Direct numerical simulations of hydrodynamics and mass
transfer in dense bubbly flows
Citation for published version (APA):
Roghair, I. (2012). Direct numerical simulations of hydrodynamics and mass transfer in dense bubbly flows. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR732585
DOI:
10.6100/IR732585
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Direct Numerical Simulations of Hydrodynamics and Mass
Transfer in Dense Bubbly Flows
Direct Numerical Simulations of Hydrodynamics and Mass
Transfer in Dense Bubbly Flows
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de
Technische Universiteit Eindhoven, op gezag van de
rector magnificus, prof.dr.ir. C. J. van Duijn, voor een
commissie aangewezen door het College voor
Promoties in het openbaar te verdedigen
op dinsdag juni om . uur
door
Ivo Roghair
Dit proefschri is goedgekeurd door de promotoren: prof.dr.ir. J. A. M. Kuipers
en
Samenstelling promotiecommissie:
Prof.dr.ir. J. C. Schouten Technische Universiteit Eindhoven, voorzier Prof.dr.ir. J. A. M. Kuipers Technische Universiteit Eindhoven, promotor Prof.dr.ir. M. Van Sint Annaland Technische Universiteit Eindhoven, promotor Prof.dr. D. Lohse Universiteit Twente
Prof.dr. R. F. Mudde Technische Universiteit Del Prof.dr. F. Toschi Technische Universiteit Eindhoven Dr.ir. N. G. Deen Technische Universiteit Eindhoven
is work is part of the industrial partnership programme: Fundamentals of heterogeneous
bubbly flows of the Foundation for Fundamental Research on Maer (FOM) and the
industrial partners AkzoNobel, DSM, Shell Global Solutions and TataSteel. FOM is financially supported by the Dutch Organisation for Scientific Research (NWO). Nederlandse titel: “Directe Numerieke Simulaties van de Hydrodynamica en Massa Transport in Dichte Bellenstromen.”
Copyright © by I. Roghair
All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without the prior permission of the author.
Printed by: Gildeprint Drukkerijen, Enschede. Typeset using LTEX, in Linux
Supplementary materials can be found at:http://goo.gl/UIO6e
A catalogue record is available from the Eindhoven University of Technology Library
T
Table of contents vii
Summary ix
Samenvatting xi
Nomenclature xiii
Introduction
. Background and motivation . . . . Multi-scale modelling of bubble column reactors . . . . Bubble behaviour . . . . esis outline . . .
e Front-Traing model
. Introduction . . . . Front-Tracking algorithm . . . . Remeshing . . . . Computing facility . . . . Concluding remarks . . .
Drag on bubbles rising in a swarm at intermediate and high Reynolds numbers
. Introduction . . . . Discrete bubble model . . . . Drag coefficient . . . . Analysis and set-up of bubble swarm simulations . . . . Simulation seings . . . . Simulation results . . . . Discussion . . . . Implementation in DBM . . . . Conclusions . . . vii
viii T
Drag on bubbles in bi-disperse swarms
. Introduction . . . . Simulations . . . . Results . . . . Conclusions . . .
Drag force and clustering in dense bubble swarms at low and high Reynolds
numbers
. Introduction . . . . Simulations . . . . Bubble clustering in periodic domains . . . . Bubble rise velocity . . . . Drag force on bubble swarms . . . . Bubble aspect ratio . . . . Conclusions . . .
Energy spectra and bubble velocity distributions in pseudo-turbulence:
numer-ical simulations vs. experiments
. Introduction . . . . Numerical seings and analysis . . . . Energy spectra . . . . Bubble velocity distribution . . . . Discussion and conclusions . . .
Mass transport modeling
. Introduction . . . . Numerical solution method . . . . Verification . . . . Validation . . . . Bubble swarm induced mixing . . . . Simulations of mass transfer in bubble swarms . . . . Concluding remarks . . .
References
List of publications
Dankwoord
S
Direct Numerical Simulations of Hydrodynamics and Mass
Transfer in Dense Bubbly Flows
Bubbly flows belong to dispersed two-phase flows involving a gas and a liquid phase, which are brought into contact with each other such that the gas forms bubbles which rise through the liquid. Bubbly flows are oen encountered in industrial environments, for instance in bubble columns in the (bio-)chemical or metallurgical industry. In chemical processes, (re-acting) components are exchanged between a multitude of bubbles (’bubble swarms’) and the continuous liquid phase, while in metallurgical applications the (argon) bubbles are mainly used to induce a mixing current in large vessels of molten steel.
e gas hold-up is an important parameter that determines the efficiency of operation in a bubble column. At high gas fractions, many bubbles are injected simultaneously, form-ing bubble swarms which adopt a behavior very different from sform-ingle, undisturbed risform-ing bubbles in an infinite liquid. Numerical studies on bubbly flows at industrial scales require detailed information on the effect of the gas fraction on the hydrodynamics and mass trans-fer characteristics, which can be obtained with dedicated numerical simulations resolving the physics occurring at small scales. is work focuses on the behaviour of bubbles rising in a swarm at small scales by numerical simulations. e results are presented in a form that can be used in larger-scale simulations which are performed both in academics and industry. is work is a part of an industrial partnership programme, to scientifically investigate the hydrodynamic and mass transfer phenomena prevalent in bubbly flows in an industrially relevant context.
A Front-Tracking model has been used to dynamically simulate the interaction between multiple phases (such as gas bubbles rising in a liquid), which tracks the interface between the phases by Lagrangian control points. e model is able to simulate multiple bubbles in a periodic domain, accounting for the surface tension force and deformations of the bubbles in great detail. In this work the model has been extended with the capability of simulat-ing bubbly flows with very high gas loadsimulat-ings via improved remeshsimulat-ing techniques ensursimulat-ing mass conservation. ereaer, a species solver using an explicit immersed boundary tech-nique was embedded in the model to investigate mass transfer between the phases, including
x S
chemical reactions. is model was employed to investigate several hydrodynamic and mass transfer characteristics of bubble swarms.
e drag force is one of the dominant forces in bubbly flows. In this study, the drag force acting on bubbles rising in a mono-disperse swarm has been numerically determined as a function of the gas hold-up. It was shown that the normalized drag linearly increases with the gas fraction (hindered rise) and a correlation for the drag coefficient as a function of the gas hold-up and the Eötvös number has been developed, in which the Eötvös number accounts for the bubble size and shape deformations. e correlation matches the simulation results in a wide range, i.e. for Eötvös numbers between 0.1 . Eo . 4.8 and Reynolds numbers between 80. Re . 1400. Additionally, it was shown that the derived correlation can also be applied to bi-disperse bubble swarms.
e computed turbulent energy spectrum induced by the rising bubbles shows an excel-lent agreement with experimental results obtained from literature. It was confirmed that the energy cascade follows a power-law with a slope of -, and is independent of the gas frac-tion. e distribution of bubble velocities shows again good agreement with experimental results, and it was shown that for high gas fractions, the distribution of horizontal velocity fluctuations converges to a Gaussian distribution.
