the measuring plane due to the passing
bubbles, however, seems to be the most significant for the present setup.
Due to the sampling bias, the calculation of the correct averaged velocity is not trivial.
The ensemble average is usually used to calculate the averaged velocity
𝒗
𝑃=〈𝒗𝑃 〉= 1
∑1
𝑖∑𝒗𝑃,𝑖(𝒙𝑃,𝑖,𝑡𝑖)
𝑖 ,
(3-10)
with 𝒗𝑃,𝑖(𝒙𝑃,𝑖,𝑡𝑖) the particle velocity. When the non-random picked sample is used, an
error occurs using the ensemble average. This might be quantified with the correlation
coefficient
𝜌(𝒗
𝑃,𝒞𝑇)=𝐶𝑜𝑣(𝒗
𝑃,𝒞𝑇)
𝜎(𝒗
𝑃)𝜎(𝒞𝑇) ,
(3-11)
with 𝐶𝑜𝑣(𝒗
𝑃,𝒞𝑇) the covariance between the averaged velocity 𝒗
𝑃 and the count of the
trajectories 𝒞𝑇 that are used to calculate this averaged velocity
𝐶𝑜𝑣(𝒗
𝑃,𝒞𝑇)=〈 (𝒗
𝑃−〈𝒗
𝑃〉) ⋅ ( 𝒞𝑇−〈𝒞𝑇〉 ) 〉 .
(3-12)
In PTV methods, the averaged velocity 𝒗
𝑃 is the average of the particle velocities in a
certain measuring area so that 𝒞𝑇 is the count of trajectories in this measuring area. The
correlation coefficients of the examples in Figure 3-19 are both around -0.4.
To overcome the sampling bias various methods exist. If the flow contains enough
particles and the velocity is only desired at one specific point, a windowed ensemble
average over time 𝒗
𝑃,𝑖 will provide reasonable results
𝒗
𝑃,𝑖 = 〈𝒗𝑃,𝑖 〉Δ𝑡= 1
𝒞𝑇Δ𝑡 ∑ 𝒗𝑃(𝒙𝑃,𝑡𝑖)
𝑡𝑖−Δ𝑡
2≤𝑡𝑖<𝑡𝑖+Δ𝑡
2 ,
(3-13)
with 𝒞𝑇Δ𝑡 the count of trajectories in the time window Δ𝑡. The averaged velocity over the
total time is then calculated by
𝒗
𝑃=(𝒗
𝑃,𝑖)
=〈〈𝒗𝑃,𝑖 〉Δ𝑡 〉= 1
∑1
𝑖∑〈𝒗𝑃,𝑖 〉Δ𝑡
𝑖 .
(3-14)
As a consequence, the covariance between 𝒗
𝑃,𝑖 and the count of the windowed averages
𝒞Δ𝑡=1 is zero because 1−〈1〉=0 and, thus, also the correlation coefficient. This
averaging is also used in single-phase flow problems using (LDA) (Edwards 1987) (Murai
et al. 2001) and is called hold processor.
3.4 Liquid velocity, turbulence & Sampling bias in bubbly flows
49
Figure 3-19 Sampling bias in bubbly flows using BTV (top) and PIV (bottom). The tracked
vertical velocity (dashed blue line) and the count of the determined trajectories
(continuous red line) are smoothed with a moving average to represent the sampling bias
clearly.
The length of the time interval Δ𝑡 is problematic; if a too long or too short interval is
used, the same sampling bias will be obtained. The problem is solved by using a variable
time interval depending on the distribution of the velocity information over the
measuring area.
In fact, the velocity information is not distributed equally over the measuring area for
the used setups. More bubbles that hinder the field of view are found in the center so that
the count of the velocity information near the wall is twice as high as in the center.
Therefore, simple hold-processors that wait in time until certain amounts of trajectories
in one area/at one point are sampled are not meaningful. Therefore, the measuring area
is discretized in grid cells and the hold processor waits the time Δ𝑡𝑖 (hold time) until all
grid cells are filled with at least one velocity information. Afterwards, the velocity
information is averaged over the time Δ𝑡𝑖 inside the grid cells. Thus, one value in each
grid cell is obtained afterwards. The averaging over the grid cell is a windowed averaging
in space. After the complete measuring time, these averaged values are arithmetic
averaged. In the following, this hold processor in space and time is written as 〈𝑣𝑃,𝑗〉Δ𝑡𝑖
.
This procedure can be formulated as
3 Experimental Methods
50
〈𝑣𝑃〉Δ𝑡𝑖
(𝒙𝑗)= 1
𝒞𝑇Δ𝑡𝑖(𝒙𝑗)∑ 𝒗𝑃(𝒙𝑃,𝑡𝑖)
𝑡𝑖−Δ𝑡𝑖
2≤𝑡𝑖<𝑡𝑖+Δ𝑡𝑖
2,𝑥𝑃
𝑘∈(𝑥𝑗𝑘−𝑑𝑘,𝑥𝑗𝑘+𝑑𝑘] ,
(3-15)
with 𝒙𝑗 the centroid of the 𝑗𝑡ℎ-grid cell and 𝒞𝑇Δ𝑡𝑖(𝒙𝑗) the count of the trajectories
collected over Δ𝑡𝑖 in the 𝑗𝑡ℎ-grid. For the present setups, the grid cells are quadratic so
that 𝑥𝑃
𝑘∈(𝑥𝑗𝑘−𝑑𝑘,𝑥𝑗𝑘+𝑑𝑘] for the 𝑘𝑡ℎ-coordinate. The algorithm is illustrated in Figure
3-20.
Figure 3-20 The algorithm for the hold processor in space and time with an example on
an Eulerian grid with illustrated velocity vectors.
Using this hold processor, the sampling bias is overcome. For investigation a test function
with an analytic solution can be defined, for example
𝑦𝑗(𝑥𝑖)=sin((𝔾(𝑛(𝑖))𝑗
√𝑠+xi
𝜋) ⋅ 𝜋) +𝑑, 𝑥𝑖=𝑥𝑖−1sgn(sin((Δ𝑥⋅𝑖)π)) ⋅ Δ𝑥 .
(3-16)
The function described by Equation (3-16) is a discretized sinus function. The
Gaussian distribution function 𝔾(𝑛(𝑖))𝑗 provides 𝑛(𝑖) points between 0 and 1. The sinus
function is meandering in time by shifting the x-axis; the time is denoted with 𝑖. The
amount of discretization points 𝑛(𝑖) is randomly distributed over time. The meandering
Gaussian distribution simulates a problem similar to the one shown in Figure 3-19 with
a positive correlation coefficient between sampling count (represented by the Gaussian
3.4 Liquid velocity, turbulence & Sampling bias in bubbly flows
51
distribution) and the measuring value (the sinus function value y) . The correct average
in continuous space over the steps 𝑖 of this test function is simply 𝑦(〈𝑥𝑖〉)=sin(𝑋𝜋+
〈𝑥𝑖〉)+𝑑 (𝑋 is the x-axis).
The above defined hold processor 〈𝑣𝑃,𝑗〉Δ𝑡𝑖
can be studied by the use of this easy test
function nicely. For example, using Δ𝑥=0.1 and 𝑑=0.5 the sinus function is
meandering in 0.1𝜋
⁄ steps around 0.5 with 𝑥𝑖∈[−0.5,0.5]. Using 𝑠=50, the normalized
averaged results obtained with the hold processor, the simple ensemble averaging and
the analytical solution are shown in Figure 3-21. Obviously, the hold processor can
represent the averaged function. The ensemble average method, in contrast, cannot
represent it.
Figure 3-21 Comparison of the hold processor with the simple ensemble averaging used
on the analytical test function.
If the time step is too large or the hold processor have to wait too long, the result will
tend to the simple ensemble-averaged result. The best results will be obtained if no
waiting time is needed so that only the information is windowed averaged in space due
to the grid definition; if the grid cells are too large, the sampling bias persists inside them.
Naturally, if no sampling bias occurs and enough velocity information is available, the
hold processor is equal to the ensemble average.
Turbulence parameters have to be formulated correctly when the hold processor is
used. For example, the turbulent kinetic energy 𝑘 is defined as
𝑘(𝑡)=12(𝑣𝑥′𝑣𝑥′+𝑣𝑦′𝑣𝑦′+𝑣𝑧′𝑣𝑧′ ) .
(3-17)
With 𝑣𝑘′ the fluctuation in the 𝑘𝑡ℎ-direction
𝑣𝑘′(𝑡)= 𝑣𝑘(𝑡) − 〈𝑣𝑘〉 .
(3-18)
The hold processor must not be simply used on the turbulent kinetic energy or the
fluctuation. The fluctuation in the 𝑗𝑡ℎ-cell in the 𝑘𝑡ℎ-direction have to be written as
𝑣𝑗,𝑘
′(𝑡)= 𝑣𝑗,𝑘(𝑡)−𝑣𝑗,𝑘=𝑣𝑗,𝑘(𝑡)−〈𝑣𝑗,𝑘〉Δ𝑡𝑖
.
(3-19)
Moreover, the fluctuation must not be simply averaged over the hold time because the
fluctuations with different sign would compensate each other. Consequently, the square
3 Experimental Methods
52
of the fluctuation is averaged with the hold processor, hence the turbulent kinetic energy
at the discrete time 𝑡𝑖 in the 𝑗𝑡ℎ-cell is calculated by
𝑘𝑗(𝑡𝑖)=12(〈𝑣𝑗,𝑥
′𝑣𝑗,𝑥
′〉Δ𝑡𝑖+〈𝑣𝑗,𝑦
′𝑣𝑗,𝑦
′〉Δ𝑡𝑖+〈𝑣𝑗,𝑧
′𝑣𝑗,𝑧
′〉Δ𝑡𝑖 ) .
(3-20)
The same treatment is needed for all statistic variables.
3.4.4 Results
The bubble column that is described in Section 3.1 is used. The liquid velocities and
turbulence parameters were determined at the centerline 0.2 m above the ground plate
as shown in Figure 3-22. The sparger that is installed level with the ground plate consists
of eight needles with an inner diameter of 1.5 mm.
Six different volume flow rates, which are given in Table 3-2, are investigated. The
volume flow rate was measured and controlled with a mass flow controller. 0.375 liter
per minute per needle was the smallest possible volume flow rate whereas 20 liter per
minute in total was the highest possible with used mass flow controllers.
Figure 3-22 Experimental setup used for the liquid velocity experiments. Left a sketch of
the facility, the measuring line is dotted red, the axis indicate the origin; right the ground
plate of the bubble column with the holes for the used needle sparger.
Case
Number
Volume flow rate
[liter/min]
Flow per needle
[liter/min]
3
3
0.375
4
4
0.5
5
5
0.625
7
7
0.875
13
13
1.625
20
20
2.5
Table 3-2 The different gas volume flow rates used for the experiments, the values are
refereed to standard conditions.
3.4 Liquid velocity, turbulence & Sampling bias in bubbly flows
53
3.4.4.1 Micro bubble tracking velocimetry
The micro bubbles were tracked by the use of a Redlake motion pro high-speed camera
and a Sigma macro objective with a focal length of 300 mm. Since a distinct magnification
was needed, only a quarter of the bubble column could be recorded so that two different
measurements had to be executed in order to get a velocity profile over half of the bubble
column. Moreover, the different volume flow rates were measured for one window
consecutively to reduce the measuring effort. Nevertheless, no significant mismatch at
the overlapping regions was observed as demonstrated in Figure 3-23. Therefore, this
error is negligible and the overlapping regions are simply averaged.
Figure 3-23 Left and the right measuring window of the left half of the column for case
13.
Micro bubble size
The size of the micro bubbles that are used for particle tracking is not uniform and
different at different positions. Bubble size distributions at two positions are
demonstrated in Figure 3-24. Near the wall, the count of smaller bubbles is higher than
towards the center. Hypothetically, if the smaller bubbles follow the flow better than the
larger bubbles, locally different turbulence parameters are obtained. Therefore, it is
essential to track bubbles in a range of size in which the capability to follow the flow is
the same. Moreover, the general capability of the micro bubbles to follow the liquid flow
fields is of interest to assess possible errors due to the larger size and smaller density of
them compared to, for example, LDA or PIV particles.
The results that are obtained by using different micro bubble sizes are compared to
each other for case 13 in Figure 3-25. They are discretized in five different groups. For
every group the liquid velocity and fluctuation is determined.
The vertical liquid velocity is similar for all bubble groups except for the group of 400-
500 µm micro bubbles; near the wall, the results are lower than the others are. Tracking
larger micro bubbles, however, must not imply a larger vertical velocity due to the higher
terminal velocity because the liquid velocity is corrected with this. The results that are
obtained by using all bubbles (bubble group 150-500 µm) is not equal to the average of
the results over all sub bubble groups because all results are obtained by using the hold
processor. In addition, the quantity of the tracked micro bubbles in the different bubble
3 Experimental Methods
54
groups is not the same, but the quantity of the smallest group is sufficient to produce a
plausible result.
Figure 3-24 Count of the tracked micro bubbles for case 13 at the wall and towards the
center; the center is at x=0.125 m.
The normal Reynolds stress tensor component v′v′(v is the vertical velocity) might
indicate a different ability to follow the flow for the different bubble sizes. Looking at b)
in Figure 3-25 a clear trend is seen. In the center v′v′ is decreasing with increasing bubble
size. The results obtained with the bubble groups ranging from 150 to 250 µm and from
200 to 300 µm, however, are almost equal.
In addition, the probability density function (PDF) of the fluctuations is shown in c)
and d); the larger and smaller micro bubbles have the same behavior in general. The PDF
of the larger bubbles, however, are smoother than the functions of the smaller bubbles,
although the quantity of the smaller size group is larger; particularly in picture c), the
smaller size group contains 7 600 tracks whereas the larger bubble size group only 6 300
tracks. Despite this, the PDFs of the small bubbles and the large bubbles are similar,
especially at the shoulders.
Overall, the results obtained using larger micro bubbles are different compared to the
results using smaller micro bubbles. The trend of decreasing Reynolds stress tensor
component v′v′ with increasing size might indicate a worse capability to follow the flow
of these. However, the PDFs of the small and large bubbles are similar at high fluctuations.
A definite conclusion cannot be drawn since the characteristic time scale of the fluid flow
is unknown. Nevertheless, the results that are obtained using bubbles between 150 and
300 µm are similar for the present setup so that this bubble group is used for further
investigations. Noteworthy, the good agreement between the PIV results and the bubble
tracking results, which is later discussed, might confirm that the chosen bubble size group
is reasonable.
3.4 Liquid velocity, turbulence & Sampling bias in bubbly flows
55
a)
b)
c)
d)
Figure 3-25 Comparing the results obtained by tracking different bubble sizes. a) The
vertical liquid velocity v for the different bubble sizes, b) the normal Reynolds stress
tensor component v′v′, c) the fluctuation probability function in 0.0375 m<x<0.05 m
for two bubble groups, b) the fluctuation probability function in 0.0875 m<x<0.1 m
for two bubble groups.
Influence of the different volume flow rates
The results for all cases that are obtained with the BTV technique are shown in Figure
3-26. The progression of the vertical liquid velocity with increasing the gas volume flow
rates is reasonable.
As expected, the normal Reynolds stress tensor components 𝑣′𝑣′ and 𝑢′𝑢′ are
increasing with increasing gas volume flow rates (𝑢 is the horizontal velocity). The graphs
of 𝑣′𝑣′ show a distinct peak between the wall and the center for all volume flow rates. In
contrast, 𝑢′𝑢′ is permanently increasing towards the center. This behavior is also
described in other work using similar experimental setups (Mudde et al. 1997) (Simiano
et al. 2006). Since the flow regime is changing, 𝑣′𝑣′ and 𝑢′𝑢′ obtained for case 20 are
distinctly higher than these for the other flow rates are. In case 20 all bubble sizes are
pulled downwards in the recirculation zone so that the bubble column is completely filled
with bubbles; whereas in case 13 bubble clusters are pulled downward occasionally and
3 Experimental Methods
56
in case 7 only few bubbles. The possible outlier in the 𝑢′𝑢′ graph obtained for case 20 at
around x=0.055 m is discussed below.
a)
b)
c)
Figure 3-26 Results of the BTV for all cases. a) Vertical liquid velocity v b) normal
Reynolds stress component v′v′ c) normal Reynolds stress u′u′ (u is the horizontal liquid
velocity).
3.4.4.2 Comparison with PIV
Influence of the sampling bias
The influence of the sampling bias on the PIV results is demonstrated for three different
volume flow rates in Figure 3-27. The measuring area is discretized in twelve areas, the
hold processor waits until all areas contain at least one velocity information as discussed
above. Looking at the liquid velocity, the sampling bias leads to a flat velocity profile for
all flow rates. The underprediction in the center is due to the bubbles which drive the
flow and ,parallel, hinder the view on the measuring plane, therefore, the velocity
information which contains the higher velocities are underrated. A large negative vertical
velocity at the wall might be connected to a larger count of bubbles that are pulled
downward so that these bubbles might block the view on the measuring plane, which
leads to an underrating of the large negative velocities.
The normal component of the Reynolds stress tensor 𝑣′𝑣′ is affected by the sampling
bias in the same way as the liquid velocity. Towards the center, the 𝑣′𝑣′ values are
3.4 Liquid velocity, turbulence & Sampling bias in bubbly flows
57
underpredicted for all volume flow rates; the underprediction increases with increasing
the volume flow rate. Surprisingly, the sampling bias has no effect near the wall in
contrast to the liquid velocity. Similar to 𝑣′𝑣′ the cross component 𝑢′𝑣′ of the Reynolds
stress tensor is affected by the sampling bias, although the effect is smaller in the center.
3 l/min
5 l/min
7 l/min
Figure 3-27 The influence of the sampling bias on the PIV results for different volume flow
rates.
The influence on the BTV results is not as strong as on the PIV results as shown in
Figure 3-28; the influence is weak near the wall for the liquid velocity. Towards the
center, however, the influence is significant. The same trend is seen for the 𝑣′𝑣′ graph.
Looking at the 𝑢′𝑣′ graph the influence is compared to the PIV results minor.
Although PIV and BTV are influenced differently by the sampling bias, the results
obtained with the hold processor are similar. For the other volume flow rates, the
agreement between PIV and BTV is very good as well, which is discussed below. Thus, the
hold processor might be reasonable and both, BTV and PIV, can represent the liquid
velocity fields.
3 Experimental Methods
58
Figure 3-28 The influence of the sampling bias on the BTV results compared to the PIV
results for case 13.
Comparison for different volume flow rates
The results that are obtained with PIV are compared to those with BTV; for the PIV
measurements 10 minutes were recorded per case whereas for the BTV measurements
3.75 minutes were recorded. All results are obtained by the use of the hold processor.
The PDFs of v′ for case 13, which are shown in Figure 3-29, are in good agreement.
At zero fluctuations near the wall between 0.0375 m<x<0.05 m, however, the results
are slightly different but difficult to compare since the BTV graph is too noisy here.
Nevertheless, the peak at around -0.12 ms
⁄ is clearly represented by both measuring
techniques. The PDFs of the fluctuation towards the center at 0.0875 m<x<0.1 m are
almost perfectly matching.
a)
b)
Figure 3-29 The probability density function of the upward liquid velocity fluctuations
obtained with PIV and BTV for case 13. a) Near the wall between 0.0375 m<x<0.05 m,
b) towards the center between 0.0875 m<x<0.1 m.
The time averaged liquid velocity profiles that are obtained with PIV and BTV and
shown in Figure 3-30 are perfectly matching until case 13. Despite a relatively high gas
void fraction for case 13 and 20, still a good result is obtained with PIV.
3.4 Liquid velocity, turbulence & Sampling bias in bubbly flows
59
Figure 3-30 The vertical liquid velocity obtained with PIV with BTV for different gas
volume flow rates.
The results for 𝑣′𝑣′ and 𝑢′𝑢′, which are shown in Figure 3-31 a), are similar up to case
20. Some deviations occur, for case 3 the peak of the 𝑣′𝑣′ graph obtained with BTV is
closer to the wall than the peak obtained with PIV. In fact, the amount of the naturally
occurring micro bubbles is very small for this case because of the small gas volume flow
rate. In combination with the small void fraction in general, the PIV technique might be
advantageous for case 3. For case 4 and case 5, however, the obtained Reynolds stresses
are very similar.
Looking at 𝑣′𝑣′ for case 7 and 𝑢′𝑢′ for case 20, possible outliers at around 0.055 m
occur. These outliers are situated at the connection between the left and the right
measurement window. The two needed separate measurements to get a half profile as
discussed above, in combination that the bubble plume sometimes tends to prefer one
side of the bubble column so that it is swinging not symmetrical for a distinct time, might
be the reasons for this possible outlier. For the PIV measurements, in contrast, a larger
measuring time was used and the complete measuring area was recorded at once so that
such problems not arose.
The first distinct differences between the PIV and BTV measurements occur for case
13. In the center, the normal Reynolds stresses 𝑣′𝑣′ that are obtained with PIV are
smaller. The higher gas volume fraction might distort the PIV measurements, the bubble
clusters that are pulled downward at the walls block the laser assembled at the side of
the bubble column. In comparison, since the amount of micro bubbles is increasing with
increasing volume flow rate the BTV measurements are easier to evaluate for case 13. In
addition, the volume illumination is not as strong disturbed by the higher gas void
fraction as the PIV laser. Nevertheless, the results are still in good agreement as also
discussed above by using the PDFs of the fluctuation shown in Figure 3-29.
3 Experimental Methods
60
a)
b)
Figure 3-31 Reynolds stress components obtained by using PIV and BTV. a) Normal
Reynolds stress components v′v′ and u′u′ (u is the horizontal liquid velocity), b) cross
Reynolds stress component u′v′.
A mismatch between both methods is obtained for case 20; here all sizes of bubbles
are pulled downward by the circulating flow so that the light sheet of the PIV laser is
barely available. In contrast, the BTV measurements are still good manageable because
of the volume illumination. Although the agreement is good for the time-averaged
velocity, the normal Reynolds stresses obtained with PIV are strongly under predicted.
These normal Reynolds stresses are in the range of the results obtained for case 13, which
is not reasonable. As expected, the results for case 20 obtained with BTV are higher than
3.4 Liquid velocity, turbulence & Sampling bias in bubbly flows
61
the results for case 13. Therefore, for case 20 the BTV results are more reasonable than
the PIV results.
The results for the cross Reynolds stress component 𝑢′𝑣′ are shown in Figure 3-31
b). The agreement between both measuring techniques is acceptable. However, the
shorter measuring time in combination with a relatively small amount of micro bubbles
might lead to an insufficient statistic in the BTV measurements, especially for case 3.
Surprisingly, for case 13 the graphs are matching almost perfectly.
3.4.5 Conclusions
The particle tracking velocimetry using micro bubbles (BTV) is investigated. Micro
bubbles in the range of 100-300 µm that naturally occur in bubbly flows are used to
determine the liquid velocity and basic turbulence parameters. The particle relaxation
time of the micro bubbles is comparable to other measurement techniques used in
multiphase flows. Also from the results obtained with different micro bubble sizes and
from the comparison with PIV measurements these micro bubbles are capable to
represent the flow for the present setup. The liquid velocity, two normal and a cross
Reynolds stress tensor component obtained with BTV are compared to PIV
measurements, very good agreement is obtained.
Moreover, the use of a volume illumination in bubbly flows is described. Bubbles in a
quasi-two dimensional plane are identified with an edge filter provided by a camera setup
with a narrow depth of field. This measurement assembly is very simple so that
measurements in difficult environments like in pilot plants or submerged oceanic
multiphase flows problems are simpler feasible. In addition, high void fraction (in the
present measurements over 15 % gas holdup) measurements up in a narrow test section
are easily possible.
The sampling bias caused by the presence of bubbles was described. It was found that
indifferent whether BTV or PIV is used a distinct sampling bias occurs. In general, the
sampling bias is important for all measuring techniques affected by the dispersed phase.
This effect might be quantified by calculating the correlation coefficient of the measured
value and the sample. For PIV and BTV measurements, the measured value is the velocity
and the sample was chosen to the count of the velocity information.
The described sampling bias is overcome by using a multidimensional hold processor
defined in the present work. This derived hold processor, which was also tested with
analytical test functions, gave reasonable results. It was found that the BTV and PIV
measurements were affected by the hold processor differently. The results obtained with
the hold processor for PIV and BTV, however, were similar; further, both measurements
are only in accordance by using the hold processor. Therefore, the quality of the velocity
measurements in bubbly flows using PIV and PTV/BTV can be improved with this
method.
62
63
4 Eulerian bubbly flow simulations with the URANS equations
In the following the method for simulating bubbly flows in the present work is developed.
As described above, the recent baseline model concept developed mainly for pipe flows
should be adopted and studied for gravity driven flows. For this purpose, the URANS
solution method described in Section 2.4.2 is used to cover the influence of large-scale
turbulence (Mudde 2005) in such flows. Such large-scale phenomenon occurs due to an
uneven aeration (Juliá et al. 2007) or in heterogeneous flow regimes (Lucas et al. 2007b).
