Physics: Fluid mechanics Books

Book SynopsisIncompressible Flow The latest edition of the classic introduction to fluid dynamics This textbook offers a detailed study of fluid dynamics. Equal emphasis is given to physical concepts, mathematical methods, and illustrative flow patterns. The book begins with a precise and careful formulation of physical concepts followed by derivations of the laws governing the motion of an arbitrary fluid, the Navier-Stokes equations. Throughout, there is an emphasis on scaling variables and dimensional analysis. Incompressible flow is presented as an asymptotic expansion of solutions to the Navier-Stokes equations with low Mach numbers and arbitrary Reynolds numbers. The different physical behaviors of flows with low, medium, and high Reynolds number are thoroughly investigated. Additionally, several special introductory chapters are provided on lubrication theory, flow stability, and turbulence. In the Fifth Edition, a chapter on gas dynamics has been added. Gas dynamics

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Book SynopsisThis book presents the physical principles of wave propagation in fluid mechanics and hydraulics. The mathematical techniques that allow the behavior of the waves to be analyzed are presented, along with existing numerical methods for the simulation of wave propagation. Particular attention is paid to discontinuous flows, such as steep fronts and shock waves, and their mathematical treatment. A number of practical examples are taken from various areas fluid mechanics and hydraulics, such as contaminant transport, the motion of immiscible hydrocarbons in aquifers, river flow, pipe transients and gas dynamics. Finite difference methods and finite volume methods are analyzed and applied to practical situations, with particular attention being given to their advantages and disadvantages. Application exercises are given at the end of each chapter, enabling readers to test their understanding of the subject.Table of ContentsIntroduction xv Chapter 1. Scalar Hyperbolic Conservation Laws in One Dimension of Space 1 1.1. Definitions 1 1.1.1. Hyperbolic scalar conservation laws 1 1.1.2. Derivation from general conservation principles 3 1.1.3. Non-conservation form 6 1.1.4. Characteristic form – Riemann invariants 7 1.2. Determination of the solution 9 1.2.1. Representation in the phase space 9 1.2.2. Initial conditions, boundary conditions 12 1.3. A linear law: the advection equation 14 1.3.1. Physical context – conservation form 14 1.3.2. Characteristic form 16 1.3.3. Example: movement of a contaminant in a river 17 1.3.4. Summary 21 1.4. A convex law: the inviscid Burgers equation 21 1.4.1. Physical context – conservation form 21 1.4.2. Characteristic form 23 1.4.3. Example: propagation of a perturbation in a fluid 24 1.4.4. Summary 28 1.5. Another convex law: the kinematic wave for free-surface hydraulics 28 1.5.1. Physical context – conservation form 28 1.5.2. Non-conservation and characteristic forms 29 1.5.3. Expression of the celerity 31 1.5.4. Specific case: flow in a rectangular channel 34 1.5.5. Summary 35 1.6. A non-convex conservation law: the Buckley-Leverett equation 36 1.6.1. Physical context – conservation form 36 1.6.2. Characteristic form 39 1.6.3. Example: decontamination of an aquifer 40 1.6.4. Summary 42 1.7. Advection with adsorption/desorption 42 1.7.1. Physical context – conservation form 42 1.7.2. Characteristic form 45 1.7.3. Summary 47 1.8. Conclusions 48 1.8.1. What you should remember 48 1.8.2. Application exercises 48 Chapter 2. Hyperbolic Systems of Conservation Laws in One Dimension of Space 55 2.1. Definitions 55 2.1.1. Hyperbolic systems of conservation laws 55 2.1.2. Hyperbolic systems of conservation laws – examples 57 2.1.3. Characteristic form – Riemann invariants 59 