Computational Thermo-Fluid Dynamics E-Book

Petr A. Nikrityuk

• Herausgeber: Wiley-VCH
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 579

Veröffentlichungsjahr: 2011

Beschreibung

Combining previously unconnected computational methods, this monograph discusses the latest basic schemes and algorithms for the solution of fluid, heat and mass transfer problems coupled with electrodynamics. It presents the necessary mathematical background of computational thermo-fluid dynamics, the numerical implementation and the application to real-world problems. Particular emphasis is placed throughout on the use of electromagnetic fields to control the heat, mass and fluid flows in melts and on phase change phenomena during the solidification of pure materials and binary alloys. However, the book provides much more than formalisms and algorithms; it also stresses the importance of good, feasible and workable models to understand complex systems, and develops these in detail. Bringing computational fluid dynamics, thermodynamics and electrodynamics together, this is a useful source for materials scientists, PhD students, solid state physicists, process engineers and mechanical engineers, as well as lecturers in mechanical engineering.

Leseprobe

Contents

Cover

Half Title page

Related Titles

Title page

Copyright page

Preface

Acknowledgments

Chapter 1: Introduction

1.1 Heat and Fluid Flows in Materials Science and Engineering

1.2 Overview of the Present Work

Chapter 2: Mathematical Description of Physical Phenomena in Thermofluid Dynamics

2.1 Conservation Equations for Continuum Media

2.2 Boundary and Initial Conditions

2.3 Conservation Equations in Electromagnetics

Chapter 3: Discretization Approaches and Numerical Methods

3.1 The Finite Difference Method

3.2 The Finite Volume Method

3.3 Solution of Linear Equation Systems

Chapter 4: Calculations of Flows with Heat and Mass Transfer

4.1 Solution of Incompressible Navier–Stokes Equations

4.2 Pressure and Velocity Coupling: SIMPLE Family

4.3 Illustrations of Schemes for Flow with Heat Transfer

4.4 Complex Geometry Problems on Fixed Cartesian Grids

Chapter 5: Convection–Diffusion Phase-Change Problems

5.1 Some Aspects of Solidification Thermodynamics

5.2 Modeling of Macroscale Phase-Change Phenomena

5.3 Turbulent Solidification

5.4 Microscale Phase-Change Phenomena

5.5 Modeling of Crystal Growth

5.6 Melting of Pure Calium under the Influence of Natural Convection

Chapter 6: Application I: Spin-Up of a Liquid Metal in Cylindrical Cavities

6.1 Spin-Up of Isothermal Flow Driven by a Rotating Magnetic Field

6.2 Impact of Buoyancy Force on Spin-Up Dynamics

Chapter 7: Application II: Laminar and Turbulent Flows Driven by an RMF

7.1 Laminar Flows: State of the Art

7.2 Turbulent Flows

Chapter 8: Application III: Contactless Mixing of Liquid Metals

8.1 Mixing under Zero-Gravity Conditions

8.2 The Impact of Gravity on Mixing

Chapter 9: Application IV: Electromagnetic Control of Binary Metal Alloys Solidification

9.1 Control of a Binary Metal Alloy Solidification by Use of Alternating Current Fields

9.2 Control of Solidification by Use of Steady Electromagnetic Fields

9.3 The Impact of a Steady Electrical Current on Unidirectional Solidification

9.4 The Impact of an Electric Field on the Shape of a Dendrite

9.5 The Impact of Parallel Applied Electric and Magnetic Fields on Dendritic Growth

References

Index

Petr A. Nikrityuk

Computational Thermo-Fluid Dynamics

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The Author

Dr. Petr A. NikrityukTU Bergakademie FreibergZIK VirtuhconFuchsmühlenweg 909596 FreibergGermany

All books published byWiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

Library of Congress Card No.: applied for

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Bibliographic information published by the Deutsche NationalbibliothekThe Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.d-nb.de.

© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany

All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.

