Practical LES

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Technical Report 1997:11ISSN 0348-467X

ISRN KTH/MEK/TR--97/11--SE

Large Eddy Simulation of Particulate

Turbulent Channel Flows

Koji Fukagata

May 1997Technical Reports from

Royal Institute of TechnologyFaxen Laboratory

S-100 44 Stockholm, Sweden


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Fukagata, K. 1997 Large Eddy Simulation of Particulate Turbulent Channel FlowsFaxen LaboratoryRoyal Institute of TechnologyS-100 44 Stockholm, Sweden

Abstract

This thesis deals with numerical simulations of particulate turbulent channel flows.Turbulent velocity field is simulated using large eddy simulation (LES) and thetrajectories of particles are tracked by solving a particle equation of motion.

The methodology used is at first assessed through comparisons between predic-tions and the available data for a turbulent deposition of small particles in a channel.

The prediction is in a good agreement with an empirical relation by Wood1 and sim-ulations by Li & Ahmadi2.

The assessment is done also with comparisons on the statistics of 70 m copperparticles in a turbulent channel flow at Re = 180. The modification of fluid flow dueto particles is neglected, i.e. one-way coupling is considered. Very good agreementis found with a direct numerical simulation by Rouson & Eaton3 and a large eddysimulation by Wang & Squires4. The development time used in Wang & Squires isfound to be short for obtaining a fully developed flow from the given initial condition.

The statistics of 70 m copper particles and 50 m glass particles are accu-mulated after the flow has fully developed, and compared with the earlier data

documented in the literature. Some discrepancies are indicated. The statistics arealso used to evaluate different terms in the averaged Eulerian particle momentumequation, based on which a simplified version of the equation is derived.

At last, simulations for the same parameters are performed taking into accountthe modification of fluid velocity field due to the presence of particles, i.e. two-waycoupling. Statistics are compared with the cases of one-way coupling. Suppressionof the turbulence is observed in the cases of 50 m glass particles and 70 m copperparticles at mass flow rates of 0.4 and 3.0, respectively. An equation describing therelation between the modulation of fluid stresses and interphase stress is derived.

The thesis consists of a survey on the development time for the flows, force

balances in the particle phase and modulations in the carrier fluid turbulence dueto the presence of particles followed by three papers, describing specific results.

1WOOD N. B., 1981, J. Inst. Energy, 76, 76-93.2LI A. & AHMADI G., 1993, Int. J. Engng Sci., 31, 435-451.3ROUSON D. W. I. & EATON J. K., 1994, Proceedings of the 7th Workshop on Two-phase

Flow Predictions, Erlangen, Germany.4WANG Q. & SQUIRES K. D., 1996, Phys. Fluids, 8, 1207-1223.


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Preface

This thesis deals with numerical simulations of particulate turbulent channel flows,using large eddy simulation and Lagrangian particle tracking methodology. Thethesis is based on the following three papers (updated from the original thesis).

Paper 1:

FUKAGATA K., ZAHRAI S. & BARK F. H., 1997, Large eddy simulation of particlemotion in a turbulent channel flow appeared partly in two papers:

Fukagata, K., Zahrai, S. & Bark, F. H., 1997. Large eddy simulationof particle motion in a turbulent channel flow. Proc. 1997 ASME Fluids Eng.Div. Summer Meeting(CD-ROM), Paper No. FEDSM97-3591, 1-6.

Fukagata, K., Zahrai, S. & Bark, F. H., 2004. Dynamics of Brownianparticles in a turbulent channel flow. Heat Mass Transfer 40, 715-726.

Paper 2:FUKAGATA K., ZAHRAI S. & BARK F. H., 1998, Force balance in a turbulentparticulate channel flow, Int. J. Multiphase Flow 24, 867-887.

Paper 3:

FUKAGATA K., ZAHRAI S. & BARK F. H., 1998, Fluid stress balance in a tur-bulent particulate channel flow. Proc. 3rd Int. Conf. Multiphase Flow(CD-ROM),Paper No. 157, 1-8.

