Fluid Mechanics and Pipe Flow: Turbulence, Simulation and Dynamics

FLUID MECHANICS AND PIPE FLOW: TURBULENCE, SIMULATION AND DYNAMICS No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

FLUID MECHANICS AND PIPE FLOW: TURBULENCE, SIMULATION AND DYNAMICS

DONALD MATOS AND

CRISTIAN VALERIO EDITORS

Nova Science Publishers, Inc. New York

Copyright © 2009 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Fluid mechanics and pipe flow : turbulence, simulation, and dynamics / editors, Donald Matos and Cristian Valerio. p. cm. Includes bibliographical references and index. ISBN 978-1-61668-990-2 (E-Book) 1. Fluid mechanics. 2. Pipe--Fluid dynamics. I. Matos, Donald. II. Valerio, Cristian. TA357.F5787 2009 620.1'06--dc22 2009017666

Published by Nova Science Publishers, Inc. Ô New York

CONTENTS Preface

vii

Chapter 1

Solute Transport, Dispersion, and Separation in Nanofluidic Channels Xiangchun Xuan

1

Chapter 2

H2O in the Mantle: From Fluid to High-Pressure Hydrous Silicates N.R. Khisina, R. Wirth and S. Matsyuk

27

Chapter 3

On the Numerical Simulation of Turbulence Modulation in TwoPhase Flows K. Mohanarangam and J.Y. Tu

41

Chapter 4

A Review of Population Balance Modelling for Multiphase Flows: Approaches, Applications and Future Aspects Sherman C.P. Cheung, G.H. Yeoh and J.Y. Tu

117

Chapter 5

Numerical Analysis of Heat Transfer and Fluid Flow for ThreeDimensional Horizontal Annuli with Open Ends Chun-Lang Yeh

171

Chapter 6

Convective Heat Transfer in the Thermal Entrance Region of Parallel Flow Double-Pipe Heat Exchangers for Non-Newtonian Fluids Ryoichi Chiba, Masaaki Izumi and Yoshihiro Sugano

205

Chapter 7

Numerical Simulation of Turbulent Pipe Flow M. Ould-Rouis and A.A. Feiz

231

Chapter 8

Pipe Flow Analysis of Uranium Nuclear Heating with Conjugate Heat Transfer G.H. Yeoh and M.K.M. Ho

269

Chapter 9

First and Second Law Thermodynamics Analysis of Pipe Flow Ahmet Z. Sahin

317

vi

Contents

Chapter 10

Single-Phase Incompressible Fluid Flow in Mini- and Micro-channels Lixin Cheng

343

Chapter 11

Experimental Study of Pulsating Turbulent Flow through a Divergent Tube Masaru Sumida

365

Chapter 12

Solution of an Airfoil Design Inverse Problem for a Viscous Flow Using a Contractive Operator Jan Šimák and Jaroslav Pelant

379

Chapter 13

Some Free Boundary Problems in Potential Flow Regime Using the Level Set Method M. Garzon, N. Bobillo-Ares and J.A. Sethian

399

Chapter 14

A New Approach for Polydispersed Turbulent Two-Phase Flows: The Case of Deposition in Pipe-Flows S. Chibbaro

441

Index

455

PREFACE Fluid mechanics is the study of how fluids move and the forces that develop as a result. Fluids include liquids and gases and fluid flow can be either laminar or turbulent. This book presents a level set based methodology that will avoid problems in potential flow models with moving boundaries. A review of the state-of-the-art population balance modelling techniques that have been adopted to describe the nature of dispersed phase in multiphase problems is presented as well. Recent works that are aimed at putting forward the main ideas behind a new theoretical approach to turbulent wall-bounded flows are examined, including a state-ofthe-art review on single-phase incompressible fluid flow. Recent breakthrough in nanofabrication has stimulated the interest of solute separation in nanofluidic channels. Since the hydraulic radius of nanochannels is comparable to the thickness of electric double layers, the enormous electric fields inherent to the latter generate transverse electromigrations causing charge-dependent solute distributions over the channel cross-section. As a consequence, the non-uniform fluid flow through nanochannels yields charge-dependent solute speeds enabling the separation of solutes by charge alone. In Chapter 1 we develop a theoretical model of solute transport, dispersion and separation in electroosmotic and pressure-driven flows through nanofluidic channels. This model provides a basis for the optimization of solute separation in nanochannels in terms of selectivity and resolution as traditionally defined. As presented in Chapter 2, infrared spectroscopic data show that nominally anhydrous olivine (Mg,Fe)2SiO4 contains traces of H2O, up to several hundred wt. ppm of H2O (Miller et al., 1987; Bell et al., 2004; Koch-Muller et al., 2006; Matsyuk & Langer, 2004) and therefore olivine is suggested to be a water carrier in the mantle (Thompson, 1992). Protonation of olivine during its crystallization from a hydrous melt resulted in the appearance of intrinsic OH-defects (Libowitsky & Beran, 1995). Mantle olivine nodules from kimberlites were investigated with FTIR and TEM methods (Khisina et al., 2001, 2002, 2008). The results are the following: (1) Water content in xenoliths is lower than water content in xenocrysts. From these data we concluded that kimberlite magma had been saturated by H2O, whereas adjacent mantle rocks had been crystallized from water-depleted melts. (2) Extrinsic water in olivine is represented by high-pressure phases, 10Å-Phase Mg3Si4O10(OH)2.nH2O and hydrous olivine n(Mg,Fe)2SiO4.(H2MgSiO4), both of which belong to the group of Dense Hydrous Magnesium Silicates (DHMS), which were synthesized in laboratory high-pressure experiments (Prewitt & Downs, 1999). The DHMS were regarded as possible mineral carriers for H2O in the mantle; however, they were not found in natural material until quite recently.

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Our observations demonstrate the first finding of the 10Å-Phase and hydrous olivine as a mantle substance. (3) 10Å-Phase, which occurred as either nanoinclusions or narrow veins in olivine, is a ubiquitous nano-mineral of kimberlite and closely related to olivine. (4) There are two different mechanisms of the 10Å-Phase formation: (a) purification of olivine from OHbearing defects resulting in transformation of olivine to the 10 Å-Phase with the liberation of water fluid; and (b) replacement of olivine for the 10Å-Phase due to hydrous metasomatism in the mantle in the presence of H2O fluid. With the increase of computational power, computational modelling of two-phase flow problems using computational fluid dynamics (CFD) techniques is gradually becoming attractive in the engineering field. The major aim of Chapter 3 is to investigate the Turbulence Modulation (TM) of dilute two phase flows. Various density regimes of the two-phase flows have been investigated in this paper, namely the dilute Gas-Particle (GP) flow, LiquidParticle (LP) flow and also the Liquid-Air (LA) flows. While the density is quite high for the dispersed phase flow for the gas-particle flow, the density ratio is almost the same for the liquid particle flow, while for the liquid-air flow the density is quite high for the carrier phase flow. The study of all these density regimes gives a clear picture of how the carrier phase behaves in the presence of the dispersed phases, which ultimately leads to better design and safety of many two-phase flow equipments and processes. In order to carry out this approach, an Eulerian-Eulerian Two-Fluid model, with additional source terms to account for the presence of the dispersed phase in the turbulence equations has been employed for particulate flows, whereas Population Balance (PB) have been employed to study the bubbly flows. For the dilute gas-particle flows, particle-turbulence interaction over a backward-facing step geometry was numerically investigated. Two different particle classes with same Stokes number and varied particle Reynolds number are considered in this study. A detailed study into the turbulent behaviour of dilute particulate flow under the influence of two carrier phases namely gas and liquid was also been carried out behind a sudden expansion geometry. The major endeavour of the study is to ascertain the response of the particles within the carrier (gas or liquid) phase. The main aim prompting the current study is the density difference between the carrier and the dispersed phase. While the ratio is quite high in terms of the dispersed phase for the gas-particle flows, the ratio is far more less in terms of the liquid-particle flows. Numerical simulations were carried out for both these classes of flows and their results were validated against their respective sets of experimental data. For the Liquid-Air flows the phenomenon of drag reduction by the injection of micro-bubbles into turbulent boundary layer has been investigated using an Eulerian-Eulerian two-fluid model. Two variants namely the Inhomogeneous and MUSIG (MUltiple-SIze-Group) based on Population balance models are investigated. The simulated results were benchmarked against the experimental findings and also against other numerical studies explaining the various aspects of drag reduction. For the two Reynolds number cases considered, the buoyancy with the plate on the bottom configuration is investigated, as from the experiments it is seen that buoyancy seem to play a role in the drag reduction. The under predictions of the MUSIG model at low flow rates was investigated and reported, their predictions seem to fair better with the decrease of the break-up tendency among the micro-bubbles. Population balance modelling is of significant importance in many scientific and industrial instances such as: fluidizations, precipitation, particles formation in aerosols, bubbly and droplet flows and so on. In population balance modelling, the solution of the population balance equation (PBE) records the number of entities in dispersed phase that

Preface

ix

always governs the overall behaviour of the practical system under consideration. For the majority of cases, the solution evolves dynamically according to the “birth” and “death” processes of which it is tightly coupled with the system operation condition. The implementation of PBE in conjunction with the Computational Fluid Dynamics (CFD) is thereby becoming ever a crucial consideration in multiphase flow simulations. Nevertheless, the inherent integrodifferential form of the PBE poses tremendous difficulties on its solution procedures where analytical solutions are rare and impossible to be achieved. In Chapter 4, we present a review of the state-of-the-art population balance modelling techniques that have been adopted to describe the phenomenological nature of dispersed phase in multiphase problems. The main focus of the review can be broadly classified into three categories: (i) Numerical approaches or solution algorithms of the PBE; (ii) Applications of the PBE in practical gas-liquid multiphase problems and (iii) Possible aspects of the future development in population balance modelling. For the first category, details of solution algorithms based on both method of moment (MOM) and discrete class method (CM) that have been proposed in the literature are provided. Advantages and drawbacks of both approaches are also discussed from the theoretical and practical viewpoints. For the second category, applications of existing population balance models in practical multiphase problems that have been proposed in the literature are summarized. Selected existing mathematical closures for modelling the “birth” and “death” rate of bubbles in gas-liquid flows are introduced. Particular attention is devoted to assess the capability of some selected models in predicting bubbly flow conditions through detail validation studies against experimental data. These studies demonstrate that good agreement can be achieved by the present model by comparing the predicted results against measured data with regards to the radial distribution of void fraction, Sauter mean bubble diameter, interfacial area concentration and liquid axial velocity. Finally, weaknesses and limitations of the existing models are revealed are suggestions for further development are discussed. Emerging topics for future population balance studies are provided as to complete the aspect of population balance modelling. Study of the heat transfer and fluid flow inside concentric or eccentric annuli can be applied in many engineering fields, e.g. solar energy collection, fire protection, underground conduit, heat dissipation for electrical equipment, etc. In the past few decades, these studies were concentrated in two-dimensional research and were mostly devoted to the investigation of the effects of convective heat transfer. However, in practical situation, this problem should be three-dimensional, except for the vertical concentric annuli which could be modeled as two-dimensional (axisymmetric). In addition, the effects of heat conduction and radiation should not be neglected unless the outer cylinder is adiabatic and the temperature of the flow field is sufficiently low. As the author knows, none of the open literature is devoted to the investigation of the conjugated heat transfer of convection, conduction and radiation for this problem. The author has worked in industrial piping design area and is experienced in this field. The author has also employed three-dimensional body-fitted coordinate system associated with zonal grid method to analyze the natural convective heat transfer and fluid flow inside three-dimensional horizontal concentric or eccentric annuli with open ends. Owing to its broad application in practical engineering problems, Chapter 5 is devoted to a detailed discussion of the simulation method for the heat transfer and fluid flow inside threedimensional horizontal concentric or eccentric annuli with open ends. Two illustrative problems are exhibited to demonstrate its practical applications.

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Donald Matos and Cristian Valerio

In Chapter 6, the conjugated Graetz problem in parallel flow double-pipe heat exchangers is analytically solved by an integral transform method—Vodicka’s method—and an analytical solution to the fluid temperatures varying along the radial and axial directions is obtained in a completely explicit form. Since the present study focuses on the range of a sufficiently large Péclet number, heat conduction along the axial direction is considered to be negligible. An important feature of the analytical method presented is that it permits arbitrary velocity distributions of the fluids as long as they are hydrodynamically fully developed. Numerical calculations are performed for the case in which a Newtonian fluid flows in the annulus of the double pipe, whereas a non-Newtonian fluid obeying a simple power law flows through the inner pipe. The numerical results demonstrate the effects of the thermal conductivity ratio of the fluids, Péclet number ratio and power-law index on the temperature distributions in the fluids and the amount of exchanged heat between the two fluids. Many experimental and numerical studies have been devoted to turbulent pipe flows due to the number of applications in which theses flows govern heat or mass transfer processes: heat exchangers, agricultural spraying machines, gasoline engines, and gas turbines for examples. The simplest case of non-rotating pipe has been extensively studied experimentally and numerically. Most of pipe flow numerical simulations have studied stability and transition. Some Direct Numerical Simulations (DNS) have been performed, with a 3-D spectral code, or using mixed finite difference and spectral methods. There is few DNS of the turbulent rotating pipe flow in the literature. Investigations devoted to Large Eddy Simulations (LES) of turbulence pipe flow are very limited. With DNS and LES, one can derive more turbulence statistics and determine a well-resolved flow field which is a prerequisite for correct predictions of heat transfer. However, the turbulent pipe flows have not been so deeply studied through DNS and LES as the plane-channel flows, due to the peculiar numerical difficulties associated with the cylindrical coordinate system used for the numerical simulation of the pipe flows. Chapter 7 presents Direct Numerical Simulations and Large Eddy Simulations of fully developed turbulent pipe flow in non-rotating and rotating cases. The governing equations are discretized on a staggered mesh in cylindrical coordinates. The numerical integration is performed by a finite difference scheme, second-order accurate in space and time. The time advancement employs a fractional step method. The aim of this study is to investigate the effects of the Reynolds number and of the rotation number on the turbulent flow characteristics. The mean velocity profiles and many turbulence statistics are compared to numerical and experimental data available in the literature, and reasonably good agreement is obtained. In particular, the results show that the axial velocity profile gradually approaches a laminar shape when increasing the rotation rate, due to the stability effect caused by the centrifugal force. Consequently, the friction factor decreases. The rotation of the wall has large effects on the root mean square (rms), these effects being more pronounced for the streamwise rms velocity. The influence of rotation is to reduce the Reynolds stress component 〈Vr'Vz'〉 and to increase the two other stresses 〈Vr'Vθ'〉 and 〈Vθ'Vz'〉. The effect of the Reynolds number on the rms of the axial velocity (〈Vz'2〉1/2) and the distributions of 〈Vr'Vz'〉 is evident, and it increases with an increase in the Reynolds number. On the other hand, the 〈Vr'Vθ'〉profiles appear to be nearly independent of the Reynolds number. The present DNS and LES predictions will be helpful for developing more accurate turbulence models for heat transfer and fluid flow in pipe flows.

Preface

xi

The field of computational fluid dynamics (CFD) has evolved from an academic curiosity to a tool of practical importance. Applications of CFD have become increasingly important in nuclear engineering and science, where exacting standards of safety and reliability are paramount. The newly-commissioned Open Pool Australian Light-water (OPAL) research reactor at the Australian Nuclear Science and Technology Organisation (ANSTO) has been designed to irradiate uranium targets to produce molybdenum medical isotopes for diagnosis and radiotherapy. During the irradiation process, a vast amount of power is generated which requires efficient heat removal. The preferred method is by light-water forced convection cooling—essentially a study of complex pipe flows with coupled conjugate heat transfer. Feasibility investigation on the use of computational fluid dynamics methodologies into various pipe flow configurations for a variety of molybdenum targets and pipe geometries are detailed in Chapter 8. Such an undertaking has been met with a number of significant modeling challenges: firstly, the complexity of the geometry that needed to be modeled. Herein, challenges in grid generation are addressed by the creation of purpose-built bodyfitted and/or unstructured meshes to map the intricacies within the geometry in order to ensure numerical accuracy as well as computational efficiency in the solution of the predicted result. Secondly, various parts of the irradiation rig that are required to be specified as composite solid materials are defined to attain the correct heat transfer characteristics. Thirdly, the use of an appropriate turbulence model is deemed to be necessary for the correct description of the fluid and heat flow through the irradiation targets, since the heat removal is forced convection and the flow regime is fully turbulent, which further adds to the complexity of the solution. As complicated as the computational fluid dynamics modeling is, numerical modeling has significantly reduced the cost and lead time in the molybdenum-target design process, and such an approach would not have been possible without the continual improvement of computational power and hardware. This chapter also addresses the importance of experimental modeling to evaluate the design and numerical results of the velocity and flow paths generated by the numerical models. Predicted results have been found to agree well with experimental observations of pipe flows through transparent models and experimental measurements via the Laser Doppler Velocimetry instrument. In Chapter 9, the entropy generation for during fluid flow in a pipe is investigated. The temperature dependence of the viscosity is taken into consideration in the analysis. Laminar and turbulent flow cases are treated separately. Two types of thermal boundary conditions are considered; uniform heat flux and constant wall temperature. In addition, various crosssectional pipe geometries were compared from the point of view of entropy generation and pumping power requirement in order to determine the possible optimum pipe geometry which minimizes the exergy losses. Chapter 10 aims to present a state-of-the-art review on single-phase incompressible fluid flow in mini- and micro-channels. First, classification of mini- and micro-channels is discussed. Then, conventional theories on laminar, laminar to turbulent transition and turbulent fluid flow in macro-channels (conventional channels) are summarized. Next, a brief review of the available studies on single-phase incompressible fluid flow in mini- and microchannels is presented. Some experimental results on single phase laminar, laminar to turbulent transition and turbulent flows are presented. The deviations from the conventional friction factor correlations for single-phase incompressible fluid flow in mini and microchannels are discussed. The effect factors on mini- and micro-channel single-phase fluid flow are analyzed. Especially, the surface roughness effect is focused on. According to this review,

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Donald Matos and Cristian Valerio

the future research needs have been identified. So far, no systematic agreed knowledge of single-phase fluid flow in mini- and micro-channels has yet been achieved. Therefore, efforts should be made to contribute to systematic theories for microscale fluid flow through very careful experiments. In Chapter 11, an experimental investigation was conducted of pulsating turbulent flow in a conically divergent tube with a total divergence angle of 12°. The experiments were carried out under the conditions of Womersley numbers of α =10∼40, mean Reynolds number of Reta =20000 and oscillatory Reynolds number of Reos =10000 (the flow rate ratio of η = 0.5). Time-dependent wall static pressure and axial velocity were measured at several longitudinal stations and the distributions were illustrated for representative phases within one cycle. The rise between the pressures at the inlet and the exit of the divergent tube does not become too large when the flow rate increases, it being moderately high in the decelerative phase. The profiles of the phase-averaged velocity and the turbulence intensity in the cross section are very different from those for steady flow. Also, they show complex changes along the tube axis in both the accelerative and decelerative phases. Chapter 12 deals with a numerical method for a solution of an airfoil design inverse problem. The presented method is intended for a design of an airfoil based on a prescribed pressure distribution along a mean camber line, especially for modifying existing airfoils. The main idea of this method is a coupling of a direct and approximate inverse operator. The goal is to find a pseudo-distribution corresponding to the desired airfoil with respect to the approximate inversion. This is done in an iterative way. The direct operator represents a solution of a flow around an airfoil, described by a system of the Navier-Stokes equations in the case of a laminar flow and by the k−ω model in the case of a turbulent flow. There is a relative freedom of choosing the model describing the flow. The system of PDEs is solved by an implicit finite volume method. The approximate inverse operator is based on a thin airfoil theory for a potential flow, equipped with some corrections according to the model used. The airfoil is constructed using a mean camber line and a thickness function. The so far developed method has several restrictions. It is applicable to a subsonic pressure distribution satisfying a certain condition for the position of a stagnation point. Numerical results are presented. Recent advances in the field of fluid mechanics with moving fronts are linked to the use of Level SetMethods, a versatile mathematical technique to follow free boundaries which undergo topological changes. A challenging class of problems in this context are those related to the solution of a partial differential equation posed on a moving domain, in which the boundary condition for the PDE solver has to be obtained from a partial differential equation defined on the front. This is the case of potential flow models with moving boundaries. Moreover, the fluid front may carry some material substance which diffuses in the front and is advected by the front velocity, as for example the use of surfactants to lower surface tension. We present a Level Set based methodology to embed this partial differential equations defined on the front in a complete Eulerian framework, fully avoiding the tracking of fluid particles and its known limitations. To show the advantages of this approach in the field of Fluid Mechanics we present in Chapter 13 one particular application: the numerical approximation of a potential flow model to simulate the evolution and breaking of a solitary wave propagating over a slopping bottom and compare the level set based algorithm with previous front tracking models.

Preface

xiii

Chapter 14 is basically a review of recent works that is aimed at putting forward the main ideas behind a new theoretical approach to turbulent wall-bounded flows, notably pipe-flows, in which complex physics is involved, such as combustion or particle transport. Pipe flows are ubiquitous in industrial applications and have been studied intensively in the last century, both from a theoretical and experimental point of view. The result of such a strong effort is a good comprehension of the physics underlying the dynamics of these flows and the proposition of reliable models for simple turbulent pipe-flows at large Reynolds number Nevertheless, the advancing of engineering frontiers casts a growing demand for models suitable for the study of more complex flows. For instance, the motion and the interaction with walls of aerosol particles, the presence of roughness on walls and the possibility of drag reduction through the introduction of few complex molecules in the flow constitute some interesting examples of pipe-flows with some new complex physics involved. A good modeling approach to these flows is yet to come and, in the commentary, we support the idea that a new angle of attack is needed with respect to present methods. In this article, we analyze which are the fundamental features of complex two-phase flows and we point out that there are two key elements to be taken into account by a suitable theoretical model: 1) These flows exhibit chaotic patterns; 2) The presence of instantaneous coherent structures radically change the flow properties. From a methodological point of view, two main theoretical approaches have been considered so far: the solution of equations based on first principles (for example, the Navier-Stokes equations for a single phase fluid) or Eulerian models based on constitutive relations. In analogy with the language of statistical physics, we consider the former as a microscopic approach and the later as a macroscopic one. We discuss why we consider both approaches unsatisfying with regard to the description of general complex turbulent flows, like two-phase flows. Hence, we argue that a significant breakthrough can be obtained by choosing a new approach based upon two main ideas: 1) The approach has to be mesoscopic (in the middle between the microscopic and the macroscopic) and statistical; 2) Some geometrical features of turbulence have to be introduced in the statistical model. We present the main characteristics of a stochastic model which respects the conditions expressed by the point 1) and a method to fulfill the point 2). These arguments are backed up with some recent numerical results of deposition onto walls in turbulent pipe-flows. Finally, some perspectives are also given.

In: Fluid Mechanics and Pipe Flow Editors: D. Matos and C. Valerio, pp. 1-26

ISBN: 978-1-60741-037-9 © 2009 Nova Science Publishers, Inc.

Chapter 1

SOLUTE TRANSPORT, DISPERSION, AND SEPARATION IN NANOFLUIDIC CHANNELS Xiangchun Xuan* Department of Mechanical Engineering, Clemson University, Clemson, SC 29634-0921, USA

Abstract Recent breakthrough in nanofabrication has stimulated the interest of solute separation in nanofluidic channels. Since the hydraulic radius of nanochannels is comparable to the thickness of electric double layers, the enormous electric fields inherent to the latter generate transverse electromigrations causing charge-dependent solute distributions over the channel cross-section. As a consequence, the non-uniform fluid flow through nanochannels yields charge-dependent solute speeds enabling the separation of solutes by charge alone. In this chapter we develop a theoretical model of solute transport, dispersion and separation in electroosmotic and pressure-driven flows through nanofluidic channels. This model provides a basis for the optimization of solute separation in nanochannels in terms of selectivity and resolution as traditionally defined.

1. Introduction Solute transport and separation in micro-columns (e.g., micro capillaries and chip-based microchannels) have been a focus of research and development in electrophoresis and chromatography communities for many years. Recent breakthrough in nanofabrication has initiated the study of these topics among others in nanofluidic channels [1-3]. Since the hydraulic radius of nanochannels is comparable to the thickness of electric double layers (EDL), the enormous electric fields inherent to the latter generate transverse electromigrations causing charge-dependent solute distributions over the channel cross-section [4-7]. As a consequence, the non-uniform fluid flow in nanochannels yields charge-dependent solute speeds enabling the separation of solutes by charge alone [8,9]. Such charge-based solute *

E-mail address: [email protected]. Tel: (864) 656-5630. Fax: (864) 656-7299

2

Xiangchun Xuan

separation was first proposed and implemented by Pennathur and Santiago [10] and Garcia et al. [11] in electroosmotic flow through nanoscale channels, termed nanochannel electrophoresis. As a matter of fact, this separation may also happen in pressure-driven flow along nanoscale channels, termed here as nanochannel chromatography for comparison, which was first demonstrated theoretically by Griffiths and Nilson [12] and Xuan and Li [13], and later experimentally verified by Liu’s group [14]. So far, a number of theoretical studies have been conducted on the transport [4-6,913,15,19], dispersion [7,9,12,15-19] and separation [4,9-13,15,19] of solutes in free solutions through nanofluidic channels. This chapter combines and unites the works from ourselves in this area [6,13,15,17-19], and is aimed to develop a general analytical model of solute transport, dispersion and separation in nanochannels. It is important to note that this model applies only to point-like solutes. For those with a finite size, one must consider the hydrodynamic and electrostatic interactions among solutes, electric field, and flow fluid, and as well the Steric interactions between solutes and channel walls etc [20].

2. Nomenclature a Bi cb ci Ci ci,0 Ci,0 Di

channel half-height defined function, = exp(−ziΨ) bulk concentration of the background electrolyte concentration of solute species i bulk concentration of solute species i concentration of solute species i at the channel centerline initial concentration of solute species i at the channel centerline solute diffusion coefficient effective solute diffusion coefficient

E Est F hi j Ki L P Pe rji R Rji t T ui

ui

axial electric field streaming potential field Faraday’s constant reduced theoretical plate height electric current density hydrodynamic dispersion channel length hydrodynamic pressure drop per unit channel length Peclet number solute selectivity Universal gas constant resolution time coordinate absolute temperature axial solute speed mean solute speed

vi Wi

solute mobility half width of the initially injected solute zone

Di′

Solute Transport, Dispersion, and Separation in Nanofluidic Channels x Xi y zi Z

3

streamwise or longitudinal coordinate the central location of the injected solute zone transverse coordinate valence of ions electrokinetic “figure of merit”

Greek Symbols β ε γ κ χi λb μ ψ Ψ Ψ0

ρe σb σi σt ζ ζ*

non-dimensional product of fluid properties, = λbμ/εRT permittivity apparent viscosity ratio reciprocal of Debye length dispersion coefficient molar conductivity of the background electrolyte fluid viscosity electrical double layer potential non-dimensional EDL potential EDL potential at the channel center net charge density bulk electric conductivity of background electrolyte standard deviation of solute peak distribution standard deviation of solute peak distribution in the time domain zeta potential non-dimensional zeta potential

Subscripts e p i

electroosmosis related pressure-driven related solute species i

3. Fluid Flow in Nanochannels Given the fact that the width (in micrometers) of state-of-the-art nanofluidic channels is usually much larger than their depth (in nanometers) [1-3], we consider the solute transport in fluid flow through a long straight nanoslit, see Figure 1 for the schematic. The flow may be electric field-driven, i.e., electroosmotic, or pressure-driven. For simplicity, the electrolyte solution is assumed symmetric with unit-charge, e.g., KCl. As the time scale for fluid flow (in the order of nanoseconds) is far less than that of solute transport (typically of tens of seconds), we assume a steady-state, fully-developed incompressible fluid motion, which in a slit channel is governed by

4

Xiangchun Xuan

y

Solute zone u + viziFE

u

a

x

Figure 1. Schematic of solute transport in a slit nanochannel (only the top half is illustrated due to the symmetry).

μ

d 2u + P + ρe E = 0 dy 2

(3-1)

where μ is the fluid viscosity, u the axial fluid velocity, y the transverse coordinate originating from the channel axis, P the pressure drop per unit channel length, and E the axial electric field either externally applied in electroosmotic flow or internally induced in pressure-driven flow (i.e., the so-called streaming potential field) [21-24]. The net charge density, ρe, is solved from the Poisson equation [25]

d 2ψ ρ e = −ε 2 dy

(3-2)

where ε is the fluid permittivity and ψ is the EDL potential. Invoking the no-slip condition for Eq. (3-1) and the zeta potential condition for Eq. (3-2) on the channel wall (i.e., y = a), one can easily obtain

u = u p + ue up =

(3-3)

a2 ⎛ y2 ⎞ ⎜1 − ⎟ P 2μ ⎝ a 2 ⎠

(3-4)

εζ μ

(3-5)

ue = −

⎛ Ψ⎞ ⎜1 − ∗ ⎟ E ⎝ ζ ⎠

where up is the pressure-driven fluid velocity, ue the electroosmotic fluid velocity, a the halfheight of the channel, and Ψ = Fψ/RT and ζ* = Fζ/RT the dimensionless forms of the EDL and wall zeta potentials with F the Faraday’s constant, R the universal gas constant and T the absolute fluid temperature. It is noted that the contribution of charged solutes to the net charge density ρe has been neglected. This is reasonable as long as the solute concentration is much lower than the ionic concentration of the background electrolyte, which is fulfilled in typical solute separations. Under such a condition, it is also safe to assume a uniform zeta potential on the channel wall.

Solute Transport, Dispersion, and Separation in Nanofluidic Channels

5

The non-dimensional EDL potential in Eq. (3-5), Ψ, may be solved from the PoissonBoltzmann equation [25]

d 2Ψ = κ 2 sinh ( Ψ ) 2 dy where

(3-6)

κ = 2 F 2 cb ε RT is the inverse of the so-called Debye screening length with cb the

bulk concentration of the background electrolyte. We recognize that the assumed Boltzmann distribution of electrolyte ions in Poisson-Boltzmann equation might be questionable in nanoscale channels, especially in those with strong EDL overlapping [26,27]. However, this equation has been successfully used to explain the experimentally measured electric conductance and streaming current in variable nanofluidic channels [28-32], and is thus still employed here. For the case of a small magnitude of ζ (e.g., |ζ| < 25 mV or |ζ*| < 1) which is actually desirable for sensitive solute separations in nanochannels as demonstrated by Griffiths and Nilson [12], one may use the Debye-Huckel approximation to simplify Eq. (3-6) as [21,25]

d 2Ψ = κ 2Ψ dy 2

(3-7)

It is then straightforward to obtain

Ψ =ζ*

cosh (κ y ) cosh (κ a )

(3-8)

where κa may be viewed as the normalized channel half-height. It is important to note that for a given fluid and channel combination, the wall zeta potential will in general vary with κa [28,31,32]. One option to address this is to use a surface-charge based potential parameter for scaling instead of zeta potential [16]. In this work and other studies [6-14], the zeta potential is used directly, because it may be readily determined through experiment and provides a direct measure of the electroosmotic mobility. The area-averaged fluid velocity

u may be written in terms of the Poiseuille and

electroosmotic components

u = u p + ue up =

a2 P 3μ

(3-9)

(3-10)

6

Xiangchun Xuan

ue = −

where " =

εζ μ

⎡ tanh (κ a ) ⎤ ⎢1 − ⎥E κ a ⎣ ⎦

(3-11)

∫ (") d ( y a ) signifies an area-averaged quantity. a

0

3.1. Electroosmotic Flow For electroosmotic flow, no pressure gradient is present, and so the fluid motion in Eq. (3-3) is described by

u p = 0 and ue = −

εζ ⎡ cosh (κ y ) ⎤ ⎢1 − ⎥E μ ⎣ cosh (κ a ) ⎦

Electroosmotic velocity profile

1 10

(3-12)

20

5

0.8 2

0.6 0.4

κa = 1 0.2 0 0

0.2

0.4

0.6

0.8

1

y/a Figure 2. Radial profile of the normalized electroosmotic fluid velocity, ue/UHS = 1 − cosh(κy)/cosh(κa), at different κa values. All symbols are referred to the nomenclature.

Figure 2 shows the profile of electroosmotic velocity normalized by the so-called HelmholtzSmoluchowski velocity UHS = −εζE/μ [25], i.e., ue/UHS = 1 − cosh(κy)/cosh(κa), in a slit channel with different κa values. When κa > 10, the curves are almost plug-like except near the channel wall and the bulk velocity is equal to UHS. These are the typical features of electroosmotic flow when there is little or zero EDL overlapping. The profiles at κa < 5 become essentially parabolic resembling the traditional pressure-driven flow. Moreover, the maximum velocity along the channel centerline is significantly lower than UHS and decreases with κa, indicating a vanishingly small electroosmotic mobility when κa approaches 0.

Solute Transport, Dispersion, and Separation in Nanofluidic Channels

7

3.2. Pressure-Driven Flow For pressure-driven flow, the downstream accumulation of counter-ions results in the development of a streaming potential field [21-24]. This induced electric field, Est, can be determined from the condition of zero electric current though the channel. If an equal mobility for the positive and negative ions of the electrolyte is assumed, the electric current density, j, in pressure-driven flow is given as [21-24]

j = ρeu + σ b cosh ( Ψ ) Est

(3-13)

where σb = cbλb is the bulk conductivity of the electrolyte with λb being the molar conductivity. Referring to Eqs. (3-2) and (3-3), one may rewrite the last equation as

j = −ε

d 2 Ψ RT ( u p + ue ) + cbλb cosh ( Ψ ) Est dy 2 F

(3-14)

Note that the surface conductance of the outer diffusion layer in the EDL has been considered through the cosine hyperbolic function in Eq. (3-14) (which reduces to 1 at Ψ = 0). The contribution of the inner Stern layer conductance [33] to the electric current is, however, ignored. Readers may be referred to Davidson and Xuan [34] for a discussion of this issue in electrokinetic streaming effects. Integrating j in Eq. (3-14) over the channel cross-section and using the zero electric current condition in a steady-state pressure-driven flow yield

g1 ζ * Est = P cb F ( g 2 + β g3 ζ *2 )

(3-15)

tanh (κa ) tanh (κa ) 1 ⎛ y⎞ , g2 = and g 3 = ∫ cosh (Ψ )d ⎜ ⎟ (3-16) − 2 κa cosh (κa ) κa ⎝a⎠ 0 a

g1 = 1 −

Therefore, the fluid motion in pressure-driven flow is characterized as

up =

εζ a2 ⎛ y2 ⎞ ⎜1 − 2 ⎟ P and ue = − μ 2μ ⎝ a ⎠

⎡ cosh (κ y ) ⎤ ⎢1 − ⎥ Est cosh κ a ( ) ⎣ ⎦

(3-17)

Area-averaging the two velocity components in the last equation and combining them with Eq. (3-15) lead to

ue up

= −Z

(3-18)

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Xiangchun Xuan

where Z is previously termed electrokinetic “figure of merit” as it gauges the efficiency of electrokinetic energy conversion [35,36], and defined as

Z=

(κ a )

2

(g

β=

3g12

(3-19)

∗2 ) 2 + β g3 ζ

λb μ εRT

(3-20)

where κ2 = 2F2cb/εRT has been invoked during the derivation, and β is a non-dimensional product of fluid properties whose reciprocal was termed Levine number by Griffiths and Nilson [37]. Apparently, Z depends on three non-dimensional parameters, β, κa and ζ*, among which β spans in the range of 2 ≤ β ≤ 10 and ζ* spans in the rage of −8 ≤ ζ* ≤ 0 [33] for typical aqueous solutions. Moreover, Z is unconditionally positive and less than unity due to the entropy generation in non-equilibrium electrokinetic flow [38]. The curves of Z at ζ* = −1 (or ζ ≈ −25 mV) and β = 2 and 10, respectively, are displayed in Figure 3 as a function of κa. One can see that Z achieves the maximum at around κa = 2, indicating that nanochannels with a strong EDL overlapping are the necessary conditions for efficient electrokinetic energy conversion. For more information about Z and its function in electrokinetic energy conversion, the reader is referred to Xuan and Li [36].

0.15

β = 10 0.12

Z

0.09 0.06

β=2

0.03 0.00 0.1

1

κa

10

100

Figure 3. The electrokinetic “figure of merit” as a function of κa at ζ* = −1 and β = 2 and 10, respectively.

Solute Transport, Dispersion, and Separation in Nanofluidic Channels

9

Based on Eq. (3-18), the effects of streaming potential on the average fluid speed, i.e.,

u in Eq. (3-9), in an otherwise pure pressure-driven flow, are characterized by

u up

= 1− Z

(3-21)

This equation also provides a measure of the so-called electro-viscous effects in micro/nanochannels [39]. If the concept of apparent viscosity is employed to characterize such retardation effects, the apparent viscosity ratio γ is readily derived as

γ = 1 (1 − Z )

(3-22)

For more information on this topic, the reader is referred to Li [39] and Xuan [40].

4. Solute Transport in Nanochannels Solute transport in nanochannels is governed by the Nernst-Planck equation in the absence of chemical reactions, which under the assumption of fully-developed fluid flow is written as

∂ci ∂c ∂ 2c ∂ 2c ∂ ⎛ ∂ψ ⎞ + ui i = Di 2i + Di 2i + vi zi F ⎜ ci ⎟ ∂t ∂x ∂x ∂y ∂y ⎝ ∂y ⎠

(4-1)

ui = ue + u p + vi zi FE

(4-2)

where ci is the concentration of solute species i, t the time coordinate, ui the local solute speed (a combination of electroosmosis, pressure-driven motion, and electrophoresis), x the axial coordinate originating from the channel inlet (see Figure 1), Di the molecular diffusion coefficient, vi the solute mobility, and zi the solute charge number. Note that the product viziF represents the solute electrophoretic mobility. Integrating Eq. (4.1) over the channel crosssection eliminates the last two terms on the right hand side due to the impermeable wall conditions

∂ ci ∂ ui ci ∂ 2 ci + = Di ∂t ∂x ∂x 2

(4-3)

where again ... indicates the area-average over the channel cross-section as defined earlier. Since the time scale for transverse solute diffusion in nanochannels (characterized by a2/Di, which is about 100 μs when a = 100 nm and Di = 1×10−10 m2/s) is much shorter than

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Xiangchun Xuan

that for longitudinal solute transport (typically of tens of seconds), it is reasonable to assume that solute species are at a quasi-steady equilibrium in the y direction, i.e.,

0 = Di

∂ 2 ci ∂ ⎛ ∂ψ ⎞ + vi zi F ⎜ ci ⎟ 2 ∂y ∂y ⎝ ∂y ⎠

(4-4)

Integrating Eq. (4.4) twice and using the Nernst-Einstein relation [33], vi = Di/RT, one obtains

ci ( x, y, t ) = ci ,0 ( x, t ) exp ⎡⎣ − zi ( Ψ − Ψ 0 ) ⎤⎦

(4-5)

where ci,0 is the solute concentration at the channel centerline where the local EDL potential is defined as Ψ0 (non-zero in the presence of EDL overlapping). Substituting Eq. (4.5) into Eq. (4.3) and considering the hydrodynamic dispersion due to the velocity non-uniformity over the channel cross-section [41,42], one may obtain

∂ci ,0 ∂t

+ ui

∂ci ,0 ∂x

= Di′

∂ 2 ci ,0 ∂x 2

ui = uip + uie + vi zi FE uip =

u p Bi Bi

and uie =

ue Bi Bi

(4-6) (4-7)

(4-8)

where ui is the mean solute speed (i.e., zone velocity) with uip and uie being its components due to pressure-driven and electroosmotic flows, Di′ the effective diffusion coefficient which is a combination of molecule diffusion and hydrodynamic dispersion and will be addressed in the next section, and Bi = exp(−ziΨ) the like-Boltzmann distribution of solutes in the crossstream direction. It is the dependence of ui on the charge number zi that enables the chargedbased solute separation in nanochannels. For an initially uniform concentration Ci,0 of solute species i along the channel axis, a closed-form solution to Eq. (4-6) is given by [5]

ci ,0 =

⎛ x − X i − ui t ⎞ ⎤ Ci ,0 ⎡ ⎛ Wi − x + X i + ui t ⎞ ⎢erf ⎜ ⎟ + erf ⎜ ⎟⎥ ⎟ ⎜ 2 D′t ⎟ ⎥ 2 ⎢⎣ ⎜⎝ 2 Di′t i ⎠ ⎝ ⎠⎦

(4-9)

where erf denotes the error function, and Wi is the half width of the initial solute zone with its center being located at Xi. As such, the electrokinetic transport of solute species in nanochannels is described by

Solute Transport, Dispersion, and Separation in Nanofluidic Channels

ci ( x, y, t ) =

Ci 2

⎡ ⎛W − x + X +u t ⎞ ⎛ x − X i − ui t ⎞ ⎤ i i ⎢erf ⎜ i ⎟ + erf ⎜ ⎟ ⎥ exp ( − zi Ψ ) ⎟ ⎜ 2 D′t ⎟ ⎥ 2 Di′t ⎢⎣ ⎜⎝ i ⎠ ⎝ ⎠⎦

11

(4-10)

where Ci = Ci ,0 exp ( zi Ψ 0 ) is the bulk solute concentration at zero potential outside the slit nanochannel, refer to Eq. (4-5). zv = +2 zv = +1 zv = 0 zv = −1 zv = −2 0.2 Cmax

0.4 Cmax

0.6 Cmax

0.8 Cmax

Cmax

Figure 4. Transport of solutes with zi = [+2, −2] through a 100 nm deep channel in nanochannel chromatography. Other parameters are referred to the text. Reprinted with permission from [13].

Figure 4 illustrates the transport of an initially Wi = 1 μm wide plug of solutes with zi = [+2, −2] (from top to bottom) through a 100 nm deep (i.e., a = 50 nm) channel 5 s after a pressure gradient P = 1×108 Pam-1 was imposed. Note that only the top half of the channel is shown due to symmetry. The ionic concentration of the background electrolyte is cb = 1 mM, corresponding to κa ≈ 5. The other two non-dimensional parameters are assumed to be β = 4 and ζ* = −2 (or ζ = −50 mV), both of which are typical to aqueous solutions as indicated above. As to the validity of the Debye-Huckel approximation at ζ* = −2, we have recently demonstrated using numerical simulation the fairly good accuracy of Eq. (3-7) in predicting the solute migration velocity [5]. As shown, positive solutes are concentrated to near the negatively charged wall due to the solute-wall electrostatic interactions [9,11], or in essence the transverse electromigration in response to the induced EDL field [5,10,12]. Moreover, the higher the charge number is, the closer the solutes are to the walls. As the fluid velocity near no-slip walls is slower than its average, positive solutes move slower than neutral solutes that are still uniformly distributed over the channel cross-section. Conversely, negative solutes are repelled by the negatively changed walls and concentrated to the region close to the channel center. Hence, they move faster than neutral solutes as seen in Figure 4. As the fluid velocity profile is available in Eq. (3-12) for electroosmotic flow and in Eq. (3-17) for pressure-driven flow, the mean speed of solutes, ui , is readily obtained from Eq. (4-7). Figure 5 compares the mean speed of solutes with zi = [−2, +2] in electroosmotic flow with an electric field of 4 kV/m. One can see that ui of all five solutes decreases when κa gets smaller. This reduction may be explained by the overall lower electroosmotic velocity at a smaller κa, as demonstrated in Figure 2. When κa > 100, negatively charged solutes move slower than positive ones due to their opposite electrophoresis to fluid electroosmosis

12

Xiangchun Xuan

(identical to the curve with zi = 0). When κa gets smaller than 100, however, negatively solutes start moving faster than positively solutes as the latter ones are concentrated to the EDL region within which the fluid has a slower speed than the bulk as explained above. The relative magnitude of ui between the solutes of like charges is, however, a complex function of both the charge number zi, which determines the velocity component due to fluid flow, i.e., uip + uie in Eq. (4-7), and the solute mobility vi, which determines the velocity component due to solute electrophoresis, i.e., the most right term in Eq. (4-7). When κa further decreases to less than 1, the EDL potential becomes nearly flat due to the strong EDL overlapping (see Figure 2), and so the order of ui for the three solutes at large κa (i.e., microchannel electrophoresis) is recovered.

Figure 5. Comparison of the mean speeds of solutes with zi = [−2, +2] as a function of κa in nanochannel electrophoresis. The solute diffusivity is assumed to be constant, Di = 5×10−11 m2/s. Other parameters are referred to the text.

Combining Eqs. (4-7) and (4-8) provides a measure of the streaming potential effects on the solute mean speed in pressure-driven flow,

u B + vi zi F Bi Est ui = 1+ e i uip u p Bi

(4-11)

It is obvious that the last equation is reduced to Eq. (3-21) for neutral solutes, i.e., zi = 0 and thus Bi = 1. Figure 6 shows the ratio ui uip as a function of κa. As expected, streaming

Solute Transport, Dispersion, and Separation in Nanofluidic Channels

13

potential effects reduce the solute speed due to the induced electroosmotic backflow. This reduction varies with the solute charge zi and attains the extreme at about κa = 3, where the figure of merit Z (refer to Figure 3) approaches its maximum indicating the largest streaming potential effects. In both the high and low limits of κa, streaming potential effects become negligible, i.e., Z → 0, see Figure 3. Accordingly, ui uip reduces to 1 for all solutes at large

κa while varying with zi at small κa because of the finite solute mobility [15]. 1.1 −2 −1

1 ui/uip

0 +1

0.9

zi = +2 0.8

0.7 0.1

1

10

100

κa Figure 6. Effects of streaming potential on the solute mean speed in nanochannel chromatography.

5. Solute Dispersion in Nanochannels As up and ue vary over the channel cross-section (refer to Eqs. (3-4) and (3-5), and Figure 2), they both contribute to the spreading of solutes along the flow direction, which is termed hydrodynamic dispersion or Taylor dispersion [41,42]. The general formula for calculating this dispersion is given by [43,44]

a2 Ki = Di

y Bi−1 ⎡ ∫ Bi ( ui − ui ) dy′⎤ ⎣⎢ 0 ⎦⎥

2

(5-1)

Bi

Referring to Eqs. (4-2) and (4-7), one may then rewrite the last equation as

Ki =

(

a2 Fip u p Di

2

+ Fipe u p ue + Fie ue

2

)

(5-2)

14

Xiangchun Xuan y Fip = Bi−1 ⎡ ∫ Bi ( u ∗p − uip∗ ) dy′⎤ ⎣⎢ 0 ⎦⎥

y Fie = Bi−1 ⎡ ∫ Bi ( ue∗ − uie∗ ) dy′⎤ ⎢⎣ 0 ⎥⎦

2

Bi

2

Bi

−1

−1

y y Fipe = 2 Bi−1 ⎡ ∫ ( u ∗p − uip∗ ) Bi dy′⎤ ⎡ ∫ ( ue∗ − uie∗ ) Bi dy′⎤ ⎢⎣ 0 ⎥⎦ ⎢⎣ 0 ⎥⎦ ∗

where um = um

(5-3a)

(5-3b)

Bi

−1

(5-3c)

um and uim∗ = uim um (m = p and e). Note that the three terms, Fip, Fipe

and Fie in Eq. (5-2) represent the contributions to dispersion due to the pressure-driven flow, the coupling between pressure-driven and electroosmotic flows, and the electroosmotic flow, respectively. Hydrodynamic dispersion is often expressed in terms of a non-dimensional dispersion coefficient χi [45],

K i = χ i Pei2 Di

(5-4)

where the Peclet number Pei in this case may be defined with respect to the mean solute speed, i.e., Pei = ui a Di [12,19], or to the area-averaged fluid velocity, i.e.,

(

)

Pei = u p + ue a Di [15-18,45]. Using the solute speed-based Peclet number, χi becomes dependent on the solute diffusivity Di which complicates the analysis. This is because the solute mobility vi in ui [see Eq. (4-7)] is coupled to Di via the Nernst-Einstein relation, vi = Di/RT. Such dependence doesn’t occur if χi is defined using the fluid velocitybased Peclet number. Here, we employ the latter definition in keeping with the dispersion studies of neutral solutes in the literature [45]. As such, the dispersion coefficient χi may be easily obtained from Eq. (5-2) as

χi =

Fip u p

2

+ Fipe u p ue + Fie ue

(

u p + ue

)

2

2

(5-5)

It is important to note that χi is independent of the solute speed or the driving force of the flow while Ki (in the unit of Di) not. Instead, χi is primarily determined by the flow type (pressure- or electric field-driven), channel structure (including shape and size) and solute charge number.