2.2. Determination of the solution 62 2.2.1. Domain of influence, domain of dependence 62 2.2.2. Existence and uniqueness of solutions – initial and boundary conditions 64 2.3. Specific case: compressible flows 65 2.3.1. Definition 65 2.3.2. Conservation form 65 2.3.3. Characteristic form 68 2.3.4. Physical interpretation 70 2.4. A 2×2 linear system: the water hammer equations 71 2.4.1. Physical context – hypotheses 71 2.4.2. Conservation form 73 2.4.3. Characteristic form – Riemann invariants 78 2.4.4. Calculation of the solution 82 2.4.5. Summary 87 2.5. A nonlinear 2×2 system: the Saint Venant equations 87 2.5.1. Physical context – hypotheses 87 2.5.2. Conservation form 88 2.5.3. Characteristic form – Riemann invariants 94 2.5.4. Calculation of solutions 105 2.5.5. Summary 112 2.6. A nonlinear 3×3 system: the Euler equations 112 2.6.1. Physical context – hypotheses 112 2.6.2. Conservation form 114 2.6.3. Characteristic form – Riemann invariants 118 2.6.4. Calculation of the solution 122 2.6.5. Summary 126 2.7. Summary of Chapter 2 127 2.7.1. What you should remember 127 2.7.2. Application exercises 128 Chapter 3. Weak Solutions and their Properties 135 3.1. Appearance of discontinuous solutions 135 3.1.1. Governing mechanisms 135 3.1.2. Local invalidity of the characteristic formulation– graphical approach 138 3.1.3. Practical examples of discontinuous flows 140 3.2. Classification of waves 143 3.2.1. Shock wave 143 3.2.2. Rarefaction wave 144 3.2.3. Contact discontinuity 145 3.2.4. Mixed/compound wave 145 3.3. Simple waves 146 3.3.1. Definition and properties 146 3.3.2. Generalized Riemann invariants 147 3.4. Weak solutions and their properties 149 3.4.1. Definitions 149 3.4.2. Non-equivalence between the formulations 150 3.4.3. Jump relationships 150 3.4.4. Non-uniqueness of weak solutions 152 3.4.5. The entropy condition 157 3.4.6. Irreversibility 159 3.4.7. Approximations for the jump relationships 160 3.5. Summary 161 3.5.1. What you should remember 161 3.5.2. Application exercises 162 Chapter 4. The Riemann Problem 165 4.1. Definitions – solution properties 165 4.1.1. The Riemann problem 165 4.1.2. The generalized Riemann problem 166 4.1.3. Solution properties 167 4.2. Solution for scalar conservation laws 167 4.2.1. The linear advection equation 167 4.2.2. The inviscid Burgers equation 168 4.2.3. The Buckley-Leverett equation 170 4.3. Solution for hyperbolic systems of conservation laws 175 4.3.1. General principle 175 4.3.2. Application to the water hammer problem: sudden valve failure 176 4.3.3. Free surface flow: the dambreak problem 179 4.3.4. The Euler equations: the shock tube problem 186 4.4. Summary 192 4.4.1. What you should remember 192 4.4.2. Application exercises 193 Chapter 5. Multidimensional Hyperbolic Systems 195 5.1. Definitions 195 5.1.1. Scalar laws 195 5.1.2. Two-dimensional hyperbolic systems 197 5.1.3. Three-dimensional hyperbolic systems 199 5.2. Derivation from conservation principles 200 5.3. Solution properties 203 5.3.1. Two-dimensional hyperbolic systems 203 5.3.2. Three-dimensional hyperbolic systems 210 5.4. Application to two-dimensional free-surface flow 211 5.4.1. Governing equations 211 5.4.2. The secant plane approach 217 5.4.3. Interpretation – determination of the solution 222 5.5. Summary 225 5.5.1. What you should remember 225 5.5.2. Application exercises 225 Chapter 6. Finite Difference Methods for Hyperbolic Systems 229 6.1. Discretization of time and space 229 6.1.1. Discretization for one-dimensional problems 229 6.1.2. Multidimensional discretization 230 6.1.3. Explicit schemes, implicit schemes 231 6.2. The method of characteristics (MOC) 232 6.2.1. MOC for scalar hyperbolic laws 232 6.2.2. MOC for hyperbolic systems of conservation laws 241 6.2.3. Application examples 246 6.3. Upwind schemes for scalar laws 250 