Other reports related to this thesis:

FUKAGATA K., 1995, Simulation of particle motion in a turbulent veloc-ity field, part I, Technical Report, SECRC/KB/TR-96/162E, ABB CorporateResearch.

FUKAGATA K. & ZAHRAI S., 1996, Simulation of particle motion in a tur-bulent velocity field, part II, Technical Report, SECRC/B/TR-96/107E, ABBCorporate Research.


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a)

z+

x+

b)

z+

x+

Figure 1: Modification of turbulent structure due to 50 m glass particles at mass flow

ratio of 0.4. See Paper 3.a) one-way coupling; b) two-way coupling.Black point, particle.Red, high fluid velocity; blue, low fluid velocity.

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a)

z+

x+

b)

z+

x+

Figure 2: Modification of turbulent structure due to 70 m copper particles at mass flow

ratio of 3.0. See Paper 3.a) one-way coupling; b) two-way coupling.Black point, particle.Red, high fluid velocity; blue, low fluid velocity.

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Contents

1 Introduction 1

1.1 Particulate turbulent flow . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Aim of this study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Overview of the papers 72.1 Paper 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Paper 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Paper 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Acknowledgement 11

Bibliography 13

A Paper 1 17

B Paper 2 47

C Paper 3 77

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vi CONTENTS


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Chapter 1

Introduction

1.1 Particulate turbulent flow

Two-phase flow can be found almost everywhere from the processes in the naturesuch as the clouds on the sky, which have existed since long time before mankindappeared on the earth, to the industrial processes such as in the pipes in nuclearreactors which were realised by assembling modern technology. As well as most ofthe single phase flows, two-phase flows are often turbulent.

Since perhaps the first scientific observation on the turbulence by da Vinci in15th century, turbulence has been an attractive research area. Numerous experi-ments were carried out to understand the characteristics of turbulence and to modelit mathematically. The effort put on turbulence research has brought two monu-mental achievements in the turbulent theory, i.e. the mixing length model and the

Kolmogorov spectrum.Despite the enormous effort put on the subject, turbulence has not fully been

understood. According to Tennekes & Lumley (1972), turbulence can be character-ized by irregularity, diffusivity, large Reynolds number, three-dimensionalvorticity fluctuations, dissipation, continuum and turbulent flows are flows.Among others it is the irregularity that makes the deterministic approach difficult.

This nature of turbulence will be a hurdle also when, for example, turbulentflows including many small particles are considered. Even if the particle equation ofmotion is simple, the motion of the particles becomes complicated due to underlyingirregularity of carrier flow. One exception is the case when the particles have in-

finitely large inertia compared to the fluid and do not interact with the carrier fluid.In that case particles can be treated similarly to the theory for gas molecules, seeSundaram & Collins (1997). In most cases which are of interest, however, particlesdo not have infinitely large inertia and do interact with the carrier fluid. Therefore,similarly to the approach for single phase turbulent flow, statistical methods are tobe relied on.

Turbulent diffusivity is known to cause rapid mixing of the containments suchas heat and particles. The diffusivity of the particles in a turbulent flow is of greatinterest when a filtering, separating or mixing apparatus is designed, see e.g. vander Velde (1996).

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2 CHAPTER 1. INTRODUCTION

After the invention of computers, many researchers examined to predict thestatistics, such as the mean velocity, of turbulent flows by solving the averagedNavier-Stokes equation. Since the Navier-Stokes equation involves an advection termwhich is non-linear, extra unknowns called Reynolds stresses appear in the averaged

equation. Therefore a relevant closure model is needed to close the equation.One of the most popular closure models is the k model (Jones & Launder,1972), which is based on the mixing-length theory. The Reynolds stress is approx-imated to be the product of the turbulent viscosity and the mean strain rate. Theturbulent viscosity is assumed to be proportional to k2/, where k is the turbulentkinetic energy and is the dissipation rate which are obtained by solving the trans-port equations for k and . Although the k model has some disadvantages suchthat it cannot predict a highly anisotropic flow or a flow under rotation, it is widelyused for industrial purposes owing to its simplicity and numerical stability.