Solute Transport, Dispersion, and Separation in Nanofluidic Channels

15

5.1. Electroosmotic Flow In electroosmotic flow, the hydrodynamic dispersion in Eq. (5-2) is reduced to

K i = Fie

a 2 ue

2

(5-6)

Di

Accordingly, the dispersion coefficient in Eq. (5.5) is simplified as

χ i = Fie

(5-7)

χi for nanochannel electrophoresis

0.1 +2

0.01 +1 zi = 0

0.001 −1 −2

0.0001 0.1

1

10

100

κa Figure 7. Illustration of dispersion coefficient χi of solutes with zi = [−2, +2] in nanochannel electrophoresis as a function of κa.

Figure 7 shows χi of solutes with zi = [−2, +2] in nanochannel electrophoresis as a function of κa. We see that in the entire range of κa, χi of positive solutes is larger than that of neutral ones while χi of negative solutes is smaller than the latter. This is because positive solutes are concentrated to near the channel walls where the velocity gradients are large while negative ones are concentrated to the channel centerline where the velocity gradients are small (refer to Figure 4). Moreover, the higher the charge number zi, the larger is χi for positive solutes and the smaller for negative ones. In the low limit of κa (i.e., the narrowest channel), the EDL potential is essentially uniform over the channel cross-section (refer to Figure 2), and so is the solute distribution regardless of the charge number. As a consequence, χi of all charged solutes approach that of neutral solutes, i.e., 2/105. Note that this value is equal to the dispersion coefficient of neutral solutes in a pure pressure-driven flow indicating the resemblance between pressure-driven and electroosmotic flows in very small

16

Xiangchun Xuan

nanochannels. This aspect will be revisited shortly. In the high limit of κa (i.e., the widest channel), the EDL thickness is so thin compared to the channel height that the solute distribution becomes once again uniform across the channel (the EDL potential is, here, uniformly zero while equal to the wall zeta potential in the low limit of κa). Therefore, the hydrodynamic dispersion in electroosmotic flow, or the so-called electrokinetic dispersion [46], decreases with the square of κa and ultimately converges to zero [47,48].

5.2. Pressure-Driven Flow In pressure-driven flow with consideration of streaming potential, the hydrodynamic dispersion is obtained from Eq. (5-2) as

K i = Fip

a2 u p

2

Di

(1 − δ

i2

Z + δi3Z 2 )

δ i 2 = Fipe Fip and δ i 3 = Fie Fip

(5-8)

(5-9)

during which Eqs. (3-17) and (3-18) have been invoked and Z is the electrokinetic “figure of merit” as defined in Eq. (3-19). It is apparent that the streaming potential induced electroosmotic backflow produces two additional dispersions in pressure-driven flow: one is the electrokinetic dispersion due to electroosmotic flow itself, the term with δi3 in Eq. (5-8), which tends to increase the total dispersion, and the other is due to the coupling of pressuredriven and electroosmotic flows, the term with δi2 in Eq. (5-8), which tends to decrease the total dispersion. The latter phenomenon has been employed previously to reduce the hydrodynamic dispersion in capillary electrophoresis where a pressure-driven backflow is intentionally introduced to partially compensate the non-uniformity in electroosmotic velocity profile [49,50]. If streaming potential effects are ignored, i.e., for a pure pressure-driven flow with Z = 0, Eq. (5-8) reduces to

K ip = Fip

2

a2 u p

(5-10)

Di

such that

Ki = 1 − δi 2 Z + δi3Z 2 K ip as

(5-11)

Similarly, the dispersion coefficient in Eq. (5-5) for real pressure-driven flow is obtained

χ i = Fip

1 − δi 2 Z + δi3Z 2

(1 − Z )

2

(5-12)

Solute Transport, Dispersion, and Separation in Nanofluidic Channels

17

where Fip = χip is the dispersion coefficient for a pure pressure-driven flow by analogy to Eq. (5-7) in a pure electroosmotic flow. We thus have

χi 1 − δ i 2 Z + δ i 3 Z 2 K = =γ2 i 2 K ip χ ip (1 − Z )

(5-13)

Therefore, χi/χip differs from Ki/Kip by only the square of the apparent viscosity ratio γ, see the definition in Eq. (3-22). As γ is independent of the solute charge number zi, it is expected that the variation of χi/χip with respect to zi will be identical to that of Ki/Kip.

χi for nanochannel chromatography

1 +2

0.1

+1 zi = 0 −1

0.01

−2

0.001 0.1

1

κa

10

100

Figure 8. Dispersion coefficient χi of solutes with zi = [−2, +2] in nanochannel chromatography as a function of κa.

Figure 8 shows χi of solutes with zi = [−2, +2] as a function of κa in nanochannel chromatography. Due to the same reason as stated above for nanochannel electrophoresis, χi of positive solutes is larger than that of neutral ones while χi of negative solutes is the smallest. In both the high and low limits of κa, the flow-induced streaming potential is negligible, see Eq. (3-18) and Figure 3. Hence, the electroosmotic back flow and the induced solute electrophoresis vanish, yielding χi = 2/105 regardless of the solute charge [51]. It is important to note that χi of neutral solutes in pressure-driven flow is not uniformly 2/105 as accepted in the literature. Due to the effects of flow-induced streaming potential, χi is increased by the electroosmotic backflow [i.e., δi3Z2 term in Eq. (5-12)] even though the coupled dispersion term [i.e., −δi2Z term in Eq. (5-12)] drops for neutral solutes [17].

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Xiangchun Xuan

1.15 1.1

χi/χip

Ki/Kip or χi/χip

1.05

zi

1 0.95 0.9 0.85 Ki/Kip

0.8

zi

0.75 0.1

1

10

κa

100 .

Figure 9. Effects of streaming potential on the ratio of solute dispersion, Ki/Kip, and the ratio of dispersion coefficient, χi/χip, in nanochannel chromatography as a function of κa. Adapted with permission from [18].

Figure 9 displays the effects of streaming potential on the ratio of solute dispersion, Ki/Kip, and the ratio of dispersion coefficient, χi/χip, in nanochannel chromatography as a function of κa. In all cases, Ki/Kip is less than 1 indicating that streaming potential effects result in a decrease in hydrodynamic dispersion. This reduction, as a consequence of the induced electroosmotic backflow, gets larger (i.e., Ki/Kip deviates further away from 1) when the solute charge zi increases. The optimum κa at which Ki/Kip achieves its extreme increases slightly with zi. In contrast to the decrease in solute dispersion, the dispersion coefficient is increased by the effects of streaming potential, i.e., χi/χip > 1. These dissimilar trends stem from the dependence of γ on κa, see Eqs. (3-19), (3-22) and (5-13). As streaming potential effects increase (or in other words, the electrokinetic “figure of merit” Z increases), the electroosmotic backflow increases causing a decrease in Ki/Kip (and Ki/Kip < 1) while an increase in γ (and γ > 1). The net result is the observed variation of χi/χip with respect to κa. The increase in χi/χip is more sensitive to zi than the decrease in Ki/Kip. However, the trend that χi/χip varies with respect to zi is consistent with Ki/Kip as pointed out earlier. In addition, χi/χip attains a maximum at a larger value of κa than that at which Ki/Kip is minimized.

5.3. Neutral Solutes For neutral solutes (zi = 0), closed-form formulae are available for the functions Fm (m = ip, ipe, ie) defined in Eq. (5-3) which in turn determine δi2 and δi3 through Eq. (5-9). Specifically, we find [17]

Solute Transport, Dispersion, and Separation in Nanofluidic Channels

Fip = 2 105 Fipe =

Fie =

ω2 2 (1 − ω )

(5-14)

ω ⎡2 2 6 ⎛ 1 − ω ⎞⎤ + ⎢ − ⎥ (1 − ω ) ⎢⎣15 (κ a )2 (κ a )4 ⎜⎝ ω ⎟⎠ ⎥⎦

(5-15)

⎡1 ⎤ 2 3 1 − − 2 ⎢ + ⎥ 2 2 2 2 ⎢⎣ 3 (κ a ) 2ω (κ a ) 2ω (κ a ) cosh (κ a ) ⎥⎦

ω=

19

tanh (κ a ) κa

(5-16)

(5-17)

Figure 10 compares the magnitude of δi2 and δi3 [see their definitions in Eq. (5-9)] for neutral solutes as a function of κa. As shown, δi2 is always larger than δi3. In the low limit of κa, δi2 approaches 2 while δi3 approaches 1, and the square bracketed terms in Eq. (5-2) thus reduces to ( + )2 reflecting the similarity of pressure-driven and electroosmotic flow profiles in very narrow nanochannels. In the high limit of κa, both δi2 and δi3 approach zero because the streaming potential is negligible and the electroosmotic velocity profile becomes essentially plug-like. Note that Eq. (5-14) gives the well-known hydrodynamic dispersion coefficient of neutral solutes in a pure pressure-driven flow between two parallel plates [51]. Moreover, Eq. (5-16) is identical to that derived by Griffiths and Nilson [47,48] which gives the electrokinetic dispersion coefficient of neutral solutes in a pure electroosmotic flow between two parallel plates. 2

δi2

δ

1.5

1

δi3

0.5

0 0.1

1

10 κa

100

1000

Figure 10. Plot of δi2 and δ i3 for neutral solutes as a function of κa. Adapted with permission from [17].

20

Xiangchun Xuan

6. Solute Separation in Nanochannels Solute separation is typically characterized by retention, selectivity, plate height (or plate number), and resolution [43], of which retention and plate height are related to only one type of solutes. In contrast, selectivity and resolution are both dependent on two types of solute species, and thus provide a direct measure of the separation performance of solutes. As plate height is involved in the definition of resolution, see Eq. (6-8), it will still be considered below along with the selectivity and resolution for a comprehensive understanding of solute separation in nanofluidic channels. In order to emphasize the advantage of electrophoresis and chromatography in nanochannels over those taking place in micro-columns, we focus on the solutes with a similar electrophoretic mobility, or specifically, viziF = constant. This is equivalent to assuming a constant charge-to-size ratio or a constant product, Dz = Dizi, of solute charge and diffusivity because solute size is inversely proportional to its diffusivity via the NernstEinstein relation [33]. Such solutes are unable to be separated in free solutions through pressure-driven or electroosmotic microchannel flows. A typical value of the solute chargediffusivity product, Dz = 1×10−10 m2/s, was selected in the following demonstrations while the solute charge number zi may be varied from −4 to +4. The ratio of channel length to halfheight was fixed at L/a = 104 for convenience even though we recognize that fixing the channel length might be a wiser option when the channel height is varied.

6.1. Selectivity Selectivity, rji, is defined as the ratio of the mean speeds of solutes i and j

rji = ui u j

(6-1)

and should be larger than 1 as traditionally defined [43]. A larger rji indicates a better separation. Figure 11 compares the selectivity, rji, of (a) positive and (b) negative solutes in nanochannel chromatography (solid lines) and nanochannel electrophoresis (dashed lines). It is important to note that the indices of rji, which indicate the charge values of the two solutes to be separated, are switched between positive and negative solutes in order that rji > 1 as traditional defined [43]. Specifically, we use r21, r32, and r43 for positive solutes (or more generally, solutes with ziζ* < 0) as those with higher charges are concentrated in a region of smaller fluid speed (i.e., closer to the channel wall) and thus move slower. Note that the solute electrophoretic mobility has been assumed to remain unvaried. In contrast, negative solutes (or solutes with ziζ* > 0) with higher charges appear predominantly in the region of larger fluid speed (closer to the channel center) and thus move faster. Therefore, we need to use r12, r23, and r34 for negative solutes. This index switch also applies to the resolution, Rji, which will be illustrated in Figure 12.

Solute Transport, Dispersion, and Separation in Nanofluidic Channels

4

21

(a)

Selectivity, rji

r43 3 r32

2

r43

r21 r32 r21

1 0.1

1

10

100

κa

Selectivity, rji

1.2

(b)

r12

1.1

r23 r12

r34

r23 r34

1.0 0.1

1

10

100

κa Figure 11: Selectivity, rji, of (a) positive and (b) negative solutes in nanochannel chromatography (solid lines) and nanochannel electrophoresis (dashed lines). Reprinted with permission from [19].

One can see in Figure 11a that the selectivity, rji, of positive solutes in nanochannel chromatography is always greater than that of the same pair of solutes in nanochannel electrophoresis. This discrepancy gets larger when the solute charge number zi increases. Meanwhile, the optimal κa value at which rji is maximized increases for both chromatography and electrophoresis though it is always smaller in the former case. The discrepancy between

22

Xiangchun Xuan

these two optimal κa also increases with the rise of zi. For negative solutes, Figure 11b shows a significantly lower rji than positive solutes in nanochannel chromatography. Moreover, rji decreases when the solute charge number increases. The optimal κa at which rji is maximized is also smaller than that for positive solutes, and decreases (but only slightly) with zi. All these results apply equally to rji of negative solutes in nanochannel electrophoresis except at around κa = 0.6 where rji varies rapidly with κa. Within this region of κa, the electrophoretic velocity of negative solutes is close to the fluid electroosmotic velocity [more accurately, uie in Eq. (4-7)] while in the opposite direction. Therefore, the real solute speed is essentially so small that even a trivial difference in the solute speed (essentially the difference in uie as the solute electrophoretic velocity is constant due to the fixed charge-to-size ratio) could yield a large rji. It is, however, important to note that the speed of negative solutes could be reversed in nanochannel electrophoresis when κa is less than a threshold value (e.g., κa = 0.6 in Figure 11b). In other words, solutes migrate to the anode side instead of the cathode side along with the electrolyte solution. In such circumstances, it is very likely that only one of the two solute species migrates toward the detector no matter the detector is placed in the cathode or the anode side of the channel. Another consequence is that the maximum rji in nanochannel electrophoresis might be achieved with a fairly long analysis time, which makes the separation practically meaningless. We therefore expect that solutes with a constant electrophoretic mobility can be better separated in nanochannel chromatography than in nanochannel electrophoresis. Moreover, solutes with ziζ* < 0 can be separated more easily than can those with ziζ* > 0.

6.2. Plate Height Plate height, Hi, is the spatial variance of the solute peak distribution,

σ i2 , divided by the

migration distance, L, within a time period of ti. It is often expressed in the following dimensionless form of a reduced plate height, hi [42,45]

H i σ i2 2 Di′ti 2 Di′ = = = a aL aL au i

(6-2)

Di′ = Di + K i = Di (1 + χ i Pei2 )

(6-3)

hi =

where Di′ is the effective diffusion coefficient due to a combination of hydrodynamic dispersion Ki [see Eq. (5-4)] and molecular diffusion Di. Note that other sources of dispersion such as injection and detection (refer to [46,52,53] for detail) have been neglected for simplicity. Following Griffiths and Nilson’s analysis [12], we may combine Eqs. (6-2) and (6-3) to rewrite the reduced plate height as

Solute Transport, Dispersion, and Separation in Nanofluidic Channels

hi = 2(1 Pei + χ i Pei )

23 (6-4)

Therefore, hi attains its minimum

hi ,min = 4 χ i at Pei ,opt = 1

χi

(6-5)

In other words, there exists an optimal value for the mean solute speed, ui ,opt = Di a

χi ,

and thus an optimal electric field in nanochannel electrophoresis or an optimal pressure gradient in nanochannel chromatography, at which the separation efficiency is maximized. As hi is a function of solely the dispersion coefficient χi that has been demonstrated in Figures 7 and 8 for nanochannel electrophoresis and chromatography, respectively, its variations with respect to zi and κa are not repeated here for brevity.

6.3. Resolution Resolution, Rji, can be defined in two different ways: the one introduced by Giddings [54], i.e., Eq. (6-6), and the one adopted by Huber [55] and Kenndler et al. [56-58], i.e., Eq. (6-7),

R ji =

t j − ti

(6-6)

2(σ t ,i + σ t , j )

R ji =

t j − ti

(6-7)

σ t ,i

where t is the migration time as defined in Eq. (6-2) and σt is the standard deviation of solute peak distribution in the time domain. Consistent with the solute selectivity rji, a larger value of Rji indicates a better separation. Substituting ti = L ui , t j = L u j and σ t ,i = σ i ui into the last equation leads to

R ji =

L

σi

(r

ji

− 1) =

La hi

(r

ji

− 1)

(6-8)

Referring back to Eq. (6-5), it is straightforward to obtain

R ji ,max =

La hi ,min

(r

ji

− 1) at Pei ,opt = 1

χi

(6-9)

24

Xiangchun Xuan

because the selectivity rji is independent of the solute Peclet number. Therefore, when the plate height of one type of solute is minimized, the separation resolution of this solute from another type of solute may reach the maximum value. Figure 12 compares the maximum resolution, Rji,max, of positive and negative solutes (as labeled) in nanochannel chromatography (solid lines) and nanochannel electrophoresis (dashed lines). The indices of Rji,max are assigned following those of the selectivity, rji, in Figure 11, to ensure Rji,max > 0. One can see that Rji,max of positive solutes in chromatography is larger than that of negative ones throughout the range of κa. In electrophoresis, the former also yield a better resolution if κa > 1. When κa < 1, Rji,max of negative solutes increases and reaches the extremes at κa = 0.6 due to the sudden rise in the selectivity (refer to Figure 11b) as explained above. Within the same range of κa, Rji,max of positive solutes continues decreasing when κa decreases and thus becomes smaller than that of negative solutes. Interestingly, chromatography and electrophoresis offer a comparable resolution for both types of solutes in nanoscale channels if κa > 1.

Positive solutes

Maximum resolution, Rji,max

100

R32

R32

R43

R43 R21 R21

10

R12 Negative solutes

R23 R34

1 0.1

1

10

100

κa Figure 12. Maximum resolution, Rji,max, of positive and negative solutes in nanochannel chromatography (solid lines) and nanochannel electrophoresis (dashed lines). Reprinted with permission from [19].

It is also noted in Figure 12 that the optimum channel size for both separation methods appears to be 1 < κa < 10. In other words, the best channel half-height for solute separation in nanochannels will be 10 nm < a < 100 nm if 1 mM electrolyte solutions are used. In this context, the optimum Peclet number to achieve the maximum resolution in a channel of κa = 5 (or a ≈ 50 nm) will be Pei,opt = O(4) because hi,min = O(1). Although this Peclet number (corresponding to the mean solute speed of the order of 8 mm/s) seems a little too high in current nanofluidics, it indicates that large fluid flows are preferred in both nanochannel

Solute Transport, Dispersion, and Separation in Nanofluidic Channels

25

chromatography and nanochannel electrophoresis for high throughputs and separation efficiencies.

7. Conclusion We have developed an analytical model to study the transport, dispersion and separation of solutes (both charged and non-charged) in electroosmotic and pressure-driven flows through nanoscale slit channels. This model explains why solutes can be separated by charge in nanochannels, and provides compact formulas for calculating the migration speed and hydrodynamic dispersion of solutes. It also presents a simple approach to optimizing the separation performance in nanochannels, which has been applied particularly to solutes with a similar electrophoretic mobility. In addition, we would like to point out that the model or the approach developed in this work can be readily extended to one-dimensional round nanotubes [12,15-17,19] and to even two-dimensional rectangular nanochannels [7,59,60].

References [1] Prakash, S.; Piruska, A.; Gatimu, E. N.; Bohn, P. W. et al. IEEE Sensor. J. 2008, 8, 441450. [2] Abgrall, P.; and Nguyen, N. T. Anal. Chem. 2008, 80, 2326-2341. [3] Schoh, R. B.; and Han, J. Y.; Renaud, P. Rev. Modern Phys. 2008, 80, 839-883. [4] Pennathur, S.; and Santiago, J. G. Anal. Chem. 2005, 77, 6772-6781. [5] Petsev, D. N. J. Chem. Phys. 2005, 123, 244907. [6] Xuan, X.; and Li, D. Electrophoresis 2006, 27, 5020-5031. [7] Dutta, D. J. Colloid. Interf. Sci. 2007, 315, 740-746. [8] Pennathur, S.; Baldessari, F.; Santiago, J. G.; Kattah, M. G. et al. Anal. Chem. 2007, 79, 8316-8322. [9] Das, S.; and Chakraborty, S. Electrophoresis, 2008, 29, 1115-1124. [10] Pennathur, S.; and Santiago, J. G. Anal. Chem. 2005, 77, 6782-6789. [11] Garcia, A. L.; Ista, L. K.; Petsev, D. N. et al. Lab Chip 2005, 5, 1271-1276. [12] Griffiths, S. K.; and Nilson, R. N. Anal. Chem. 2006, 78, 8134-8141. [13] Xuan, X.; and Li, D. Electrophoresis 2007, 28, 627-634. [14] Wang, X.; Kang , J.; Wang, S.; Lu, J.; and Liu, S. J Chromatography A 2008, 1200, 108-113. [15] Xuan, X. J. Chromatography A 2008, 1187, 289-292. [16] De Leebeeck, A.; and Sinton, D. Electrophoresis 2006, 27, 4999-5008. [17] Xuan, X.; and Sinton, D. Microfluid. Nanofluid. 2007, 3, 723-728. [18] Xuan, X. Anal. Chem. 2007, 79, 7928-7932. [19] Xuan, X. Electrophoresis 2008, 29, 3737-3743. [20] Israelachvili, J. Intermolecular & Surface Forces, Academic Press, 2nd edition, San Diego, CA, 1991. [21] Burgreen, D., and Nakache, F. R., J. Phys. Chem. 1964, 68, 1084-1091. [22] Rice, C. L. and Whitehead, R., J. Phys. Chem. 1965, 69, 4017-4024. [23] Hildreth, D., J. Phys. Chem. 1970, 74, 2006-2015.

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[24] Li, D. Colloid. Surf. A 2001, 191, 35-57. [25] Hunter, R. J. Zeta potential in colloid science, principles and applications, Academic Press, New York, 1981. [26] Qu, W.; and Li, D. J. Colloid and Interface Sci. 2000, 224, 397-407. [27] Taylor, J.; and Ren, C. L. Microfluid. Nanofluid. 2005, 1, 356-363. [28] Stein, D.; Kruithof, M.; and Dekker, C. Phys. Rev. Lett. 2004, 93, 035901. [29] Karnik, R.; Fan, R.; Yue, M. et al. Nano Lett. 2005, 5, 943-948. [30] Fan, R.; Yue, M.; Karnik, R. et al. Phys. Rev. Lett. 2005, 95, 086607. [31] Van der Heyden, F. H.J.; Stein, D.; and Dekker, C. Phys. Rev. Lett. 2005, 95, 116104. [32] Van Der Hayden, F. H. J.; Bonthius, D. J.; Stein, D.; Meyer, C.; and Dekker, C. Nano Lett. 2007, 7, 1022-1025. [33] Probstein, R. F. Physicochemical hydrodynamics, John Willey & Sons, New York, 1995. [34] Davidson, C., and Xuan, X., Electrophoresis 2008, 29, 1125-1130. [35] Morrison, F. A.; and Osterle, J. F. J. Chem. Phys. 1965, 43, 2111-2115. [36] X. Xuan, and D. Li, J. Power Source, 156 (2006) 677-684. [37] Griffiths, S. K.; and Nilson, R. H. Electrophoresis 2005, 26, 351-361. [38] Xuan, X.; and Li, D J Micromech. Microeng. 2004, 14, 290-298. [39] Li, D., Electrokinetics in Microfluidics, Elsevier Academic Press, Burlington, MA 2004. [40] Xuan, X. Microfluid. Nanofluid. 2008, 4, 457-462. [41] Taylor, G. I. Proc. Roy. Soc. London A 1953, 219, 186-203. [42] Aris, R. Proc. Roy. Soc. London A 1956, 235, 67-77. [43] Giddings, J. C. Unified separation science, John Wiley & Sons, Inc., New York, 1991. [44] Martin, M.; Giddings, J. C. J. Phys. Chem. 1981, 85, 727-733. [45] Dutta, D.; Ramachandran, A.; and Leighton, D. T. Microfluid. Nanofluid. 2006, 2, 275-290. [46] Ghosal, S., Annu. Rev. Fluid Mech. 2006, 38, 309-338. [47] Griffiths, S. K., Nilson, R. H. Anal. Chem. 1999, 71, 5522-5529. [48] Griffiths, S. K., and Nilson, R. N., Anal. Chem. 2000, 72, 4767-4777. [49] Datta, R. Biotechnol. Prog. 1990, 6, 485-493. [50] Datta, R.; and Kotamarthi, V. R AICHE J. 1990, 36. 916-926. [51] Wooding, R. A., J. Fluid. Mech. 1960, 7, 501-515. [52] Gas, B.; Stedry, M.; and Kenndler, E. Electrophoresis 1997, 18, 2123-2133. [53] Gas, B.; Kenndler, E. Electrophoresis 2000, 21, 3888-3897. [54] Giddings, J. C. Sep. Sci. 1969, 4, 181-189. [55] Huber, J. F. K. Fresenius' Z. Anal. Chem. 1975, 277, 341-347. [56] Kenndler, E. J Cap. Elec. 1996, 3, 191-198. [57] Schwer, C.; and Kenndler, E. Chromatographia 1992, 33, 331-335. [58] Kenndler, E.; and Fridel, W. J Chromatography 1992, 608, 161-170. [59] Dutta, D. Electrophoresis 2007, 28, 4552-4560. [60] Dutta, D. Anal. Chem. 2008, 80, 4723-4730.

In: Fluid Mechanics and Pipe Flow Editors: D. Matos and C. Valerio, pp. 27-39

ISBN: 978-1-60741-037-9 © 2009 Nova Science Publishers, Inc.

Chapter 2

H2O IN THE MANTLE: FROM FLUID TO HIGH-PRESSURE HYDROUS SILICATES N.R. Khisina1,*, R. Wirth2 and S. Matsyuk3 1

Institute of Geochemistry and Analytical Chemistry of Russian Academy of Sciences, Kosygin st. 19, 119991 Moscow, Russia 2 GeoForschungZentrum Potsdam, Germany 3 Institute of Geochemistry, Mineralogy and Ore Formation, National Academy of Sciences of Ukraine, Paladin Ave., 34, 03680 Kiev-142, Ukraine

Abstract Infrared spectroscopic data show that nominally anhydrous olivine (Mg,Fe)2SiO4 contains traces of H2O, up to several hundred wt. ppm of H2O (Miller et al., 1987; Bell et al., 2004; Koch-Muller et al., 2006; Matsyuk & Langer, 2004) and therefore olivine is suggested to be a water carrier in the mantle (Thompson, 1992). Protonation of olivine during its crystallization from a hydrous melt resulted in the appearance of intrinsic OH-defects (Libowitsky & Beran, 1995). Mantle olivine nodules from kimberlites were investigated with FTIR and TEM methods (Khisina et al., 2001, 2002, 2008). The results are the following: (1) Water content in xenoliths is lower than water content in xenocrysts. From these data we concluded that kimberlite magma had been saturated by H2O, whereas adjacent mantle rocks had been crystallized from water-depleted melts. (2) Extrinsic water in olivine is represented by highand hydrous olivine pressure phases, 10Å-Phase Mg3Si4O10(OH)2.nH2O n(Mg,Fe)2SiO4.(H2MgSiO4), both of which belong to the group of Dense Hydrous Magnesium Silicates (DHMS), which were synthesized in laboratory high-pressure experiments (Prewitt & Downs, 1999). The DHMS were regarded as possible mineral carriers for H2O in the mantle; however, they were not found in natural material until quite recently. Our observations demonstrate the first finding of the 10Å-Phase and hydrous olivine as a mantle substance. (3) 10Å-Phase, which occurred as either nanoinclusions or narrow veins in olivine, is a ubiquitous nano-mineral of kimberlite and closely related to olivine. (4) There are two different mechanisms of the 10Å-Phase formation: (a) purification of olivine from OHbearing defects resulting in transformation of olivine to the 10 Å-Phase with the liberation of water fluid; and (b) replacement of olivine for the 10Å-Phase due to hydrous metasomatism in the mantle in the presence of H2O fluid. *

E-mail address: [email protected]

28

N.R. Khisina, R. Wirth and S. Matsyuk

Introduction The presence of water in the mantle, either as H2O molecules or OH- groups, has been the subject of long-term interest in geochemistry and geophysics because of the dramatic H2O influence on the melting and the physical properties of mantle rocks. Nowdays the concept of “wet” and heterogeneous mantle is universally accepted among petrologists (Kamenetsky et al., 2004; Sobolev & Chaussidon, 1996; Katayama et al., 2005). However, which minerals could serve as deep water storage in the mantle is yet to be a widely-discussed topic. The problem is that the mantle material available for a direct investigation is very limited and restricted by mantle nodules trapped by kimberlitic or basaltic magma from the depth. Among the presumed candidates considered for water storage in the mantle were the so-called DHMS phases (dense hydrous magnesium silicates) synthesized in laboratory experiments at P-T conditions of the mantle (Prewitt & Downs, 1999); however, DHMS phases have not yet been found as macroscopic minerals in mantle material. The main mineral of the mantle is olivine (Mg,Fe)2SiO4, which is stable at mantle pressures up to ~ 15 GPa; with increasing pressure the olivine ( α-phase) transforms to higher density structure of wadsleite (β-phase). The P-Tconditions of α-β transition in olivine correspond to the depth of ~ 400 km; according to geophysical data, the mantle has a discontinuity at this depth, specified as a boundary between the upper mantle and transition zone. Infrared spectroscopic data show that olivine contains traces of H2O up to several hundreds wt. ppm of H2O (Bell et al., 2003; Koch-Müller et al., 2006; Kurosawa et al., 1997; Matsyuk & Langer, 2004; Miller et al., 1987); therefore, nominally anhydrous olivine is considered as water storage in the mantle (Thompson, 1992). The highest water content, such as about 400 wt. ppm of H2O, was registered for olivine samples from mantle peridotite nodules in kimberlites. Water in olivine occurs as either OHor H2O. There are two modes of “water” occurrence in olivine: intrinsic and extrinsic. An intrinsic mode represents the OH- incorporated into the olivine structure and is considered a water-derived defect complex either associated with a metal vacancy {vMe, 2OH-} or by a vacancy at the Si site {vSi,4OH-} (Beran & Putnis, 1983; Beran & Libovitzky, 2006; Lemaire et al., 2004). Extrinsic water is possessed by inclusions, either solid or fluid, and occurs as OH- or H2O. Recent TEM investigations of olivine nodules from Udachnaya kimberlite (Yakuyia) revealed the nanoinclusions of hydrous magnesium silicates represented by highpressure phases, 10Å-Phase and hydrous olivine, partially replaced by low-pressure serpentine + talc assemblage (Wirth & Khisina, 1998; Khisina et al., 2001; Khisina & Wirth, 2002, 2008a, 2008b). 10Å-Phase and hydrous olivine belong to the group of DHMS phases. This first finding of DHMS phases in mantle material (Wirth & Khisina, 1998; Khisina et al., 2001; Khisina & Wirth, 2002, 2008a, 2008b) provides direct evidence of the DHMS occurrence in the mantle and specifies them as nanominerals of non-magmatic origin, closely related to olivine. The amount of water incorporated into the olivine structure depends on the P-T conditions as well as on the chemical environment and olivine composition (Fe/Mg ratio in olivine), and increases with increasing water activity, oxygen fugacity, pressure and temperature (Kohlstedt et al., 1996; Bai & Kohlstedt, 1993; Zhao et al., 2004). Diffusion of hydrogen in olivine is very fast (Kohlstedt & Mackwell, 1998); therefore, due to interaction of olivine and surrounding melt the initial water content in olivine can be changed under changes of either P, or T, or fO2 or fH2O during a post-crystallization stage. Olivine remained

H2O in the Mantle: From Fluid to High-Pressure Hydrous Silicates

29

in a host melt can loose water due to decompression; olivine trapped by a foreign melt can become either deprotonated or secondarily protonated, depending on whether the foreign melt is lower or higher by water concentration (by water activity) in comparison to the host melt, correspondingly. Experiments by Peslier and Luhr (2006) and Mosenfelder et al. (2006a, 2006b) demonstrate rapid processes of deprotonation and secondary protonation of olivine. FTIR and TEM data provide the information about H2O content and a mode of water occurrence in olivine. Here we suggest a way to reconstruct the P-history of the kimberlite process and elucidate the water behavior at different stages of the kimberlite process. The collected data on the H2O occurrence in mantle olivine nodules represented by xenoliths, xenocrysts and phenocrysts from Yakutian kimberlites (Udachnaya, Obnazennaya, Mir, Kievlyanka, Slyudyanka, Vtorogodnitza and Bazovayua pipes) are used here as a guide for tracing H2O behavior from fluid to high-pressure DHMS phases in the mantle. We show here that nominally anhydrous olivine is a carrier of water in the mantle and can be used as indicator of P-f(H2O) regime in the mantle.

Samples and Collected Data Sample Description Typical kimberlites сontain xenocrysts and xenoliths of mantle and crustal origin embedded into a fine- to coarse-grained groundmass of crystallized kimberlite melt (Sobolev V.S. et al, 1972; Sobolev N.V., 1974; Pokhilenko et al., 1993; Ukhanov et al., 1988; Matsyuk et al., 1995). We collected the mantle olivine samples represented by xenoliths, xenocrysts and phenocrysts from kimberlite pipes of Yakutian kimberlite province (Udachnaya, Obnazennaya, Mir, Slyudyanka, Vtorogodniza and Kievlyanka). Olivine samples are classified as xenoliths, xenocrysts and phenocrysts on the base of petrographic examination. Phenocrysts are olivine single grains disintegrated from groundmass of kimberlite rock. Xenoliths are fragments of adjacent mantle and crustal rocks trapped by kimberlite magma and transported from the depths during kimberlite eruption. Xenocrysts are olivine singlegrains disintegrated from adjacent rocks, either mantle or crustal, and lifted from the depths during kimberlite eruption. Xenoliths are of several cm in size. Xenocrysts are less than 1 cm in size and comparable by size with phenocrysts. The samples were studied with optical microscopy, FTIR and TEM.

H2O Content in the Olivine Samples The highest H2O content in the mantle olivine samples from Yakutian kimberlites was measured as 400–420 wt. ppm of H2O (Koch-Muller et al., 2006; Matsyuk & Langer, 2004). H2O contents in xenoliths from Udachnaya and Obnazennaya pipes vary between 14 and 246 wt. ppm (Table 1). Present FTIR study on the H2O content in xenoliths from Udachnaya and Obnazennaya, together with previous Infrared spectroscopic data on xenocrystic and phenocrystic olivine samples (Matsyuk and Langer, 2004; Koch-Muller et al., 2006) show the wide variation of

30

N.R. Khisina, R. Wirth and S. Matsyuk

the H2O content in the olivine samples, such as between 0–3 and 400 wt. ppm.of H2O. The results are shown in histograms (Figure 1). Summarized FTIR data on the H2O content in the all studied mantle olivine samples including xenoliths, xenocrysts and phenocrysts from Yakutian kimberlites (Matsyuk and Langer, 2004; Koch-Müller et al., 2006; present study) are represented at histogram (Figure 1a). The same data is plotted individually for phenocrysts (Figure 1b) and xenoliths together with xenocrysts (Figure 1c). For comparison, the H2O contents in mantle-derived olivine megacrysts from the Monastery kimberlite, South Africa (Bell et al., 2004) are plotted on the histogram in Figure 1d. The most frequent values of H2O in olivine samples from Monastery kimberlite are 150–200 wt. ppm of H2O. Table 1. Water contents in olivine from mantle xenoliths

Ob-152

Obnazennaya pipe

Garnet lerzolite

H2O content, wt. ppm 246

Ob-105

″

Garnet lerzolite

14–65

Ob-174

″

Garnet lerzolite

143(15)

Ob-312

″

Garnet lerzolite

45

U-47/76

Udachnaya pipe

18

9206

″

9.5 cm in size megacrystalline . diamond-bearing xenolith. Harzburgite-dunite 5.2 cm in size coarse-grained xenolith. Garnet-free harzburgite

Xenolith sample

Locality

Xenolith description

12

12

8

8

4

4

0

0 0

100

200 300 H2O, wt. ppm

400

0

100

(A)

200 H2O, wt. ppm

(B) Figure 1. Continued on next page.

29

300

400

H2O in the Mantle: From Fluid to High-Pressure Hydrous Silicates 12

31

10

8

8 6

4

4 2

0

0 0

100

200 H2O, wt. ppm

300

(C)

400

0

50 100 150 200 250 H2O, wt. ppmÐ2Ùá öåþ ççüÐ2Ùá öåþ ççü

300

(D)

Figure 1. Histograms of the H2O contents in mantle olivine samples from kimberlites. a – c: The FTIR data for Yakutiyan olivine samples from Udachnaya, Obnazennaya, Mir, Vtorogodnitza, Kievlyanka and Sluydyanka kimberlites (present data; Matsyuk & Langer, 2004; Koch-Müller et al., 2006); a – xenoliths, xenocrysts and phenocrysts all together; b – phenocrysts; c – xenoliths and xenocrysts; d – Xenoliths from Monastery kimberlite, South Africa (Bell et al., 2003).

Extrinsic H2O in Olivine Samples TEM examination of the olivine samples (Khisina et al., 2001; Khisina & Wirth, 2002; Khisina et al., 2008) revealed the OH- segregation resulted in nano-heterogeneity of several types: (i) nanoinclusions; (ii) lamellar precipitates; (iii) veins developed along healed microcracks. All kinds of heterogeneity are associated with deformation slip bands in the samples understudy (Khisina et al., 2008). (i) Nanoinclusions were observed in both xenoliths and xenocrysts. They are several tens of nanometers in size and have a shape of pseudohexagonal negative crystals. The nanoinclusions are often arranged in arrays along [100], [011], [101] and [-101] crystallographic directions of the olivine host. The phase constituents of nanoinclusions were identified from TEM data (Wirth & Khisina, 1998; Khisina et al., 2001; Khisina & Wirth, 2002; Khisina et al., 2008) as 10Å-Phase Mg3Si4O10(OH)2.nH2O, where n = 0.65, 1.0, and 2.0, and hydrous olivine (MgH2SiO4).n(Mg2SiO4), both of them represent high-pressure DHMS phases (Bauer & Sclar, 1981; Khisina & Wirth, 2002; Churakov et al., 2003). 10Å-Phase is a typical constituent of nanoinclusions in xenocrysts, while hydrous olivine is more common for nanoinclusions in xenoliths. Both 10Å-Phase and hydrous olivine are replaced often by a low-pressure assemblage of serpentine Mg3Si2O5(OH)4 + talc Mg3Si4O10(OH)2. High-pressure phases in nanoinclusions as well as their replacement products are strictly aligned relative to the crystallographic directions of the olivine matrics, with aol ║ ahy ║ c10Å ║ ctc ║ cserp (hy is abbreviation of hydrous

32

N.R. Khisina, R. Wirth and S. Matsyuk

Figure 2. Array of nanoinclusions in olivine xenocryst from Udachnaya kimberlite. Nanoinclusions are composed of the 10Å-Phase in the middle part of the inclusions, and filled by H2O fluid in the regions bordered the adjacent olivine (white areas of nanoinclusions).

Figure 3. Healed microcrack filled by 10Å-Phase in olivine xenolite sample from Udachnaya kimberlite.

olivine), which is indicative of the topotaxic character of these intergrowths. A characteristic feature of nanoinclusions in xenocrystic olivine samples is the presence of voids unfilled with solid material (Figure 2). Solid phase fills the equatorial area of the inclusions parallel to the (100) plane of the olivine host, and voids in nanoinclusions are observed at the polar areas bordering the olivine matrix (Figure

H2O in the Mantle: From Fluid to High-Pressure Hydrous Silicates

33

2). The observations led to the conclusion that these voids are not artifacts and were not produced during sample preparation. According to TEM observations, the nanoinclusions are not connected with the surface of the crystals through channels, which could have served as pathways for the transport of H2O from the external medium during the formation of inclusions. (ii) (100) Lamellar precipitates of hydrous olivine, several nanometers in thickness were observed in xenolithic olivine (Khisina et al., 2001; Khisina & Wirth, 2002; Churakov et al., 2003). Veins filled by 10Å-Phase + talc, were observed in xenolithic olivine (Figure 3). These veins were developed along (100) healed microcracks.

Discussion Olivine as Water Storage in the Mantle Histogram of the all H2O content measurements for olivine samples, as summarized for xenoliths, xenocrysts and phenocrysts together (Figure 1a) reveals two maximums, at 0–50 wt. ppm of H2O and 200–250 wt. ppm of H2O. The same maximums are pronounced at the histogram represented the H2O contents in xenoliths and xenocrysts together (Figure 1c), whereas only one maximum at 200–250 wt. ppm of H2O is observed at the histogram represented the H2O data for phenocrystic olivine (Figure 1b). Histograms at Figure 1b,c show that the maximum between 0–50 wt. ppm of H2O corresponds to the most frequent H2O contents in xenoliths and xenocrysts, while the maximum between 200–250 wt. ppm of H2O corresponds to the most frequent H2O values in phenocrysts. We suggest that the most frequent H2O contents correspond to the initial H2O contents incorporated by the olivine samples during crystallization. Hence, the initial H2O content in olivine derived from adjacent and kimberlite rocks has been different. We conclude that compared to xenolithic and xenocrystic samples, the olivine phenocrysts have incorporated more amount of water during crystallization. According to the experimental data on the P-f(H2O)–dependence of the OHsolubility in olivine (Kohlstedt et al, 1996), this could mean that the trapped fragments of adjacent and kimberlite rocks have been crystallized either at different depths or from different melts. On the basis of the data by Kohlstedt (1996) the pressure conditions of olivine crystallization from water-saturated melt can be estimated as < 1 GPa for xenocrysts and xenoliths and as 4–4.5 GPa for phenocrysts. The pressure of 4–4.5 GPa estimated as the pressure of crystallization of phenocrysts corresponds to the depth of 120–135 km that is consistent with the upper mantle depths and is in good agreement with estimations of the depths of kimberlite formation as between 120 and 230 km (Pochilenko et al., 1993). The pressure < 1 GPa estimated as the pressure of crystallization of mantle olivine samples represented by xenoliths and xenocrysts, is less than the lowest pressures in the mantle and corresponds to the depth of α max ) max ⎩ * g

On the Numerical Simulation of Turbulence Modulation in Two-Phase Flows

51

The non-dimensional inter-phase transfer coefficients can be correlated in terms of the particle Reynolds number and is given by

Re lg =

ρl U g − U l d g μl

where µl is the viscosity of the liquid phase.

3.2.2. MUSIG Model To account for non-uniform bubble size distribution, the MUSIG model employs multiple discrete bubble size groups to represent the population balance of bubbles. Assuming each bubble class travel at the same mean algebraic velocity, individual number density of bubble class i based on Kumar and Ramkrishna (1996a) can be expressed as:

⎛ ⎞ ∂ni K + ∇ ⋅ (u g ni ) = ⎜⎜ ∑ R j ⎟⎟ ∂t ⎝ j ⎠i where

(∑ R ) j

j

i

(3.26)

represents the net change in the number density distribution due to

coalescence and break-up processes. The discrete bubble class between bubble volumes vi and vi +1 is represented by the centre point of a fixed non-uniform volume distributed grid interval. The interaction term

(∑ R ) = (P j

j

i

C

+ PB − DC − DB ) contains the source rate of

PC , PB , DC and DB , which are, respectively, the production rates due to coalescence and break-up and the death rate due to coalescence and break-up of bubbles. 3.2.2.1. MUSIG Break-up Rate The production and death rate of bubbles due to the turbulent induced breakage is formulated as:

PB =

∑ Ω (v j : vi )n j N

j =i +1

N

DB = Ωi ni with Ωi = ∑ Ωki

(3.27)

k =1

Here, the break-up rate of bubbles of volume v j into volume vi is modelled according to the model developed by Luo and Svendsen (1996). The model is developed based on the assumption of bubble binary break-up under isotropic turbulence situation. The major difference is the daughter size distribution which has been taken account using a stochastic breakage volume fraction fBV. By incorporating the increase coefficient of surface area, cf =

52

K. Mohanarangam and J.Y. Tu 2/3

[ f BV +(1-fBV)2/3-1], into the breakage efficient, the break-up rate of bubbles can be obtained as:

Ω (v j : vi )

⎛ ε ⎞ = FB C ⎜ 2 ⎟ ⎜d ⎟ (1 − α g )n j ⎝ j ⎠ where

1/ 3 1

∫

ξ min

(1 + ξ )2 × exp⎛⎜ − ξ

⎜ ⎝

11 / 3

12c f σ βρl ε

2/3

d

ξ

5/3

11 / 3

⎞ ⎟dξ ⎟ ⎠

ξ = λ / d j is the size ratio between an eddy and a particle in the inertial sub-range

and consequently

ξ min = λmin / d j and C and β are determined, respectively, from

fundamental consideration of drops or bubbles breakage in turbulent dispersion systems to be 0.923 and 2.0. 3.2.2.2. MUSIG Coalescence Rate The number density of individual bubble groups governed by coalescence can be expressed as:

PC =

1 i i ∑∑η jki χ ij ni n j 2 k =1 l =1

η jki =

(ν j + ν k ) − ν i −1 /(ν i − ν i −1 )

if ν i −1 < ν j + ν k < ν i

ν i +1 − (ν j + ν k ) /(ν i +1 − ν i )

if ν i < ν j + ν k < ν i +1

0

otherwise

N

DC = ∑ χ ij ni n j j =1

From the physical point of view, bubble coalescence occurs via collision of two bubbles which may be caused by wake entrainment, random turbulence and buoyancy. However, only turbulence random collision is considered in the present study as all bubbles are assumed to be spherical (wake entrainment becomes negligible). Furthermore, as all bubbles travel at the same velocity in the MUSIG model, buoyancy effect is also eliminated. The coalescence rate considering turbulent collision taken from Prince and Blanch (1990) can be expressed as:

χ ij = FC

π

[d 4

i

+dj

] (u 2

2 ti

+ u tj2

)

0.5

⎛ t ij exp⎜ − ⎜ τ ⎝ ij

where τ ij is the contact time for two bubbles given by ( d ij / 2)

2/3

⎞ ⎟ ⎟ ⎠

/ ε 1 / 3 and t ij is the time

required for two bubbles to coalesce having diameter di and dj estimated to be

[(dij / 2)3 ρl / 16σ ]0.5 ln(h0 / h f ) . The equivalent diameter dij is calculated as suggested by

On the Numerical Simulation of Turbulence Modulation in Two-Phase Flows

53

−1

Chesters and Hoffman (1982): (d ij = ( 2 / d i + 2 / d j ) . According to Prince and Blanch (1990), for air-water systems, experiments have determined the initial film thickness ho and −4

−8

critical film thickness hf at which rupture occurs as 1 × 10 and 1 × 10 m respectively. The turbulent velocity ut in the inertial subrange of isotropic turbulence (Rotta, 1972) is given by: u t =

2ε 1 / 3 d 1 / 3 .