Finally, simulations on mass transfer in bubble swarms have been performed. For ac-curate modelling of mass transfer in large-scale bubble column models, detailed knowledge on the liquid side mass transfer coefficient is required. Several interpretation methods to determine the liquid-side mass transfer coefficients in bubbly flows have been compared, and a closure relation for mass transfer in bubbly flows has been formulated.
e insight and closure relations developed in this work can be used to improve existing numerical models, used by academics and industry, and allow for more efficient design and operation of bubble column reactors.
S
Directe Numerieke Simulaties van de Hydrodynamica en
Massa Transport in Dichte Bellenstromen
Bellen ontstaan bij het injecteren van een gas in een vloeistof, waarbij de bellen zullen stijgen terwijl zij interacties met elkaar en met de omringende vloeistof hebben. De stromingen die hierbij ontstaan hebben zeer karakteristieke eigenschappen. Naast de alledaagse situaties waarbij we bellenstromen tegenkomen, worden bellenstromen ook vaak toegepast in de (bio)chemische en metaalindustrie. In sommige chemische processen vindt een reactie plaats tussen een gasvormige en een vloeibare component, waarbij de reagerende stoffen tussen beide fasen worden uitgewisseld via het beloppervlak (massaoverdracht). Bij de productie van staal wordt het vloeibare metaal in beweging gehouden door inerte argon bellen in te blazen, welke een stroming en de gewenste menging in het reservoir veroorzaken.
De gasfractie is een karakteristieke eigenschap van een gas-vloeistofmengsel waarmee in belangrijke mate de efficientie van het gehele proces bepaald wordt, of dit nu de mas-saoverdracht of menging is. Bij deze processen wordt vaak gebruik gemaakt van relatief grote gasstromen, waarbij ook grote hoeveelheden bellen ontstaan. Samen vormen deze bellen zwermen. Het gedrag van bellen in zwermen verschilt sterk van het gedrag van een enkele afzonderlijke bel die stijgt in een schijnbaar oneindige, oorspronkelijk stilstaande vloeistof. Juist deze zwermeffecten zijn van belang bij het beschrijven van het gedrag van belstromen in industriële toepassingen. Om deze effecten te onderzoeken zijn numerieke modellen ontwikkeld waarmee gas-vloeistof stromingen op industriële schaal bestudeerd kunnen worden. Deze grote-schaal modellen werken echter alleen goed wanneer zij wor-den voorzien van relaties die de effecten die op de kleinste schaal plaatsvinwor-den beschrijven. Met zeer gedetailleerde modellen, op kleine schaal, kunnen deze relaties worden bepaald. Het onderzoek beschreven in dit proefschri richt zich op het beschrijven van de fysica van het gedrag van bellen in zwermen op kleine schaal. De resultaten kunnen vervolgens wor-den gebruikt in modellen die de fenomenen op een grotere schaal beschrijven, en waarmee industriële processen veiliger, zuiniger en schoner kunnen worden gemaakt. Dit werk is onderdeel van een onderzoeksprogramma waarbij de krachten van de academische en in-dustriële wereld worden gebundeld om belzwermen nader onder de loep te nemen.
xii S
Een Front-Tracking model is gebruikt om de dynamische interacties tussen meerdere fasen (zoals bellen in een vloeisto) in detail te kunnen beschrijven. Het model volgt het grensvlak tussen gas en vloeistof met behulp van Lagrangiaanse punten die zijn verdeeld over het grensvlak. Met dit model is het mogelijk om meerdere bellen in een periodiek domein te simuleren, waarbij rekening gehouden wordt met het effect van de oppervlak-tespanning en vervorming van de bellen tot in de kleinste details. In dit proefschri staat beschreven hoe het model is aangepast zodat de bellen gedurende de simulatie hun vol-ume exact behouden zonder concessies te doen aan de nauwkeurigheid van de beschrijving van de vorm van de bellen. Deze methode is zodanig opgesteld dat zeer hoge gasfracties (> 40%) gesimuleerd kunnen worden. Daarnaast is het Front-Tracking model uitgebreid met een convectie-diffusie vergelijking met een immersed boundary techniek die de mas-saoverdracht tussen de bellen en de vloeistof in aanwezigheid van chemische reacties in de vloeistof kan beschrijven. Het model is vervolgens gebruikt om de hydrodynamica en massaoverdracht in belzwermen te karakteriseren.
De weerstand van de vloeistof op een stijgende bel is van grote invloed op de uitein-delijke stijgsnelheid van de bel. Voor een zwerm met bellen van gelijke grooe is de weer-stand van de vloeistof op de bellen bepaald. Hierbij blijkt dat de weerweer-stand, genormaliseerd met de weerstand op een enkele bel in een schijnbaar oneindige vloeistof, lineair toeneemt met de gasfractie, waarbij de helling van deze relatie beschreven kan worden met het getal van Eötvös —de ratio tussen de opwaartse kracht en de oppervlaktespanning. Bovendien is aangetoond dat ook bellen in zwermen die bestaan uit grote en kleine bellen een weerstand ondervinden die met deze relatie beschreven kan worden.
Het turbulente energiespectrum van de vloeistof, dat wordt veroorzaakt door de stij-gende bellen, is bepaald door middel van numerieke simulaties en laat een grote overeenkomst zien met experimentele resultaten uit de literatuur. Hierbij laat het energiespectrum een verval zien met helling−3, onaankelijk van de gasfractie. De berekende belsnelheden
vertonen ook grote overeenkomsten met experimentele resultaten, waarbij is aangetoond dat de distributie van de snelheden steeds meer tot een normale verdeling nadert naarmate de gasfractie toeneemt.
Tensloe zijn er simulaties uitgevoerd die het massatransport tussen bellen en de om-ringende vloeistof beschrijven waarbij gebruik is gemaakt van diverse nieuwe technieken om het concentratieveld gedetailleerd op te lossen. Om de resultaten te interpreteren zijn diverse methoden gebruikt, die allen wijzen op een geringe toename van de massaover-drachtscoëfficient bij toename van de gasfractie.
De inzichten en sluitingsrelaties die worden beschreven in dit werk kunnen worden ge-bruikt om de resultaten van grote-schaal modellen, in dienst van zowel academische als industriële onderzoekers, te verbeteren, en maken een beter ontwerp en procesoperatie van gas-vloeistofsystemen mogelijk.