This book is based on the Habilitation submitted to Technische Universität Bergakademie Freiberg 2010.Specialization is Fluid Dynamics. Reviewers are Prof. Dr.-Ing. BMeyer, Prof. Dr.-Ing. habil. C. Brücker and Prof. (em.) Dr.-Ing. R.Grundmann.

ISBN Print 978-3-527-33101-7

ISBN oBook 978-3-527-63607-5ISBN ePub 978-3-527-63608-2ISBN ePDF 978-3-527-63609-9ISBN Mobi 978-3-527-63610-5

Preface

“Success consists of going from failure to failure without loss of enthusiasm.”

Winston Churchill (1874–1965)

This work is about modeling and simulations of different physical processes related to materials science and engineering. Today, the design of innovative, multifunctional materials, including their optimal use in different engineering applications, is impossible without computer modeling. The impressive development of computers, in particular the invention of multicore processors, and the reduction in their prices over the last 20 years have made it possible to perform sophisticated scientific computations on a standard PC. In parallel with the development of hardware, increasingly efficient commercial software (ANSYS-FLUENT, ANSYS-CFX, StarCD, Flow-3D, and so on) and noncommercial software (openFOAM, ELMER, etc.) has appeared that is used in many industrial applications for the development and optimization of different processes in materials science and engineering. In particular, the role of open source codes in the development of new models and their validation against experimental data is very important. First of all, the use of commercial programs for these goals is sometimes a nontrivial task, especially for phase-change-related problems, and second, open source codes are transparent in the sense of the models used, are easy to debug, and have more possibilities when it comes to tuning models.

This monograph was written as a specialized textbook for master’s or Ph.D. courses in the area of computational thermal and materials science, and engineering. The main aims of this monograph are as follows: first, to give some main insights into the latest basic mathematical models and numerical methods commonly used by engineers to solve coupled-heat and mass-transfer problems linked to materials processing, and second, to demonstrate the new results gained by the author in materials-science-related applications. In particular, the emphasis in applications is made on the use of different electromagnetic fields to control the heat, mass, and fluid flows in melts. The finite-volume method is favored in this work due to its perfect suitability for solving different multiphysical and multiscale problems.

This work is divided into nine major chapters. The first five chapters are devoted to a short description of the latest numerical methods and models used by engineers and researchers in applied materials science and engineering. Chapter 1 summarizes the role of thermofluid dynamics in materials science and engineering and gives a short overview of the present work. Chapter 2 introduces the basic conservation equations for continuum media and electromagnetic fields. Chapter 3 explains the basic discretization approaches and numerical methods used in thermofluid dynamics. Particular attention is paid to finite-volume methods as the most popular in the computational-heat and mass-transfer community. After each chapter illustrative examples are given to demonstrate the advantages and disadvantages of different methods. Chapter 4 contains a short introduction to basic algorithms used in the simulation of fluid flows coupled with heat and mass transfer. The most popular immersed-boundary methods are discussed. Some of them are illustrated by benchmark tests. Chapter 5, which comprises the principal part of the book, is devoted to a short description of existing models for the simulation of phase-change phenomena applied to pure materials and binary alloys. After this chapter a short benchmark example is given in order to demonstrate the accuracy of the numerical methods and models discussed in the chapter. The last four chapters (Chapters 6–7) are concerned with the numerical modeling of a variety of specific applications solved using the finite-volume method. The main purpose of these chapters is to show the main insights into the physics of rotating fluids driven by electromagnetic fields and of phase-change phenomena under the influence of electromagnetic fields. All these examples are concerned with heat and mass transfer coupled with fluid flow.

It should be noted that due to the various fields of computational science and engineering proposed in this book, the text provides a rather introductory view of numerical and physical models. Readers who wish to gain more insights into one of the fields discussed here are referred to the specialized literature cited in the text.

Acknowledgments

“We make a living by what we get, but we make a life by what we give.”

Winston Churchill (1874–1965)

The theoretical part of this manuscript was basically done during my work at CIC Virtuhcon, Department of Energy Process Engineering and Chemical Engineering, Technische Universität Bergakademie Freiberg, where I am currently working. The part of the manuscript devoted to applications is based on the important achievements of my research performed at Technische Universität Dresden, Institute for Aerospace Engineering, within the framework of Collaborative Research Program SFB609, financed by the Deutsche Forschungsgemeinschaft (German Research Foundation).