During the passed few decades, due to the rapid development of computationalability, numerical simulations of turbulent flows without high degree of modeling has

been made possible. Direct numerical simulation (DNS) uses very fine computationalmeshes such that the smallest eddies comparable with the Kolmogorov length scalecan be captured. In DNS, the Navier-Stokes equation is solved as it is and thereis no need for any closure models. Large eddy simulation (LES) uses also finemeshes, but coarser than those in DNS, such that relatively large energetic eddiescan be resolved according to the averaged Navier-Stokes equation. The relativelylarge eddies in this context designate the same sizes or larger than those comparablewith the integral length scale in the flow. The smaller, subgrid scale, eddies areconsidered to be independent of the geometry of the flow and isotropic. Thereforethey can be modeled using subgrid scale model.

DNS and LES have frequently been used to investigate pure turbulence. How-ever, recently, these methods are also applied to more complex flows such as turbu-lent two-phase flows.

Two-phase flows consist of two different phases, e.g. gas and liquid, gas andparticle or liquid and particle. In general these different phases interact each otherand change their shape of the surface and transit from one flow pattern to another.For this reason, two-phase flows are usually divided into different flow regimes,which are treated separately. For example, gas-liquid two-phase flows in a pipe canbe classified depending on the gas flow rate and the liquid flow rate, e.g. bubbleflow, slug flow, churn flow, annular flow, dispersed flow etc..

The interest in this thesis is focused on the dispersed flow, to be more specific,fluid-particle flows consisting of a fluid as a continuous phase and spherical solidparticles as dispersed phase. As the carrier flow considered here is turbulent, theparticulate flows considered in this study are often called particulate turbulent flows.


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1.2. PREVIOUS WORK 3

1.2 Previous work

To understand the behavior of an ensemble of particles in a flow of a continuousphase, it is at first necessary to have a knowledge about the motion of one singleparticle in the flow.

Basset (1888), Boussinesq (1903) and Oseen (1927) separately examined themotion of a sphere in a fluid settling down under gravitational force. The particleequation of motion developed by them is called Basset-Boussinesq-Oseen (BBO)equation. Tchen (1947) extended the BBO equation such that it could describe theparticle motion in an unsteady flow. Extension was also examined for non-uniformflows, however, inconsistency was pointed out by Corrsin & Lumley (1956).

Later, Buevich (1966) & Riley (1971) made a correction in the Tchens expressionand Maxey & Riley (1983) presented a particle equation of motion, which is oftenused today, consisting of the following terms:

1. drag force

2. gravitational force

3. fluid acceleration

4. added mass

5. history effect

A correction term due to the velocity curvature of the carrier fluid, Faxen correction(Faxen, 1922) is included in the drag force, the added mass and the history effect

appearing in the expression by Maxey & Riley (1983). Under certain conditions,contributions from the following terms may be taken into account:

6. lift force

7. Brownian motion

8. Cunningham slip correction

The lift force may be included when the shear rate of carrier flow is large, i.e.Saffmans lift force (Saffman, 1965), or when the particles are rotating quickly. TheBrownian motion and the Cunningham slip correction are of importance in the cases

where the diameter of particles is comparable to the mean free path of carrier fluidmolecules.

When the particle to be considered is much heavier than fluid e.g. p/f 103, the particle equation of motion can reduce from dimensional analysis to thatconsisting of only drag force and gravitational force. The reduced particle equationof motion normalized by the Stokes number, includes only two parameters, theStokes number, which is the ratio of particle time scale to fluid time scale, andthe drift parameter, which reflects the gravitational effect. The effects of thesetwo parameters on the diffusivity of particles are studied and summerized by Stock(1996).