A proper turbulence modeling including large-scale structures in dispersed
multiphase flows is essential for a correct prediction of the momentum exchange
between the phases. Especially for bubbly flows the break-up and coalescence processes,
which are responsible for the bubble size distribution, are dominated by turbulence (Liao
& Lucas 2009). Since all modeled forces depend on the bubble size, the importance of a
reliable turbulence prediction is underlined. In bubble columns the large scale structures
are also very important for mixing in technical apparatuses, as described for example by
Joshi et al. (2002). Mixing might be underpredicted if these large-scale fluctuations are
suppressed by a steady solution method.
In contrast to the consistently stated conclusion, e.g. by (Tabib et al. 2008) or (Masood
et al. 2014), it is shown that the virtual mass force is not negligible in bubbly flows with
distinct large-scale turbulence structures. In addition, with the aid of a developed
convergence criteria it is shown that a solution time, time step length and mesh size
independent solution for the bubbly flow URANS simulations exists.
4.1 Modelling, setup and convergence criteria
As an experimental reference, the results of Mohd Akbar et al. (2012) are used. The
experiments were executed in a rectangular water/air bubble column at ambient
conditions. The ground plate is a rectangle of 240 × 72 mm and the water level is at 700
mm. The inlet is realized through needles at the bottom. Measurements were performed
for two superficial velocities, 3 mm/s and 13 mm/s, the integral void fraction for both
conditions is below 10%. The measurement plane is 500 mm above the inlet. A sketch of
the experimental setup is shown in Figure 4-1.
The measured quantities are the liquid velocity, gas volume fraction and the
turbulence intensity in the upward direction. Additionally, the bubble size distributions
at the inlet and at the measurement plane were measured. The bubble size distributions
are reproduced in Figure 4-2.
In contrast to the case with 3 mm/s superficial velocity, which is treated as
monodisperse using 4.3 mm as bubble size, for the case with 13 mm/s superficial velocity
a distinct amount of bubbles is above and below 5.83 mm. Therefore, two bubble classes
with its own velocity field as described in Section 2.3.3 are used. In particular, the first
bubble class has a bubble diameter of 5.3 mm, the second 6.3 mm. The inlet gas volume
flow is split up to 63 % and 37 %, respectively. As indicated in Figure 4-2 coalescence and
break-up processes are not dominant for the present setup, thus these processes are
neglected.
4 Eulerian bubbly flow simulations with the URANS equations
64
Figure 4-1 Experimental setup
Figure 4-2 Number density function of the bubble diameter in the experiment of Mohd
Akbar et al. (2012)
The rectangular bubble column is discretized in structured rectangular volumes. The
size of the volumes is determined after a mesh study, which is shown below. The inlet is
defined as surfaces at the bottom of the domain, representing the experimental needle
setup. The surface that represents one needle is rectangular with an edge length of 4 × 4
mm. The gas volume flow is divided equally over all needles.
4.2 Mesh and time step study
65
To determine whether the results are independent of the total simulation time a
convergence criterion is needed. Often a fixed total simulation time is taken as a
convergence criterion. If this fixed simulation time is reached, the simulation is defined
as convergent. This simple method makes the assumption that a convergent state exists
and that this state is reliably reached after the defined time. Therefore, this method is
insufficient to investigate the convergence behavior of a simulation. In addition, this
method is insufficient if it is unknown if the convergence is reliably reached after this
time. Therefore, another convergence criterion is needed for the present investigations.
The convergence criterion is defined in a way that averages 𝑓 taken over the
simulation time 𝑇 of a function 𝑓do not change significantly anymore when 𝑇 is
increased. The average over a finite time 𝜁 is defined as
𝑓(𝜁)=1𝜁∫𝑓(𝑡)𝑑𝑡
𝜁
0 .
(4-1)
In particular, the averages 𝑓 tend to be a constant asymptote as 𝜁 is increased. A
reasonable convergence criterion can be defined by analyzing the distance between 𝑓(𝜁)
and this constant asymptote.
Nevertheless, the constant asymptote that 𝑓(𝜁) is tending to is not known. However,
if 𝑓(𝜁) is tending to be a constant asymptote, the values of 𝑓(𝜁) will change less with
increasing 𝜁. For example, the difference between 𝑓(𝑇−Δ𝜁) and 𝑓(𝜁) tends to zero with
increasing simulation time 𝑇. If the difference between all values of 𝑓 in the interval
between 𝑇− Δ𝜁 and 𝑇 is evaluated, a trustworthy convergence criterion is obtained. The
effort of this procedure is reduced by comparing each value of 𝑓 in this interval to an
average of 𝑓 over this interval. This is expressed mathematically by requiring that
|1
Δ𝜁∫ 𝑓(𝜁)𝑑𝜁
𝑇
𝑇−Δ𝜁 −𝑓(𝜁)|≤𝜖; 𝑇− Δ𝜁≤𝜁≤𝑇 .
(4-2)
As function 𝑓 the upward liquid velocity is chosen. Based on experience, Δ𝜁=150𝑠
and 𝜖 to half of the experimental uncertainty (1.5% of the experimental value) is chosen
to obtain a good approximation without consuming excessive CPU-time. The convergence
criterion is evaluated at two points, 𝑥1 and 𝑥2, which are chosen symmetric. Therefore, a
criterion evaluating the symmetry of the obtained result can be defined
|𝑓(𝑥1,𝜁)−𝑓(𝑥2,𝜁)|≤2𝜖; 𝑇 − Δ𝜁≤𝜁≤𝑇 .
(4-3)
This criterion is meaningful because the setup is symmetrical and a symmetric result is
expected. It will be used in the further discussion.
4.2 Mesh and time step study
To obtain a mesh independent solution an intensive mesh study was performed. An
extract of this study for the case with a superficial velocity of 13 mm/s is shown in Figure
4-3. All simulations are converged using the defined convergence criterion. Four meshes
are presented:
4 Eulerian bubbly flow simulations with the URANS equations
66
an isotropic mesh with 4 mm edge length of each cell, which contains around
200 000 cells,
two anisotropic meshes, one with an edge length of 3 mm in depth and vertical
direction and 4 mm in the width direction, which contains around 300 000 cells
and the other with an edge length of 5 mm in depth and vertical direction and 4
mm in the width direction which contains around 140 000 cells,
a dilation in stream wise direction with 10 mm edge length in the vertical direction
and 4 mm edge length in depth and width direction with 80 000 cells.
The mesh study is conducted by investigating the gas volume fraction, the upward
liquid velocity and the root mean square of the normal component of the Reynolds stress
tensor 𝑅𝑀𝑆(𝑤’𝑤’). Comparing the obtained values for the gas volume fraction and the
upward liquid velocity even the coarse grid with 80 000 cells gives similar results as the
finest mesh with 300 000 cells. The resolved turbulence intensity is a little bit different,
but a clear trend with mesh size is not observable.
The 𝑅𝑀𝑆(𝑤’𝑤’) diagram consists of three curves that correspond to the URANS
modeling discussed in Section 2.4.2. The curve marked with ‘resolved’ corresponds to the
resolved part of the Reynolds stress component 𝑤′
𝑤′
, the curve marked with
‘unresolved’ to the modeled component 𝑤′′𝑤′′
=23
⁄𝑘𝑚𝑜𝑑
and the curve marked with
‘total’ to the total component 𝑤′𝑤′
. The unresolved curve is the result that would be
obtained if a stationary simulation would be performed. Further, the resolved curve
represents the amount, which is added through the transient simulation. The total curve
represents the 𝑅𝑀𝑆(𝑤’𝑤’) as it is obtained in the experiment.
The differences in the 𝑅𝑀𝑆(𝑤’𝑤’) graphs occur close to the wall. Using the isotropic
and the finest mesh, two peaks are noticeable at the walls. Using the two coarser meshes,
these wall peaks are less pronounced. With the coarsest mesh, a slightly higher value
overall is obtained. Nevertheless, deviations are quantitatively small.
Summarizing, the solution is mesh-independent already for the isotropic mesh;
hence, this is used for the further calculations. It should be noted that a mesh study is only
possible if the solution is independent of the time step and vice versa. This circumstance
was considered and the mesh study was performed with sufficiently minor steps, which
is discussed in the following.
To find conditions under which the solution becomes independent of the time step a
study is performed for 13 mm/s superficial velocity. Since it turns out that the time step
is connected with the virtual mass force, both model variations including the virtual mass
force and not including the virtual mass force are investigated. The difference between
both model setups is discussed in detail in the next section.
To characterize the discretization of the problem in time and space the Courant–
Friedrichs–Lewy number (CFL number, 𝐶𝐹𝐿=|𝑢|(|Δ𝑥|/Δ𝑡)) is used. Because the
velocity is a function of position and time so is the CFL number. To get a characteristic
value, the root mean square of all CFL numbers in the computational domain is calculated.
Further, the maximum and minimum 𝑅𝑀𝑆(𝐶𝐹𝐿) numbers over time are given.
4.2 Mesh and time step study
67
a)
b)
c)
Figure 4-3 Mesh study for four different meshes.
4 Eulerian bubbly flow simulations with the URANS equations
68
4.3 Influence of the virtual mass force
The time step study with the virtual mass force was performed in the range of
𝑅𝑀𝑆(𝐶𝐹𝐿) = 0.8 up to 𝑅𝑀𝑆(𝐶𝐹𝐿)=2.6. The results are shown in Figure 4-4 for the 13
mm/s case. For both simulations the convergence and symmetry criteria are reached.
a)
b)
c)
Figure 4-4 Time step study for different CFL numbers using the virtual mass force for
the 13 mm/s case
Comparing the gas volume fraction and the liquid velocity profile for both time steps
good accordance is reached. The volume fraction profile is nearly the same for both time
steps. The liquid velocity profile differs a little bit for the different time steps. The
resolved upward turbulence profiles for the different time steps are slightly different, the
peak near the wall being slightly higher for the larger value of 𝑅𝑀𝑆(𝐶𝐹𝐿). The unresolved
turbulence profiles are equal for both time steps. Since the unresolved contribution
constitutes a major part of the total turbulence intensity, the curves for this quantity are
in good agreement as well.
In conclusion, when the virtual mass force is included, the solution becomes
independent of the time step for 𝑅𝑀𝑆(𝐶𝐹𝐿)≲2.6.
The time step study without using the virtual mass force was performed in the range
of 𝑅𝑀𝑆(𝐶𝐹𝐿)=0.6 up to 𝑅𝑀𝑆(𝐶𝐹𝐿)=8. In Figure 4-5 selected results of the time step
4.3 Influence of the virtual mass force
69
study are shown. All simulations are convergent using the convergence criterion defined
in Section 4.1.
a)
b)
c)
d)
Figure 4-5 Time study for different 𝑅𝑀𝑆(𝐶𝐹𝐿)-numbers without using the virtual mass
force for the 13 mm/s case
Comparing the gas volume fraction, the upward liquid velocity and 𝑅𝑀𝑆(𝑤′𝑤′)
significant differences can be seen. In particular, the simulations using 𝑅𝑀𝑆(𝐶𝐹𝐿) above
1 do not fulfill the expected symmetry according to the criterion given in Section 4.1. In
contrast, the simulation using 𝑅𝑀𝑆(𝐶𝐹𝐿) below 1 does fulfill this criterion. Also, in
contrast to the simulations using 𝑅𝑀𝑆(𝐶𝐹𝐿) above 1 the simulation using 𝑅𝑀𝑆(𝐶𝐹𝐿)
below 1 gives two peaks in all three quantities. Comparing the simulation using
𝑅𝑀𝑆(𝐶𝐹𝐿) below 1 with the simulations including the virtual mass force in Figure 4-4,
very small differences are seen.
In conclusion, when the virtual mass force is neglected, a solution that is independent
of the time step is achieved if the condition 𝑅𝑀𝑆(𝐶𝐹𝐿)<1 is satisfied.
Besides the discussed influence on a reliable 𝐶𝐹𝐿 number, the virtual mass force
influences also the other values. In Figure 4-6 the results of the simulations with and
without virtual mass force are shown for both superficial velocities 13 mm/s and 3 mm/s.
For 3 mm/s superficial velocity the results obtained with and without virtual mass force
4 Eulerian bubbly flow simulations with the URANS equations
70
are the same. This is different for 13 mm/s superficial velocity. Therefore, the following
discussion is only related to the 13 mm/s case.
a)
b)
c)
d)
Figure 4-6 Comparison between using the virtual mass force and not using the virtual
mass force for a superficial velocity of 13 mm/s and 3 mm/s. The curves for using the
virtual mass force and not using the virtual mass force for the 3 mm/s case are on the top
of each other.
Looking at the liquid velocity profiles for the case with 13 m/s superficial velocity, no
differences between the model variants with and without virtual mass force are seen.
Distinct peaks at each side can be observed in the profiles. At the same positions as in the
liquid velocity profile, broad maxima can be observed in the gas volume fraction profile
for both model variants. In addition, if the model variant including the virtual mass force
is used, the gas volume fraction profile will exhibit sharp peaks almost at the wall. In
contrast, if the model variant neglecting the virtual mass force is used, these sharp peaks
will nearly vanish.
The broad maxima near the center in the gas volume fraction profile can be explained
by the stability criterion of Lucas et al. (2005), which is derived analytically from the force
balance, depending on the volume fraction of big and small bubbles. This stability
criterion is based on the change of sign in the lift force coefficient and is, therefore,
4.3 Influence of the virtual mass force
71
connected to the gradient of the liquid velocity. By solving separate momentum equations
for big and small bubbles this effect is also taken into account in the present simulations.
Since the liquid velocity gradient and the volume fractions of big and small bubbles
depend on the local position, the stability criterion of Lucas et al. (2005) has to be
evaluated locally.
In the lower section of the column, the big and small bubbles are not separated. Due
to the wall shear stress and the resulting liquid velocity gradient, the big and small
bubbles separate with increasing height. The big bubbles move to the center, the small
bubbles move to the wall. Consequently, the local concentration of the big bubbles rises
from the wall towards the center of the column. Further, away from the wall the lateral
movement of the big bubbles is slowed down, because of the decreasing liquid velocity
gradient. As a result, the big bubbles accumulate and the local void fraction of the big
bubbles increases at the same point. Due to buoyancy, this is accompanied by an increase
of the local liquid velocity. If the stability criterion described in Lucas et al. (2005) is
exceeded, a distinct liquid velocity peak will be formed at this point. Once this has
happened the large bubbles cannot move further towards the center because of the
negative lift coefficient. This means that steady profiles with peaks in the liquid velocity
and gas fraction are established.
The near wall peak in the gas fraction graphs is also caused by the described
separation of small and big bubbles. The small bubbles move to the wall due to the liquid
velocity gradient; however, near the wall the wall force push the bubbles away from the
wall. The bubbles accumulate where both forces have the same quantity and,
consequently, a wall peak occurs.
Figure 4-6 also shows the upward 𝑅𝑀𝑆(𝑤′𝑤′) values. Here, for the case with 13
mm/s superficial velocity also peaks near the wall can be observed. These peaks are not
at the same position as the peaks in the liquid velocity profile and might be less affected
by the separation of big and small bubbles. The near wall peaks in the 𝑅𝑀𝑆(𝑤′𝑤′) profile
are nearly at the point where the liquid velocity passes through the zero line, which is the
point of the highest liquid velocity gradient. In addition, the resolved 𝑅𝑀𝑆(𝑤′𝑤′) profile
is higher in general for the simulation without using the virtual mass force. Therefore, by
using the virtual mass force a damping of the liquid velocity fluctuations is introduced.
Overall, neglecting the virtual mass force leads to different results for the case with
13 mm/s superficial velocity. The gas volume fraction profile is quantitatively almost the
same for both model variants. However, not using the virtual mass force the near wall
peak in the gas volume fraction profile nearly vanishes. The resolved 𝑅𝑀𝑆(𝑤′𝑤′) profiles
have the same shape, but quantitatively the resolved 𝑅𝑀𝑆(𝑤′𝑤′) is higher if the virtual
mass force is neglected. While the gas volume fraction and the upward turbulence profiles
are different, the liquid velocity profile is nearly the same for both model variants. In
contrast to the case with 13 mm/s superficial velocity, all profiles obtained for the 3
mm/s superficial velocity are the same. The equality might be explained by the fact that
the resolved upward turbulence intensity at the measurement plane is nearly zero.
Consequently, nearly no fluctuation is resolved and the acceleration is nearly zero. As a
result, the virtual mass force is nearly zero.
4 Eulerian bubbly flow simulations with the URANS equations
72
4.4 Influence of the bubble induced turbulence
The influence of the bubble induced turbulence (BIT) model on the URANS simulations is
discussed in this section. For this purpose, the bubble induced turbulence (BIT) modeling
used in the baseline model with source terms (Rzehak & Krepper 2013a), with an
additional viscosity (Sato et al. 1981), both described in Section 2.4.4 and a model
neglecting the bubble induced turbulence are compared.
The results for 13 mm/s superficial velocity are shown in Figure 4-7. The gas hold up
is quantitatively very similar for all considered models, but the sharp near wall peak is
pronounced only for turbulence modeling with source terms. The liquid velocity using
the Sato model and using the no BIT model is lower than the experiments and the profile
obtained with the baseline model. Qualitatively the different model approaches show the
same behavior. However, using the Sato model and using the no BIT model both peaks in
the liquid velocity profile are shifted towards the center and are smaller.
a)
b)
c)
d)
Figure 4-7 Comparison of different bubble induced turbulence modeling approaches for
13 mm/s superficial velocity.
Remarkably, the quantity of the resolved turbulence intensity is very similar for all
used BIT models. Concerning the shape of the profiles, the peaks are shifted to the center
and are smaller for the models not using the source terms. The total 𝑅𝑀𝑆(𝑤′𝑤′) values
4.4 Influence of the bubble induced turbulence
73
are underpredicted by all models but significantly closer to the data for the models with
source terms. Differences between the Sato model and neglecting BIT are small in
comparison.
The differences between the different approaches to BIT modeling can be explained
by considering the turbulent viscosity, which is shown in Figure 4-8. As the resolved
𝑅𝑀𝑆(𝑤′𝑤′) values are comparable for all models, only the unresolved part of the
turbulent viscosity is shown. It can be seen from Figure 4-8 that for the BIT modeling
using source terms the turbulent viscosity is the lowest. This is caused by a higher
turbulence dissipation rate (not shown). Looking only at the turbulent kinetic energy that
is the highest for the modeling using source terms, the opposite effect on the turbulence
viscosity may have been expected. The reason for the behavior observed in the
simulations must be sought in the bubble induced source term of the turbulence
dissipation rate.
Figure 4-8 Unresolved turbulent viscosity
for different modeling approaches for 13
mm/s superficial velocity.
Figure 4-9 Comparison of the total upward
turbulence intensity for different bubble
induced turbulence modeling approaches
for 3 mm/s superficial velocity.
Further, as expected, the turbulent viscosity using a BIT model with additional
viscosity is the highest. Using the no BIT model the level of the turbulent viscosity is
between the other approaches.
The higher turbulent viscosity obtained with the Sato model and using the no BIT
model is causing a reduced amplitude in the lower liquid velocity profile compared to the
BIT modeling with source terms, as shown in Figure 4-7. In particular, using the Sato
model and using the no BIT model the liquid velocity gradient near the wall is smaller
compared to the experiment and the BIT modeling with source terms. Consequently, the
instability caused by the separation by the big and small bubbles, as discussed above, is
also shifted to the center. Therefore, the observed velocity peaks using the Sato model
and using the no BIT model observed in Figure 4-7 are shifted to the center.
Another effect of the higher turbulent viscosity that is obtained with the Sato model
and using the no BIT model is a higher turbulent dispersion of the bubbles. As described
in Section 2.3, the turbulent dispersion force is proportional to the turbulent viscosity
and to the gradient of the gas volume fraction. It acts towards a uniform distribution of
4 Eulerian bubbly flow simulations with the URANS equations
74
gas. As a result, the peaks in the gas volume fraction profiles shown in Figure 4-7 are
flatter when using the Sato model or using the no BIT model compared to the BIT
modeling with source terms. Consequently, the liquid velocity peak is also flatten when
using the Sato model or using the no BIT model. In particular, the near wall peak of the
small bubbles that can be observed for the BIT modeling with source terms in Figure 4-7
nearly vanishes when using the Sato model or using no BIT model.
For the case with 3 mm /s superficial velocity, the liquid velocity and the gas volume
fraction profiles obtained by using the different BIT model approaches are nearly the
same. Therefore, only the total 𝑅𝑀𝑆(𝑤′𝑤′) values are discussed in the following. The
results are shown in Figure 4-9.
It can be seen from Figure 4-9 that the 𝑅𝑀𝑆(𝑤′𝑤′) values are quite well predicted by
the BIT modeling with source terms. In contrast, using the Sato model or using the no BIT
model they are considerably underpredicted. This is the same trend seen for the case with
13 mm/s superficial velocity.
Summarizing, the best prediction of 𝑅𝑀𝑆(𝑤′𝑤′) is obtained by the turbulence
modeling with source terms, using the baseline model. The position of the peak in the
𝑅𝑀𝑆(𝑤′𝑤′) graphs for 13 mm/s superficial velocity is well reproduced. Using the Sato
model or the no BIT model 𝑅𝑀𝑆(𝑤′𝑤′) is considerably underpredicted compared to the
experimental data. Furthermore, there is no peak in the 𝑅𝑀𝑆(𝑤′𝑤′) graphs for 13 mm/s
superficial velocity.
The turbulent viscosity obtained for this case with the Sato model or using the no BIT
model is significantly higher than the obtained turbulent viscosity using the BIT modeling
with source terms using the formulation of Rzehak and Krepper (2013b). Consequently,
the liquid velocity profiles are less steep using the Sato model and using no BIT model
compared to the BIT modeling with source terms. Compared to the experimental data the
liquid velocity is underpredicted using the Sato model or the no BIT model, but predicted
well by the modeling using source terms.
4.5 Conclusions
It was shown that for transient simulations with RANS-based turbulence modeling
(URANS) an independent solution concerning simulation time, time step length and mesh
size is reachable with the two fluid model. For this purpose, the defined convergence and
a symmetry criterion give reliable information.
The resolved flow structures that are obtained due to the transient simulation give
important contributions to the turbulence, indifferent of the used bubble induced
turbulence (BIT) model. As expected, the resolved turbulence is very low for the low gas
volume flow rate; here the turbulence might be dominated by the BIT. Thus, the URANS
simulations are capable to reproduce bubbly flows dominated by large-scale structures
and dominated by BIT.
Moreover, it was found that the virtual mass force is not negligible, especially for the
higher gas volume flow rate in which the resolved turbulence is significant. Despite the
simulation results reproduce the experiments better without the virtual mass force, the
force has to be included towards a reliable modelling.
The baseline model gives very good results, especially with the baseline BIT model
the turbulent kinetic energy for both volume flow rates can be reproduced. The baseline
4.5 Conclusions
75
BIT model was also compared to other BIT models for the present experimental setup in
(Ziegenhein et al. 2013); regarding to the root mean square of the normal Reynolds stress
tensor component the baseline model gives the best results.
The further studies will be performed with the here developed method. From the
findings above, it is justified to identify two leading turbulence scales: The scales that are
in the size of the apparatus and the small scales, which might be dominated by the bubble,
induced turbulence. For the here investigated experimental setup these scales are both
present for the 13 mm/s superficial velocity case. In contrast, for the 3 mm/s superficial
velocity case large scales might not be present. In the following section, the capability of
the shown URANS approach with the baseline model is investigated to reproduce the
large-scale flow structures in a bubble column.
76
77
5 Prediction of the large liquid structures of bubbly flows with
the URANS equations
In this section, especially the fluid dynamics on the large scale is of interest, which is often
generated due to a partial aeration or a heterogeneous bubbly flow regime. Such partial
aeration is often obtained with common spargers like the ring or spider sparger (Kulkarni
et al. 2009). Moreover, the heterogeneous flow regime produced by large bubbles created
at the inlet or due to coalescence might be also a common flow regime. Thus, this topic is
of general interest.
The local high gas void fraction generated by a partial aeration is in general called
bubble plume. Besides the bubble plumes in bubble columns this effect is found for a wide
variety of applications. In the field of bioengineering partial aeration is used to drive the
flow inside different vessels (Wang et al. 2014) (Bitog et al. 2011) and wastewater pools
(Garcia & Garcia 2006). Without confining walls bubble plumes are used to mix complete
lakes (Wuest et al. 1992) (Boegman & Sleep 2012), helps to build barriers for oil on water
(McClimans et al. 2012) or even fishing (Grimaldo et al. 2011). Moreover, bubble plumes
are naturally found in oceans (Schmale et al. 2015) (Nauw et al. 2015) or occur due to
mining processes or pipe ruptures under water (Cloete et al. 2009).
In the following, the simulations are focused on bubble columns for chemical or
biological engineering since here comprehensive experimental data exist.
Notwithstanding the large amount of experiments and simulations conducted in the
recent years on this topic, the research is ongoing experimental (Besbes et al. 2015) as
well as theoretically (Masood & Delgado 2014).