N
Variables
a specific surface area, [m−] A surface area, [m]
c concentration, [mol/L] or a.u. C coefficient, [-]
d diameter, [m]
D diffusion coefficient, [m/s]
e deviation between small and large time step, [-] E bubble aspect ratio (height vs. width) [-] Fs concentration forcing term, [mol/m/s]
F force, [N]
h, ∆x, ∆y, ∆z mesh size, [m]
H Henry coefficient, [-] or a.u. kL mass transfer coefficient, [m/s] k1 reaction rate constant, [s−]
nb number of bubbles, [-]
nx, ny, nz number of cells in x, y, z direction, [-] N number of connected vertices, [-]
p pressure, [Pa] r, R radius, [m] rij roughness, [-] R refinement ratio of Γsvs. Γh, [-] s semiperimeter, [m] S surface, [m] t time, [s] u fluid velocity, [m/s] v bubble velocity, [m/s] V volume, [m] x position, [m] xiii
xiv N
xi composition parameter for bidisperse swarms; number of small/large
bubbles vs. average diameter, [-]
yi composition parameter for bidisperse swarms; fraction of small/large
bubbles vs. total fraction, [-]
Greek leers
α gas fraction, [-]
Γ background mesh
ε average phase fraction parameter for , [m] θ angle, [◦]
µ viscosity, [Pa· s] ρ density, [kg/m]
σ surface tension coefficient, [N/m] τ stress tensor, [Pa]
φ phase fraction, [-]
χ bubble aspect ratio (major axis vs. minor axis), [-]
Ω interface mesh
Remeshing
A mean normal on point
c centroid e edge I identity matrix ` edge length, [m] n normal ˆ
n optimal marker displacement
t tangential
Subscripts and superscripts
g gas phase l liquid phase b bubble x, y, z in x, y, z direction eq sphere equivalent G gravity P pressure
N xv
L li
V M virtual mass
D drag
m marker
∞ single rising bubble in an infinite liquid
G gravity
Abbreviations
adaptive time stepping computational fluid dynamics coupled level-set and volume-of-fluid constant temperature anemometry discrete bubble model
direct numerical simulations divergence theorem algorithm front-tracking
incomplete Cholesky conjugate gradient laice-Boltzmann method
laser doppler anemometry level-set
probability density function particle image velocimetry particle tracking velocimetry two-fluid model
volume-of-fluid
Dimensionless numbers
AL Achterland number, Vl,total/Vl,film
Eo Eötvös number, gzd2bρl/σ Fo Fourier number,Dt/R2 Ha Haa number, k1D/ ( k2 L ) Mo Morton number, gzµ4l∆ρ/ ( ρ2 lσ3 ) Pe Péclet number,kvkdeq/D = Re · Sc Re Reynolds number, ρkvkdb/µl Sc Schmidt number, µ/ (ρD) Sh Sherwood number, kLdb/D We Weber number, dbkvk2ρl/σ
C
I
. Background and motivation
Bubbles are encountered in many occasions in daily life, and we are all well accustomed by their presence. For centuries, bubbles have aracted interest of people and many have engaged the challenging task to study and describe the intricate behaviour of immersed gas bubbles rising in a liquid. However, the physics involved are very complex, and even a single phenomenon may take years of dedicated research to be understood. Leonardo da Vinci (—) for instance, aempted to understand the spiralling motion of a single rising bubble, and his theories were revisited several times until only very recently, a more definite explanation has been provided (Prosperei, ).
ere are many different types of bubbles, each adopting their own characteristic be-haviour. In a glass of beer or soda, for instance, the bubbles are very small, spherically shaped and relatively low in number. If one bubble detaches from the glass, it will generally rise unaccompanied, without being affected by other bubbles. In a whirlpool, however, the bubbles are much larger and more deformed and are rising in great numbers. Hence, they are rising together and are experiencing all kinds of interactions with neighboring bubbles directly, but also via the liquid in which they are immersed. ese interactions are collec-tively referred to as the “bubble swarm effects”. As a result, bubbles rising in a swarm adopt a behaviour very different from single, undisturbed rising bubbles. An important factor de-termining the extent of bubble swarm effects is the gas hold-up (or gas fraction), denoted by
α, which is defined as the ratio between the volume occupied by the gas bubbles over the
C . I
Figure .: Different flow regimes emerge as a function of the superficial gas velocity or the reactor size (adapted from Shah et al. ()).
total volume:
α = Vgas
Vgas+ Vliquid (.)
Considering the fact that for single rising bubbles, centuries of research have been required to build up to our current understanding, it looks a daunting task to understand the be-haviour of multiple rising bubbles. Nevertheless, a good reason to pursue this research (sheer curiosity aside) is that bubbly flows are oen encountered and exploited in industrial processes. e metallurgical industry uses the injection of bubbles to generate a mixing current in ladles with molten steel without sacrificing equipment. In oil recovery, pumped up gas may form slugs in pipelines, which is detrimental for the production process. In the chemical industry, many processes involve the exchange between (possibly reacting) com-ponents between a gas and liquid phase, for instance in phosgenation, oxidation, hydro-genation or alkylation. Such processes are typically performed in bubble column reactors, a column filled with liquid (with continuous feeding of liquid reactants and extraction of liquid products) in which gaseous reactants are introduced at the boom or via immersed spargers, usually at a large rate so that dense bubble clouds (i.e. high gas hold-up) rise through the liquid. At these conditions, the bubbles may coalesce, resulting in larger bubbles. Depend-ing on the reactor dimensions, gaseous slugs may form decreasDepend-ing the specific surface area of the bubbles, oen limiting the product yield, or result in a heterogeneous bubble flow. Shah et al. () have outlined the different flow regimes that can be encountered in bubble column reactors, reproduced in Figure .. Clearly, not only the flow regimes depicted in this figure are of importance, but the entire spectrum of phenomena that prevail in bubbly flows. is includes, but is not limited to, the motion, positioning and orientation of the bubbles within the swarm (clustering), the bubble rise velocity, turbulence induced by the
.. M
bubbles, liquid phase mixing, shape deformations and rate of inter-phase mass transfer, all of which are depending strongly on the gas fraction and other physical parameters. A good and detailed understanding of the phenomena exhibited by bubbles rising in a swarm, is of utmost importance for industrial engineers seeking to increase product quality and yield. A detailed quantitative description is required for optimal equipment design, to operate the process in a more reliable and safer way and it will help to save energy and resources.
