Frontiers of Computational Fluid Dynamics 2006.

By: Caughey, D. AContributor(s): Hafez, M. MMaterial type: TextTextPublisher: Singapore : World Scientific Publishing Co Pte Ltd, 2005Copyright date: ©2005Description: 1 online resource (466 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9789812703187Subject(s): Fluid dynamics -- Data processing -- Congresses | Fluid mechanicsGenre/Form: Electronic books.Additional physical formats: Print version:: Frontiers of Computational Fluid Dynamics 2006DDC classification: 532 LOC classification: QA911.F776 2005Online resources: Click to View
Contents:
Intro -- Contents -- Dedication -- Chapter 1 The Contributions of David Caughey to Computational Fluid Dynamics Mohamed M. Hafez -- 1.1 Introduction -- 1.2 Shock Wave Structure and Sonic Boom -- 1.3 Potential Flow Simulations -- 1.4 Solutions of Euler Equations -- 1.5 Solutions of Navier-Stokes Equations -- 1.6 Simulation of Turbulent Reactive Flows -- 1.7 Special Topics -- 1.8 Review Articles -- 1.9 Fluid Mechanics: An Interactive Text -- 1.10 Concluding Remarks -- 1-A Ph.D. Students Supervised by David A. Caughey -- 1-B Publications of David A. Caughey -- I. Design and Optimization -- Chapter 2 Computational Fluid Dynamics in the Analysis and Design of Engineered Systems M. Damodaran, S. Alil and S. Dayanandan -- 2.1 Introduction -- 2.2 Flow Modeling for Fire Control Strategies and Scenario Planning in an Underground Road Tunnel -- 2.3 Flow Modeling in a Hard Disk Drive Enclosure -- 2.4 Concluding Remarks -- 2.5 Bibliography -- Chapter 3 Advances in Aerodynamic Shape Optimization Antony Jameson -- 3.1 Introduction -- 3.2 Formulation of the Optimization Procedure -- 3.2.1 Gradient Calculation -- 3.3 Design using the Euler Equations -- 3.4 The Reduced Gradient Formulation -- 3.5 Optimization Procedure -- 3.5.1 The Need for a Sobolev Inner Product in the Definition of the Gradient -- 3.5.2 Sobolev Gradient for Shape Optimization -- 3.5.3 Outline of the Design Procedure -- 3.6 Case Studies -- 3.6.1 Two-Dimensional Studies of Transonic Airfoil Design -- 3.6.2 B747 Euler Planform Result -- 3.6.3 Super B747 -- 3.7 Super P51 Racer -- 3.7.1 Shape Optimization for a Transonic Business Jet -- 3.8 Conclusion -- 3.9 Acknowledgment -- 3.10 Bibliography -- Chapter 4 Design Optirnization of Propeller Blades Luigi Martinellil and James. Dreyer -- 4.1 Introduction -- 4.2 Formulation as a Control Problem -- 4.2.1 Cost Functions for Propeller Blades.
4.2.2 Search Procedure -- 4.3 Implement at ion -- 4.4 Optimization of a Blade Section for Low Cav- it at ion -- 4.4.1 Comparisons with Water Tunnel Measurements -- 4.5 Conclusions -- 4.6 Bibliography -- Chapter 5 Flow Boundary Conditions Modeling in 4D for Optimized, Adaptive and Unsteady Configurations Helmut Sobieczky -- 5.1 Introduction -- 5.2 Geometry concept for 4-dimensional problems -- 5.3 Optimization -- 5.4 Adaptive configurations -- 5.5 Unsteady boundary conditions -- 5.6 Bio-fluidmechanic applications -- 5.7 Conclusion -- 5.8 Bibliography -- II. Algorithms and Accuracy -- Chapter 6 Stability and Efficiency of Implicit Residual-Based Compact Schemes C. Corre & A. Lerat -- 6.1 Introduction -- 6.2 Implicit schemes description -- 6.3 Direct solver efficiency -- 6.4 Implicit treatment description -- 6.5 Iterative solver efficiency and stability -- 6.6 Concluding remarks -- 6.7 Bibliography -- Chapter 7 Higher- Order Time-Integration Schemes for Dynamic Unstructured Mesh CFD Simulations Dimitri J. Mavriplis and Zhi Yang -- 7.1 Abstract -- 7.2 Introduction -- 7.3 Governing Equations in Arbitrary-Lagrangian-Eulerian (ALE) Form and Base Flow Solver -- 7.4 Higher-order Time Integration and the Discrete Geometric Conservation Law -- 7.5 Mesh Motion Strategies -- 7.5.1 Tension spring analogy -- 7.5.2 Linear elasticity analogy -- 7.6 Acceleration Strategies -- 7.7 Mesh Motion Results -- 7.7.1 Convergence of the mesh motion equations -- 7.8 Unsteady Flow Simulations using Backwards Difference Schemes -- 7.8.1 Multigrid Convergence Efficiency -- 7.8.2 Time-Accuracy Validation -- 7.9 Implicit-Runge-Kutta Methods for Dynamic Mesh Problems -- 7.10 Conclusions -- 7.11 Acknowledgments -- 7.12 Bibliography -- 7-A The Geometric Convervation Law for BDF3.