Numerical Procedure All the transport equations are discretized using a finite volume formulation in a generalized coordinate space, with metric information expressed in terms of area vectors. The equations are solved on a nonstaggered grid system, wherein all primitive variables are stored at the centroids of the mass control volumes. Third-order QUICK scheme is used to approximate the convective terms, while second-order accurate central difference scheme is adopted for the diffusion terms. The velocity correction is realized to satisfy continuity through SIMPLE algorithm, which couples velocity and pressure. At the inlet boundary the particulate phase velocity is taken to be the same as the gas velocity. The concentration of the particulate phase is set to be uniform at the inlet. At the outlet the zero streamwise gradients are used for all variables. The wall boundary conditions are based on the model of Tu and Fletcher (1995). All the governing equations for both the carrier and dispersed phases are solved sequentially at each iteration, the solution process is started by solving the momentum equations for the gas phase followed by the pressure-correction through the continuity equation, turbulence equations for the gas phase, are solved in succession. While the solution process for the particle phase starts by the solution of momentum equations followed by the concentration then gas-particle turbulence interaction to reflect the two-way coupling, the process ends by the solution of turbulence equation for the particulate phase. At each global iteration, each equation is iterated, typically 3 to 5 times, using a strongly implicit procedure (SIP).The above solution process is marched towards a steady state and is repeated until a converged solution is obtained.

Numerical Predictions Gas –Particle Flow 4.1. Code Verification In this section the code is validated for mean streamwise velocities and fluctuations for both the carrier and dispersed phases against the benchmark experimental data of Fessler and Eaton (1995). This task is undertaken to verify the fact that these two classes of particles, which share the same Stokes number but varied particle Reynolds number can be handled by the code. Figure 4.1 show the backward facing step geometry used in this study, which is similar to the one used in the experiments of Fessler and Eaton (1995), which has got a step height (h) of 26.7mm. As the span wise z-direction perpendicular to the paper is much larger

54

K. Mohanarangam and J.Y. Tu

than the y-direction used in the experiments, the flow is considered to be essentially twodimensional. The backward facing step has an expansion ratio of 5:3. The Reynolds number over the step works out to be 18,400 calculated based on the centerline velocity and step height (h). The independency of grid over the converged solution was checked by refining the mesh system through doubling the number of grid points along the streamwise and the lateral directions. Simulations results revealed that the difference of the reattachment length between the two mesh schemes is less than 3%.

H

y

H=40mm h=26.7mm

x h

x=35h Figure 4.1. Backward facing step geometry.

4.1.1. Mean Streamwise Velocities The mean streamwise velocities for the gas phase have been shown in Figure 4.2 for various stations along the backward-facing step geometry, it can be generally seen that there is fairly good agreement with experimental findings of Fessler and Eaton (1995). This is then followed by the streamwise velocities for the two classes of particles considered in this study, whose properties are tabulated in Table 1. The broad varying characteristics of different particle sizes and material properties can be unified by a single dimensionless parameter; it is also used to quantify the particles responsivity to fluid motions and this non-dimensional parameter is the Stokes number (St) and is given by the ratio of particle relaxation time to time that of the appropriate fluid time scale, St = tp/ts. In choosing the appropriate fluid time scale, the reattachment length has not been considered as it varies due to addition of particles and is not constant in this study, rather a constant length scale of five step heights, which is in accordance to the reattachment length is used. The resulting time scale is given by ts=5h/Uo, in accordance with the experiments of Fessler and Eaton (1995). Table 4.1 Properties of the dispersed phase particles Nominal Diameter(μm)

150

70

Material

Glass

Copper

Density(kg/m3)

2500

8800

Stokes Number (St)

7.4

7.1

Particle Reynolds number (Rep)

9.0

4.0

On the Numerical Simulation of Turbulence Modulation in Two-Phase Flows 0

u/Ub 1.5

2.50

Height (y/h)

2.00 x/h=

x/h=

x/h=

x/h=

x/h=9

x/h=1

x/h=1

1.50 1.00 0.50 0.00 Distance along the step (x/h) ○ Experimental

Numerical

Figure 4.2. Mean streamwise gas velocities.

0

u/Ub 1.5

2.50

Height (y/h)

2.00 1.50

x/h=

x/h=5

x/h=

x/h=9

1.00 0.50 0.00

Distance along the step (x/h) ○ Experimental

Numerical

Figure 4.3.a. Streamwise mean velocity for 70μm copper particles.

x/h=1

55

56

K. Mohanarangam and J.Y. Tu

The significance of the Stokes number is that a particle with a small Stokes number (St1) particles are found be no longer in equilibrium with the surrounding fluid phase as they are unresponsive to fluid velocity fluctuations and they will pass unaffected through eddies and other flow structures. Figures 4.3a & 4.3b shows the mean streamwise velocity profiles for the two particle classes considered in this study, it can be seen that there is generally a fairly good agreement with the experimental results. From the particle mean velocity graphs of the carrier and dispersed phases it can be inferred, that the particle streamwise velocity at the first station x/h=2 is lower than the corresponding gas velocities, this is in lines with the fully developed channel flow reaching the step as described in the experiments of Kulick et al (1994), wherein the particles at the channel centerline have lower streamwise velocities than that of the fluid as a result of cross-stream mixing. However the gas velocity lags behind the particle velocities aft of the sudden expansion as the particles inertia makes them slower to respond to the adverse pressure gradient than the fluid. 0

u/Ub 1.5

2.50

Height (y/h)

2.00 1.50

x/h=

x/h=

x/h=7

x/h=9

x/h=1

1.00 0.50 0.00 Distance along the step (x/h) ○ Experimental

Numerical

Figure 4.3.b. Streamwise mean velocity for 150μm glass particles.

4.1.2. Mean Streamwise Fluctuations The code has been further validated for mean streamwise fluctuations and Figure 4.4 shows the mean streamwise fluctuations for the gas phase against the experimental data. It is seen that there is a general under prediction of the simulated data with the experimental

On the Numerical Simulation of Turbulence Modulation in Two-Phase Flows

57

results and this is more pronounced towards the lower wall for a height of up to y/h≤2, however the pattern of the simulated results have been found to be in tune with its experimental counterpart. 0

u'/Ub

0.2

2.50

Height (y/h)

2.00 x/h=

x/h

x/h=

x/h

x/h=1

1.50

1.00

0.50

0.00 Distance along the step (x/h) Figure 4.4. Fluctuating streamwise gas velocities particles.

0

u'/Ub 0.3

2.50

Height (y/h)

2.00

x/h=2

x/h=5

x/h=7

x/h=9

x/h=12

1.50 1.00 0.50 0.00

Distance along the step (x/h) ○ Experimental

Numerical

Figure 4.5.a. Fluctuating streamwise particle velocities for 70μm copper particles.

58

K. Mohanarangam and J.Y. Tu 0

u'/Ub

0.3

2.50

Height (y/h)

2.00 x/h=5

x/h=

x/h=7

x/h=9

1.50

x/h=1

1.00 0.50 0.00 Distance along the step (x/h) ○ Experimental

Numerical

Figure 4.5.b. Fluctuating streamwise particle velocities for 150μm glass particles.

Figures 4.5a & 4.5b show the streamwise fluctuating particle velocities for the two different classes of particles considered, there has been a minor under-prediction until stations y/h ≤ 1, over all a fairly good agreement can be observed. It can be also seen that for y/h > 1.5, the particle fluctuating velocities are considerably larger than those of the fluid. This again is in accordance with channel flow inlet conditions, where the particles have higher fluctuating velocities than those of the fluid owing to cross-stream mixing. All the experimental results used for comparison of particle fluctuating velocities correspond to maximum mass loadings of particles as reported in the experiments of Fessler & Eaton (1999).

4.2. Results and Discussion 4.2.1. Turbulence Modulation (TM) The plot depicted in the following sections to represent the Turbulent Modulation (TM) of the carrier gas phase is given by the ratio of the laden flow r.m.s streamwise velocity to the unladen r.m.s streamwise velocity. These plots signify that any turbulence modulation felt in the carrier phase is reflected as an exit of the ratio from unity. 4.2.1.1. Analysis of Experimental Data Plots 4.6a and 4.6b depict the experimental data as obtained from the experiments of Fessler and Eaton (1995). Figures 4.6a&b represent the mean streamwise particle velocities

On the Numerical Simulation of Turbulence Modulation in Two-Phase Flows

59

and fluctuations respectively for the two sets of dispersed phase particles i.e., copper and glass particles. It can be well seen that the mean velocities and to a similar extent the fluctuations for these two classes seem to behave analogous to each other. However from figures 4.7a-c, which shows the experimental TM for the carrier phase in the presence of the dispersed phase at the same mass loading of 40% seem to behave in contrary to the above findings. In all these stations considered in this study the glass particle seems to attenuate the carrier phase more than copper particles. It is also be seen at the station x/h=2, the attenuation 0

u/Ub

1.5

2.50 x/h=2

x/h=5

x/h=7

x/h=9

Height (y/h)

2.00

1.50

1.00

0.50

0.00 Distance along the step (x/h)

70μm copper

150μm glass

Figure 4.6.a. Experimental mean streamwise particle velocity.

of the turbulence for glass particle is totally opposite in relation to copper particle for location y/h>1.25. At station x/h=7 for certain regions along the height of the step the turbulence attenuation for the two particle classes seems to be in phase, whereas before and after this small region of unison the glass particle seem to attenuate more than the copper particles. At the station x/h=14, it can be clearly seen, that there is a uniform degree of difference in attenuation all along the step, this is more attributed to the uniform distribution of the particles seen along the step. The maximum turbulence attenuation can be seen for the 150μm particles, which up to 35% as reported by Fessler and Eaton (1999). From the above analysis, it is quite clear that particles with the same Stokes number modulate the carrier phase turbulence in a totally different fashion. This makes us conclude that although Stokes number can be generalized to account for mean values of velocity and fluctuations, it cannot be generalized when it comes to TM, in which case something more than Stokes number is required to define one’s particle response to surrounding carrier phase turbulence either to attenuate or enhance it.

60

K. Mohanarangam and J.Y. Tu 0

u'/Ub

0.3

2.50 x/h=

x/h=

x/h=

x/h=7

Height (y/h)

2.00

1.50

1.00

0.50

0.00 Distance along the step (x/h) Figure 4.6.b. Experimental fluctuating streamwise particle velocity. 1.10

Turbulence Modulation (TM)

x/h=2 1.00

0.90

0.80

0.70

0.60 0.00

0.50

1.00

70μm copper

1.50 Height (y/h)

2.00

150μm glass

Figure 4.7.a. Experimental Turbulence Modulation at x/h=2.

2.50

On the Numerical Simulation of Turbulence Modulation in Two-Phase Flows

1.10

Turbulence Modulation (TM)

x/h= 1.00

0.90

0.80

0.70

0.60 0.00

0.50

1.00 1.50 Height (y/h)

2.00

2.50

Figure 4.7.b. Experimental Turbulence Modulation at x/h=7.

1.10 Turbulence Modulation (TM)

x/h=14

1.00

0.90

0.80

0.70

0.60 0.00

0.50 70μm copper

1.00 1.50 Height (y/h)

2.00 150μm glass

Figure 4.7.c. Experimental Turbulence Modulation at x/h=14.

2.50

61

62

K. Mohanarangam and J.Y. Tu

4.2.2. TM & (Particle Number Density) PND Results

12.00

1.20

10.00

1.00

8.00

0.80

6.00

0.60

4.00

0.40

2.00

0.20

0.00 0.00 ●Experimental-TM

0.50

1.00 1.50 Height (y/h)

2.00

Numerical-TM ▲Experimental-PND

Turbulence modification (TM)

Particle Number density (PND)

In this section we have tried to derive an understanding for the Turbulence Modulation (TM) of the carrier gas phase in the presence of particles using our turbulent formulation, along with its corresponding PND results for the two classes we have considered in this study. The simulated results of the above two parameters are plotted along side the experimental findings of Fessler and Eaton (1995). The experimental values for the modulation are plotted with error bars, as significant scatter are apparent in these plots, thereby making any variations on the order of ±5% insignificant (Fessler & Eaton, 1995).

0.00 2.50

Numerical-PND

Figure 4.8.a. Turbulence Modulation & Particle Number Density for 70μm copper particles at x/h=2.

The plots 4.8a-c shows the combined numerical and experimental results of TM (secondary axis) and PND (primary axis) for three sections along the step viz. x/H=2, 7 &14 for copper particles. It can be noted at station x/h=2 just aft of the step, for y/h1.5, while the PND seem to vary from under predicting to over predicting the experimental data. Figures 4.9a-c shows plots of the glass particles for the same three sections along the step. At section x/h=2, the simulated values seem to over predict the experimental data for y/h>1, however this distinct behavior of maximum attenuation as reported by Fessler and Eaton (1995) occurs here, this under prediction is not quite in terms with the PND results which seem to show a uniform distribution. A fairly good

On the Numerical Simulation of Turbulence Modulation in Two-Phase Flows

63

12.00

1.20

10.00

1.00

8.00

0.80

6.00

0.60 `

4.00

0.40

2.00

0.20

0.00 0.00 ●Experimental-TM

0.50

1.00 1.50 Height (y/h)

2.00

Numerical-TM ▲Experimental-PND

Turbulence modification (TM)

Particle Number density (PND)

agreement is with both turbulence modulation and PND can be seen for stations x/h=7 and 14.

0.00 2.50 Numerical-PND

Figure 4.8.b. Turbulence Modulation & Particle Number Density for 70μm copper particles at x/h=7.

It is generally seen that for the two classes of particles considered, there exist considerably more particles in the region y/h>1 for locations x/h=2 and 7, and the particles exhibit a uniform distribution by the time it reaches the location x/h=14 due to its uniform spreading action, but despite a uniform distribution the turbulence attenuation is still small for y/h1, while the showing an increase with respect to liquid in the lower part. At the middle section considered, for a height of about y/h>1 they exhibit a homogenous flow behaviour, whereas at the lower part the particles again seem to exceed its liquid counterpart. Whereas near the exit, both the continuous and dispersed phases have almost the same pattern prompting to the fact that the particle ‘catch up’ with the liquid phase, mimicking a homogenous flow pattern. Figures 5.5a-c show the fluctuation results for the GP flow, at the entry section x/h=2, it can be seen that the particulate phase has a higher fluctuation in comparison to the gas phase, while at the middle section x/h=7, the gas phase seem to catch up with the particulate phase, while at the exit at x/h=14, it is observed that both the dispersed and the continuous phase seem to fluctuate in unison. From the above experimental results of the fluctuation, it can be ascertained that the particle ‘lag’ behind with respect to the continuous phase in terms of LP flows, whereas they ‘lead’ in terms of the gas-particle flows.

2.00 1.75 x/h=0.7 1.50

y/h

1.25 1.00 0.75 0.50 0.25 0.00 -0.20 0.00 0.20 0.40 0.60 0.80 1.00 u/Uo Liquid

Particle

Figure 5.2.a. Experimental mean streamwise velocities at x/h=0.7 for liquid-particle flow.

70

K. Mohanarangam and J.Y. Tu

2.00 1.75

x/h=7.8

1.50

y/h

1.25 1.00 0.75 0.50 0.25 0.00 -0.20 0.00

0.20

0.40 0.60 u/Uo

0.80

1.00

Figure 5.2.b. Experimental mean streamwise velocities at x/h=7.8 for liquid-particle flow.

2.00 1.75 x/h=15.7

1.50

y/h

1.25 1.00 0.75 0.50 0.25 0.00 -0.20 0.00 0.20 0.40 0.60 0.80 1.00 u/Uo Liquid

Particle

Figure 5.2.c. Experimental mean streamwise velocities at x/h=15.7 for liquid-particle flow.

On the Numerical Simulation of Turbulence Modulation in Two-Phase Flows 2.50

2.00

x/h=2

y/h

1.50

1.00

0.50

0.00 -0.25

0.00

0.25 0.50 u/Uo

0.75

1.00

Figure 5.3.a. Experimental mean streamwise velocities at x/h=2 for gas-particle flow. 2.50

2.00 x/h=7

y/h

1.50

1.00

0.50

0.00 0.00

0.25

Gas

0.50 u/Uo

0.75

1.00

Particle

Figure 5.3.b. Experimental mean streamwise velocities at x/h=7 for gas-particle flow.

71

72

K. Mohanarangam and J.Y. Tu

2.50

2.00 x/h=14

y/h

1.50

1.00

0.50

0.00 0.00

0.20

0.40

Gas

0.60 u/Uo

0.80

1.00

Particle

Figure 5.3.c. Experimental mean streamwise velocities at x/h=14 for gas-particle flow.

2.00 1.75 1.50

x/h=0.7

y/h

1.25 1.00 0.75 0.50 0.25 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 u'/Uo

Liquid

Particle

Figure 5.4.a. Experimental mean fluctuating velocities at x/h=0.7 for liquid-particle flow.

On the Numerical Simulation of Turbulence Modulation in Two-Phase Flows

2.00 1.75 1.50

x/h=7.8

y/h

1.25 1.00 0.75 0.50 0.25 0.00 0.02

0.04

0.06

0.08 0.10 u'/Uo

0.12

0.14

0.16

Figure 5.4.b. Experimental mean fluctuating velocities at x/h=7.8 for liquid-particle flow.

2.00 1.75

x/h=15.7

1.50

y/h

1.25 1.00 0.75 0.50 0.25 0.00 0.02 0.04

0.06 0.08 0.10 0.12 u'/Uo

Liquid

0.14 0.16

Particle

Figure 5.4.c. Experimental mean fluctuating velocities at x/h=15.7 for liquid-particle flow.

73

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K. Mohanarangam and J.Y. Tu

2.50

2.00

x/h=2

y/h

1.50

1.00

0.50

0.00 0.00

0.05

0.10 u'/Uo

0.15

0.20

Figure 5.5.a. Experimental mean fluctuating velocities at x/h=2 for gas-particle flow. 2.50

x/h=7

2.00

y/h

1.50

1.00

0.50

0.00 0.00

0.05

Gas

0.10 u'/Uo

0.15

0.20

Particle

Figure 5.5.b. Experimental mean fluctuating velocities at x/h=7 for gas-particle flow.

On the Numerical Simulation of Turbulence Modulation in Two-Phase Flows

75

2.50

2.00

x/h=14

y/h

1.50

1.00

0.50

0.00 0.00

0.05

0.10 u'/Uo

Gas

0.15

0.20

Particle

Figure 5.5.c. Experimental mean fluctuating velocities at x/h=14 for gas-particle flow.

5.2. Numerical Code Validation In this section the code is validated for mean streamwise velocities and fluctuations for both the carrier and dispersed phases against the benchmark experimental data of Fessler and Eaton (1995) for GP flow and the experimental data of Founti and Klipfel (1998) for the LP flow. This task is undertaken to verify the fact that particulate flows with two varied carrier phases can be handled by the code. The ability of the numerical code to replicate the experimental results of GP (Fessler and Eaton, 1999) and LP (Founti and Klipfel, 1998) flows are tested. Figure 5.6a shows the numerical findings of single phase (Diesel oil) mean velocities against the experimental data, although the overall behaviour is replicated numerically there have been some under prediction for a height of y/h>1 for mid-section of the geometry, while a minor over prediction is felt along the entire height at section x/h=15.7. Figure 5.6b shows the fluctuating liquid velocities along the step compared against the experimental findings, there have been some minor under prediction for a height of y/h y0+

(19)

The above relationship is often called the log-law and the layer where the wall distance +

y lies between the range of 30 < y + < 500 is known as the log-law layer. Values of κ (~0.4) and E (~9.8) in equation (19) are universal constants valid for all turbulent flows past +

smooth walls at high Reynolds numbers. The cross-over point y0 can be ascertained by computing the intersection between the viscous sub-layer and the logarithmic region based on the upper root of

y0+ =

1

κ

ln ( E y0+ )

(20)

A similar universal, non-dimensional function can also be constructed to the heat transfer. The enthalpy in the wall layer is assumed to be:

H + = Pry + for y + < yH+ H+ =

PrT

κ

ln ( FH y + ) for y + > yH+

where FH is determined by using the empirical formula of Jayatilleke (1969):

21)

(22)

Pipe Flow Analysis of Uranium Nuclear Heating with Conjugate Heat Transfer

⎡⎛ Pr ⎞0.75 ⎤ ⎡ ⎛ Pr ⎪⎧ FH = E exp ⎨9.0κ ⎢⎜ ⎟ − 1⎥ ⎢1 + 0.28exp ⎜ −0.007 Prt ⎢⎣⎝ Prt ⎠ ⎥⎦ ⎣ ⎝ ⎪⎩ By definition, the dimensionless enthalpy H

H

+

(H =

w

+

279

⎞ ⎤ ⎪⎫ ⎟⎥ ⎬ ⎠ ⎦ ⎪⎭

(23)

is given by:

− H ) ρ Cμ0.25 k 0.5

(24)

JH

where H w is the value of enthalpy at the wall and the diffusion flux JH is equivalent to the

(

normal gradient of the enthalpy ∂H ∂n

)

wall

perpendicular to the wall. The thickness of the

thermal conduction layer is usually different from the thickness of the viscous sub-layer, and +

changes from fluid to fluid. As demonstrated in equation (20), the cross-over point yH can also be similarly computed through the intersection between the thermal conduction layer and the logarithmic region based on the upper root of

Pr yH+ = PrT

1

κ

ln ( FH yH+ )

(25)

For the rest of the boundary conditions at a solid wall, all wall temperatures are determined using the energy balance. Boundary conditions for the turbulent kinetic energy and the dissipation have its normal derivative at the wall equal to zero and obtained through the relation

ε=

Cμ3 4 k 3 2

κd

(26)

where d is the distance of the nearest grid point from the wall boundary.

Computational Procedure The algebraic forms of the governing equations to be solved by matrix solvers can be formed by integrating the system of equations using the finite volume method over small control volumes. By employing the general variable φ , the generic form of the governing equations can be written initially in the form as

∂ ( ρφ ) + ∇ ⋅ ( ρ Vφ ) = ∇ ⋅ ⎣⎡Γφ ∇φ ⎦⎤ + Sφ ∂t

(27)

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G.H. Yeoh and M.K.M. Ho

In order to bring forth the common features, terms that are not shared between the equations are placed into the source term Sφ . By setting the transport variable φ equal to 1,

u , v , w , H , k and ε and selecting appropriate values for the diffusion coefficient Γφ and source term Sφ , special forms of each of the partial differential equations for the continuity, momentum and energy as well as for the turbulent scalars can thus be obtained. The cornerstone of the finite volume method is the control volume integration. In order to numerically solve the approximate forms of equation (27), it is convenient to consider its integral form of this generic transport equation over a finite control volume. Integration of the equation over a three-dimensional control volume ΔV yields:

∂ ( ρφ ) dV + ∫ ∇ ⋅ ( ρ Vφ ) dV = ∫ ∇ ⋅ ⎡⎣Γφ ∇φ ⎤⎦ dV + ∫ Sφ dV ∂t ΔV ΔV ΔV ΔV

∫

(28)

By applying the Gauss’ divergence theorem to the volume integral of the advection and diffusion terms, equation (28) can now be expressed in terms of the elemental dA as

∂ ( ρφ ) dV + ∫ ( ρ Vφ ) ⋅ n dA = ∫ ⎡⎣Γφ ∇φ ⎤⎦ ⋅ n dA + ∫ Sφ dV ∂t ΔV ΔA ΔA ΔV

∫

(29)

Equation (29) needs also to be further augmented with an integration over a finite time step Δt. By changing the order of integration in the time derivative terms, t +Δt ⎛ t +Δt ∂ ( ρφ ) ⎞ ⎛ ⎞ ∫ΔV ⎜⎝ ∫t ∂t dt ⎟⎠ dV + ∫t ⎜⎝ Δ∫A ( ρ Vφ ) ⋅ n dA ⎟⎠ dt = t +Δt t +Δt ⎛ ⎞ ∫t ⎜⎝ Δ∫A ⎡⎣Γφ ∇φ ⎤⎦ ⋅ n dA ⎟⎠ dt + ∫t Δ∫V Sφ dVdt

(30)

In essence, the finite volume method discretises the integral forms of the transport equations directly in the physical space. If the physical domain is considered to be subdivided into a number of finite contiguous control volumes, the resulting statements express the exact conservation of property φk from equation (21) for each of the control volumes. In a control volume, the bounding surface areas of the element are, in general, directly linked to the discretisation of the advection and diffusion terms. The discretised forms of these terms are:

∫ ( ρ Vφ ) ⋅ n dA ≈ ∑ ( ρ V ⋅ nφ ) k

ΔA

f

f

ΔA f

(31)

Pipe Flow Analysis of Uranium Nuclear Heating with Conjugate Heat Transfer

∫ ⎡⎣Γφ ∇φ ⎤⎦ ⋅ n dA ≈ ∑ ⎡⎣Γφ ∇φ ⋅ n ⎤⎦ f

ΔA

f

ΔA f

281

(32)

where the summation in equations (31) and (32) is over the number of faces of the element and ΔAf is the area of the face of the control volume. The source term can be subsequently approximated by:

∫ Sφ dV ≈ Sφ ΔV

(33)

ΔV

For the time derivative term, the commonly adopted first order accurate approximation entails: t +Δt

∫ t

∂ ( ρφ ) ( ρφ ) − ( ρφ ) dt ≈ ∂t Δt n +1

n

(34)

where Δt is the incremental time step and the superscripts n and n + 1 denote the previous and current time levels respectively. Equation (21) can then be iteratively solved accordingly to the fully implicit procedure by

( ρφ )

n +1

− ( ρφ )

n

Δt

⎛ ⎞ + ⎜ ∑ ( ρ V ⋅ nφ ) f ΔAf ⎟ ⎝ f ⎠

⎛ ⎞ ⎜ ∑ ⎡⎣Γφ ∇φ ⋅ n ⎤⎦ f ΔAf ⎟ ⎝ f ⎠

n +1

= (35)

n +1

+ Sφn +1ΔV

Consider the particular control volume element in question of which point P is taken to represent the centriod of the control volume, which is connected with the respective centroids of other surrounding control volumes. Equation (35) can thus be expressed in terms of the transport quantities at point P and surrounding nodal points with a suitable prescription of normal vectors at each control volume face and dropping the superscript n +1 which by default denotes the current time level as:

aφ

n +1 P P

= ∑a φ

n +1 nb nb

( ρφ ) P ΔVP n

+S

n +1 off

+S

n +1 non

+S

n +1 u

ΔVP +

nb

Δt

(36)

where

aP = ∑ anb + S P ΔVP + ∑ F nb

( ρφ ) P

n +1

n +1 f

+

Δt

ΔVP

(37)

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G.H. Yeoh and M.K.M. Ho n +1

and the added contribution due to non-orthogonality of the mesh is given by S non , which is required to be ascertained especially for body-fitted and unstructured meshes. Note that the

(

convective flux is given by Ffn +1 = ρ V ⋅ n φ

)

n +1 f

ΔAf For the sake of numerical treatment, the

source term for the control volume in equation (35) has been treated by

Sφn +1ΔV = ( Sun +1 − S PφPn +1 ) ΔVP In equation (36), aP is the diagonal matrix coefficient of

(38)

φPn +1 ,

∑F

n +1 f

are the mass

imbalances over all faces of the control volume and S P is the coefficient that is extracted from the treatment of the source term in order to further increase the diagonal dominance. The k coefficients of any neighboring nodes for any surrounding control volumes anb in equation (36) can be expressed by

anb = D nf +1 + max ( − Ffn +1 , 0 ) n +1

where D f

(39)

is the diffusive flux containing the diffusion coefficient Γφ along with the

geometrical quantities of the particular element within the mesh system. The treatment of the advection term which results in the form presented in equation (39) is known as upwind differencing to guarantee diagonal dominance. In order to reduce the effect of false diffusion caused by upwind differencing, the well-known deferred correction approach is adopted to n +1

treat the off-diagonal contributions Soff

in equation (36) due to higher resolution

differencing schemes. In this study, the coupled solution approach, a more robust alternative to the segregated approach, is adopted to solve the velocity and pressure equations simultaneously. Algebraic Multigrid accelerated Incomplete Lower Upper (ILU) factorization technique is employed to resolve each of the discrete system of linearized algebraic equations in the form of equation (36). The advantages of a coupled treatment over a segregated approach are: robustness, efficiency, generality and simplicity. Nevertheless, the principal drawback is the high storage requirements for all the non-zero matrix entries.

Results and Discussion For the analysis of reactor thermo-hydraulic safety, the central focus often befalls on a ‘bounding case study’ which demarks what can be loosely regarded as the operational envelop of the reactor cooling system. This bounding case study is a simulation of all extreme operational conditions that may occur simultaneously during the life of the reactor. The meaningful operands of these bounding operational conditions are represented by input parameters (such as coolant temperatures, mass flow, etc.) of the system which become the parametric specifications of the computer model.

Pipe Flow Analysis of Uranium Nuclear Heating with Conjugate Heat Transfer

283

However, before these inputs are specified, a computer model of the irradiation rig must first be generated; complete with flow passages, internal structures and irradiation target(s). Components of the model rig assembly must be built in accordance to engineering blueprints and where possible, minor geometric simplifications are introduced into the model so that simulations remained physically representative without having to waste computational resources or prolong solution times over unnecessarily detailed control volumes. After the model is generated, it is geometrically discretised by the generation of a ‘mesh’ which further segregates each control volumes into sub-volumes. The physical properties of each geometric feature – such as the aluminium in the rig structure are then specified in the corresponding volumes of the model. It is standard practice to conduct a grid-convergence test before the modelled geometry is deemed representative of the physical prototype. To perform this, the amount of control volumes is increased in all three axes until the increase in mesh resolution does not result in any appreciable difference in final key results, such as in the model’s maximum attainable temperature. Finally, the results are analysed and compared to the safety limit. This procedure for this safety analysis can be summarised in five steps: 1. 2. 3. 4.

Determining the bounding case scenario Modelling the geometry, the physics and physical properties Solving the CFD model by numerical approximation techniques Checking the validity of our solution by mesh sensitivity analysis as well as by comparisons with other simpler numerical models such as from one-dimensional simulations reactor nuclear thermo-hydraulic code 5. Analysing and comparing results with safety requirement

‘Rocket-Can’ Design as Used in the HIFAR Reactor The first pipe-cooling irradiation system to be examined is the ‘rocket-can’ design used in HIFAR. This study investigated the maximum temperature of 2.2% 235U enriched UO2 pellets during irradiation. The recently decommissioned HIFAR reactor, a 10 MW nuclear research reactor at the Australian Nuclear Science and Technology Organisation (ANSTO), produced a steady supply of technicium-99 (99Tc) radiopharmaceutical for domestic and international use. Technicium-99 is formed by the radioactive-decay of Molybdenum-99 (99Mo), which itself is a fission product of Uranium-235 (235U). The process of generating technicium-99 started with the loading of seventeen UO2 pellets into a thick aluminium tube shaped like a ‘rocket-can’ (Figure 2). Granulated magnesium oxide (MgO) was packed in with the pellets to assist heat conduction and to control pellet spacing. The rocket-cans were then sealed by welding on a cap and inserted into slots inside a long vertical ‘stringer’ assembly which looked much like a ‘cake-stand’. In turn, the stringer was inserted into a ‘liner’, a hollow tube with flow by-pass inlets at its conical tip. Finally the liner was inserted into the centre of four concentric annular fuel plates which formed the fuel assembly (Figure 1). The purpose of the liner was to separate high speed flow passing through the four outer concentric fuel plates from the slower flow going through the liner cooling the cans. All components: cans, stringer and liner were fabricated

284

G.H. Yeoh and M.K.M. Ho

using aluminium on account of its unique qualities of neutron transparency, anti-corrosion and good thermal conductivity.

SALVAGE COUPLING PHALANGE FOR ASSEMBLY TO SHIELD PLUG IDENTIFICATION NUMBER

EMERGENCY COOLING WEIR AND SPRAY RING INTERMEDIATE SECTION

THERMOCOUPLE TUBE

COOLANT FLOW

PERFORATED EXTENSION

THERMOCOUPLE TUBE

DOWEL

EMERGENCY COOLING WATER TUBE (NOT USED IN HIFAR)

THERMOCOUPLE COMB

VIEW OF ARROW A

FUEL TUBES

OUTLINE OF LINER LOWER COMB

GUIDE NOSE SPHERICAL SEAT

SKIRT COOLANT FLOW

Figure 1. Schematic drawing of the HIFAR fuel element.

When the HIFAR reactor was operating, 235U in fuel elements underwent fission which absorbed and generated neutrons in a self-sustaining process. The nuclear reaction produced large amounts of heat that was removed by upwardly flowing heavy water (deuterium oxide, D2O) through the fuel element. One and a half percent of coolant flow bypassed into the centre liner for rocket-can cooling. Also, the neutrons produced by the nuclear process in the fuel were absorbed by the UO2 pellets to produce molybdenum that had a short half-life of 2.7 days before beta-decaying into technicium-99. After irradiation, the molybdenum-99 was chemically separated and quickly packaged so that it arrived at clinics in its usable betadecayed form as technicium.

Pipe Flow Analysis of Uranium Nuclear Heating with Conjugate Heat Transfer

285

Figure 2. Schematic drawing of the inadiation rig and placement of target cans in the liner.

During irradiation, a vast amount of heat was generated in the uranium pellets which must be evacuated. For safety and licensing purposes, it was necessary to demonstrate that the pellet would not melt during irradiation. Thus, neutronic and thermal-hydraulic analyses using numerical methods were used to determine the pellet maximum temperature. This method directly solved the three conservation equations of mass, momentum and energy in the coolant flow domain which was coupled with the conduction physics of the solid domains of the can, magnesium oxide and uranium oxide pellets. The result of this work was critical to the licensing requirements of HIFAR as bounded by operation licence conditions (OLCs) and evaluated safety limits.

Grid Generation of Rocket-Can The complete geometry of can, stringer, liner, fuel plates and flow passages was generated simultaneously and their respective volumes patched as separate logical entities as shown in Figure 3. The simultaneous volume generation meant that separate component surfaces were logically connected and no surface patching was required between components.

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G.H. Yeoh and M.K.M. Ho

On the outset, minor geometric details were simplified in regions of little flow-dynamic consequence in-order to accelerate the time to solution convergence. Some work was needed to organise the arrangement of each entity with respects to other parts but the effort expended was outweighed by the benefit of not needing to specify intra-component surface patching. Note in Figure 3 that hexahedral body-fitted volumes were used in the geometry to properly model the thin conduction regions of the liner and fuel plates. The thin solid region of the liner and stringer could only be practically constructed by using a structured mesh because of the high thermal flux across the thin stretch of aluminium. Numerical diffusion over such a thin area of high thermal flux given if we were using unstructured mesh would have been exceedingly pronounced. The additional benefit of using hexahedral control volumes was in their superiority over tetrahedral elements for the modelling of near-wall flows where the correlation-model of wall flow-profiles work best with body-fitted rectangular mesh. This was also represented an economy in the number of rectangular mesh required as compared to tetrahedral mesh to attain the same level of solution accuracy. The pellet-stack geometry consisted of a 119.5 mm vertical cylinder of MgO with a radius of 5 mm. The pellets modelled inside the MgO stack was 3.5mm high with a radius 4.5mm as shown in Figure 4a. In Figure 5a, the rocket-can and stringer are shown simultaneously. Notice the relief windows at the top of the stringer which allowed the coolant flow to pass through as was identical to the real physical prototype. Finally, five volumes of the same model as shown in Figure 3b were stacked on each other to produce the complete stringer assembly such as shown in Figure 9.

(a)

(b)

Figure 3. The modelled ‘block’ of multiple components is shown with the (a) rocket-can and (b) slotted stringer highlighted.

Pipe Flow Analysis of Uranium Nuclear Heating with Conjugate Heat Transfer

(a)

287

(b)

Figure 4. The modelled stack of (a) UO2 pellets in MgO was built to fit inside (b) the rocket-can’s internal cavity.

(a)

(b)

Figure 5. The model of (a) rocket-can inserted inside stringer and (b) multiple stringers were assembled to produce a complete stringer assembly.

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G.H. Yeoh and M.K.M. Ho

Specification of Material Properties The total D2O mass flow through the rig was 16 kg/s with 98.5% passing through the fuel and 1.5% passing through the liner. The material properties of aluminium and heavy water are summarised in Table 1 and the variable conductivities of magnesium oxide and uranium dioxide are shown in Figure 6. Table 1. Material Properties of Aluminium and Heavy Water MATERIAL PROPERTIES -3

Density

kg.m

Specific Heat Capacity

2702

1094.92

-1

903

4.12849

-1

-1

273

0.614259

J.kg .K

W.m .K

Dynamic Viscosity μ

Pa.s

0.000712

Uranium Oxide conductivity vs. Temp.

Magnesium Oxide conductivity vs. Temp. 2.0

Thermal Conductivity [W/(m.K)]

Thermal Conductivity [W/(m.K)]

D2O

-1

Thermal Conductivity

1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 300

Al

8 7 6 5 4 3 2 200

3000

418

1800

3000

Tem p. [K]

Tem p. [K]

(a)

(b)

Touloukian 1970.

Figure 6. Temperature dependent thermal conductivities of (a) magnesium oxide and (b) uranium dioxide.

Computational Geometry The domains of the simulation were specified as follows (the default solid-domain material was aluminium): • • • • • • •

‘Fuel1’, ‘Fuel2’, ‘Fuel3’, ‘Fuel4’– Solid Domain ‘Liner’ – Solid Domain ‘Stringer’ – Solid Domain ‘Can’ – Solid Domain ‘MgO’ – Solid Domain, material: Magnesium Oxide ‘UO2’ – Solid Domain, material: Uranium Oxide ‘Fluid’ – Fluid Domain, fluid: heavy water

Pipe Flow Analysis of Uranium Nuclear Heating with Conjugate Heat Transfer

289

In summary the CFD problem can be simplified as a list of input and outputs: Inputs • Geometry • Material Properties • Heat Source • Boundary Conditions

Outputs • Temperature Profile T (x,y,z) • Wall Heat Flux q (x,y,z)

UO2 pellets producde heat at a steady rate. The heat was conducted through the MgO into the aluminium rocket can then into the flowing heavy water D2O. The solutions of interest given by CFD included: the temperature profile T (x,y,z) of the whole region and wall heat flux q (x,y,z) over the surface of the rocket can. The subdomain ‘Pellets’ was defined for the entire UO2 domain as an energy source with a volumetric power of 8.4556 x 108 W.m-3. The fuel plates, the structural material (aluminium) and the coolant (heavy water) would also produce heat but the heating from fuel plates was not modeled because it was known that they were cooled efficiently by the faster flowing coolant outside the liner and heating from other aluminium components was negligible. For the boundary conditions, there were two inlets, one at ‘INLET FUEL’ with Normal Speed 2.58816 m.s-1, another at ‘INLET LINER’ with Normal Speed 0.0717642m.s-1 (these were calculated from coolant mass flow value, inlet areas and bypass ratios), both with a static temperature of 318K (45°C) in the 16-pellet simulation. In these simulations, there were two separate flow regions.

Power Density Calculation The power emitted from 2.2% enriched UO2 pellets were calculated by using the Monte Carlo N-Particle (MCNP) transport code for simulating neutronic reactions. For this safety analysis a conservative reactor power of 11MW was assumed. The 16 pellets of a can were modeled discretely (Dia. 9mm × 3.5mm high). Total power per m-3 for 16 pellets = 3204.7 W.m-3 / 3.79 × 10-6 m3 (Volume of 16 Pellets) = 8.4556 × 108 W.m-3. This power density signified the source term for the conduction equation.

Turbulence Solver and Convergence Criteria To account for the turbulent pipe flow, the standard k-ε model was used. Computational predictions were deemed to be converged when the normalized residual mass was less than 1 x 10-4.

Computational Predictions Simulation for the heating of sixteen pellets in a rocket can had been previously modeled by Yeoh and Storr (2000) which was interested in the rocket-can’s maximum surface heat

290

G.H. Yeoh and M.K.M. Ho

flux to determine the margin for Onset of Nucleate Boiling (ONB). The CFD model of Yeoh & Storr’s simulation had a lesser degree of resolution and modeled the stack of pellets as a single entity instead of discrete pellets. This was partly attributed to the more limited computational resources available at the time. Yeoh & Storr’s CFD result’s were validated to a degree by simple irradiation experiments where metal samples were introduced inside the pellet stack and subsequently examined for evidence of melting in order to estimate maximum temperatures. In this investigation calculated by CFX-10, we were primarily interested in the maximum temperature of the UO2 pellets. Temperature contours in Figure 7a indicate a maximum pellet temperature of 2656°C which remained below the pellet melting temperature of 2847°C. For clarity, the four concentric annular fuel plates, liner, stringer and can in Figure 7 are shown in silhouette. This result was consistent with the maximum temperature of Yeoh & Storr’s large-volume pellet stack at 2277°C. The temperature was understandably lower in Yeoh & Storr’s simulation because the same power was distributed over a larger volumetric space.

(a)

(b)

Figure 7. Section view of temperature contours showing maximum temperatures of (a) UO2 pellets: 2656°C and (b) MgO: 2439°C.

Other maximum temperatures of interest include: the magnesium oxide at 2439°C (mp. 2800°C) in Figure 7b; the aluminium can at 307°C (mp. 660°C) in Figure 8a and the heavy water D2O at 79°C (bp. 120°C at 2 atm) in Figure 8b. The temperatures were highest in the middle of the pellets, decreasing steeply across the width of the pellet and still more steeply across the MgO powder, reflecting its low thermal conductivity. This was in contrast with the low thermal gradient across the aluminium rocket can, due to the high thermal conductivity of aluminium. Figures 9 and 10 show the wall heat flux at the surface of rocket cans, the higher fluxes were colored blue, due to outward flux being defined as negative. There were four high heat flux regions corresponding to the slot opening. Opposite the slot opening in Figure 10, forced convection cooling was restricted by the stringer enclosure and as a consequence caused a lower heat-flux as indicated by the green region of the can surface. The sudden expansion of the open slot at the side of the stringer created a highly turbulent region which assists in

Pipe Flow Analysis of Uranium Nuclear Heating with Conjugate Heat Transfer

291

forced convection cooling and this resulted in the highest thermal heat flux achieved as indicated in Figure 10 by the dark blue region of the can.

(a)

(b)

Figure 8. Section view of temperature contours showing maximum temperatures of (a) rocket can: 307°C and (b) D2O: 79°C.

Figure 9. Heat-flux contour at the surface of the rocket cans. The view is through the opening in the stringer. The silhouette of the stringer is made out in transparent purple.

Inside the liner, the mean flow accelerated when squeezed into tighter paths between the can and the stringer, as indicated by yellow and orange vectors in Figure 11 and decelerated when entering an expansion, as shown by blue vectors near the tip of the rocket can. The deceleration was basically due to reasons of mass conservation as explained by the Bernoulli equation. The length of an arrow in Figure 11 is proportional to the flow velocity at that point. The long red arrows belonged to the faster flow outside the liner, with velocities of up to 4m.s-1.

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G.H. Yeoh and M.K.M. Ho

Figure 10. Heat-flux contour at the surface of the rocket cans. The view is through the enclosed back of the stringer.

Figure 11. Flow velocity vector plot of coolant channels inside outside the liner.

Proposed ‘Annular Can’ Replacement Design for HIFAR Flow visualization and LDV measurements were performed to better understand the fluid flow around the narrow spaces within the X216 irradiation rig, prototypes of annular target cans and liner. A three-dimensional computational fluid dynamics model was used to

Pipe Flow Analysis of Uranium Nuclear Heating with Conjugate Heat Transfer

293

investigate the hydraulics behavior within a HIFAR fuel element liner model. An interesting feature of the computational model was the use of unstructured meshing, which consisted of triangular elements and tetrahedrons within the flow space (Figure 13), to model the “scaledup” experimental model. This present investigation focused on the evaluation of the CFD model in its capability to predict the complex flow structures inside the liner containing the mock-up X216 rig with two targets. The reliability of the model was validated against experimental observations and measurements.

Description of the Water Tunnel Experimental Apparatus and Methods Figure 12 describes the transparent model of the prototype design of the rig and annular target cans that were placed in the water tunnel facility. In this test section, the liner was sealed so that the flow path was only through the liner inlet holes. An orifice plate was used to measure the flow rates within the facility. Flow visualization of the fluid flow was performed using an Argon Ion laser light sheet, high-resolution digital camera and standard video equipment. The flow was seeded with Iriodin powder concentrations of 0.5 to 1 gram per 3000 liters of water. Particles in the flow were illuminated while they were in the field of the laser light sheet. With the digital images and video footage the flow field was clearly visible.

Figure 12. Points of velocity measurements in X216 rig.

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G.H. Yeoh and M.K.M. Ho

Figure 13. Unstructured mesh of the X-216 annular target.

Velocities were measured using a two-dimensional (2D) Dantec LDV system, operated in 1D mode measuring axial velocity components. Data were stored on the computer attached to the LDV hardware. Uncertainty in measurement using the LDV equipment was determined to be a maximum of ± 3.5% for a particular optics configuration but generally less than ±1.8%. Points of velocity measurement are shown in Figure 12. The points measured in the plane shown had been taken with the laser entering at the open side of the mock-up rig and the forward scatter detector viewing through the closed side of the rig. Measurements were also taken using the LDV probe in backscatter mode, and were verified at a number of points by using the forward scatter mode. Backscatter mode was used only for measurements in the grid pattern between the heights of 185 mm and 360 mm, since at the other locations the beams were sufficiently attenuated due to the additional influences of the acrylic interfaces, giving very low data rates.

Computational Details A three-dimensional CFD program ANSYS-CFX5.6 has been employed to simulate the complex thermal-hydraulics behavior in the space within a HIFAR fuel element liner model in the water tunnel. The CFD code solved the conservation equations of mass, momentum and energy. Turbulence of the fluid flow was accounted via a standard k-ε model. Buoyancy

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effect was included for regions where low velocities were present. This effect was included within the source terms of the momentum equations as well as in the turbulence models. The unstructured meshing was adopted for the construction of the computational model of the mock-up because of the inherent intricate details of the irradiation rig and the placements of the annular targets within the rig. The mesh was prepared in two stages. A surface mesh of triangular mesh elements was initially generated on all the surfaces covering the model components. The volume mesh consisted of tetrahedral elements was then subsequently generated within the fluid flow domain from the surface mesh elements. Figure 13 shows the mesh layout of triangular surface elements around the pin and support tube nose cone of the irradiation rig. For the entire geometrical structure that included the assembly of two annular target cans placed within the prototype rig in the liner and mock-up fuel element, a volume mesh of 605158 tetrahedrons and a surface mesh of 42030 triangular elements were allocated. The governing equations were solved by matrix solution techniques formulated by integrating the system of equations using the finite volume method over small elemental volumes. For each elemental volume, relevant quantity (mass, momentum and turbulence) was conserved in a discrete sense for each control volume. Here, a coupled solver, which solved the hydrodynamic equations (for velocities and pressure) as a single system, was employed. It has been found in the segregated approach that the strategy to first solve the momentum equations using a guessed pressure and an equation for a pressure correction resulted in a large number of iterations to achieve reasonable convergence. By adopting the coupled solver, it has been established that such a coupled treatment significantly outweighed the segregated approach in terms of robustness, efficiency, generality and simplicity. To accelerate convergence for each of the discretised algebraic equations, the Algebraic Multigrid solver was adopted.