First I would like to thank Prof. Igor P. Nazarenko, my doctorate advisor from Moscow Aviation Institute (MAI), for his encouragement and financial support during my Ph.D. study at MAI 10 years ago.

I would also like to express my deep appreciation to Prof. Dr.-Ing. Roger Grundmann and Dr. Kerstin Eckert, who gave me many opportunities in my research, steadily supported me in my work, and shared their vast expertise with me. I am especially grateful to Prof. Grundmann for his support during the writing of this manuscript.

Some original data presented in this monograph are the result of collaborative work. I would like to make special mention of the contribution of Sergey Ananiev to the implementation of the modified cellular automaton model in the computer code in the context of Chinese–German cooperation in electromagnetic processing of materials, financed by the Deutsche Forschungsgemeinschaft. I was very glad that Sergey agreed to work on our team. Our productive discussions in the field of scientific computing have made possible the development of new physical models (catastrophic elastic remelting, see [114]) and numerical models (Cartesian grid matrix-cut method, see [1]).

Professor D.M. Stefanescu from The Ohio State University, USA, and Professor M. Ungarish from The Israel Institute of Technology, Israel, are gratefully acknowledged for their help and productive cooperation in the fields of microscale modeling of solidification and numerical simulation of spin-up processes, respectively. I would also like to single out Prof. D. Eskin from Delft University for his help related to understanding solidification.

I am very grateful to Drs. S. Eckert and G. Gerbeth as well as to Bernd Willers and Dirk Räbiger from the Research Center Dresden-Rossendorf for close cooperation and productive discussions that resulted in joint publications.

I extend thanks to all of my colleagues from the Institute for Aerospace Engineering, Technische Universität Dresden for the friendly atmosphere and their help, in one way or another, in my research. In particular I would like to thank Dr. Thomas Albrecht, Dr. Kristina Koal, Dr. Andreas Richter, and Armin Heinze for sharing some Unix scripts to produce a visualization and animation of the simulations.

Professor M. Peric is gratefully acknowledged for his open source code for the solution of Navier-Stokes equations.

I am grateful to Professor Bernd Meyer from Technische Universität Bergakademie Freiberg, where I am currently working, for his financial support, academic freedom, and encouragement in writing this manuscript. The administrators at Technische Universität Bergakademie Freiberg are acknowledged for their support of my research.

I am thankful to Dr. A. Richter, Frank Dierich, Robin Schmidt, Kay Wittig, and Anne Ellen Koth for their comments and careful reading of the manuscript.

I hope that any colleagues whose work has not been mentioned by me in this acknowledgment will forgive me since any omissions are unintentional.

And, finally, I would like to thank my wife, children, and parents for their love, encouragement, and support. Without their support, this work would never have come into existence.

Freiberg 2011

Petr Nikrityuk

Chapter 1

Introduction

“In CFD there are no non-solvable problems, there is only the lack of computing time to solve them.”

CFD community

1.1 Heat and Fluid Flows in Materials Science and Engineering

Materials science and engineering is one of the most important and active areas of research in computational heat transfer today. The development of novel materials and innovative processing technologies today is impossible without the assistance of computational thermofluid dynamics (TFD).1 For example, fluid flow and heat transfer are extremely important in materials processing techniques such as crystal growing, casting, chemical vapor deposition, spray coating, and welding. For instance, the flows that occur in melts during crystal growing due to temperature and concentration differences can modify the quality of the crystal and, thus, of the semiconductors made from this crystal. The buoyancy-driven flows generated in a melt by casting processes strongly influence micro- and macrosegregation and, ultimately, the microstructure of solidified alloys. As a result, it is important to understand these flows and develop technologies to control such effects. One way to gain such control is through the use of electromagnetic fields [2]. For instance, over the last 30 years electromagnetic fields have become an important part of materials processing technologies [3]. Nowadays the electromagnetic processing of materials (EPM) is one area of engineering where electromagnetic fields are used to process innovative materials such as semiconductors, pure metals, multicomponent alloys, and electrolytes. The background required for this field of engineering is interdisciplinary, basically combining materials science and magnetohydrodynamics.2

