Title The application of the Chebyshev-Spectral method in transport phenomena / Weidong Guo, Gérard Labrosse, Ranga Narayanan

Published Berlin ; New York : Springer, ©2012

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Description 1 online resource Series Lecture notes in applied and computational mechanics, 1613-7736 ; 68 Lecture notes in applied and computational mechanics ; 68. Contents 880-01 An Introduction to the Book and a Road Map -- An Introduction to the Spectral Method -- Steady One-Dimensional (1D) Heat Conduction Problems -- Unsteady 1D Heat Conduction Problems -- Steady Two-Dimensional (2D) Heat Conduction Problems -- 2D Closed Flow Problems: The Driven Cavity -- Applications to Transport Instabilities -- Exercises for the Reader 880-01/(S Machine generated contents note: 1. Introduction to the Book and a Road Map -- 1.1. Introduction -- 1.2. Road Map -- 2. Introduction to the Spectral Method -- 2.1. Method -- 2.1.1. Chebyshev Gauss-Lobatto Collocation Differentiation Matrices -- 2.1.2. Why We Use Non-uniform Grids -- 2.1.3. Chebyshev Polynomials and the Pseudo-Spectrum -- 2.1.4. Differentiation Matrices in the Pseudo-Spectral Space -- 2.1.5. Gauss-Radau Collocation Differentiation Matrices -- 2.1.6. Boundary Conditions -- 2.2. What Lies Ahead -- 2.3. Endnotes -- 2.3.1. Accurate Evaluation of Definite Integrals -- 2.3.2. Some Useful Relationships Involving the Chebyshev Polynomials -- References -- 3. Steady One-Dimensional (ID) Heat Conduction Problems -- 3.1. One-Domain Problems -- 3.1.1. Inhomogeneous Problem with Dirichlet-Dirichlet (DD) Conditions -- 3.1.2. Inhomogeneous Problem with Neumann-Dirichlet (ND) Conditions -- 3.1.3. Inhomogeneous Problem with Neumann-Neumann (NN) Conditions -- 3.1.4. Homogeneous Problems with Dirichlet and/or Neumann Conditions -- 3.1.5. Robin Boundary Conditions -- 3.1.6. Why We are Interested in Homogeneous Differential Problems -- 3.1.7. Homogeneous Problems with Radial Coordinates -- 3.2. Two-Medium/Two-Domain Problems -- 3.2.1. Two-Medium Inhomogeneous Problem with Interface Conditions -- 3.2.2. Two-Domain Inhomogeneous Problem and Numerical-Accuracy Considerations -- 3.2.3. Homogeneous Problem with Interface Conditions -- 3.3. Endnotes -- 3.3.1. Partial Inverse of A and Solvability of A U = F -- 3.3.2. Transforming the Generalized Eigenvalue Problem into an Ordinary Eigenvalue Problem -- 3.3.3. Leading Eigenmodes and Non-leading Eigenmodes -- 3.3.4. Ellipticity of an Operator -- 3.3.5. Inhomogeneous Problem in Polar Coordinates -- 3.3.6. Transforming the Two-Domain Coupled Inhomogeneous Problem into a Set of Uncoupled Problems and Extension to the Multi-Domain Case -- 3.3.7. Transforming the Two-Domain Coupled Homogeneous Problem into an Ordinary Eigenvalue Problem -- 3.3.8. Distinction Between the Error and the Residual -- 3.3.9. Particular Caution When Solving Coupled Differential Problems -- References -- 4. Unsteady ID Heat Conduction Problems -- 4.1. Inhomogeneous Problem with Neumann Conditions -- 4.1.1. Continuous Problem -- 4.1.2. Time Discretization -- 4.1.3. Fully Discretized System -- 4.1.4. Note on the Flux Solvability Condition -- 4.2. Inhomogeneous Problem with a Non-linear Source Term -- 4.3. ID Convection-Diffusion Heat Equation -- 4.4. Some Closing Thoughts on Time Marching -- 4.5. Endnotes -- 4.5.1. From Taylor Expansions to Time Discretization -- 4.5.2. Destiny of a Solution to a Transient Problem Whose Steady Counterpart is not Solvable -- 4.5.3. Comment About the Accuracy of the Time Discretization Approach -- 4.5.4. Time Integration can be Spectrally Accurate -- References -- 5. Steady Two-Dimensional (2D) Heat Conduction Problems -- 5.1. One-Domain Problems -- 5.1.1. Inhomogeneous Problem in Cartesian Coordinates -- 5.1.2. Inhomogeneous Problems in (r, z) Cylindrical Coordinates -- 5.1.3. Homogeneous Problems -- 5.2. Two-Medium Inhomogeneous Problems with Interface Conditions -- 5.3. Endnotes -- 5.3.1. Kronecker Product -- 5.3.2. Multi-dimensional Matrix Multiplication -- 5.3.3. Optimized Solver for 2u = f Using Successive Diagonalization -- 5.3.4. Solving the 2D Two-Medium Inhomogeneous Diffusion Problems by Successive Diagonalization -- References -- 6. 