Validation Against Water Tunnel Observations and Measurements CFD simulation of the fluid flow through the various components of the fuel element model that included the mock-up rig and annular target cans was performed. Figure 14 illustrates the computed flow distribution inside and outside of the liner nose cone. Based on the experimental flow rate of 1.6965 kgs-1 and a base diameter of 0.3 m of the mock-up fuel element model, there observed a very low flow velocity outside of the liner nose cone in Figure 14(b). Nevertheless, the fluid after being squeezed through the small size bottom and side holes of the liner nose cone caused these interacting merging flows to yield a very highly complicated flow structure consisting of multiple vortices of recirculating flows (see Figure 14(a)). It was also evident that due to the significant acceleration of the flow found near the liner holes, the velocities increased dramatically to a magnitude of 5.0 ms-1 and resulted in large pressure drops. Near the bottom hole of the liner nose cone, the CFD model predicted a normal velocity of approximately of 3 ms-1. This predicted value has been found to be in good agreement with experimental LDV measurement of 2.6 ms-1, which provided confidence to the reliability of the models in the CFD computer program. As the fluid moved vertically upwards, the flow gradually became more uniform.

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(a)

(b)

Figure 14. (a) Velocity vectors and (b) velocity contours around perforated liner.

(a)

(b)

Figure 15. Flow separation near the pin as examined by (a) CFD and (b) PIV capture.

Another important consideration for the rig and target specification was the incorporation of a pin situated at some distance below the placement of the rig as can be seen in Figure 15. This design feature was implemented in the liner because of safety concerns in the event of a possible accident scenario of the rig falling through to the bottom and impacting on the liner nose cone. The selection of pin size was an important requirement for the rig and target specification. From the flow predictions in Figure 15(a), it could be ascertained that the pin size chosen contributed to only minor flow disturbances in the area between the pin and the bottom surface of the rig nose cone. It was also observed that the majority of the bulk fluid flow was unperturbed and diverged smoothly as it approached the rig. Flow visualization

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performed during experiments (see Figure 15(b)) projected a similar flow pattern, which further confirmed the reliable predictions of the CFD model.

(a)

(b)

Figure 16. CFD study of flow distribution around annular can.

Figure 16 presents the flow distribution of the fluid traveling between the inner annular can wall and rig outer surface designated for the purpose of illustration as region 1 and the area between the outer annular can wall and the liner inner surface designated as region 2. The fluid flowing within these spaces was found to be rather uniform. These favorable flow structures indicated that axial cooling in the reactor rig along the length of the target could remove the heat effectively for the design where uranium foils are embedded in the sealed annular targets during the irradiation process. An interesting aspect of the model predictions through the velocity contours in Figure 16(b) showed succinctly more fluid moving vertically upwards in region 2 than in region 1. Based on the LDV flow measurements at the discrete locations in regions 1 and 2 in Figure 12, the experiments confirmed the CFD predictions of the different velocities in the two regions. Velocity values of 0.44 ms-1 and 0.502 ms-1 were measured during experiments for regions 1 and 2 respectively. The predicted velocities as depicted by the velocity contours in Figure 16(b) demonstrated the similar trend predicted through the CFD model where region 1 yielded lower velocities compared to the higher velocities in region 2. Figure 17 illustrates the LDV measurements for the unmodified and modified designs of the can flutes affecting the axial velocity distributions for two volumetric flow rates. Changes introduced to the flute designs through the removal of any sharp edges significantly altered the axial velocity distributions. The profiles were flatter thereby resulting in more uniform and lesser wake flow structures. In the same figure, comparison between predicted and measured vertical velocities is also presented. The predicted velocity profiles through the CFD model showed similar encouraging distributions with the measured profiles in the inner

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and outer channels. Good qualitative agreement was achieved between the predicted and measured velocities. X216 Rig Channel Axial Velocities 0 degree plane 1.61 l/s Inner Channel 1.61 l/s Outer Channel 1.61 l/s IC;RMS 1.61 l/s OC;RMS 1.68 l/s IC 1.68 l/s OC 1.68 l/s IC;RMS 1.68 l/s OC;RMS 1.61 l/s mod IC 1.61 l/s mod OC 1.61 l/s mod IC;RMS 1.61 l/s mod OC;RMS 1.61 l/s IC CFD prediction 1.61 l/s OC CFD prediction

0.7

0.6

Fluid Velocity m/s

0.5

0.4

0.3

0.2

0.1

0 0

50

100

150

200

250

300

350

400

450

500

NOTES: 1) Run @ 1.61 l/s was performed before fin smoothing 2) Run @ 1.61 l/s mod was performed after fin smoothing

Axial Distance From Stem Tip (mm)

Figure 17. Inner and outer channel LDV velocity measurements along X216 Rig vertical axis.

Current ‘Molybdenum-Plate’ Design Used in the New OPAL Reactor The third pipe-cooling irradiation system to be examined is the ‘molybdenum-plate’ design used in the new Open Pool Australian Light-water (OPAL) research reactor. This study investigated the maximum temperature of 20% 235U enriched U-Al compound sealed in aluminium cladding plates. It will also demonstrate the incremental development of the model and the affect this has on the maximum temperature, as more physics and boundary conditions were introduced. The OPAL research reactor is an open-pool design constructed by INVAP, Argentina. The core rests thirteen meters under an open pool of light-water which provides both cooling and radiation protection. Surrounding the core is a Heavy Water Reflector Vessel that is physically isolated from the bulk light water coolant. Its primary purpose is to reflect neutrons back into the core to maintain the critical (nuclear) conditions necessary for steady fission reaction rates during normal reactor operation. The secondary purpose of the reflector vessel is to provide a relatively large volume with high flux for irradiation and neutron beam facilities. A compact reactor core measuring only 0.35 m x 0.35 m x 0.62 m (width, breadth, height) is located 13m below the reactor pool surface. The Reflector Vessel surrounding the Core contains heavy water and accommodates facilities such as irradiation rigs and the cold Neutron Source (see Figure 18). A number of irradiation positions penetrating the Reflector Vessel provide facilitates for the production of radioisotopes. These positions are sealed to

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prevent the H2O and the D2O from mixing. Moreover, the irradiation positions contain builtin mechanisms for cooling which draws cold H2O from the main pool downwards into the irradiation targets which is then conveyed away through pipes.

Figure 18. OPAL reactor facilities (top view).

Molybdenum-Plate Life Cycle A primary aspect of the OPAL reactor is the production of Technetium-99 (99mTc) for the medical supply of local and international markets. To manufacture Molybdenum-99, a product made of the compound U-Al contained in an aluminium matrix (called meat) is rolled in between two parallel plates of aluminium alloy A96061. The result is a simple, selfcontained plate of aluminium measuring 230 mm x 28 mm x 1.64 mm that contains a sandwich of U-Al ‘meat’ in the centre (see Figure 19). It is of note that the isotopic content of fission uranium is high, at an enrichment of 20% 235U but low enough to abide by international guidelines for LEU (Low Enriched Uranium) fuels and irradiation targets. Assembly of the irradiation rigs are performed in a hot-cell situated at pool level. Fresh, un-irradiated molybdenum plates are loaded—four at a time—into a holder known as the ‘target’ (see Figures 20 and 21). A rig stem is then inserted through the target’s hollow centre to form an assembled irradiation ‘rig’. The rig incorporates two to three irradiation targets along with other flow control devices like nozzles and flow restrictors (see Figure 22). Once assembled, the rigs are transported from the hot-cells into the service pool (underwater) via a hermetically sealed elevator. These rigs could then be remotely lowered to the reflector vessel’s bulk irradiation facilities (see Figure 18).

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Figure 19. One irradiation target with four molybdenum plates.

Figure 20. Isometric view of inadiation target model.

Figure 21. Cross sectional view of the irradiation target with four molybdenum plates inside.

Before the reactor is brought to critical, all primary cooling and secondary cooling systems are switched on. The rigs cooling flow draws light water from the main pool down

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into the irradiation rigs before being passed onto radiation decay tanks and heat exchangers after. The fast neutrons radiating from the fuel plates are slowed by the H2O core coolant and are completely thermalised by the D2O between the core and the facility. Having been slowed, these neutrons could then be captured by the Uranium-235 meat resulting in fission. A maximum number of three targets could be loaded in one irradiation rig. This constituted the bounding case to be modeled.

Figure 22. Molybdenum plate target and rig (shown on its side but loaded vertically).

Design Characteristics Compared to the previous method of molybdenum manufacture, the philosophy of this purpose-built system was for high molybdenum yield and low radio-active waste output. The previous design of molybdenum irradiation in the recently decommissioned HIFAR reactor (an old British design of the DIDO class), utilized 2.2% enriched UO2 reactor power fuel pellets packed in bulky aluminium cans. These cans were centrally inserted into concentrically arranged annular fuel plates and differed from the new irradiation rigs that lie outside the core itself. Consequently, the two main changes in molybdenum targets as compared to the previous HIFAR (High Flux Australian Research) reactor are: (1) the geometry, which has changed from a can to a plate, (2) the enrichment, increasing from 2.2% to 20% and (3) the location of the irradiation rig outside the core. This new configuration allows the irradiation of more 235U but also uses a substantially lower mass of aluminium and 238U, the result of which is an increase in yield and a decrease in aluminium waste. Despite these gains in efficiency, the increase in power generated by such a high concentration of enriched uranium poses challenging conditions for heat removal. The thin layer of aluminium gives little impedance to the flow of heat away from the U-Al centre but at the same time, any interruption to the forced convection flow of coolant could cause the cladding to blister or at the very worst melt, as there is very little aluminium mass to which the very large amounts of heat produced can conduct away. Thus, the prediction of flow on and over the molybdenum plates is crucial to ensure no isolated heat spots could develop that would result in the blistering of the clad or in the most extreme case, the melting of clad during irradiation (burn out). Thus, the aim of this numerical study was to show the thermo-hydraulic design of the molybdenum holder was sufficient to remove the heat generated by fission with a sizable margin between bounding operational conditions and extreme (i.e. almost impossible) conditions. After a maximum of 14 days of irradiation, the rig assembly is removed from the reflector vessel and placed in the service pool storage racks to ‘cool’. Loading and unloading

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of irradiation rigs can be performed whilst the reactor is at full power. The removal of the rig from its irradiation position halts the neutron-irradiation heating, leaving only the nucleardecay heat to be removed by means of natural convection. The decay-heat produced by weaknuclear reactions is much lower than the fission heat—initially at approximately 8% of the power produced during irradiation and decreases exponentially with time.

Methods of Calculations and Model Set-Up The maximum steady state temperatures of the molybdenum-99 rigs as a result of the coupled action of nuclear heating and light-water forced convection cooling were determined. These values could then be used to determine two important margins: (1) the difference between the maximum attainable temperature in the plate and the aluminium cladding blister temperature and (2) the difference between the operational maximum heat flux and the heat flux at the onset of nuclear boiling. For the simulation of heat transfer during irradiation, different parts of the irradiation assembly were separately modeled, meshed and combined by what is termed as ‘patching’. Afterwards, material properties were explicitly specified in each domain, along with the relevant boundary conditions. The built model incorporated nothing more than the necessary areas under examination to optimize the use of computational resources. Other fundamental features such as the setting of appropriate turbulence model, the solution’s mass-residual convergence criteria and the time & space discretisation schemes for the iterative solution process were then set (Tu et al., 2008). Some features of the numerical model, such as the modeling of buoyancy were deemed to have a negligible effect to the overall flow characteristics and were thus not modeled. In order to validate the CFD model, a comparison with a set of independent results was necessary. Available INVAP results using a more simplified one-dimensional code were used as comparative data against the CFD results. In this way, we were able to evaluate the consistency of the numerical models. For those cases which have no corresponding INVAP study, the only method for validation was by means of grid convergence testing and by the comparison of results using different initial values or different convergence criteria. The approach with this numerical study was to start with the most simplified assumptions in the CFD model and to introduce increasing complexities to examine the relative importance of each effect. A series of CFD simulations, utilizing a three-dimensional CFD program ANSYS-CFX10.0, were thus conducted to attain a final validation-case which included all the identified bounding conditions.

Power Distribution of Molybdenum-Plate Targets To undergo irradiation, assembled rigs loaded with two or three targets are inserted vertically inside the reflector vessel. When the reactor is at power, nuclear reactions occur in the molybdenum-plates as thermal neutrons are absorbed by uranium with the effect of producing heat. Since the horizontal neutron flux from the core radiates uniformly in the near vicinity of the circular reflector pool, there is little flux variation in the horizontal plane of the irradiation rig. However, the irradiation rig is very long, so the neutron flux distribution varies

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appreciably in the vertical direction, resulting in a cosine power distribution along the vertical axis of the molybdenum-rig. Using the Monte Carlo N-Particle (MCNP) transport code for simulating neutronic reactions, it was found that for a three target configuration, the middle position of three targets generated the most power. A separate and independent neutronic calculation conducted by INVAP (MOLY-0100-3OEIN-004, 2003) found that the maximum total power loading was 40.2 kW for one target and 79.6 kW for two targets. Since these results indicate a trend for proportionately lower powers for each additional target, it was deemed that each target could generate no more than 40.2kW by itself during normal reactor operations. However, to increase the safety margin for this analysis, it was specified that each of the three targets would generate a power of 60 kW. The power distribution is defined in terms of the ratio between the maximum and the average power along the vertical position of the plates in the irradiation rig. This can be defined either in terms of the plate surface heat flux or in terms of the volumetric power density, with a ratio known as the Power Peaking Factor (PPF). In ref. Moly-0100, the PPF has been shown to be equal to 1.1 for a single target and 1.3 for a two-target loading. It was thus assumed that a PPF of 1.3 could be used for three targets. This assumption has been validated later by an MCNP model of the three targets loaded for irradiation with a core of fresh fuel. Thus, the bounding case was defined as an irradiation rig loaded with three targets, each containing four plates. The power density was defined as follows:

q = PPF

Q plate V

cos ( B ( z − z0 ) ) = 1.3

15000 cos ( 3.134453 ( z − 0.413) ) (40) 5188.8 10-9

where q PPF Qplate V B z z0

: : : : : : :

Power density, [W.m-3] Power Peaking Factor, [-] Total power per plate, [W] Meat volume, [m3] Adjustable constant, [m-1] Coordinate on the z axis, [m] offset of the centre, [m]

Under nominal conditions, the average flow velocity through most of the rig’s cross section along the plates is 3.8 m.s-1. It was decided to set a more conservative limit with a reduced average flow velocity of 3 m.s-1 for this study.

Boundary Conditions and Domain Restriction Since coolant for the molybdenum-rigs is drawn from the pool, the temperature is essentially the pool water temperature which is regulated at a steady 37°C. To establish a safety margin of 3°C, a value of 40°C was conservatively applied for the inlet coolant boundary condition. It is difficult to ascertain the coolant velocity profile entering the rig

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because the initial pool-water flow conditions could not be reliably predicted. But seeing as the purpose of this investigation was to examine the heat transfer characteristics of the rig, the exact flow pattern entering the rig was not of primary importance. Furthermore, the reducerexpansion arrangement of the nozzle preceding the three targets had the intended effect of straightening the flow before it passed over the molybdenum-plates. Therefore the flow can be assumed to have a developed turbulent velocity profile before entering the targets’ coolant channels. By these considerations, the total control volume modeled in these simulations only extended from the rig inlet nozzle to the outlet restrictor (see Figure 22) and a fully developed turbulent flow profile was assumed as the inlet condition. To allow the average velocity to develop into a turbulent velocity flow profile, an artificial pipe extension of fifteen hydraulic diameters was modeled before the inlet. Also, lest the flow after the restriction nozzle affect the preceding flow domain, an artificial pipe of 15 hydraulic diameters was also attached to the end of the restrictor. Finally, the outer surface of the rig was conservatively assumed to be adiabatic to attain the highest possible temperature in the coolant, now set to be solely responsible for the removal of heat from the rig.

Figure 23. Cross-sectional view of control volumes representing the irradiation rig.

Figure 24. The control volumes grouped in their respective materials shows each separate entity more clearly.

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Since it is obvious that the rig geometry is symmetrical in two planes (see Figures 23 and 24), only one-quarter of the irradiation rig needed to be modeled with symmetrical boundaries applied to the ‘cut planes’ to limit the number of mesh elements. In truth, a one-eighth model would have sufficed but it was decided a complete coolant channel would be modeled so that no assumptions need be made of what was subsequently demonstrated as a symmetrical turbulent velocity profile through the pipe. The inlet condition was defined at mass flow rate of 0.6259 kg.s-1, corresponding to the 3 m/s average velocity in the major channel (see Figure 25). The inlet coolant temperature was also conservatively set at a higher temperature of 40°C, providing a 3°C margin above the nominal 37°C pool temperature. At the outlet, a 0 Pa gauge pressure was specified and the cylindrical outer wall was defined as adiabatic. The standard k-ε model was chosen for this highly forced convective problems with automatic adjustments for kinetic and dissipation values as calculated by local velocities.

(a)

(b)

(c)

Figure 25. Quarter-sections of the: (a) nozzle (b) target (c) restrictor; blue indicates flow areas.

Grid Generation For the sake of accuracy and computational efficiency, a body fitted orthogonal mesh was selected to model the irradiation rig. Building quadrilateral surfaces for the body fitted mesh required more time than automatically generated grids but this resulted in the use of less nodes and reduced the amount of mesh-induced solution diffusivity when compared to tetrahedral mesh. The occurrence of this artificial diffusion is a result of interpolation errors arising from highly skewed mesh contact angles of tetrahedral-meshes. In areas where large gradients were expected, the volumes were meshed with relatively high density to capture sharp temperature changes. Similarly, grid optimization was performed in those areas where the temperature and velocity gradients were small by a reduction in mesh density. Within the flow domain, the change of mesh sizes was gradual to prevent numerical instability in the solution. Whilst mesh size transitions in the solid domain were not required to be as stringent because the harmonic-mean interpolation for thermal conduction between solid nodes is not as geometrically sensitive as the solution for convection and diffusion in fluid nodes. As mentioned, highly skewed elements were avoided where possible to minimize numerical diffusion. In Figure 26, the circled area in (a) indicates the mesh is too skewed and has been re-meshed in (b) so that all cell face angles stood between 38° and 142°. This optimization improved the solution maximum temperature result by a difference of 6°C.

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(a)

(b)

Figure 26. Circled area re-meshed from (a) to (b) to reduce the presence of skewed mesh.

Mesh refinement was checked using three different meshes. Due to limits in computational resources, it was not possible to double the mesh seeding in every direction at the same time. Thus, two new meshes were created, one with a doubling of mesh in the streamwise directions and one with a doubling in the crosswise. In each case, a simulation was run and the maximum temperatures did not vary by more than 1°C. Thus it was shown that grid convergence had been established, with a final node element count of 432,667.

Validation Case This simulation compared the temperatures attained in INVAP’s correlation-based onedimensional code with our CFD model under the same flow and power conditions. The nominal condition study conducted by INVAP evaluated a two target loading scenario with a total power output of 79.4 kW, a cooling flow of 3.6 m.s-1 and an oxide layer thickness of 7 μm as calculated by the Argonne National Lab (ANL) model for oxide layer growth. Since INVAP’s model calculated for two targets whilst the CFD model was built with three, one four-plate target was completely ‘switched off’, so that only two targets produced power as simulated in the INVAP scenario. The inlet mass flow of the CFD model was increased from the safety scenario of 0.6258 kg.s-1 to 0.7510 kg.s-1 to correspond with the average flow velocity increase from 3 m.s-1 (safety study) to 3.6m.s-1 (validation study). INVAP’s input parameters of solid physical properties and oxide layer thickness were also adopted in this validation study. A summary of settings is displayed in Tables 2 and 3. Details of the model can have effects on the criterion maximum temperature. These details include: 1. modeling of the oxide layer 2. modeling of contact conditions between the Plate and the Target Holder 3. power density distribution in the U-Al meat To appreciate differences between model details, a simple CFD model was first built with the assumptions:

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1. without oxide layer. 2. perfect contact between the plate and the holder 3. a homogeneous power profile distribution across the U – Al meat Table 2. Summary of the settings for the validation case Oxide thickness Number of targets Total Power Inlet flow rate Inlet temperature Outlet gauge pressure

7 μm 2 79.4 kW 0.7510 kg.s-1 37°C 0 Pa

Table 3. Summary of the material properties for the validation case MATERIAL PROPERTIES -1

Molar Mass Density

g.mol kg.m-3

Specific Heat Capacity Thermal Conductivity

J.kg-1.K-1 W.m-1.K-1

Dynamic Viscosity μ

Al

U2Al and Al matrix Oxide

Water

26.98 2702

270 4625.34

18.02 997

903 165

900 148

4181.7 0.6069

-1 -1

kg.m .s

2.25

0.0008899

Effect of Oxide Layer Modelling During irradiation, the layer of aluminium oxide that forms on the molybdenum-plate is relatively thin at 7μm. Physically, this thin layer could not be directly represented in the computer model because the mesh was not fine enough to virtually interpret what was physically there. The oxide layer could not be ignored either because though small, its reduced conductivity had a significant effect on the molybdenum-plate’s maximum temperature.

Figure 27. Step-function thermal conductivity profile.

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Figure 28. Graphical representation of distance vs. temperature for realistic and modeled thermal conductivity for aluminum and aluminum oxide.

A first attempt was made to simply prescribe a thermal conductivity profile onto the cladding elements as shown in Figure 27. The thermal conductivity of the aluminium varies with respects to cladding depth, with the circled regions of aluminium-oxide possessing a much lower thermal conductivity. However, the result of this simulation was proven a failure as the maximum temperature proved no different from a separate identical simulation run with no oxide-layer. To circumvent this problem without resorting to the use of more computationally expensive mesh, the effect of the oxide layer was mathematically modeled as an integral part of the cladding. This was done by calculating an equivalent conductivity via the formula below which was derived from flux conservation principles (Patankar, 1980):

⎛ 1− f f ⎞ k =⎜ + ⎟ kox ⎠ ⎝ kAl where f =

−1

(41)

δ ox . The oxide layer thickness was assumed to be 7 μm corresponding with δ Al + δ ox

INVAP’s assumption. The equivalent conductivity of the cladding result was thus: ⎧δ ox = 7 μ m ⎪δ + δ = 350 μ m −1 ox 7 ⎪ Al ⎛ 1 − 0.02 0.02 ⎞ −1 −1 f k ⇒ = = 0.02 ⇒ = + ⎨ ⎜ ⎟ = 67.44 W .m .K −1 −1 k W m K 165 . . = 350 165 2.25 ⎝ ⎠ ⎪ Al ⎪k = 2.25W .m −1.K −1 ⎩ ox

This formula modified the aluminium conductivity to attain the correct temperature at the aluminium boundary but at the expense of correct temperatures inside the aluminium itself, as graphically explained in Figure 28. Maximum temperatures from the step-conductivity profile and the average conductivity profile technique are displayed in detail in Table 4. The largest difference in temperature

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proved to be between the no-oxide and average-conductivity oxide simulations, with a difference of 3.9°C in the U-Al meat. However, the stepwise-conductivity profile technique resulted in only a 0.5°C gain in UAl meat maximum temperature. It is obvious from these results that the step-wise conductivity profile technique was unsuitable. The average conductivity profile technique was adopted as the method for simulating the effect of aluminium oxide on cladding conductivity. Table 4. Comparison of models Max. Temperatures (°C) Reference

Oxide model

Oxide layer

absent

step

average

Contact type

perfect

perfect

perfect

Power distribution

uniform

uniform

uniform

Surface (water side)

59.2

59.2

59.0

Surface (cladding side)

99.1

99.3

100.5

Cladding – Meat interface

100.9

101.3

104.7

Meat – centre line

102.0

102.5

105.9

Effect of Contact Surface Since the thermal conductivity of water was much less than aluminium, a break in the solid conduction path between the molybdenum-plate and the target holder would be detrimental for the conduction of heat away from the plate. Also, as there was a large tolerance between the plate and the recess in which the plate is held, a water gap existed between one side of the plate and the holder. This study examined the maximum temperature difference caused by the presence of this water gap (see Figure 29). During the creation of this geometry, a space between the plate and its holder was made and patched separately to allow different contact conditions to be simulated. Since the inner channel was slightly larger than the outer channel, slight difference in local velocities on either side of the plate would produce a minute pressure difference that forces the plate to one side. To model the reduced thermal-conduction effects of the water gap, solids with the thermal conductivity of water were patched in between the plate and the holder. Effectively, the water gap acted as a low conductivity solid with no coolant advection. This was a reasonable and conservative assumption because the very restricted flow path of the gap would result in very small flow rates that would provide negligible amounts of forced convection cooling. The alternative was to mesh the gap at a higher resolution to adequately solve for the fluid advection which was in practical terms, unnecessary.

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Figure 29. Detailed view of contact realistic conditions between the plate and its holder.

Table 5. Effect of the contact surface Temperatures (°C) Thermal conduction profile type: Contact type Power distribution Surface (water side) Surface (cladding side)

average perfect uniform 59.0 100.5

average partial uniform 62.1 103.6

Cladding - Meat interface

104.7

107.6

Meat - centre line

105.9

108.8

The difference in contact conditions only proved to be marginal as can be seen in Table 5. The reduction from ‘full contact’ to ‘half contact’ between the plate and the holder only increased the overall maximum temperature by 3°C. This proved to be positive from an engineer’s standpoint because it was undesirable to have heat transfer characteristics that are too sensitive to unpredictable contact conditions. The half-contact condition between plate and holder was thus adopted for future simulations for its more conservative assumption.

Effect of Power Distribution In this study, the effect of a uniform and cosine power distribution on the maximum temperature was examined. A simpler study by INVAP using a one-dimensional heat transfer code provided a simple comparative assessment. According to MCNP calculations conducted by ANSTO (Geoff, 2006), a power density profile distributing a total power of 79.4 kW over eight plates with a power peaking factor (PPF) of 1.3 was calculated as:

79400 8 q = PPF cos ( B ( z − z0 ) ) = 1.3 cos ( 4.97424 ( z − 0.288 ) ) (41) V 5188.8 10-9 Q plate

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where : Power density, [W.m-3] : Power Peaking Factor, [-] : Total power per plate, [W] : Meat volume, [m3] : Adjustable constant, [m-1] : Coordinate on the z axis, [m] : Offset of the centre, [m]

q PPF Qplate V B z z0

Power Density vs. Vertical Position 2 Targets 2.60E+09 2.40E+09

q [W.m^-3]

2.20E+09 2.00E+09 1.80E+09 1.60E+09 1.40E+09 1.20E+09 1.00E+09 8.00E+08 0

0.2

0.4

0.6

z [m]

Figure 30. Power distribution along the plates (two positions).

Table 6. Power distribution effect on temperatures Temperatures (°C) Validation case

INVAP study

Oxide layer

average

average

—

Contact type

partial

partial

—

uniform

cosine

—

Surface (water side)

62.1

61.9

Surface (cladding side)

103.6

114.9

Oxide - Cladding interface

—

—

109

Cladding - Meat interface

107.6

120.2

111

Meat - centre line

108.8

121.6

113

Power distribution

105

This definition allowed the use of the same maximum power density (i.e., 2.5 109 W.m-3) as the one used by INVAP to calculate the maximum meat temperature, and also the same

312

G.H. Yeoh and M.K.M. Ho

total power in two plates. It also respected the PPF of 1.3 for the two position case. Applying the cosine power distribution to the model gave the simulation more realism than if a constant power was applied throughout the plate. Two simulations were run to observe the effect of the difference between a constant power and a cosine distribution power. Using a cosine-shape distribution for the power increased the maximum meat temperature by 12.8°C as shown in Table 5. Since a cosine power distribution (see Figure 30) is more conservatively representative, this model feature was retained for the “bounding case” scenarios.

Conclusive Remarks for the Validation Case Beginning from a simple model and making successive improvements with increasing detailed features, a final validation case was attained which has been proven to be consistent and conservative against INVAP’s one-dimensional heat-transfer study. This course of action has also allowed an appreciation of the sensitivity of temperature changes with respect to varying engineering assumptions. The temperatures on the interface between the cladding and the oxide layer could not be attained in the CFD model, and therefore has not been compared. The CFX commercial software appears to quote the temperature value held at the centre of the control volume on either side of the plate-water interface, so that these results could not be directly compared against INVAP’s study. However, the cladding side surface temperature yields valid and comparable results to INVAP’s plate surface temperature results. Table 7. Summary of the simulations for the validation case. Temperatures (°C) Validation Case

INVAP study

Oxide layer

absent

step

average

average

average

—

Contact type

perfect

perfect

perfect

partial

partial

—

Power distribution

uniform

uniform

uniform

uniform

cosine

—

Surface (water side)

59.2

59.2

59.0

62.1

61.9

Surface (cladding side)

99.1

99.3

100.5

103.6

114.9

Oxide - Cladding interface

—

—

—

—

—

109

Cladding - Meat interface

100.9

101.3

104.7

107.6

120.2

111

102

102.5

105.9

108.8

121.6

113

Meat - centre line

105

Various hypotheses were tested on the validation case. Several simulations were run which revealed: •

The application of an average thermal conductivity over the whole cladding was more realistic than the definition of a step function to lower the conductivity in the oxide layer.

Pipe Flow Analysis of Uranium Nuclear Heating with Conjugate Heat Transfer • •

313

The assumption of one-side contact between the plate and its holder was more realistic than assuming full aluminium contact on both sides of the plate. The application of a variable power density was more realistic than a simple definition of the average power density.

In conclusion, these results were in good agreement with INVAP’s calculations and as such, this model could now be used to examine even more conservative flow conditions for the purpose of safety analysis. Summary of results are presented in Table 7.

Bounding Case Having demonstrated the consistency of the CFD model against INVAP’s verification data, the CFD model could now be used to investigate the flow conditions of the bounding case scenario. A summary of the material properties are detailed in Table 8. Similar to the validation simulation, a cosine power profile (see Figure 31) for three targets was set for this bounding simulation and a ‘high resolution’ advection scheme with automatic timescale was adopted for the solution process in-order to accelerate solution convergence. A summary of temperature results for oxide thicknesses of 20 μm and 10 μm are displayed in Table 9 below: Table 8. Summary of physical properties MATERIAL PROPERTIES g.mol kg.m-3 J.kg-1.K-1 W.m-1.K-1 kg.m-1.s-1

26.98 2700 903 165

270 4625.34 900 148

4.00E+09 3.50E+09 3.00E+09 2.50E+09 2.00E+09 1.50E+09 0

0.2

0.4

0.6

oxide

2.25

Pow er Density vs. Vertical Position 3 Targets

q [W.m^-3]

Molar Mass Density Specific Heat Capacity Thermal Conductivity Dynamic Viscosity μ

Aluminium U2Al and Al matrix

-1

0.8

z [m ]

Figure 31. Power distribution over 3 plates.

Water 18.02 996.2 4178.8 0.609 0.0008327

314

G.H. Yeoh and M.K.M. Ho Table 9. Calculated temperatures for the bounding case Temperatures (°C)

Oxide layer Surface (water side) Surface (cladding side) Cladding - Meat interface Meat - centre line

20 μm 92.2 183.1 199.5 201.6

10 μm 92.5 181.1 190.9 193

Thus, assuming an average flow velocity of 3 m.s-1 at 40°C, an oxide thickness of 20 μm, and a PPF of 1.3 for a total power of 45 kW over three plates, the main results were obtained: • •

Maximum water temperature: 93°C Maximum meat temperature: 202°C

These results showed a sufficient margin from plate melting and water boiling.

Conclusion The cooling systems for three separate molybdenum irradiation facilities have been solved for maximum mean temperatures using the computational fluid dynamics methodology. In order to accomplish this, the geometry of all three systems were first modeled and geometrically discretised in a variety of structured and unstructured mesh. Secondly, accurate material properties were attributed to corresponding areas of the numeric model. Uranium volumes were then attributed Power densities as calculated by MCNP to simulate the power produced within irradiated targets. The high velocities and thus high Reynolds numbers of all three case studies placed the flow regime within a turbulent setting. As the study was concerned with time-averaged results, the RANS approach for turbulence modeling was most suited and the simple standard k-ε turbulence model was selected over other models for its applicability to fully turbulent flows. Other types of flows, such as large swirling flows or flows with large amounts of laminar-turbulent blending, would have required more sophisticated RANS modeling like the RNG k-ε model, reliazable k-ε model and SST Menter’s model. However, this level of sophistication was unnecessary for the solution of these pipe flow systems and was thus not employed. Computational results of all three case studies have been demonstrated to agree well with independent experimental and numerical studies. The success of these studies further confirms the robustness and versatility of CFD methods in the field of nuclear engineering and will remain a continual feature in the field of thermo-hydraulics.

Pipe Flow Analysis of Uranium Nuclear Heating with Conjugate Heat Transfer

315

Acknowledgements The authors would like to thank Dr. George Braoudakis (Head of Nuclear Analysis Section) for providing the MCNP power input parameters in these simulations and for proofreading this document; past internship students Mr. Tony Chung (UNSW, Sydney) and Mr. Guillaume Bois (INSA, Lyon) for their assistance in attaining the computational simulations; Mr. David Wassink (Water Tunnel Manager) for his experimental validation data; and, finally, to ANSTO for allowing this work to be published.

References [1] Durance, G. (2006), Private communication, ANSTO. [2] Gumbert, C., Lohner, R., Parikh, P. & Pirzadeh, S. (1989), A package for unstructured grid generation and finite element flow solvers, AIAA Paper 89-2175. [3] Jayatilleke, C. L. V. (1969), The influence of Prandtl number and surface roughness on the resistance of the laminar sublayer to momentum and heat transfer, Prog. Heat Mass Transfer, 1, 193 -321. [4] Launder, B. E. & Spalding, D. B. (1974), The numerical computation of turbulent flows, Comp. Meth. Appl. Mech. Eng., 3, 269-289. [5] Lo, S. H. (1985), A new mesh generation scheme for arbitrary planar domains, Int. J. Numer. Methods Eng., 21, 1403-1426. [6] Marcum, D. L. & Weatherill, N. P. (1995), Unstructured grid generation using iterative point insertion and local reconnection, AIAA Paper 94-1926. [7] Mavriplis, D. J. (1997), Unstructured grid techniques, Ann. Rev. Fluid Mech., 29, 473514. [8] Menter, F. R. (1993), Zonal two equation k-ω turbulence models for aerodynamics flows, AIAA paper 93-2906. [9] Menter, F. R. (1996), A comparison of some recent eddy-viscosity turbulence models, J. Fluids Eng., 118, 514-519. [10] MOLY-0100-3OEIN-004 (2003), Heating calculation of Molybdenum targets, INVAP Report. [11] Patankar, S. V. (1980), Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, New York. [12] Shepard, M. S. & Georges, M. K. (1991), Three-dimensional mesh generation by finite octree technique, Int. J. Numer. Methods Eng., 32, 709-749. [13] Shewchuk, J. S. (2002), Delaunay refinement algorithms for triangular mesh generation, Computational Geometry: Theory and Applications, 22, 21–74. [14] Shih, T.-H., Liou, W. W., Shabbir, A., Yang, Z. & Zhu, J. (1995). A new k-ε eddy viscosity model for high Reynolds number turbulent flows, Comp. Fluids, 24, 227-238. [15] Smith, R. E. (1982), Algebraic grid generation, Numerical Grid Generation, Thompson, J. F. (Ed.), North-Holland, Amsterdam, 137. [16] Thompson, J. F. (1982), General curvilinear coordinate systems, Numerical Grid Generation, Thompson, J. F. (Ed.), North-Holland, Amsterdam, 1-30. [17] Touloukian, Y.S., Powell, R.W., Ho, C.Y., Klemens, P.G. (1970). Thermophysical Properties of Matter, 2.

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[18] Tu, J. Y., Yeoh G. H. & Liu C. Q. (2008), Computational Fluid Dynamics – A Practical approach. Butterworth-Heinemann, Oxford. [19] Wilcox, D. C. (1998), Turbulence Modelling for CFD, DCW Industries, Inc. [20] Yakhot, V., Prszag, S. A., Tangham, S., Gatski, T. B. & Speciale, C. G. (1992), Development of turbulence models for shear flows by a double expansion technique, Phys Fluids A: Fluid Dynamics, 4, 1510-1520. [21] Yeoh, G. H. and Storr, G. J. (2000), A three-dimensional study of heat and mass transfer within the irradiation space of the HIFAR fuel element, Advance Computational Methods in Heat Transfer VI, Sunden, B. and Brebbia, C. A. (Eds.), WIT Press, Southampton, 343-351. [22] Yerry, M. A. & Shepard, M. S. (1984), Automatic three-dimensional mesh generation by the modified-octree technique, Int. J. Numer. Methods Eng., 20, 1965-1990.

In: Fluid Mechanics and Pipe Flow Editors: D. Matos and C. Valerio, pp. 317-342

ISBN: 978-1-60741-037-9 © 2009 Nova Science Publishers, Inc.

Chapter 9

FIRST AND SECOND LAW THERMODYNAMICS ANALYSIS OF PIPE FLOW Ahmet Z. Sahin* Mechanical Engineering Department King Fahd University of Petroleum and Minerals Dhahran 31261, Saudi Arabia

Introduction In a fully developed laminar flow through a pipe, the velocity profile at any cross section is parabolic when there is no heat transfer. But when a considerable heat transfer occurs and the thermo-physical properties of the fluid vary with temperature, the velocity profile is distorted. If the thermo-physical properties of the fluids in a heat exchanger vary substantially with temperature, the velocity and the temperature profiles become interrelated and, thus, the heat transfer is affected. Viscosity of a fluid is one of the properties which are most sensitive to temperature. In the majority of cases, viscosity becomes the only property which may have considerable effect on the heat transfer and temperature variation and dependence of other thermo-physical properties to temperature is often negligible. The viscosity of the liquids decreases with increasing temperature, while the reverse trend is observed in gases (Kreith and Bohn, 1993). Heat transfer and pressure drop characteristics are affected significantly with variations in the fluid viscosity. When the temperature is increased from 20 to 80 oC, the viscosity of engine oil decreases 24 times, the viscosity of water decreases 2.7 times and the viscosity of air increases 1.4 times. Therefore, selection of the type of fluid and the range of operating temperatures are very important in the design and performance calculations of a heat exchanger. In the process of designing a heat exchanger, there are two main considerations. These are the heat transfer rates between the fluids and the pumping power requirement to overcome the fluid friction and move the fluids through the heat exchanger. Although the effect of viscous dissipation is negligible for low-velocity gas flows, it is important for high-velocity *

E-mail address: [email protected]

318

Ahmet Z. Sahin

gas flows and liquids even at very moderate velocities (Shah, 1981; Kays and Crawford, 1993). Heat transfer rates and pumping power requirements can become comparable especially for gas-to-gas heat exchangers in which considerably large surface areas are required when compared with liquid to liquid heat exchangers such as condensers and evaporators. In addition, the mechanical energy spent as pumping power to overcome the fluid friction is worth 4 to 10 times as high as its equivalent heat (Kays and London, 1984). Therefore, viscous friction which is the primary responsible cause for the pressure drop and pumping power requirements is an important consideration in heat exchanger design. On the other hand, efficient utilization of energy is a primary objective in designing a thermodynamic system. This can be achieved by the minimization of entropy generation in processes of the thermodynamic system. The irreversibilities associated with fluid flow through a pipe are usually related to heat transfer and viscous friction. Various mechanisms and design features contribute to the irreversibility terms (Bejan, 1988). There may exist an optimal thermodynamic design which minimizes the amount of entropy generation. For a given system, a set of thermodynamic parameters which optimizes the operating conditions may be obtained. The irreversibility associated with viscous friction is directly proportional to the viscosity of the fluid in laminar flow. Therefore, it is necessary to investigate the effect of a change of viscosity during a heating process for an accurate determination of entropy generation and of the required pumping power. Heat transfer and fluid pumping power in a piping system are strongly dependent upon the type of fluid flowing through the system. It is important to know the fluid properties and their dependence to temperature for a heat exchanger analysis. As a result of heat transfer, the temperature changes in the direction of flow and the fluid properties are affected. The temperature variation across the individual flow passages influences the velocity and temperature profiles, and thereby influences the friction factor and the convective heat transfer coefficient. If the thermo-physical properties of the fluids in a piping system vary substantially with temperature, the velocity and the temperature profiles become interrelated and, thus, the heat transfer is affected. Viscosity of a fluid is one of the properties which are most sensitive to temperature. Therefore, selection of the type of fluid and the range of operating temperatures are very important in the design and performance calculations of a piping system. On the other hand, the exergy losses associated with fluid flow through a pipe are usually related to heat transfer and viscous friction. Dependence of the thermo-physical properties on the temperature affects not only the viscous friction and pressure drop, but also the heat transfer. This implies that the exergy losses associated with the heat transfer and the viscous friction are also affected. The contributions of various mechanisms and design features on the different irreversibility terms often compete with one another. Therefore, an optimal thermodynamic design which minimizes the amount of entropy generation may exist. In other words, a set of thermodynamic parameters which optimize operating conditions could be obtained for a given thermodynamic system. In this chapter, the entropy generation for during fluid flow in a pipe is investigated. The temperature dependence of the viscosity is taken into consideration in the analysis. Laminar and turbulent flow cases are treated separately. Two types of thermal boundary conditions are considered; uniform heat flux and constant wall temperature. In addition, various crosssectional pipe geometries were compared from the point of view of entropy generation and

First and Second Law Thermodynamics Analysis of Pipe Flow

319

pumping power requirement in order to determine the possible optimum pipe geometry which minimizes the exergy losses.

Pipe Subjected to Uniform Wall Heat Flux We consider the smooth pipe with constant cross section shown in Figure 1. A constant  and inlet heat flux q′′ is imposed on its surface. An viscous fluid with mass flow rate m temperature T0 enters the pipe of length L . Density

ρ , thermal conductivity k , and

specific heat C p of the fluid are assumed to be constant. Heat transfer to the bulk of the fluid occurs at the inner surface with an average heat-transfer coefficient h , which is a function of temperature dependent viscosity. The effect of viscosity on the average heat transfer coefficient is given by Kays and Crawford (1993)

⎛μ ⎞ h Nu = =⎜ b ⎟ hc. p . Nu c. p . ⎝ μw ⎠

n

(1)

where the exponent n is equal to 0.14 for laminar flow. In the case of turbulent flow the exponent n is equal to 0.11 for heating and 0.25 for cooling. For fully developed laminar flow, constant property heat transfer coefficient hc. p. is given by Incropera and DeWitt (1996).

hc. p. =

k 48 k Nu c. p. = D 11 D

Figure 1. Sketch of constant cross sectional area pipe subjected to uniform wall heat flux.

(2)

320

Ahmet Z. Sahin And for fully developed turbulent flow, constant property heat transfer coefficient hc. p. is

given by

hc. p. =

k k ( f / 8) ( Re − 1000 ) Pr Nu c. p. = D D 1 + 12.7 ( Pr 2 / 3 − 1) f / 8

(3)

where the Reynolds number is

Re =

ρUD . μb

(4)

The average Darcy friction factor, f , for a smooth pipe is also considered to be a function of temperature dependent viscosity and is given by Kays and Crawford (1993) m

f f c. p. f f c. p.

⎛μ ⎞ = ⎜⎜ b ⎟⎟ for liquids ⎝ μw ⎠

(5a)

m

⎛T ⎞ = ⎜⎜ b ⎟⎟ for gases ⎝ Tw ⎠

(5b)

where the exponent m = −0.14 for laminar flow and m = −0.25 for turbulent flow case. The friction factor for constant properties for laminar flow is given by Incropera and DeWitt (1996) as

f c. p . = and for turbulent flow as

64 Re

f c. p . = [0.79 ln(Re) − 1.64]

(6)

−2

To account for the variation of the bulk temperature along the pipe length,

(7)

μb and

therefore Re and Pr, in Eqs. (3) – (7), are related to the bulk fluid temperature halfway between the inlet and outlet of the pipe, as suggested by Kreith and Bohn (1993). Since the temperature variation along the pipe is initially unknown and depends on h , a trial and error procedure is followed to determine both h and f .

First and Second Law Thermodynamics Analysis of Pipe Flow

321

Total Heat Transfer Rate The rate of heat transfer to the fluid inside the control volume shown in Figure 1 is

 p dT = q′′π Ddx δ Q = mC

(8)

 = ρUπ D / 4 . In writing Eq. (8), the pipe is assumed to have a circular cross where m 2

section. However, the analysis is not affected by assuming cross-sectional areas other than circular. Integrating Eq. (8), the bulk-temperature variation of the fluid along the pipe is obtained as

T − T0 =

4 q′′ x. ρUDC p

(9)

The temperature variation along the pipe is linear when the viscous dissipation and axial conduction effects are neglected. The temperature gradient for the fluid in this case depends mainly on the magnitude of the heat flux. For a constant heat flux q′′ and average heat transfer coefficient h evaluated at the bulk temperature halfway between inlet and outlet of pipe, the temperature difference between the wall surface and bulk of the fluid is given as

Tw − T =

q′′ . h

(10)

Eq. (9) may be rewritten as

θ =4 where

St x, D

(11)

θ is the dimensionless temperature defined as

θ=

T − T0 q′′ / h

(12)

St =

h . ρUC p

(13)

and the Stanton number is

322

Ahmet Z. Sahin

Total Entropy Generation The entropy generation within the control volume of Figure 1 can be written as (Bejan, 1996)

δ Q  − , dS gen = mds Tw

(14)

dT dP − . T ρT

(15)

where the entropy change is

ds = C p Substituting Eq. (8) into Eq. (14),

⎡T − T ⎤ 1  p⎢ w dS gen = mC dT − dP ⎥ . ρ C pT ⎦ ⎣ TTw

(16)

The pressure drop is (Kreith and Bohn, 1993)

f ρU 2 dP = − dx , 2D

(17)

where f is the Darcy friction factor. Integrating Eq. (16) along the pipe length L , using Eqs. (9) and (17), the total entropy generation becomes (Sahin, 1999 and Sahin, 2002)

⎧⎪ ⎡ (1 + 4 Stτλ )(1 + τ ) ⎤ 1 f ρU 3 ⎫⎪  p ⎨ln ⎢ τλ ln 1 4 S gen = mC St + + ( ) ⎬, ⎥ ⎩⎪ ⎣ 1 + τ + 4 Stτλ ⎦ 8 q′′ ⎭⎪

(18)

where the dimensionless wall heat flux is

τ=

q′′ / h T0

(19)

and the dimensionless length of the pipe is

λ=

L . D

(20)

First and Second Law Thermodynamics Analysis of Pipe Flow

323

Eq. (18) can be written in a dimensionless form as

S

ψ =  gen Q / T0

(21)

 pT0 ( 4 Stτλ ) Q = mC

(22)

where the total heat transfer is

Thus, the entropy generation per unit heat transfer rate becomes

ψ=

⎫ 1 ⎧ ⎡ (1 + 4 Stτλ )(1 + τ ) ⎤ 1 Ec + f ln(1 + 4 Stτλ )⎬ ⎨ln ⎢ ⎥ 4 Stτλ ⎩ ⎣ 1 + τ + 4 Stτλ ⎦ 8 St ⎭

(23)

where the Eckert number is defined as

Ec =

U2 U2 . = C p (Tw − T ) C p ( q′′ / h )

(24)

Two dimensionless groups arise naturally in Eq. (23), namely,

Π1 = Stλ

(25)

and

Π2 = f

Ec St

(26)

Thus, Eq. (23) becomes in compact form

ψ=

⎫ 1 ⎧ ⎡ (1 + 4τΠ1 )(1 + τ ) ⎤ 1 + Π 2 ln(1 + 4τΠ1 )⎬ ⎨ln ⎢ ⎥ 4τΠ1 ⎩ ⎣ 1 + τ + 4τΠ1 ⎦ 8 ⎭

(27)

1 ⎡ 1+τ ⎤ 1 (1 + 4τΠ1 )1+ 8 Π2 ⎥ . ln ⎢ 4τΠ1 ⎣1 + τ + 4τΠ1 ⎦

(28)

or

ψ=

The first and second terms in Eq. (27) are related to entropy generation due to heat transfer and due to viscous friction respectively. Eq. (27) contains three non-dimensional parameters, namely, τ , Π1 and Π 2 . Among these parameters, τ represents the heat flux imposed on the wall of the pipe q′′ and Π1 represents the pipe length L . Once the type of the fluid and the mass flow rate are fixed, the parameter Π 2 can be calculated on the basis of

324

Ahmet Z. Sahin

temperature analysis. Thus,

τ and Π1 are the two design parameters that can be varied for

determining the effects of pipe length and/or wall heat flux on the entropy generation. For small values of the wall heat flux ( τ 3 mm. Mini-channels: 200 μ m < dh ≤ 3 mm. Micro-channels: 10 μ m < dh ≤ 200μ m. Transitional micro-channels: 1 μ m < dh ≤ 10μ m. Transitional nano-hannels: 0.1 μ m < dh ≤ 1μ m. Nano-channels: dh ≤ 0.1 μ m. In the case of non-circular channels, it is recommended that the minimum channel dimension, for example, the short side of a rectangular cross-section should be used in place of the diameter d. In the available studies on fluid flow in mini- and micro-channels, some researchers have concluded that the conventional theories work for mini- and micro-channels while others have showed that the conventional theories do not work well. Therefore, these controversies should be clarified. As the channel size becomes smaller, some of the conventional theories for (bulk) fluid, energy and mass transport need to be revisited for validation. There are two

Single-Phase Incompressible Fluid Flo in Mini- and Micro-channels

345

fundamental elements responsible for departure from the conventional theories at mini- and micro-scale. For example, differences in modeling fluid flow in mini- and micro-channels may arise as a result of (i) a change in the fundamental process, such as a deviation from the continuum assumption for fluid flow, or an increase influence of some additional forces, such as electrokinetic forces, etc. (ii) uncertainty regarding the applicability of empirical factors derived from experiments conducted at larger scales, such as entrance and exit loss coefficients for fluid flow in pipes, etc., (iii) uncertainty in measurements at mini- and microscale, including geometrical dimensions and operating parameters. In this chapter, the available studies of single-phase incompressible fluid flow in miniand micro-channels are reviewed to address these relevant important issues.