Such large-scale structures that are generated by a bubble plume dominate the flow
field in the vessel, thus the prediction of the correct transient behavior of such is essential.
This is studied with the above-discussed URANS approach. Often a distinct bubble plume
frequency can be determined from the experiment which is easily compared to
simulations; many studies regarding this topic exist, e.g. (Rensen & Roig 2001) (Buwa &
Ranade 2004) (Juliá et al. 2007).
Three experiments dominated by bubble plumes with different operating conditions
are chosen from the literature. Namely, the experiment by Becker et al. (1994) with an
asymmetric gas aeration, the experiment by Pfleger et al. (1999) with a central gas
aeration and the experiments by Julia et al. (2007) using different aeration widths and
different gas volume flow rates.
The experiments of Becker et al. (1994) and Pfleger et al. (1999) are well known and
often simulated. The experiment of Becker et al. (1994) might be one of the most
simulated bubble plume experiments ranging from Euler-Euler two fluid model methods
(Sokolichin & Eigenberger 1999) (Mudde & Simonin 1999) to Euler-Lagrange methods
(Delnoij et al. 1997) (Hu & Celik 2008). The experiment of Pfleger et al. (1999) is also
used for many different model validation e.g. by (Bech 2005) or (Ma et al. 2015). These
two experiments are standard experiments and are used to show that the here used
simulation methods are capable to give the same good results as other methods. The
experiments of Julia et al. (2007) are more complex since different sparger conditions
5 Prediction of the large liquid structures of bubbly flows with the URANS equations
78
and high superficial velocities are realized. Simulations for these experiments are not
published yet.
5.1 Simulation setup and experimental data
The experimental setups of the three investigated cases of Becker et al. (1994), Pfleger et
al. (1999) and Julia et al. (2007) are shown in Figure 5-1. All setups are narrow bubble
columns of different sizes and operate in water with air supply. The largest bubble
column is used by Becker et al. with a water level of 1.5 m a width of 0.5 m and a deep of
0.08 m. The gas is aerated with 1.6 l/min through a porous frit at the left side 0.15 m away
from the wall. The liquid velocity was measured with laser Doppler anemometry (LDA)
at the points A and B as well over the width 0.75 m above the ground plate. The bubble
size was measured using a photoelectrical suction probe and was determined to 3 mm.
The bubble column used by Pfleger et al. (1999) is smaller and is centrally aerated
with 0.8 l/min through eight holes. The liquid velocity was measured using LDA as well.
The bubble size were not measured explicitly but are given by Pfleger et al. between 1
and 5 mm; however, here a bubble size of 2 mm is used, which is also used by Pfleger et
al. for their own simulations. The velocity profile is measured 0.13 m, 0.25 m and 0.37 m
above the ground plate. In addition, the velocity is measured over time in the center of
the column 0.25 m above the ground plate.
By Julia et al. (2007) a flat bubble column is used with a variable needle sparger
consisting up to 137 needles. The bubbles are uniformly aerated over the 0.031 m column
depth with different aeration width. The liquid velocities are determined with LDA on
different heights. The bubble plume dynamics are measured mainly on a line 0.3 m above
the ground plate. The used experiments for comparison with simulations are summarized
in Table 5-1. Three different aeration pattern aerating 59 % to 18 % of the ground plate
are used for the simulations from the experiments. The bubble sizes are determined with
digital imaging from extra experiments in which only a single needle is operating. The
holdup is determined by measuring the bed expansion for the smallest and the lowest gas
volume flow rate.
The bubble columns are discretized to a regular grid. The grid size was determined
after conducting a mesh study using a minimum of three different grid sizes. All
simulations are conducted with the baseline model as described in the previous section
and in Section 2.5. In particular, the column of Becker et al. is discretized with
100x150x16 (width x height x depth) rectangular cells. The column of Pfleger et al. is
discretized with 40x90x16 cells. Finally, a rectangular mesh composed of 50x300x6 was
found to be sufficient for the bubble column of Julia et al. The total simulation time varies
from case to case; nevertheless, the convergence criteria defined in the previous section
is fulfilled every time.
5.2 Results
79
Figure 5-1 Setup of the experiments by Becker et al. (1994) (left), Pfleger et al. (1999)
(middle) and Julia et al. (2007) (right). The measurement planes are dotted and points
with measurements over time are marked red.
Flow pattern (Aerated
area/ Used needles)
Superficial velocity
[mm/s]
Bubble size
[mm]
Holdup
[%]
F8 (59 % / 81)
16
3.78
4.7
29
4.20
40
4.56
49
4.85
17.1
F11 (44 % / 59)
12
3.79
2.5
16
3.97
24
4.33
29
4.56
36
4.87
11
F16 (18 % / 25)
5
3.78
0.9
12
4.52
16
4.94
29
6.31
7.6
Table 5-1 Setups of the experiments by Julia et al. (2007) used for simulations.
5.2 Results
At first, the simulation results regarding the experiments of Becker et al. (1994) and
Pfleger et al. (1999) are compared to the experimental observations. These two
experiments were executed at relatively low gas volume flow rates with small bubbles.
After that, the simulation results regarding to the experiments of Julia et al. (2007), which
are summarized in Table 5-1, are compared to the experimental results. In contrast to the
other two experiments, the experiments by Julia et al. are conducted over a wide range of
bubble sizes and volume flow rates.
5.2.1 Experiments of Becker et al. (1994)
In the experiments of Becker et al. (1994) a large vortex is built near the wall due to
asymmetric aeration. The vortex is wandering from the top to the bottom as can be seen
in Figure 5-2. Another vortex is built on the other side pressing the bubble plume to the
5 Prediction of the large liquid structures of bubbly flows with the URANS equations
80
wall in the bottom section. This behavior is very good reproduced by the simulations. In
addition, the frequency of the meandering vortex is very good reproduced.
Figure 5-2 Qualitatively results of the experiments (top) of Becker et al. (1994) and the
simulations (bottom). Pictures are taken every 5 seconds; the crosses in the simulation
pictures mark the measuring points Point A and Point B, respectively.
Looking at the velocity at Point A and Point B over time shown in Figure 5-3 the
frequency of the vortex is very well seen. The meandering of the bubble plume is more
regular in the experiments than in the simulations. The first three passages of the vortex
structures, however, are very similar to the experiments. The simulated meandering
bubble plume tends sometimes to stay at one position as is seen at the last two passages.
The absolute velocity peaks obtained by the LDA measurements are a little bit higher
compared to the simulations; this is explained because the URANS modeling is not
resolving the large velocity peaks.
The averaged results of the simulation are compared to the experiments in Figure
5-4, the experimental results are taken from (Deen 2001) since they are not given by
Becker et al. in their published work. The vertical liquid velocity profile is reproduced
qualitatively and quantitatively by the simulations. The root mean square of the normal
components of the Reynolds stress tensor 𝑣′𝑣′ and 𝑢′𝑢′ (𝑅𝑀𝑆(𝑢′𝑢′),𝑅𝑀𝑆(𝑣′𝑣′) (𝑢 the
horizontal and 𝑣 the vertical velocity) are under predicted. Nevertheless, simulations
qualitatively reproduce the experiments. A maximum in the 𝑅𝑀𝑆(𝑣′𝑣′) graphs at the wall
is observed, they are falling below the 𝑅𝑀𝑆(𝑢′𝑢′) graph to a minimum around 𝑋 𝑑
⁄=0.4
to rise again above the 𝑅𝑀𝑆(𝑢′𝑢′) graph towards the right wall. The 𝑅𝑀𝑆(𝑢′𝑢′) graphs,
in contrast, show a maximum around 𝑋 𝑑
⁄=0.4. Moreover, the unresolved part of the
liquid velocity fluctuations in both directions is negligible with the baseline model.
5.2 Results
81
Figure 5-3 Liquid upward velocity at two different points obtained from experiments of
Becker et al. (1994) and simulations.
Figure 5-4 Comparison of the averaged results with the experiment at 0.75 m above the
ground plate.
5.2.2 Experiments of Pfleger et al. (1999)
In contrast to the above-discussed bubble column, the bubble column of Pfleger et al.
(1999) is aerated symmetric in the center. Here, also large vortexes near the wall are
observed but symmetrical at both sides. Thus, a symmetrical swinging motion of the
bubble plume is observed.
5 Prediction of the large liquid structures of bubbly flows with the URANS equations
82
This swinging motion is seen at the sideward velocities determined with LDA in
Figure 5-5 which can be reproduced by the simulations with nearly the same frequency.
In contrast to the above discussed experiment, here the bubble plume obtained in the
experiment tend to swing uneven and the bubble plume obtained from the simulations
has a more regular motion. The absolute value of the measured velocity in Figure 5-5 is
not comparable to the simulations since the experiments are moving averaged over time
with an unknown filter length.
Figure 5-5 Liquid velocity in sideward direction at 0.25 m above the ground plate.
Figure 5-6 Liquid upward velocity profiles at three different heights.
5.2 Results
83
Nevertheless, the vertical velocity profiles are also measured for three different heights,
which are compared to the simulation results in Figure 5-6. The simulation results are
in good agreement with the experimental data. Overall, the simulations can reproduce
the qualitatively observed swinging bubble plume with nearly the same frequency as
well as the quantitatively measured velocity profiles on three different heights.
5.2.3 Experiments of Julia et al. (2007)
Julia et al. (2007) conducted experiments over a wide range of gas volume flow rates and
for three different aeration pattern as given in Table 5-1. In contrast to the previous
experiments, the bubble plume is spreading after a certain height over the complete area
of the bubble column.
The spreading is seen for most aeration pattern as shown in Figure 5-7. The vortex
structures for pattern F11 and F8 are vanishing towards the top of the column, whereas
the vortex structures of F16 are propagating until the top. For F8 and F11, the gas clusters
detached from the bubble plume in the bottom are spreading over the column cross
section and rise up as a front. At the bottom, the bubbles pulled downward in the large
vortexes or even retained by them.
Figure 5-7 Simulation results for three different cases velocity vectors and void fraction
profiles at the center plane. The measuring lines at 0.15 m and 0.3 m height are marked
black.
Before looking at the transient behavior of the bubble plume, the averaged simulation
results are compared to the experiments in Figure 5-8 for pattern F8 and F11 at four
5 Prediction of the large liquid structures of bubbly flows with the URANS equations
84
different heights with 29 mm/s superficial velocity. For pattern F8 at a height of 0.15 m,
the simulations clearly over predict the vertical velocity in the center and the downward
velocity near the sidewalls. Due to the very high velocities in the simulations near the
wall, more bubbles might be pulled downwards at the wall compared to the experiments.
The experimental results at 0.15 m and 0.3 m are almost equal; in contrast, a flat velocity
profile is obtained by the simulations at 0.3 m. In fact, the velocity fluctuations obtained
by the simulations at 0.3 m have an amplitude of around 0.2 m/s but are symmetric
leading to the flat velocity profile. Looking at the profiles at 0.6 m and 1.125 m, the
measured velocity profiles show a wall peak while the simulations show a center peak.
The velocity wall peaks might be indicating a coalescence process for the F8 pattern in
the experiments since the small bubbles, which are obtained from the single needle
experiments and used in the simulations, would migrate away from velocity peaks due to
the lift force and a flat velocity profile would be obtained. Consequently, a wall peak in
the velocity profile cannot be seen in the simulations since the bubble size distribution is
fixed and the used lift force model forbids such peaks at the wall.
F8 - 29mm/s
F11 - 29 mm/s
Figure 5-8 Velocity profiles at four different heights for pattern F8 and F11 with 29
mm/s superficial velocity compared with the experiments.
5.2 Results
85
The F11 pattern at 29 mm/s superficial velocity, in contrast, is well reproduced by
the simulations shown in Figure 5-8. Nevertheless, at 0.15 m the simulations also
overpredict the downward velocity near the walls and the upward velocity at the center.
At the next measurement line at 0.3 m, the simulations are in agreement with the
experimental data. The discrepancy between the simulations and the measurements are
also small for the other two heights at 0.6 m and 1.125 m. However, the measured velocity
profiles tend to have a more flat profile in the center than the profiles obtained from the
simulation, in general.
The trend from pattern F11 to F8 is different for the simulations. First, the velocity
profiles measured at 0.15 m are nearly the same; in contrast, the profiles obtained from
the simulations are flatter for the F11 pattern. It would be expected that the center peak
for the broader aeration pattern is also broader. Vortex structures that compress the
bubble plume at the bottom, which is seen in Figure 5-7, however, lead to a narrow
velocity center peak. The vortex structures in the experiments for both patterns might
have the same influence on the bubble plume since the velocity profiles are almost equal,
which might also explain the same bubble plume swinging frequency as discussed below.
Nevertheless, at a height of 0.3 m this trend of a flatter velocity profile obtained with the
narrower aeration patter F11 is also seen in the experiments. The wall peaks in the
measured velocity profiles at 0.6 m and 1.125 m for F8, which might indicate a bubble
coalescence, are not present for F11.
Figure 5-9 Vertical velocity over time in the center of the column for pattern F11 at a
superficial velocity of 29 mm/s. Simulation results are shown at two heights.
5 Prediction of the large liquid structures of bubbly flows with the URANS equations
86
Despite the good agreement between the simulations and the experiments for the
F11 pattern at 29 mm/s superficial velocity, distinct differences occur looking at the
velocities over time at the center at a height of 0.3 m as shown in Figure 5-9. Clearly, the
simulation results at 0.3 m height do not coincide with the experiments whereas the
simulation results at 0.15 m height are perfectly matching in frequency and amplitude.
This might be explained by an earlier spreading bubble plume than in the experiments.
Indeed, the void fraction profiles obtained from the simulations are at 0.3 m already
homogenized and a distinct plume is not remaining visible as indicated in Figure 5-7. A
transition of the flow regime is obtained by the simulation shown in Figure 5-10. From
the void fraction profiles at 0.15 m, a transition is seen from 16 mm/s to 24 mm/s. At 24
mm/s, more bubbles are pulled downwards at the walls than are found above the sparger.
This effect significantly changes the velocity profiles at 0.3 m. The downward movement
of the bubbles near the wall requires a strong downward flow; however, as seen in Figure
5-8 the averaged absolute vertical velocities near the wall from the experiments are
distinctly smaller than in the simulations. Therefore, the downward movement of the
bubbles could be different in the experiments; this might explain the differences in the
transient behavior seen in Figure 5-9.
Figure 5-10 Velocity and gas void fraction profiles for pattern F11 for different
superficial velocities at two different heights.
5.2 Results
87
The bubble plume frequency is determined from the simulations by analyzing the
vertical velocities at 0.15 m and not at 0.3 m as in the experiments. The plume frequency
is changing over time in the simulations; in Figure 5-9 simulation and measurement are
compared at a time at which the accordance between them is good. Time sections are
found in which the simulations are not matching the experimental data. Therefore, the
frequency was determined with the windowed Fourier transformation. The windows
have the same length as the measurement time, namely 60 s and are overlapping 50
percentage; the obtained frequencies were averaged over time. The real time of the
simulations is between 1000 and 2000 s.
Despite the differences between simulations and experiments discussed above, for
pattern F8 and F11 the agreement regarding the bubble plume frequencies is very good
as shown in Figure 5-11. The almost linear increase of the measured frequencies for the
F8 and F11 pattern is reproduced by the simulations. The accuracy of the determined
frequencies, especially for the experiments, is 160
⁄𝑠=0.0166 𝑠 (60 s measuring time).
Figure 5-11 Comparison of the bubble plume frequency obtained by experiments and
simulations. Left the Frequencies for the F11 pattern; right the frequencies for the F16
and F8 pattern.
Large differences are obtained for the F16 pattern in general and for the F11 pattern
at 36 mm/s superficial velocity. The bubble plume tends to stabilize and is not moving
anymore. The stabilized bubble plume for F11 at 36 mm/s superficial velocity and F16 at
16 mm/s superficial velocity is demonstrated in Figure 5-12. The averaging time was
around 2000 seconds for the F11 pattern and around 1000 seconds for the F16 pattern.
The bubble plume is meandering at the beginning but slowly stabilize after several
minutes. Then, the vortex zones that would move downwards and dissipate at the bottom
are stabilizing as is seen from the vector plots.
Such stabilized bubble plumes were also observed in previous simulations (Lucas et
al. 2007b) in which the lift force stabilizes or destabilizes the plume (Lucas et al. 2005).
The influence of the lift force on the present setup is demonstrated in Figure 5-13 for
pattern F11 with a superficial velocity of 29 mm/s. Since the lift force model of Tomiyama
et al. (2002) is used with a bubble diameter of 4.56 mm (see Table 5-1), the lift force
coefficient is positive with a value of 0.25 in the original setup. Thus, the bubbles are
moving towards the walls of the bubble column and away from positive vertical velocity
peaks. These stabilizing effect leads to a homogenizing towards the top. Without the lift
force, the bubble plume stands on one side, after approximately 400 seconds the plume
5 Prediction of the large liquid structures of bubbly flows with the URANS equations
88
swings to the other side. This leads to a wall-peaked velocity profile near the sparger as
seen in Figure 5-13. Moreover, towards the top of the column an inhomogeneous flow
pattern is observed with a distinct upward flow in the center and downward flow at the
walls.
Figure 5-12 Averaged upward velocities and vector plots for pattern F11 at 36 mm/s
superficial velocity and F16 at 16 mm/s superficial velocity.
Figure 5-13 Sensitivity regarding lift force for pattern F11, superficial velocity 29 mm/s.
5.2 Results
89
Since the bubble sizes in the experiment are determined from single needle
experiments, another bubble size in the fully aerated bubble column is possible. Since the
flow is stabilized for smaller bubbles due to the lift force (Lucas et al. 2005), the bubble
size is decreased to 4 mm; among other effects, the lift force coefficient is increased to
in 𝐶𝑙=0.288. The influence of the bubble size is shown in Figure 5-14. The cases F11 –
36 mm/s and F16 – 29 mm/s do not show a swinging motion with the original setup (see
Figure 5-11). A different behavior is seen for F11 with reduced bubble size, the stabilized
vortex structures are broken up and the plume shows a swinging motion, occasionally.
Nevertheless, the standing bubble plume persists for F16. However, the flow is
homogenizing towards the center as is expected due to the lateral motion of the small
bubbles away from the center, which is induced by the lift force.
F11 with 36 mm/s
F16 with 16 mm/s
Figure 5-14 Influence of the bubble size for two different flow patterns.
From experience, also the bubble induced turbulence model influences the swinging
motion of the bubble plume. This is demonstrated in Figure 5-15 for pattern F16 with 16
mm/s superficial velocity by comparing the baseline BIT model with the Sato model (Sato
et al. 1981) described in Section 2.4.4. The stabilized vortex structures are not found with
the Sato model, but the bubble plume is swinging slower compared to the experiments.
The velocity profiles are symmetric as would be expected. These effects are due to the
higher eddy viscosity shown on the right hand side of Figure 5-15. The highest eddy
viscosity is found at the position where the bubbles are redirected and occasionally
pulled downward. Nevertheless, using the Sato model leads to a non-swinging motion of
the bubble plume for the lower superficial velocities (not shown).
5 Prediction of the large liquid structures of bubbly flows with the URANS equations
90
Figure 5-15 Influence of the turbulence model for flow pattern F16 and a superficial
velocity of 16 mm/s.
5.3 Conclusions
The swinging bubble plumes are reproduced by the simulations in general. For the
Becker and Pfleger cases, the experiments were reproduced qualitatively and
quantitatively for the time-averaged results as well as for the transient results. For the
Julia cases several deviations occur. The time averaged results are well reproduced for
some cases; however, for some other cases the experiments could not be reproduced
qualitatively as well as quantitatively.
Surprisingly, the bubble plume frequencies are similar to the experiments for most of
the cases for the F8 and F11 flow pattern. Nevertheless, for the highest gas volume flow
rate of the F11 pattern and for almost all cases of the F16 pattern the bubble plumes
obtained from the simulations are more or less swinging asymmetrically; asymmetric
time averaged velocity profiles were obtained as well.
The influence of the lift force, bubble size and turbulence model on the asymmetric
swinging bubble plumes is significantly depending on the gas volume flow rate and
aeration pattern. Changing these properties improve some cases but also has no effect on
or worsen the result for other cases. Changing the lift force or bubble sizes in a way that
the stability criterion by Lucas et al. (2005) is fulfilled, the asymmetric plumes are mostly
maintained but are homogenizing faster towards the top of the column. Using the Sato
BIT model, which is resulting in a significant higher turbulent viscosity, the asymmetric
plumes tend to be symmetric, but the swinging motion is suppressed and for lower gas
volume flow rates even no swinging motion is obtained.
The asymmetric bubble plume position is a stable or meta-stable state in the
simulations. The comparison with experiments, however, is difficult especially since the
5.3 Conclusions
91
precise bubble size for the Julia experiments is not known. Nevertheless, from own
experimental experience a bubble plume tends occasionally to stay at one side of the
bubble column for an apparently arbitrary long time, especially if the bubble column is
not exactly in absolute straight alignment. Since it might be difficult to align large facilities
exactly straight, the effect of an asymmetric bubble plume is important for practical use,
especially if flow situations exists that amplify such asymmetric behaviors.
It should be noted, that locally very high void fractions occur in the bubble plume.
Swarm effects, however, are not minded in the present baseline model, but might be
significant.
Nevertheless, the URANS approach with the present baseline model is capable to
reproduce the large-scale structures in a bubble column at least for lower void fraction.
The frequency of the bubble plume is reproduced for these cases as well as the time-
averaged values.
92
93
6 Turbulence in bubbly flows using the URANS equations with
separation in large and small turbulence structures
In bubbly flows, two flow regimes with different characteristics might be identified,
especially in bubble columns. In one of them, there is a uniform flow pattern with a
uniform distribution of the gas content over the cross section. The other flow regime is
characterized by large-scale flow structures with a non-uniform flow pattern, which arise
from partially distributed gas content and/or because of bubble coalescence and break-
up processes (Mudde et al. 2009). Therefore, the bubble sizes in this regime are often
ranging from very small to very large bubbles. In the uniform flow regime in contrast
coalescence and breakup is absent. In this context, they are also refereed as
heterogeneous and homogeneous regimes, respectively. Because of the different flow
patterns, the dominant turbulent structures in heterogeneous and homogenous regimes
might be different.
Partially aerated bubble columns like that of Becker et al. (1994), Pfleger et al., (1999)
or Deen et al. (2001) are frequently used to validate simulation models for flows with a
non-uniform pattern. The outcome of such studies is in general that the hydrodynamics
can be well predicted even by neglecting the bubble induced turbulence (Sokolichin et al.
2004) or bubble forces like the lift force (Diaz et al. 2009). More recently, Ojima et al.
(2014) concluded that even without any turbulence modeling non-uniform bubbly flows
with large vortex structures could be well described.
Whereas in the previous section the ability of the URANS to reproduce the transient
behavior in the heterogeneous regime in general was discussed, in the present it is shown
that by using the URANS baseline model, both regimes, non-uniform and uniform bubbly
flows, can be reproduced. An approach that is capable to model both regimes is essential
for reliable CFD calculations since geometrical changes of the facilities can lead to a
regime transition or just bubble coalescence and break up (Lucas et al. 2007b).
The heterogeneous regime is investigated by using the experimental data of Deen et
al. (2001), the homogenous regime by using the experimental data of Julia et al. (2007) .
At this point, the heterogeneous regime is simplified by using a partially aerated bubble
column with a fixed bubble size distribution, which bypasses the additional complexity
and uncertainty of modeling bubble coalescence and breakup.
6.1 Simulation Setup and experimental data
A sketch of both experimental setups, Deen et al. (2001) and Julia et al. (2007), is shown
in Figure 6-1. The gas volume flow rates and bubble sizes are summarized in Table 6-1.
The bubble column used by Deen et al. has a quadratic ground plate and an initial water
level of 0.45 m. The sparger consists of an array of 7 by 7 holes. The mean and fluctuation
of the liquid velocity were measured using LDA and PIV on a line 0.25 m above the ground
plate. The bubble sizes were estimated as 4 mm by not further specified visual
observations. Water was used in which 0.04 weight % of kitchen salt were added to
suppress coalescence.
The bubble column used by Julia et al. (2007) is a flat bubble column with a
homogenous distributed needle sparger that consists of 137 needles. The liquid velocities
6 Turbulence in bubbly flows using the URANS equations with separation in large and
small turbulence structures
94
and turbulence properties were measured with LDA on a line 0.6 m above the ground
plate. The bubble sizes were obtained by single needle experiments using video cameras.
Figure 6-1 Sketch of the experimental setups. Right the setup of Julia et al. (2007); left
the setup of Deen et al. (2001). The measurement positions are shown as dotted lines.
Author
Superficial velocity
Bubble size
Deen et al. (2001)
4.9 mm/s
4 mm
Julia et al. (2007)
21 mm/s
3.66 mm
29 mm/s
3.82 mm
43 mm/s
4.09 mm
58 mm/s
5.6 mm
Table 6-1 The superficial velocities and bubbles sizes.