2D Closed Flow Problems: The Driven Cavity -- 6.1. Driven Cavity and Navier-Stokes Equations -- 6.2. Stokes Problem -- 6.3. Outline of the Remaining Part of the Chapter -- 6.4. 2D Stokes Eigenproblem -- 6.4.1. Numerical Procedure -- 6.4.2. Spurious Pressure Modes (SPM) -- 6.5. Stokes Steady Flow Problem -- 6.5.1. Regularizing the Model -- 6.5.2. Numerical Procedure -- 6.6. Unsteady Stokes Flow Problem -- 6.6.1. (v, p) Uncoupling by Projection-Diffusion -- 6.6.2. Time Discretization -- 6.6.3. Space Discretization of the Projection Step -- 6.6.4. Comment on Why We Do not Use 2p to Determine the Pressure Field -- 6.6.5. Other Stokes Solvers -- 6.7. Navier-Stokes Flow Problem -- 6.7.1. Non-linear Equations -- 6.7.2. Numerical Procedure -- 6.7.3. Numerical Stability Considerations -- 6.8. Endnotes -- 6.8.1. Ellipticity of the Stokes Operator -- 6.8.2. Constructing the 2D/3D Pressure Operator from Projection-Diffusion -- 6.8.3. Projection-Diffusion "toy" Problem Posed in a Two-domain Configuration -- 6.8.4. Projection-Diffusion in 2D Multi-domain -- References -- 7. Applications to Transport Instabilities -- 7.1. Ignition in a Solid -- 7.1.1. Method (a): Time Marching -- 7.1.2. Method (b): Steady-Solution Determination and Continuation Methods -- 7.1.3. Method (c): The Leading Eigenvalue of the Problem Linearized About a Base State -- 7.1.4. Some Closing Thoughts -- 7.2. Rayleigh-Benard Problem in a Porous Medium -- 7.2.1. Scaled Equations -- 7.2.2. Method (a): Time Marching with the Non-linear Problem -- 7.2.3. Method (b): Steady-Solution Determination -- 7.2.4. Method (c): The Leading Eigenvalue of the Problem Linearized About a Base State -- 7.2.5. Method (d): The Time Marching with the Problem Linearized About a Base State -- 7.3. Solidification Front -- 7.3.1. Model and Its Base State -- 7.3.2. Front Instability and the Linearized Equations -- 7.3.3. Numerical Evaluation of the Time Growth Rate, σ -- 7.3.4. Results of the Calculation -- References -- 8. Exercises for the Reader -- References Summary Transport phenomena problems that occur in engineering and physics are often multi-dimensional and multi-phase in character. When taking recourse to numerical methods the spectral method is particularly useful and efficient. The book is meant principally to train students and non-specialists to use the spectral method for solving problems that model fluid flow in closed geometries with heat or mass transfer. To this aim the reader should bring a working knowledge of fluid mechanics and heat transfer and should be readily conversant with simple concepts of linear algebra including spectral decomposition of matrices as well as solvability conditions for inhomogeneous problems. The book is neither meant to supply a ready-to-use program that is all-purpose nor to go through all manners of mathematical proofs. The focus in this tutorial is on the use of the spectral methods for space discretization, because this is where most of the difficulty lies. While time dependent problems are also of great interest, time marching procedures are dealt with by briefly introducing and providing a simple, direct, and efficient method. Many examples are provided in the text as well as numerous exercises for each chapter. Several of the examples are attended by subtle points which the reader will face while working them out. Some of these points are deliberated upon in endnotes to the various chapters, others are touched upon in the book itself Analysis Engineering Computer science Hydraulic engineering Engineering Thermodynamics, Heat and Mass Transfer Numerical and Computational Physics Engineering Fluid Dynamics Fluid- and Aerodynamics Computational Science and Engineering fysica physics computational science thermodynamica thermodynamics vloeistoffen (fluids) fluids aerodynamics Engineering (General) Techniek (algemeen) Bibliography Includes bibliographical references and index Subject Transport theory -- Mathematical models Spectral theory (Mathematics) SCIENCE -- Physics -- General. Ingénierie. Spectral theory (Mathematics) Transport theory -- Mathematical models Form Electronic book Author Labrosse, Gérard Narayanan, Ranga. LC no. 2012952021 ISBN 9783642340888 3642340881 3642340873 9783642340871 1299336760 9781299336766

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