2. Single-Phase Frictional Pressure Drop Methods in Macro-channels 2.1. Laminar and Turbulent Flow The flow of a fluid in a pipe may be laminar or turbulent flow, or in between transitional flow. Figure 1 shows the x component of the fluid velocity as a function of time t at a point A in the flow for laminar, transitional and turbulent flows in a pipe. For laminar flow, there is only one component of velocity. For turbulent flow, the predominant component of the velocity is also along the pipe, but it is unsteady (random) and accompanied by random components normal to the pipe axis. For transitional flow, both laminar and turbulent features occur.

Q

A

x

VA Turbulent Transitional Laminar t Figure 1. Time dependence of fluid velocity at a point.

346

Lixin Cheng

For pipe flow, the most important dimensionless parameter is the Reynolds number Re, the ratio of the inertia to viscous effects in the flow, which is defined as Re =

ρVd μ

(1)

where V is the average velocity in the pipe. The flow in a pipe is laminar, transitional or turbulent provided the Reynolds number Re is small enough, intermediate or large enough. It should be pointed out here that the Reynolds number ranges for which laminar, transitional or turbulent pipe flows are obtained cannot be precisely given. The actual transition from laminar to turbulent pipe flow may take place at various Reynolds numbers, depending on how much the flow is distributed by vibrations of the pipe, roughness of the entrance region, and the like. For general engineering purposes (i.e. without undue precautions to eliminate such disturbances), the following values are appropriate: the flow in a round pipe is laminar if the Reynolds number Re is less than approximately 2100 and the flow in is turbulent if the Reynolds number Re is greater than approximately 4000. For the Reynolds numbers between these two limits, the flow may switch between laminar and turbulent conditions in an apparently random fashion (transitional flow).

2.2. Entrance Region (Developing Flow) and Fully Developed Flow The region of flow near where the fluid enters a pipe is termed the entrance region (developing flow) and is illustrated in Figure 2. The fluid typically enters the pipe with a nearly uniform velocity profile at section 1. As the fluid moves through the pipe, viscous effects cause it to stick to the pipe wall (the no-slip boundary condition). Thus, a boundary layer in which viscous effects are important is produced along the pipe wall such that the initial velocity profile changes with distance along the pipe, x, until the fluid reaches the end of the entrance length, section 2, beyond which the velocity profile does not vary with x. The boundary layer has grown in thickness to completely fill the pipe. Viscous effects are of considerable importance within the boundary layer. For fluid outside the boundary layer (within the inviscid core surrounding the centerline from 1 to 2), viscous effects are negligible. The shape of the velocity profile in the pipe depends on whether the flow is laminar or turbulent, as does the length of the entrance region, le. As with many other properties of pipe flow, the dimensionless entrance length le/d, correlates quite well with the Reynolds number Re. Typical entrance lengths are given by le = 0.06 Re for laminar flow d

(2)

le = 4.4 Re1/ 6 for turbulent flow d

(3)

Calculation of the velocity profile and pressure distribution within the entrance region (developing flow) is quite complex. However, once the fluid reaches the end of the entrance region, section 2 in Figure 2, the flow is simpler to describe because the velocity is a function

Single-Phase Incompressible Fluid Flo in Mini- and Micro-channels

347

of only the distance from the pipe centerline, r, and independent of x. The flow after section 2 is termed fully developed flow.

Entrance region flow le Fully developed flow Inviscid core r V

x 1

2 d

Boundary layer

Figure 2. Entrance region, developing flow and fully developed flow in a pipe.

2.3. Single-Phase Friction Pressure Drop Methods The nature of the pipe flow is strongly dependent on weather the flow is laminar or turbulent. This is a direct consequence of the differences in the nature of the shear stress in laminar and turbulent flows. The shear tress in laminar flow is a direct result of momentum transfer among the randomly moving molecules (a microscopic phenomenon). The shear stress in turbulent flow is largely a result of momentum transfer among the randomly moving, finite-sized bundles of fluid particles (a macroscopic phenomenon). The net result is that the physical properties of the shear stress are quite different for laminar than for turbulent flow. The friction pressure drop for both laminar and turbulent flow can be expressed as Δp = f

l ρV 2 d 2

(4)

For fully developed laminar flow in a circular pipe, the friction factor f is simply as f =

64 Re

(5)

ε

. d For turbulent flow, the dependence of the friction factor on the Reynolds number Re is much more complex than that given by Eq. (5) for laminar flow. For fully developed turbulent flow and transition from laminar to turbulent flow. The Moody chart [7] shown in Figure 3 provides the friction factor f, which can be expressed as

and the value of f is independent of the relative roughness

ε⎞ ⎛ f = φ ⎜ Re, ⎟ d⎠ ⎝

(6)

Figure 3. Friction factor as a function of the Reynolds number and relative roughness for circular pipes-the Moody chart [7].

Single-Phase Incompressible Fluid Flo in Mini- and Micro-channels

349

For turbulent flow, the friction factor f is a function of the Reynolds number Re and ε relative roughness . The results are obtained from an exhaustive set of experiments and d usually presented in terms of a curve-fitting formulae or the equivalent graphical form. In commercially available pipes, the roughness is not as uniform and well defined as in the artificially roughened pipes. However, it is possible to obtain a measure of the effective relative roughness of typical pipes and thus to obtain the friction factor. Typical roughness values for various pipe surfaces are also given in Figure 3. It should be noted that the values ε do not necessarily correspond to the actual values obtained by a microscopic of d determination of the average height of the roughness of the surface. They do, however, provide the correct correlation for friction factor f. The following characteristics are observed from Figure 3 for laminar flow, friction factor is calculated by Eq. (2), which is independent of relative roughness. For very large Reynolds number Re, the friction factor is a function of ε the relative roughness as d ⎛ε ⎞ f =φ⎜ ⎟ ⎝d ⎠

(7)

which is independent of the Reynolds number Re. For such flows, commonly termed completely turbulent flow (or wholly turbulent flow), the laminar sub-layer is so thin (its thickness decreases with increasing the Reynolds number Re) that the surface roughness completely dominates the character of the flow near the wall. Hence, the pressure drop required is a result of an inertia-dominated turbulent shear stress rather than the viscositydominated laminar shear stress normally found in the viscous sublayer. For flows with moderate values of Re, the friction factor is indeed dependent on both the ε . Flow in the range of 2100< Re 12, the velocity profile overshoots the value of the velocity ue (x). If Λ < −12, a separation occurs and the velocity profile is not nonnegative. The boundary layer thickness δ is obtained from the von Kármán’s momentum equation and relations for the boundary layer parameters: due dθ τwall = (2θ + δ ∗ ) ue + u2e , ρ dx dx   2 3 1 due δ δ∗ = δ − , 10 120 dx ν

1 due δ 2 1 37 − − θ=δ 315 945 dx ν 9072   2 µue 1 due δ 2+ . τwall = ρ 6 dx ν

(3.25) (3.26) 

due δ 2 dx ν

2 !

,

(3.27) (3.28)

The new thickness tvis is computed using the formula (3.22). After that, the mean camber line is modified to eliminate viscous effects. To do so, a correction function ∆(x) =

∗ − δ∗ δup lo 2

is subtracted from the mean camber line function s(x). The function s(x) is then transformed to satisfy s(0) = s(b) = 0.

3.6.

Finding the Mean Camber Line of a General Airfoil

As was mentioned earlier, the pressure distribution is given along a mean camber line. The inverse operator evaluates the function describing this line, so the mean camber line of the designed airfoil is known. But if the given pressure distribution is based on a known airfoil, it is useful to have a way how to get its mean camber line. Assume the airfoil coordinates are known. The process of the airfoil construction (3.17)–(3.18) can be rewritten as ! 1 s′ (x) ψup x − t(x) p = s(x) + t(x) p , ′2 1 + s (x) 1 + s′2 (x) ! s′ (x) 1 ψlo x + t(x) p = s(x) − t(x) p , x ∈ h0, bi (3.29) ′2 1 + s (x) 1 + s′2 (x)

under the assumptions s(x) ∈ C 1 h0, bi, t(x) ∈ C h0, bi. The symbols ψup (˜ x), ψlo (˜ x) denote the y-coordinates of the upper and lower part of the airfoil (corresponding to xcoordinate x ˜). Linearization of the left hand sides and rearrangement leads to a differential equation for an unknown function s(x), 
 ′ ′ ′ ′ ′ s (x) ψup (x)ψlo (x) + ψup (x)ψlo (x) − s(x) ψup (x) + ψlo (x) = 2s(x) − ψup (x) − ψlo (x),

x ∈ (0, b) .

(3.30)

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This equation is equipped with boundary conditions s(0) = 0,

s(b) = 0.

′ (x) 6= −1 and The equation was derived under the assumptions that s′ (x)ψup ′ (x) 6= −1. In other words, the tangential vectors of the airfoil shape and the s′ (x)ψlo mean camber line are not perpendicular to each other. This is true with an admissible geometry. The expression enclosed by parentheses in (3.30) can be zero at some points from the interval h0, bi. Thus the derivative s′ (x) at these points is undefined by (3.30). However, the equation then says s(x) = (ψup (x) + ψlo (x))/2, which corresponds to the idea of a mean camber line. If the airfoil is closed and the curves representing upper and lower ′ (x) = ψ ′ (x). parts are smooth, then there exists at least one point x ∈ h0, bi such that ψup lo ′ (x) = −1 or the term in parentheses in (3.30) is zero. From At this point, either s′ (x)ψup that reason it is necessary to be careful when solving the differential equation and utilize both boundary conditions. Generally, the existence and uniqueness of the solution is not guaranteed, but in the common cases the solution is unique.

4.

Direct Operator

This operator represents the solution of the flow problem. Depending on the model of flow used, its formulation can vary. In this case, where the viscous compressible flow is assumed, the model is described by the system of the Navier-Stokes equations.

4.1.

Mathematical Formulation

The Navier-Stokes equations are given by 2

∂ρ X ∂(ρvj ) + = 0, ∂t ∂xj

(4.31)

j=1

2 ∂(ρvi vj + p δij ) X ∂τij = , i = 1, 2, (4.32) ∂xj ∂xj j=1 j=1
    2 2 X ∂E X ∂ (E + p)vj ∂ µ µT ∂e + = τj1 v1 + τj2 v2 + + γ . ∂t ∂xj ∂xj Pr PrT ∂xj

∂(ρvi ) + ∂t

2 X

j=1

j=1

(4.33)

Since most of the flow in a real situation is turbulent, the laminar model seems insufficient. To improve the quality of the predicted flow and also the stability of the method, a model of turbulence is included (hence the term with the subscript T in (4.33)). In the case described in this chapter the k − ω turbulence model is used [6], [7]. New variables describing the turbulence properties are introduced, a turbulent kinetic energy k and a specific

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turbulent dissipation ω. These two variables are linked together by equations 2

2

j=1

j=1

2

2

j=1

j=1

X ∂ ∂ρk X ∂ρkvj + = ∂t ∂xj ∂xj

  ∂k (µ + σk µT ) + Pk − β ∗ ρωk, ∂xj

(4.34)

X ∂ ∂ρω X ∂ρωvj + = ∂t ∂xj ∂xj

  ∂ω (µ + σω µT ) + Pω − βρω 2 + CD . ∂xj

(4.35)

For the simplicity, the system can be rewritten into the vector form 2

2

∂w X ∂Fj (w) X ∂Gj (w, ∇w) + = + S (w, ∇w) . ∂t ∂xj ∂xj j=1

(4.36)

j=1

By µT an eddy viscosity coefficient is denoted. This coefficient is given by the formula µT =

ρk . ω

The stress tensor in the N.-S. equations is given by relations   2 ∂v2 4 ∂v1 − − τ11 = (µ + µT ) 3 ∂x1 3 ∂x2   2 ∂v1 4 ∂v2 τ22 = (µ + µT ) − + − 3 ∂x1 3 ∂x2   ∂v2 ∂v1 . + τ12 = τ21 = (µ + µT ) ∂x2 ∂x1

2ρk , 3 2ρk , 3

The production of turbulence Pk on the right hand side of the eq. (4.34) and the production of dissipation Pω in eq. (4.35) are expressed as   ∂v1 ∂v2 ∂v1 ∂v2 Pk = τ 11 + τ 12 + , + τ 22 ∂x1 ∂x2 ∂x1 ∂x2 Pk Pω = αω ω , k where τ ij = τij for µ = 0. Finally, the cross-diffusion term CD is given by the relation   ρ ∂k ∂ω ∂k ∂ω CD = σD max + ,0 . ω ∂x1 ∂x1 ∂x2 ∂x2 The turbulence model is closed by parameters β ∗ = 0.09, β = 5β ∗ /6, αω = β/β ∗ − √ σω κ2 / β ∗ (where κ = 0.41 is the von Kármán constant), σk = 2/3, σω = 0.5 a σD = 0.5. This choice of parameters resolves the dependence of the k − ω model on the free stream values [7]. In the standard Wilcox model cross diffusion the parameters are √ without ∗ ∗ ∗ 2 ∗ β = 0.09, β = 5β /6, αω = β/β − σω κ / β , σk = 0.5, σω = 0.5 a σD = 0. If the turbulent kinetic energy k is set to zero, the turbulence model has no influence upon the N.-S. equations and the laminar model can be solved.

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Jan Šimák and Jaroslav Pelant

Numerical Treatment

In the numerical method, a dimensionless system of equations is solved. The variables p, ρ, v1 , v2 , k and ω are normalized with respect to the critical values of density ρ∗ , velocity c∗ and pressure p∗ and to the characteristic length l∗ . The critical values are the values corresponding to a unit Mach number. The dimensionless system has the same form as (4.31)–(4.35). The relations between the original and normalized variables are the following: vˆ1 =

v1 , c∗

vˆ2 =

k kˆ = 2 , c∗

v2 , c∗ ω ˆ=

ρˆ = ωl∗ , c∗

ρ , ρ∗

p , ρ∗ c2∗ µT µ ˆT = . ρ ∗ c∗ l ∗ pˆ =

(4.37)

The problem is solved by an implicit finite volume method. Details about this method and the numerical solution of a flow in general can be found in many textbooks, for example [8]. Since the coupling between the equations describing the flow and the equations describing the turbulence is only by the viscous terms, it is possible to solve the problem in two parts [6]. When the flow variables p, ρ, v1 , v2 in the time tk+1 are computed using (4.31)–(4.33), the turbulent variables k and ω are assumed to be constant in time with values corresponding to the time tk . On the contrary, when computing the turbulent variables k, ω in a time tk+1 using (4.34) and (4.35), the flow variables p, ρ, v1 , v2 are assumed to be constant in the time tk . That means the solution of the problem consists of two systems, 1. (v1k+1 , v2k+1 , ρk+1 , pk+1 ) = NS(v1k+1 , v2k+1 , ρk+1 , pk+1 , k k , ω k ), 2. (k k+1 , ω k+1 ) = Turb(v1k , v2k , ρk , pk , k k+1 , ω k+1 ). These systems of equations can be solved independently of each other. The solution of the system of equations (4.31)-(4.33) is similar to the way how a laminar problem is solved. The equations are identical with the pure N.-S. equations except the stress tensor τij in viscous terms and the heat flux, which depend on k and ω. Since k and ω are taken as parameters in the system, it is quite easy to modify the existing laminar solver. By wh will be denoted a vector of 6-dimensional blocks wi of the values of an approximate solution on finite volumes Di ∈ Dh . For wh ∈ Rn the vector Φ(wh ) consists of 6-dimensional blocks Φi (wh ) given by ! 2 2 X 1 X X Φi (wh ) = ns Fs,h (wh ; i, j) |Γij | − ns Gs,h (wh ; i, j) |Γij | |Di | j∈S(i)

s=1

− Sh (wh ; i, j) ,

s=1

(4.38)

where |Di | denotes the cell area, |Γij | denotes the length of the edge between Di and Dj , nij = (n1 , n2 ) denotes the outer normal to Di . Functions Fs,h (wh ; i, j), Gs,h (wh ; i, j) and Sh (wh ; i, j) are approximations of F(w), G(w, ∇w) and S(w, ∇w) on the grid Dh . In order to have a higher order method we apply the Van Leer κ-scheme or the Van Albada limiter inside the functions Fs,h . By S(i) is denoted a set of indices of neighbouring elements and by the symbol τ will be denoted the time step. Thus the implicit finite volume

Solution of an Airfoil Design Inverse Problem...

393

scheme in a cell Di can be written as wik+1 = wik − τ k Φi (whk+1 ).

(4.39)

The nonlinear equation above is linearized by the Newton method. The arising system of linear algebraic equations is solved by the GMRES method (using software SPARSKIT2 [9]). The convective terms Fi are evaluated using the Osher-Solomon numerical flux in the case of the flow part and by the Vijayasundaram numerical flux in the turbulent part. The numerical evaluation of a gradient on the edge V2 V3 on a boundary (Fig. 4) is done by the following formulae (the index denotes the value in the corresponding vertex) − (V3,y − V2,y ) (f1 − fwall ) ∂f ≈ , ∂x1 V2 V3 |(V2,x − V1,x )(V3,y − V1,y ) − (V3,x − V1,x )(V2,y − V1,y )| (V3,x − V2,x ) (f1 − fwall ) ∂f ≈ . (4.40) ∂x2 V2 V3 |(V2,x − V1,x )(V3,y − V1,y ) − (V3,x − V1,x )(V2,y − V1,y )| If the edge is inside the domain, another scheme according to Fig. 5 is used,

[V1,x , V1,y ]

[V3,x , V3,y ]

[V2,x , V2,y ]

Figure 4. Scheme for a derivative on a wall. [V4,x , V4,y ]

[V1,x , V1,y ]

[V3,x , V3,y ] [V2,x , V2,y ]

Figure 5. Scheme for a derivative inside the domain. (f3 − f1 )(V4,y − V2,y ) − (f4 − f2 )(V3,y − V1,y ) ∂f = , ∂x1 V1 V3 |(V3,x − V1,x )(V4,y − V2,y ) − (V4,x − V2,x )(V3,y − V1,y )| (f3 − f1 )(V4,x − V2,x ) − (f4 − f2 )(V3,x − V1,x ) ∂f =− . ∂x2 V1 V3 |(V3,x − V1,x )(V4,y − V2,y ) − (V4,x − V2,x )(V3,y − V1,y )|

(4.41)

The points V2 and V4 are centres of corresponding cells, f2 and f4 are values in these cells. The values f1 and f3 are obtained as an arithmetic mean of values of the four neighboring cells.

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4.3.

Jan Šimák and Jaroslav Pelant

Boundary Conditions

In our problem, three types of boundary conditions occur: a condition on a wall, a condition at an inlet boundary and a condition at an outlet boundary. Due to the viscosity, the zero velocity on the wall is prescribed, further the zero turbulent kinetic energy and a static temperature are prescribed. The value of the specific turbulent dissipation ω is obtained by the formula 120µ ωwall = , ρyc2 where yc is the distance between the wall and the centre of a cell in the first row. At the inlet part of the boundary, the velocity vector (v1 , v2 ), the density ρ, turbulent energy k and dissipation ω are prescribed. At the outlet part of the boundary, three variables are prescribed, the static pressure p, turbulent energy k and dissipation ω. The other variables are evaluated from values inside the domain. The values of k and ω on the boundary are values of the free stream and are given in the form of a turbulent intensity I and a viscosity ratio ReT = µT /µ. The turbulent intensity is defined as r 2 k . (4.42) I= 3 v∞ Following this, the relations for k∞ and ω∞ are obtained,

4.4.

2 (v∞ I)2 , 3   ρk µT −1 = . µ µ

k∞ =

(4.43)

ω∞

(4.44)

Mesh Deformation

During the inverse method iterations, the direct operator is applied a number of time. This is of course true, a new airfoil shape is designed in each iteration. Moreover, there is a need to find an appropriate angle of attack, which ensures the required position of the stagnation point on the leading edge. This is done by the rotation of the airfoil. This all results in changes of the computational domain and of the mesh, of course. In order to remove the dependency on the mesh generator used, the actual mesh is deformed to fit the new geometry. The deformation is based on the linear elasticity model which is described in many textbooks. The grid cells are stretched or shrinked and moved to fit the new domain. No grid points are created or deleted, the neighbouring cells are the same in the new grid as in the original grid. The linear elasticity model is described by the equation div σ = f

in Ω,

(4.45)

where σ is a stress tensor and f is some body force. The symbol Ω denotes the domain, its boundary will be denoted by Γ. The stress tensor is expressed using a strain tensor ǫ as σ = λ Tr(ǫ)I + 2µǫ

Solution of an Airfoil Design Inverse Problem...

395

and the components of the strain tensor can be expressed using a displacement function u as   ∂uj 1 ∂ui ǫij = + . 2 ∂xj ∂xi The symbols λ and µ are the Lamé constants which describe the physical properties of the solid. They can be expressed using the Young’s modulus E and Poisson’s ratio ν, λ=

νE , (1 + ν)(1 − 2ν)

µ=

E . 2(1 + ν)

Substituting the above mentioned relations into (4.45), the following problem for the displacement is obtained: Find an unknown function u : Ω → R2 such that (λ + µ)∇(div u) + µ∆u = f u = uD

in Ω, on Γ.

(4.46)

The function uD is the displacement on the boundary, which is known. This problem is reformulated into the weak sense. Thus the problem for the mesh deformation can be formulated in the form: Find a function u ∈ H 1 (Ω)2 such that u−u∗ ∈ H01 (Ω)2 , where u∗ represents Dirichlet boundary conditions (that means u∗ ∈ H 1 (Ω)2 , u∗ |Γ = uD ) and the function u satisfies the equation Z Z Z −µ ∇u : ∇ϕT dx − (λ + µ) div u · div ϕ dx = f · ϕ dx (4.47) Ω

for all ϕ ∈

H01 (Ω)2 .

Ω

Ω

The colon operator is defined by the relation A:B =

2 X 2 X i=1 j=1

aij bji ,

A, B ∈ R2×2 .

The weak problem described above can be solved by a finite element method. The domain is discretized by a triangular mesh with nodal points, which are the same as in the mesh for the flow problem. This leads to the fact, that the solution, which represents the movement of the nodal points of the original mesh, is evaluated at the appropriate points. The parameter E is set proportional to the reciprocal values of the cell volumes. This ensures that the most deformation is carried out on large cells instead of the small ones.

5. 5.1.

Numerical Examples Example 1

In this example the given pressure distribution is obtained as a result of a flow around the NACA4412 airfoil. The inlet Mach number is M∞ = 0.6, angle of attack α∞ = 1.57◦ and the Reynolds number Re = 6 · 106 . The flow is described by the use of the turbulence k − ω model. The resulting airfoil is compared to the original one and thus shown the correctness of the method. The results are in Figure 6. The relative error of pressure distribution (measured in L2 -norm) after 40 iterations is 8.32 · 10−4 .

396

Jan Šimák and Jaroslav Pelant 0.0003

1.4 0.0002

0.6

0.0001

||e||

0.8 1.2

0

1

0

0.2

0.4 X 0.6

0.8

1

0

0.2

0.4 X 0.6

0.8

1

P

0.4 0.01

0.8

0.2 p-f

0

Y

0.005 0.6

0 0.4

0

0.2

0.4

X

0.6

0.8

1

-0.005

Figure 6. Example 1. Pressure distribution and resulting airfoil shape, error of the resulting airfoil measured as a norm kψresult − ψN ACA4412 ke , difference between the prescribed and resulting pressure distribution on the chord (values are normalized).

5.2.

Example 2

In this example a laminar flow with low Reynolds number is computed. The starting pressure distribution is computed on the NACA3210 airfoil with parameters Re = 1000, M∞ = 0.6, α∞ = 4.56◦ . At first, the problem is computed without any viscous correction and then the mentioned correction based on the Pohlhausen’s method is used. From the results is evident, that for very low Reynolds numbers the correction is necessary. The comparisons are in Figure. 7.

1.4

0.2 y

0.8

1.2

0

0.6

P

0.4

0

0.2

0.4 X 0.6

0.8

1

0

0.2

0.4 X 0.6

0.8

1

1 0.02 0.2

0.01 p-f

0

Y

0.8 0

-0.01 0.6

0

0.2

0.4

X

0.6

0.8

1

-0.02

Figure 7. Example 2. Pressure distributions and resulting airfoil shapes (solid - correction, dashed - without correction), comparison of the airfoils with the NACA3210 (dotted), difference between the prescribed and resulting pressure distribution along the chord (values are normalized).

Solution of an Airfoil Design Inverse Problem...

6.

397

Conclusion

A numerical method for a solution of an inverse problem of flow around an airfoil was described. The advantage of this method is the weak dependency on the model describing the flow. It is easy to modify the method by assuming a different model. The correction due to viscosity is necessary only with low Reynolds numbers. The presented method has still some drawbacks. These are the dependency of the angle of attack on the prescribed distribution and the applicability on the subsonic regimes only. The method can be improved in the future.

Acknowledgment This work was supported by the Grant MSM 0001066902 of the Ministry of Education, Youth and Sports of the Czech Republic.

References [1] Pelant, J. Inverse Problem for Two-dimensional Flow around a Profile, Report No. Z–69; VZLÚ, Prague, 1998. [2] Šimák J.; Pelant J. A contractive operator solution of an airfoil design inverse problem; PAMM Vol. 7, No. 1 (ICIAM07), pp. 2100023–2100024. [3] Šimák, J.; Pelant, J. Solution of an Airfoil Design Problem With Respect to a Given Pressure Distribution for a Viscous Laminar Flow, Report No. R–4186; VZLÚ, Prague, 2007. [4] Michlin, S. G. Integral Equations; Pergamon Press, Oxford, 1964. [5] Schlichting, H. Boundary-Layer Theory; McGraw-Hill, New York, 1979. [6] Wilcox, D. C. Turbulence Modeling for CFD; DCW Industries Inc., 1998; 2nd ed. [7] Kok, J. C. Resolving the Dependence on Freestream Values for the k − ω Turbulence Model; AIAA Journal 2000, vol. 38, No. 7, pp. 1292–1295. [8] Feistauer, M.; Felcman, J.; Straškraba, I. Mathematical and Computational Methods for Compressible Flow; Clarendon Press, Oxford, 2003. [9] Saad, Y. Iterative Methods for Sparse Linear Systems; SIAM, 2003; 2nd ed.

In: Fluid Mechanics and Pipe Flow Editors: D. Matos and C. Valerio, pp. 399-440

ISBN 978-1-60741-037-9 c 2009 Nova Science Publishers, Inc.

Chapter 13

S OME F REE B OUNDARY P ROBLEMS IN P OTENTIAL F LOW R EGIME U SING THE L EVEL S ET M ETHOD M. Garzon1 , N. Bobillo-Ares1 and J.A. Sethian2 1 Dept. de Matem´aticas, Univ. of Oviedo, Spain 2 Dept. of Mathematics, University of California, Berkeley, and Mathematics Department, Lawrence Berkeley National Laboratory.

Abstract Recent advances in the field of fluid mechanics with moving fronts are linked to the use of Level Set Methods, a versatile mathematical technique to follow free boundaries which undergo topological changes. A challenging class of problems in this context are those related to the solution of a partial differential equation posed on a moving domain, in which the boundary condition for the PDE solver has to be obtained from a partial differential equation defined on the front. This is the case of potential flow models with moving boundaries. Moreover, the fluid front may carry some material substance which diffuses in the front and is advected by the front velocity, as for example the use of surfactants to lower surface tension. We present a Level Set based methodology to embed this partial differential equations defined on the front in a complete Eulerian framework, fully avoiding the tracking of fluid particles and its known limitations. To show the advantages of this approach in the field of Fluid Mechanics we present in this work one particular application: the numerical approximation of a potential flow model to simulate the evolution and breaking of a solitary wave propagating over a slopping bottom and compare the level set based algorithm with previous front tracking models.

1.

Introduction

In this chapter we present a class of problems in the field of fluid mechanics that can be modeled using the potential flow assumptions, that is, inviscid and incompressible fluids moving under an irrotational velocity field. While these are significant assumptions, in the presence of moving boundaries, the resulting equations is a non linear partial differential equation, which adds considerable complexity to the computational problem. In the literature this model is often called the fully non linear potential flow model (FNPFM). Several

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M. Garzon, N. Bobillo-Ares and J.A. Sethian

interesting and rather complicated phenomenon are described using the FNPFM, as for example, Helle-Shaw flows, jet evolution and drop formation, sprays and electrosprays, wave propagation and breaking mechanisms, etc, see [21], [22], [30], [13]. Level Set Methods (LSM) [31], [33], [34] [37] are widely used in fluid mechanics, as well as other fields such as medical imaging, semiconductor manufacturing, ink jet printing, and seismology. The LSM is a powerful mathematical tool to move interfaces, once the velocity is known. In many physical problems, the interface velocity is obtained by solving the partial differential equations system used to model the fluid/fluids flow. The LSM is based on embedding the moving front as the zero level set of one higher dimensional function. By doing so, the problem can be formulated in a complete Eulerian description and topological changes of the free surface are automatically included. The equation for the motion of the level set function is an initial value hyperbolic partial differential equation, which can be easily approximated using upwind finite differences schemes. Recently, the LSM has been extended to formulate problems involving the transport and diffusion of material quantities, see [3]. In [3] model equations and algorithms are presented together with the corresponding test examples and convergence studies. This led to the realization that the nonlinear boundary conditions in potential flow problems could also be embedded using level set based methods. As a result, the FNPFM can also be formulated with an Eulerian description with the associated computational advantages. Two difficult problems that have been already approximated using this novel algorithm are wave breaking over sloping beaches [16], [17] and the Rayleigh taylor instability of a water jet [20]. Moreover, related to drop formation and wave breaking, it has been recently reported in the literature [46], [45] that the presence of surfactants on the fluid surface affects the flow patterns. The models described in this chapter are the groundwork for solving these complex problems. This chapter is organized as follows: in section 2. we have made an effort to obtain dynamic equations valid for any spatial coordinate system. To do so, we derive the equations using only objects defined in an intrinsic way (i.e., independent of any coordinate system). At the same time, in accordance with the level set perspective, we have avoided as much as possible, the “ material description” (Lagrangian coordinates). Geometric quantities are defined using the level sets and tensor fields in the space. In section 3. a brief description of the Levels Set Method is given using this intrinsic approach. Section 4. is devoted to describe two particular potential flow models, the first one related to drop formation in the presence of surfactants, which combines all the models derived in section 2.. The wave breaking problem is modeled in 2D, code development in 3D is underway. In section 5., we present the numerical approximation and algorithm for the wave breaking problem. Numerical results and accuracy tests are also presented in section 6.. Precise definitions of certain needed geometrical tools, throughout used in this chapter, are shown in Appendix III.

2.

Some Physical Models

Here, we discuss the derivations of fluid problems and their corresponding reformulation using the Level Set Method (LSM) techniques. The brief derivation of known physical laws is used also as a pretext to introduce some preliminary concepts and notation.

Some Free Boundary Problems in Potential Flow Regime...

2.1.

401

Kinematic Relationships

Reference configuration. The configuration of a continuous medium at certain time t is known when the position of each particle is specified. We name Ωt the space region occupied by the continuous medium at that time. Kinematics require the movement description of each particle. To this aim, we must: i. Label the particles. ii. Specify the movement of each particle. The first step is done considering the configuration at an arbitrary instant t0 (reference configuration). Particles are marked by the point P0 ∈ Ωt0 they occupy. Points in Ωt0 are good labels because they are in a 1 to 1 correspondence with the particles (“particles can not penetrate each other”). In what follows we will abbreviate the phrase “particle with label P0 ” by “particle P0 ”. Once all the particles are labeled, it is now possible to undertake the second step. Let P0 ∈ Ωt0 be a particle. Its position P at instant t is given by the function: P = R(P0 , t),

P ∈ Ωt , P0 ∈ Ωt0 .

(1)

According to the reference configuration definition, we have: R(P0 , t0 ) = P0 .

(2)

The mapping Rt , Rt (P0 ) := R(P0 , t) = P , must be invertible: P0 = Rt−1 (P ) ∈ Ωt0 , P ∈ Ωt .

(3)

Lagrangian/Eulerian descriptions. Any tensor field w may be described in two ways, using (1): w = w(P, t) = w(R(P0 , t), t) = w0 (P0 , t). (4) Function w(P, t) corresponds to the so called Eulerian description and w0 (P0 , t) corresponds to the Lagrangian description. As a consequence any tensorial field w admits two time partial derivatives. The “spatial” derivative, corresponding to the Eulerian description: d ∂t w := w(P, t + ǫ) , (5) dǫ ǫ=0

measures the variation rate with time of w from a fixed point in the space. The “convective” derivative, corresponding to the Lagrangian description: d Dt w := w0 (P0 , t + ǫ) , (6) dǫ ǫ=0

gives the variation rate of w following the particle P0 .

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Velocity field The velocity u = u(P0 , t) of the particle P0 is obtained using the convective derivative (“following the particle”) of the position P = R(P0 , t): u = Dt P,

P = R(P0 , t).

(7)

Obviously, u admits both descriptions: u = u(P, t) = u0 (P0 , t),

P = R(P0 , t).

(8)

Given an arbitrary tensor field w, its spatial and convective derivatives are related using the calculus chain rule and the definition (7): Dt w = ∂t w + ∂u w.

(9)

Here, ∂u w designates the directional derivative of w along u (see Appendix III). The acceleration of particle P0 is obtained by the convective derivative of the velocity field. Using (9), we have: Dt u = ∂t u + u · ∇u. (10) Transport of a vector due to a moving medium. A fluid particle is located at point1 P at time t. After a time ∆t, the same particle is at point R(P, t + ∆t). Clearly, the function R must verify that R(P, t + 0) = P . A nearby particle at same time t is located at P + ǫa, and at t + ∆t is at point R(P + ǫa, t + ∆t). We have again P + ǫa = R(P + ǫa, t + 0). The vector ǫa that connects both particles varies as they move. Denote by Dt ǫa its rate of change with time: 1 [(R(P + ǫa, t + ∆t) − R(P, t + ∆t)) − (R(P + ǫa, t) − R(P, t))] ∆t R(P + ǫa, t + ∆t) − R(P + ǫa, t) R(P, t + ∆t) − R(P, t) = lim − lim . ∆t→0 ∆t→0 ∆t ∆t

Dt ǫa =

lim

∆t→0

The first term of the right hand side of previous equation is, by definition, the particle velocity at P + ǫa, u(P + ǫa, t), and the second term the particle velocity at P , u(P, t). Thus we have: Dt ǫa = u(P + ǫa, t) − u(P, t). Letting ǫ → 0, we obtain the rate of change with time of an infinitesimal vector dragged by the medium: 1 u(P + ǫa, t) − u(P, t) d Dt a := lim Dt ǫa = lim = u(P + ǫa, t) = ∂a u. (11) ǫ→0 ǫ ǫ→0 ǫ dǫ ǫ=0 We denote ∂a the operator that performs the directional derivative along the vector a (see Appendix III). 1

For this calculation we use here the configuration at t as the reference configuration.

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Fluid volume change as it is transported by the velocity field. Let a, b and c be three small vectors with origin at point P . The volume of the parallelepiped spanned by vectors a, b, c is given by the trilinear alternate form δV = [a, b, c] = a · b × c. The rate of change of this volume, when particles located on its vertices move, is given by Dt δV , and thus we have Dt δV = Dt [a, b, c] = [Dt a, b, c] + [a, Dt b, c] + [a, b, Dt c]. Using now (11) we get Dt δV = [∂a u, b, c] + [a, ∂b u, c] + [a, b, ∂c u], which is also a trilinear alternate form. As in the tridimensional space all these forms are proportional, we can set Dt [a, b, c] = (div u)[a, b, c], (12) which gives us an intrinsic definition of the divergence of the field u. If the continuous medium is incompressible, the volume δV does not change, Dt δV = 0, and we arrive at the incompressibility condition div u = 0. (13)

2.2.

Dynamic Relationships

Conservation of mass. Denote by ρ = ̺(P, t) the volumetric mass density of the continuous medium at point P and at time t. The rate of change of the mass in a small volume δV dragged by the velocity field is, using definition (12), Dt (ρδV ) = (Dt ρ)δV + ρDt δV = (Dt ρ + ρ div u)δV. The mass conservation law is thus Dt ρ + ρ div u = 0.

(14)

Applying general formula (9) to ρ, we have Dt ρ = ∂t ρ + ∂u ρ. In the case of an homogeneous and incompressible medium with uniform initial density ρ0 , using equations (14) and (13), we have Dt ρ = 0 which gives ̺(P, t) = ρ0 . Conservation of the momentum associated with a small piece of continuous medium. From Newton’s second law applied to a fluid volume V we get the relation Z Z Z Dt u ρdV = g ρdV + τ (ds). (15) V

V

∂V

The term in left hand side of this equation is the rate of change with time of the momentum associated with volume V when dragged by the continuous medium. The first term in the right hand side corresponds to the volumetric forces inside V , generated by a vector field

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per unit mass g, usually the gravitational field. The second term represents the “contact” forces applied by the rest of the medium over the part in V . The Cauchy’s tensor τ is a linear operator field that is obtained from specific relationships which depend on the material, the so called constitutive relations. We are interested in inviscid fluids which verify the Pascal’s law: τ (ds) = −pds, where p is the pressure scalar field. Green’s formula, Z Z −p ds = −∇p dV, ∂V

V

shows that contact forces may be computed as a kind of volume forces with density −∇p. For a small volume δV dragged by the fluid, equation (15) can be written: Dt (u ρδV ) = (g ρ − ∇p)δV.

(16)

Due to the mass conservation law, Dt (ρδV ) = 0, equation (16) leads to the Euler equation: 1 Dt u = ∂t u + ∂u u = g − ∇p. ρ

(17)

If g is a uniform field it comes from the gradient of a potential function: g = −∇U (P ), U (P ) = −g · (P − O), where P − O is the position vector of the point P .

2.3.

Potential Flow

Assuming an irrotational flow regime, curl u = 0, there exists an scalar field φ such that u = ∇φ.

(18)

Outside of the fluid domain, and separated by a free boundary, there is a gas at pressure pa that is assumed to be constant. This means that, within the gas, the time needed to restore the equilibrium is very small compared with the time evolution of the fluid. Therefore, at the fluid free boundary, the boundary condition is just: p = pa .

(19)

Using the vectorial relationship ∇u2 /2 = ∂u u + u × (curl u), and relations (18) and (13) we have   1 2 p ∇ ∂t φ + u + + U = 0. 2 ρ Performing the first integration, 1 p ∂t φ + u2 + + U = C(t), 2 ρ

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where C(t) is an arbitrary function of time, which can be chosen in such a way that the previous relation can be written: 1 p − pa ∂t φ + u2 + + U = 0. 2 ρ Now using the obvious relation ∂t φ + u2 = ∂t φ + ∂u φ = Dt φ, we finally obtain p − pa 1 + U = 0. Dt φ − u2 + 2 ρ

2.4.

(20)

Advection

On the surface of a continuous medium with a known movement, a certain substance is distributed, which will be named as “charge”. This is adhered to the fluid particles and it is transported by them. In this way a set of particles will always carry the same amount of “charge”. This phenomenon is called advection. The continuous medium surface is implicitly described as the zero level set of a certain scalar function Ψ = ψ(P, t): Γt = {Q|ψ(Q, t) = 0}.

(21)

Vectors a tangent to the surface are characterized by the condition ∂a Ψ = a · ∇Ψ = 0; thus, the tangent vectorial plane at each point of the surface is given by the normal unit vector2 ∇Ψ n= . |∇Ψ|

The function ψ by itself does not specify the particle movement on the surface, just its shape. We need to add the information about how these particles move, e.g., specifying the velocity field on the surface Q ∈ Γt , u = u(Q, t). A small vector a connecting two nearby particles on the surface and dragged by them as they move, has a rate of change given by (11), Dt a = ∂a u.

(22)

Note that a is a tangent vector, n · a = 0. Surface areas. Using the normal vector to the surface, n, a 2–form to calculate surface areas can be constructed:3 ω(a, b) := [n, a, b] = n · a × b, 2

Ψ must increase from the interior to the exterior of the surface to get n outwards. The surface area definition is not made using the Gram determinant of two tangent vectors, because this procedure involves a particular parametrization of the surface. Instead, a 2-form is defined from the volume form in space (“the parallelogram area is the volume of a rectangular prism of unit height”). 3

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where ω(a, b) is the area spanned by tangent vectors a, b, and [n, a, b] the volume form in the 3D space. As the tangent vectors a, b are transported by the surface movement, the parallelogram area associated to them changes. The rate of change with time is easily obtained: Dt ω(a, b) = (Dt n) · a × b + n · (Dt a) × b + n · a × Dt b. First term of the right hand side of previous equation is zero since a × b is a normal vector and Dt n is tangent: indeed, as n2 = 1, we have Dt n2 = 2n · Dt n = 0. Using (22) we have Dt ω(a, b) = ∂a u · b × n + ∂b u · n × a. This expression is bilinear and alternate with respect the tangent vectors. It must be, at each point on the surface, proportional to the 2–form ω. We denote by Div u, “surface divergence”, the proportionality coefficient: ∂a u · b × n + ∂b u · n × a := (Div u) ω(a, b)

(23)

This definition of Div u does not depend upon the choice of tangent vectors a and b. In Appendix I, the expression for the surface divergence of an arbitrary vector field w using rectangular coordinates is shown. Advection law. Now, let be σ = σ(Q, t), Q ∈ Γt the “charge” surface density. The “charge” δq carried by a small parallelogram, spanned by two small tangent vectors (a, b), of area ω(a, b) is δq = σ ω(a, b). As the “charge” is conserved, the advection law is Dt δq = 0. Now, by definition (23), we have Dt (σω) = (Dt σ)ω + σDt ω = (Dt σ + σ Div u)ω. Hence we arrive to the intrinsic equation for the advection phenomena: Dt σ + σ Div u = 0.

(24)

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2.5.

407

Advection-Diffusion

Next, we are going to assume that the “charge” diffuses along particles on the surface according to the Fick’s law:4 j = −α∇σ, where α is the diffusion coefficient, j is the “charge” flux and ∇σ is the “charge” surface density gradient. As σ is only defined on the surface Γt , ∇σ is only defined for tangent vectors: ∇σ · a := ∂a σ, a tangent vector. On the surface Γt let us consider a surface region S, bounded by a curve ∂S. Let be ν the unit vector field tangent to Γt and orthogonal to the curve ∂S at each point. The “charge” that leaves the surface per unit time is the outward flux through the boundary ∂S: Z Z Z j · ν dl = − j · n × dl = n × j · dl. δS

∂S

∂S

Applying now Stokes’ theorem, we have Z Z n × j · dl = A ω(d1 P, d2 P ). ∂S

(25)

S

The 2-form of the surface integral is obtained using the intrinsic formula A ω(a, b) = ∂a (n × j · b) − ∂b (n × j · a).

(26)

We interpret A ω(a, b) as the “charge” per unit time that, by diffusion, leaves the small parallelogram spanned by the tangent vectors (a, b). Now it is straightforward to set the condition for the advection-diffusion mechanism     “charge” rate of change “charge” that leaves within the tangent the parallelogram =− ,     parallelogram (a, b) by diffusion

that is

Dt (σω(a, b)) = −A ω(a, b). In Appendix II the following expression for A is obtained: A = Div j − (Div n) j · n. Hence, using (24) we arrive at the general equation for the advection-diffusion model:  Dt σ + σ Div u = − Div j + (Div n) j · n j = −α ∇σ or Dt σ + σ Div u = α Div ∇σ − α(Div n) ∇σ · n (27) 4

Fick’s diffusion law applies when the “charge” particles move randomly without any preferential direction (Brownian movement).

408

M. Garzon, N. Bobillo-Ares and J.A. Sethian The Cartesian expressions5 for Div u, Div ∇σ and Div n are:  1  2 (∇Ψ) ∂ ∂ Ψ − ∂ Ψ∂ Ψ∂ ∂ Ψ , i i j i j i |∇Ψ|3   (∇Ψ)2 ∂i ∂i σ − ∂i Ψ∂j Ψ∂i ∂j σ ,

Div n = (δij − ni nj )∂j ni = Div ∇σ = Div u =

1 |∇Ψ|2  1  (∇Ψ)2 ∂i ui − ∂i Ψ∂j Ψ∂i uj . 2 |∇Ψ|

(28) (29) (30)

Expanding the implicit summands, we obtain the following expressions for the 3D space (i, j = 1, 2, 3): 1  (∂1 Ψ)2 (∂22 Ψ + ∂32 Ψ) + (∂2 Ψ)2 (∂12 Ψ + ∂32 Ψ)+ |∇Ψ|3 +(∂3 Ψ)2 (∂12 Ψ + ∂22 Ψ) − 2∂1 Ψ∂2 Ψ∂1 ∂2 Ψ −  − 2∂1 Ψ∂3 Ψ∂1 ∂3 Ψ − 2∂2 Ψ∂3 Ψ∂2 ∂3 Ψ , 1  Div ∇σ = (∂1 Ψ)2 (∂22 σ + ∂32 σ) + (∂2 Ψ)2 (∂12 σ + ∂32 σ)+ |∇Ψ|2 +(∂3 Ψ)2 (∂12 σ + ∂22 σ) − 2∂1 Ψ∂2 Ψ∂1 ∂2 σ −  − 2∂1 Ψ∂3 Ψ∂1 ∂3 σ − 2∂2 Ψ∂3 Ψ∂2 ∂3 σ , 1  Div u = (∂1 Ψ)2 (∂2 u2 + ∂3 u3 ) + (∂2 Ψ)2 (∂1 u1 + ∂3 u3 )+ |∇Ψ|2 +(∂3 Ψ)2 (∂1 u1 + ∂2 u2 ) − ∂1 Ψ∂2 Ψ(∂1 u2 + ∂2 u1 ) −  − ∂1 Ψ∂3 Ψ(∂1 u3 + ∂3 u1 ) − ∂2 Ψ∂3 Ψ(∂2 u3 + ∂3 u2 ) . Div n =

(31)

(32)

(33)

To obtain the formulas for the plane we assume axial symmetry in the direction 3: ∂3 Ψ = 0,

∂32 Ψ = 0,

∂3 σ = 0,

u3 = 0, ∂3 ui = 0,

|∇Ψ|2 = (∂1 Ψ)2 + (∂2 Ψ)2 . Inserting these values in (31), (32) and (33) we get: Div n = Div ∇σ = Div u =

5

 1  (∂1 Ψ)2 (∂22 Ψ) + (∂2 Ψ)2 (∂12 Ψ) − 2∂1 Ψ∂2 Ψ∂1 ∂2 Ψ , 3 |∇Ψ|  1  (∂1 Ψ)2 (∂22 σ) + (∂2 Ψ)2 (∂12 σ) − 2∂1 Ψ∂2 Ψ∂1 ∂2 σ , 2 |∇Ψ| 1  (∂1 Ψ)2 (∂2 u2 ) + (∂2 Ψ)2 (∂1 u1 )− |∇Ψ|2  − 2∂1 Ψ∂2 Ψ(∂1 u2 + ∂2 u1 ) .