For both cases, grid studies were conducted. The bubble column of Deen et al. (2001)
is discretized using a uniform cubic mesh of 5 mm cell size consisting of 81 000 cells. The
bubble column of Julia et al. (2007) is, likewise, discretized using a uniform rectangular
mesh of Δx=5mm,Δy=8.2 mm,Δz=5.16 size, consisting of 90 000 cells.
For the Deen column, the inlet is modeled as a region of size of 0.03 x 0.03 m in the
center of the column bottom as in the original work (Deen et al 2001). For the Julia
column, the inlet corresponds with the full column bottom. Otherwise, the setup and
methods are used as defined in Section 2.5 and Section 4.
6.2 Simulation results for both regimes
95
6.2 Simulation results for both regimes
6.3 Heterogeneous regime
The flow in the bubble column of Deen et al. (2001) is characterized by large-scale flow
structures. The liquid velocities were measured using particle image velocimetry (PIV)
and laser Doppler anemometry (LDA). In the following, the PIV results are plotted with
the LDA results as error bars to indicate the uncertainty of the experimental data. The
measurement technique to determine the bubble size is not described by Deen et al., so
the reported value may not be reliable.
Therefore three different bubble sizes 3, 4, and 5 mm were tested. The results are
shown in Figure 6-2. The vertical liquid velocity varies strongly with the bubble size; the
simulation with 4 mm gives here the best results compared to experiments. In contrast,
the root mean square of the normal Reynolds stress tensor components v′v′ and 𝑢′𝑢′
(RMS(v′v′) and RMS(u′u′), respectively) are not as much influenced by the bubble size
as the vertical liquid velocity. Nevertheless, the RMS(v′v′) and RMS(u′u′) results shown
in Figure 6-2 are the summation of the unresolved and resolved turbulence modeling,
both contributions are varying with the bubble sizes but the total amount is nearly the
same. Since the results for 4 mm bubble size are in fact closest to the experimental data,
this value will be used from hereon.
Figure 6-2 Simulation results using different bubble sizes compared to experiments
(Deen et al. 2001). Left the vertical liquid velocity is shown; right the root mean square
of the normal components of the Reynolds shear stress tensor (𝑣 vertical, 𝑢 horizontal).
The contribution of the unresolved as well as the resolved parts of RMS(v′v′)
and RMS(u′u′), both with and without BIT modeling are shown in Figure 6-3. For the
mean liquid velocity, there is no significant difference whether BIT is included or not,
therefore no additional figure is shown. Considering the measurement uncertainty, the
total values of both RMS(v′v′) and RMS(u′u′) obtained by the simulations both with and
without BIT are in reasonable agreement with the data. The most prominent difference
between the models with and without BIT is the shape of the total RMS(v′v′) profiles.
With BIT, a profile with a single center-peak is obtained with a trend to overpredict the
measured values in the center of the column. Without BIT, a double-peaked profile is
found which is somewhat closer to the data. The nature of these differences is situated in
6 Turbulence in bubbly flows using the URANS equations with separation in large and
small turbulence structures
96
the resolved contribution. For both models, the predicted variations are too pronounced
compared with the experimental profiles, which have a rather flat shape. For the
total RMS(u′u′), both models give similar results.
Such double peaked RMS(v′v′) profiles that are obtained without the BIT modeling
have been observed in flat bubble columns with partial aeration (Simiano et al.
2006)(Section 3.4 and Section 7). The experimental data of Deen et al. in the square
shaped bubble column might indicate also such profile shape; however, due to the
measurement uncertainty this is speculative.
RMS(v′v′) is dominated by the resolved contribution for both models with and
without BIT. In contrast, for RMS(u′u′) the resolved and unresolved contributions have
similar values for the model with BIT whereas for the model without BIT the modeled
contribution is much larger than the resolved contribution.
The resolved contribution to the Reynolds stresses is higher with BIT model than
without a BIT model in general. This is caused by the eight times lower eddy viscosity
obtained with the BIT model. Accordingly, the eddy dissipation rate is distinctly higher
with the BIT model. Nevertheless, the liquid velocity profiles (not shown) obtained with
BIT und without BIT modeling are nearly the same.
a)
b)
c)
d)
6.4 Homogenous regime
97
Figure 6-3 Resolved and unresolved parts of RMS(v′v′) and RMS(u′u′) with and without
BIT at 0.25 m above sparger compared to experimental data by Deen et al. (2001). a)
And b) using BIT; c) and d) without a BIT model.
Summarizing, the non-uniform regime represented here by the experiments of Deen
et al. (2001) using a monodisperse bubble size distribution in a partially aerated bubble
column can be reproduced satisfactorily using a two equation turbulence model either
with or without BIT source terms. Moreover, the URANS modeling gives nearly the same
results as large eddy simulations and scale adaptive simulations (Ma et al. 2015) or
Reynolds stress models (Masood et al. 2014).
6.4 Homogenous regime
The uniform regime is characterized by a uniform distribution of the bubbles over the
cross section of the column. When the bubbles are small enough this situation is stabilized
by the lift force as discussed by Lucas et al. (2005) (2007b). Hence, no larger vorticities
are expected in this regime as well as the turbulence is dominated by the BIT part. As
discussed above, the experiments of Julia et al. (2007) are chosen for the investigation of
this regime.
Profiles of the liquid velocity, which were measured with LDA for the cases with gas
superficial velocities of 29 mm/s and 43 mm/s, are compared with the URANS modeling
in Figure 6-4 a), b) and c), d) respectively. Simulation results obtained both with and
without BIT modeling are shown. As expected, the resolved part of the fluctuations in
both directions RMS(v′v′) and RMS(u′u′) turns out to be zero in the simulations for both
model variants, thus only the modeled part is shown which equals the total fluctuations.
As a consequence the calculated values for RMS(v′v′) and RMS(u′u′) are equal since the
modeled fluctuations are isotropic.
In comparison to the non-uniform regime the liquid velocity profiles are much more
plug flow like with the liquid flowing downwards only much closer to the sidewalls.
Similarly the profiles of RMS(v′v′) and RMS(u′u′) here are almost constant over the
column width and drop to zero steeply near the walls.
Comparing experiment and simulations, the vertical liquid velocity is underpredicted
in the simulations for both values of the superficial velocity. As seen from the comparison
of LDA and PIV data by Deen et al. (2001) a possible systematic error of the LDA method,
which gives consistently higher values than obtained by PIV, may contribute to this
deviation. From the modeling, the neglect of swarm effects and the use of a drag
correlation for contaminated rather than clean systems are likely to play a role. The data
show characteristic peaks near the walls, which are in principle also indicated in the
simulations. Simulations with and without BIT modeling give quite similar results for the
vertical liquid velocity.
Concerning the fluctuations the RMS(v′v′) and RMS(u′u′) which have both been
measured for the lower superficial velocity of 29 mm/s a certain anisotropy is seen that
is not covered by the present two-equation turbulence model. A Reynolds stress model is
needed to capture this effect. Without a BIT model the turbulence is underpredicted by a
factor of 4 or more whereas with BIT model it is overpredicted by a factor of
approximately 2. The profile shape with BIT model shows peculiar peaks at the wall
which are absent in the data. For the higher superficial velocity of 43 mm/s only data for
6 Turbulence in bubbly flows using the URANS equations with separation in large and
small turbulence structures
98
RMS(v′v′) are available which show slightly higher values than for 29 mm/s. This trend
is also seen in the simulations with BIT model but not in those without BIT model.
a)
b)
c)
d)
Figure 6-4 Vertical liquid velocity and RMS values of the normal Reynolds stress
components obtained with the URANS modeling compared to experiment (Juliá et al.
2007). a) and b) superficial velocity of 29 mm/s. c) And d) superficial velocity of 43
mm/s.
For all four superficial velocities, averages of the RMS(v′v′) and RMS(u′u′) values
along the measurement line have been reported by Julia et al. (2007) (without the near
wall region). These are shown in Figure 6-5 together with the corresponding simulation
results for the model with BIT. As can be seen, the overprediction of the values by the
simulations is present for all superficial velocities with the deviations increasing with
decreasing superficial velocity.
In addition, the data from another experiment for uniform flow in a bubble column
by bin Mohd Akbar et al. (2012) and corresponding URANS simulations from Section 4
using the same modeling as in the present study with BIT is included. The results obtained
with the URANS simulations for the latter case are in good agreement with the
measured RMS(v′v′).
6.5 Comparison with Large Eddy simulation
99
Since the data of Julia et al. (2007) and bin Mohd Akbar et al. (2012) do not fall on a
single curve, it appears that besides the superficial velocity some other relevant
parameter must exist. The bubble sizes in both experiments are about 4 mm and both
have been conducted in an air/water system. Also the geometries are rather similar, the
column of bin Mohd Akbar et al. (2012) being about twice as thick as that of Julia et al.
(2007) or in other words almost twenty and ten times the bubble diameter respectively.
The integral gas hold up differs strongly between the two experiments, for the
experiments of bin Mohd Akbar et al. (2012) is about 1.5 % while for the experiments by
and Julia et al. (2007) ranges from 5.4 % to 20.2 %. However, this difference corresponds
with the different superficial velocity. Therefore, it is not clear where the mismatch
between both experiments comes from.
In Julia et al. (2007) it is mentioned that a turbulence model for bubbly flows based
on the pseudo turbulence obtained from potential flow (de Bertodano et al. 1990) with a
dependency on the void fraction as 𝛼23
⁄ (Lance et al. 1991) can reproduce their
experimental data. Additional simulations performed using this model, however, have
shown heavy underprediction for the experiments by bin Mohd Akbar et al. (2012).
Hence, such a model adaptation is not useful in general. Moreover, introducing a similar
void fraction dependence in the prefactor 𝐶𝜖𝐵 of the present BIT model would also
worsen the agreement for the experiments by bin Mohd Akbar et al. (2012) and probably
also for the pipe flow tests by Rzehak & Krepper (2013b) (2015) and Rzehak &
Kriebitzsch (2014).
Figure 6-5 Reynolds stresses in homogenous bubble columns for different superficial
velocities. The four superficial velocities from the present study (Juliá et al. 2007) and the
experiments by bin Mohd Akbar et al. (2012) with the corresponding URANS simulations
by Ziegenhein et al. (2015) are shown.
Summarizing, the uniform regime represented by the experiments of Julia et al.
(2007) can be roughly reproduced by the URANS approach. The resolved part of the
turbulence modeling is zero as would be expected and the simulated RMS values of the
normal Reynolds stress components are completely dominated by the used BIT model.
6.5 Comparison with Large Eddy simulation
6 Turbulence in bubbly flows using the URANS equations with separation in large and
small turbulence structures
100
Besides the URANS approach also large eddy simulations (LES) were performed in
corporation with the group of Prof. Jochen Fröhlich from TU-Dresden. The LES were
executed by Dipl.-Ing. Tian Ma (HZDR/TU-Dresden), the results are described in several
publications (Ma et al. 2015) (Ma et al. 2015) (Ma et al. 2015). At this point, the results
are shortly compared to the URANS results by using the results of the last two mentioned
papers. The results from the LES are reproduced with permission of Dipl.-Ing. Tian Ma,
which is not repeated in the following. The results are taken from the LES with the
Smagorinsky model using 𝐶𝑠=0.12 and the Sato BIT model (see Section 2.4.3). Further,
the same force model set was taken as used for the baseline URANS except the turbulent
dispersion force since these turbulence structures are resolved in the LES.
For the experiment of Deen et al. (2001) which is investigated in the previous section,
the LES from Ma et al. and URANS simulations are compared in Figure 6-6. The vertical
liquid velocity is stronger over-predicted than by the URANS simulations compared to
the experiments. The obtained root mean square of the normal components of the
Reynolds stress tensor 𝑢′𝑢′ and 𝑣′𝑣′ are similar for both approaches. The 𝑅𝑀𝑆(𝑣′𝑣′)
obtained from the LES, however, shows a very strong peak in the center. Likewise as the
liquid velocity, the void fraction is higher in the center compared to the URANS
simulations. The eddy viscosity obtained from the LES is distinctly smaller compared to
that obtained from the URANS simulations, as would be expected. For comparison, the
eddy viscosity obtained from the URANS simulations without a BIT model is shown. The
lower the modeled eddy viscosity the higher the resolved part of the turbulence.
6.5 Comparison with Large Eddy simulation
101
Figure 6-6 LES compared with the URANS simulations for experiments of Deen et al.
(2001).
It should be noted, the LES result obtained without the Sato BIT model, which is
leading to a lower eddy viscosity, fits better to the experiment. Therefore, LES are capable
to reproduce the bubbly flow for this experiment in general.
For homogenous regimes that might be dominated by the BIT, the LES, however,
cannot represent the turbulence as shown in Figure 6-7. Here, the experiments of Mohd
bin Akbar et al. (2012), which are discussed in Section 4, are compared to LES and URANS
simulations. For the 3 mm/s superficial velocity it was previously found that the resolved
turbulence scales obtained from the URANS simulations are almost zero. The same result
is obtained from the LES resulting in a strong underprediction of the 𝑅𝑀𝑆(𝑤′𝑤′) (𝑤 is
the upward velocity to stay in the notation of the experiments) values shown in Figure
6-7 d). Nevertheless, vertical liquid velocity and gas volume fraction are similar to the
URANS simulations.
The results of the LES for the 13 mm/s case are similar to the URANS results. Looking
at the 𝑅𝑀𝑆(𝑤′𝑤′) profiles an improvement is obtained compared to the experiments; the
distinct near wall peak is better reproduced by the LES.
a)
b)
6 Turbulence in bubbly flows using the URANS equations with separation in large and
small turbulence structures
102
c)
d)
Figure 6-7 LES compared with the URANS simulations for the experiments of bin Mohd
Akbar et al. (2012) (cf. Section 4). a) and b) vertical liquid velocity and gas volume fraction
for 3 mm/s and 13 mm/s superficial velocity, c) 𝑅𝑀𝑆(𝑤′𝑤′) for 13 mm/s superficial
velocity (𝑤 is the upward velocity), d) 𝑅𝑀𝑆(𝑤′𝑤′) for 3 mm/s superficial velocity.
Summarizing, the baseline URANS model approach and the LES give similar results
in the inhomogeneous, large-scale flow dominated bubbly flow regime. Therefore, the
URANS simulations are capable to reproduce such structures in general. The URANS
approach is capable to reproduce the homogenous, BIT dominated regime, which is not
possible with the present BIT modeling for the LES. Therefore, towards a general
approach to model bubbly flows the URANS approach is advantageous.
6.6 Conclusions
The resolved structures obtained from the simulations give an important
contribution to the turbulence in general. Further, the anisotropic character of the
turbulence in this regime is reproduced in this way. Comparing the URANS results to
other turbulence model approaches like LES (Section 6.5) or a Reynolds stress model
(RSM) (Masood et al. 2014) similar results are obtained. Therefore, in the heterogeneous
bubbly flow regime the URANS approach is capable to reproduce the large-scale flow
structures in the same way as approaches that are more sophisticated.
The total turbulent kinetic energy is similar with and without BIT modeling; the
unresolved and resolved parts, however, are different. Since also the LES with and
without BIT and the RSM simulations give almost the same results for the Deen
experiments, it is reasonable to assume that the BIT modelling has a minor influence in
the heterogeneous regime. Differences between the approaches are small and are in the
range of the experimental uncertainty; especially the not measured bubble size
distribution is problematic.
A different situation is found in the homogeneous regime. The liquid velocity is small
and no large-scale flow structures are expected. This expectation is consistent with the
results obtained from the URANS simulations since the resolved turbulence part is nearly
zero. Therefore, the turbulence is completely characterized by the unresolved turbulence
model. Simulations without BIT modeling heavily underpredict the turbulence in this
regime. The results obtained from the simulations with BIT in contrast overpredict the
6.6 Conclusions
103
experimental data to somewhat lesser degree with deviations increasing with decreasing
superficial velocity. Despite the necessity to improve the modeling further, the URANS
approach with the BIT model used here is capable to describe the uniform bubbly flow
regime as well.
Therefore, the present URANS approach with a BIT modeling using source terms for
𝑘 and 𝜖/𝜔 is able to reproduce both regimes. To improve the Eulerian LES it might be
necessary to define at least a one-equation turbulence transport with a filter width
addicted to the bubble size and not to the mesh size. To cover isotropic turbulence in the
unresolved part, especially in the BIT modeling, a RSM approach might be the only
reasonable approach.
The nature of the bubble-induced turbulence remains an issue of active research and
up to now, no model is sufficient for a general formulation. Besides fundamental
investigations to this topic, also reliable experimental data for a validation is needed. For
this purpose, a comprehensive set of locally measured values is needed in different flow
conditions. Indeed, the above discussed experiments, except of the one used in Section 4,
do not provide such data since often information about the gas phase (gas void fraction)
or liquid velocity field (basic turbulence parameters) are missing. In particular, the
bubble sizes are insufficient determined since the measuring method is often not given
or the bubble sizes are estimated from simplified flow conditions. Furthermore, the
bubble size distribution at different positions is needed to evaluate if coalescence and
break up processes are negligible or not. Unfortunately, such comprehensive
experimental data that can be used for CFD validation are rare in the literature. Thus, an
own experimental study is performed in the next section covering all the needs of a CFD
validation.
In parallel, the modeling of the different closure models in the momentum equation
(see Section 2.3) has to be evolved. From the discussion from the last sections it appears
that the lift force plays an important role. Reliable measurements, however, were only
performed under high Morton numbers and low Reynolds numbers, which are far away
from the here used conditions. Moreover, the significance of the lift force in turbulent
air/water bubbly flows is an open discussion in the community. Therefore, new methods
are developed to measure the lift force in the after next section.
104
105
7 A complex validation case for CFD simulations: Airlift reactor
In the bubble columns that are discussed in the previous sections the gas bubbles drive
the flow and usually the liquid is rising in the center and falling near the wall. The up- and
downward flow are next to each other and can interact. Alternatively, internal walls can
be placed in bubble columns to separate the up and downward flow; these reactors are
called internal airlift reactors.
For many applications that use airlift reactors it is desired to know the exact fluid
dynamics. For example, the light exposure of microorganisms in airlift photo bioreactors
can be optimized by knowing the fluid dynamics (Fernandes et al. 2010). Moreover, the
shear rate and turbulence parameters are important for all process with microorganisms
(Liu & Tay 2002) (Miron et al. 200) (Oliver-Salvador et al. 2013) and for mass transfer
modelling (Korpijarvi et al. 1999) (Lu et al. 2000). Nevertheless, such detailed
information of the fluid dynamics are rarely accessible by the use of experiments.
A better understanding of the fluid dynamics is gained by using the methods of the
computational fluid dynamics (CFD). A lot of work was done simulating airlift reactors in
the past with the Eulerian two-fluid approach. However, in general the bubble sizes were
not known (Huang et al. 2010) or only known in the downcomer (Luo & Al-Dahhan 2011).
Further, often only integral measured values were available (Simcik 2011) (Ghasemi &
Hosseini 2012). Hence, a validation of the closure models in airlift reactors is limited with
the existing experimental data.
In the present section, an internal airlift reactor is experimental studied to provide a
comprehensive set of locally measured data for CFD validation. To the best of the author’s
knowledge, those measurements were not published in the past and are urgently needed.
Moreover, the measured data provide a complete picture of the flow in an internal airlift
reactor.
7.1 Setup
The bubble column that is described in Section 3.1 is used with internal walls as shown
in Figure 7-1. The 5 mm thick internal walls separate the 0.12 m wide riser from the
downcomers. Each downcomer has a width of 0.06 m so that the riser and the sum of both
downcomers have the same cross section. The distance from the ground plate to the
beginning of the internal walls is 0.06 m, which is equal to the width of a downcomer. In
addition, the distance from the top of the internal walls to the water surface (the top
clearance) is held constant to 0.06 m for all gas volume flow rates. Thus, the liquid level
is at 0.72 m above the ground plate for all setups.
Liquid velocity, turbulent kinetic energy, available Reynolds stress tensor
components and bubble sizes are determined at a height of 0.2 m and 0.6 m in the riser
and the downcomer, which is indicated with red lines in Figure 7-1. The void fraction is
measured at a height of 0.6 m in the riser. In addition, the bubble size distribution and the
void fraction are determined along the downcomer.
7 A complex validation case for CFD simulations: Airlift reactor
106
Figure 7-1 Experimental setup and the used ground plate setup. The red lines label the
measuring positions.
Rubber seals that are attached at the side of the internal walls hold them in place.
Therefore, no interaction between the riser and the downcomer is possible and no flow
disturbing installations are needed to hold the walls in place. The Gas is injected through
the ground plate, which is shown in Figure 7-1 on the right hand side, by using up to eight
needles with an inner diameter of 0.6 mm. The volume flow rate per needle is held
constant for all cases to get a similar bubble size distribution. The total gas volume flow
rate is regulated by changing the needle count. A summary of the important parameters
is given in Table 7-1.
case
cumber
Volume
flow
Sparger
needle
Needle
count
Volume flow
rate per needle
S
W
(gas on)
4
3 l/min
0.6 mm
4
0.75 l/min
35 mm
60 mm
6
4.5 l/min
0.6 mm
6
0.75 l/min
60 mm
60 mm
8
6 l/min
0.6 mm
8
0.75 l/min
85 mm
60 mm
Table 7-1 Experimental parameters at standard conditions.
7.2 Results
The bubble size distribution is determined with videography at several positions as
discussed in Section 3.2. The volume fraction in the riser is measured with a conductivity
needle probe as described in Section 3.3.1. In contrast, the volume fraction in the
downcomer is determined by the use of videography as described in Section 3.3.2. The
7.2 Results
107
liquid velocity and the turbulent kinetic energy are measured with particle-tracking
velocimetry using micro bubbles (BTV), which is discussed in Section 3.4.
7.2.1 Bubble size distribution
The bubble size distributions in the riser for the different volume flow rates are shown in
Figure 7-2. Bubbles that are smaller than 1.5 mm are not evaluated in order to reduce the
measuring effort since the count of the small bubbles is large whereas such small bubbles
are not significant for the Sauter diameter, which is used for CFD calculations. In the riser,
the bubble sizes are determined at a height of 0.2 m and 0.6 m to evaluate possible break-
up and coalescence effects. The number density function is identical at both heights for
case 6 and case 4 (case 4 is not shown). For case 8, the number density function is shifted
slightly towards smaller bubbles. Nevertheless, comparing the area density and volume
density function of case 8 with the results of case 6 and case 4 no large differences are
seen so that coalescence and break up might be negligible.
a)
b)
c)
d)
Figure 7-2 Bubble size distributions in the riser. a) Number density for case 6 at two
different heights b) number density for case 8 at two different heights; c) averaged area
density function at 0.2 m and 0.6 m; d) averaged volume density function at 0.2 m and 0.6
m.
7 A complex validation case for CFD simulations: Airlift reactor
108
The bubbles in the riser are shown in Figure 7-3. Bubble cluster are seen clearly for
all volume flow rates. The bubbles inside the clusters are identified by chasing the bubble
clusters over several images until the bubbles are seen distinctly as described in Section
3.2.3.
a)
b)
c)
Figure 7-3 Pictures of the bubbly flow in the riser at a height of y=0.2 m. a) case 4; b)
case 6; c) case 8.
The automatically determined bubble sizes along the downcomer are shown in
Figure 7-4 picture a). The bubble sizes are averaged over the cross section of the
downcomer. The bubble size at the top of the downcomer for case 6 and case 8 are
determined by hand because the void fraction is too high for an automated evaluation. A
separation of the bubble sizes along the downcomer occurs.
Besides a separation over the height, also a separation of the bubble sizes over the
width of the downcomer is seen as demonstrated in Figure 7-4 picture b) for case 8. The
bubble sizes are averaged over height from y=0.3 m to y=0.4 m and are plotted
against the horizontal coordinate from the airlift reactor wall at x=0 m to the internal
wall at x=0.06 m. Near the reactor wall, larger bubbles are situated compared to the
bubbles that are found near the internal walls. As will be discussed below, the liquid
velocity near the reactor wall is higher than near the internal walls so that this separation
occurs.
The situation in the downcomer is shown in Figure 7-5 for the three flow rates. The
bubble count for case 4 is very low, although the size of the bubbles might be comparable
to case 6. Many large bubbles are seen for case 8, which might be the reason for the larger
Sauter diameter.
7.2 Results
109
a)
b)
Figure 7-4 Bubble sizes in the downcomer. a) Bubble sizes along the downcomer. b)
Bubble sizes over the width of the downcomer for case 8 averaged over height from y=
0.3 m to y =0.4 m
a)
b)
c)
Figure 7-5 Situation in the downcomer, a) case 4 b) case 6 c) case 8.
7.2.2 Liquid velocity and turbulence
The described sampling bias for BTV is also present in the airlift reactor as demonstrated
in Figure 7-6. The count of the trajectories is plotted with the velocity over time; a moving
average over 2 s is used to show the sampling bias clearly. The count of the trajectories is
low when the velocity is high and vice versa over 80 s. Therefore, the count of the tracked
micro bubbles is correlated to the velocity, which is leading to a sampling bias. To
overcome the bias the hold processor derived in Section 3.4.3 is used as described in
Section 3.4.4.