Despite a considerable scientific interest in bubble column reactors, and growing since the s (Deen et al., ), at this point this knowledge, most notably effects featuring at high gas fractions, is insufficient. One of the most important reasons is that it is very difficult, if not (yet) impossible to obtain this information experimentally. At high gas fractions, optical techniques such as Particle Tracking Velocimetry (), Particle Image Velocimetry () or Laser-Doppler Anemometry (), are unfeasible due to the large number of bubbles that prevent optical access. Moreover, the extreme conditions of the processes mentioned before (molten steel, corrosive environments) are not suitable for use of hot wire or optical probes, tracing particles or even looking glasses. is tampers not only a detailed study on the hydrodynamics of gas and liquid flows, but also the mass exchange between the phases. Finally, simultaneous measurements of liquid flow and bubble tracking (with or without mass transfer) are tedious and expensive to perform.
One way to overcome these limitations is via numerical simulations, which is the main topic of this thesis. While numerical simulations of bubbly flows are oen plagued by other difficulties, such as accurate treatment of the large pressure jump and the surface tension force at the gas-liquid interface or the simulation of bubbly flows at very high gas frac-tions, these issues can be overcome by techniques developed and outlined in this thesis, and are consequently capable of describing the wide range of phenomena that prevail in dense bubble swarms.
Note that a wealth of experimental data is available for bubbly flows at (very) low gas fractions. Experiments, if performed with outstanding technical skills, advanced equipment and eye for detail, are invaluable for insight, validation, and inspiration for scientists. Once set up, experiments are easily run for extended periods of time, yielding extremely good statistics and a great and essential way to validate numerical simulations.
is introduction will continue with an outline of used numerical techniques to simulate bubbly flows, discussing several candidate modelling techniques for our specific demands. Next, a discussion of bubble characteristics is given which will prove useful throughout this thesis. e Chapter closes with an outline of the thesis.
. Multi-scale modelling of bubble column reactors
Numerical modelling of bubble columns using Computational Fluid Dynamics () has re-ceived increasingly more aention since the s (Deen et al., ; Unverdi and Tryggva-son, ). While in old times costs for scientific computational facilities were prohibitively expensive, currently almost everyone can buy a multi-core workstation at the price of a
de- C . I
cent second-hand bicycle. Needless to say, computational sciences have flourished in the last decades, which has also resulted in the development of advanced numerical solution methods, which have become more and more accurate and efficient. On the other hand, models also tend to resolve the simulation in more and more detail, for instance by increas-ing the domain size or the grid resolution, or solvincreas-ing multiple physical processes in parallel, so that still large computational times are required.
An industrial bubble column is typically tens of meters high. At such scales, the simu-lation of an entire bubble column is still impossible if the fluid flow is to be resolved in full detail. erefore, the multi-scale modeling approach (Deen et al., ; van Sint Annaland et al., ) has been adopted, where large-scale models are provided with accurate, yet easy to compute, closure correlations derived from more-detailed but smaller-scale models. e key idea here is that the closure correlations incorporate the (temporal and/or spatial aver-aged) physics at very small scales, to improve the accuracy of the larger-scale model without having to perform the actual detailed calculations. Together, the models form a hierarchy (see Figure .).
.. Model hierarchy
At the smallest scale (typically in the order of centimeters), closures are derived using Direct Numerical Simulations (, of which a variety of modelling techniques are available (see Section ..)). In , no assumptions are required and the gas/liquid behaviour follows immediately from first principles, generally the Navier-Stokes equations. A key issue for on bubbly flows is the calculation of the surface tension force and the associated description of the gas-liquid interface, especially in dense bubble swarms.
e simulation of intermediate scales (in the order of meters) is done using the Discrete Bubble Model (). In this Euler-Lagrangian approach, the bubbles are tracked by La-grangian points which represent bubbles by finite-size spheres, while the liquid flow field is calculated with the volume-averaged Navier-Stokes equations (Deen et al., ). e dy-namics of the interface is not resolved, but the effects are taken into account via closure correlations, thereby saving enormous amounts of computation time. is model is espe-cially suited to investigate large-scale swarm effects and effects of coalescence and break-up of bubbles since bubble-bubble encounters are treated in a deterministic manner.
For even larger scales than the , the Two-Fluid Model () is used, which is an Euler-Euler method that simulates the two-phase flow at industrial scales, i.e. tens of meters, using phase fraction tracking. e effect of break-up and coalescence, distilled from , is provided as additional input for this type of model.
e current work, however, focuses on the description of physical phenomena using , including the forces acting on a bubble, its shape and positioning in the swarm, bubble induced turbulence and rate of mass exchange. ese descriptions can consequently be used to improve the results from and other higher-level models. In a related project, these descriptions, called closures, are actually implemented to study the bubble column hydrodynamics at larger scales (e.g. Lau et al. ()). For our required model type, ,
.. M
(a) DNS (b) E-L (c) E-E
Figure .: Gas-liquid multi-scale modelling hierarchy with le: Fully resolved Direct Nu-merical Simulations (here: Front-Tracking model), center: Euler-Lagrangian method (also referred to as the Discrete Bubble Model) and right: Euler-Euler method (also indicated by the Continuum or Two-Fluid Model). From le to right, the amount of detail decreases while the scale of the simulation increases.
several techniques are available, each with their own strengths and weaknesses. An outline of the different models is provided below, with a discussion on the selection of the model used in this work.
.. Overview of DNS techniques for bubbly flows
Direct Numerical Simulations are computer simulations that resolve the physics based on first principles, without any assumptions or simplifications. Since, for bubbly flows, the flow phenomena take place at small scales, the spatial and temporal resolution of such computa-tions must be relatively high. is makes the simulacomputa-tions computationally very expensive, and a reasonable wall clock time can only be achieved by limiting the domain size. How-ever, provides full insight in the dynamics of the flow, including the deformation of the
C . I
interface and the micro-structure of the flow field.
A variety of techniques that have been developed in the literature to simulate multiphase flows with deformable interfaces are shortly discussed below. In these simulations, the sur-face tension force calculation is a very important feature of these models. As such, the repre-sentation of the gas-liquid interface is of crucial importance. In the different techniques the interface between the bubble and the surrounding liquid is either captured (reconstructed, e.g. from the phase fraction in a computational cell) or tracked (e.g. using tracking points). Apart from the methods discussed in the following sections, which are considered the most important modelling techniques currently used for the direct numerical simulation of bub-bly flows, also other techniques such as Shock Capturing and Marker Particle exist, which are discussed in van Sint Annaland et al. (, ). e key characteristics of the different techniques have been summarized in Table ..