(34) (35)

(36)

In the following expressions we use only subscripts because orthonormal bases coincides with their corresponding reciprocal ones. Then, the position of the indices becomes irrelevant.

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3.

409

The Level Set Method

The Level Set method is a mathematical tool developed by Osher and Sethian [31] to follow interfaces which move with a given velocity field. The key idea is to view the moving front as the zero level set of one higher dimensional function called the level set function. One main advantage of this approach comes when the moving boundary changes topology, and thus a simple connected domain splits into separated disconnected domains. Let be Γt the set of points lying in the surface boundary at time t. This surface is defined through the zero level set of the scalar field Ψ = ψ(P, t): Γt = {Q|ψ(Q, t) = 0}.

(37)

To identify the fluid particles, the configuration at t0 (reference configuration) is used: Γt0 = {Q0 |ψ(Q0 , t0 ) = 0}.

(38)

The particle movement is specified through the function Q = R(Q0 , t),

(39)

which gives the position Q ∈ Γt of the fluid particle Q0 ∈ Γt0 . The particle Q0 velocity is calculated using the convective derivative Dt (“following the particle”): d u = Dt Q = R(Q0 , t + ǫ) . (40) dǫ ǫ=0 According to definition (37), we have ψ(R(Q0 , t), t) = 0. Deriving with respect to time and applying the chain rule, we obtain ∂t Ψ + u · ∇Ψ = 0.

(41)

which has to be completed with the value of the level set function at time t = 0, usually set to be the signed distance function to the initial front, Ψ(P, 0) = s(P )d(P ), being d(P ) the distance from the point P to the surface at the initial configuration Γ0 , s(P ) = −1 if P ∈ Ω0 and s(P ) = +1 if P ∈ / Ω0 . Now, if we take a fixed 3D domain ΩD that contains the free surface for all times, we can define the initial value problem for the level set function Ψ posed on ΩD : ∂t Ψ + u · ∇Ψ = 0 in ΩD

Ψ(P, 0) = s(P )d(P ) in ΩD

(42) (43)

A graphical interpretation of the level set function evolution is depicted in figure 1 Equation (42) moves all the level set of Ψ, not just the zero level set, and in many physical applications the front velocity is just defined for points lying on the free boundary. Therefore for this equation to be valid on the whole domain we have to extend the velocity u off the front. There exist several extension procedures which will be briefly commented below.

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t

y (P, 0)

y (P, t)

G0

Gt

Figure 1. Evolution of the level set function.

3.1.

Extension of Functions Defined on the Front

Let us consider a classical result from functional analysis: suppose a domain Ω bounded by a closed surface ∂Ω. If for k ≥ 1 the surface ∂Ω ∈ C k , then for all functions F (x) ∈ ¯ such that Fext |∂Ω = F (x). C k (∂Ω) there exists a function Fext (x) ∈ C k (Ω) In practice, there are several ways to extend any magnitude F defined on the front onto ΩD . As shown in [10] for the numerical stability of the level set equation it is convenient to preserve Ψ as a signed distance function, which is characterized by the property |∇Ψ| = 1. One way is to perform reinitializations of the level set function at chosen times. If this is done periodically, it will smooth the level set function. However, done too often, especially using poor reinitialization techniques, spurious mass loss/gain will occur. Thus, it is important to perform reinitialization both sparingly and accurately. For the potential flow problems presented in this chapter we follow the strategy introduced in [2]. The idea is to extrapolate F given at the front along its gradient. Mathematically the extended variable Fext is the solution of ∇Fext · ∇Ψ = 0. (44) It is straightforward to show that this choice maintains the signed distance function for the level sets of Ψ for all times. For the numerical approximation we proceed as follows: ˆn given a level set function Ψ at time n, namely Ψn , one first obtains a distance function Ψ around the zero level set. Simultaneous with this construction, the extended quantity Fext is obtained satisfying Eq. (44). For a complete explanation of this extension method see [2].

4.

Examples of Potential Flow Models with Moving Boundaries

In this section the governing equations of two interesting physical problems will be formulated using a level set framework. First, drop formation is a complex 3D phenomena driven mainly by capillary forces, which can be modeled using the potential flow assump-

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tions. It is well-known that the presence of surface surfactants lowers the surface tension affecting drops shape. Secondly, propagation and wave breaking over sloping beaches can also be modeled with the potential flow equations, which are valid until the jet of the wave impinges against the flat water surface. In this case we can formulate the equations in 2D taking a vertical section of the beach, which facilitates the algorithm and code development. The wave numerical simulations will be presented in section 6..

4.1.

Governing Equations for Surface Tension Driven Flows with Material Advection-Diffusion

Let Ωt be the 3D closed fluid domain surrounded by air and Γt the free surface boundary at time t. Suppose that initially a certain amount of surfactants, which are assumed to be insoluble in water, are uniformly distributed on the surface (see Figure 2).

Figure 2. A fluid volume with surface surfactants. For an incompressible and inviscid fluid, the governing equations are the Euler equations (17). On the free boundary the following partial differential equations apply: • The advection-diffusion equation for the surfactant is (27): Dt σ + σDiv u = α(Div∇σ − κ ∇σ · n) on Γt , where σ is the surface density of the surfactant, u is the free boundary velocity, α is the surface diffusion coefficient, κ = Div n = R11 + R12 and R1 , R2 the principal radii of curvature of Γt at each point. • Continuity of the stress tensor between water and air leads to the balance of the surface tension forces, p = pa +γ( R11 + R12 ), where γ is the surface tension coefficient

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M. Garzon, N. Bobillo-Ares and J.A. Sethian that may depend on the surfactant concentration σ. Thus Eq. (20) becomes 1 γ ∂t φ + (∇φ · ∇φ) + κ + U = 0 on Γt . 2 ρ

• Finally, if Q = R(Q0 , t) is the position of a fluid particle Q0 on the free surface, the definition (40) states Dt Q = u(Q, t), Q ∈ Γt . (45) The complete model equations in 3D are therefore, u = ∇φ in Ωt

∆φ = 0 in Ωt

Dt Q = u on Γt γ 1 Dt φ = −U + (∇φ · ∇φ) − κ on Γt 2 ρ Dt σ = −σDiv u + α(Div∇σ − κ ∇σ · n) on Γt .

(46) (47) (48) (49) (50)

This is the Lagrangian-Eulerian formulation of the model equations. Classical methods to approximate this set of equations are the so-called “front tracking methods”, in which a fixed number of markers are chosen initially and the trajectories of this markers are followed as time evolves. This method suffers difficulties when the free boundary changes topology: these problems are avoided by a level set formulation. Level Set Framework Equation (48) can be directly formulated as the level set Eq. (41). For the velocity field u(Q, t), the trajectory of a fluid particle at initial position Q0 is given by the solution of Dt Q = u(R(Q0 , t), t), R(Q0 , 0) = Q0 .

(51)

Next, let ΩD be a fixed 3D domain that contains the free surface for all times and let G(P, t) and S(P, t) be two functions defined on ΩD such that for every Q ∈ Γt G(Q, t) = φ(Q, t) ,

(52)

S(Q, t) = σ(Q, t) .

(53)

It is important to remark here that G(P, t) and S(P, t) are auxiliary functions defined in ΩD that can be chosen arbitrarily, the only restriction is that they equal φ(Q, t) and σ(Q, t) on Γt respectively. Figure 3 gives an interpretation of this property for a moving curve in 2D. Applying the chain rule in both identities (52) and (53) we obtain γ 1 ∂t G + u · ∇G = −U + (∇φ · ∇φ) − κ, 2 ρ ∂t S + u · ∇S = −σDiv u + α(Div∇σ − κ ∇σ · n).

(54) (55)

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413

t G(P, 0)

G(P, t)

f(Q , t)

f(Q , 0)

G0

Q

Gt

Q

Figure 3. Extension of the velocity potential off the front. which holds on Γt . Note that u and the right hand side of Eq. (54) and Eq. (55) are only defined on Γt , and thus, in order to solve these equations over the fixed domain ΩD , these variables must be extended off the front. This strategy has been discussed in Section 3.. Naming 1 γ f = −U + (∇φ · ∇φ) − κ, 2 ρ h = −σDiv u + α(Div∇σ − κ ∇σ · n), the system of equations, written in a complete Eulerian framework, is u = ∇φ in Ωt

(56)

Ψt + uext · ∇Ψ = 0 in ΩD .

(58)

∆φ = 0 in Ωt

Gt + uext · ∇G = fext in ΩD St + uext · ∇S = hext in ΩD

(57) (59) (60)

Here the subscript “ext” denotes the extension of f , h and u onto ΩD .

4.2.

Governing Equations for the Wave Breaking Problem

We now derive our coupled level set/extension potential equations for breaking waves in two dimensions for which a numerical approximation will be also presented. Let Ωt be the 2D fluid domain in the vertical plane (x, z) at time t, with z the vertical upward direction (and z = 0 at the undisturbed free surface), and Γt the free boundary at time t (see Figure 4). We assume also an inviscid and incompressible fluid, and capillary forces are disregarded on the free boundary curve. The model equations in the Lagrangian-Eulerian formulation are therefore:

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6

z Γt

Q Γ1

6

h

Ωt

?

Γb Figure 4. The domain.

   Γb

u = ∇φ in Ωt

∆φ = 0 in Ωt

Dt Q = u on Γt 1 Dt φ = −gz + (∇φ · ∇φ) on Γt 2 φn = 0 on Γb ∪ Γ1 ∪ Γ2 ,

Γ2

x

-

(61) (62) (63) (64) (65)

Let Ω1 be a fixed 2D domain that contains Γt for all times. Following the same embedding procedure for the potential function as in previous section, we obtain the complete 2D Eulerian formulation: u = ∇φ in Ωt

∆φ = 0 in Ωt

Ψt + uext · ∇Ψ = 0 in Ω1 .

Gt + uext · ∇G = fext in Ω1 φn = 0 on Γb ∪ Γ1 ∪ Γ2

(66) (67) (68) (69) (70)

being here f = 12 (∇φ · ∇φ) − gz and fext the extension of f onto Ω1 .

5.

Numerical Approximations and Algorithms

In this section, we provide overviews of the numerical schemes used to approximate the wave model equations. The integral formulation of Eq. (66) is approximated using a liner boundary element method (BEM), which will provide the velocity of the front node representation. More detailed discussions of the various components may be found in the cited references.

5.1.

Initialization

The initial front position Γ0 and initial velocity potential φ(Q, 0), Q ∈ Γ0 , are needed to solve equations (68) and (69) respectively. Given an initial solitary wave amplitude (H0 )

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and the physical length of the domain (L), Tanaka’s method gives a way of calculating these quantities. Here, we briefly discuss the theoretical basis of this method. Assuming constant depth, the flow field can be reduced to steady state by using a coordinate system that moves horizontally with speed equal to the wave celerity c. The stream function ψ(x, z) is also harmonic and takes constant values at the bottom and at the free surface of the domain. ¿From the definition of stream function and velocity potential we have φx = ψy , φy = −ψx . Under sensible assumptions about the smoothness of φ and ψ, these are just the CauchyRiemann equations which are satisfied by the real and imaginary parts of the function W = φ+iψ, which is called the complex potential and it is a an analytical function of the complex variable Z = x + iz in the domain occupied by the fluid. By interchanging the role of the variables Z and W , we can take φ and ψ as independent variables, since W = φ + iψ provides a one to one correspondence between the physical and complex potential planes. With this transformation, the fluid region is mapped into the strip 0 < ψ < 1, −∞ < φ < ∞ in the W plane with ψ = 1 on the free surface, ψ = 0 on the bottom and φ = 0 at the wave crest. Denote by u, v the horizontal and vertical components of the velocity u, q = |u| and θ the angle between the velocity and the x axis. The complex velocity is defined by dW = φx + iφy = u − iv = qeiθ dZ and it is also analytic in the flow domain. Therefore, the quantity ω = ln(

dW ) = ln q − iθ, dZ

is an analytic function of W , so τ = ln q must be harmonic in the strip 0 < ψ < 1, −∞ < φ < ∞. The Bernoulli condition at the free surface and the bottom condition can be expressed in terms of q and θ as: dq 3 3 = − 2 sin θ on ψ = 1 dφ F θ = 0 on ψ = 0,

(71) (72)

where F is the Froude number defined by F = √cgh . The problem of finding a solitary wave solution can thus be transformed into the problem of finding a complex function ω that is analytic with respect to W within the region of the unit strip 0 < ψ < 1, decays at infinity, and satisfies the boundary conditions (71) and (72). Tanaka’s method provides a way to solve the previous outlined equations in terms of the new variables τ , θ and a full description of the algorithm can be found in [40].

5.2.

The Level Set and Velocity Potential Updating

We use the standard Narrow Band Level Method, introduced by Adalsteinsson and Sethian [2], which limits computation to a thin band around the front of interest. Following the algorithm discussed in [31], we use second order in space upwind differences to

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approximate the gradient in the level set equation, and a first order time scheme to update the solution. For boundary conditions, homogeneous flux boundary conditions are usually chosen, which are implemented by creating an extra layer of ghost cells around the domain whose values are simply direct copies of the Ψ values along the actual boundary. The level set function is built from the initial position of the front by computing the signed distance function. This is done using the Fast Marching Method [36], which is a Dijkstra-like finite difference method for computing the solution to the Eikonal equation in O(N log N ), where N is the total number of points in the computational domain. The velocity and the velocity potential are both initially defined only on the interface. In order to create values throughout the narrow band, which are required to update the fixed grid Eulerian partial differential equations, we use the extension methodology developed by Adalsteinsson and Sethian in [2] to construct appropriate extensions. The idea of building extension velocities was first introduced in [26]; in that approach, the extension velocity at any grid point in the domain was taken as equal to the velocity on the closest point on the front itself. As shown in [7], this is equivalent to solving the equations ∇u · ∇Ψ = 0, ∇v · ∇Ψ = 0 for the velocity components, and in that paper, the equation was solved using a finite difference iteration. In [2], Adalsteinsson and Sethian present a technique for computing this extension velocity using the very efficient Fast Marching methodology. Finally, in [3], this approach was developed to build extension values for arbitrary material quantities whose evolution affects the underlying interface dynamics. The potential equation (69) is a convection equation with a strong non-linear source term, and homogeneous Newmann boundary conditions are imposed on the boundary of Ω1 . To update in time this equation, note that it is similar to (68) except that it has a nonlinear source term, and therefore we use similar schemes. For example a straightforward first order scheme is −x +x n n n Gn+1 i,j = Gi,j − ∆t(max(ui,j , 0)Di,j + min(ui,j , 0)Di,j + −z +z n n n max(vi,j , 0)Di,j + min(vi,j , 0)Di,j ) + ∆tfi,j

where Gni,j − Gni−1,j ∆x n n G i+1,j − Gi,j +x +x n Di,j = Di,j Gi,j = ∆x are the backward and forward finite approximation for the derivative in the x direction (we −z +z have the same expressions for for Di,j and Di,j .) Note that for simplicity we have written u, v, G, f instead of uext , vext , Gext , fext , and we describe a first order explicit scheme with a centered source term. Initial values of G0i,j are obtained by extending φ(x, z, 0)|Γ0 as previously discussed. However, at any time step n it is always possible to perform a new extension of φn (x, z, n∆t) to obtain a better value of Gni,j . A key issue is how one obtains fext in the grid points of Ω1 . There are several ways of doing so. Here we calculate f = 12 (∇φ · ∇φ) − gz on free surface nodes, and use these values together with the condition ∇f · ∇Ψ = 0 to obtain fext . This algorithm for extending quantities defined on the front off the front works very well for the velocity −x −x n Di,j = Di,j Gi,j =

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field in the case of equation (68), because it maintains the signed distance function for the level sets of Ψ. However, regarding equation (69) for this particular wave problem, and due to the high variations of f along the front together with its topological structure when overturning, the previous method creates strong G and f gradients in Ω1 . This fact limits the grid spacing in Ω1 and the time step needed to maintain accuracy (see the section on numerical experiments).

5.3.

The Boundary Integral Equation and the BEM Approximation

A first order boundary element method is used to approximate equation (66). Boundary integral equations are well suited to moving boundary problems for two principal reasons. First, determining the surface velocity generally requires computing function derivatives on this boundary, which are accurately evaluated within this formulation. Second, remeshing the moving boundary is clearly simpler than remeshing the entire domain. The Laplace equation for the velocity potential (67) is solved by approximating the corresponding boundary integral equation. Boundary conditions are given by (70) and, on the free boundary, at each time step, by the updated potential velocity given by equation (69). The approximation of the integral equation is done by the BEM, which calculates the potential and the potential gradient on the free surface, that is, its velocity u. The boundary integral equation for the potential φ(P ), in a domain Ω(t) having boundary Σ = ∂Ω(t), can be written as  Z  ∂φ ∂G (73) P(P ) = φ(P ) + lim φ(Q) (PI , Q) − G(PI , Q) (Q) dQ = 0 , PI →P Σ ∂n ∂n where n = n(Q) denotes the unit outward normal on the boundary surface and {PI } are interior points converging to the boundary point P . The Green’s function or fundamental solution (in two dimensions) is G(P, Q) = −

1 log(r) . 4π

(74)

The integral equation is usually written with the ∂G ∂n singular integral evaluated as a Cauchy Principal Value (CPV), resulting in a ‘interior angle’ coefficient c(P ) multiplying the leading φ(P ) term [5, 6]. The reason for employing the seemingly more complicated limit process will become clear in the discussion of gradient evaluation. The exterior limit equation  Z  ∂φ ∂G lim φ(Q) (PE , Q) − G(PE , Q) (Q) dQ = 0 . (75) PE →P Σ ∂n ∂n yields precisely the same equation: the jump in the CPV integral as one crosses the boundary accounts for the ‘free term’ difference. In this work, a Galerkin (weak form) approximation of Eq. (73) has been employed, and the boundary and boundary functions are interpolated using the simplest approximation, linear shape functions. Thus, the equations that are solved are of the form Z ψk (P )P(P ) dP = 0 , (76) Σ

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where the weight functions ψˆk (P ) are comprised of all shape functions which are nonzero at a particular node Pk [5]. The calculations reported herein employed the simplest approximation, linear shape functions. These approximations reduce the integral equation to a finite system of linear equations, and invoking the boundary conditions allows the solution of the unknown values of potential and flux on the boundary. Details concerning the limit evaluation of the singular integrals can be found in [14]. As noted above, for the wave problem, and moving boundary problems in general, knowledge of the normal flux is not sufficient, the complete gradient of φ is required to compute the surface velocity. The remainder of this section will present the algorithm for computing this gradient. ¿From Eq. (73) a gradient component can be expressed as ∂φ(P ) = lim PI →P ∂Ek

Z  Σ

 ∂G ∂2G ∂φ (PI , Q) dQ . (PI , Q) (Q) − φ(Q) ∂Ek ∂n ∂Ek ∂n

(77)

Once the boundary value problem has been solved, all quantities on the right hand side are known: a direct evaluation of nodal derivatives would therefore be easy were it not for wellknown difficulties with the hypersingular (two derivatives of the Green’s function) integral [28, 29, 27]. As described in [15], a Galerkin approximation of this equation, Z ∂φ(P ) dP = (78) ψˆk (P ) ∂Ek Σ  Z  Z ∂G ∂φ ∂2G ψˆk (P ) (PI , Q) (Q) − φ(Q) (PI , Q) dQ dP lim PI →P Σ ∂n ∂Ek ∂n Σ ∂Ek allows a treatment of the hypersingular integral using standard continuous elements. Interpolating ∂φ(P )/∂Ek as a linear combination of the shape functions results in a simple system of linear equations for nodal values of the derivative everywhere on Σ; the coefficient matrix is obtained by simply integrating products of two shape functions. However, the complete boundary integrations required to compute the right hand side are quite expensive. The computational cost of this procedure can be significantly reduced by exploiting the exterior limit equation, Eq. (75). It appears to be useless for computing tangential derivatives, since, lacking the free term, the corresponding derivative equation takes the form  Z  ∂G ∂φ ∂2G 0 = lim (PE , Q) (Q) − φ(Q) (PE , Q) dQ , (79) PE →P Σ ∂Ek ∂n ∂Ek ∂n and the derivatives obviously do not appear. However, subtracting this equation from Eq. (77) yields (with shorthand notation)  Z   ∂φ(P ) ∂2G ∂G ∂φ = lim − lim (Q) − φ(Q) dQ . (80) PI →P PE →P ∂Ek ∂Ek ∂n Σ ∂Ek ∂n The advantage of this formulation is that now only the terms that are discontinuous crossing boundary contribute to the integral. In particular, all non-singular integrations, by far the most time consuming, drop out. The calculation of the right hand side in Eq. (80) reduces

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to a few ‘local’ singular integrations, and as these integrations are carried out partially analytically, this produces an accurate algorithm. Further details about the evaluation of Eq. (80) can be found in [15].

5.4.

Regridding of the Free Surface

In a level set formulation the position of the front is only known implicitly through the node values of the level set function Ψ. In order to extract the front, it is possible to construct first order and second order approximations of the interface using local data of Ψ on the mesh (see [9] for example.) Here we use a first order linear approximation of the free surface, which yields a polygonal interface formed by unevenly distributed nodes, which we call LS nodes. As a result of this extraction technique, occasionally one gets front nodes which are very close together, and this can cause difficulties and instabilities for boundary element calculations. To overcome this problem, and also to achieve more front resolution when needed, we employed a front node regridding technique. An initialization point on the front is selected according to a particular criterion, such as maximum value of height, velocity modulus, or front curvature. This point divides the front in two halves and new nodes are chosen so that, lying in the same polygon, they are redistributed by arclength according to the formula: si+1 − si = d0 (1 + si (f0 − 1)) where si denotes the arclength distance from node i to the initialization point (i = 0) and d0 , f0 are user selected parameters. These regridded nodes on the front are used to create the input file for the BEM calculations and are denoted by BEM nodes.

5.5.

The Algorithm

To initialize the position of the front and the velocity potential on the front, we use Tanaka’s method for computing numerical exact solitary waves. The basic algorithm can be summarized as follows: 1. Compute initial front position and velocity potential φ(Q, 0) on Γ0 . 2. Extend φ(Q, 0) onto the grid points of Ω1 to initialize G. 3. Generate Ωt and solve (67), using the Boundary Element Method. This yields the velocity u and source term f on the front nodes. 4. Extend u and f off the front onto Ω1 . 5. Update G using (69) in Ω1 . 6. Move the front with velocity u using (68) in Ω1 7. Interpolate (bi-cubic interpolation) G from grid points of Ω1 to the front nodes to obtain new boundary conditions for (67). Go back to step 3 and repeat forward in time.

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A more detailed algorithm including regridding is: Initialization: Given Γ0 = Γ0 , φ0 = φ(Q, 0) 1. Calculate Ψ0 and LS nodes. 2. Extend φ0 to obtain G0 . 3. Redistribute LS nodes to obtain BEM nodes. 4. Calculate u0 at BEM nodes. 5. Find u0 and f 0 at LS nodes and extend onto Ω1 . Steps: Given Ψn , φn , un 1. Calculate Ψn+1 and LS nodes. 2. Calculate Gn+1 in Ω1 grid points. 3. Redistribute LS nodes to obtain BEM nodes. 4. Interpolate G on BEM nodes to find φn+1 . 5. Calculate un+1 at BEM nodes. 6. Find un+1 and f n+1 at LS nodes and extend onto Ω1 . Go to step 1 and repeat. 7. If reinitialization (a) Take LS nodes and reinitialize Ψn+1 . (b) Take BEM nodes and extend φn+1 .

5.6.

Numerical Accuracy

The model equations imply that the wave mass and its total energy should be conserved as the wave evolves in time. One way to check the numerical accuracy of the discretized equations is to compute these quantities each time step. The wave mass above z = 0 is given by Z Z Z m(t) =

dΩ =

Ωt

znz ds =

∂Ωt

znz ds

Γt

and the total energy is E(t) = Ep (t) + Ek (t), where Ep (t), Ek (t) denotes the potential and kinetic wave energy respectively. They can be calculated using the expressions Z Z 1 1 Ep (t) = ρg zdΩ = ρg z 2 nz ds, 2 2 Ωt Γt which is the potential energy with respect z = 0, and Z Z Z 1 ∂φ 1 ∂φ 1 Ek (t) = ρ ∇φ · ∇φdΩ = ρ φ ds = ρ φ ds, 2 Ωt 2 ∂Ωt ∂n 2 Γt ∂n

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where the divergence theorem has been applied to the three formulas and we have used the ∂φ fact ∂n = 0 on Γb , Γ1 , Γ2 for the kinetic energy formula. These integrals are approximated by a composite trapezoidal rule, using the values of the quantities on the free boundary BEM nodes. Note that LS nodes could have been used for m(t) and Ep (t) approximations but we also used BEM nodes for simplicity. The components of the normal vector to the free surface are computed using the level set embedding function to obtain surface geometrical variables. A common procedure to study the accuracy and convergence properties of the discretized equations with respect the mesh sizes and the time step is by means of an analytical solution. A solitary wave propagating over a constant depth is a traveling wave that moves in the x direction with speed equal to the celerity of the wave (c). The velocity potential and the velocity on the front as functions of x are also translated with the same speed c. Therefore, in this case, by calculating initial wave data with Tanaka’s method and translating it, we are able to compute the L2 norms of the errors for the various magnitudes. For the case of a solitary wave shoaling over a sloping bottom, the accuracy can only be checked looking at the mass and energy conservation properties and comparing breaking wave characteristic obtained here with those reported elsewhere, for example in [22].

6.

Numerical Results

The system of equations to be discretized is a non-linear system of strongly coupled partial differential equations. First order in time and second order in space schemes are used for equation (68); first order in time and in space schemes are used for equation (69); and a first order BEM solver is used for the velocity updating. To study the convergence properties of this method and its capability to predict wave breaking characteristics, the numerical results corresponding to the following physical settings are presented: A solitary wave propagating over a constant depth and the shoaling and breaking of a solitary wave propagating over various sloping bottoms.

6.1.

Constant Depth Test

In order to tune the discretization parameters and see how they affect numerical accuracy we performed a series of numerical tests with a solitary wave of H0 = 0.5 m (wave height at the crest) propagating over a constant depth of 1 m. The wave crest is initially located at x = 6.5 m and the domain has L = 15 m of length. In what follows, the units are taken as meters and seconds for length and time, respectively. Let Ω1 = [0, 15] × [−0.3, 1] be the fictitious domain that contains the free boundary for all t ∈ [0, 0.5], ∆x = ∆z the grid size and ∆t the time step. To discretize ∂Ωt , in order to generate the input BEM file, a variable mesh size is used: ∆l = 0.1 for Γ1 and Γ2 , ∆l = 0.2 for Γb , and the regridding parameters for Γt are chosen to be d0 = 0.005, f0 = 10. This gives 193 BEM nodes on the moving front and 98 nodes on the fixed boundaries. The mesh size ∆x = ∆z for Ω1 should be chosen in order to achieve accurate interpolated values of front position and potential on the front. For the time step selection, a first limitation is the CFL condition. While this condition is enough for the stability of the numerical approximation of equations (68) and (69), the accuracy in the numerical solution of

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equation (69) requires a smaller time step. This is due to the fact that G and the source term f , for this particular wave problem, develops high gradients in Ω1 . Therefore we present the results for the following test cases: • (a) ∆x = 0.1, ∆t = 0.01. • (b) ∆x = 0.1, ∆t = 0.001. • (c) ∆x = 0.01, ∆t = 0.001. • (d) ∆x = 0.01, ∆t = 0.0001. For a given solitary wave parameters (H0 and length L in the x direction) Tanaka’s method gives us the initial wave magnitudes, front location, velocity potential, velocity components at front points and wave celerity c. At any time t, let (xex , zex ), φex , uex , vex be the values of these variables obtained by translating initial values a distance ct along the x direction and spline interpolating in LS nodes. Denote by (xc , zc ), φc , uc , vc the computed values at LS nodes, L2 (z) =k zc − zex kL2 (Γt ) , L2 (φ) =k φc − φex kL2 (Γt ) , L2 (u) =k uc − uex kL2 (Γt ) and L2 (v) =k vc − vex kL2 (Γt ) the L2 norm of the errors. Table 1 shows these errors at the final time t = 0.5 for the various test cases. Table 1. Values of the L2 error norms at t = 0.5 Test (a) (b) (c) (d)

L2 (z) 0.007239 0.009762 0.001476 0.001699

L2 (φ) 0.095254 0.021451 0.011363 0.00424601

L2 (u) 0.025147 0.039635 0.0099744 0.0106674

L2 (v) 0.025856 0.035685 0.009356 0.010188

Figures 5 and 6 show L2 (z), L2 (φ), L2 (u), L2 (v) versus time for cases (c) and (d) respectively. As observed from these results, the L2 error norm in front location and velocity components decreases with mesh size (∆x) but not with the time step. Only the velocity potential gains accuracy when ∆t is reduced accordingly to the above mentioned facts. Regarding wave mass and energy conservation, at each time step we calculate m(t) and E(t) as explained in 5.6. Figures 7 and 8 show the values of |m(t)−m(0)| and |E(t)−E(0)| versus time and same behavior of these quantities with respect discretization parameters is observed. Next, to see if we gain accuracy in the velocity calculations by increasing the number of BEM nodes, we take ∆l = 0.05 on Γ1 and Γ2 , ∆l = 0.1 on Γb , and d0 = 0.001, f0 = 5 on Γt . This gives 1720 BEM nodes on the moving front and 196 nodes for the fixed boundaries. For this discretization of the bEM boundary we run two more cases: • (e) ∆x = 0.01, ∆t = 0.001. • (f) ∆x = 0.01, ∆t = 0.0001.

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Values of the L2 error norms for case (e) and (f) are almost identical to those obtained for case (c) and (d) respectively, which means that accuracy in velocity is not gained by increasing the number of bEM nodes. However, as is shown in Figure 7, |m(t) − m(0)| has decreased by almost an order of magnitude due to the accuracy in front position and the improvement in the integral approximation to calculate m(t). Figure 9 shows for case (e) the absolute errors in Ep (t), Ek (t), E(t) versus time and, in agreement with the previous discussion, the kinetic energy is much less accurate than the potential energy. From these numerical experiments we conclude that the proposed algorithm converges, but we do not achieve exactly first order convergence with respect discretization parameters. This is due to the strong interdependence of the equations. Note that f depends nonlinearly on u and linearly on z and that the boundary condition imposed on Γt for the bEM solver builds up numerical and round off error as we step forward in time; we note that the level set approach is stable and robust with respect to these small sawtooth instabilities resulting from velocity calculations on very closely spaced nodes, and the use of filtering or smoothing was not required. Case (c) discretization parameters give sufficient accuracy and we show wave profiles, velocity potential and velocity components for various times in Figures 10, 11 and 12 respectively.

6.2.

Sloping Bottom Test

A solitary wave propagating over a sloping bed changes its shape gradually, slightly increasing maximum height and front steepness, till a point where a vertical front tangent is reached. This is usually called the breaking point BP=(tbp , xbp , zbp ), where xbp represents the x coordinate, zbp the height at xbp and tbp the time of occurrence. Beyond the BP, the wave tip develops, with velocities much bigger than the wave celerity, causing the wave overturning and the subsequent falling of the jet toward the flat water surface. Denote this endpoint as EP=(tep , xep , zep ). Total wave mass and total energy should be, theoretically, conserved until EP. However beyond the BP a lost in potential energy and the corresponding gain in kinetic energy is expected, due to the large velocities on the wave jet. Wave breaking characteristics change, mainly according to initial wave amplitude (H0 ) and bottom topography. To study how our numerical method predicts wave breaking we run the following test cases: • (a) H0 = 0.6, L = 25, slope=1 : 22, xc = 6.05, xs = 6 • (b) H0 = 0.6, L = 18, slope=1 : 15, xc = 5.55, xs = 5.4 and compare the results obtained here for case (b) with those reported in [21]. Here xc denotes the x coordinate at the crest for the initial wave and xs the x coordinate where the bottom slope starts. A series of numerical experiments have been made, and optimal discretization parameters found are: ∆x = 0.01, ∆t = 0.0001 and d0 = 0.005, f0 = 10 (approximately 193 BEM nodes) for all cases. Front regridding has been made according to maximum height before the BP and according to maximum velocity modulus beyond BP. Beyond the BP, and due to the complex topography of the wave front, reinitialization of Ψ and new φ(x, z, t) extension has been performed every 1000 time steps.

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Table 2 shows the breaking characteristics for the test cases. Grilli et all reported in [21] for test (b) values of tbp = 2.41, xbp = 15.64 and zbp = 0.67. The discrepancies can be attributed to the slightly different position of the initial wave (xc = 5.5) and the higher order approximations used in their Lagrangian-Eulerian formulation. Table 2. Breaking characteristics Test (a) (b)

tbp 2.76 2.34

xbp 17.39 15.20

zbp 0.674 0.662

tep 3.36 2.90

xep 20.2 17.8

In Figure 13 we show m(t) versus time for case (a) and (b) and Figures 14 and 15 show the evolution of Ep , Ek and E with time for cases (a) and (b) respectively. Maximum absolute error in wave mass is 0.01 before BP and 0.02 beyond BP and maximum absolute error in total wave energy is 0.02 near the BP. Although this errors could be improved by increasing the number of BEM nodes on the free boundary (as shown in the constant depth cases), it would require considerably more CPU time per run due to the high cost of the BEM solver. Regarding the evolution of the potential and kinetic energy of the wave we observe the expected behavior beyond the BP. Figure 16 shows wave shape for case (a) at t = 0, 1, 2, 2.76, 2.94, 2.14, 3.34 and Figure 17 shows wave shape for case (b) at t = 0, 1, 2, 2.34, 2, 48, 2.68, 2.90. In Figures 18 and 19 we show in more detail the wave profiles from the BP to the EP for cases (a) and (b) respectively. Finally in Figure 20 the front BEM nodes for case (a) and time 3.34 are shown. ¿From these numerical experiments we conclude that the numerical method presented here is capable of reproducing wave shoaling and breaking till the touchdown of the wave jet. Considering that we use only first order approximations of the model equations, a piecewise linear approximation of the free boundary, and a first order linear BEM, the results are quite accurate. The absolute errors in mass and energy seem to be higher than those reported in [21]. This is not surprising due to the fact that in [21] a higher order BEM is used (both higher order elements to define local interpolation between nodes and spline approximation of the free boundary geometry) and time integration for the free boundary conditions is at least second order in time.

6.3.

Sinusoidal Bottom Test

To see how wave shape and breaking characteristics change with bottom topography, we consider two more tests, this time with a sinusoidal shape bottom: • (c) H0 = 0.6, L = 25, xc = 6.05, Ab = 0.5, hmin = 0.5 • (d) H0 = 0.6, L = 25, xc = 6.05, Ab = 0.8, hmin = 0.2 where Ab denote the amplitude of the sinusoidal function that represents the bottom and hmin the minimum depth.

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As can be seen in Table 3, the breaking characteristics are considerably different for these simulations, and, in particular, case (c) behaves like a spilling breaker rather than the plunging breaker of case (a) and (b). In Figures 21 and 22 we show wave profiles for various Table 3. Breaking characteristics Test (c) (d)

tbp 1.6 1.0

xbp 12.5 10.5

zbp 0.71 0.55

tep 1.96 1.38

xep 14.1 13.6

times corresponding to case (c) and (d) respectively. Measurements for the mass and total energy conservation behave similar to previous cases. In Figure 23 we show the evolution of wave mass for cases (c) and (d). Finally, Figures 24 and 25 show the evolution of Ep , Ek and E corresponding to cases (c) and (d) respectively. These results show that, in response to the bottom topography, wave height follows a sinusoidal curve, as does the potential and kinetic wave energies, with an amplitude related to the sinusoidal bottom amplitude.

7.

Conclusion

To summarize, in this chapter we have derived some physical models related to moving interfaces in an intrinsic way, that is, independent of any coordinate system. Based on these models a complete Eulerian description of potential flow problems for a single fluid, with or without advection-diffusion of material quantities on the front has been stablished. For the case of two-dimensional breaking waves over sloping beaches a coupled level setboundary integral algorithm has been developed. Numerical results and convergence tests show that even first order level set schemes produce quantitative results in a robust and efficient fashion.

Acknowledgements All work was performed at the Lawrence Berkeley National Laboratory, and the Mathematics Dept. of the University of California at Berkeley. First author was partially supported by the Spanish Project MTM2007-65088. Second author was supported by Spanish CGL2006-06401-BTE and CGL2008-03786-BTE projects, both funded by Ministerio de Educaci´on y Ciencia and Fondo Europeo de Desarrollo Regional (FEDER). We want to thank E. Su´arez D´ıaz for his help with some of the figures.

Appendix I. The Surface Divergence in Rectilinear Coordinates Given a vector field w, we want to find an expression for the surface divergence Div w using rectilinear coordinates. We start from the definition: ω(a, b) Div w := ∂a w · b × n + ∂b w · n × a,

(81)

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being a and b arbitrary tangent vectors to the surface, ω(a, b) the area of the corresponding parallelogram and n the unit vector normal to the surface at the same point. To abbreviate the computations we use indices for basis vectors:6 a1 = a,

a2 = b,

a3 = n.

(82)

The reciprocal basis, designed by ai (i = 1, 2, 3), ai · aj = δji ,

i, j = 1, 2, 3,

(83)

is calculated by the formulae: a1 =

a3 × a1 a1 × a2 a2 × a3 , a2 = , a3 = = n. [a1 , a2 , a3 ] [a1 , a2 , a3 ] [a1 , a2 , a3 ]

(84)

According to definition (81), we have for Div w: a2 × a3 a3 × a1 · ∂ a1 w + · ∂ a2 w [a1 , a2 , a3 ] [a1 , a2 , a3 ] = a1 · ∂a1 w + a2 · ∂a2 w = aα · ∂aα w.

Div w =

(85)

In the last expression and below we have used the summation convection: when in a monomial expression we have two repeated indices it must be interpreted as a summation, from 1 to 2 for greek indices and from 1 to 3 for latin indices. Notice that the basis ai is in general different in each surface point. We want now to express Div w using the components and coordinates in a fixed basis (global) ei and the reciprocal one ej , defined7 by the nine equations ej · ei = δij . We set: ai = hji ej , ai = fki ek , w = wj ej ; (fki hkj = δji ). (86) Substituting this expressions in the last term of (85), we have: Div w = fkα ek · ∂hiα ei wj ej = fjα hiα ∂i wj .

(87)

Considering that nj = a3 · ej = fj3 and ni = a3 · ei = hi3 , the coefficient in the previous result becomes: fjα hiα = fjk hik − fj3 hi3 = δji − nj ni . (88) Substituting this result in equation (87), the searched expression is obtained:
Div w = δji − nj ni ∂i wj .

(89)

Notice that, as we have anticipated, the final result does not depend on the selected tangent vectors a and b. 6

Latin indices i, j,... go through the values 1, 2 y 3 and greek indices α, β,... go through 1 y 2. The vectors ai (i = 1, 2, 3) must accomplish the condition [a1 , a2 , a3 ] > 0. 7 When ei is an orthonormal basis (Cartesian coordinates) then it coincides with the corresponding reciprocal basis: ei = ei .

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427

Appendix II. The Differential Operator A We are going to show that the differential operator A=

1 (∂a (n × j · b) − ∂b (n × j · a)) , ω(a, b)

(90)

appearing in (25), may be written using surface divergences of j and n. To do that, we perform in the previous definition the indicated derivatives, A=

1 (b × n · ∂a j + n × a · ∂b j) + ω(a, b) 1 + (b × ∂a n + ∂b n × a) · j. ω(a, b)

According to definition (81), the first term is Div j. Let be: A = Div j + B.

(91)

In order to identify de second term B, we select the basis ai , following the specified notation in (82). As ∂aα n (α = 1, 2) are tangent vectors to the surface, we can set ∂aα n = Nαβ aβ ,

α = 1, 2.

(92)

Also, for the two terms in B we obtain: b × ∂a n ω(a, b) ∂b n × a j· ω(a, b)

j·

= j·

a2 × N11 a1 = −N11 j · n, ω(a1 , a2 )

= −N22 j · n.

Therefore: B = −Nαα j · n.

(93)

On the other hand, as Nαβ = aβ · ∂aα n, making α = β, summing and using the result (85), we have: Nαα = aα · ∂aα n = Div n. (94) Finally, substituting this result in (93) we arrive to the searched expression: A = Div j − (Div n)j · n.

(95)

Appendix III. Useful Definitions Points and vectors. In our euclidean space we can define two useful operations. Given a point P and a displacement vector a, we define P + a as the point that results translating point P by vector a. Also, given two points A and B, we define A − B as a vector c, so that: A − B := c ⇔ B + c = A. (96)

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M. Garzon, N. Bobillo-Ares and J.A. Sethian

Directional derivative. Given a tensor field w = w(P, t), function of the position P (a point of our Euclidean space) and the time t, we define the directional derivative along the vector a as d w(P + ǫa, t) . (97) ∂a w := dǫ ǫ=0 The result ∂a w is a tensor of the same rank that w. Differentiability of the field w implies that ∂a w is a linear function of the vector a. When w is a vector field, we use, as customary, the special notation: a · ∇w := ∂a w.

(98)

In this case, ∂a w is a linear operator acting on the vector a.

Gradient of a scalar field. Let us consider a scalar field φ = ϕ(P, t). As ∂a φ is a real valued linear function on the argument a, we can define the gradient vector field, ∇φ, a · ∇φ := ∂a φ.

(99)

L2 Errors. (H0=0.5, depth=1) 0.012 front potential u v

0.01

L2 error

0.008

0.006

0.004

0.002

0

0

0.05

0.1

0.15

0.2

0.25 time

0.3

0.35

0.4

0.45

0.5

Figure 5. L2 (z), L2 (φ), L2 (u), L2 (v) vs time for case (c).

Some Free Boundary Problems in Potential Flow Regime... L2 Errors. (H0=0.5, depth=1) 0.012 front potential u v

0.01

L2 error

0.008

0.006

0.004

0.002

0

0

0.05

0.1

0.15

0.2

0.25 time

0.3

0.35

0.4

0.45

0.5

Figure 6. L2 (z), L2 (φ), L2 (u), L2 (v) vs time for case (d).

−3

3

Absolute error in wave mass. (H0=0.5, depth=1)

x 10

(a) (b) (c) (d) (e)

2.5

abs(m(t)− m(0))

2

1.5

1

0.5

0

0

0.05

0.1

0.15

0.2

0.25 time

0.3

0.35

0.4

Figure 7. Absolute error in wave mass.

0.45

0.5

429

430

M. Garzon, N. Bobillo-Ares and J.A. Sethian −3

7

Absolute error in total wave energy. (H0=0.5, depth=1)

x 10

(a) (b) (c) (d)

6

abs(E(t)− E(0))

5

4

3

2

1

0

0

0.05

0.1

0.15

0.2

0.25 time

0.3

0.35

0.4

0.45

0.5

Figure 8. Absolute error in wave total energy. −4

8

Wave Energy. (H0=0.5 depth=1)

x 10

Ep Ek E

abs(E(t)−E(0)

6

4

2

0

0

0.05

0.1

0.15

0.2

0.25 time

0.3

0.35

0.4

0.45

0.5

Figure 9. Absolute error in potential, kinetic and total energy. Case (e). wave shape at several times. (H0=0.5, depth=1) 1

z

0.5

0

−0.5

−1

0

5

10

15

x

Figure 10. Front location at t = 0, 0.1, 0.2, 0.3, 0.4, 0.5. Case (c).

Some Free Boundary Problems in Potential Flow Regime... velocity potential. (H0=0.5, depth=1) 3

2

potential

1

0

−1

−2

−3

0

0.1

0.2

0.3

0.4

0.5 s

0.6

0.7

0.8

0.9

1

Figure 11. Velocity potential at t = 0, 0.25, 0.5. Case (c). u velocity. (H0=0.5, depth=1) 2 1.5

u

1 0.5 0 −0.5

0

0.1

0.2

0.3

0.4

0.5 s

0.6

0.7

0.8

0.9

1

0.7

0.8

0.9

1

v velocity. (H0=0.5, depth=1) 0.6 0.4 0.2 v

0 −0.2 −0.4 −0.6 −0.8

0

0.1

0.2

0.3

0.4

0.5 s

0.6

Figure 12. Velocity components at t = 0, 0.25, 0.5. Case (c). Wave mass 2 slope 1:22 slope 1:15

1.98

mass

1.96 1.94 1.92 1.9 1.88 0

0.5

1

1.5

2

2.5

3

time

Figure 13. Wave mass vs time. Case (a) and (b).

431

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M. Garzon, N. Bobillo-Ares and J.A. Sethian Wave Energy. (H0=0.6 slope=1:22) 0.8 Ep Ek E

0.75

0.7

0.65

E

0.6

0.55

0.5

0.45

0.4

0.35

0.3

0

0.5

1

1.5

2

2.5

3

3.5

time

Figure 14. Wave energy. Case (a).

Wave Energy. (H0=0.6 slope=1:15) 0.9 Ep Ek E 0.8

0.7

E

0.6

0.5

0.4

0.3

0.2

0

0.5

1

1.5 time

2

Figure 15. Wave energy. Case (b).

2.5

3

Some Free Boundary Problems in Potential Flow Regime...

433

H0=0.6 slope1:22 4

z

2

0

−2

−4

0

5

10

15

20

25

x

Figure 16. Wave shape at various times. Case (a) H0=0.6 slope1:15 4 3 2

z

1 0 −1 −2 −3 −4

0

2

4

6

8

10

12

14

16

18

x

Figure 17. Wave shape at various times. Case (a). H0=0.6 slope1:22 2

1.5

1

z

0.5

0

−0.5

−1

−1.5

−2 16

16.5

17

17.5

18

18.5 x

19

19.5

20

20.5

Figure 18. Wave shape at various times. Case (a).