7 A complex validation case for CFD simulations: Airlift reactor
110
Figure 7-6 Sampling bias in the center of the riser for case 8 at y=0.2 m.
The liquid velocity profiles in the riser are obtained from four single measurements
with a distinct time between them. In total, 48 000 bursts, equivalent to eight minutes
measuring time, are evaluated in the riser at a height of 0.2 and 0.6 m. In the upper part
of the downcomer at a height of 0.6 m 36 000 bursts are recorded and in the lower part
24 000 bursts. The long measuring time is necessary because a bubble plume with a very
long time scale occurs; especially in the lower part of the riser, this effect is noticeable.
The liquid velocities at two different heights in the riser and the downcomer for the
investigated volume flow rates are shown in Figure 7-7. At a height of 0.2 m, the velocity
profiles in the downcomer are flat and are nearly the same for all three flow-rates.
Surprisingly, the integral averaged velocity for case 6 and case 8 along this measuring line
are both almost exactly -0.2 m/s; this is nearly the exact value obtained for both cases at
the height of 0.6 m in the downcomer. For case 4 a slightly lower averaged velocity of -
0.18 m/s at y=0.2 m and y=0.6 m is obtained.
The similar results obtained in the riser at y=0.2 m for all cases are due to a distinct
bubble plume created by the circulating liquid that constricts the bubbles developed at
the sparger. This bubble plume swings from one side to the other. An occasionally
asymmetric stabilization at the internal walls of the bubble plume was observed.
y=0.2 m
y=0.6 m
Figure 7-7 Liquid velocity profiles measured at two different heights.
7.2 Results
111
The transient liquid velocity results in Figure 7-8 demonstrate the occasionally
stabilization of the bubble plume at one side. From the transient results in the riser at a
height y=0.2 m in the left quarter at x=0.09 m (upper plot), it is seen that the bubble plume
is standing 40 seconds at the left wall before going to the right wall. However, between
60 s and 110 s a steady bubble plume swinging motion is not seen as well. Nevertheless,
a steady swinging motion was dominant during the experiments. This motion is observed
in all areas of the reactor as demonstrated in the lower transient vertical velocity plot,
which was recorded in the downcomer, in Figure 7-8. Here, a more or less steady
frequency over 120 s is observed.
Figure 7-8 Vertical velocity over time at two different positions for case 6, the time scale
is arbitrarily set to zero for both and is not synchronized. Top: The vertical velocity over
time in the left quarter of the riser at x=0.095 m and y=0.2 m; every measuring point
is moving averaged over 0.08 s. Bottom: The vertical velocity over time in the center of
the left downcomer at x=0.03 m and y=0.6 m; every measuring point is moving
averaged over 2 s.
The behavior of a standing bubble plume at one side of the riser for a distinct quantity
of time was observed for every case. The time that the bubble plume stood at one side
seemed to be arbitrary in the range of several seconds to minutes. In addition, the
switching between the situations of a permanently swinging motion to a standing one at
one wall seemed to be arbitrary. Deviations between the four consecutively conducted
7 A complex validation case for CFD simulations: Airlift reactor
112
liquid velocity measurements, as discussed in the method section, were observed,
particularly, in the bottom part of the riser.
A more continuous situation is obtained in the upper part of the column. The bubble
plume is spreading towards the top. The results that are obtained from the single
measurements are not deviating much. The same is found in the downcomer.
Noteworthy, the vertical liquid velocity in the downcomer at a height of y=0.6 m is
zero near the internal walls and distinctly negative towards the reactor walls.
Consequently, a large standing vortex in this region is observed; the liquid in the top
clearance is forced to the side because of the driving force of the bubbles in the riser.
Reaching the reactor wall the liquid is pulled downward in the downcomer. From visual
observations, bubbles are dragged in the downcomer by the same mechanism. However,
many bubbles that are pulled in the downcomer at the reactor walls migrate to the
internal walls and rise up at them because of a lower vertical liquid velocity there. This
lower vertical liquid velocity at the internal walls was observed along the complete
downcomer and can be still observed at y=0.2 m for all volume flow rates.
Despite the averaged liquid velocities are similar for all investigated cases, the normal
components of the Reynolds stress tensor u′u′ and v′v′ shown in Figure 7-9 are very
different among the volume flow rates. In general, an increasing of u′u′ and v′v′ with
increasing gas volume flow rate is seen. Looking at the bottom of the riser at a height
of y=0.2 m, the v’v’ graphs show clear maxima located beside the center for all cases.
Such maxima are consistent to previous measurements in the bubble plume regime
(Section 3.4) (Simiano et al. 2006) (Mudde et al. 1997). In contrast, the u′u′ graphs show
maxima in the center, which is also observed in the previously mentioned work.
Averaged over the cross section of the riser at y=0.2 m, the v′v′ values are for all
cases almost exactly twice as high as the u′u′ values. The averaged normal components of
the Reynolds stress tensor along the centerline seem to increase linearly with the volume
flow rate, for example for v’v’ 0.01 m2/s2 from case 4 to case 6 and 0.011 m2/s2 from
case 6 to case 8.
With increasing height u′u′ and v′v′ are decreasing in the riser as shown in Figure 7-9.
The averaged v′v′ values, however, are remaining twice as high as the u′u′ values.
Looking at the results obtained for case 8 distinct maxima are seen in the u′u′ and v′v′
graphs at x=0.095 m. These maxima can be found in every measurement and are,
therefore, no outlier. The origin of this effect is unknown.
Along the downcomer, u′u′ and v′v′ are decreasing, compared to the riser, the values
in the downcomer are low. At a height of y=0.6 m, the obtained profiles for case 4 and
case 6 are, surprisingly, very similar. In contrast, for case 8 the values are distinctly
higher. Nevertheless, the ratio between u′u′ and v′v′ is 1.5 for all cases, compared to a
ratio of two in the riser. The turbulence intensity is almost zero at the bottom.
The cross component of the Reynolds stress tensor u′v′ is shown in Figure 7-10.
Similar to the normal components, large values are obtained at a height of y=0.2 m in
the riser that are decreasing with increasing height. Although, the u′v′ values are similar
for case 4 and case 6 at y=0.2 m in the riser, the values for case 8 are distinctly larger.
In the downcomer at y=0.6 m u′v′ is very similar for all cases. Parallel to the normal
components, the u′v′ values are decreasing along the downcomer to almost zero.
7.2 Results
113
y=0.2 m
y=0.6 m
y=0.2 m
y=0.6 m
Figure 7-9 Normal Reynolds stresses in the vertical (v’v’) and horizontal (u’u’) direction
at two different heights.
y=0.2 m
y=0.6 m
Figure 7-10 Cross Reynolds stress u′v′ at two different heights.
7 A complex validation case for CFD simulations: Airlift reactor
114
7.2.3 Void fraction
The void fraction was determined inside the riser with a needle probe and along the
downcomer by using videography. The void fraction that is measured with a needle probe
is a local value. In contrast, the values that are obtained in the downcomer by using
videography are the quantity of gas inside a specific measuring volume. The measuring
volume is composed of the cross section of the downcomer (0.06 m width and 0.05 m
depth) and a height of Δy=0.025 m. The given values are placed in the middle of these
volumes.
The void fraction results for the upper region of the riser and along the lower region
of the downcomer are shown in Figure 7-11. The void fraction is measured only in the
upper region of the riser in order not to disturb the bubble plume at the bottom. The void
fraction inside the downcomer is determined only up to a height of 0.45 m because a
flange is blocking the view. Above the flange the void fraction inside the vortex structure
at the top of the internal walls, which is described above, was too high for reliable
measurements with the videography method.
Figure 7-11 Void fraction in the riser at y=0.6 m (left) and along the downcomer
(right).
The void fraction inside the riser is increasing with the volume flow rate. A center
peak is observed for all cases, with a maximum void fraction of around 7 % for case 8 at
the center.
Along the downcomer, surprisingly, the profiles obtained for case 6 and case 8 are
very similar. The void fraction is steadily decreasing with decreasing height in general.
Moreover near the end of the internal walls (the bottom edge of the internal walls is
at 𝑦=0.06 𝑚) the void fraction is rapidly decreasing due to an increasing liquid velocity
at this point. The rising bubbles in the riser pull the liquid from the downcomer into the
riser and, therefore, also the bubbles from the downcomer.
The steadily decreasing void fraction along the downcomer in general is explained by
the liquid velocity field. As a strong downward flow at the reactor wall is observed, a
positive horizontal liquid velocity in the downcomer towards the internal walls occurs,
as shown in Figure 7-12 a). Naturally, this horizontal velocity is decreasing along the
downcomer, surprisingly, in the bottom region the strongest horizontal velocity is seen
for case 4 whereas it is for case 8 almost zero. This behavior might be due to the higher
void fraction and/or larger bubble sizes for case 6 and case 8. Nevertheless, a liquid
7.3 CFD Simulations
115
velocity that pushes the bubbles towards the internal walls is observed in the downcomer
in general.
a)
b)
Figure 7-12 Flow situation in the downcomer. a) Horizontal liquid velocity at two
different heights, b) void fraction profiles for case 6 at three different heights.
The bubbles are pulled into the downcomer near the reactor wall so that the void
fraction profile in the upper part of the downcomer at y=0.415 m has a peak near the
reactor walls at x=0.02 m, as shown in Figure 7-12 b). Below this measuring position at
y=0.375 m the void fraction near the reactor wall is distinctly smaller but towards the
internal walls almost the same. This indicates a migration of the bubbles away from the
reactor wall to the internal walls where a lower downward liquid velocity is observed so
that the bubbles rise up (see Figure 7-7). Nevertheless, the peak is still near x=0.02 m.
Looking at the void fraction profile further downstream at y=0.215 m, the gas void
fraction at the internal walls is still at the same level, further the maximum gas fraction is
seen near the internal walls at x=0.04 m. Thus, more and more bubbles had moved to
the internal walls and risen up.
7.3 CFD Simulations
The URANS baseline setup as discussed in Section 2.5 and in Section 4 is used to simulate
the hydrodynamics in the investigated airlift reactor. For this purpose, four bubble size
classes with their own velocity field were used. The volume weighted bubble size
distribution was used to distribute the gas volume on the single bubbles classes at the
inlet. Since coalescence and break up might be negligible, the bubble size distribution is
fixed. The inlet conditions are summarized in Table 7-2.
The gas volume flow rate of the bubble classes 1 and 2 is very small at the inlet, thus
they are negligible in the riser but are important in the downcomer. The bubble class 1
contains all bubbles 𝑑𝐵<3 𝑚𝑚, bubble class 2: 3 𝑚𝑚≤𝑑𝐵<4.5 𝑚𝑚. These two bubble
classes are chosen somewhat arbitrary by evaluating the occurring bubble sizes in the
downcomer from the experiment. Bubble class 3 was chosen to cover the remaining
bubble diameters up to 5.83 mm, which is the zero of the Tomiyama lift force coefficient.
Bubble class 4 includes the remaining from that diameter.
7 A complex validation case for CFD simulations: Airlift reactor
116
Figure 7-13 The complete computational mesh and a magnification of the top region.
Bubble class
Sauter diameter [mm]
Void fraction [-]
Case 4
1
2.15
0.0085
2
3.85
0.0263
3
5.37
0.1321
4
7.50
0.8331
Case 6
1
2.24
0.0120
2
3.79
0.0226
3
5.38
0.1325
4
7.63
0.8329
Case 8
1
2.30
0.0145
2
3.87
0.0312
3
5.32
0.1623
4
7.59
0.7920
Table 7-2 Inlet conditions.
7.3 CFD Simulations
117
The computational domain is the complete airlift reactor in three dimensions as
shown in Figure 7-1. A structured mesh was used as shown in Figure 7-13. After a mesh
study, the base mesh size was chosen to 3.75 mm, cells at the wall were refined to 1.875
mm. In addition, above and below the internal walls the mesh was locally refined to cover
large gradients at these points. Since the top clearance was held constant in the
experiments, the same mesh was used for all cases. The internal walls are treated in the
same way as the reactor wall, namely, no-slip condition for the liquid phase and slip
condition for the gas phase are applied.
7.3.1 Results
After a problem time of 400 seconds the simulations had converged by using the local
integral convergence criteria described in Section 4.1, although the symmetric
convergence criteria is not fulfilled. Thus, an asymmetric flow behavior for all cases is
obtained, which is demonstrated in Figure 7-14. The shown averaged vertical liquid
velocity tends to the right-hand side internal wall for case 4 and case 6 and to the left-
hand side internal wall for case 8. Indeed, this behavior of an asymmetric bubble plume
was also observed in the experiments as described above.
Figure 7-14 Asymmetric flow behavior in the airlift reactor. From left to right: Case 4, case
6, case 8.
If the bubble plume in the experiments shows a symmetric state, it is still swinging
asymmetric rather than standing at one wall. In addition, the bubble plume often tends
to restore a symmetric swinging motion, but the time of the asymmetric behavior was
sometimes distinctly long; the liquid velocity measurements during these times were
discarded. During the experiments, it was assumed that this asymmetric behavior was
7 A complex validation case for CFD simulations: Airlift reactor
118
due to not exactly placed internal walls or not exactly equal mass flow controllers; two
mass flow controllers were used to control the gas volume flow rate of four needles of the
eight used needles each. However, since the simulations show an equal behavior, at least
three metastable conditions that contain the asymmetric bubble plume at each side and
the symmetric bubble plume swinging might exist. A systematic error in the simulations
is unlikely because a perfectly symmetrical regular mesh was used and the position
where the plume stands is different for case 8 compared to case 4 and case 6. In addition,
this behavior was seen in all cases of the mesh study. The bubble plume in the simulations
is asymmetric swinging at one side and not steady standing, which is the same behavior
as observed in the experiments.
Since the state when the bubble plume swings asymmetric was discarded in the
experiments, a sampling bias might be the result because the symmetric state was
actively chosen, especially in the lower part of the riser. Nevertheless, a comparison of
the simulations with the experiments is meaningful because towards the top of the riser
all quantities are more or less homogenized over the cross section. Moreover, the
simulation results in the left and the right downcomer are almost equal, for all cases.
In Figure 7-15 the void fraction profiles are compared to the experimental data in the
riser at y=0.6 m. Still, an asymmetrical void fraction profile from the simulations is seen.
Considering this, the simulation results are in agreement with the experimental results.
However, for case 8 the void fraction profile is underpredicted in the center in general.
Figure 7-15 Void fraction at y=0.6 m in the riser.
In Figure 7-16 the obtained liquid velocity and the normal components of the
Reynolds stress tensor 𝑢′𝑢′ and 𝑣′𝑣′ are compared with the experiments. As expected, at
the bottom of the riser at y=0.2 m the liquid velocity profiles are asymmetrical, so a
comparison with the experiments is difficult. At y=0.6 m, the liquid velocity profiles are
becoming symmetrical but all simulations over-predict the liquid velocity.
The obtained 𝑣′𝑣′ profiles are more or less symmetrical in the riser at y=0.2 m. For
case 6 and case 8, the 𝑣′𝑣′ profiles are in good agreement with the experimental data. The
minima in the center as well as the maxima towards the inner walls are reproduced; the
7.3 CFD Simulations
119
heights of the maxima are almost equal compared to the experimental data whereas the
positions of the maxima are slightly displaced. The 𝑣′𝑣′ profile for case 4 is asymmetrical,
so this result might be difficult to compare with the experiment; nevertheless, the
quantity of the results is similar to the experiments in general. In the top region of the
riser at y=0.6 m, the simulations for case 6 and case 8 over-predict the 𝑣′𝑣′ profiles.
Case 4 is better reproduced quantitatively. Nevertheless, all simulation results show a
distinct peak towards the internal walls that is only seen for case 8 in the experiments.
y=0.2 m
y=0.6 m
Figure 7-16 Liquid velocity and normal components of the Reynolds stress tensor v′v′
and u′u′ (u is the horizontal liquid velocity) at y=0.2 m (left) and y=0.6 m (right) with
delineated internal walls.
The obtained u′u′ values at y=0.2 m are consistently too small in the riser compared
to the experiments for all cases. At y=0.6 m, however, the simulations are, despite a
slight underprediction, in agreement with the experimental data. In contrast to the
experiments, the obtained u′u′ values are almost equal at y=0.2 m and y=0.6 m.
In the downcomer at y=0.6 m, the u′u′ and v′v′ profiles are underpredicted in
general. The decreasing turbulence intensity to almost zero at y=0.2 m, however, is
reproduced in the simulations.
7 A complex validation case for CFD simulations: Airlift reactor
120
The vertical liquid velocity is overpredicted in the downcomer at y=0.6 m for case 6 and
case 8. However, at y=0.2 m the velocity obtained from the simulation is in agreement
with the experiments and for all cases almost equal as in the experiment.
The resolved and unresolved contribution of the turbulence modelling to
u′u′ and v′v′ are shown in Figure 7-17. In this context, the summation of the contributions
gives the total value. In the riser, the double-peaked v′v′ profiles are produced by the
resolved part at both heights; further, the resolved part is distinctly larger than the
unresolved part. In contrast, the influence of the resolved part is minor for the u′u′
profiles at both heights. The same trend is found in the downcomer at y=0.6 m, whereas
the resolved and unresolved part for v′v′ are similar.
y=0.2 m
y=0.6m
Figure 7-17 Resolved and unresolved normal components of the Reynolds stress tensor
v′v′ and u′u′ (u is the horizontal liquid velocity) for case 6 at y=0.2 m (left) and y=
0.6 m (right) with delineated internal walls.
The obtained gas void fraction along the downcomer is compared to the experiments
in Figure 7-18. Despite of case 4, the qualitative behavior is not well reproduced by the
simulations. In the experiments, the void fraction is slowly decreasing over the height, in
the simulations the void fraction is rapidly falling from very large values to almost zero
at the bottom of the downcomer. In addition, the void fraction profiles for case 6 and case
8 are not similar as observed in the experiments.
7.3 CFD Simulations
121
Figure 7-18 Void fraction along the downcomer.
The Sauter diameters are compared to the experiments in Figure 7-19. The Sauter
diameters are determined by averaging the (fixed) bubble sizes of the four bubble classes
with their void fraction by weight. Thus, the separation of the bubble classes along the
downcomer is seen. For case 4, the simulation underpredicts the Sauter diameter in the
upper region of the downcomer, since the void fraction becomes zero towards the bottom
region no bubble size could be calculated for the simulation here. The experimental
determined Sauter diameters for case 6 are well reproduced by the simulation; again, the
void fraction becomes zero towards the bottom region. For case 8, the bubble sizes are
distinctly underpredicted over the complete downcomer.
Figure 7-19 Bubble size seperation along the downcomer.
7 A complex validation case for CFD simulations: Airlift reactor
122
Despite the acceptable agreement of the Sauter diameter especially for case 6, the
void fraction obtained from the simulations are distinctly different. Several reason can be
identified; on the one hand, the discretization of the bubble size distribution might be too
coarse, on the other hand horizontal liquid velocity in the downcomer over-predicted as
shown in Figure 7-20. This higher velocity pushes the bubbles towards the internal walls
where they can rise up in the downcomer as discussed below (as well as discussed above
in the experimental section). Consequently, a rapid decrease of the void fraction along the
downcomer occurs as shown in Figure 7-20 on the right hand side.
Figure 7-20 Horizontal liquid velocity and gas void fraction in the downcomer compared
to experiments for case 6.
The Situation of the gas phase in the downcomer is shown in Figure 7-21 for case 6.
A strong vortex at the top of the downcomer is observed in the simulations as is seen from
the two dimensional gas fraction profile. In this, the bubbles are pulled deeper into the
downcomer but a majority is already rising up and leaving the downcomer, which is also
observed in the experiments. The positions where bubbles rise up are seen in the gas
velocity profile, the red color indicate a positive (upward) gas velocity. The bubbles
clearly rise up at the internal walls of the downcomer where they are pushed by the liquid
velocity. The liquid velocity at 𝑦=0.415 m is shown in the vector plot in which a large
vortex structure directed towards the internal walls is seen. This structure is propelled
by the bubbles themselves as seen in the horizontal slip velocity plot. The positive
(directed to the internal walls) horizontal slip velocity means that a positive bubble force
acts. From the baseline model, this force is the summation of the lift and turbulent
dispersion force, in which the lift force might be dominating.
7.4 Conclusion
123
Figure 7-21 Flow situation of the gas phase in the downcomer for case 6. From right to
left: The gas void fraction and the gas velocity in the center plane; the liquid velocity
vectors from the aerial view in the downcomer at 𝑦=0.415 m, the horizontal slip
velocity in the downcomer at 𝑦=0.415 m. All shown values are time averaged.
Consequently, the situation in the downcomer is very complex and not only the drag
force regulate the void fraction in the downcomer. The void fraction here might be very
sensitive to the bubble size since the coefficient of the lift force, which propels the
horizontal flow in the downcomer, strongly depends on it. Moreover, the turbulent
dispersion force might be important since bubbles that rise up lead to a gradient in the
void fraction profile that the dispersion counteracts.
7.4 Conclusion
The experiments are situated in the regime of a constant velocity in the downcomer as
described e.g. by van Benthum et al. (1999) or Law et al. (2013). Indeed, for case 6 and
case 8 the mean velocity in the downcomer along the centerline is 0.2 m/s and for case 4
around 0.18 m/s. Therefore, case 4 might be at the beginning of this regime. The velocity
profiles in the riser are similar for all cases.
Although similar velocity profiles obtained for the different flow rates, distinct
differences are found for the normal and cross components of the Reynolds stress tensor,
especially in the riser. The turbulence parameters might be a summation of larger scale
structures induced by the swinging bubble plume and bubble induced turbulence
phenomena. The CFD simulations reproduce the similar velocity profiles for the different
flow rates as well as the different normal components of the Reynolds stress tensor.
Although the vertical normal components are dominated by the resolved large-scale
structures, the horizontal normal component is dominated by the unresolved part.
Moreover, the influence of the resolved large-scale structures is smaller towards the top
7 A complex validation case for CFD simulations: Airlift reactor
124
region of the riser. In the downcomer, however, both parts are almost equal for both
directions. The resolved structures as well as the unresolved part, which might be
dominated by BIT, are important for the present setup. As discussed above, a steady state
RANS model as well as a LES or RSM model without a proper BIT modeling might not be
able to reproduce the experiments.
For all used sparger setups a distinct bubble plume is build up at the bottom of the
riser so that a distinct swinging motion is observed. In the experiments, the bubble plume
swings asymmetric on one side occasionally for an arbitrary time, but the swinging
motion is restored in general. In the CFD simulations, however, this metastable state of
an asymmetric bubble plume is overrated so that asymmetric time averaged results are
obtained, especially in the bottom of the riser. The origin of this effect is unclear, as
already discussed in Section 5. Moreover, asymmetric time averaged profiles can be
observed in many work dealing with CFD simulations of bubble plumes, e.g. (Masood &
Delgado 2014) and (Pourtousi et al. 2015).
A large vortex structure is seen at the top of the downcomer so that bubbles are pulled
in the downcomer at the reactor wall. Due to a horizontal liquid velocity, these bubbles
move to the internal wall where they can rise up because the vertical velocity is smaller
there. Both effects are well reproduced by the CFD simulations; from the simulations, the
horizontal liquid velocity is accelerated by the lateral bubble movement due to lift and
turbulent dispersion force. However, the resulting bubble movement to the internal walls
might be overpredicted in the simulations, because the void fraction is rapidly decreasing
along the downcomer, in contrast to a slow decreasing in the experiments. Since it is a
complex phenomenon, the reasons for this deviation are manifold. One likely reason is a
too coarse discretization of the bubble size distribution. In general, discrete bubble
classes might be critical for such cases in which a bubble size separation takes place since
locally very different bubble size distributions are seen in the experiment. At least,
besides a mesh size and time step study, a study for the discretization of the bubble size
distribution is needed. Another reason might be the missing consistency regarding clean
and contaminated systems in the bubble forces. Indeed, the water used in the present
experiment is reliably pure by using the naturally occurring micro bubbles as tracer for
particle tracking instead of additional tracer particles.
125
8 Lift force measurements in very low Morton number
systems and high bubble Reynolds number flows
Measuring the lift force on bubbles in a shear field is a very challenging task. Up to now,
the only working measurement concept that allows the direct measurement of the lift
force consists of a submerged rotating belt confined by walls (Kariyasaki 1987). The
rotating belt drives the flow and produces a shear field between the belt and a wall.
Tomiyama et al. (2002) used this experimental setup to measure the lift force on bubbles
in water/glycerin systems. With these experiments, it was shown that the lift force
coefficient changes it sign with increasing bubble diameter, also a well-known empirical
lift force correlation was obtained. Moreover, such experiments were repeated by
Dijkhuizen et al. (2010b) for polluted systems, in which a distinctly shear rate
dependency of the lift force was observed. Nevertheless, all experiments are conducted
in fluids with a large Morton number of around 𝑀𝑜>10−5, which results in a very low
bubble Reynolds number in the rage of 101 and laminar flow conditions. In contrast,
bubbly flows are often investigated in air/water like systems having a very low Morton
number of 𝑀𝑜=2.63×10−11 with bubble Reynolds numbers in the range of 103 and,
additionally, with a turbulent background flow. Therefore, determining the lift force in
such systems is highly desirable. Nevertheless, the change of sign of the lift force for large
bubbles in air/water systems was already shown by Lucas & Tomiyama (2011) by using
a large database of bubbly pipe flow experiments.