Level-Set
e Level-Set () method (a.o. Sussman et al. (, )) is used in many different fields, including computational fluid dynamics. It is based on the tracking of a zero level set func-tion through the flow field. Since it explicitly tracks the interface, the interface is sharp. However, the method is intrinsically not mass-conservative, and even higher-order advec-tion schemes cannot prevent this behaviour. It is therefore oen coupled to another popular method; Volume-of-Fluid (see below), which results in the Coupled Level-Set and Volume of Fluid () (see e.g. Sussman and Pucke, ). van der Pijl et al. () have described a different approach of a mass-conservative method, which shares its roots with , and also Smolianski et al. () proposed a mass conserving level-set correction step, with which they performed 2D bubbly flow computations.
Volume of Fluid
e Volume of Fluid () method (Hirt and Nichols, ; Youngs, ) is a true front-capturing method. e parameterisation of the interface takes place by evaluating the mass fraction of a specific phase in each computational cell. Since the resolution of the grid dic-tates the accuracy with which the volume fraction can be tracked, the interface is relatively diffuse. e elongation of a dispersed element may lead to artificial break-up due to the lack of resolution, if the volume enclosed by the interface gets thin enough. Because the inter-face is not explicitly tracked, problems associated with inaccuracies in the computation of the surface tension force may arise (e.g. artificial parasitic currents at the interface), making it difficult to simulate small air bubbles (db< 2mm) in water, due to the high density ratio
and a high surface tension force (Dijkhuizen et al., ). It is, however, intrinsically volume conservative. Novel improvements in the numerical implementations can reduce spurious currents due to inaccurate surface tension calculation considerably. e mentioned methods therefore get the best out of both methods to simulate multiphase flows with sharp interfaces and volume conservation. In both and , the density and viscosity values
.. M Table .: O ver vie w of te chniques for multi-fluid flo w s (r epr oduce d fr om van Sint Annaland et al. ( )). Metho d A dvantag es Disadvantag es Le vel-Set Conceptually simple Limite d accuracy Easy to implement Loss of mass (v olume ) Sho ck-capturing Straightfor ward implementation Numerical diffusion Abundance of adv ection schemes ar e available Fine grids re quir ed Limite d to small discontinuities Mark er particle Extr emely accurate Computationally exp ensiv e Robust Re distribution of mark er particles re quir ed A ccounts for substantial top olog y chang es in interface SLIC V OF Conceptually simple Numerical diffusion Straightfor ward extension to thr ee dimensions Limite d accuracy Mer ging and br eak ag e of interfaces occur s automatically PLIC V OF Relativ ely simple Difficult to implement in thr ee dimensions A ccurate Mer ging and br eak ag e of interfaces occur s automatically A ccounts for substantial top olog y chang es in interface Laice Boltzmann A ccurate Difficult to implement A ccounts for substantial top olog y chang es in interface Mer ging and br eak ag e of interfaces occur s automatically Fr ont-Tracking Extr emely accurate Mapping of interface mesh onto Eulerian mesh Robust D ynamic remeshing re quir ed A ccounts for substantial top olog y chang es in interface Co alescence and br eak-up of interfaces re quir es subgrid No artificial mer ging or br eak-up mo del
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are regularized according to the colour function/mass fraction. e concepts are relatively simple to implement (van Sint Annaland et al., , ), and is used for a wide variety of multiphase flow problems (see Osher and Fedkiw, ).
Laice-Boltzmann
Another popular technique to simulate bubbly flows is the Laice-Boltzmann Method (), which is based on the Boltzmann equation instead of the Navier-Stokes equations. Since is highly parallellizable, it is in theory capable of simulating bubbly flows at relatively large scales in an efficient manner. e downside of the is that low viscosities for the liquid phase (as seen in the air-water regime) and very small bubbles generally causes numerical difficulties (Sankaranarayanan et al., ). Recently, Yu and Fan () have proposed a multi-relaxation-time method that enhances the numerical stability at low viscosities for small bubbles, showing that the intensity of spurious currents dropped by more than a factor 10, making such simulations possible. However, to the best knowledge of the author, no work using has appeared yet discussing the hydrodynamics of bubble swarms for gas-liquid dispersed systems.
Front-Tracking
Finally, the Front-Tracking method is a well-known technique (a.o. Unverdi and Tryg-gvason, ; van Sint Annaland et al., ; Dijkhuizen et al., ; Tryggvason et al., ; Popinet and Zaleski, ) to simulate multiphase flows with deformable interfaces in full detail. As the name suggests, the Front-Tracking method explicitly tracks the interface be-tween two phases (“front”), doing so using Lagrangian control points distributed over the surface, which is the main difference with the other methods outlined here. As a conse-quence, the , and models exhibit automatic, potentially unphysical, coalescence when two bubbles are in close proximity. Additional measures in such codes can prevent this behaviour. In contrast, the Front-Tracking model needs a sub grid model to allow bub-bles to merge (or break up). For example, Singh and Shyy () can change the topology of two separate interface meshes to a single mesh (coalescence) based on the phase fraction that is observed by two probes normal to each mesh cell. However, without such a feature, the Front-Tracking method is uniquely qualified for studying swarm effects without coales-cence and break-up effects. Secondly, the superior interface resolution of the Front-Tracking model is potentially more accurate in the description of surface tension forces. is is es-pecially important in the (oen used) air-water system, where large density jumps at the interface and high Reynolds numbers are inclined to produce spurious currents.
It is because of these characteristics that the Front-Tracking modeling technique has been chosen to perform simulations to study bubble swarms in great detail. Of course, Front-Tracking also has some drawbacks, most notably the fact that the volume of the dispersed phases is not intrinsically conserved. Also, the need for restructuring the interface mesh is
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generally regarded as a drawback in literature (van Sint Annaland et al., ). is thesis outlines numerical methods to properly handle and even eliminate these disadvantages.
.. On break-up and coalescence
e focus in this thesis lies on the simulation of bubble swarms. For an unambiguous inter-pretation of the simulation results, it is imperative to have a precisely defined framework for the simulations. is means that the composition of the swarm, i.e. number of bubbles, the volume of the bubbles and the gas fraction, should not change during the simulation, e.g. as a result of coalescence and break-up.
e main consideration of not including interface merging and break-up in this work is that first the elementary hydrodynamics of a bubble swarm without these effects should be understood. Subsequently, sub-grid (semi-empirical) models to describe break-up and coalescence could be implemented to study their influence but this is outside the scope of this thesis.
. Bubble behaviour
Before studying the behaviour of bubbles rising in swarms, it is very important to first sum-marize the behaviour and classification of single rising bubbles in unbounded liquids. ese aspects will provide a context along which to present the results for bubble swarms. More details can be found in literature, for instance the work of Dijkhuizen () and Darmana ().