21

434

M. Garzon, N. Bobillo-Ares and J.A. Sethian H0=0.6 slope1:15 2

1.5

1

z

0.5

0

−0.5

−1

−1.5

−2 14

14.5

15

15.5

16 x

16.5

17

17.5

18

Figure 19. Wave shape at various times. Case (b). H0=0.6 slope1:22 2

1.5

1

z

0.5

0

−0.5

−1

−1.5

−2 18

18.5

19

19.5

20 x

20.5

21

21.5

Figure 20. Front BEM nodes at t=3.34. Case (a).

22

Some Free Boundary Problems in Potential Flow Regime...

435

H0=0.6 , sinusoidal bottom 4 3 2

z

1 0 −1 −2 −3 −4

2

4

6

8

10

12

14

16

18

20

x

Figure 21. Wave shape at various times. Case (c).

H0=0.6 , sinusoidal bottom 4 3 2 1 z

0 −1 −2 −3 −4

2

4

6

8

10

12

14

16

18

20

x

Figure 22. Wave shape at various times. Case (d).

Wave mass 2

1.98 (c) (d)

mass

1.96

1.94

1.92

1.9

1.88 0

0.2

0.4

0.6

0.8

1 time

1.2

1.4

1.6

1.8

Figure 23. Wave mass vs time. Case (c) and (d).

2

436

M. Garzon, N. Bobillo-Ares and J.A. Sethian Wave Energy 0.9 Ep Ek E

0.8

0.7

E

0.6

0.5

0.4

0.3

0.2

0

0.2

0.4

0.6

0.8

1 time

1.2

1.4

1.6

1.8

2

Figure 24. Wave energy. Case (c). Wave Energy 0.75 Ep Ek E

0.7

0.65

0.6

E

0.55

0.5

0.45

0.4

0.35

0.3

0.25

0

0.2

0.4

0.6

0.8

1

time

Figure 25. Wave energy. Case (d).

1.2

1.4

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References [1] Adalsteinsson, D., and Sethian, J.A., A Fast Level Set Method for Propagating Interfaces, J. Comp. Phys., 118, 2, pp. 269–277, 1995. [2] Adalsteinsson, D., and Sethian, J.A., The Fast Construction of Extension Velocities in Level Set Methods, 148, J. Comp. Phys., 1999, pp. 2-22. [3] Adalsteinsson, D., and Sethian, J.A., Transport and Diffusion of Material Quantities on Propagating Interfaces via Level Set Methods, J. Comp. Phys, 185, 1, pp. 271-288, 2002. [4] Beale, J. Thomas, Hou, Thomas Y., Lowengrub, John, Convergence of a boundary integral method for water waves, SIAM J. Numer. Anal., 33, 5, pp.1797-1843, 1996. [5] Bonnet M. Boundary Integral Equation Methods for Solids and Fluids, Wiley and Sons, England, 1995. [6] Brebbia C. A.,Telles J. C. F. and Wrobel L. C., Boundary Element Techniques, SV, BNY, 1984. [7] Chen, S., Merriman, B., Osher, S., Smereka P., A simple level set method for solving Stefan problems. J. Comput. Phys.135, pp. 8–29, 1997. [8] Chang, Y.C., Hou, T.Y., Merriman, B., Osher, S.J., A level set formulation of Eulerian interface capturing methods for incompressible fluid flows, J. Comput. Phys., 124, pp. 449–64, 1996. [9] Chopp, D.L., Some Improvements of the Fast Marching Method. SIAM J.Sci Comput. 23, pp. 230–244, 2001. [10] Chopp, D.L., Computing minimal surfaces via level set curvature flow. J. Comput. Phys. 106, pp. 77–91, 1993. [11] Chorin, A.J., Numerical solution of the Navier-Stokes equations. Math. Comput. 22, pp. 745–62, 1968. [12] Christensen, E.D., Deigaard, R., Large Eddy Simulation of Breaking Waves. Coastal Engineering 42 (2001) 53-86. [13] Eggers, Jen, Nonlinear dynamics and breakup of free-surface flows, Rev. Mod. Phys., 69, 3, pp.865-929, 1997. [14] Gray L. J., Evaluation of singular and hypersingular Galerkin boundary integrals: direct limits and symbolic computation, Singular Integrals in the Boundary Element Method, V. Sladek and J. Sladek, Computational Mechanics Publishers, chapter 2, pp 33-84, 1998. [15] Gray L. J., Phan A. -V and Kaplan T., Boundary Integral Evaluation of Surface Derivatives, SIAM J. Sci. Comput.,in press, 2004.

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[16] M. Garzon, D. Adalsteinsson, L. J Gray and J. A Sethian, A coupled level setboundary integral method for moving boundaries simulations, Interfaces and Free Boundaries, 7, 277-302, 2005. [17] M. Garzon, J.A. Sethian, Wave breaking over sloping beaches using a coupled boundary integral-level set method, International Series of Numerical methods, 154, 189198, 2006. [18] L. J. Gray, M. Garzon, On a Hermite boundary integral approximation, Computers and Structures, 83, 889-894 (2005). [19] Integral analysis for the axisymmetric laplace equation, L. J. Gray, M. Garzon, V. Mantic, and E. Graciani, International Journal For Numerical Methods in Engineering, 66, 2014-2034, 2005. [20] M. Garzon, J.A. Sethian, L. Gray, Numerical solution of non-viscous pinch off using a coupled level set boundary integral method, Proceedings in Applied Mathematics and Mechanics, 2007. [21] Grilli, S.T., Guyenne, P., and Dias, F., A Fully Non-linear Model for Three Dimensional Overturning Waves Over an Arbitrary Bottom. International Journal for Numerical Methods in Fluids 35:829-867pp (2001). [22] Grilli, S.T., Svendsen, I.A., and Subramanya, R., Breaking Criterion and Characteristics For Solitary Waves On Slopes. Journal Of Waterway, Port, Coastal, and Ocean Engineering (June 1997). [23] Grilli, S.T., Modeling Of Non-linear Wave Motion In Shallow Water. In Computational Methods for Free and Moving Boundary Problems in Heat and Fluid Flow. Wrobel LC, Brebbia CA (eds.). Computational Mechanics Publishers: Southampton, 1995:91-122. [24] Grilli, S.T., Subramanya, R., Numerical Modeling of Wave Breaking Induced by Fixed or Moving Boundaries. Computational Mechanics 1996; 17:374-391. [25] Lin, P., Chang, K., and Liu, P.L., Runup and Rundown of Solitary Waves on Sloping Beaches. Journal Of Waterway, Port, Coastal, and Ocean Engineering (Sep/Oct 1999). [26] Malladi R., Sethian J.A., Vemuri B.C., Shape Modeling with Front Propagation: A Level Set Approach IEEE Trans. on Pattern Analysis and Machine Intelligence, 17, 2, pp. 158–175, 1995. [27] Martin P. A., Rizzo F. J. and Cruse T. A., Smoothness-relaxation strategies for singular and hypersingular integral equations, Int. J. Numer. Meth. Engrg., Vol 42, pp 885-906, 1998. [28] Martin P. A, and Rizzo F. J., On boundary integral equations for crack problems, Proc. R. Soc. Lond., Vol A421, pp 341-355, 1989.

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[29] Martin P. A, and Rizzo F. J., Hypersingular integrals: how smooth must the density be?, Int. J. Numer. Meth. Engrg., Vol 39, pp 687-704, 1996. [30] Notz, Patrick K., and Basaran, Osman A., Dynamics of drop formation in an electric field, J. Colloid Interface Sci., 213, p.. 218-237, 1999. [31] Osher, S., and Sethian, J.A., Fronts Propagating with Curvature-Dependent Speed: Algorithms Based on Hamilton–Jacobi Formulations, Journal of Computational Physics, 79, pp. 12–49, 1988. [32] Peregrine, D.H., Breaking Waves on Beaches. Annual Review in Fluid Mechanics 1983; 15:149-178. [33] Sethian, J.A., An Analysis of Flame Propagation, Ph.D. Dissertation, Dept. of Mathematics, University of California, Berkeley, CA, 1982. [34] Sethian, J.A., Curvature and the Evolution of Fronts, Comm. in Math. Phys., 101, pp. 487–499, 1985. [35] Sethian, J.A., Numerical Methods for Propagating Fronts, in Variational Methods for Free Surface Interfaces, Eds.. P. Concus and R. Finn, Springer-Verlag, NY, 1987. [36] Sethian, J.A., A Fast Marching Level Set Method for Monotonically Advancing Fronts, Proc. Nat. Acad. Sci., 93, 4, pp.1591–1595, 1996. [37] Sethian, J.A., Level Set Methods and Fast Marching Methods. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press (1999). [38] Sethian, J.A., and Smereka, P., Level Set Methods for Fluid Interfaces, Annual Review of Fluid Mechanics, 35, pp.341-372, 2003. [39] Sussman, M., Smereka, P., Osher, S.J., A level set approach to computing solutions to incompressible two-phase flow , J. Comput. Phys., 114, pp. 146–159, 1994. [40] Tanaka, M., The stability of solitary waves , Phys. Fluids, 29 (3), pp. 650–655, 1986. [41] Yan, Fang, Farouk, Baktier, and Ko, Frank, Numerical modeling of an electrostatically driven liquid meniscus in the cone-jet mode, Aerosol Science, 34, pp. 99-116, 2003, [42] Yu, J-D., Sakai, S., and Sethian, J.A., A Coupled Level Set Projection Method Applied to Ink Jet Simulation, in press, Interfaces and Free Boundaries, 2003. [43] Zelt. J.A., The Run-up of Non-breaking and Breaking Solitary Waves. Coastal Engineering, 15 (1991) 205-246. [44] Zhu, J., Sethian, J.A., Projection Methods Coupled to Level Set Interface Techniques, J. Comp. Phys., 102, pp. 128–138, 1992. [45] Xinan Liu, J. H. Duncan The effects of surfactants on spilling breaking waves Nature International weekly journal of science, 421, pp. 520-523, 2003.

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[46] Maik J. Geerkena, Rob G.H. Lammertink and Matthias Wessling Interfacial aspects of water drop formation at micro-engineered orifices Journal of Colloid Interface Science, 312, pp. 460-469, 2007.

In: Fluid Mechanics and Pipe Flow Editors: D. Matos and C. Valerio, pp. 441-454

ISBN 978-1-60741-037-9 c 2009 Nova Science Publishers, Inc.

Chapter 14

A N EW A PPROACH FOR P OLYDISPERSED T URBULENT T WO -P HASE F LOWS : T HE C ASE OF D EPOSITION IN P IPE -F LOWS S. Chibbaro∗ Dept. of Mechanical Engineering, University of “Tor Vergata”, via del politecnico 1 00133, Rome, Italy

Abstract This article is basically a review of recent works that is aimed at putting forward the main ideas behind a new theoretical approach to turbulent wall-bounded flows, notably pipe-flows, in which complex physics is involved, such as combustion or particle transport. Pipe flows are ubiquitous in industrial applications and have been studied intensively in the last century, both from a theoretical and experimental point of view. The result of such a strong effort is a good comprehension of the physics underlying the dynamics of these flows and the proposition of reliable models for simple turbulent pipe-flows at large Reynolds number Nevertheless, the advancing of engineering frontiers casts a growing demand for models suitable for the study of more complex flows. For instance, the motion and the interaction with walls of aerosol particles, the presence of roughness on walls and the possibility of drag reduction through the introduction of few complex molecules in the flow constitute some interesting examples of pipe-flows with some new complex physics involved. A good modeling approach to these flows is yet to come and, in the commentary, we support the idea that a new angle of attack is needed with respect to present methods. In this article, we analyze which are the fundamental features of complex two-phase flows and we point out that there are two key elements to be taken into account by a suitable theoretical model: 1) These flows exhibit chaotic patterns; 2) The presence of instantaneous coherent structures radically change the flow properties. From a methodological point of view, two main theoretical approaches have been considered so far: the solution of equations based on first principles (for example, the Navier-Stokes equations for a single phase fluid) or Eulerian models based on constitutive relations. In analogy with the language of statistical physics, we consider the former as a microscopic approach and the later as a macroscopic one. We discuss why we consider both approaches unsatisfying with regard to the description of general complex turbulent flows, like two-phase ∗

E-mail address: xxxx

442

S. Chibbaro flows. Hence, we argue that a significant breakthrough can be obtained by choosing a new approach based upon two main ideas: 1) The approach has to be mesoscopic (in the middle between the microscopic and the macroscopic) and statistical; 2) Some geometrical features of turbulence have to be introduced in the statistical model. We present the main characteristics of a stochastic model which respects the conditions expressed by the point 1) and a method to fulfill the point 2). These arguments are backed up with some recent numerical results of deposition onto walls in turbulent pipe-flows. Finally, some perspectives are also given.

1.

Introduction

Turbulent flows are ubiquitous in nature. The boundary layer in the earth’s atmosphere, rivers and canals, the photosphere of the sun, the interstellar medium, most combustion processes, the flow of natural gas and oil in pipelines are just a few examples of turbulent motions. Most turbulent flows are bounded (at least in part) by one or more solid surfaces. Examples include internal flows such as the flow through pipes and ducts; external flows such as the flow around aircraft. Since Reynolds’ experiment in 1883, pipe flow has played an important role in the development of our understanding of turbulent flows. In particular, it is quite simple to measure the drop in pressure over a length of fully developed turbulent pipe flow and hence to determine the skin-friction coefficient. Laminar Poiseuille flow occurs when a fluid in a straight channel, or pipe, is driven by a constant upstream pressure gradient, yielding a symmetric parabolic stream-wise velocity profile. In turbulent states, the mean stream-wise velocity profile remains symmetric, but is flattened in the center because of the increase in velocity fluctuations. A lot of research has been carried out for turbulent wall flows, [1, 3, 4, 5] and, in particular, in the case of pipe flow, experiments for measuring the mean-velocity profile have been successfully performed at moderate to high Reynolds numbers [6, 7]. Thus, we can say that the basic physics of these flows is well-understood, even though the fundamental understanding of how these profiles change as functions of the Reynolds number and of the dissipative mechanisms have yet to be assessed. However, this is not at all such cases where some complex phenomena are added like combustion [35], particle dispersion (two-phase flows) [42], presence of wall-roughness or of complex molecules which cause a drag reduction [8, 9, 10]. In these cases physical understanding remains limited and appears to be scarce compared to that obtained for simpler turbulent flows. The purpose of the present work is to analyse a suitable modeling approach, which has simplified rules compared to the real phenomena, and which is used to simulate the overall and collective behaviour of a complex system. The question is therefore whether the model contains the right physics (thus the need to understand clearly the important phenomena) and then how to reach an acceptable compromise between the simplicity of the model versus its physical realism (thus the need of an appropriate formalism). In this commentary, which tries to propose an overview of recent modeling developments [42, 44, 47], we analyze the case of two-phase flows and, in particular, particle deposition onto walls.

A New Approach for Polydispersed Turbulent Two-Phase Flows

2.

443

A Sketch of the Physics of Turbulent Two-Phase Wall Flows

In this section, we would like to outline the present knowledge about near-wall physics and, more specifically, the physical mechanisms which can be considered as important for particle deposition. Generally speaking, two elements constitute the most significant signatures of those turbulent flows: i) the flow is chaotic ii) quasi-coherent structures are present. It is important to underline that the first point gives information on the statistical nature of these flows, while the second concerns the geometrical one. For the first point, Navier-Stokes equations, which describe accurately a turbulent flow [11], represent a dynamical system with a very large number of degree of freedom [13, 14, 15, 16, 17]. Turbulence is characterized by non-Gaussian velocity fluctuations on a wide range of scales and frequencies. The number of degrees of freedom is of the order of Re9/4 for a Reynolds number Re that is typically 105 ÷ 108 . The existence of such of wide range of scales, and of the acute sensitivity of turbulent flows to small perturbations in initial and boundary conditions (which are never known absolutely) explain the search of a statistical description of such flows. Turbulent structures are identified by flow visualization, by conditional sampling techniques, or by other eduction methodologies; but they are difficult to define precisely. The idea is that they are regions of space and time (significantly larger than the smallest flow or turbulence scales) within which the flow field has a characteristic coherent pattern. Kline and Robinson [18] and Robinson [19] provide a useful categorization of quasi-coherent structures in channel flow and boundary layers. The eight categories identified are the following: 1. Low-speed streaks in the region (0 < y + < 10). 2. Ejections of low-speed fluid outward from the wall. 3. Sweeps of high-speed fluid toward the wall. 4. Vortical structures of several proposed forms. 5. Strong internal shear layers in the wall zone (y + < 80). 6. Near-wall pockets, observed as areas clear of marked fluid in certain types of flow visualizations. 7. Backs: surfaces (of scale S) across which the stream wise velocity changes abruptly. 8. Large-scale motions in the outer layers. The deposition of very small particles is mainly led by diffusion process and Brownian motion. At the same time, it is largely accepted that particle transfer in the wall region and also deposition onto walls are processes which are dominated by near-wall turbulent coherent structures [20, 21, 22]. As seen above, there are many different quasi-coherent structures in wall-flows. Among all these structures, four appear to be determinant for particle deposition: the low-speed streaks in the region 0 < y + ≤ 10, the Ejections of low-speed fluid outward from the wall, the Sweeps of high-speed fluid towards the wall

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and Vortical structures of various size and intensity [23, 24, 25, 26, 27, 28, 29]. Moreover, several experiments have investigated particle transfer in near-wall region and they have found that particles tend to remain trapped along the streaks when in viscous-layer. This migration phenomenon results in overall mean fluxes and it has been referred to as Turbophoresis [30]. This name may be, however, rather misleading since it suggests the existence of some hidden force or mechanism which bring particles towards walls. Instead, a net migration happens because the transfer of particles towards the wall is more efficient than the transfer, due to entrainment, of particles from the wall into outer flow. Furthermore, this net particle flux to the wall is related to quasi-coherent phenomena, which are instantaneous realizations of the Reynolds stress [29, 31]. Sweeps events have been found to be strongly correlated with particle trapping in lowspeed streaks, while ejections events are correlated with particle entrainment in the outer flow [27, 21]. Ejections topology has been found more efficient than sweeps events and this explains particle accumulation in near-wall region. Furthermore, the importance of these mechanisms for particle deposition depends on particle inertia. In particular, light particles follow much more closely sweeps and ejections and their motion towards the wall appears to be very well-correlated with turbulent structures. On the contrary, heavy particles are not so well-correlated with turbulent structures and their motion is less influenced by them in the near-wall region. Given the physical picture, one first important conclusion can be drawn: a model which aims to be appropriate for the description of particle deposition in turbulent two-phase flows has to be (i) a statistical model, to be capable to describe the solutions of the basic equations as being random or stochastic processes. It is necessary to cope with a reduced or contracted description of continuous fields. (ii) To take into account the effect of the most important geometrical structures, we believe that a model which does not consider this step is likely to fail a proper and general description of particle deposition in turbulent flows.

3.

Modeling

Given the framework put forward in the previous section, that is a statistical one which can include some geometrical information, it is necessary to determine which kind of approach can be included in this framework. Let us introduce briefly the basic equations for turbulent two-phase flows [33]. For heavy particles where ρp ≫ ρf , the drag and gravity forces are the dominant forces and the particle equation of motion is reduced to dUp 1 = (Us − Up ) + g. dt τp

(1)

The drag force has been written in this form to bring out the particle relaxation time scale τp =

ρp 4 d p . ρf 3 CD |Ur |

(2)

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In Eq. (1), τp appears as the only scale, and is the time necessary for a particle to adjust to fluid velocities. In the limit when Rep ≪ 1, it is seen from the expression of CD in that range that ρp d2p τp = , (3) ρf 18 νf which is the Stokes value. The drag coefficient is an empirical coefficient that can be estimated through experiments. Various expressions have been put forward, cf. Clift et al. [32], among which an often retained form is   24  1 + 0.15Re0.687  if Rep ≤ 1000, p CD = Rep (4)  0.44 if Rep ≥ 1000. The particle relaxation time scale is a non-linear function of particle properties. In the Stokes regime, it is quadratic in the particle diameter dp . Outside the Stokes regime, the dependence of τp on particle properties and variables, such as Us and Up , is more complicated. Broadly speaking, there are three classes of approaches to compute (two-phase) turbulent flows, which we classify with regard to the level of reduction : 1. Direct numerical simulation (DNS) 2. Large eddy simulations (LES) 3. Probability density function (PDF) 4. Reynolds average Navier Stokes (RANS) methods Using an analogy with the statistical physics language, we can say that the two first approaches can be considered “microscopic”, in the sense that they a have a least degree of modeling. DNS is model-free and can be thought to describe correctly turbulence [12]. LES approach is based upon the idea of modeling only some degree of freedoms, with the purpose of describing accurately the largest part of the degree of freedoms of the problem. The PDF approach is “mesoscopic”. The construction of a reduced state vector can be achieved by a coarse-graining procedure where the system is described on a large enough scale to eliminate some degrees of freedom. Information is therefore lost and this lack of complete knowledge will be reflected by the use of a stochastic description for the remaining degrees of freedom. In this method, the model is not directly written in terms of macroscopic variables but it is introduced at a mesoscopic level: the idea is to build a modeled equation for the pdf. Finally, the last approach is “macroscopic”, it starts by applying some averaging or filtering operator to the exact equations and, hence, we obtain exact but unclosed mean equations in which closure relations are then introduced. Closed mean equations result. Therefore, closure is attempted directly at the macroscopic level, and when it is performed available information is of course strictly limited to the very macroscopic variables that have been explicitly retained in the second step of the procedure, that is a limited number of moments at each point (usually not more than two). It is important to underline here that the PDF approach is different. Here, the exact instantaneous equations are replaced by models

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but still at the instantaneous level. Thus, in the PDF approach, the introduction of a model is made at an upstream level, where far more information is still available (since we model a probability density function) and has not yet been eliminated. From a practical point of view, in PDF approach to two-phase flows mean-field equations (Rij − ǫ ) are used for the fluid, whereas a particle PDF equation is solved by a Monte Carlo method using a trajectory point of view. The PDF model is therefore formulated as a particle stochastic Lagrangian model (a set of Langevin SDEs) [35, 34]. In this formulation, this approach corresponds to a Eulerian/Lagrangian method. There are, finally, some other techniques like Proper orthogonal (POD) [36] or group (SO(3)- SO(2)) [37] decomposition, which have been conceived specifically for the description of turbulent structures and which can be linked to the above methods. These are eduction techniques, which accessing directly to some actual information can rebuild the velocity field and identify the structure components. All these approaches are statistical and, therefore, belong to the suitable framework. Nevertheless, the computational cost is very different among them. More specifically, microscopic approaches are very demanding and, in practice, are both not of great help in engineering applications. Indeed, in the case of a large number of particles and/or of turbulent flows at high Reynolds numbers, the number of degrees of freedom is huge and one has to resort to a contracted probabilistic description. At this level, we can already point out an important drawback which concerns the macroscopic approach. When particle diameters vary considerably from particle to particle or when we are confronted with a situation where particles have completely different histories (highly complicated but local laws), deriving partial differential equations for mean quantities is a thorny issue. The case of complicated source terms happens whenever we want to have particle evaporation or combustion with complex expressions in terms of individual particle properties. When dealing with a distribution of particle diameters, one is faced with the problem of expressing, as a function of mean velocities hUs i, hUp i and the mean particle diameter hdp i, quantities such as h

Us i, τp

h

Up i. τp

(5)

These are complicated functions, due to the complex dependence of τp on particle diameters dp and also on particle and fluid velocities, Eq. (2) and Eq. (4). In theses cases, the Lagrangian PDF (mesoscopic) approach is particularly attractive since it treats these phenomena without approximation while the derivation of closed moment equations is next to impossible unless very crude simplifications are introduced. Concerning the turbulent structure one consideration and two questions will help us to determine which is the best-suited approach, in our opinion. The consideration is that the macroscopic approach is not able to represent properly turbulent structures. Beyond the technical difficulties to achieve physical meaningful closure of two-phase turbulent equations, as said above, this approach is placed at the level of mean equations and the effect of many disparate scales are modeled at the same time through some constitutive relations. In such methods, it appears at least courageous any tentative to add some physical-sound term which take into account of the statistical effect of instantaneous and zero-mean phenomena like quasi-coherent structures. Ad-hoc terms are neither

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general neither under complete control and, therefore, should be considered very carefully as a possible solution of this issue. Therefore, our first main conclusion is that macroscopic RANS approaches should be discarded as appropriate models for polydispersed two-phase turbulent flows and, notably, for particle deposition in wall-bounded flows. Two questions arise about turbulent structures: 1. Which structures are really important? 2. How does one must include these structures? Actually, it is very hard to answer to the first question and it is necessary to analyse it case-by-case. In the example discussed here, DNS simulations and experimental results seem to indicate that sweeps and ejections are particularly relevant. However, much smaller structures, like worms or point-vortex, might be much more effective for explaining internal intermittency in isotropic turbulence [38]. The second question is more probing. In principle, two ways can be explored. The first, the most usual, is to compute directly the turbulent structures, that is to resolve all the scales which are responsible for those geometrical features. Since quasi-coherent structures are fluid-velocity structures, this approach consists in obtaining (in some way) an instantaneous velocity field which contains such information. It is worth emphasising that this approach is based upon the idea of calculating a very accurate instantaneous fluid-velocity field and, therefore, can be pursued only within a microscopic approach. The second possible route is a mesoscopic one. The modeling issue is transferred to the particle phase and, notably, to the problem of building a suitable model for the velocity of the fluid seen. The general form of the Langevin model chosen for the velocity of the fluid seen consists in writing dUs,i = As,i (t, Z) dt + Bs,ij (t, Z) dWj ,

(6)

where the drift vector A and the diffusion matrix B have to be modelled. The complete Langevin equation model can therefore be written dxp,i = Up,i dt,

(7a)

dUp,i = Ap,i dt,

(7b)

dUs,i = As,i (t, Z) dt + Bs,ij (t, Z) dWj ,

(7c)

where the particle acceleration is Ap,i = (Us,i −Up,i )/τp +gi . This formulation is equivalent to a Fokker-Planck equation given in closed form for the corresponding pdf p(t; yp , Vp , Vs ) which is, in sample space.   ∂p ∂ ∂ ∂ 1 ∂2 + [Vp,i p ] + [Ap,i p ] + [As,i p ] = (Bs BsT )ij p . (8) ∂t ∂yp,i ∂Vp,i ∂Vs,i 2 ∂Vs,i ∂Vs,j

In this approach, the statistical effect of the most important turbulent structures should be included in eq. (7)c via appropriate terms. Some comments are in order. Although the first way can appear attractive for its accuracy and conceptual simplicity, some major drawbacks undermines its use for engineering

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applications. DNS is certainly a very accurate approach since it is able to reproduce all the scales for both phases and, therefore, together with the experimental studies should be viewed as the preferred method to investigate the fundamentals of turbulent flows [39, 40]. Further, since it is difficult to analyse experimentally many important quantities (frequency and strength of intermittent phenomena, geometrical features of small scales just to make few examples), it appear as an unavoidable searching tool mainly for modeling purposes. Indeed, physical models are based upon such information to guarantee some physical sound basis. However, DNS is a very demanding tool and its practical use in high-Reynolds number and/or geometrical complex flows is at least doubtful. LES is very similar to DNS for wall-bounded flows in terms of computational cost and thus the same remark applies as well [41]. Moreover, LES is a model, even though the model concerns only a part of the energy spectrum. In such sense, it is questionable if this approach is able to reproduce accurately quasi-coherent structures. Of course, it does reproduce some quasi-coherent structures but not all. In particular, small scale structures, which are modeled, are probably not present or, at least, not necessarily well treated. POD and SO(3)-SO(2) techniques are not predictive methods, because they need some information which is provided by previous DNS computations. In particular, SO(3) needs some ”a priori” knowledge of the whole statistical properties of the velocity field at all scales. After getting it, geometry-by-geometry, one may hope for a reduction of the important degrees of freedom by keeping track only of those statistical correlations, isotropic or anisotropic, which are more relevant in the SO(3) decomposition These approaches still aim to compute, even though indirectly, the complete velocity field and, therefore, they join the same category of microscopic approaches. Moreover, they are based upon a projection on a finite (often small for practical purposes) number of chosen eigenstates and, thus, they constitute a contracted model of the complete field. In conclusion, none of these microscopic models appear to be adequate for engineering application and in particular two-phase turbulent pipe flows. On the other hand, mesoscopic approach is radical different. We can say that it is an “active” approach. It is based on “a-priori” choice and not on “a-posteriori” analysis. Indeed, if we know in advance the problem we want to tackle, we can analyse the physical problem on the basis of experimental and DNS results. On this basis, we shall choose which structures appear to be particularly relevant for our own problem. In this way, we avoid the problem which affects LES, POD and SO(3) approaches. Then, we shall try to include the statistical effect of those (and only those) geometrical features which seem more important. Eventually, the model will be written in terms of the known typical statistical signatures (time-frequency, characteristic length scales, life-time...). For this step, DNS simulations will result particularly interesting.

4.

Numerical Results

In this section, we try to illustrate this view with few examples describing particle deposition in a turbulent pipe-flow. The point of depart is the Langevin PDF approach developed

A New Approach for Polydispersed Turbulent Two-Phase Flows

449

in the last decade by Minier and his collaborators [42, 34, 43]: dxp,i = Up,i dt 1 dUp,i = (Us,i − Up,i )dt τp ∂hUf,i i 1 ∂hP i dUs,i = − dt + (hUp,j i − hUf,j i) dt ρf ∂xi ∂xj 1 − ∗ (Us,i − hUf,i i) dt TL,i s   2 ˜ ˜ + hǫi C0 bi k/k + (bi k/k − 1) dWi . 3 10

10

(10)

(11)

0

-1

-2

kp/u*

10

(9)

10

10

10

-3

exp.

-4

Standard Stand. + structures

-5

10

-2

10

-1

10

0

+

10

1

10

2

10

3

τp

Figure 1. Deposition rate velocity for the different model used. In all numerical cases the continuous phase is solved via standard k − ǫ model. Experimental results are given for reference (triangle down). The standard results are indicated by the curve labeled with “Standard model” (circles). The results obtained with the new phenomenological model are shown by the curve indicated by “Stand. + structures” (crosses). The results obtained with the phenomenological model derived from DNS are in good agreement with experimental results, in particular small particles deposit only rarely. This model was developed without taking into account quasi-coherent structures and, therefore, it fails to describe correctly the particle deposition, see fig. 1a. Thus, as simple and first step toward considering some of the signatures of coherent structures, a new phenomenological model, built on the basis of DNS results, has been proposed and introduced

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S. Chibbaro

in the numerical simulations [44]. The results obtained with this model are in good agreement with experiments [45] and, hence, show that to take into account some geometrical features of the flow improves significantly the statistical description of particle deposition. Following this suggestion, a more systematic introduction of geometrical features in statistical PDF approach, where coherent structures are introduced as new stochastic terms in the modeled equations, has been recently attempted [47]. The sketch of that model is given in figure 2a, for more details refer to the papers [46, 47]. Standard Langevin Model Core flow y+ = 100 1

0.1

Sweep Ejection

Diffusion

+ p

0.01

k

Outer zone

0.001

0.0001

Possible reentrainment

1e-05

Interface Inner zone

Diffusion

1e-06 0.0001

0.001

0.01

0.1

τp

+

1

10

100

1000

Figure 2. (a) Sketch of the stochastic model of sweeps and ejections structures. (b)Deposition rate for the standard model (down-triangle), the new stochastic model (Full circle) and experimental results (star). The new model is used in a large range of diameters. The deposition rate surges with particle inertia in the range of 1 < τp+ < 70 and slightly decreases for greater inertia. (Courtesy from Physics of fluids). The deposition velocity computed by the new stochastic model for 12 classes of particles is represented in Fig. 2b; it is compared with the experimental data gathered by Papavergos and Hedley [48] and with the results provided by the standard Langevin model. It can be observed that the deposition rate computed by the present model is in fair agreement with the experiments.

5.

Perspectives

There are many complex flows in which geometrical features play a major role and whose modeling is still lacking. A recent stochastic description of geometrical asperities for problems of particle resuspension joins to this category [49]. Nevertheless, in author’s

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opinion, the most intriguing direction to pursue is to put forward a suitable model for bubbly flows. These flows have an enormous importance in a vast spectrum of applications and are very hard to simulate within present methods (microscopic and macroscopic). In these flows the interactions between quasi-coherent structures and particles is particularly relevant since it is well known that particles tend to concentrate within the vortexes [50], at variance with heavy particles, showing a strong preferential-concentration behaviour. The fact that bubbly particles pass most of their time within turbulent structures should make certainly difficult to propose an accurate model without taking them into account. While, for heavy particles, it appears essential to take into account geometrical features in certain situations (like in particle deposition), for bubbly flows it might be true for almost all flows (even for isotropic symmetry). A recent hard effort in DNS simulation of these flows will be of great help for the modeling.

6.

Conclusions

In this commentary, which is based upon recent modeling developments, we have discussed what features should characterize a suitable model for complex turbulent two-phase flows. The most of the attention has been devoted to the case of particle deposition onto walls in turbulent pipe-flows. However, we believe that the rationale can be applied to many other situations and in particular to all those situations in which geometrical features have a not negligible role. We have discussed why a suitable model should be statistical and should be easily linked to geometrical characteristics of the flows. Then, we have analyzed the different statistical approaches available for the description of turbulent two-phase flows and we have tried to explain why the “mesoscopic” PDF approach should be preferred to the others. In particular, we have discussed in some details the issue of modeling quasi-coherent structures and we have emphasized that an active choice of the modeler permits to avoid both too much demanding numerical simulations and a misleading treatment of such structures. Finally, we have shown some recent results obtained through this PDF approach for particle deposition in a turbulent pipe-flow which lend support to this point of view.

Acknowledgments S. Chibbaro’s work is supported by a ERG EU grant. He greatly acknowledges the financial support given also by the consortium SCIRE. More information is available at http://www.consorzio-cometa.it. The author desires to thank in the most particular way Jean-Pierre Minier, since the present author’s point of view is incommensurately related to his highest scientific and pedagogical lessons. Furthermore, I would like to thank him for interesting suggestions for the present manuscript. I thank Dott. Mathieu Guingo for his courtesy in giving me figure 2.

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[45] B. Liu.and K. Agarwal, Aerosol Science, 5, 145 (1974). [46] M. Guingo and J-P. Minier ICMF proceedings, Leipzig (2007). [47] M. Guingo and J-P. Minier Phys. Fluids, 20 053303 (2008). [48] P. G. Papavergos and A. B. Hedley, Chem. Eng. Res. Des. 62, 275 1984. [49] M. Guingo, and J.-P. Minier, J. Aerosol Science 2008, doi: j.jaerosci.2008.06.007.

10.1016/

[50] E. Calzavarini, M. Kerscher, D. Lohse, and F. Toschi. J. Fluid Mech., 607:1324, 2008. [51] R. Volk, E. Calzavarini, G. Verhille, D. Lohse, N. Mordant, J.-F. Pinton, F. Toschi Physica D 237 14-17 (2008) 2084-2089. Reviewed By Prof. Luca Biferale Professor of Theoretical Physics Dept. of Physics and INFN, University of Roma, Tor Vergata Via della Ricerca Scientifica 1, 00133, Roma, Italy ph +39 067259.4595, fax +39 062023507 http://www.fisica.uniroma2.it/ biferale/ skype callto://lucabiferale email [email protected]; [email protected]

INDEX A absorption, 208, 229 academic, xi, 269 accidents, 366 accounting, 127, 141, 277 accuracy, xi, 121, 185, 214, 238, 244, 249, 259, 269, 286, 305, 357, 400, 417, 420, 421, 422, 423, 447 acetylene, 166 actuation, 156 acute, 443 adiabatic, ix, 171, 173, 177, 178, 180, 184, 188, 189, 190, 192, 193, 195, 201, 304, 305, 381 advection-diffusion, 407, 411, 425 AEA, 114, 166 aerosol, xiii, 162, 164, 167, 441 aerosols, viii, 117, 163 aerospace, 344 AFM, 186 Africa, 30, 31, 36, 37 agent, 38 aggregation, 36, 166 agricultural, x, 231 aid, 42, 44 air, 43, 80, 86, 103, 105, 106, 108, 110, 114, 167, 179, 182, 184, 185, 186, 271, 317, 344, 366, 411 air pollution, 167 algorithm, xii, 53, 122, 163, 186, 188, 208, 228, 399, 400, 411, 415, 416, 418, 419, 420, 423, 425 alkali, 38 alternative, 211, 272, 282, 309 alters, 131 aluminium, 283, 284, 286, 288, 289, 290, 298, 299, 301, 302, 307, 308, 309, 313 aluminum oxide, 308 ambiguity, 157 amplitude, 368, 370, 414, 423, 424, 425 Amsterdam, 315, 452 anisotropy, 252 anode, 22 appendix, 212, 213, 222

application, ix, xii, 113, 126, 127, 128, 152, 156, 158, 162, 164, 171, 175, 185, 270, 271, 272, 312, 313, 344, 361, 399, 448 applied mathematics, 163 aqueous solution, 8, 11 aqueous solutions, 8, 11 Arabia, 317 Argentina, 298 argument, 43, 108, 428 arithmetic, 393 aspect ratio, 327, 328, 336, 337, 351, 355, 360, 362 aspiration, 156 assessment, 146, 310 assignment, 207 assumptions, 45, 207, 209, 238, 302, 305, 306, 312, 389, 390, 399, 415 asymptotic, 207, 208, 219 asymptotically, 324 atmosphere, 442 atmospheric pressure, 173 Australia, 41, 117, 269, 270 Australian Research Council, 162 availability, 43, 120, 142, 271 averaging, 45, 46, 99, 238, 271, 368, 445 azimuthal angle, 173

B baths, 167 beaches, 400, 411, 425, 438 beams, 270, 294 behavior, 29, 34, 62, 63, 64, 65, 110, 111, 168, 214, 219, 249, 257, 259, 293, 294, 334, 349, 422, 424 behaviours, 129, 157 benchmark, 53, 75, 111 Bessel, 206, 212, 226 bioreactors, 118 birth, ix, 117, 118, 121, 124 blocks, 185, 366, 392 boiling, 129, 131, 133, 136, 137, 139, 142, 143, 144, 145, 150, 152, 153, 154, 156, 158, 159, 162, 163, 164, 165, 166, 168, 169, 302, 314, 361, 362 Boston, 164

456

Index

bottleneck, 159 boundary conditions, xi, 53, 84, 95, 96, 143, 144, 173, 177, 178, 179, 185, 188, 189, 207, 210, 211, 212, 224, 273, 279, 289, 302, 318, 383, 388, 390, 394, 395, 400, 415, 416, 418, 419, 424, 443 boundary surface, 273, 417 boundary value problem, 211, 418 bounds, 164 Boussinesq, 180 Brazil, 166, 224 breakage rate, 121, 135, 145, 149, 162 bubble, ix, 43, 44, 50, 51, 52, 104, 105, 106, 108, 109, 110, 112, 113, 114, 118, 123, 126, 127, 128, 129, 130, 131, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169 bubbles, ix, 44, 49, 50, 51, 52, 95, 97, 103, 104, 105, 106, 108, 109, 110, 112, 113, 114, 118, 127, 129, 130, 131, 132, 133, 135, 136, 137, 138, 139, 143, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 161, 163, 164, 166, 168, 169, 453 buffer, 98, 101, 103, 105, 244, 246, 261 bun, 406 burn, 301 bypass, 289

C cables, 185 calculus, 402 calibration, 38, 126, 136, 145, 160 Canada, 362 canals, 442 candidates, 28 capacity, 119, 120, 139, 172, 206, 227, 270, 288, 307, 313, 324, 334 capillary, 16, 353, 362, 410, 413 carbonates, 38 carrier, vii, viii, 27, 29, 41, 42, 43, 44, 45, 46, 53, 56, 58, 59, 62, 64, 65, 67, 68, 75, 80, 83, 84, 88, 94, 95, 100, 101, 104, 110, 111 Cartesian coordinates, 172, 179, 273, 426 case study, 282 cast, 186 categorization, 443 cathode, 22 cation, 36, 114 cavities, 131, 140, 152, 185 C-C, 189, 190, 191, 195, 196 cell, 36, 164, 167, 305, 392, 393, 394, 395 CFD, viii, xi, 41, 43, 95, 120, 121, 123, 125, 132, 143, 144, 162, 166, 167, 186, 269, 270, 271, 272, 274, 283, 289, 290, 293, 294, 295, 296, 297, 298, 302, 306, 312, 313, 314, 316, 397

channels, vii, xi, 1, 2, 3, 5, 20, 24, 25, 33, 35, 207, 210, 292, 298, 304, 343, 344, 351, 352, 353, 355, 361, 362 chaos, 173, 174 charge density, 3, 4 chemical composition, 36 chemical reactions, 9 chromatography, 1, 2, 11, 13, 17, 18, 20, 21, 22, 23, 24, 25 circulation, 381, 382 cladding, 298, 301, 302, 308, 309, 310, 311, 312, 314 classes, viii, 41, 42, 44, 53, 54, 56, 58, 59, 62, 63, 64, 111, 125, 126, 143, 144, 149, 153, 154, 155, 156, 160, 163, 445, 450 classical, 259, 410 classification, xi, 114, 343, 369 climate change, 167 clinics, 284 closure, 44, 113, 122, 158, 164, 165, 233, 234, 445, 446 clustering, 97 coagulation, 163, 166 coagulation process, 163 codes, 166 collisions, 46 colon, 395 combustion, xiii, 129, 159, 164, 165, 441, 442, 446 combustion processes, 129 communication, 315 communities, 1 competence, 158 complex systems, 270 complexity, xi, 44, 68, 95, 119, 150, 158, 269, 272, 276, 399 components, 5, 7, 10, 121, 139, 152, 177, 178, 232, 234, 245, 246, 249, 252, 254, 263, 265, 270, 275, 283, 285, 286, 289, 294, 295, 344, 345, 368, 382, 395, 414, 415, 416, 421, 422, 423, 426, 431, 446 composition, 28, 36, 38, 39, 118 comprehension, xiii, 441 computation, 95, 97, 115, 122, 245, 271, 315, 415, 437 Computational Fluid Dynamics (CFD), viii, ix, xi, 41, 117, 118, 129, 185, 186, 269, 292, 314, 316, 366 computational grid, 234, 238 computer technology, 127 computing, 118, 122, 185, 271, 273, 278, 392, 416, 417, 418, 419, 439 concentration, ix, 2, 4, 5, 9, 10, 11, 29, 35, 49, 53, 118, 126, 148, 149, 150, 153, 158, 159, 164, 301, 412 concentric annuli, ix, 171, 173, 174, 175, 229 concrete, 180, 181, 182, 183, 185 condensation, 131, 133, 136, 152, 153, 154, 155, 158, 161 conductance, 5, 7, 185

Index conduction, ix, x, 137, 138, 139, 160, 171, 173, 174, 176, 180, 181, 182, 183, 184, 185, 205, 208, 209, 211, 219, 229, 273, 279, 283, 285, 286, 289, 305, 309, 310, 321 conductivity, x, 3, 7, 172, 182, 205, 206, 208, 209, 217, 224, 275, 284, 288, 290, 307, 308, 309, 312, 319, 330, 340 confidence, 295 configuration, viii, 42, 99, 145, 175, 189, 294, 301, 303, 401, 402, 409 conformity, 64 conservation, 36, 127, 143, 185, 188, 271, 274, 280, 285, 291, 294, 308, 403, 404, 421, 422, 425 constraints, 127 construction, 271, 272, 273, 295, 381, 389, 410, 445 contact time, 52, 136, 162 continuity, 45, 50, 53, 188, 210, 211, 212, 224, 225, 235, 280 control, 53, 136, 145, 186, 208, 228, 236, 279, 280, 281, 282, 283, 286, 295, 299, 304, 312, 321, 322, 330, 331, 385, 447 convection, ix, xi, 118, 138, 160, 165, 171, 173, 174, 175, 176, 177, 180, 181, 182, 184, 185, 189, 201, 205, 207, 209, 229, 269, 290, 291, 301, 302, 305, 309, 416, 426 convective, ix, 53, 137, 171, 172, 173, 176, 179, 206, 282, 305, 318, 362, 393, 401, 402, 409 convergence, 49, 96, 143, 179, 188, 214, 273, 286, 295, 302, 306, 313, 400, 421, 423, 425 convergence criteria, 302 conversion, 8 cooling, xi, 35, 269, 270, 282, 283, 284, 290, 291, 297, 298, 299, 300, 302, 306, 309, 314, 319, 330, 344 copper, 55, 57, 59, 60, 61, 62, 63, 64, 65, 111 correlation, 47, 48, 50, 152, 339, 349, 351, 354, 356, 362 correlations, xi, 42, 152, 156, 343, 353, 354, 448 cosine, 7, 303, 310, 311, 312, 313 costs, 42 couples, 53 coupling, xii, 14, 16, 43, 44, 45, 49, 53, 147, 185, 207, 379, 392 covering, 105, 126, 143, 152, 271, 295 CPU, 424 crack, 438 CRC, 341 critical value, 122, 156, 180, 392 cross-sectional, 137, 227, 251, 321, 351, 360, 368, 371, 372 cryogenic, 344 crystalline, 39 crystallization, vii, 27, 33, 34 crystals, 31, 33 curiosity, xi, 269 curve-fitting, 349 cycles, 368 cyclone, 113 Czech Republic, 379, 397

457

D damping, 239 Darcy, 320, 322, 330, 340 data set, 241 data structure, 273 death, ix, 51, 117, 118, 121, 124, 128, 159 death rate, 51, 121, 128, 159 decay, 301, 302, 376 decomposition, 446, 448 decompression, 29, 34, 35, 37 decoupling, 120 deduction, 110 defects, viii, 27, 36, 186 defense, 344 definition, 14, 17, 20, 68, 138, 184, 272, 276, 279, 311, 312, 313, 351, 388, 401, 402, 403, 405, 406, 409, 412, 415, 425, 426, 427 deformation, 31, 34, 35, 37, 38, 43, 234, 237, 265, 276, 394, 395 degrees of freedom, 443, 445, 446, 448 dehydration, 34, 35 delivery, 42 demand, xiii, 441 Denmark, 163 densitometry, 164 density, viii, 2, 7, 28, 35, 36, 37, 41, 42, 43, 44, 45, 46, 50, 51, 52, 62, 63, 64, 65, 66, 95, 110, 113, 122, 125, 127, 129, 133, 137, 138, 152, 154, 160, 162, 165, 166, 172, 180, 275, 289, 303, 305, 306, 310, 311, 313, 361, 370, 380, 387, 392, 394, 403, 404, 406, 407, 411, 439, 445 dependent variable, 118, 173, 188, 189 deposition, xiii, 42, 442, 443, 444, 447, 448, 449, 450, 451 deposition rate, 450 derivatives, 178, 401, 402, 417, 418, 427 destruction, 35, 43, 324 detachment, 131, 169 detection, 22 deuterium oxide, 284 deviation, 143, 345, 353, 355 diesel, 80, 111 differential equations, 208 differentiation, 142, 225, 226 diffusion, 2, 7, 9, 10, 22, 35, 36, 53, 118, 142, 172, 209, 278, 279, 280, 282, 286, 305, 391, 400, 407, 411, 443, 447 diffusion process, 36, 443 diffusivity, 12, 14, 20, 142, 161, 172, 206, 209, 210, 305 digital images, 293 dilute gas, viii, 41, 44 direct measure, 5, 20 discontinuity, 28 discretization, 421, 422, 423 dislocations, 35