In the following section, a measuring concept is developed for measuring the lift force
in systems with a very low Morton number. The focus during the present work was on
the development of the new techniques and assembling the experimental setup.
Nevertheless, preliminary results are shown at which the measurement concept is
demonstrated.
8.1 Experimental setup
Measuring the lift force in low viscosity systems like water under turbulent conditions
poses some problems. Moreover, using a belt to produce a shear field is connected to
several extra problems. In the first place, the moving of the belt generates abrasion debris
that pollutes the measurement system (Dijkhuizen et al. 2010b). Furthermore, producing
a linear shear field with a moving belt in low viscosity systems over a necessary wide
channel under turbulent conditions might be hardly possible.
Another serious problem is that bubbles in turbulent low Morton number systems
rise in a non-straight motion so that a clear direction of movement cannot be seen. In
addition, the turbulent background distributes the bubbles over the channel, which has
to be separated from the migration induced by a lift force.
In the present work, a suitable shear flow is produced with an uneven aerated bubbly
flow, as shown in Figure 8-1, to overcome the problems that arise by using belts in low
viscosity systems. A circulation flow is induced by aerating the bubble column at one side
with large bubbles. Particularly, a linear shear field in a large vortex over a wider range
is obtained in the center. The lift force is determined for single bubbles generated at the
other side of the bubble column. They are pulled into the vortex where they experience
8 Lift force measurements in very low Morton number systems and high bubble
Reynolds number flows
126
the shear field. Here, the lift force is determined by investigating the movement of the
single bubbles in the vortex structure.
Figure 8-1 Setup to determine the lift force coefficient with the measuring area delineated
form 𝑧=0.5 𝑚 to 𝑧=0.65 𝑚 (left) and the flow structure from PIV measurements
(right).
Nevertheless, the bubbles still rise up in an arbitrary swinging motion overlaying the
lateral migration. To determine this migration clearly, time averaged three dimensional
void fraction distributions are determined. For this purpose, the rising bubbles are
photographed from the front and the side view. The three dimensional shape and position
of the bubbles are calculated from these two projections. Afterwards, the reconstructed
bubble is transferred to a grid discretizing the used bubble column. A representative gas
void fraction at one grid point is obtained by counting the times when gas is present at
this point. Assuming that the swinging rising motion of the bubbles is truly random, the
time averaged bubble trace is calculated by connecting the maxima of the void fraction in
the horizontal planes along the vertical Axis.
Furthermore, assuming that the turbulent dispersion force is symmetrical along the
obtained bubble trace, in particular that no turbophoresis effects occur, the bubble force
balance along the trace reads
0=𝑭
𝐵𝑢𝑜𝑦𝑎𝑛𝑐𝑦+𝑭
𝑉𝑖𝑟𝑡𝑢𝑎𝑙𝑀𝑎𝑠𝑠+𝑭
𝐷𝑟𝑎𝑔+𝑭
𝐿𝑖𝑓𝑡 .
(8-1)
The wall force in the 𝑥 and 𝑧 direction is neglected, moreover, the wall force is
assumed to be symmetrical in the y direction along the trace. The bar over the forces
indicates that the time-averaged forces fulfill this balance.
The time average of the buoyancy force and virtual mass force are trivial since all
quantities are constant for the present setup over time and statistically independent,
8.1 Experimental setup
127
respectively. Nevertheless, to calculate the time average values of the other forces some
assumptions have to be made. The time average is calculated by using the ensemble
average 〈⋅〉
𝑭
𝐷𝑟𝑎𝑔=〈𝑭𝐷𝑟𝑎𝑔〉=3
4𝑑𝐵𝜌𝑙〈𝐶𝐷|𝑢
𝐺−𝑢
𝐿|(𝑢
𝐺−𝑢
𝐿)〉
=3
4𝑑𝐵𝜌𝑙𝐶𝐷 |〈𝑢
𝐺〉−〈𝑢
𝐿〉| (〈𝑢
𝐺〉−〈𝑢
𝐿〉) .
(8-2)
The last equal sign is only valid if 𝐶𝐷, |𝑢
𝐺−𝑢
𝐿| and (𝑢
𝐺−𝑢
𝐿)𝑖 are statistically
independent. In particular, this can be interpreted that the drag force is statistically
independent of the flow condition. During the further procedure, this will be assumed.
Looking at the averaged lift force similar problems arise
𝑭
𝐿𝑖𝑓𝑡=〈𝑭𝐿𝑖𝑓𝑡〉=−𝜌𝑙〈𝐶𝐿(𝑢
𝐺−𝑢
𝐿)×𝑟𝑜𝑡(𝑢
𝐿)〉
=−𝜌𝑙𝐶𝐿(〈𝑢
𝐺〉−〈𝑢
𝐿〉)×𝑟𝑜𝑡(〈𝑢
𝐿〉) .
(8-3)
Again, it is assumed that 𝐶𝐿 and (𝑢
𝐺−𝑢
𝐿)×𝑟𝑜𝑡(𝑢
𝐿) are statistically independent during
one measurement. This might be fulfilled since the flow conditions are less changing
during the same experiment due to a low turbulence level as will be discussed below. In
addition, it is assumed that the slip velocity of the bubble is independent from the liquid
velocity shear field ((𝑢
𝐺−𝑢
𝐿) and 𝑟𝑜𝑡(𝑢
𝐿)), which was shown valid by Dijkhuizen et al.
(2010b) using DNS.
From these simplifications, the velocity field of the liquid and gas phase can be
determined separately. In addition, since only single bubbles are generated at the left side
of the bubble column (Figure 8-1) it is assumed that these single bubbles do not influence
the time averaged liquid velocity field. This assumption was found to be valid by
comparing velocity profiles obtained with and without the generated small bubbles.
Therefore, the liquid velocity is measured only once without the single bubbles in the test
section.
As will be discussed below the liquid velocity is measured with particle image
velocimetry (PIV), therefore the separation of the liquid and gas velocity is necessary
because the lift force experiments cannot be executed with PIV tracer particles since they
contaminate the bubble surface. This, in turn, implies the assumption that the generated
flow by the large bubbles on the right hand side is not influenced by the tracer particles.
After the liquid and gas velocity fields are determined, the lift force can be directly
calculated by using the force balance in Equation (8-1). In particular, only the horizontal
vector component is used for the calculation because the vertical direction is
superimposed with the relatively strong buoyancy force.
The direct calculation is an improvement compared to the method that is usually used
estimating the bubble velocity by calculating the derivation of the bubble trajectory in
vertical direction (Tomiyama 2002) (Bothe et al. 2006) (Dijkhuizen et al. 2010b). The
disadvantageous of such a method is that a bubble trajectory has to be smoothly
reconstructed to obtain a reliable derivation. Especially if an arbitrary movement is
superimposed on the rising bubble, this reconstruction is difficult and not well defined.
The entire bubble column is divided in five measuring areas in which the three-
dimensional void fraction is determined and three areas in which the velocity is
measured with PIV. However, only the results from a height of 0.5 m to 0.65 m including
8 Lift force measurements in very low Morton number systems and high bubble
Reynolds number flows
128
two void measuring areas and one PIV area will be used for the lift force calculations.
Three different gas volume flow rates for the large bubbles, which drive the flow, are used
to generate different shear fields. The lift force is evaluated for several bubble sizes that
are generated by using eight different single needles. The experimental inlet conditions
are summarized in Table 8-1. The used drilled out needle is different compared to the
other needles, which are described in Section 3.1. A flat needle with 1.5 mm inner
diameter is drilled out to a cone that flares to 3 mm diameter.
Needle inner diameter
Volume flow rate
Bubble diameter
Driving volume flow rate
0.2 mm
1 ml/min
2.22 mm
800 ml/min
-
-
0.3 mm
3 ml/min
2.25 mm
800 ml/min
1000 ml/min
-
0.5 mm
6 ml/min
2.76 mm
800 ml/min
-
-
0.6 mm
10 ml/min
2.95 mm
800 ml/min
1000 ml/min
-
0.7 mm
10 ml/min
3.16 mm
800 ml/min
-
-
0.9 mm
10 ml/min
3.77 mm
800 ml/min
1000 ml/min
1200 ml/min
1.5 mm
13 ml/min
6.17 mm
-
1000 ml/min
-
Drilled out 3 mm
10 ml/min
4.34 mm
800 ml/min
1000 ml/min
-
Table 8-1 Experimental conditions of the lift force experiments.
8.2 Methods
8.2.1 3D-Videometry
Approaches to determine the three dimensional void fraction with videometry are well
known and are used in various fields ranging from 3D particle tracking (Pereira et al.
2006) to 3D bubble identification with stereo cameras (Murai et al. 2001). Two
synchronized cameras are used and the bubble is reconstructed from the projected front
view of the bubble. The side view is used to position the recorded bubbles on the y-axis.
The bubble is reconstructed to three dimensions by defining an ellipse with the major
and minor axis from the front view of the bubble.
8.2 Methods
129
For the present setup, the identification of the bubbles in the side view is problematic
because the bubbles from the driving bubble plume interfere the view on the single
bubbles. To overcome this problem the driving bubble plume is not illuminated by the
backlight as shown in Figure 8-2.
Figure 8-2 Back illumination (yellow) of the experimental setup.
Nevertheless, the driving bubble plume is not completely dark since at least one side
is always open and scattered light illuminates the bubbles. The single bubbles that are
directly illuminated by the backlight, however, are reflecting the light very bright on the
backlight facing bubble tip as shown in Figure 8-3. This reflection is used to identify the
bubbles in the side view.
The position of the bubble in the side view is not directly accessible since only the left
tip is seen. Here, the position is determined by adding half the length of the major axis
(𝑑𝑀 in Figure 8-4), which is determined from the front view, to the left illuminated tip as
shown in Figure 8-4. This procedure is reasonable since the obtained void fraction
profiles in y-direction are symmetrical with a peak in the center (not shown).
After determining the position of the bubble, an ellipse defined by the minor and
major axis from the front view is discretized to a three dimensional grid of the measuring
area. If a grid point is inside the ellipse, the value of the grid point is increased by one.
This is repeated for every bubble. Over time, a three dimensional representation of the
void fraction in the bubble column is obtained. The recording time for every measuring
area for all used needle sizes was 15 minutes, resulting in 100 000 up to 400 000 recorded
bubbles for the 1.5 mm and the 0.3 mm inner diameter needle, respectively. The gird size
on which the bubbles are discretized is chosen to 0.1 mm.
8 Lift force measurements in very low Morton number systems and high bubble
Reynolds number flows
130
Figure 8-3 Side view (left), which is illuminated from left, and front view (right), which is
illuminated from the back. The red rectangles mark the same bubble and edge the
pictures used in Figure 8-4.
Figure 8-4 Determining the position of the bubble in the side view (left) by adding half
of the major axis dM determined from the front view (right) to the left tip.
8.2.2 Averaged bubble trace and spline interpolation
After a representative three dimensional void fraction function on a grid is determined,
the averaged bubble trace is calculated. For this purpose, the maximum of the
representative void fraction function for every horizontal plane on the grid is calculated
by determining the local minima of the absolute value of the derivative. To find the
correct local minima of the derivation, the neighborhood is defined as a region with a
sufficient count of bubbles. Particularly, if not enough bubbles are in the neighborhood of
a grid point, the derivation is not calculated. Moreover, the void fraction function is
smoothed for this purpose by using a constant convolution function with a size
8.2 Methods
131
of 3x3x3 mm. The determined maxima in the void fraction function are shown in Figure
8-5 on the left hand side. Indeed, a simplification was done for further calculation by
reducing the calculation area to a two dimensional cut through the center as shown in
Figure 8-5. This simplification reduced the amount of used grid points from 109 to 106.
This, however, implies the assumption that the bubble path is situated in the center of the
bubble column which was found to be reasonable.
As mentioned above, the experimental setup was divided in five measuring sections
and only two sections were used to evaluate the lift force coefficient. The border of the
sections is at 𝑦=0.6 𝑚; the void fraction function is averaged in the overlapping region.
In Figure 8-5 the 0.6 m line is delineated with a straight dashed line and the overlapping
region is marked with a red arrow.
Figure 8-5 The void fraction function on a two dimensional cut in the center between z=
0.4 m and z=0.7 m for the drilled out needle and 800 ml/min driving volume flow rate.
The color from red to blue indicates the void fraction function from high to low values,
respectively. The left picture shows the determined local maxima and the right picture
the spline that is taken as bubble trace.
In general, a discontinuous trace is obtained if the determined maxima in the void
fraction function are connected along the vertical axis. To get a continuous and
sufficiently smooth trace, a B-Spline curve is determined with the algorithm proposed by
Liu et al. (2005). The algorithm is simplified by introducing a minimum energy criterion,
y=0.6 m
8 Lift force measurements in very low Morton number systems and high bubble
Reynolds number flows
132
in which the second order derivation of the B-Spline is minimized. The performance of
the B-Spline approximation is demonstrated in Figure 8-5 on the right hand side.
It should be noted that in fact a great effort was done to reconstruct the bubble trace
with B-Spline functions in a way that the first derivation is meaningful to compare the
present method with the method used in previous work. Promisingly, by minimizing the
energy represented by the second derivation of the B-Spline, very good results were
obtained. Nevertheless, calculating the lift force directly with the velocity of the gas phase
is much more reliable compared to using the angle of the bubble trace as done in previous
work for single bubble trajectories. Especially, the horizontal bubble velocity is very small
compared to the vertical velocity so that small deviations or uncertainties have a big
effect on the first derivation. In addition, the method that uses the first derivation
includes the assumption of a simple shear flow without a gradient in vertical direction,
which is not fulfilled in the present setup satisfactorily.
8.2.3 Liquid phase velocity
The liquid velocity is determined with PIV. The velocity is measured once using only the
driving flow. The different generated shear rates 𝜕𝑣𝑧𝜕𝑥
⁄ are around 2.45 1/s for 800
ml/min, around 2.65 1/s for 1000 ml/min and around 2.9 1/s for 1200 ml/min.
The averaged velocity field that is generated by the 800 ml/min flow is shown
simplified in Figure 8-6. Clearly, the shear field ranging from the left wall at x=0 m to
the driving flow at the right wall, which is situated between x=0.2 m and x=0.25 m, is
seen. The measuring window shows the lower part of the vortex created along the bubble
column. The size of the vortex can be adjusted by the water level of the bubble column,
which was preliminary optimized with CFD simulations.
Figure 8-6 Liquid velocity field determined with PIV for 800 ml/min. The left reactor wall
is at x=0 m, the driving flow at the right wall is between x=0.2 m and x=0.25 m.
8.2 Methods
133
The normal components of the Reynolds stress tensor 𝑢′𝑢′ and 𝑣′𝑣′ are shown in
Figure 8-7. From the two dimensional distribution for 800 ml/min gas volume flow rate
it is seen that larger values are present in the downflow region. Moreover, the 𝑣′𝑣′ values
are high at the left wall and near the driving bubble plume on the right-hand side; in these
regions, however, no bubble trajectories are obtained. The values are increasing with
increasing gas volume flow rate, which drives the flow.
Figure 8-7 Normal components of the Reynolds stress tensor u′u′ and v′v′. Top: Two
dimensional distribution of u′u′ (left) and v′v′ (right) for 800 ml/min gas volume flow
rate. Bottom: Profile at y=0.6 m for all gas volume flow rates.
In the vortex, the vertical velocity is almost constant over height as demonstrated in
Figure 8-8 on the left-hand side for the 800 ml/min case. The shear rate 𝜕𝑣𝑧𝜕𝑥
⁄ is also
almost constant over height in the measuring section between x=0.1 m and x=0.15 m.
The horizontal velocity, however, is changing over height because of the round shape of
the generated vortex, which is not avoidable.
8 Lift force measurements in very low Morton number systems and high bubble
Reynolds number flows
134
Figure 8-8 Vertical and horizontal liquid velocity along the x-axis for different heights
generated by the 800 ml/min driving flow. Left the vertical velocity and right the
horizontal velocity.
The bubble traces for the different bubble sizes are not the same. Thus, the different
bubble sizes run through the vortex on different paths and, therefore, experience a
slightly other flow field of the vortex each. The velocities and shear rates along the traces
over height for the 0.3 mm inner diameter needle (2.25 mm bubble size) and the drilled
out needle (4.13 mm bubbles size) are shown in Figure 8-9.
Looking at the vertical velocity, the smaller bubbles experience a higher negative
velocity compared to the larger bubbles because the smaller bubbles are situated more
left in the vortex. Indeed, the bubble trace has a parabolic shape as indicated in Figure 8-5
so the vertical velocity is for both bubble sizes not constant over height. Furthermore, the
vortex is slightly inclined what is seen by the shifting of the zero of the vertical velocity
with increasing height in Figure 8-8. The experienced shear rate 𝜕𝑣 𝜕𝑥
⁄ is slightly different
for the different bubble sizes as shown in Figure 8-9. The shear rate 𝜕𝑣 𝜕𝑧
⁄ (not shown) is
very small for both traces and constant over height.
Looking at the horizontal velocity, the experienced velocity and largest shear rate
𝜕𝑣 𝜕𝑥
⁄ component is almost equal for both traces as well as the shear rate 𝜕𝑢 𝜕𝑧
⁄ (not
shown).
8.2.4 Bubble Size and gas phase velocity
The bubble sizes as well as the major and minor axis are determined from the front view
by using edge-detecting algorithms as described in Section 3.2. Tomiyama et al. (2002)
formulated an empirical lift force coefficient as a function of the maximum axis of the
bubble perpendicular to the flow, in the following this is stated as major axis or is denoted
with ⊥. This axis, however, is in general not known, so the empirical correlation by Wellek
et al. (1966) is used
𝑑𝑀𝑎𝑗𝑜𝑟
𝑑𝑀𝑖𝑛𝑜𝑟=1+0.163𝐸𝑜0.757 ,
(8-4)
𝑑𝑀𝑎𝑗𝑜𝑟=𝑑𝐵√1+0.163𝐸𝑜0.757
3 ,
(8-5)
with 𝑑𝐵 the spherical equivalent diameter.
8.2 Methods
135
a)
b)
c)
d)
Figure 8-9 Fluid velocities along two bubble traces over height for 800 ml/min driving
flow.
The empirical Wellek correlation, however, is derived for fully contaminated flow
without a turbulent background flow. The results from the lift force experiments are
compared to the Wellek correlation in Figure 8-10. The ratio of major to minor axis is
clearly underpredicted by the correlation; the bubbles have a stronger ellipsoid shape.
8 Lift force measurements in very low Morton number systems and high bubble
Reynolds number flows
136
Figure 8-10 Comparison of the empirical Wellek correlation with the results in
air/water with turbulent background flow.
The bubbles from different experiments are shown in Figure 8-11. The shape of small
bubbles is clearly ellipsoidal in contrast to the Wellek correlation, which predicts an
almost spherical shape with 𝑑𝑀𝑎𝑗𝑜𝑟≈𝑑𝑀𝑖𝑛𝑜𝑟. A significant difference among the different
driving gas volume flow rates was not observed.
𝑑𝐵=2.22 𝑚𝑚;𝐸𝑜=0.7 (ID 0.2 mm)
𝑑𝐵=2.95 𝑚𝑚;𝐸𝑜=1.2 (ID 0.6 mm)
𝑑𝐵=4.43;𝐸𝑜=2.5 (Drilled out needle)
𝑑𝐵=6.18 𝑚𝑚;𝐸𝑜=5.21 (ID 1.5 mm)
Figure 8-11 Different bubble sizes in the lift force experiment (All pictures have the same
scale).
The number density functions of the determined spherical equivalent bubble
diameter and the major axis are plotted in Figure 8-12 for 0.8 l/min driving gas flow rate.
The used bubble sizes are averaged along the single bubble tracks. The density functions
are discretized with 0.25 mm. Except for the 1.5 mm needle, the generated bubbles have
a low variance in the range of 0.25 mm.
The density function of the large 6.18 mm bubbles generated by the 1.5 mm needle is
clearly asymmetric for both values. In the experiments, it was observed that the bubbles
3 mm
8.2 Methods
137
sometimes break up in the shear field and sometimes the bubble generation at the needle
was instable.
Figure 8-12 Number density function of the spherical equivalent diameter and major
axis for 0.8 𝑙/𝑚𝑖𝑛 driving flow, except for the 1.5 mm needle 1.0 𝑙/𝑚𝑖𝑛 is plotted.
The velocity of the gas phase is calculated by tracking the bubbles in the same way as
described in Section 3.4.2. The used measuring frequency was 25 Hz. The calculated
relative velocities are compared to results from literature in Figure 8-13.
The relative velocities that are determined from experiments are situated between
the terminal velocities obtained from the drag force model of Bozzano & Dente (2001)
and Tomiyama et al. (1998) for pure systems. Therefore, it is reasonable to assume that
the used water is pure. The drag force coefficient, which is needed to solve the force
balance equation, is calculated by the experimental determined relative velocity. For the
use in the force balance, it is assumed that these obtained drag coefficients are isotropic
as also assumed in previous work, e.g. by Tomiyama et al. (2002).
Furthermore, the velocity field of the gas phase has to be determined. Indeed, the gas
velocity used for calculating the lift force coefficient is only calculated along the
determined averaged bubble traces with a round averaging area. The results for the
determined (absolute) gas phase velocity field on a rectangular mesh, however, are
demonstrated in Figure 8-14 for the 0.3 mm needle and the drilled out needle at a driving
flow of 800 ml/min.
8 Lift force measurements in very low Morton number systems and high bubble
Reynolds number flows
138
Figure 8-13 Determined terminal velocity.
Figure 8-14 Absolute gas phase velocity field with vertical velocity as color for 800
ml/min driving flow. The left reactor wall is at x=0 m, the driving flow at the right wall
is between x=0.2 m and x=0.25 m. Left: The velocity field obtained with the 0.3 mm
inner diameter needle generating 2.27 mm bubbles. Right: The velocity field obtained
with the drilled out needle generating 4.13 mm bubbles.
8.3 Results for air/water
139
In Figure 8-14 only the velocity vectors that are calculated by a sufficient amount of
tracked bubbles are shown. Clearly, the shear field of the underlying liquid velocity is
seen. The bubbles follow the velocity field shown in Figure 8-6. However, comparing the
velocity field of the small bubbles (2.25 mm) generated by the 0.3 mm needle and the
large bubbles (4.3 mm) generated by the drilled out needle, some differences are seen. At
first, the large bubbles are dragged further towards the right wall. Moreover, the small
bubbles move towards the left wall (towards x=0 m) whereas the large bubbles are
rising straight up. Nevertheless, since along both traces a positive liquid velocity is
recorded (see Figure 8-9) the relative horizontal gas velocity is negative (in the direction
of 𝑥=0 𝑚) for both traces. This horizontal velocity of the gas phase, which is directed
opposite to the positive liquid velocity, is the result of a lateral force that is assumed the
lift force. The distinctly higher negative horizontal velocity of the smaller bubbles, which
is indicated by the velocity vectors pointing towards the negative x-axis, might be the
result from a higher lift force. In contrast, plotting the (absolute) gas phase velocity field
of the largest bubbles (6.17 mm) (not shown) the velocity vectors point towards the
positive x-axis.
8.3 Results for air/water
The lift force coefficient is calculated with the velocity data of the liquid and gas phase
along the averaged traces, as discussed in the previous section. Using the force balance
given in Equation (8-1) the lift force coefficient is the only unknown remaining. The lift
force coefficient is calculated by using the horizontal component of the force balance.
The lift force coefficients are averaged between a height of 𝑦=0.5 𝑚 and 𝑦=0.65 𝑚.
In this area, the bubbles are less accelerated due to small liquid velocity gradients 𝑑𝑣𝑙,𝑖/
𝑑𝑥𝑖. In the lower and upper sections, these are large; the lift force coefficients obtained in
these sections might not be comparable to the lift force coefficients obtained in other
work with a not accelerated bubbles motion due to the flow. The results are shown in
Figure 8-15. Clearly, the lift force coefficient is not constant along the bubble traces, which
might indicate an insufficient amount of tracked bubbles in general.
The available data do not allow an obvious statement regarding the dependency of
the lift force coefficient of the shear rate strength as shown in Figure 8-16. The lift force
coefficient is plotted against the modified Eötvös number
𝐸𝑜⊥=Δ𝜌𝑔𝑑⊥
𝜎 ,
(8-6)
with 𝑑⊥ the axis perpendicular to the flow, which is assumed equal with the major axis.
The amount of experiments is too small in general so that the lift force coefficients are
scattered and no clear trend is found. From previous findings in high Morton number
systems under laminar conditions (Tomiyama et al. 2002) (Dijkhuizen et al. 2010b),
however, also no dependency is found. Nevertheless, a high shear rate is connected to a
high turbulence intensity in the present setup. Thus, no clear dependency of the lift force
regarding the turbulence intensity is seen as well, or possible effects cancel each other
out.