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There are mainly two different ways to deal with the behavior of large numberof particles in a flow, the Eulerian-Eulerian approach and the Eulerian-Lagrangianapproach. In the Eulerian-Eulerian approach, the particle phase is treated as acontinuum and the equations for all phases are solved at the same time. In this

case, closure models are needed also for the particle phase equation similarly tothe equation for fluid. In the Eulerian-Lagrangian approach, the trajectories ofparticles are tracked separately according to the particle equation of motion, whichis expressed in the Lagrangian frame.

The simplest case to be studied is a particulate flow with a homogeneous turbu-lence. Shih & Lumley (1986) reported on second-order closure modeling of particledispersion in a decaying homogeneous turbulence and the simulation was found tobe in good agreement with the experimental data by Wells & Stock (1983). Directnumerical simulation and Lagrangian tracking of motion of particles in a decayinghomogeneous turbulence was performed by Elghobashi & Truesdell (1992) and theimportant quantities such as Lagrangian velocity auto-correlations and diffusivityof the particles were studied.

Shear flows and jet flows are the next simplest cases because solid boundariesneed not to be taken into account. Simonin, Deutsch & Boivin (1995) developed asecond-order closure model of particle fluctuating motion in a turbulent shear flowand the time evaluation of the various correlations were found in good agreementwith the data from large eddy simulation carried out by them. A turbulent partic-ulate jet flow was simulated by Chen & Wood (1985) using a k model. Effectof the particle size and particle loading on the dispersion was studied by comparingthe simulation data with available experimental data by, e.g. Boguslawski & Popiel(1979), Wall, Subramanian & Howley (1982) and Moderrass, Tan & Elghobashi

(1984).For the practical applications, such as predictions of behaviors of particulate

flows appearing in industrial systems, it is necessary to understand the motion ofparticles in geometries with rigid walls. The simplest examples with such geometriesare straight pipes or channels bounded with two parallel walls. In fact most ofthe experiments, except jet, are carried out in such pipes or channels. To observeparticle motion, laser Dopper anemometry, by e.g. Maeda, Hishida & Furutani(1980), Lee & Durst (1981), Tsuji, Morikawa & Shiomi (1984), Kulick, Fessler &Eaton (1994) and Kaftori, Hetsroni & Banerjee (1995), or high-speed video, by e.g.Rashidi, Hetsroni & Banerjee (1990) and Nino & Garcia (1996), has been used. It

was observed in the experiments that the carrier fluid turbulence are suppressedby suspension of small particles and enhanced by large particles. Hetsroni (1989)and Gore & Crowe (1989) summerized the data from the available experiments anddiscussed the criteria between the enhancement and the suppression of turbulence.It was also found that the particle with finite, small inertia tend to concentrate inthe regions with low shear in the streaky turbulent structure near the wall.

Theoretical studies about these wall-bounded particulate flows using numericalsimulations have also been reported. The main research interest may be 1) to clar-ify the phenomena of deposition and entrainment of particles near the wall and 2)to provide statistical moments of particles. Ounis, Ahmadi & McLaughlin (1991)


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1.3. AIM OF THIS STUDY 5

reported on the dispersion and the deposition of very small particles using DNS andLagrangian particle tracking. The Browinan motion due to interaction between par-ticles and fluid molecules was also taken into account. With the same methodology,the entrainment process was studied by Soltani & Ahmadi (1995). Deposition in

a pipe flow was simulated by Wijttewaal & Oliemans (1996). In Wang & Squires(1996a), LES was used for simulations of particle deposition in a channel insteadof DNS. Li & Ahmadi (1993) studied the deposition rate of particles in a channelusing a quasi-turbulent fluid velocity field and Lagrangian particle tracking. Thepredicted deposition velocity was in good agreement with the empirical relation byWood (1981). The method was also applied to the prediction of deposition rate ina complex geometry (Li, Ahmadi, Bayer & Gaynes, 1993). Swailes & Reeks (1994)used a kinetic theory for high inertia particles to simulate the particle deposition.