As a consequence of the importance of fluid flow and heat and mass transfer in materials processing, extensive work has been carried out, presently directed at numerical modeling; see reviews [4, 5]. Following these reviews computer modeling became one of the most crucial elements in the design and optimization of novel technologies in the field of engineering and materials science. However, numerical simulations of flows relevant to materials science and engineering often include complex physical and chemical phenomena. And what is often lacking is a proper mathematical model capable of adequately describing the physical processes. But what does it mean to develop a model of any physical process? As was mentioned at the beginning, practical processes and systems are very often complicated. Thus, to be able to solve a problem, basically we have to simplify some phenomena within this problem through idealizations and approximations. This process of simplifying a given problem is termed model development. Once a mathematical model is produced, it has to be implemented in computer code and then validated against experimental data.3 If the model is a good representation of the actual system under consideration, it can be used to study the behavior of the system. This information may be used in the design of new processes or in tuning the performance of existing processes to obtain an optimal design.

One advantage of computer modeling is that the behavior and characteristics of a system may be investigated without actually fabricating a prototype. Thus, the total costs of product development can be reduced. In addition, it should be noted that the simplifications and approximations that lead to a mathematical model also indicate the dominant variables in a problem. This helps in developing efficient physical or experimental models. The best strategy to develop a good working model is to start from a simple model and then to add complexity as the solution proceeds. Then, comparisons with experimental data may provide ways of improvement. By contrast, if one starts from a sophisticated model, then not even a converging solution may be obtained. However, even if computational results are obtained (after a long debugging procedure), it would be problematic to identify possible improvements to a such complex model; for example, see [6].

The basic conservation equations describing fluid flow were already available at the end of the eighteenth century. Major contributions were made by Newton, Euler, Lagrange, Navier, and Stokes [7]. However, the numerical methods to solve these equations for engineering applications were developed in the second half of the twentieth century due to the appearance of computers. A historical record of scientists contributing to the development of fluid mechanics can be found in the review written by Durst et al. [7]. Since this review, computational fluid dynamics (CFD) has already accumulated the so-called critical mass of computational methods and computational resources such that one can say that the golden age of fluid mechanics lies ahead of us [7]. This statement has been demonstrated by the rapid increase of publications devoted to numerical simulations of flow-related problems in all engineering areas from bioengineering to materials science engineering.

It is true that the invention of the computer made it possible to obtain particular solutions for typical flows in different engineering applications including phase-change phenomena. Today, a wide range of commercial software is available on the market allowing engineers to predict and optimize heat and fluid flow in various industrial applications. However, there are still many uncertainties in predicting multiphase and phase-change flow problems, for example, gas–liquid or solid–liquid–gas system behavior. At the same time, the use of so-called direct numerical simulations is limited by the lack of computing power to perform direct numerical simulations of natural multiscale processes including turbulent flow problems for high Re numbers or even the solidification of alloys. Thus, the development of novel mathematical models covering the multiscale and multiphysical nature of many fluid-flow-related problems remains a current task for engineers engaged in CFD. As a result, in spite of the “golden age of fluid mechanics,” much remains to be done for the next generation of CFD scientists.

The main goal of the present work is to sketch out the role of fluid mechanics in phase-change phenomena by way of a combined theory of numerics and solidification including some illustrative examples. Finally, it should be noted that no attempt has been made in this monograph to explore all aspects of solidification and computational TFD. In particular, numerous books dealing with CFD have already been published. Some of the best, by subject matter, are cited below:

Mathematical fluid mechanics [8],Physics of fluid mechanics [9],Numerical aproaches to heat and fluid flow for beginners [10],Computational methods for fluid dynamics (incompressible flows) [11, 12],Computational methods for fluid dynamics (compressible flows) [13, 14],Fluid flows in magnetohydrodynamics [15, 16].