8 Lift force measurements in very low Morton number systems and high bubble
Reynolds number flows
140
ID=0.2 mm 𝑑𝐵=2.22 𝑚𝑚 𝑑⊥=2.86 𝑚𝑚
ID=0.3 mm 𝑑𝐵=2.25 𝑚𝑚 𝑑⊥=2.90 𝑚𝑚
ID=0.5 mm 𝑑𝐵=2.76 𝑚𝑚 𝑑⊥=3.61 𝑚𝑚
ID=0.6 mm 𝑑𝐵=2.95 𝑚𝑚 𝑑⊥=3.96 𝑚𝑚
ID=0.7 mm 𝑑𝐵=3.16 𝑚𝑚 𝑑⊥=4.28 𝑚𝑚
ID=0.9 mm 𝑑𝐵=3.77 𝑚𝑚 𝑑⊥=5.27 𝑚𝑚
Drilled 𝑑𝐵=4.34 𝑚𝑚 𝑑⊥=6.19 𝑚𝑚
ID=1.5 mm 𝑑𝐵=6.17 𝑚𝑚 𝑑⊥=9.27 𝑚𝑚
Figure 8-15 Lift force coefficient along the averaged traces.
8.3 Results for air/water
141
Figure 8-16 Averaged lift force coefficients along the averaged bubble traces for different
shear rates. The needle diameter is written at the points.
The results are compared to results from literature (Tomiyama et al. 2002)
(Dijkhuizen et al. 2010b) in Figure 8-17 for the Eötvös number calculated with the
spherical equivalent diameter and with the perpendicular flow axis. The lift force
coefficients obtained from the present experiments are falling to negative values with
rising Eötvös numbers. This falling is consistent to the results obtained in the other work.
Apparently, from the actual measurements a peak is seen at an Eötvös number of around
1.2, after the peak the lift force is tending to smaller values with decreasing Eötvös
numbers.
Comparing the results of the present study with the results of the well-known
Tomiyama lift force model, distinct differences are observed for the Eötvös number graph
but similarities for the modified Eötvös number graph. In contrast, the present
experiments fit the DNS of Dijkhuizen et al. (2010b) very well for both, the normal and
modified Eötvös number. Overall, the DNS, the present experiments and the empirical
Tomiyama lift force model agree for the modified Eötvös number.
Whereas the graph for the normal Eötvös number was obtained for the present
experiments and the DNS from directly measured variables, the correlation of Wellek et
al. (1966) was used by Tomiyama et al. (2002) to calculate the spherical equivalent
diameter from the (measured) perpendicular flow axis. In Section 8.2.4 it was shown that
the Wellek correlation might not be applicable to pure and/or turbulent air/water flows.
Thus, it is reasonable that such large deviations regarding the lift force coefficient occur
for the normal Eötvös number. Moreover, since the present experiments agree well with
the DNS of Dijkhuizen et al., which were also done for pure systems, for both Eötvös
numbers it is reasonable to assume that the Wellek correlation is incorrect. The rotating
belt generating the shear field in the experiments of Tomiyama et al. might create
abrasion that lead to a contamination of the liquid as pointed out by Dijkhuizen et al.
(2010b). Speculatively, this contamination might be the reason why Tomiyama et al. used
the Wellek correlation formulated for contaminated flows (Tomiyama et al. 2002).
8 Lift force measurements in very low Morton number systems and high bubble
Reynolds number flows
142
Figure 8-17 Results of the present lift force measurements in turbulent air/water flow
(Morton number of around 2.63⋅10−11) compared to results from the literature for
different Morton numbers. The DNS by Dijkhuizen et al. (2010b) and the experiments of
Tomiyama et al. (2002) are conducted under laminar conditions.
Nevertheless, the agreement regarding the modified Eötvös number of the present
experiment, the DNS and the empirical Tomiyama model is suprisingly good considering
the very different flow conditions. In the first place, the Morton numbers used in the
Tomiyama experiment are distinctly higher (in the range of 10−3 to 10−6) compared to
the present experiment in air/water (Mo=2.63⋅10−11). In addition, the experiments of
Tomiyama et al. (2002) are conducted under laminar flow conditions with bubble
Reynolds numbers below 102 whereas turbulent flow conditions with bubble Reynolds
numbers ranging from 680 to 1500. The shear rates used in the present study are
comparable with the shear rates used by Tomiyama et al.
The used bubble Reynolds numbers for the DNS of Dijkhuizen et al. (2010b) are in
the range of 10 to 1609 (1609 only for very large bubbles, which are not shown here) and
8.4 Discussion and Conclusions
143
comparable to the present experiment. In addition, the used Morton numbers are in the
range of an air/water system. The background flow, however, was laminar. The amount
of DNS is limited, especially in the distorted regime in which the bubbles are wobbling.
This leads to a poor statistic since the wobbling motion is random. Therefore, the DNS lift
force coefficients are scattering because many DNS are needed for reliable results (Bothe
et al. 2006).
8.4 Discussion and Conclusions
With the developed measuring concept, the lift force was successfully determined in a
system with a very low Morton number under turbulent conditions by using long-term
measurements. This is the first time that the lift force is direct measureable in such
systems. In contrast to other measuring concepts, the here presented concept do not have
any moving parts like moving belts that tend to contaminate the measuring system.
Therefore, it is a very simple and reliable method and, further, can be applied to every
chemical system.
Comparing to results from the literature (Tomiyama et al. 2002) (Dijkhuizen et al.
2010b) a very good agreement is reached. From the present measurements, it is
confirmed that the lift force coefficient is significant and is likely changing its sign in
turbulent flow and in very low Morton number systems.
Nevertheless, more experiments are needed to confirm the findings quantitatively,
especially the gap between 𝐸𝑜=2.5 and 𝐸𝑜=5.2 has to be filled up. In addition, the
measuring effort has to be decreased; now, the complete bubble column is investigated,
which results in a measuring time of around 80 hours. Focusing only on one measuring
area might be sufficient. Moreover, the scattering of the determined lift force coefficients
has to be decreased, which is caused by a poor statistic on the gas phase side. This is a
general problem because the lift force can only be evaluated on the center plane since the
liquid velocity is only known there. As a result, the bubbles not situated in this plane
cannot be taken into account. Assuming an equal velocity, an extension to a narrow three-
dimensional region around the center plane might be justified to increase the bubble
count. Nonetheless, the measuring time has to be increased in general to get accurate
data.
144
145
9 Summary
The complex nature of bubbly flows limits by now the predictive capabilities of two-fluid
CFD approaches. Applying such CFD simulations to bubbly flows for unknown
experimental setups, however, is highly preferable for process design. Since the physical
phenomena are independent of the application, an important step toward predictive CFD
simulations are baseline models in which all closure models and all constants are fixed.
Such models have to reflect the underlying physics. Based on this idea, a baseline model
developed at the Helmholtz-Zentrum Dresden-Rossendorf mainly basing on
experimental data for bubbly pipe flows is validated to bubble columns, bubble plumes
and airlift reactors that are relevant in chemical and biological engineering applications.
Such applications comprise buoyancy driven bubbly flows that often show dynamics
on the scale of the used facility. It was shown that this large-scale flow phenomena can be
well described with the baseline model in combination with the unsteady Reynolds-
averaged Navier–Stokes equations (URANS) approach. The advantage of the URANS
approach is that problems in which such large scales appear (mainly heterogeneous
bubbly flows and partial aerated reactors) as well as problems that are dominated by the
small scales (mainly homogenous bubbly flows) can be well predicted. This was shown
by validating a large variety of experimental setups with respect to time averaged but
also transient experimental results. However, in literature there is a lack of experimental
data that are suitable for comprehensive CFD validation, in particular the bubble size
distribution and reliable liquid velocity measurements in higher void fractions are
missing.
In order to conduct own CFD-grade experiments measuring techniques were
developed to determine the bubble size distribution in bubble clusters and to measure
the liquid velocities without contaminating the flow by tracer particles for high void
fractions. The main idea for the bubble size measurements is to follow the bubbles over
a distinct period so that bubbles that are overlapped by others can be clearly seen in one
of the images since the shape of a bubble cluster is steadily changing. In combination with
an edge detector, this method gave reproducible and reliable results in complex bubbly
flows.
Measuring the liquid velocity in bubbly flows is very difficult in general. Independent
of the used measuring technique a sampling bias that had not been yet published is
described in the present work. In particular, the sampling bias was shown to be distinct
in bubbly flows for particle image and particle tracking velocimetry. Using the developed
hold processor, very similar averaged liquid velocities and Reynolds-stresses could be
obtained with both measuring methods. Moreover, common tracer particles, which are
necessary for measuring the liquid velocities, contaminate the bubble interface.
Contaminated bubbles can show a different behavior compared to clean bubbles so that
a modeling of such is problematic since the grade of contamination due to the presence
of tracer particles is not known. In order to obtain experimental data in reliable clean
systems, micro bubbles that are naturally occurring in bubbly flows were used as tracer
particles. It was shown that micro bubbles up to 300 µm are feasible for liquid velocity
measurements for the setups used in the present work.
9 Summary
146
A CFD-grade airlift experiment was conducted, for which the bubble size, void
fraction, liquid velocities and normal Reynolds-stress components were determined at
two different heights in the riser and the downcomer as well as along the downcomer. In
addition, the transient behavior of the airlift was studied since a distinct bubble plume
was observed. The complexity of the experiments is intentional reduced by using gas
volume flow rates and a sparger with which break-up and coalescence processes can be
neglected. The URANS CFD-simulations gave good results in the riser; however, the void
fraction in the downcomer does not fit the experimental observations. The reason for that
might be a too strong predicted migration of the bubbles toward the internal walls at
which the bubbles can rise up and escape the downcomer.
Besides a validation of closure models, a new measuring concept to determine the lift
force on bubbles in systems with a very low Morton number and under turbulent
conditions was developed. In particular, the new concept does not have any moving parts,
which other measuring concepts usually have, that causes some problems with
contamination due to abbreviation reported in the literature. Preliminary studies with
the new concept in air/water confirmed that the lift force changes its sign for larger
bubbles and the lift force coefficient is in the range of experiments conducted in high
Morton number systems under laminar conditions. In comparison with DNS results from
the literature a good agreement was reached.
147
References
Adoua, R., Legendre, D. & Magnaudet, J., 2009. Reversal of the lift force on an oblate
bubble in a weakly viscous linear shear flow. J. Fluid Mech., Volume 628, p. 23.
Ansys, 2013. CFX 14.5 Manual: CFX-Solver Modeling Guide.. Canonsburg: Ansys, Inc..
Aris, R., 1962. Vectors, Tensors and the Basic Equations of Fluid Mechanics. Englewood
Cliffs: Prentice-Hall.
Auton, T. R., 1987. The lift force on a spherical body in a rotational flow. J. Fluid Mech.,
Volume 183, p. 199.
Badreddine, H., Sato, Y., Niceno, B. & Prasser, H.-M., 2015. Finite size Lagrangian particle
tracking approach to simulate dispersed bubbly flows. Chemical Engineering Science ,
122(0), pp. 321-335.
Bech, K., 2005. Dynamic simulation of a 2D bubble column. Chemical Engineering
Science, 60(19), pp. 5294-5304.
Becker, S., Sokolichin, A. & Eigenberger, G., 1994. Gas-liquid flow in bubble columns and
loop reactors: Part II. Comparison of detailed experiments and flow simulations.
Chemical Engineering Science, 49(24, Part 2), pp. 5747-5762.
Besbes, S. et al., 2015. PIV measurements and Eulerian-agrangian simulations of the
unsteady gas-liquid flow in a needle sparger rectangular bubble column. Chemical
Engineering Science, 126(0), pp. 560-572.
bin Mohd Akbar , M. H., Hayashi , K., Hosokawa , S. & Tomiyama, A., 2012. Bubble
tracking simulation of bubble-induced pseudoturbulence. Multiphase Science and
Technology, 24(3), pp. 197-222.
Bitog, J. et al., 2011. Application of computational fluid dynamics for modeling and
designing photobioreactors for microalgae production: A review. Computers and
Electronics in Agriculture , 76(2), pp. 131-147.
Boegman, L. & Sleep, S., 2012. Feasibility of Bubble Plume Destratification of Central
Lake Erie. Journal of Hydraulic Engineering, 138(11), pp. 985-989.
Bolotnov, I. A. et al., 2011. Detached direct numerical simulations of turbulent two-
phase bubbly channel flow. International Journal of Multiphase Flow , 37(6), pp. 647-
659.
Bothe, D., Schmidtke, M. & Warnecke, H.-J., 2006. VOF-Simulation of the Lift Force for
Single Bubbles in a Simple Shear Flow. Chem. Eng. Technol., Volume 29, p. 1048.
Bourgoin, M. et al., 2011. Turbulent transport of finite sized material particles. Journal of
Physics: Conference Series, Volume 318.
Bozzano, G. & Dente, M., 2001. Shape and terminal velocity of single bubble motion: a
novel approach. Computers and Chemical Engineering, Volume 25, p. 571–576.
148
Bröder, D. & Sommerfeld, M., 2007. Planar shadow image velocimetry for the analysis of
the hydrodynamics in bubbly flows. Meas. Sci. Technol., Volume 18, p. 2513–2528.
Brücker, C., 2000. PIV IN TWO-PHASE FLOWS. Von Karman Institute for Fluid Lecture
Series 2000-01, 17-21 January.
Buckingham, E., 1914. On Physically Similar Systems; Illustrations of the Use of
Dimensional Equations. Phys. Rev., Volume 4, p. 345–376.
Burns, A., Frank, T., Hamill, I. & Shi, J.-M., 2004. The Favre Averaged Drag Model for
Turbulent Dispersion in Eulerian Multi-Phase Flows. Yokohama, Japan,, 5th International
Conference on Multiphase Flow.
Buwa, V. V. & Ranade, V. V., 2004. Characterization of dynamics of gas-liquid flows in
rectangular bubble columns. AIChE Journal, 50(10), pp. 2394-2407.
Calzavarini, E., Cencini, M., Lohse, D. & Toschi, F., 2008. Quantifying Turbulence-Induced
Segregation of Inertial Particles. Phys. Rev. Lett., Aug, Volume 101, pp. 1-5.
Calzavarini, E. et al., 2011. Impact of trailing wake drag on the statistical properties and
dynamics of finite-sized particle in turbulence. Physica D: Nonlinear Phenomena, 241(3),
pp. 237-244.
Canny, J., 1986. A Computational Approach To Edge Detection. IEEE Trans. Pattern
Analysis and Machine Intelligence, 8(6), p. 679–698.
Cloete, S., Olsen, J. E. & Skjetne, P., 2009. CFDmodeling of plume and free surface
behavior resulting from a sub-sea gas release. Applied Ocean Research, 31(3), pp. 220-
225.
da Silva, M. J., 2008. Impedance Sensors for Fast Multiphase Flow. Technische Universität
Dresden: Fakultät Elektrotechnik und Informationstechnik der Technischen Universität
Dresden .
Da Silva, M., Schleicher, E. & Hampel, U., 2007. A Novel Needle Probe Based on High-
Speed Complex Permittivity Measurements for Investigation of Dynamic Fluid Flows.
Instrumentation and Measurement, IEEE Transactions on, Aug, 56(4), pp. 1249-1256.
Dabiri, S., Lu, J. & Tryggvason, G., 2013. Transition between regimes of a vertical channel
bubbly upflow due to bubble. Physics of Fluids, Volume 25.
de Bertodano, M. L., Lahey, R. T. & Jones, O. C., 1994. Development of a k-epsilon Model
for Bubbly Two-Phase Flow. Journal of Fluids Engineering, Volume 116, p. 128.
de Bertodano, M. L., Lee, S.-J., R. T. Lahey, J. & Drew, D. A., 1990. The Prediction of Two-
Phase Turbulence and Phase Distribution Phenomena Using a Reynolds Stress Model. J.
Fluids Eng., Volume 112, p. 107.
Deen, N., 2001. PHD thesis: An Experimental and Computational Study of Fluid Dynamics
in Gas-Liquid Chemical Reactors. Esbjerg, Denmark: The Faculty of Engineering and
Science, Aalborg University.
149
Deen, N. G., Solberg, T. & Hjertager, B., 2001. Large eddy simulation of the Gas–Liquid
flow in a square cross-sectioned bubble column. Chemical Engineering Science, 56(21-
22), p. 6341–6349.
Deen, N. G., Westerweel, J. & Delnoij, E., 2002. Two-Phase PIV in Bubbly Flows: Status
and Trends. Chem. Eng. Technol. , Volume 25.
Deen, N., Solberg, T. & Hjertager, B., 2001. Large eddy simulation of the Gas-Liquid flow
in a square cross-sectioned bubble column. Chemical Engineering Science, Volume 56,
pp. 6341-6349.
Delhaye, J. M., 1974. Jump conditions and entropy sources in two-phase systems. Local
instant formulation. International Journal of Multiphase Flow, 1(3), pp. 395-409.
Delnoij, E., Kuipers, J. A. M., van Swaaij, W. P. M. & Westerweel, J., 2000. Measurement of
gas-liquid two-phase flow in bubble columns using ensemble correlation PIV. Chemical
Engineering Science , Volume 55, pp. 3385 - 3395.
Delnoij, E., Lammers, F., Kuipers, J. & van Swaaij, W., 1997. Dynamic simulation of
dispersed gas-liquid two-phase flow using a discrete bubble model. Chemical
Engineering Science , 52(9), pp. 1429-1458.
Diaz, M., Montes, F. & Galan, M., 2009. Influence of the lift force closures on the
numerical simulation of bubble plumes in a rectangular bubble column. Chemical
Engineering Science, Volume 64, pp. 930-944.
Dijkhuizen, W., Roghair, I., Van Sint Annaland, M. & Kuipers, J. A., 2010a. DNS of gas
bubbles behaviour using an improved 3D front tracking. Chemical Engineering Science,
Volume 65, p. 1415–1426.
Dijkhuizen, W., van Sint Annaland, M. & Kuipers, J. A., 2010b. Numerical and
experimental investigation of the lift force on single bubbles. Chemical Engineering
Science, Volume 65, p. 1274–1287.
Dimotakis, P., Debussy, F. D. & Koochesfahani, M. H., 1981. Particle streak velocity field
measurements in a two-dimensional mixing layer. pp. 995-999.
Edwards, R. V., 1987. Report of the Special Panel on Statistical Particle Bias Problems in
Laser Anemometry. J. Fluids Eng., 109(2), pp. 89-93.
Ervin, E. A. & Tryggvason, G., 1997. The Rise of Bubbles in a Vertical Shear Flow. J. Fluids
Eng., Volume 119, p. 443.
Feng, Y., Goree, J. & Liu, B., 2007. Accurate particle position measurement from images.
Review of Scientific Instruments, 78(5).
Feng, Y., Goree, J. & Liu, B., 2011. Errors in particle tracking velocimetry with high-speed
cameras. Rev. Sci. Instrum., Volume 82.
Fernandes, B. D., Dragone, G. M., Teixeira, J. A. & Vicente, A. A., 2010. Light Regime
Characterization in an Airlift Photobioreactor for Production of Microalgae with High
Starch Content. Appl Biochem Biotechnol, pp. 218-226.
150
Ferziger, J. H. & Peric, M., 2002. Computational Methods for Fluid Dynamics. 3rd ed.
Berlin Heidelberg New York: Springer-Verlag.
Fish, F. E., Legac, P., Williams, T. M. & Wei, T., 2014. Measurement of hydrodynamic force
generation by swimming dolphins using bubble DPIV. The Journal of Experimental
Biology, 217(2), pp. 252-260.
Frank, T. et al., 2008. Validation of CFD models for mono- and polydisperse air–water
two-phase flows in pipes. Nuclear Engineering and Design, Volume 238, pp. 647-659.
Fujiwara, A., Minato, D. & Hishida, K., 2004. Effect of bubble diameter on modification of
turbulence in an upward pipe flow. International Journal of Heat and Fluid Flow, 25(3),
p. 481–488.
Fukuta, M., Takagi, S. & Matsumoto, Y., 2008. Numerical study on the shear-induced lift
force acting on a spherical bubble in aqueous surfactant solutions. Phys. Fluids, Volume
20, p. 040704.
Garcia, C. & Garcia, M., 2006. Characterization of flow turbulence in large-scale bubble-
plume experiments. Experiments in Fluids, 41(1), pp. 91-101.
Ghasemi, H. & Hosseini, S. H., 2012. Investigation of hydrodynamics and transition
regime in an internal loop airlift reactor using CFD. Brazilian Journal of Chemical
Engineering, 12, Volume 29, pp. 821-833.
Grace, J. R., Wairegi, T. & Nguyen, T. H., 1976. Shapes and Velocities of Single Drops and
Bubbles Moving Freely Through Immiscible Liquids. Transactions of the Institute of the
Chemical Engineers, Volume 54, p. 167.
Graff, E. C., Pereira, F. & Gharib, M., 2008. Defocusing Digital Particle Image Velocimetry:
A Volumetric DPIV Technique for Dual and Single Phase Flows. Lisbon, Portugal, 14th Int
Symp on Applications of Laser Techniques to Fluid Mechanics, pp. 1-12.
Grimaldo, E. et al., 2011. Field demonstration of a novel towed, area bubble-plume
zooplankton (Calanus sp.) harvester. Fisheries Research, 107(13), pp. 147-158.
Gruber, M., Radl, S. & Khinast, J., 2013. Coalescence and Break-Up in Bubble Columns:
Euler-Lagrange Simulations Using a Stochastic Approach. Chemie Ingenieur Technik,
85(7), pp. 1118-1130.
Haberman, W. L. & Morton, R. K., 1953. An experimental investigation of the drag and
shape of air bubbles rising in various liquids, Bethesda, MD.: David W. Taylor Model
Basin.
Hadamard, J. S., 1911. Mouvement permanent lent d'une sphere liquide et visqueuse
dans un liquide visqueux". CR Acad. Sci. (in French), Volume 152, p. 1735–1738.
Hampel, U. et al., 2012. Multiphase flow investigations with ultrafast electron beam X-
ray tomography. AIP Conference Proceedings, Volume 1428, pp. 167-174.
151
Hänsch, S., Lucas, D., Krepper, E. & Höhne, T., 2012. A multi-field two-fluid concept for
transitions between different scales of interfacial structures. International Journal of
Multiphase Flow , Volume 47, pp. 171-182.
Hoesel, W. & Rodi, W., 1977. New biasing elimination method for laser‐Doppler
velocimeter counter. Rev. Sci. Instrum., Volume 48, pp. 910-919.
Hosokawa, S. & Tomiyama, A., 2004. Turbulence modification in gas-liquid and solid-
liquid dispersed two-phase pipe flows. Int. J. Heat and Fluid Flow, Volume 25, p. 489.
Hosokawa, S. & Tomiyama, A., 2013. Bubble-induced pseudo turbulence in laminar pipe
flows. Int. J. Heat Fluid Flow, Volume 40, pp. 97-105.
Hosokawa, S., Tomiyama, A., Misaki, S. & Hamada, T., 2002. Lateral Migration of Single
Bubbles Due to the Presence of Wall. Montreal, Quebec, ASME 2002 Joint U.S.-European
Fluids Engineering Division Conference, p. 855.
Hua, J., 2015. CFDsimulations of the effects of small dispersed bubbles on the rising of a
single large bubble in 2D vertical channels. Chemical Engineering Science, 123(0), pp.
99-115.
Huang, Q., Yang, C., Yu, G. & Mao, Z.-S., 2010. CFD simulation of hydrodynamics and mass
transfer in an internal airlift loop reactor using a steady two-fluid model. Chemical
Engineering Science , 65(20), pp. 5527-5536.
Hu, G. & Celik, I., 2008. Eulerian-Lagrangian based large-eddy simulation of a partially
aerated flat bubble column. Chemical Engineering Science, 63(1), pp. 253-271.
Ishii, M. & Hibiki, T., 2006. THERMO-FLUID DYNAMICS OF TWO-PHASE FLOW. 2nd ed.
New York: Springer Science+Business Media, Inc..
Ishii, M. & Zuber, N., 1979. Drag Coefficient and Relative Velocity in Bubbly, Droplet or
Particulate Flows. AlChE Journal, Volume 25, p. 843.
Ishikawa, M., Irabu, K., Teruya, I. & Nitta, M., 2009. PIV Measurement of a Contraction
Flow using Micro-Bubble Tracer. Journal of Physics: Conference Series, Volume 147, p.
012010.
Jain, D., Kuipers, J. & Deen, N. G., 2014. Numerical study of coalescence and breakup in a
bubble column using a hybrid volume of fluid and discrete bubble model approach.
Chemical Engineering Science , 119(0), pp. 134-146.
Jain, D., Lau, Y. M., Kuipers, J. & Deen, N. G., 2013. Discrete bubble modeling for a micro-
structured bubble column. Chemical Engineering Science , 100(0), pp. 496-505.