458

Index

dispersion, vii, 1, 2, 3, 10, 13, 14, 15, 16, 17, 18, 19, 22, 23, 25, 44, 52, 105, 112, 113, 114, 118, 133, 136, 146, 160, 166, 442 displacement, 388, 395, 427 distribution, ix, xii, 3, 5, 10, 15, 16, 22, 23, 51, 59, 62, 63, 97, 108, 109, 110, 112, 118, 119, 120, 121, 125, 126, 129, 130, 131, 135, 143, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 162, 165, 166, 167, 174, 177, 188, 189, 208, 210, 228, 234, 244, 245, 246, 257, 259, 263, 295, 297, 302, 303, 306, 307, 309, 310, 311, 312, 313, 346, 365, 366, 370, 371, 372, 377, 379, 380, 381, 382, 383, 385, 386, 387, 389, 395, 396, 397, 446 distribution function, 109, 110 divergence, xii, 102, 280, 365, 366, 368, 373, 374, 375, 376, 377, 378, 403, 406, 421, 425 diversity, 44, 118, 270 dominance, 282 Doppler, xi, 56, 254, 270 dust, 42, 164 dynamic viscosity, 275, 360 dynamical system, 443

E earth, 442 eddies, 56, 271, 272, 375 elasticity, 394 electric conductivity, 3 electric current, 2, 7 electric field, vii, 1, 2, 3, 7, 11, 14, 23 electrical power, 176 electrical resistance, 176, 178 electrolyte, 2, 3, 4, 5, 7, 11, 22, 24 electromigration, 11 electroosmosis, 3, 9, 11 electrophoresis, 1, 2, 9, 11, 12, 15, 16, 17, 20, 21, 22, 23, 24, 25 electrostatic force, 352 electrostatic interactions, 2, 11 email, 454 energy, ix, 8, 43, 44, 46, 47, 48, 88, 94, 102, 115, 132, 139, 144, 160, 171, 173, 185, 210, 238, 241, 242, 263, 271, 274, 275, 276, 279, 280, 285, 289, 294, 318, 324, 344, 365, 370, 372, 390, 391, 394, 420, 421, 422, 423, 424, 425, 430, 432, 436, 448 energy transfer, 238 engines, x, 42, 231 England, 437 enlargement, 148 entropy, xi, 8, 318, 322, 323, 324, 325, 326, 327, 328, 329, 331, 332, 333, 334, 335, 336, 337, 338, 341 environment, 28, 38, 43, 86 equilibrium, 10, 35, 56, 68, 104, 164, 182, 404 equipment, ix, 42, 171, 173, 293, 294 erosion, 114 estimating, 142

ethane, 166 ethylene, 166 Euclidean space, 428 Euler equations, 411 Eulerian, xii, xiii, 43, 45, 49, 111, 113, 164, 399, 400, 401, 413, 414, 416, 425, 437, 441, 446 evaporation, 133, 137, 151, 152, 158, 160, 165, 446 evolution, xii, 118, 120, 122, 124, 126, 129, 131, 149, 163, 164, 399, 400, 404, 409, 416, 424, 425 exaggeration, 375 exercise, 111 expansions, 208 experimental condition, 68, 95 extraction, 113, 419

F failure, 308 fauna, 43 fax, 454 feedback, 44 feeding, 112 film, 53, 106, 136, 160, 245 film thickness, 53, 105, 136, 160 financial support, 162, 451 finite differences, 400 finite element method, 395 finite volume, xii, 53, 185, 279, 280, 295, 379, 392 finite volume method, xii, 185, 279, 280, 295, 392 fire, ix, 171, 173 first principles, xiii, 441 fission, 283, 284, 298, 299, 301, 302 flatness, 234, 254, 257, 259, 263, 264, 265 flexibility, 120, 121, 124, 127, 273 flooding, 140, 160 flora, 43 flora and fauna, 43 flow, vii, viii, ix, x, xi, xii, xiii, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 19, 35, 41, 42, 43, 44, 45, 47, 49, 54, 56, 58, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 80, 82, 83, 84, 85, 86, 87, 94, 95, 96, 97, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 111, 112, 113, 114, 115, 118, 119, 123, 124, 126, 127, 129, 130, 131, 134, 136, 137, 141, 143, 144, 145, 146, 147, 149, 150, 152, 154, 156, 157, 158, 159, 160, 162, 163, 164, 165, 166, 167, 168, 171, 173, 174, 175, 176, 177, 178, 179, 180, 184, 185, 186, 188, 189, 194, 195, 197, 201, 205, 206, 207, 208, 209, 212, 214, 215, 219, 221, 223, 224, 227, 228, 229, 231, 232, 233, 234, 235, 237, 238, 239, 241, 244, 249, 251, 252, 254, 257, 259, 260, 263, 265, 269, 270, 271, 272, 273, 276, 277, 282, 283, 284, 285, 286, 288, 289, 291, 292, 293, 294, 295, 296, 297, 299, 300, 301, 302, 303, 304, 305, 306, 307, 309, 313, 314, 315, 317, 318, 319, 320, 323, 324, 325, 326, 327, 330, 333, 334, 336, 337, 340, 343, 344, 345, 346, 347, 349, 350, 351, 352, 353, 355, 356, 357, 358, 360, 361, 362, 363,

Index 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 379, 380, 381, 382, 383, 386, 388, 390, 392, 393, 395, 396, 397, 399, 400, 404, 410, 411, 415, 425, 437, 439, 441, 442, 443, 444, 450 flow behaviour, 118 flow field, x, 43, 44, 45, 68, 124, 171, 174, 176, 179, 180, 184, 189, 195, 231, 233, 244, 257, 260, 272, 293, 377, 415, 443 flow rate, viii, xii, 42, 96, 99, 101, 103, 104, 105, 106, 107, 108, 112, 185, 206, 227, 277, 293, 295, 297, 305, 307, 309, 319, 323, 325, 330, 333, 334, 340, 357, 358, 365, 366, 368, 369, 370, 371, 372, 373, 374, 375, 377, 383 flow value, 289 fluctuations, 53, 56, 59, 68, 69, 75, 82, 95, 112, 241, 243, 246, 247, 248, 249, 252, 255, 256, 257, 258, 261, 262, 263, 271, 272, 442, 443 fluid, vii, viii, ix, x, xi, xii, xiii, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 20, 22, 24, 27, 28, 29, 32, 34, 35, 36, 37, 41, 43, 54, 56, 58, 65, 68, 80, 95, 97, 102, 106, 112, 115, 120, 129, 132, 134, 142, 143, 159, 162, 163, 164, 171, 173, 174, 175, 176, 177, 185, 186, 201, 205, 207, 208, 209, 210, 211, 214, 217, 218, 219, 221, 224, 232, 269, 271, 275, 277, 279, 288, 292, 293, 294, 295, 296, 297, 305, 309, 314, 317, 318, 319, 320, 321, 323, 324, 325, 326, 330, 332, 333, 334, 335, 336, 337, 339, 340, 341, 343, 344, 345, 346, 347, 349, 352, 353, 355, 356, 357, 358, 360, 361, 362, 363, 365, 366, 368, 370, 372, 373, 375, 376, 377, 399, 400, 402, 403, 404, 405, 409, 411, 412, 413, 415, 425, 437, 441, 442, 443, 445, 446, 447 fluid mechanics, xii, 399, 400 fluid transport, 366 fluidized bed, 112, 118, 129, 164, 165 focusing, 120, 130, 372 foils, 297 Fourier, 368, 371 Fox, 123, 164, 166, 167 fragmentation, 166 France, 164, 168, 231 freedom, xii, 379, 443, 445, 446, 448 freedoms, 445 friction, x, xi, 95, 98, 99, 101, 105, 106, 108, 109, 114, 161, 231, 232, 233, 234, 241, 242, 244, 245, 249, 261, 265, 278, 317, 318, 320, 322, 323, 327, 328, 330, 334, 336, 337, 340, 343, 344, 347, 349, 350, 351, 352, 353, 354, 355, 356, 358, 359, 360, 361, 362, 363 FTIR, vii, 27, 29, 30, 31, 37, 38 fuel, 283, 284, 285, 286, 288, 289, 290, 293, 294, 295, 301, 303, 316 functional analysis, 410 funding, 344

459

G gas, viii, x, 2, 4, 41, 42, 45, 46, 47, 48, 49, 50, 53, 54, 55, 56, 57, 58, 62, 65, 68, 69, 78, 80, 94, 95, 96, 97, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 111, 112, 115, 118, 126, 127, 130, 131, 133, 134, 135, 140, 143, 144, 145, 146, 147, 148, 154, 156, 157, 158, 159, 163, 164, 180, 185, 208, 229, 231, 233, 317, 318, 353, 404, 442 gas phase, 45, 46, 47, 48, 49, 50, 53, 54, 56, 58, 62, 65, 68, 69, 80, 131, 133, 135, 143, 147 gas turbine, x, 42, 231 gases, vii, 43, 179, 317, 320 gasoline, x, 231 gauge, 305, 307 Gaussian, 122, 257, 263 gel, 213 generation, xi, 8, 34, 69, 133, 137, 139, 152, 154, 155, 158, 160, 178, 185, 283, 285, 315, 316, 318, 322, 323, 324, 325, 326, 327, 328, 329, 331, 332, 333, 334, 335, 336, 337, 338, 340, 341 geochemical, 39 geochemistry, 28 geophysical, 28 Germany, 27, 114, 165, 168, 266 glass, 56, 58, 59, 60, 61, 62, 64, 65, 66, 79, 80, 111, 366 glycerol, 339 grain, 35 grains, 29 gravitational constant, 160 gravitational field, 404 gravity, 46, 50, 277, 444 greek, 426 grid generation, xi, 269, 272, 273, 315 grid resolution, 238, 239, 241, 259, 260 grids, 115, 232, 238, 254, 273, 305 groups, 44, 51, 52, 106, 112, 125, 126, 143, 156, 323, 333 growth, 118, 119, 130, 138, 139, 140, 141, 142, 160, 161, 164, 166, 306, 344 growth rate, 142 growth time, 140 guidelines, 299

H half-life, 284 handling, 97, 119, 120, 132, 134, 156, 158, 186, 272 haze, 164 heat, ix, x, xi, 115, 118, 120, 129, 131, 133, 137, 138, 139, 142, 143, 144, 150, 152, 158, 160, 163, 164, 165, 167, 168, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 189, 201, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 218, 219, 220, 221, 222, 223, 224, 227, 228, 229, 231, 232, 233, 234, 269, 270, 271, 273,

460

Index

275, 277, 278, 283, 284, 285, 289, 290, 291, 297, 301, 302, 303, 304, 309, 310, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 330, 331, 332, 333, 334, 335, 336, 337, 339, 340, 341, 344, 361, 362, 363, 392 heat capacity, 139, 206, 227, 334 Heat Exchangers, 228, 341 heat removal, xi, 269, 270, 301 heat transfer, ix, x, xi, 115, 131, 133, 137, 152, 160, 164, 165, 167, 168, 171, 172, 173, 174, 175, 176, 180, 182, 183, 184, 185, 201, 205, 206, 207, 208, 212, 222, 223, 227, 228, 229, 231, 232, 233, 269, 270, 271, 273, 278, 302, 304, 310, 315, 317, 318, 319, 320, 321, 323, 324, 325, 326, 327, 330, 331, 332, 333, 334, 335, 336, 337, 339, 340, 361, 362, 363 heating, 137, 140, 150, 174, 175, 207, 219, 222, 289, 302, 318, 319, 330 heavy particle, 113, 444, 451 heavy water, 284, 288, 289, 298 height, 2, 4, 16, 20, 22, 24, 53, 54, 57, 59, 67, 68, 69, 75, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 111, 143, 298, 349, 350, 353, 354, 405, 419, 421, 423, 425 helicopters, 42 helium, 186 heterogeneity, 31 heterogeneous, 28 hexane, 353 high pressure, 183 high resolution, 313 high temperature, 180 high-speed, 443 histogram, 30, 33 Holland, 204 homogenous, 44, 69, 114, 157, 158 host, 29, 31, 32, 34, 35, 36 human, 42, 43 hybrid, 97, 134, 273 hydration, 34, 37 hydrodynamic, 2, 10, 13, 15, 16, 18, 19, 22, 25, 141, 160, 295, 352 hydrodynamics, 26, 154, 158, 165, 353, 362 hydrogen, 28, 36, 38, 39 hydroxide, 39 hyperbolic, 7, 238, 400 hypothesis, 276

I IAC, 126, 143, 149, 150, 152, 155, 156, 157, 158 identification, 33 identity, 238, 385 images, 293, 358, 359 imaging, 400 imbalances, 282 implementation, ix, 117, 120, 125, 127, 385 inclusion, 36, 37, 167

incompressible, vii, xi, 3, 343, 345, 350, 352, 383, 388, 399, 403, 411, 413, 437, 439 independence, 145 independent variable, 415 indication, 257, 259 indices, 20, 24, 392, 408, 426 industrial, viii, ix, xiii, 42, 44, 114, 117, 118, 120, 121, 122, 171, 186, 441 industrial application, xiii, 441 industry, 344 inert, 43 inertia, 43, 56, 80, 95, 346, 351, 444, 450 infinite, 212, 213, 214, 222 infrared, vii, 38, 39 injection, viii, 22, 42, 43, 95, 96, 99, 100, 101, 102, 103, 104, 106, 107, 108, 110, 112, 131, 163 injections, 42 insertion, 273, 315 insight, 234, 368 instabilities, 419, 423 instability, 122, 157, 173, 174, 175, 305, 400 institutions, 344 instruments, 44 insulation, 143 integration, x, 231, 238, 280, 383, 385, 404, 424 intensity, xii, 102, 147, 149, 249, 259, 265, 365, 366, 368, 372, 373, 374, 375, 376, 377, 381, 383, 394, 444 interaction, viii, xiii, 28, 34, 41, 47, 48, 51, 53, 111, 113, 174, 207, 441 interactions, 2, 11, 43, 110, 115, 131, 156, 158, 186, 451 interdependence, 423 interdisciplinary, 42 interface, 133, 177, 188, 211, 309, 310, 311, 312, 314, 400, 416, 419, 437 interference, 85 international markets, 299 internship, 315 interphase, 42 interpretation, 237, 409, 412 interrelations, 259 interval, 51, 380, 381, 384, 385, 389, 390 intrinsic, vii, 27, 28, 35, 38, 400, 403, 406, 407, 425 inversion, xii, 124, 379, 380, 386, 387 Investigations, x, 231 investigative, 159 ionic, 4, 11 ions, 3, 5, 7, 95, 123, 143, 194, 197, 302, 417, 418, 422, 423 iron, 39 irradiation, xi, 269, 270, 272, 277, 283, 284, 285, 290, 292, 295, 297, 298, 299, 300, 301, 302, 303, 304, 305, 307, 314, 316 isothermal, 45, 49, 113, 129, 131, 133, 143, 144, 145, 156, 158, 163, 177, 189, 191, 192, 193, 194, 195, 201 isotopes, xi, 269, 270 isotropic, 51, 53, 135, 136, 237, 249, 447, 448, 451

Index Italy, 113, 164, 441, 454 iteration, 53, 188, 189, 386, 394, 416 iterative solution, 302

J Jacobian, 122, 172, 187 Japan, 203, 205, 266 Japanese, 378 judge, 179 Jung, 184, 204

K kernel, 168 kinetic energy, 44, 46, 48, 88, 94, 102, 160, 241, 275, 276, 279, 365, 370, 372, 390, 391, 394, 421, 423, 424 King, 317, 343, 361 Kolmogorov, 452

L Lagrangian, 43, 104, 115, 400, 401, 446 Lagrangian approach, 104 lamellar, 31 lamina, vii, x, xi, xii, 45, 162, 163, 166, 174, 179, 185, 207, 208, 214, 223, 228, 229, 231, 232, 233, 235, 276, 315, 317, 318, 319, 320, 326, 330, 336, 343, 344, 345, 346, 347, 349, 350, 351, 352, 353, 355, 356, 358, 362, 379, 388, 390, 391, 392, 396 laminar, vii, x, xi, xii, 45, 162, 163, 166, 174, 179, 185, 207, 208, 214, 223, 228, 229, 231, 232, 233, 235, 276, 315, 317, 318, 319, 320, 326, 330, 336, 341, 342, 343, 344, 345, 346, 347, 349, 350, 351, 352, 353, 355, 356, 358, 361, 362, 379, 388, 390, 391, 392, 396, 397, 442 laminated, 229 land, 183 language, xiii, 441, 445 large-scale, 249 laser, 254, 293, 294 law, x, 97, 98, 180, 205, 208, 214, 224, 233, 324, 354, 375, 403, 404, 406, 407 laws, 400, 446 lead, xi, 7, 42, 68, 69, 94, 111, 195, 255, 269 liberation, viii, 27, 34, 35, 37 licensing, 285 life forms, 43 lifetime, 176, 201 likelihood, 148 limitation, 156, 158, 359, 421 limitations, ix, xii, 44, 118, 120, 123, 350, 399 linear, 124, 139, 175, 186, 213, 244, 249, 278, 321, 327, 336, 339, 340, 393, 394, 399, 404, 417, 418, 419, 424, 428

461

linear function, 428 liquid film, 105, 106 liquid nitrogen, 362 liquid phase, 49, 50, 51, 68, 69, 101, 103, 104, 126, 133, 135 liquids, vii, 113, 163, 228, 317, 318, 320, 339 lithosphere, 39 location, 3, 59, 63, 65, 102, 103, 118, 146, 148, 149, 219, 220, 227, 301, 377, 388, 422, 430 London, 26, 228, 318, 326, 330, 336, 341, 351, 353, 361 longevity, 42 long-term, 28 losses, xi, 43, 318, 319, 362, 366 LSM, 400 lubrication, 133, 146, 149, 157, 160 lying, 409, 419

M machinery, 365 machines, x, 231, 366 magma, vii, 27, 28, 29, 34, 35, 37, 39 magnesium, 28, 37, 143, 283, 285, 288, 290 maintenance, 42 manipulation, 225 mantle, vii, viii, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39 manufacturing, 400 mapping, 380, 385, 386, 401 markets, 299 mass loss, 410 mass transfer, x, 50, 118, 120, 128, 129, 131, 132, 133, 150, 158, 162, 168, 208, 228, 231, 316 mass transfer process, x, 150, 158, 231 matrix, 32, 36, 80, 111, 123, 124, 161, 225, 279, 282, 295, 299, 307, 313, 418, 447 measurement, 148, 152, 163, 294, 295, 352, 358, 368, 372 measures, 401 meat, 299, 301, 306, 307, 309, 311, 312, 314 mechanical energy, 318 media, 185 medications, 42 medicine, 270 melt, vii, 27, 28, 29, 33, 34, 35, 37, 285, 301 melting, 28, 290, 301, 314 melts, vii, 27, 33, 34, 37, 38, 39 MEMS, 344 mesoscopic, xiii, 442, 445, 446, 447, 448, 451 metals, 356 metric, 53, 172, 187 Mexico, 39 Mg2+, 36 microbial, 118 microelectronics, 344 microscope, 358 microscopy, 29

462

Index

microstructures, 38 microtubes, 353, 355, 357, 359, 362, 363 migration, 11, 22, 23, 25, 444 military, 344 mimicking, 69 mineralogy, 39 minerals, 28, 38 miniaturization, 344 Ministry of Education, 397 misleading, 337, 444, 451 MIT, 266 mixing, 56, 58, 80, 95, 113, 114, 186, 233, 299 mobility, 2, 5, 6, 7, 9, 12, 13, 14, 20, 22, 25 modeling, xi, xiii, 47, 115, 122, 164, 167, 186, 233, 234, 269, 270, 277, 302, 306, 314, 345, 439, 441, 442, 445, 447, 448, 450, 451 models, vii, viii, ix, x, xi, xii, xiii, 42, 43, 44, 46, 95, 97, 107, 108, 112, 113, 114, 118, 127, 130, 131, 134, 146, 149, 156, 158, 159, 164, 166, 167, 185, 186, 232, 233, 234, 259, 260, 265, 270, 271, 272, 283, 295, 302, 309, 314, 315, 316, 340, 399, 400, 425, 441, 445, 447, 448 modulation, 46, 58, 62, 63, 64, 65, 103, 108, 118 modulus, 251, 395, 419, 423 molar conduct, 3 molar volume, 36 molecules, xiii, 28, 119, 347, 441, 442 molybdenum, xi, 269, 284, 299, 300, 301, 314 MOM, ix, 117, 121, 122, 124, 126 momentum, 42, 43, 44, 45, 49, 50, 53, 87, 95, 118, 132, 143, 144, 147, 148, 159, 209, 234, 235, 238, 271, 274, 275, 280, 285, 294, 295, 315, 347, 389, 403 monotone, 220 Monte Carlo, 113, 121, 185, 303, 446 Monte Carlo method, 185, 446 Moscow, 27, 37, 39 motion, xiii, 3, 6, 7, 9, 42, 43, 44, 68, 86, 95, 104, 112, 118, 131, 137, 159, 185, 190, 197, 236, 239, 265, 368, 400, 441, 443, 444 motivation, 260 movement, 395, 401, 405, 406, 407, 409 multidimensional, 113, 121 multiphase flow, ix, 113, 117, 118, 119, 120, 122, 125, 127, 128, 134, 164, 167, 168

N nanofabrication, vii, 1 nanometers, 3, 31, 33 nanotubes, 25 NASA, 229 national, 270 natural, vii, ix, 27, 38, 39, 42, 119, 171, 173, 174, 175, 176, 177, 179, 180, 184, 185, 189, 201, 209, 302, 385, 442 natural gas, 442

Navier-Stokes, xii, xiii, 44, 271, 379, 390, 437, 441, 443 Navier-Stokes equation, xii, xiii, 379, 390, 437, 441, 443 neglect, 182 negotiating, 271 net migration, 444 neutrons, 284, 298, 301, 302 New York, 26, 167, 203, 228, 229, 315, 341, 342, 361, 397, 452, 453 Newton, 214, 393, 403 Newtonian, x, 205, 208, 214, 221, 224, 229 nitrogen, 353, 362 nitrogen gas, 353 nodes, 273, 282, 305, 387, 416, 419, 420, 421, 422, 423, 424, 434 nodules, vii, 27, 28, 29 nonlinear, 186, 238, 271, 393, 400, 416, 421, 445 non-Newtonian, x, 205, 208, 214, 224, 229 non-Newtonian fluid, x, 205, 208, 214, 224, 229 non-uniform, vii, 1, 10, 16, 51, 174, 185 non-uniformity, 10, 16 normal, 36, 37, 101, 102, 103, 112, 139, 140, 142, 145, 161, 188, 246, 277, 279, 281, 295, 298, 303, 345, 380, 392, 405, 406, 417, 418, 421, 426 normalization, 383 norms, 421, 422, 423 nuclear, xi, 113, 164, 269, 270, 283, 284, 298, 302, 314 nucleation, 34, 35, 118, 119, 131, 133, 137, 138, 139, 142, 152, 158, 160, 161, 164, 165, 166 nuclei, 118 nucleus, 139 numerical analysis, 185 Nusselt, 133, 152, 174, 175, 182, 185, 206, 211, 213, 214, 220, 221, 222, 228, 340

O observations, viii, xi, 27, 33, 35, 37, 104, 108, 131, 134, 145, 151, 152, 179, 234, 244, 248, 249, 263, 270, 293 oceans, 43 OECD, 168 OH-groups, 28 oil, 75, 80, 111, 317, 442 operator, xii, 225, 379, 380, 381, 386, 387, 388, 389, 390, 394, 395, 397, 402, 404, 427, 428, 445 optical, 29 optical microscopy, 29 optics, 294 optimization, vii, 1, 114, 305 order statistic, 265 organization, 252 orientation, 257, 265 orthogonality, 226 oscillation, 34, 35, 37, 366 oscillations, 249

Index oxide, 143, 283, 285, 288, 290, 306, 307, 308, 309, 312, 313, 314 oxide thickness, 313, 314 oxygen, 28

P pairing, 112 paper, viii, 41, 43, 49, 53, 67, 110, 113, 120, 203, 234, 270, 315, 363, 416 parabolic, 6, 234, 239, 317, 442 parameter, 5, 44, 54, 65, 68, 111, 150, 323, 333, 346, 380, 383, 385, 389, 395 Paris, 113 partial differential equations, xii, 179, 273, 280, 379, 399, 400, 411, 416, 421, 446 particle nucleation, 164 particles, viii, xii, xiii, 41, 42, 43, 44, 45, 53, 54, 55, 56, 57, 58, 59, 62, 63, 64, 65, 66, 67, 68, 69, 79, 80, 83, 85, 86, 87, 88, 94, 95, 111, 113, 114, 117, 118, 119, 121, 122, 125, 128, 159, 162, 164, 347, 368, 399, 401, 402, 403, 405, 407, 409, 441, 443, 444, 446, 449, 450, 451 partition, 133, 137, 139, 142, 152, 158, 168 pathways, 33 Peclet number, 2, 14, 24 pedagogical, 451 performance, 20, 25, 186, 207, 233, 259, 317, 318, 366 periodic, 238, 366, 370, 375 permeability, 96 permittivity, 3, 4 personal, 366, 367 perturbations, 443 petrographic, 29, 33 Petroleum, 317 pharmaceutical, 343 philosophy, 301 physical mechanisms, 352, 443 physical properties, 28, 209, 283, 306, 313, 347, 395 physics, xiii, 42, 110, 112, 167, 185, 283, 285, 298, 441, 442, 443, 445 pipelines, 442 planar, 315 plants, 42 plausibility, 36 play, viii, 42, 450 point defects, 35 Poisson, 4, 5, 381, 395 Poisson equation, 4 Poisson-Boltzmann equation, 5 pollen, 42 pollution, 43, 167 polymer, 42 polymers, 43 polynomial, 387, 388 polystyrene, 368 poor, 112, 410

463

population, vii, viii, ix, 51, 113, 114, 117, 118, 119, 120, 121, 122, 124, 125, 126, 128, 129, 131, 149, 152, 158, 159, 163, 164, 165, 166, 167, 168 porous, 95 Portugal, 203, 266 potential energy, 420, 423 powder, 143, 290, 293 power, viii, x, xi, 41, 42, 43, 118, 143, 176, 178, 180, 181, 182, 183, 184, 185, 205, 208, 214, 224, 269, 270, 271, 289, 290, 301, 302, 303, 306, 307, 310, 311, 312, 313, 314, 315, 317, 318, 319, 325, 326, 328, 329, 335, 336, 337, 339, 340, 344 power plant, 42 power plants, 42 power-law, x, 205, 206, 208, 214, 217, 218, 222, 224, 229 powers, 303 pragmatic, 43, 271 Prandtl, 46, 135, 162, 172, 174, 175, 275, 276, 315, 340 precipitation, viii, 117 prediction, 56, 62, 75, 80, 97, 112, 134, 145, 147, 148, 149, 152, 156, 158, 159, 162, 168, 169, 298, 301, 361, 366 pressure, xii, 2, 4, 6, 11, 14, 16, 23, 27, 28, 33, 34, 35, 37, 45, 49, 50, 53, 56, 80, 94, 95, 96, 141, 143, 148, 160, 165, 172, 173, 183, 236, 270, 275, 277, 282, 295, 305, 307, 309, 317, 318, 322, 332, 346, 347, 349, 350, 352, 353, 355, 357, 359, 360, 362, 365, 366, 367, 368, 369, 370, 371, 372, 375, 376, 377, 379, 380, 381, 387, 388, 389, 392, 394, 395, 396, 404, 442 printing, 400 probability, 121, 263, 446 probability density function, 121, 446 probability distribution, 263 probable cause, 146 probe, 43, 143, 245, 294, 366 production, 47, 48, 51, 102, 128, 160, 270, 276, 277, 298, 299, 344, 391 program, 121, 294, 295, 302 propagation, 400, 411 property, 273, 280, 317, 319, 320, 326, 330, 336, 387, 410, 412 proportionality, 406 proposition, xiii, 441 protection, ix, 171, 173, 298 prototype, 270, 283, 286, 293, 295 PSD, 120, 122, 123, 124, 125, 126, 129, 132 pseudo, 385 pumping, xi, 317, 318, 319, 325, 326, 328, 335, 336, 337, 362 pumps, 344, 366 purification, viii, 27, 36

Q quadtree, 273

464

Index

R radial distribution, ix, 118, 143, 158, 227, 245 radiation, ix, 171, 172, 173, 176, 180, 183, 184, 185, 298, 301 radical, 156, 157, 360, 448 radiopharmaceutical, 270, 283 radiotherapy, xi, 269 radius, vii, 1, 139, 140, 142, 160, 161, 182, 183, 206, 209, 211, 232, 235, 286, 377 random, 52, 136, 157, 229, 345, 346, 444 range, x, 8, 15, 24, 42, 43, 44, 80, 102, 103, 110, 112, 125, 126, 131, 142, 143, 145, 152, 156, 158, 179, 205, 208, 211, 214, 233, 259, 271, 277, 278, 317, 318, 337, 349, 350, 353, 355, 372, 443, 445, 450 Rayleigh, 172, 173, 174, 175, 179, 180, 183, 184, 195, 400 reactants, 36, 42 reaction rate, 298 realism, 312, 442 recovery, 272, 366 recrystallization, 37 rectilinear, 425 recycling, 39 reduction, viii, xiii, 11, 13, 18, 42, 43, 98, 99, 101, 105, 108, 110, 112, 114, 138, 233, 234, 239, 244, 245, 249, 251, 265, 305, 310, 373, 441, 442, 445, 448 refining, 54, 110 reflection, 257 reforms, 137 refrigeration, 344 refrigeration industry, 344 regulatory requirements, 271 relationship, 152, 213, 238, 278, 366, 404 relationships, 35, 404 relaxation, 46, 54, 68, 444, 445 relaxation time, 46, 54, 68, 444, 445 relevance, 121 reliability, xi, 244, 269, 270, 293, 295 research, ix, xi, xii, 1, 42, 119, 131, 152, 159, 171, 173, 185, 269, 270, 283, 298, 343, 344, 356, 369, 442 research and development, 1 researchers, 44, 344, 352, 356, 360 reservoir, 143 resistance, 176, 178, 209, 315 resolution, vii, 1, 2, 20, 24, 98, 126, 156, 158, 246, 260, 271, 282, 283, 290, 309, 313, 419 resources, 119, 126, 185, 271, 283, 290, 302, 306 response time, 80 retardation, 9 retention, 20 Reynolds, viii, x, xii, xiii, 41, 42, 46, 50, 51, 53, 54, 64, 67, 101, 102, 105, 111, 161, 174, 175, 185, 186, 207, 223, 229, 231, 232, 233, 234, 236, 238, 239, 244, 245, 249, 250, 254, 259, 260, 261, 262,

263, 264, 265, 275, 276, 278, 314, 315, 320, 340, 346, 347, 348, 349, 351, 352, 353, 354, 355, 356, 360, 363, 365, 368, 370, 373, 374, 395, 396, 397, 441, 442, 443, 444, 445, 446 Reynolds number, viii, x, xii, xiii, 41, 42, 50, 51, 53, 54, 64, 67, 101, 105, 111, 161, 174, 175, 185, 207, 223, 231, 232, 233, 234, 236, 238, 239, 244, 245, 249, 254, 259, 260, 261, 263, 264, 265, 278, 314, 315, 320, 340, 346, 347, 348, 349, 351, 352, 353, 354, 355, 356, 360, 363, 365, 368, 370, 373, 374, 395, 396, 397, 441, 442, 443, 446 Reynolds stress model, 185, 186 rheology, 208, 221 rivers, 442 robustness, 158, 282, 295, 314 Rome, 441 room temperature, 173, 339 roughness, xi, xiii, 315, 343, 346, 347, 348, 349, 350, 351, 352, 353, 355, 356, 357, 358, 359, 360, 363, 441 rubber, 183 Russia, 27, 37 Russian, 27, 37, 39 Russian Academy of Sciences, 27, 37

S safety, viii, xi, 41, 45, 113, 164, 269, 282, 283, 285, 289, 296, 303, 306, 313 sample, 30, 32, 33, 37, 447 sampling, 443 sand, 42 saturation, 131, 162 Saudi Arabia, 317 scalar, 125, 126, 133, 160, 185, 404, 405, 409, 428 scalar field, 404, 409, 428 scaling, 5, 68, 276 scatter, 62, 294, 368 scientists, 43 SCP, 113, 163, 168 search, 273, 443 searching, 448 seeding, 306 segregation, 31, 35 selecting, 280 selectivity, vii, 1, 2, 20, 21, 23, 24 SEM, 358, 359 semiconductor, 270, 367, 400 sensitivity, 283, 312, 443 sensors, 344 separation, vii, 1, 2, 10, 20, 22, 23, 24, 25, 26, 34, 35, 37, 156, 272, 296, 344, 366, 376, 389 series, 119, 125, 126, 143, 213, 214, 302, 368, 371, 421, 423 shape, x, 14, 31, 43, 136, 149, 159, 231, 239, 251, 346, 351, 358, 381, 387, 388, 390, 394, 396, 405, 411, 417, 418, 423, 424, 430, 433, 434, 435 sharing, 111

Index shear, 50, 68, 69, 94, 95, 106, 111, 112, 115, 134, 141, 142, 153, 159, 160, 161, 163, 164, 214, 232, 234, 249, 250, 262, 263, 265, 276, 278, 316, 347, 349, 350, 375, 443 shear rates, 214 sign, 48, 386 silica, 39, 352, 353, 355, 359 silicate, 35, 38 silicates, 28, 37 silicon, 270 similarity, 19, 64, 214 simulation, ix, x, 11, 42, 43, 44, 45, 80, 99, 104, 106, 110, 112, 114, 123, 126, 163, 166, 167, 169, 171, 174, 175, 185, 186, 201, 231, 233, 237, 244, 245, 257, 259, 270, 271, 282, 288, 289, 290, 295, 302, 306, 308, 312, 313, 445, 451 simulations, viii, ix, x, 41, 49, 95, 96, 99, 106, 108, 117, 121, 123, 126, 127, 129, 130, 143, 144, 145, 150, 156, 159, 177, 231, 233, 234, 238, 241, 244, 249, 252, 254, 271, 283, 289, 302, 304, 309, 310, 312, 315, 411, 425, 438, 445, 447, 448, 450, 451 singular, 417, 418, 419, 437, 438 singularities, 387 sites, 131, 137, 138, 161 skewness, 234, 254, 255, 257, 259, 263, 264, 265 skin, 98, 99, 101, 105, 106, 108, 109, 114, 233 smoothing, 298, 423 smoothness, 415 software, 125, 312, 393 soil, 181, 182, 183, 185 solar, ix, 171, 173 solar energy, ix, 171, 173 solid phase, 277 solubility, 33, 38 solutions, ix, 2, 20, 24, 43, 117, 120, 121, 144, 188, 208, 214, 215, 228, 229, 270, 272, 273, 289, 439, 444 soot, 128, 159, 164, 166 South Africa, 30, 31, 36, 37 Southampton, 316, 438 Spain, 399 spatial, 22, 118, 121, 132, 207, 236, 238, 400, 401, 402 spatial location, 118 species, 2, 3, 9, 10, 20, 22 specific heat, 159, 172, 319, 330 Specific Heat, 288, 307, 313 spectrum, 118, 143, 448, 451 speed, 2, 9, 10, 11, 12, 13, 14, 20, 22, 23, 24, 25, 156, 233, 283, 377, 380, 415, 421, 444 stability, x, 175, 186, 231, 233, 234, 390, 410, 421, 439 stages, 29, 119, 295 stainless steel, 352, 355, 356, 357, 359 standard deviation, 3, 23 standard model, 450 standards, xi, 269 statistical mechanics, 164 statistics, x, 231, 233, 234, 238, 254, 260, 265

465

steady state, 53, 96, 143, 179, 302, 415 steel, 143, 352, 353, 355, 356, 357, 359 stochastic, xiii, 51, 112, 135, 163, 229, 442, 444, 445, 446, 450 stochastic model, xiii, 442, 450 stochastic processes, 444 stoichiometry, 36 storage, 28, 39, 43, 185, 238, 282, 301 strain, 47, 186, 232, 237, 394, 395 strategies, 438 streams, 207, 208, 213, 221, 228 strength, 448 stress, x, 45, 102, 106, 115, 153, 161, 185, 186, 231, 232, 233, 234, 238, 249, 250, 265, 275, 276, 278, 347, 349, 350, 391, 392, 394, 411, 444 stroke, 366 students, 315 subsonic, xii, 379, 386, 397 Sun, 126, 157, 166, 168, 186, 204 superiority, 286 supply, 143, 271, 283, 299 suppression, 167, 233, 239, 265 surface area, 50, 51, 118, 121, 122, 135, 280, 318, 327, 337, 405 surface diffusion, 411 surface region, 407 surface roughness, xi, 315, 343, 349, 350, 352, 355, 357, 358, 360 surface tension, xii, 141, 159, 160, 162, 361, 399, 411 surfactant, 42, 411, 412 surfactants, xii, 43, 399, 400, 411, 439 swarm, 156, 168 swelling, 377 symbolic, 437 symbols, 6, 242, 243, 255, 360, 389, 395 symmetry, 4, 11, 143, 255, 257, 383, 408, 451 synchronous, 373 systems, 42, 52, 53, 118, 119, 120, 121, 122, 128, 136, 159, 165, 166, 167, 169, 270, 271, 273, 300, 314, 315, 343, 344, 365, 392

T Taiwan, 171 talc, 28, 31, 33, 34, 35, 38 tangible, 42 tanks, 118, 271, 301 targets, xi, 269, 293, 295, 297, 299, 301, 302, 303, 304, 306, 307, 313, 314, 315 technology, 165 TEM, vii, 27, 28, 29, 31, 33, 35, 38 temperature, ix, x, xi, 2, 4, 28, 39, 131, 137, 139, 143, 161, 169, 171, 172, 173, 174, 175, 178, 179, 180, 183, 184, 185, 188, 194, 195, 197, 200, 201, 205, 206, 207, 208, 209, 211, 212, 213, 217, 218, 219, 220, 221, 224, 227, 275, 277, 283, 285, 289, 290, 291, 298, 302, 303, 304, 305, 306, 307, 308,

466

Index

309, 310, 311, 312, 313, 314, 317, 318, 319, 320, 321, 324, 325, 326, 330, 331, 332, 333, 334, 335, 336, 337, 339, 340, 341, 394 temperature dependence, xi, 318 temperature gradient, 188, 219, 275, 321 tensor field, 400, 401, 402, 428 Texas, 224 textbooks, 119, 392, 394 theory, xii, 46, 47, 115, 233, 352, 353, 355, 356, 379, 381 Thermal Conductivity, 279, 288, 307, 313 thermal equilibrium, 182 thermal expansion, 172 thermodynamic, 318 thermodynamic parameters, 318 thermo-mechanical, 154, 158 thin film, 105 three-dimensional, ix, 132, 144, 158, 171, 173, 174, 175, 177, 179, 184, 189, 201, 233, 280, 292, 294, 302, 316 threshold, 22 time, x, xi, 2, 3, 9, 22, 23, 34, 37, 43, 44, 46, 52, 54, 63, 68, 85, 96, 105, 114, 121, 126, 132, 136, 138, 140, 142, 143, 161, 185, 231, 238, 269, 271, 273, 275, 280, 281, 286, 290, 299, 301, 302, 305, 306, 345, 368, 369, 370, 372, 373, 383, 392, 394, 400, 401, 402, 403, 404, 405, 406, 407, 409, 410, 411, 412, 413, 416, 417, 418, 419, 420, 421, 422, 423, 424, 428, 429, 430, 431, 432, 435, 436, 443, 444, 445, 446, 451 time consuming, 418 tolerance, 309 topological, xii, 399, 400, 417 topology, 409, 412, 444 total energy, 420, 423, 430 tracers, 80 tracking, xii, 43, 44, 115, 122, 126, 149, 399, 412 trajectory, 123, 412, 446 transducer, 367 transfer, ix, x, xi, 42, 46, 50, 51, 115, 118, 126, 131, 133, 137, 147, 152, 159, 160, 164, 165, 167, 168, 171, 172, 173, 174, 175, 176, 180, 181, 182, 183, 184, 185, 201, 205, 206, 207, 208, 212, 222, 223, 227, 228, 229, 231, 232, 233, 238, 269, 270, 271, 273, 278, 302, 304, 310, 315, 317, 318, 319, 320, 321, 323, 324, 325, 326, 327, 330, 331, 332, 333, 334, 335, 336, 337, 339, 340, 347, 361, 362, 363, 443, 444 transformation, viii, 27, 35, 38, 121, 122, 380, 381, 415 transition, x, xi, 28, 143, 146, 156, 157, 158, 168, 173, 174, 175, 179, 185, 231, 233, 343, 344, 346, 347, 349, 352, 353, 355, 356, 358, 362 transitions, 163, 305 transparency, 284 transparent, xi, 270, 291, 293, 366 transport, vii, xiii, 1, 2, 3, 4, 9, 10, 11, 25, 33, 42, 44, 46, 47, 48, 53, 123, 124, 126, 132, 134, 144, 157,

164, 165, 166, 168, 185, 209, 233, 234, 271, 276, 280, 281, 289, 303, 344, 362, 366, 400, 441 transpose, 225 transverse section, 178, 189, 194, 197, 200 travel, 51, 52, 87, 123, 124, 131 trend, 18, 80, 87, 94, 101, 102, 221, 249, 257, 263, 297, 303, 317 trial, 320 trial and error, 320 triangulation, 273 turbulence, viii, x, xi, xii, xiii, 41, 42, 44, 46, 47, 48, 49, 51, 52, 53, 58, 59, 63, 64, 65, 69, 94, 95, 97, 102, 104, 105, 108, 111, 112, 113, 114, 115, 118, 122, 134, 135, 136, 145, 146, 147, 149, 166, 167, 175, 185, 186, 201, 231, 232, 233, 234, 237, 239, 249, 251, 259, 261, 263, 265, 269, 271, 272, 276, 295, 302, 314, 315, 316, 350, 365, 368, 372, 373, 374, 375, 376, 377, 379, 390, 391, 392, 395, 442, 443, 445, 447 turbulent, vii, viii, x, xi, xii, xiii, 41, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 58, 62, 66, 88, 94, 95, 97, 99, 102, 103, 110, 111, 112, 113, 114, 115, 118, 129, 131, 133, 134, 135, 136, 138, 143, 145, 146, 149, 153, 156, 157, 158, 160, 161, 162, 164, 166, 168, 174, 179, 185, 186, 207, 209, 229, 231, 233, 234, 235, 236, 237, 239, 241, 243, 244, 245, 247, 249, 253, 254, 255, 257, 259, 260, 261, 263, 265, 266, 267, 269, 271, 272, 275, 276, 277, 278, 279, 280, 289, 290, 304, 305, 314, 315, 318, 319, 320, 330, 342, 343, 344, 345, 346, 347, 349, 352, 353, 355, 358, 362, 365, 366, 367, 369, 371, 373, 375, 377, 379, 390, 391, 392, 393, 394, 441, 442, 443, 444, 445, 446, 447, 448, 449, 451, 453 turbulent flows, xi, xiii, 110, 179, 186, 207, 260, 277, 278, 314, 315, 343, 344, 345, 347, 353, 355, 441, 442, 443, 444, 445, 446, 447, 448 turbulent mixing, 377 two-dimensional (2D), ix, 25, 126, 130, 171, 173, 174, 175, 179, 180, 181, 184, 185, 188, 294, 362, 366, 425 two-way, 43, 44, 53

U ubiquitous, viii, xiii, 27, 37, 441, 442 Ukraine, 27 uncertainty, 146, 345 uniform, xi, 4, 10, 15, 16, 53, 59, 62, 63, 96, 144, 186, 207, 214, 295, 297, 309, 310, 311, 312, 318, 319, 331, 346, 349, 403, 404 universal gas constant, 4 updating, 421 uranium, xi, 269, 270, 285, 288, 297, 299, 301, 302 uranium oxide, 285 uti, 52, 136

Index

V vacancies, 36 valence, 3 validation, ix, 111, 118, 120, 125, 134, 163, 234, 241, 265, 270, 302, 306, 307, 312, 313, 315, 344 validity, 11, 43, 283 values, 6, 20, 30, 33, 59, 62, 63, 80, 81, 98, 135, 188, 212, 214, 217, 218, 222, 223, 227, 233, 238, 239, 245, 249, 252, 255, 257, 263, 265, 273, 276, 277, 280, 297, 302, 305, 324, 334, 335, 346, 349, 351, 352, 356, 360, 368, 370, 372, 377, 380, 388, 391, 392, 393, 394, 395, 396, 408, 415, 416, 418, 419, 421, 422, 424, 426 vapor, 159, 160, 161, 162, 169 variable, 5, 38, 125, 131, 173, 189, 210, 229, 280, 288, 313, 410, 415, 421 variables, 49, 50, 53, 118, 120, 121, 123, 125, 132, 156, 235, 237, 277, 387, 390, 391, 392, 394, 413, 415, 421, 422, 445 variance, 22, 451 variation, 17, 18, 29, 119, 148, 195, 201, 217, 219, 223, 244, 263, 272, 302, 317, 318, 320, 321, 327, 330, 331, 335, 336, 339, 340, 352, 369, 370, 371, 373, 375, 377, 401 vector, 50, 121, 124, 160, 161, 189, 190, 191, 192, 195, 197, 198, 275, 277, 292, 382, 383, 391, 392, 394, 402, 403, 404, 405, 406, 407, 421, 425, 426, 427, 428, 445, 447 velocity, ix, x, xi, xii, 4, 5, 6, 7, 10, 11, 12, 14, 15, 16, 19, 22, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 67, 68, 79, 80, 84, 88, 94, 95, 96, 97, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108, 111, 114, 118, 121, 123, 124, 127, 129, 131, 136, 138, 142, 143, 144, 147, 154, 156, 157, 158, 161, 166, 168, 172, 177, 178, 187, 188, 189, 190, 195, 197, 205, 206, 208, 209, 210, 214, 217, 224, 227, 231, 232, 233, 234, 235, 236, 237, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 251, 252, 253, 254, 255, 256, 257, 258, 259, 261, 262, 263, 264, 265, 270, 275, 276, 277, 278, 282, 291, 292, 293, 294, 295, 296, 297, 298, 303, 304, 305, 306, 314, 317, 318, 340, 345, 346, 360, 365, 366, 367, 368, 371, 372, 373, 374, 375, 377, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 392, 394,

467

399, 400, 402, 403, 405, 409, 411, 412, 413, 414, 415, 416, 417, 418, 419, 421, 422, 423, 431, 442, 443, 446, 447, 448, 449, 450 ventilation, 176 versatility, 314 Victoria, 117 viscosity, xi, 3, 4, 9, 17, 44, 45, 46, 48, 49, 51, 95, 97, 104, 114, 134, 135, 145, 161, 167, 172, 185, 207, 214, 232, 237, 272, 275, 276, 315, 317, 318, 319, 320, 324, 325, 330, 333, 334, 335, 339, 340, 349, 350, 368, 387, 391, 394, 397 visible, 35, 293, 368 visualization, 108, 292, 293, 296, 368, 376, 443 voids, 32, 33, 35 vortex, 381 vortices, 43, 63, 65, 104, 112, 175, 295, 375, 376, 377, 381, 382, 383

W wall temperature, xi, 173, 177, 185, 189, 201, 214, 229, 279, 318, 331, 334, 335, 341 warfare, 344 water, vii, viii, 27, 28, 29, 33, 34, 35, 37, 38, 39, 65, 102, 103, 112, 142, 143, 166, 183, 271, 290, 293, 294, 298, 300, 303, 309, 310, 311, 312, 314, 317, 339, 352, 353, 355, 359, 362, 368, 375, 400, 411, 423, 437, 440 weakness, 120 welding, 283 wettability, 140 wind, 42 windows, 286 worms, 447 writing, 321, 447

Y yield, 7, 22, 24, 44, 126, 130, 131, 136, 156, 273, 295, 301, 327, 328, 337

Z zeta potential, 3, 4, 5, 16