Chapter 8 Explicit Time Domain Finite Element Methods for Electromagnetics Kenneth Morgan, Mohamed El hachemi, Oubay Hassan, and Nigel Weatherill -- 8.1 Introduction -- 8.2 Electromagnetic Scattering -- 8.2.1 Governing Equations -- 8.2.2 Boundary conditions -- Perfect electrical conductor surface -- Far field boundary and the perfectly matched layer -- 8.3 Mesh generation -- 8.4 Numerical solution algorithm -- 8.4.1 Time discretisation -- 8.4.2 Discretisation in space -- 8.4.3 Computational details -- 8.5 Numerical examples -- 8.5.1 PEC sphere -- 8.5.2 PEC almond -- 8.6 Dealing with electrically larger scatterers -- 8.6.1 Higher order Taylor-Galerkin time stepping schemes -- 8.6.2 Higher order spatial discretisation -- 8.7 Conclusions -- 8.8 Bibliography -- Chapter 9 Estimating Grid-Induced Errors in CFD Solutions T. I-P. Shih -- 9.1 Introduction -- 9.2 Classification of Methods -- 9.3 Overview of the Discrete Error Transport Equation -- 9.4 DETEs for FV Solutions of the Euler Equations -- 9.4.1 Finite-Volume Method of Solution -- 9.4.2 DETE for the FV Method -- 9.5 Usefulness of the DETEs -- 9.5.1 Test Problem 1: Inviscid Flow over an Airfoil -- 9.5.2 Test Problem 2: Viscous Flow over an Iced Airfoil -- 9.6 Final Remarks -- 9.7 Bibliography -- Chapter 10 Treatment of Vortical Flow Using Vorticity Confinement John Steinhoff & Nicholas Lynn -- 10.1 Abstract -- 10.2 Introduction -- 10.2.1 Basic Concepts -- 10.3 Illustrative One-Dimensional Example -- 10.4 Vorticity Confinement -- 10.4.1 Basic Formulation -- 10.4.1.1 VC1 Formulation -- 10.4.1.2 VC2 Formulation -- 10.4.1.3 Boundary Conditions -- 10.4.2 Comparison of the VC2 Formulation to Conventional Discontinuity Steepening Schemes -- 10.4.3 Computational Details for the VC2 Formulation -- 10.5 Results -- 10.5.1 Wing Tip Vortices -- 10.5.2 Cylinder Wake -- 10.5.3 Dynamic Stall -- 10.6 Other Studies.