The same is true of books related to descriptions of solidification. Currently, several books have been published that are devoted to different aspects of solidification modeling including:

Phenomenological description of solidification processes [17],Fundamentals of solidification with numerous examples [18, 19],Solidification theory for engineers [20],Numerical modeling in material science and engineering including fracture mechanics [21],Modeling of moving boundaries with reference to solidification [24, 26].

However, none of these books fully discusses the computational schemes and algorithms for the solution of the governing conservation equations for fluid flow and heat and mass transfer coupled with electrodynamics equations. As a result, the theoretical part of this work presents only those aspects of numerical algorithms that are primarily related to fluid flow magnetohydrodynamics and phase-change problems with reference to materials science and engineering applications.

1.2 Overview of the Present Work

This work is about modeling and simulations of different physical processes related to materials science and engineering. In particular, the goal of writing this monograph is to present recent developments in the modeling of heat- and mass-transfer applications related to phase-change phenomena under the influence of electromagnetic fields. In order to supply the information required for the reader to gain a basic understanding of the methods used in this work for solving fluid-flow-related problems, a summary of the numerical schemes and pressure-based algorithms for the solution of Navier–Stokes equations is provided. In parallel, to illustrate the computational and theoretical issues involved, examples arising from materials processing and fluid-flow-control applications are chosen to give a detailed description of the author’s findings. In the context of each physical phenomenon discussed in this work, the entire scope of the computational setup (including problem and model formulation, code and model validation, scaling, and physical interpretation) is described systematically.

The monograph aims to accomplish the following objectives:

Present basic conservation equations and boundary conditions used in flow-related problems in materials science and engineering.Show basic discretization schemes and algorithms for the numerical solution of convection- and diffusion-related problems including some methods for the solution of a linear equation system.Present recent developments in CFD for the treatment of complex geometry problems using fixed Cartesian grids.Present the basic aspects of macro- and microscale modeling of pure and binary metal alloy solidification including the control of phase-change phenomena by application of electromagnetic fields.Show comparisons between present simulations and experimental data published in the literature.Illustrate an interpretation of simulation results devoted to the control of fluids and heat and mass transfer using different combinations of electromagnetic fields related to materials science applications.

In what follows, an overview of the chapters and their content is given.

Chapter 2 briefly reviews basic conservation equations such as the conservation of mass, of momentum, of energy, and of solute. In addition, the standard boundary and initial conditions required for the solution of conservation equations are given and their physical meaning is discussed. Additionally, the equations of electromagnetism are covered in this chapter as clearly as possible. Finally, there is an illustrative example of the calculation of the Lorentz force induced by a rotating magnetic field applied to nonhomogeneous electroconducting media.

Chapter 3 explains the basic discretization approaches and numerical methods used in TFD. Particular attention is paid to finite volume methods as the most popular in the computational heat- and mass-transfer community. After each section illustrative examples are given to demonstrate the advantages and disadvantages of different numerical schemes such as the central difference scheme (CDS), the upwind first-order scheme (UDS), the linear upwind difference scheme (LUDS), the upstream weighted differencing scheme (UWDS), the total variation diminishing differencing scheme (TVD), the power-law scheme (PDS), and the upwind third-order scheme (QUICK). Finally, an example is introduced to illustrate different iterative methods for the solution of the heat-transfer equation.

Chapter 5 describes basic algorithms used when simulating incompressible fluid flows coupled with heat and mass transfer. There is a demonstration of the accuracy of different discretization schemes (UDS, LUDS, QUICK, PDS, CDS-DC) modeling convective terms in solving steady incompressible flow and heat transfer in a two-dimensional lid-driven cavity. Recent novelties in the field of fixed Cartesian grid methods, including immersed boundary methods, are discussed. Some of them are illustrated by benchmark tests.