.. Dimensionless numbers for bubble classification
e behaviour of a single rising bubble in an infinite, initially quiescent liquid is influenced by a number of independent physical properties; the densities of the liquid and gas phase ρl
and ρg, the viscosities of the liquid and gas phase µland µg, the surface tension coefficient σ,
the gravitational acceleration vector g (or gz). Additionally, the average, equivalent bubble
diameter dband terminal rise velocity v are important. With these variables four important
dimensionless numbers can be obtained, that control the bubble behaviour:
• e Reynolds number represents the ratio of inertial forces over viscous forces, using the bubble diameter dbas a characteristic length scale.
Re = ρlkvkdb µl
(.)
• e Morton number is the ratio between viscous forces and interface forces, essen-tially defining the type of liquid in which the bubble is rising. e Morton number can take values in a very wide range, as in highly viscous oils it can exceed 105, while in
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as the ¹⁰log instead of the actual number (as seen in Figure .).
Mo = gzµ4l∆ρ ρ2
lσ3
(.)
• e Eötvös number represents the force of gravity (buoyancy) on a bubble compared to the surface tension (curvature). It characterises the shape of a bubble.
Eo = gzd2bρl
σ (.)
• Similar to the Eötvös number is the Weber number, defining a dimensionless number by taking the ratio of the inertial force over the surface tension of a bubble:
We = dbkvk2ρl
σ (.)
Note that the gas phase viscosity is not included in these numbers; it however does play a significant role in high-Morton liquids as well as in very pure (surfactant free) systems (Cli et al., ).
Since bubbles are deformable, the bubble size is represented as the equivalent bubble diameter deqor dbobtained using the bubble volume Vbvia equation ..
deq= 3 √ 6Vb π (.)
.. Bubble shape
Bubbles can have various (dynamic) shapes depending on the regime dictated by the Rey-nolds number, the Eötvös number and the Morton number as depicted in Figure .. A good description of the bubble shape can be obtained using . It is important to have a correct description of the shape since it, for instance, determines the rise velocity and the surface area through which mass transfer processes take place.
A gas bubble and surrounding liquid are separated by an immaterialistic, deformable interface. e molecules on the interface aract each other (cohesion), which gives an in-terface its surface tension: the ability to resist an external force. Inside the bubble, an ex-cess pressure balances the surface tension. As described by the Young-Laplace equation, this pressure difference is proportional to the interface curvature (the inverse of the bubble size): ∆p = σ ( 2 dxy + 2 dz ) = 4σ dsphere (.)
In Equation ., σ is the surface tension, dxy the horizontal diameter (perpendicular to the
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this leads to the equation on the right hand side. e resulting pressure difference is very high for highly curved interfaces (i.e. small bubbles). For larger bubbles, the pressure dif-ference per unit area is much smaller, making the bubble more capable to deform from a spherical shape.
e key factors involved in determining the eventual shape of a rising bubble are the non-dimensional numbers mentioned in Section .., based on which Cli et al. () and Grace () provide a number of categories in which a single, freely rising bubble can ap-pear (see Figure .). A bubble will remain spherical if the interfacial forces and/or viscous forces are large with respect to the inertial forces (low Reynolds and/or Eötvös numbers). Bubbles are typically referred to as ellipsoidal when the major and minor axis differ more than 10% in magnitude, which occurs at intermediate Reynolds numbers. In relatively in-viscid liquids (e.g. water) and intermediate Eötvös numbers, bubbles are wobbling and oen lack symmetry in the horizontal plane, at least when they are relatively large. Finally, large bubbles show a flat or dimpled base with a spherical cap. in pockets of air along the sides of a bubble are oen referred to as skirts.
To quantify the bubble shape, one may calculate the bubble aspect ratio. A slight in-convenience is that some sources in the literature use either the height vs. width ratio E, whereas some others use the major vs. minor axis χ (effectively the reciprocal value of the former for oblate spheroidal bubbles).
E = √dz dxdy
χ = major axis
minor axis = E−1 (.)
.. Hydrodynamic forces
A gas bubble in a bubble column moves under the influence of multiple forces, which in total determine the rise velocity of the bubbles. erefore, the forces effectively determine the residence time in the column, which can be used to evaluate various aspects of reacting flows, such as the conversion of a specific component.
Gas bubbles rise due to the buoyancy force, which results from the density difference between the phases. Yet there are more forces acting on a rising gas bubble, and this topic has gained significant scientific aention, especially in the field of single rising bubbles. e different forces contributing to the force balance on a bubble are the force of gravity
FG, pressure gradient FP, drag FD, li FL, and virtual mass FV M(also termed added mass).
Ftotal=FG+FP +FD+FL+FV M (.)
Using detailed simulation, dedicated experiments or analytical evaluation, the effect of these forces can be investigated. As a result, many correlations have been proposed, for instance, for the drag Mei et al. (); Tomiyama (); Dijkhuizen et al. () and li forces (Leg-endre and Magnaudet, ; Tomiyama et al., b; Dijkhuizen et al., ). e majority of these correlations, however, are only valid for a single, freely rising bubble in an infinite, initially quiescent liquid. e effect of swarms has gained more aention recently (Bunner
C . I 105 104 103 102 10 10-1 10-2 10-1 1 10 102 103 1 8 7 6 5 43 2 1 0 -1 -2-3 -4 -5 -6 -7 -8-9 -10 -11 -12 -13 -14 LOG M R e yn o ld s n u m b e r, R e Eötvös number, Eo Spherical Spherical cap Ellipsoid Dimpled Ellipsoidal-cap Skirted Wobbling
Figure .: e Grace (Grace, ) diagram shows bubble shapes as a function of the Mor-ton, Eötvös and Reynolds dimensionless numbers for single rising bubbles rising in infinite, initially quiescent liquids.
and Tryggvason, a; Simonnet et al., ; Smolianski et al., ; Dijkhuizen et al., ), but especially dense swarms have received lile aention. Also, how to account for differ-ent swarm compositions (differdiffer-ent bubble sizes) has only been touched upon, e.g. Göz and Sommerfeld (), who focus on the microstructure of the bubbles and the energy spec-trum in the liquid. Due to the dominance of the drag force with respect to the rise velocity calculation, this is a major topic in this thesis.
.. Mass transfer in gas-liquid flows
Many bubbly flows used in industry involve the exchange of mass between the gas and the liquid phase. Much research has been dedicated in describing the rate of gas-liquid mass transfer using simplified models with semi-empirical parameters. A short outline will be given in this section.