Jakobsen, M. L., Easson, W. J., Greated, C. A. & Glass, D. H., 1996. Particle image
velocimetry: simultaneous two-phase flow measurements. Meas. Sci. Technol. , Volume
7, p. 1270–1280.
Jay Sanyal, ., Marchisio, D. L., Fox, R. O. & Dhanasekharana , K., 2005. On the Comparison
between Population Balance Models for CFD Simulation of Bubble Columns. Industrial &
Engineering Chemistry Research, 44(14), pp. 5063-5072.
152
Jones, W. P. & Launder, B. E., 1972. The Predicition of Laminarization with a two-
equation model of turbulence. Int. J. Heat Mass Transfer, Volume 15, pp. 301-334.
Juliá, J. E., Hernandez, L., Chiva, S. & Vela, A., 2007. Hydrodynamic characterization of a
needle sparger rectangular bubble column: Homogeneous flow, static bubble plume and
oscillating bubble plume. Chemical Engineering Science , 62(22), pp. 6361-6377.
Kariyasaki, A., 1987. Behavior of a single gas bubble in a liquid flow with a linear velocity
profile. , Proceedings of the 1987 ASME/JSME Thermal Engineering Conference. No. 384.
Kataoka, I. & Serizawa, A., 1989. Basic equations of turbulence in gas-liquid two-phase
flow. International Journal of Multiphase Flow, Volume 15, p. 843.
Kitagawa, A., Murai, Y. & Yamamoto, F., 2001. Two-way coupling of Eulerian-Lagrangian
model for dispersed multiphase flows using filtering functions. International Journal of
Multiphase Flow, 27(12), pp. 2129-2153.
Korpijarvi, J., Oinas, P. & Reunanen, J., 1999. Hydrodynamics and mass transfer in an
airlift reactor. Chemical Engineering Science, Volume 54, pp. 2255-2262.
Krepper, E. et al., 2008. The inhomogeneous MUSIG model for the simulation of
polydispersed flows. Nuclear Engineering and Design, Volume 238, p. 1690–1702.
Krepper, E., Lucas, D. & Prasser, H.-M., 2005. On the modelling of bubbly flow in vertical
pipes. Nuclear Engineering and Design, Volume 235, p. 597.
Kulkarni, A. A., 2008. Lift force on bubbles in a bubble column reactor: Experimental
analysis. Chemical Engineering Science, Volume 63, pp. 1710-1723.
Kulkarni, A. V., Badgandi, S. V. & Joshi, J. B., 2009. Design of ring and spider type spargers
for bubble column reactor: Experimental measurements and \{CFD\} simulation of flow
and weeping. Chemical Engineering Research and Design , 87(12), pp. 1612-1630.
Lakshmanan, P. & Ehrhard, P., 2010. Marangoni effects caused by contaminants
adsorbed on bubble surfaces. Journal of Fluid Mechanics, 3, Volume 647, pp. 143-161.
Lance, M. & de Bertodano, M. L., 1994. Phase Distribution Phenomena and Wall Effects
in Bubbly Two-phase Flows. Multiphase Scuience and Technology, Volume 8, p. 69.
Lance, M., Mariè, J. & Bataille, J., 1991. Homogeneous turbulence in bubbly flows. Journal
of Fluids Engineering, Volume 113, p. 295.
Lau, Y., Bai, W., Deen, N. & Kuipers, J., 2014. Numerical study of bubble break-up in
bubbly flows using a deterministic Euler-Lagrange framework. Chemical Engineering
Science, 108(0), pp. 9-22.
Lau, Y. et al., 2013. Experimental study of the bubble size distribution in a pseudo-2D
bubble column. Chemical Engineering Science , 98(0), pp. 203-211.
Law, D. & Battaglia, F., 2013. Numerical Simulations for Hydrodynamics of Air-Water
External Loop Airlift Reactor Flows With Bubble Break-Up and Coalescence Effects.
Journal of Fluids Engineering, Volume 135, pp. 081302-1 - 081302-8.
153
Lawler, M. T., 1971. The role of lift in the radial migration of particles in a pipe flow.
Advances in Solid–Liquid Flow in Pipes and Its Application, pp. 39-57.
Legendre, D. & Magnaudet, J., 1997. A note on the lift force on a spherical bubble or drop
in a low-Reynolds-number shear flow. Phys. Fluids, Volume 9, p. 3572.
Legendre, D. & Magnaudet, J., 1998. The lift-force on a spherical bubble in a viscous
linear shear flow. J. Fluid Mech., Volume 368, p. 81.
Lesieur, M., Métais, O. & Comte, P., 2005. Large-Eddy Simulations of Turbulence. 1st ed.
Cambridge: Cambridge University Press.
Liao, Y. & Lucas, D., 2009. A literature review of theoretical models for drop and bubble
breakup in turbulent dispersions. Chemical Engineering Science, Volume 64, p. 3389.
Lindken, R., Gui, L. & Merzkirch, W., 1999. Velocity Measurements in Multiphase Flow by
Means of Particle Image Velocimetry. Chem. Eng. Technol. , 22(3).
Lindken, R. & Merzkirch, W., 2002. A novel PIV technique for measurements in
multiphase flows and its application to two-phase bubbly flows. Experiments in Fluids,
Volume 33, pp. 814-825.
Liu, Y. & Tay, J.-H., 2002. The essential role of hydrodynamic shear force in the
formation of biofilm and granular sludge. Water research, Issue 1653-1665.
Liu, Y., Yang, H. & Wang, W., 2005. Reconstructing B-spline Curves from Point Clouds--A
Tangential Flow Approach Using Least Squares Minimization. Cambridge, MA, Shape
Modeling and Applications, 2005 International Conference, pp. 4-12.
Loth, E., 2008. Quasi-steady shape and drag of deformable bubbles and drops.
International Journal of Multiphase Flow, 34(6), p. 523–546.
Lucas, D., Krepper, E. & Prasser, H.-M., 2007a. Use of models for lift,wall and turbulent
dispersion forces acting on bubbles for poly-disperse flows. Chemical Engineering
Science, Volume 62, p. 4146.
Lucas, D., Krepper, E., Prasser, H.-M. & Manera, A., 2007b. Stability effect of the lateral lift
force in bubbly flows. Leipzig, Germany, 6th International Conference on Multiphase
Flow, ICMF 2007, p. S1_Mon_C_9.
Lucas, D., Prasser, H.-M. & Manera, A., 2005. Influence of the lift force on the stability of a
bubble column. Chemical Engineering Science, Volume 60, p. 3609 – 3619.
Lucas, D. & Tomiyama, A., 2011. On the role of the lateral lift force in poly-dispersed
bubbly flows. International Journal of Multiphase Flow, Volume 37, p. 1178.
Luo, H.-P. & Al-Dahhan, M. H., 2008. Local characteristics of hydrodynamics in draft tube
airlift bioreactor. Chemical Engineering Science , 63(11), pp. 3057-3068.
Luo, H.-P. & Al-Dahhan, M. H., 2011. Verification and validation of CFD simulations for
local flow dynamics in a draft tube airlift bioreactors. Chemical Engineering Science,
66(5), pp. 907-923.
154
Lu, X., Ding, J., Wang, Y. & Shi, J., 2000. Comparison of the hydrodynamics and mass
transfer characteristics of a modified square airlift reactor with common airlift reactors.
Chemical Engineering Science, Volume 55, pp. 2257-2263.
Manera, A. et al., 2009. Comparison between wire-mesh sensors and conductive needle-
probes for measurements of two-phase flow parameters. Nuclear Engineering and
Design , 239(9), pp. 1718-1724.
Maric, T., Marschall, H. & Bother, D., 2015. lentFoam – A hybrid Level Set/Front Tracking
method on unstructured. Computers & Fluids, Volume 113, pp. 20-31.
Masood, R. & Delgado, A., 2014. Numerical investigation of the interphase forces and
turbulence closure in 3D square bubble columns. Chemical Engineering Science , Issue 0,
pp. - .
Masood, R., Rauh, C. & Delgado, A., 2014. CFD simulation of bubble column flows: An
explicit algebraic Reynolds stress model approach. International Journal of Multiphase
Flow, 66(0), pp. 11-25.
Ma, T. et al., 2015. Scale-Adaptive Simulation of a square cross-sectional bubble column.
Chem. Eng. Sci., Volume 131, p. 101–108.
Ma, T., Ziegenhein, T., Lucas, D. & Fröhlich, J., 2015. Large eddy simulations of the gas-
liquid flow in a rectangular bubble column. Nuclear Engineering and Design , pp. - .
Ma, T. et al., 2015. Euler-Euler large eddy simulations for dispersed turbulent bubbly
flows. International Journal of Heat and Fluid Flow , Volume 56, pp. 51-59.
Maxey, M. R., 1987. The gravitational settling of aerosol particles in homogeneous
turbulence and random flow fields. Journal of Fluid Mechanics, Volume 174, pp. 441-
465.
McClimans, T. et al., 2012. Pneumatic oil barriers: The promise of area bubble plumes.
Journal of Engineering for the maritime enviroment, 0(0), pp. 1-17.
Mei, R. & Klausner, J., 1994. Shear lift force on spherical bubbles. International Journal of
Heat and Fluid Flow, Volume 15, p. 62.
Menter, F. R., Kuntz, M. & Langtry, R., 2003. Ten Years of Industrial Experience with the
SST Turbulence Model. Turbulence, Heat and Mass Transfer, Volume 4.
Mercado, J. M. et al., 2012. Lagrangian statistics of light particles in turbulence. Physics of
Fluids (1994-present), 24(5).
Miron, A. S. et al., 200. Bubble-Column and Airlift Photobioreactors for Algal Culture.
AIChE Journal, 49(9), pp. 1872-1887.
Morel, C., 1997. Turbulence modeling and first numerical simulations in turbulent two-
phase flows, Grenoble, France: SMTH/LDMS/97-023.
Mougin, G. & Magnaudet, J., 2002. The generalized Kirchhoff equations and their
application to the interaction between a rigid body and an arbitrary time-dependent
viscous flow. International Journal of Multiphase Flow, Volume 28, p. 1837.
155
Mudde, R. F., 2005. Gravity-driven Bubbly Flows. Annual Review of Fluid Mechanics,
37(1), pp. 393-423.
Mudde, R. F., Harteveld, W. K. & van den Akker, H. E. A., 2009. Uniform Flow in bubble-
Columns. Ind. Eng. Chem. Res., Volume 48, p. 148–158.
Mudde, R. F., Lee, D. J., Reese, J. & Fan, L.-S., 1997. Role of Coherent Structures on
Reynolds Stresses in a 2-D Bubble Column. Particle Technology and Fluidization, 43(4),
pp. 913-926.
Mudde, R. F. & Simonin, O., 1999. Two- and three-dimensional simulations of a bubble
plume using a two-fluid model. Chemical Engineering Science , 54(21), pp. 5061-5069.
Muñoz-Cobo, J. L., Chiva, S., El Aziz Essa, M. A. A. & Mendes, S., 2012. Tracking of bubble
trajectories in vertical pipes in bubbly flow regime by coupling lagrangian, eulerian and
3D random walks models: Validation with experimental data. Journal of Computational
Multiphase Flows, 4(3), p. 309.
Murai, Y., Matsumoto, Y. & Yamamoto, F., 2001. Three-dimensional measurement of void
fraction in a bubble plume using statistic stereoscopic image processing. Experiments in
Fluids, 30(1), pp. 11-21.
Murai, Y., Oshi, Y., Takeda, Y. & Yammamoto, F., 2006. Turbulent shear stress profiles in
a bubbly channel flow assessed by particle tracking velocimetry. Experiments in Fluids,
Issue 41, pp. 343-352.
Naciri, A., 1992. Contribution `a l’´etude des forces exerc´ees par un liquide sur une bulle de
gaz: portance, masse ajout´ee et interactions hydrodynamiques. Ec. Centrale Lyon,
France.: These de Doctorat.
Nauw, J., Linke, P. & Leifer, I., 2015. Bubble momentum plume as a possible mechanism
for an early breakdown of the seasonal stratification in the northern North Sea. Marine
and Petroleum Geology , Issue 0, pp. - .
Oakley, T. R., Loth, E. & Adrian, R. J., 1997. A Two-Phase Cinematic PIV Method for
Bubbly Flows. J. Fluids Eng., 119(3), pp. 707-712.
Ojima, S., Hayashi, K., Hosokawa, S. & Tomiyama, A., 2014. Distributions of void fraction
and liquid velocity in air–water bubble column. International Journal of Multiphase Flow,
Volume 67, pp. 111-121.
Oliver-Salvador, M. d. C. et al., 2013. Shear rate and microturbulence effects on the
synthesis of proteases by Jacaratia mexicana cells cultured in a bubble column, airlift,
and stirred tank bioreactors. Biotechnology and Bioprocess Engineering, pp. 808-818.
Pang, M. & Wei, J., 2013. Experimental investigation on the turbulence channel flow
laden with small bubbles by PIV. Chemical Engineering Science, Volume 94, pp. 302-315.
Parker, J. R., 2010. Algorithms for Image Processing and Computer Vision. Second ed.
Indianapolis: John Wiley & Sons.
156
Pereira, F., Stüer, H., Graff, E. C. & Gharib, M., 2006. Two-frame 3D particle tracking.
Measurement Science and Technology, 17(7), p. 1680.
Petrou, M. & Petrou, C., 2010. Image Processing: The Fundamentals. Second ed.
Chichester: John Wiley & Sons.
Pfleger, D., Gomes, S., Gilbert, N. & Wagner, H.-G., 1999. Hydrodynamic simulations of
laboratory scale bubble columns fundamental studies of the Eulerian-Eulerian
modelling approach. Chemical Engineering Science, 54(21), pp. 5091-5099.
Politano, M., Carrica, P. & Converti, J., 2003. A model for turbulent polydisperse two-
phase flow in vertical channels. International Journal of Multiphase Flow, Volume 29, p.
1153.
Pope, S. B., 2000. Turbulent Flows. 1st ed. Cambridge: Cambridge University Press.
Pourtousi, M., Zeinali, M., Ganesan, P. & SAHU, P. J. N., 2015. Prediction of multiphase
flow pattern inside a 3D bubble column reactor using a combination of CFD and ANFIS.
RSC Adv.
Prakash, V. N., 2013. PHD thesis : LIGHT PARTICLES IN TURBULENCE. Twente:
Universiteit Twente.
Prandtl, L., 1961. Ludwig Prandtl Gesammelte Abhandlungen. Berlin: Springer Berlin
Heidelberg.
Rabha, S. & Buwa, V. V., 2009. Lift Force Acting on Mono-dispersed Bubbles Rising in
Sheared Liquids. Montreal, Canada , GLS-9/8th World Congress of Chemical Engineering.
Raffel, M., Willert, C. E., Wereley, S. & Kompenhans, J., 2007. Particle Image Velocimetry.
2nd ed. Berlin Heidelberg: Springer-Verlag.
Rensen, J. & Roig, V., 2001. Experimental study of the unsteady structure of a confined
bubble plume. International Journal of Multiphase Flow , 27(8), pp. 1431-1449.
Riboux, G., Risso, F. & Legendre, D., 2010. Experimental characterization of the agitation
generated by bubbles rising at high Reynolds number. Journal of Fluid Mechanics,
Volume 643, p. 509.
Risso, F. et al., 2008. Wake attenuation in large Reynolds number dispersed two-phase
flows. Phil. Trans. R. Soc. A, Volume 366, p. 2177.
Roghair, I., M.W.Baltussen, M.VanSintAnnaland & J.A.M.Kuipers, 2013. Direct Numerical
Simulations of the drag force of bi-disperse bubble swarms. Chemical Engineering
Science, Volume 95, p. 48–53.
Rosenberg, B., 1950. The drag and shape of air bubbles moving in, Bethesda, MD: David
W. Taylor Model Basin.
Ryu, Y., Chang, K.-A. & Lim, H.-J., 2005. Use of bubble image velocimetry for
measurement of plunging wave impinging on structure and associated greenwater.
Meas. Sci. Technol., Volume 16, p. 1945–1953.
157
Rzehak, R. & Krepper, E., 2012. Bubble-induced Turbulence: Comparison of CFD Models.
Nuclear Engineering and Desig, Volume 13, pp. 57-65.
Rzehak, R. & Krepper, E., 2013a. CFD modeling of bubble-induced turbulence.
International Journal of Multiphase Flow, Volume 55, p. 138–155.
Rzehak, R. & Krepper, E., 2013b. Closure models for turbulent bubbly flows: A {CFD}
study. Nuclear Engineering and Design, Volume 265, pp. 701-711.
Rzehak, R. & Krepper, E., 2015. Bubbly flows with fixed polydispersity: Validation of a
baseline closure model. Nuclear Engineering and Design, Volume 287, pp. 108-118.
Rzehak, R., Krepper, E. & Lifante, C., 2012. Comparative study of wall-force models for
the simulation of bubbly flows. Nuclear Engineering and Design, Volume 253, pp. 41-49.
Rzehak, R. & Kriebitzsch, S., 2014. Multiphase CFD-simulation of Bubbly Pipe Flow: A
Code Comparison. Int J Multiphase Flow.
Saffman, P. G., 1965. The lift on a smalll sphere in a slow shear flow. J. Fluid Mech.,
Volume 22, pp. 385-400.
Santarelli, C. & Fröhlich, J., 2012. Simulation of bubbly flow in a vertical turbulent
channel. PAMM, 12(1), pp. 503-504.
Sathe, M. J., Thaker, I. H., Strand, T. E. & Joshi, J. B., 2010. Advanced PIV/LIF and
shadowgraphy system to visualize flow structure in two-phase bubbly flows. Chemical
Engineering Science, Volume 65, p. 2431–2442.
Sato, Y., Sadatomi, M. & Sekoguchi, K., 1981. Momentum and heat transfer in two-phase
bubble flow I: Theory. Int. J. Multiphase Flow, Volume 7, p. 167.
Saunter, C. D., 2010. Quantifying Subpixel Accuracy: An Experimental Method for
Measuring Accuracy in Image-Correlation-Based, Single-Particle Tracking. Biophysical
Journal, Volume 98, pp. 1566-1570.
Schleicher, E., Da Silva, M. & Hampel, U., 2008. Enhanced Local Void and Temperature
Measurements for Highly Transient Multiphase Flows. Instrumentation and
Measurement, IEEE Transactions on, Feb, 57(2), pp. 401-405.
Schmale, O. et al., 2015. Bubble Transport Mechanism: Indications for a gas bubble-
mediated inoculation of benthic methanotrophs into the water column. Continental Shelf
Research , 103(0), pp. 70-78.
Schmidt, E., 1934. Ähnlichkeitstheorie der Bewegung von Flüssigkeitsgasgemischen.
Forschungsheft, 365(VDI, Berlin, 1–3).
Serizawa, A. & Kataoka, I., 1990. Turbulence suppression in bubbly two-phase flow.
Nuclear Engineering and Design, Volume 122, p. 1.
Serizawa, A., Kataoka, I. & Michiyoshi, I., 1975. Turbulence structure of air-water bubbly
flow - II. local properties. Int. J. Multiphase flow, Volume 2, pp. 235-246.
158
Simcik, M., 2011. CFD simulationandexperimentalmeasurementofgasholdupandliquid
interstitialvelocityininternalloopairliftreactor. Chemical EngineeringScience, Volume 66,
p. 3268–3279.
Simiano, M. et al., 2006. Comprehensive experimental investigation of the
hydrodynamics of large-scale, 3D, oscillating bubble plumes. International Journal of
Multiphase Flow, Volume 32, pp. 1160-1181.
Sokolichin, A. & Eigenberger, G., 1999. Applicability of the standard k epsilon turbulence
model to the dynamic simulation of bubble columns: Part I. Detailed numerical
simulations. Chemical Engineering Science, 54(13-14), pp. 2273-2284.
Sokolichin, A., Eigenberger, G. & Lapin, A., 2004. Simulation of Buoyancy Driven Bubbly
Flow: Established Simplifications and Open Questions. Fluid Mechanics and Transport
Phenomena, Volume 50, pp. 24-45.
Spalart, P., 2000. Strategies for turbulence modelling and simulations. International
Journal of Heat and Fluid Flow, 21(3), pp. 252-263.
Squires , K. D., 1990. Particle response and turbulence modification in isotropic
turbulence. Physics of Fluids A: Fluid Dynamics, Volume 2, pp. 1191-1203.
Sugioka, K. & Tsukada, T., 2015. Direct numerical simulations of drag and lift forces
acting on a spherical. International Journal of Multiphase Flow, Volume 71, pp. 32-37.
Tabib, M. V., Roy, A. & Joshi, J., 2008. CFD simulation of bubble column - An analysis of
interphase forces and turbulence models. Chemical Engineering Journal, Volume 139, pp.
589-614.
Tennekes, H. & Lumley, J. L., 1972. A First Course in Turbulence. 1st ed. Cambride, MA:
The MIT Press.
Tomiyama, A., 2002. Single Bubbles in Stagnant Liquids and in Linear Shear Flows.
Dresden, Germany, Workshop on Measurement Technology (MTWS5).
Tomiyama, A., 2004. Drag, lift and virtual mass forces acting on a single bubble. Lyon,
France, 3rd International Symposium on Two-Phase Flow.
Tomiyama, A., Kataoka, I., Zun, I. & Sakaguchi, T., 1998. Drag Coefficients of single
bubbles under normal and micro gravity conditions. JSME International Journal Series B ,
41(2), p. 472.
Tomiyama, A. et al., 1995. Effects of Eötvös number and dimensionless liquid volumetric
flux on lateral motion of a bubble in a laminar duct flow. In: A. Serizawa, T. Fukano & J.
Bataille, eds. Advances in multiphase flow. : Elsevier, pp. 3-15.
Tomiyama, A., Tamai, H., Zun, I. & Hosokawa, S., 1999. Measurement of Transverse
Migration of Single Bubbles in a Couette Flow. Pisa, 2nd Int. Symposium on Two-Phase
Flow Modelling and Experimentation, p. 941.
Tomiyama, A., Tamai, H., Zun, I. & Hosokawa, S., 2002. Transverse migration of single
bubbles in simple shear flows. Chemical Engineering Science , 57(11), pp. 1849-1858.
159
Tripathi, M. K., Sahu, K. C. & Govindarajan, R., 2015. Dynamics of an initially spherical
bubble rising in quiescent liquid. NATURE COMMUNICATIONS, Volume 6.
Troshko, A. A. & Hassan, Y. A., 2001. A two-equation turbulence model of turbulent
bubbly flows. International Journal of Multiphase Flow, Volume 27, p. 1965.
Tryggvason, G., Dabiri, S., Aboulhasanzadeh, B. & Lu, J., 2013. Multiscale considerations
in direct numerical simulations of multiphase flows. PHYSICS OF FLUIDS, Volume 25.
Tryggvason, G., Esmaeeli, A., Lu, J. & Biswas, S., 2006. Direct numerical simulations of
gas/liquid multiphase flows. Fluid Dyn. Res., Volume 38, p. 660.
van Benthum, W., van der Lans, R., van Loosdrecht, M. & Heijnen, J., 1999. Bubble
recirculation regimes in an internal-loop airlift reactor. Chemical Engineering Science ,
54(18), pp. 3995-4006.
Volk, R. et al., 2008. Acceleration of heavy and light particles in turbulence: comparison
between experiments and direct numerical simulations. Physica D: Nonlinear
Phenomena, 237(14-17), p. 2084–2089.
Wang, A., Marashdeh, Q. & Fan, L.-S., 2014. ECVT imaging of 3D spiral bubble plume
structures in gas-liquid bubble columns. The Canadian Journal of Chemical Engineering,
92(12), pp. 2078-2087.
Wellek, R. M., Agrawal, A. K. & Skelland, A. H. P., 1966. Shape of Liquid Drops Moving in
Liquid Media. AlChE Journal, Volume 12, p. 854.
Wilcox, D. C., 1994. Turbulence Modeling for CFD. 2nd ed. La Canada, California: DCW
Industries, Inc..
Wuest, A., Brooks, N. H. & Imboden, D. M., 1992. Bubble plume modeling for lake
restoration. Water Resources Research, 28(12), pp. 3235-3250.
Xu, H. & Bodenschatz, E., 2008. Motion of inertial particles with size larger than
Kolmogorov scale in turbulent flows. Physica D: Nonlinear Phenomena, 237(14-17), pp.
2095-2100.
Zhongchuna, L., Xiaoming, S. & Jiyang, Y., 2014. Numerical investigation on lateral
migration and lift force of single. Nuclear Engineering and Design, Volume 274, p. 154–
163.
Ziegenhein, T., Lucas, D., Rzehak, R. & Krepper, E., 2013. Closure relations for CFD
simulation of bubble columns. Jeju, Korea, ICMF 2013.
Ziegenhein, T., Rzehak, R. & Lucas, D., 2015. Transient simulation for large scale flow in
bubble columns. Chemical Engineering Science , 122(0), pp. 1-13.
Zun, I., 1980. The transverse migration of bubbles influenced by walls in vertical bubbly
flow. International Journal of Multiphase Flow, Volume 6, p. 583.