Chapter 6 introduces existing models for the simulation of phase-change phenomena on the macro- and microscales applied to pure materials and binary metal alloys. The so-called single-domain mixture model for the macroscale prediction of solidification and the modified cellular automaton model for microscale modeling are favored in this work. The modeling of turbulent solidification is reviewed and described. Following the chapter a short benchmark example is given to demonstrate the accuracy of the fixed grid technique, where the solid–liquid interface is treated implicitly with the two-phase region modeled as a porous medium.

Chapter 7 illustrates the performance of the numerical schemes given in previous chapters on the basis of a numerical study of the spin-up of liquid metal driven by a rotating magnetic field. In particular, the transient axisymmetric swirling flow in a closed cylindrical cavity, driven by a rotating magnetic field (RMF), has been studied by means of numerical simulations. Based on the time histories of the volume-averaged azimuthal and meridional velocities, it has been shown that RMF-driven spin-up can be divided into two phases. The spin-up starts with an initial adjustment (i.a.) phase in which a secondary meridional flow in the form of two toroidal vortices is established. The i.a. phase is generally completed on achieving the first local maximum in the volume-averaged kinetic energy of the secondary flow. The second phase has been referred to as inertial, where the establishment of Bödewadt layers at the horizontal walls plays a major role. Additionally, the influence of stable thermal stratification on the spin-up dynamics is studied numerically. It is found that a stable thermal stratification damps the inertial waves and significatly reduces the magnitude of the meridional flow velocities. However, an RMF-driven flow under the action of a stable thermal stratification became unstable earlier in comparison to the isothermal flows. An increase in the Grashof number leads to the occurrence of axisymmetric instability along the side wall in the form of Taylor–Görtler vortices.

Chapter 7 explores the different flow regimes of laminar and weak turbulent rotating fluid flows driven by a rotating magnetic field. Both two-dimensional direct numerical simulations and RANS-based simulations are performed to answer the following questions. How can we define a transition between a viscous Stokes flow and an inertial regime? How relevant is the aspect ratio for the secondary flow intensity? What is the influence of magnetic forcing and the aspect ratio of the cavity on the side wall, and on the top and bottom torques? What is the influence of Taylor–Görtler vortices induced by the rotation of a melt on a time-averaged flow structure?

Chapter 8 addresses the numerical modeling of transient mass and momentum transport until the homogenization of two miscible fluids is achieved under the action of externally imposed rotating and traveling magnetic fields. The main aim of the study is to investigate the physical mechanisms responsible for enhancing a mixing process using alternating magnetic fields and to explore the role of buoyancy in rotary mixing. Finally, different combinations of a TMF and an RMF are considered in terms of the effectiveness of the mixing. It is shown that the time of the initial adjustment phase is the key parameter for enhancing the mixing processes by the periodic superposition of different electromagnetic fields.

Chapter 9 presents different ways to control binary metal alloy solidification using combinations of electromagnetic fields. In particular, in this chapter two types of electromagnetic stirring (EMS) are considered: contactless EMS and contact EMS. Contactless EMS is demonstrated through the application of rotating and traveling magnetic fields by the unidirectional solidification of an Al-Si alloy and the side-cooled solidification of an Al-Cu alloy. Contact EMS is illustrated by considering the unidirectional solidification of a Pb-Sn alloy under the influence of superimposed steady external magnetic fields and steady electrical currents applied directly to a melt by means of electrodes that have direct contact with the melt. The main purpose of this chapter is to demonstrate the influence of the so-called Lorentz force induced by electromagnetic fields on phase-change phenomena on a micro- and macroscale.

Almost all chapters include examples. They are intended to illustrate the properties of numerical schemes or to explore the role of fluid flows in heat and mass transfer coupled by means of fluid flow. The methods of analysis of the numerical results presented in this monograph can be used for a wide variety of fluid-flow problems encountered in materials science and engineering.

1) The field of TFD includes the complete set of governing equations of fluid dynamics coupled with energy and mass conservation equations.

2) Magnetofluiddynamics, or magnetohydrodynamics (MHD), describes phenomena occurring at the frontier separating fluid mechanics and electromagnetics.

3) A mathematical model is one that represents the performance and behavior of a given system in terms of mathematical equations.