10.6.1 Missile Base Drag Computation -- 10.6.2 Blade Vortex Interaction (BVI) -- 10.6.3 Turbulent Flow Simulations for Special Effects -- 10.7 Conclusions -- 10.8 Acknowledgements -- 10.9 Bibliography -- III. Flow Stability and Control -- Chapter 11 Flow Control Applications with Synthetic and Pulsed Jets R. Agarwal, J. Vadillo, Y. Tan, J. Cui, D. Guo, H. Jain A. W. Cary & W. W. Bower -- 11.1 Abstract -- 11.2 Introduction -- 11.3 CFD Flow-solvers Employed -- 11.4 Results & Discussion -- 11.5 Conclusions -- 11.6 Acknowledgments -- 11.7 Bibliography -- Chapter 12 Control of Flow Separation over a Circular Cylinder Using Electro-Magnetic Fields: Numerical Simulation Brian H. Dennis and George S. Dulikravich -- 12.1 Nomenclature -- 12.2 Introduction -- 12.3 Second Order Analytical Model of EMHD -- 12.4 Least-Squares Finite Element Method -- 12.4.1 Nondimensional First Order Form for Simplified EMHD -- 12.4.2 Verification of Accuracy -- 12.5 Numerical Results -- 12.6 Conclusion -- 12.7 Acknowledgements -- 12.8 Bibliography -- Chapter 13 Bifurcation of Transonic Flow Over a Flattened Airfoil Alexander G. Kuz'min -- 13.1 Introduction -- 13.2 Problem statement and a numerical method -- 13.3 Analysis of the lift coefficient as a function of M -- 13.4 Analysis of stability with respect to variation of -- 13.5 Summary of the results -- 13.6 Conclusion -- 13.7 Bibliography -- Chapter 14 Study of Stability of Vortex Pairs over a Slender Conical Body by Euler Computations Jinsheng Cai, Her-Mann Tsai, Shijun Luo, and Feng Liu -- 14.1 Abstract -- 14.2 Introduction -- 14.3 The Euler Solver and the Flow Model -- 14.4 Computational Grid and Boundary Conditions -- 14.5 Stationary Symmetric and Asymmetric Solutions and Their Stability -- 14.5.1 Temporal Asymmetric Perturbations -- 14.5.2 Stationary Symmetric Vortex Flow.
14.5.3 Stability of the Stationary Symmetric Vortex Flow -- 14.5.4 Stability of the Stationary Asymmetric Vortex Flow -- 14.5.5 A Mirror-Image of the Asymmetric Vortex Flow -- 14.5.6 Symmetry Nature of the Present Euler Solver -- 14.5.7 Comparison with Theoretical Predictions on Stability -- 14.5.8 Comparison with Experimental Data on Stability -- 14.6 Structure of the Vortex Core -- 14.6.1 Computational Result -- 14.6.2 Comparison with Experimental Data -- 14.7 Summary and Conclusions -- 14.8 Bibliography -- Chapter 15 Effect of Upstream Conditions on Velocity Deficit Profiles of the Turbulent Boundary Layer at Global Separation Oleg S. Ryzhov -- 15.1 Introduction -- 15.2 Singular inviscid pressure gradient -- 15.3 Governing equations -- 15.4 Inviscid sublayer 1 -- 15.5 Outer turbulent sublayer 2 -- 15.6 Outer turbulent sublayer 3 -- 15.7 Pressure-dominated flow pattern -- 15.8 Comparison with experiment -- 15.9 Conclusion -- 15.10 Bibliography -- Chapter 16 Hypersonic Magnet o-Fluid-Dynamic Interact ions J. S. Shang -- 16.1 Abstract -- 16.2 Nomenclature -- 16.3 Introduction -- 16.4 Governing equations -- 16.5 Plasma models -- 16.6 Elect ro-Fluid-Dynamic Interact ion -- 16.7 Magnet o-Fluid-Dynamic Interact ion -- 16.8 Concluding Remarks -- 16.9 Acknowledgment -- 16.10 Bibliography -- IV. Multiphase and Reacting Flows -- Chapter 17 Computing Multiphase Flows Using AUSM+-up Scheme Meng-Sing Liou and Chih-Hao Chang -- 17.1 Abstract -- 17.2 Introduction -- 17.3 Governing Equations (Models) for Multiphase Flows -- 17.3.1 Thermodynamic Equilibrium Model [14] -- 17.3.2 Two-fluid Model -- 17.3.3 Multiphase Stratified Fluid Model -- 17.3.4 Convection fluxes -- 17.3.5 Pressure fluxes, -- 17.3.6 The interfacial pressure correction term -- 17.4 Calculated Examples and Discussion -- 17.4.1 Ransom's faucet problem -- 17.4.2 Air-water shock tube problem.
17.4.3 Shock-bubble interaction problem.
Summary: The series of volumes to which this book belongs honors contributors who have made a major impact in computational fluid dynamics. This fourth volume in the series is dedicated to David Caughey on the occasion of his 60th birthday. The first volume was published in 1994 and was dedicated to Prof Antony Jameson. The second, dedicated to Earl Murman, was published in 1998. The third volume was dedicated to Robert MacCormack in 2002. Written by leading researchers from academia, government laboratories, and industry, the contributions in this volume present descriptions of the latest developments in techniques for numerical analysis of fluid flow problems, as well as applications to important problems in industry.