The statistics of relatively high inertia particles obtained from DNS and LEShave been reported by several researchers. Pedinotti, Mariotti & Banerjee (1992)carried out a simulation of motion of light particles in a horizontal channel using

DNS and Lagrangian particle tracking. The simulation data was compared with theexperimental data by Rashidi, Hetsroni & Banerjee (1990). Pan & Banerjee (1996)took into account two-way coupling, i.e. the particle motion modifies the fluidvelocity field, and the further comparison with the experiment by Rashidi et al.(1990) was done. Rouson & Eaton (1994) and Wang & Squires (1996b) performedDNS and LES, respectively, to simulate the experiments by Kulick et al..

Two-equation models, i.e. Eulerian-Eulerian approach, of the particulate turbu-lent channel flow have also reported by Rizk & Elghobashi (1989), Bolio & Sinclair(1995) and Cao & Ahmadi (1995). As an example, comparison between the modelby Rizk & Elghobashi and experiments by Maeda et al. (1980) and Tsuji et al.

(1984) shows that the model by Rizk & Elghobashi has predicted the behavior ofparticles and fluid reasonably well. For more accurate prediction, however, furtherimprovement of the existing models is likely to be needed.

1.3 Aim of this study

The final goal of this work is to study the effect of presence of particles on the turbu-lent structure near the wall and to make a simple, more accurate model to predictsuch flows. For these purposes, a computational code to calculate the motion ofparticles in a turbulent channel flow was developed based on the large eddy simula-tion code developed by Zahrai, Bark & Karlsson (1995). The code was prepared toaccount for the modulation of the turbulent fluid velocity field due to the presenceof particles. Various statistics both of the fluid and of the particles are calculated,which will provide the necessary data for the modeling.

This thesis is organized in the following way: Overviews of the papers, Paper1, Paper 2 and Paper 3, are presented in Chapter 2. Specific topics are discussedin these papers. Each paper is in a complete form and can be read also as anindependent article. The papers are attached to this introductory note.


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Chapter 2

Overview of the papers

2.1 Paper 1

The methodology was validated through comparisons with available data in litera-ture.

In the first part in the paper, statistics of very small particles in a turbulent chan-nel flow were presented. The nondimensional particle relaxation time, +p , rangedfrom 104 to 101.

The deposition velocity was compared with the result of quasi-turbulence simu-lation by Li & Ahmadi (1993) and the empirical relation suggested by Wood (1981).While the deposition velocities obtained using a time-dependent turbulent veloc-ity field were in good agreement with those predicted by Li & Ahmadi (1993) andWood (1981), a simulation using a frozen velocity field could not predict reasonable

deposition velocities except in the ranges of particle diameter where the Brownianmotion was dominant or where the inertia of particle was large.

In the second part, similarly to the direct numerical simulation carried out byRouson & Eaton (1994) and the large eddy simulation by Wang & Squires (1996b),70 m copper particles were considered. The particle relaxation time nondimension-alized by the viscous scale, +p , was 810. The Reynolds number of the carrier fluidbased on the friction velocity and the channel half-width, Re, was 180. Simulationwas started distributing 250000 particles homogeneously in the channel. The initialvelocities of the particle were set to the local fluid velocities. Similarly to Wang &Squires, a nondimensional time of 6 /u was taken for the development time, i.e.

the time necessary to obtain a statistically steady particulate flow. The statisticssuch as mean velocity and RMS levels of the velocity fluctuations were accumulatedfor a time period of 6 /u after the development time mentioned above.

Excellent agreement was found between this study and the data in the literature,both in the mean particle velocity profile and the velocity fluctuations. However, acontinuous build-up of particles was observed during the accumulation of statistics.As a simple interpretation of the latter phenomena, the development time used inthe literature was considered to be short.

Simulations were also carried out for 50 m glass particles. The particle relax-ation time, +p , was 117. Since the development time was not reported in Wang &

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8 CHAPTER 2. OVERVIEW OF THE PAPERS

Squires, an estimated value was used in this simulation. Again, excellent agreementwas found in the statistics compared with Wang & Squires. The simulations weredone both with 250000 samples and with 22000 samples. Although Wang & Squiresreported that at least 250000 particles were necessary to obtain a continuous rep-

resentation of statistics in this problem, these two simulations in this study showedthat there were no quantitative difference in the obtained statistics between the casewith 250000 samples and 22000 samples.