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mechanisms always includes molecular diffusivity, based on Fick’s law. Based on the inclu-sion of additional influences on mass transfer, different models on gas-liquid mass trans-port have been developed (Versteeg et al., ). e first type only considers stationary molecular diffusion through a liquid film to the well-mixed bulk. Whitman () and Lewis and Whitman () were the first to formulate the stagnant film theory. Other models, so called surface renewal theories, describe unsteady molecular diffusion into an infinite liquid medium (Higbie, ; Danckwerts, ), mixing the liquid elements in the film through turbulent eddies. irdly, multi-parameter models combine both molecular diffusion and interfacial turbulence (Toor and Marchello, ; King, ). Experiments performed by Colombet et al. () indicate that, while liquid fluctuations are omnipresent, the boundary layer does not get affected by the hydrodynamical interactions between the bubbles, but these experiments are only the first that provide this information in such detail and account for gas fractions up to 16.5%.
e mass transfer rate between a gas and a liquid is oen represented by the overall mass transfer coefficient kLa, with kLthe true mass transfer coefficient and a the specific surface
area where mass exchange takes place. While a can be estimated using several heuristics (e.g. from gas hold-up, bubble size distribution), kLoen cannot and several correlations have
been proposed. While no single work can comprehensively discuss all previous research in this area, Kulkarni () provides a good survey of a large number of relations for kL.
Although many semi-empirical correlations for gas-liquid mass transfer are available for various different conditions, many details on the effects of multiple bubbles on the efficiency of mass transfer are still unknown. Consider for example the mass transfer from a bubble in a swarm; a bubble that enters a high-concentration wake of a preceding bubble will show a lower mass transfer rate, due to the lower gradient in concentration at the bubble surface. On the other hand, the increased fluctuations in the liquid may refresh the boundary layer of a bubble more quickly, which may therefore increase its mass transfer as compared to a single rising bubble. While liquid probing experiments are only able to roughly estimate the mass transfer over a specific reactor volume, including mass transfer is promising to help gaining detailed insight in the different aspects of mass transfer in bubble swarms. Unfortunately, such numerical tools are not available for dense bubble swarms. In this thesis, the development and validation of such a tool is outlined.
. Thesis outline
A study of the hydrodynamics and mass transfer prevailing in bubbly flows is performed using Direct Numerical Simulations, and for this purpose the Front-Tracking modeling tech-nique has been selected. e Front-Tracking model has been extended in its capabilities of simulating bubble swarms at high gas fraction and to prevent the volume loss encountered by a traditional implementation of the model. e numerical approach is outlined in Chap-ter .
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rise velocity, and hence its residence time in the column: the drag force. For relatively large, wobbling bubbles, a correlation for the drag coefficient is derived in Chapter . Additionally, the drag on bubbles rising in a swarm consisting of bubbles of multiple sizes is discussed. In Chapter , the applicability of the developed drag closure for bubbles in a bi-disperse bubble swarm is demonstrated. e drag closure is extended to a larger range of physical properties (viz. spherical bubbles and highly viscous liquids) in Chapter . Additionally, the effect of bubble clustering within a bubble swarm is discussed.
Bubble velocity distributions are compared to experiments in Chapter . is chapter also shows the effect of rising bubbles on the turbulent energy spectrum.
e implementation and derivation of correlations of a mass transfer model combined with the Front-Tracking model is described in Chapter . is chapter wraps up with a discussion concerning mass transfer in bubble swarms, based on simulations with the newly implemented model.
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Abstract
In this apter a detailed overview is provided of the Front-Traing model and novel features in its nu-merical implementation. e flow field is described by the incompressible Navier-Stokes equations, whi include an additional term for the surface tension force and a sophisticated manner of treating the large pressure jump at the interface.
e traditional interface mesh handling routines cause (artificial) anges in the volume of the dispersed elements due to the movement of the interface marker points and restructuring of the interface mesh. An improved remeshing tenique to correct this without disturbing the shape of the interface is proposed and implemented.
e four elementary remeshing operations viz. edge splitsing, collapsing, swapping and smoothing, have been implemented in su a way that every operation is performed in a fully conservative manner fol-lowing the work by Kuprat et al. (). To account for the volume anges due to the movement of the interface (repositioning of the Lagrangian marker points), a global volume correction method is proposed, moving ea interface marker edge in the direction with minimal marker displacement. To prevent in-tersection of different interface meshes in dense bubbly flows, a tree sear algorithm efficiently traces intersecting marker points, repositions them and corrects for volume differences.
e new teniques have been validated by comparing the drag coefficient of single rising bubbles to re-sults that have been obtained and described in previous work (Dijkhuizen et al., ), whi have been validated in its turn by experiments. At the end of this apter, the computational facility on whi the production runs are performed have been described shortly.
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. Introduction
Front-Tracking is a powerful numerical technique to simulate multi-fluid systems in which the interface between the phases is explicitly tracked by Lagrangian marker points. e fluid flow is resolved on a background mesh, a Eulerian grid, including the effect of surface tension forces from the bubble interface. e interface is, in turn, affected by the flow field, since the interface markers are displaced by the interpolated fluid velocity. In short, this is the Front-Tracking method, and some of these features are illustrated in Figure .. It can in principle be used to study the hydrodynamics of multiple separate interfaces as encountered in dense bubbly flows, however the traditional implementation of the Front-Tracking model requires further improvement and addition of several features.
In the first place, the traditional Front-Tracking model is not capable of simulating very dense bubble swarms. While exactly these high gas hold-up conditions are relevant for industrial applications, the numerical stability of this model has to be improved to cope with the close vicinity of multiple interfaces and their interactions via the liquid phase.
Secondly, the handling of the interface, which may constitute of more than marker points per dispersed element, is not intrinsically mass conserving. In this chapter new remeshing techniques, based on the work by Kuprat et al. (), are described in detail, which ensure volume conservation of the dispersed elements.
is chapter will start with a short description of the most important aspects of the Front-Tracking implementation in its elementary form (based on the description in Roghair et al. (), Dijkhuizen et al. () and Roghair et al. (a)). is includes a description of hydrodynamic modelling, numerical solution techniques, surface tension calculation and interface pressure drop handling and interface displacement routines. Next, the issues con-cerning volume changes are discussed and a new implementation of the interface meshing techniques is discussed, aiming to overcome these problems with volume conservation and multiple interfaces. e chapter closes with a short contribution about the programming and computational resources.
. Front-Tracking algorithm
e Front-Tracking model used in this work has been in development for about years. e hydrodynamics discretisation and implementation using the Finite Volume Method is also described in detail in Dijkhuizen et al. (). e basic routines of the algorithm are displayed in Figure ., of which the mass transfer part is discussed separately in Chapter .