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Intro -- Contents -- Dedication -- Chapter 1 The Contributions of David Caughey to Computational Fluid Dynamics Mohamed M. Hafez -- 1.1 Introduction -- 1.2 Shock Wave Structure and Sonic Boom -- 1.3 Potential Flow Simulations -- 1.4 Solutions of Euler Equations -- 1.5 Solutions of Navier-Stokes Equations -- 1.6 Simulation of Turbulent Reactive Flows -- 1.7 Special Topics -- 1.8 Review Articles -- 1.9 Fluid Mechanics: An Interactive Text -- 1.10 Concluding Remarks -- 1-A Ph.D. Students Supervised by David A. Caughey -- 1-B Publications of David A. Caughey -- I. Design and Optimization -- Chapter 2 Computational Fluid Dynamics in the Analysis and Design of Engineered Systems M. Damodaran, S. Alil and S. Dayanandan -- 2.1 Introduction -- 2.2 Flow Modeling for Fire Control Strategies and Scenario Planning in an Underground Road Tunnel -- 2.3 Flow Modeling in a Hard Disk Drive Enclosure -- 2.4 Concluding Remarks -- 2.5 Bibliography -- Chapter 3 Advances in Aerodynamic Shape Optimization Antony Jameson -- 3.1 Introduction -- 3.2 Formulation of the Optimization Procedure -- 3.2.1 Gradient Calculation -- 3.3 Design using the Euler Equations -- 3.4 The Reduced Gradient Formulation -- 3.5 Optimization Procedure -- 3.5.1 The Need for a Sobolev Inner Product in the Definition of the Gradient -- 3.5.2 Sobolev Gradient for Shape Optimization -- 3.5.3 Outline of the Design Procedure -- 3.6 Case Studies -- 3.6.1 Two-Dimensional Studies of Transonic Airfoil Design -- 3.6.2 B747 Euler Planform Result -- 3.6.3 Super B747 -- 3.7 Super P51 Racer -- 3.7.1 Shape Optimization for a Transonic Business Jet -- 3.8 Conclusion -- 3.9 Acknowledgment -- 3.10 Bibliography -- Chapter 4 Design Optirnization of Propeller Blades Luigi Martinellil and James. Dreyer -- 4.1 Introduction -- 4.2 Formulation as a Control Problem -- 4.2.1 Cost Functions for Propeller Blades.

4.2.2 Search Procedure -- 4.3 Implement at ion -- 4.4 Optimization of a Blade Section for Low Cav- it at ion -- 4.4.1 Comparisons with Water Tunnel Measurements -- 4.5 Conclusions -- 4.6 Bibliography -- Chapter 5 Flow Boundary Conditions Modeling in 4D for Optimized, Adaptive and Unsteady Configurations Helmut Sobieczky -- 5.1 Introduction -- 5.2 Geometry concept for 4-dimensional problems -- 5.3 Optimization -- 5.4 Adaptive configurations -- 5.5 Unsteady boundary conditions -- 5.6 Bio-fluidmechanic applications -- 5.7 Conclusion -- 5.8 Bibliography -- II. Algorithms and Accuracy -- Chapter 6 Stability and Efficiency of Implicit Residual-Based Compact Schemes C. Corre & A. Lerat -- 6.1 Introduction -- 6.2 Implicit schemes description -- 6.3 Direct solver efficiency -- 6.4 Implicit treatment description -- 6.5 Iterative solver efficiency and stability -- 6.6 Concluding remarks -- 6.7 Bibliography -- Chapter 7 Higher- Order Time-Integration Schemes for Dynamic Unstructured Mesh CFD Simulations Dimitri J. Mavriplis and Zhi Yang -- 7.1 Abstract -- 7.2 Introduction -- 7.3 Governing Equations in Arbitrary-Lagrangian-Eulerian (ALE) Form and Base Flow Solver -- 7.4 Higher-order Time Integration and the Discrete Geometric Conservation Law -- 7.5 Mesh Motion Strategies -- 7.5.1 Tension spring analogy -- 7.5.2 Linear elasticity analogy -- 7.6 Acceleration Strategies -- 7.7 Mesh Motion Results -- 7.7.1 Convergence of the mesh motion equations -- 7.8 Unsteady Flow Simulations using Backwards Difference Schemes -- 7.8.1 Multigrid Convergence Efficiency -- 7.8.2 Time-Accuracy Validation -- 7.9 Implicit-Runge-Kutta Methods for Dynamic Mesh Problems -- 7.10 Conclusions -- 7.11 Acknowledgments -- 7.12 Bibliography -- 7-A The Geometric Convervation Law for BDF3.