2.2 Paper 2

The statistics of fully developed turbulent flows with 70 m copper particles and 50m glass particles in a channel were calculated using large eddy simulations.

In order to obtain fully developed flows, the development time for the 70 mcopper particles and 50 m glass particles were chosen as 12 /u and 8.4 /u,respectively, which were longer than those used in earlier studies.

The statistics of particles were compared with the data presented in Paper 1and in Wang & Squires (1996b) which used a shorter development time. Both inthe cases of 70 m copper particles and 50 m glass particles, slight decrease ofthe mean particle velocity and slight increase of the particle number density wereobserved near the wall.

The statistics were used for a detail study of force balance and interphase forces.The Reynolds decomposition based on ensemble average was applied to the Eulerianparticle momentum equation. Each term in the decomposed equation was obtainedusing the statistics.

In the direction normal to the wall, the mean drag force was found to be balancedby the transport of particle flux due to turbulence. The mean drag force was mainlycontributed from the term due to the drift velocity and the term arising from drag -velocity correlation. In the case of 50 m glass particles, where the particle Reynoldsnumber was low, the drag - velocity correlation term was also very small.

In the streamwise direction, the mean drag force was considered to be balancedby a sum of the turbulent transport of particle flux and the gravitational force. Themean drag force could be approximated by the term due to mean relative velocitybetween particles and fluid.

Based on these investigations mentioned above, simplified versions of averaged

momentum equations for particle phase were presented.

2.3 Paper 3

Two-way coupling simulations, i.e. simulations where the modification of carrierfluid flow due to the presence of particles are taken into account, were performedfor the same parameters considered in Paper 2. The forces from particles to fluidwere modeled as the sum of drag forces acting on particles with negative sign, andadded as a body force to the Navier-Stokes equation.


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2.3. PAPER 3 9

Equations were integrated during the development time before the accumulationof statistics in order to ensure that the data were representing fully developed flows.

The accumulated statistics were compared with the data for the one-way cou-pling simulations presented in Paper 2. The presence of 50 m glass particles at

mass flow rate of 0.4 did not change the mean fluid velocity. The streamwise RMSvelocity decreased only near the wall, but the other RMS velocity components andthe Reynolds stress decreased in the whole channel. The streaky structure of near-wall turbulence was modulated. The two-point velocity correlations showed thatthe streaks had become longer than those in undisturbed flows.

With 70 m copper particles at mass flow rate of 3.0, the centerline fluid velocitywas found to be double as that of undisturbed fluid. The mean fluid velocity profiledeviated significantly from the logarithmic law. The streamwise RMS velocity wasat comparable level with that of undisturbed flow. The RMS velocities in the otherdirections and the Reynolds stress were found to have values close to zero. Theturbulent structure was also destroyed.

A relation between the fluid stresses and the interphase stress were derived fol-lowing the treatment for deriving the linear relation of stresses in a pure turbulentchannel flow (Tennekes & Lumley, 1975). The resulting equations were studied termby term using the accumulated statistics.


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10 CHAPTER 2. OVERVIEW OF THE PAPERS


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Acknowledgement

I wish to thank my supervisors, Professor Fritz H. Bark and Dr. Said Zahrai, forthe kind and strict supervisions.

I also appreciate the opponent at the presentation of this thesis, Professor AlainLine at Institut National des Sciences appliquees de Toulouse, France.

For the economical support, Faxen Laboratory, which is a cooperative research

center supported by KTH, NUTEK and a number of Swedish companies includingABB Corporate Research, and Axel och Margaret Ax:son Johnsons Stiftelse aregreatly acknowledged.

Finally, I would like to thank all the colleagues at KTH, ABB Corporate Researchand University of Tokyo. I owe all of them for my fruitful time in Sweden.

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12 ACKNOWLEDGEMENT


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