.. Hydrodynamics modelling
Consider a two-phase flow in a rectangular computational domain using a Cartesian coor-dinate system with directions x, y and z. e governing equations of the fluid flow field are given by the incompressible Navier-Stokes equation and the continuity equation, where the
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(a) (b)
Figure .: Snapshots of the Front-Tracking model; (a) a typical simulation of a bubble swarm with streamlines indicating the fluid velocity. (b) A zoomed snapshot of a rising Front-Tracking bubble (at a very low resolution for illustration purposes), showing the tracking points and surface mesh, and the background grid with staggered velocity vectors. e colors of the background grid indicate the pressure profile, and the colors of the velocity vectors represent the magnitude.
physical properties depend on the local phase fractions.
ρ∂u ∂t + ρ∇ · (uu) = −∇p + ρg + ∇ · µ [ ∇u + (∇u)T]+F σ (.a) ∇ · u = 0 (.b)
where u is the fluid velocity and Fσ representing a singular source-term for the surface
tension force at the interface. Since the velocity field is continuous even across interfaces, a one-fluid formulation is used to describe the fluid flow for all phases at once. However, the shear stress and pressure are not continuous across the interface and the respective jump condition is given by Eq. ..
[−pI − τ ] · n = Fσ· n (.)
e equations are solved with a finite volume technique using a staggered discretisation, i.e. the velocities are computed at the cell faces, while the scalar quantities are evaluated at the cell centres. e flow field is solved using a two-stage projection-correction method. Both the convection terms and the non-diagonal terms in the stress tensor are calculated in an explicit manner. e diagonal components of the stress tensor are discretised implicitly, which eventually yields a semi-implicit treatment of the stress terms in the Navier-Stokes equations. is enhances the numerical stability for high-viscosity flows and allow using
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..
. . . .
. Initialisation... . .
. t < t...end Finalisation... .
. Calculate new time step (Section ..)... . .
. . . .
. Calculate physical properties... . .
. Estimate new velocity field... . .
. Calculate pressure corrections... . .
. Correct velocity field... . .
. Front advection... . .
. Surface remeshing (Section .)... . .
. Mass transfer (Chapter )... . .
. Save solution... . . . . . . . yes . no . t := t + ∆t . Fluid flow . Interface .
Figure .: Front-Tracking global algorithm
larger time steps (Dijkhuizen et al., ). Due to the semi-implicit treatment, the three velocity components ux, uy and uz can be solved separately in the projection step using
an optimised Incomplete Cholesky Conjugate Gradient () matrix solver. e surface tension force is taken into account explicitly (see Section ..). Subsequently, the total mass balance is enforced via a pressure correction step, again using the matrix solver. e boundary conditions with which the problem is solved can be adjusted between free-slip, no-slip and periodic boundary conditions. e matrix solver is adjusted to efficiently handle the bands required to solve a periodic system, while only bands are needed for no-slip or free-slip boundaries.
.. Surface mesh
e interface is parameterised by Lagrangian tracking points. e connectivity of the points build a mesh with triangular cells, called markers (Figure .b). is section will cover the calculation of the surface tension force, handling of the pressure drop across the interfaces, the mesh advection and physical properties calculation, while the remeshing will get detailed aention in section ..
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Surface tension force
In Eq. .a, Fσrepresents the surface tension force, a vector quantity that can be directly
calculated from the positions of the interface markers which is one of the major advantages of the Front-Tracking method. e individual pull-force of neighbouring marker i acting on marker m can be computed from their normal vectors and joint tangent as illustrated in Figure .a:
Fσ,i→m= σ (tmi× nmi) (.a)
e shared tangent tmiis known from the control point locations, and the shared normal
vector nmiis obtained by averaging, from which we can discard one term due to
orthogo-nality: tmi× nmi= 1 2 (t|mi{z× nm}) =0 +(tmi× ni) (.b)
Hence, the total surface tension force on a marker m is obtained by summing Eq. .a for all three neighbouring markers:
Fσ,m= 12σ ∑ i=a,b,c (tmi× ni) = 12 ∑ i=a,b,c Fσ,i→m (.c)
As a result, three pull forces on each marker are defined which yield a net force inward, opposing the pressure jump. For a closed surface, the net surface tension force on the entire object will be zero. is force Fσis then mapped to the Eulerian cells closest to marker m
(within a window h), using mass-weighing (Deen et al., ) (regularized Dirac function
D): Fσ(x) = ∑ mρ(x)D (x− xm)Fσ,m ∑ mρ(x)D (x− xm) (.a) D (r) = dx(rx) dy(ry) dz(rz) (.b) dx(rx) = 1−|rx| h if |rx| ≤ h 0 if |rx| > h (.c) Pressure jump
A difficult aspect of involving small bubbles (e.g. db = 1.0mm air bubble in water)
is the large pressure jump at the gas-liquid interface, which may cause artificial parasitic currents that may affect the final solution significantly. While these spurious currents are decreased by the mass weighing implementation (explained above), Popinet and Zaleski
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(a) Surface tension calculation on marker m with neighbouring
markers a, b and c. (b) Spurious currents in the central plane of a1.0mm bubble in a zero-gravity environment.
Figure .: (a) e surface tension calculation on marker involves the calculation of three pull-forces using the tangent and normal vectors shared with the neighbouring marker. (b) e resulting spurious currents (strongly magnified for visualization) show a maximum of
kuk = 1.8 × 10−3m/s for a db= 1.0mm bubble.
() demonstrated that the coupling between the surface forces and the pressure jump is crucial to further minimize them. e Front-Tracking model uses a method similar to Renardy and Renardy () and Francois et al. (), where the pressure forces will be extracted from the surface forces at the interface, only mapping the resulting net force.
First note that the partial pressure drop (i.e. the pressure jump [p] resulting from the surface tension force on a single marker), can be calculated using Eq. ., if the shear stress in the normal direction is neglected.
∫ ∂S [p] dS = ∫ ∂S Fσ· n [p] = ∫ ∂S∫ Fσ· n ∂SdS = ∑ m∑Fσ,m· nm mSm (.)
e sum of the surface forces of all markers yields the pressure jump of the bubble as a whole. By distributing the total pressure jump equally back to the Eulerian mesh, the pressure jump is incorporated in the right-hand side of the momentum equations. For interfaces with a con-stant curvature (i.e. a sphere), the pressure jump and surface tension cancel each other out exactly, and if the curvature varies over the interface, only a relatively small ne force will be transmied to the Euler grid. e advantage of using this technique with Front-Tracking is that the location of the interface is exactly known, and the surface tension force and pres-sure jump terms are mapped to this exact location (as opposed to front-capturing techniques as used by Francois et al. () or Renardy and Renardy ()). is is much more