Chapter 8 Explicit Time Domain Finite Element Methods for Electromagnetics Kenneth Morgan, Mohamed El hachemi, Oubay Hassan, and Nigel Weatherill -- 8.1 Introduction -- 8.2 Electromagnetic Scattering -- 8.2.1 Governing Equations -- 8.2.2 Boundary conditions -- Perfect electrical conductor surface -- Far field boundary and the perfectly matched layer -- 8.3 Mesh generation -- 8.4 Numerical solution algorithm -- 8.4.1 Time discretisation -- 8.4.2 Discretisation in space -- 8.4.3 Computational details -- 8.5 Numerical examples -- 8.5.1 PEC sphere -- 8.5.2 PEC almond -- 8.6 Dealing with electrically larger scatterers -- 8.6.1 Higher order Taylor-Galerkin time stepping schemes -- 8.6.2 Higher order spatial discretisation -- 8.7 Conclusions -- 8.8 Bibliography -- Chapter 9 Estimating Grid-Induced Errors in CFD Solutions T. I-P. Shih -- 9.1 Introduction -- 9.2 Classification of Methods -- 9.3 Overview of the Discrete Error Transport Equation -- 9.4 DETEs for FV Solutions of the Euler Equations -- 9.4.1 Finite-Volume Method of Solution -- 9.4.2 DETE for the FV Method -- 9.5 Usefulness of the DETEs -- 9.5.1 Test Problem 1: Inviscid Flow over an Airfoil -- 9.5.2 Test Problem 2: Viscous Flow over an Iced Airfoil -- 9.6 Final Remarks -- 9.7 Bibliography -- Chapter 10 Treatment of Vortical Flow Using Vorticity Confinement John Steinhoff & Nicholas Lynn -- 10.1 Abstract -- 10.2 Introduction -- 10.2.1 Basic Concepts -- 10.3 Illustrative One-Dimensional Example -- 10.4 Vorticity Confinement -- 10.4.1 Basic Formulation -- 10.4.1.1 VC1 Formulation -- 10.4.1.2 VC2 Formulation -- 10.4.1.3 Boundary Conditions -- 10.4.2 Comparison of the VC2 Formulation to Conventional Discontinuity Steepening Schemes -- 10.4.3 Computational Details for the VC2 Formulation -- 10.5 Results -- 10.5.1 Wing Tip Vortices -- 10.5.2 Cylinder Wake -- 10.5.3 Dynamic Stall -- 10.6 Other Studies.

10.6.1 Missile Base Drag Computation -- 10.6.2 Blade Vortex Interaction (BVI) -- 10.6.3 Turbulent Flow Simulations for Special Effects -- 10.7 Conclusions -- 10.8 Acknowledgements -- 10.9 Bibliography -- III. Flow Stability and Control -- Chapter 11 Flow Control Applications with Synthetic and Pulsed Jets R. Agarwal, J. Vadillo, Y. Tan, J. Cui, D. Guo, H. Jain A. W. Cary & W. W. Bower -- 11.1 Abstract -- 11.2 Introduction -- 11.3 CFD Flow-solvers Employed -- 11.4 Results & Discussion -- 11.5 Conclusions -- 11.6 Acknowledgments -- 11.7 Bibliography -- Chapter 12 Control of Flow Separation over a Circular Cylinder Using Electro-Magnetic Fields: Numerical Simulation Brian H. Dennis and George S. Dulikravich -- 12.1 Nomenclature -- 12.2 Introduction -- 12.3 Second Order Analytical Model of EMHD -- 12.4 Least-Squares Finite Element Method -- 12.4.1 Nondimensional First Order Form for Simplified EMHD -- 12.4.2 Verification of Accuracy -- 12.5 Numerical Results -- 12.6 Conclusion -- 12.7 Acknowledgements -- 12.8 Bibliography -- Chapter 13 Bifurcation of Transonic Flow Over a Flattened Airfoil Alexander G. Kuz'min -- 13.1 Introduction -- 13.2 Problem statement and a numerical method -- 13.3 Analysis of the lift coefficient as a function of M -- 13.4 Analysis of stability with respect to variation of -- 13.5 Summary of the results -- 13.6 Conclusion -- 13.7 Bibliography -- Chapter 14 Study of Stability of Vortex Pairs over a Slender Conical Body by Euler Computations Jinsheng Cai, Her-Mann Tsai, Shijun Luo, and Feng Liu -- 14.1 Abstract -- 14.2 Introduction -- 14.3 The Euler Solver and the Flow Model -- 14.4 Computational Grid and Boundary Conditions -- 14.5 Stationary Symmetric and Asymmetric Solutions and Their Stability -- 14.5.1 Temporal Asymmetric Perturbations -- 14.5.2 Stationary Symmetric Vortex Flow.

14.5.3 Stability of the Stationary Symmetric Vortex Flow -- 14.5.4 Stability of the Stationary Asymmetric Vortex Flow -- 14.5.5 A Mirror-Image of the Asymmetric Vortex Flow -- 14.5.6 Symmetry Nature of the Present Euler Solver -- 14.5.7 Comparison with Theoretical Predictions on Stability -- 14.5.8 Comparison with Experimental Data on Stability -- 14.6 Structure of the Vortex Core -- 14.6.1 Computational Result -- 14.6.2 Comparison with Experimental Data -- 14.7 Summary and Conclusions -- 14.8 Bibliography -- Chapter 15 Effect of Upstream Conditions on Velocity Deficit Profiles of the Turbulent Boundary Layer at Global Separation Oleg S. Ryzhov -- 15.1 Introduction -- 15.2 Singular inviscid pressure gradient -- 15.3 Governing equations -- 15.4 Inviscid sublayer 1 -- 15.5 Outer turbulent sublayer 2 -- 15.6 Outer turbulent sublayer 3 -- 15.7 Pressure-dominated flow pattern -- 15.8 Comparison with experiment -- 15.9 Conclusion -- 15.10 Bibliography -- Chapter 16 Hypersonic Magnet o-Fluid-Dynamic Interact ions J. S. Shang -- 16.1 Abstract -- 16.2 Nomenclature -- 16.3 Introduction -- 16.4 Governing equations -- 16.5 Plasma models -- 16.6 Elect ro-Fluid-Dynamic Interact ion -- 16.7 Magnet o-Fluid-Dynamic Interact ion -- 16.8 Concluding Remarks -- 16.9 Acknowledgment -- 16.10 Bibliography -- IV. Multiphase and Reacting Flows -- Chapter 17 Computing Multiphase Flows Using AUSM+-up Scheme Meng-Sing Liou and Chih-Hao Chang -- 17.1 Abstract -- 17.2 Introduction -- 17.3 Governing Equations (Models) for Multiphase Flows -- 17.3.1 Thermodynamic Equilibrium Model [14] -- 17.3.2 Two-fluid Model -- 17.3.3 Multiphase Stratified Fluid Model -- 17.3.4 Convection fluxes -- 17.3.5 Pressure fluxes, -- 17.3.6 The interfacial pressure correction term -- 17.4 Calculated Examples and Discussion -- 17.4.1 Ransom's faucet problem -- 17.4.2 Air-water shock tube problem.

17.4.3 Shock-bubble interaction problem.

The series of volumes to which this book belongs honors contributors who have made a major impact in computational fluid dynamics. This fourth volume in the series is dedicated to David Caughey on the occasion of his 60th birthday. The first volume was published in 1994 and was dedicated to Prof Antony Jameson. The second, dedicated to Earl Murman, was published in 1998. The third volume was dedicated to Robert MacCormack in 2002. Written by leading researchers from academia, government laboratories, and industry, the contributions in this volume present descriptions of the latest developments in techniques for numerical analysis of fluid flow problems, as well as applications to important problems in industry.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2018. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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