6.3.1. The explicit upwind scheme (non-conservation version) 250 6.3.2. The implicit upwind scheme (non-conservation version) 252 6.3.3. Conservative versions of the implicit upwind scheme 253 6.3.4. Application examples 255 6.4. The Preissmann scheme 257 6.4.1. Formulation 257 6.4.2. Estimation of nonlinear terms – algorithmic aspects 260 6.4.3. Numerical applications 261 6.5. Centered schemes 267 6.5.1. The Crank-Nicholson scheme 267 6.5.2. Centered schemes with Runge-Kutta time stepping 268 6.6. TVD schemes 270 6.6.1. Definitions 270 6.6.2. General formulation of TVD schemes 271 6.6.3. Harten’s and Sweby’s criteria 274 6.6.4. Traditional limiters 276 6.6.5. Calculation example 277 6.7. The flux splitting technique 280 6.7.1. Principle of the approach 280 6.7.2. Application to traditional schemes 283 6.8. Conservative discretizations: Roe’s matrix 289 6.8.1. Motivation and principle of the approach 289 6.8.2. Expression of Roe’s matrix 290 6.9. Multidimensional problems 293 6.9.1. Explicit alternate directions293 6.9.2. The ADI method 296 6.9.3. Multidimensional schemes 298 6.10. Summary 299 6.10.1. What you should remember 299 6.10.2. Application exercises 301 Chapter 7. Finite Volume Methods for Hyperbolic Systems 303 7.1. Principle 303 7.1.1. One-dimensional conservation laws 303 7.1.2. Multidimensional conservation laws 305 7.1.3. Application to the two-dimensional shallow water equations 308 7.2. Godunov’s scheme 310 7.2.1. Principle 310 7.2.2. Application to the scalar advection equation 311 7.2.3. Application to the inviscid Burgers equation 316 7.2.4. Application to the water hammer equations 319 7.3. Higher-order Godunov-type schemes 324 7.3.1. Rationale and principle 324 7.3.2. Example: the MUSCL scheme 328 7.4. Summary 330 7.4.1. What you should remember 330 7.4.2. Suggested exercises 331 Appendix A. Linear Algebra 333 A.1. Definitions 333 A.2. Operations on matrices and vectors 335 A.2.1. Addition 335 A.2.2. Multiplication by a scalar 335 A.2.3. Matrix product 336 A.2.4. Determinant of a matrix 336 A.2.5. Inverse of a matrix 337 A.3. Differential operations using matrices and vectors 337 A.3.1. Differentiation 337 A.3.2. Jacobian matrix 338 A.4. Eigenvalues, eigenvectors 338 A.4.1. Definitions 338 A.4.2. Example 339 Appendix B. Numerical Analysis 341 B.1. Consistency 341 B.1.1. Definitions 341 B.1.2. Principle of a consistency analysis 341 B.1.3. Numerical diffusion, numerical dispersion 343 B.2. Stability 345 B.2.1. Definition 345 B.2.2. Principle of a stability analysis 346 B.2.3. Harmonic analysis of analytical solutions 348 B.2.4. Harmonic analysis of numerical solutions 352 B.2.5. Amplitude and phase portraits 355 B.2.6. Extension to systems of equations 357 B.3. Convergence 359 B.3.1. Definition 359 B.3.2. Lax’s theorem 359 Appendix C. Approximate Riemann Solvers 361 C.1. HLL and HLLC solvers 361 C.1.1. HLL solver 361 C.1.2. HLLC solver 363 C.2. Roe’s solver 366 Appendix D. Summary of the Formulae 369 References 375 Index 379

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Book SynopsisThis book examines the phenomena of fluid flow and transfer as governed by mechanics and thermodynamics. Part 1 concentrates on equations coming from balance laws and also discusses transportation phenomena and propagation of shock waves. Part 2 explains the basic methods of metrology, signal processing, and system modeling, using a selection of examples of fluid and thermal mechanics.Table of ContentsPreface xi Chapter 1. Thermodynamics of Discrete Systems 1 1.1. The representational bases of a material system 1 1.1.1. Introduction 1 1.1.2. Systems analysis and thermodynamics 8 1.1.3. The notion of state 11 1.1.4. Processes and systems 13 1.2. Axioms of thermostatics 15 1.2.1. Introduction 15 1.2.2. Extensive quantities 16 1.2.3. Energy, work and heat 20 1.3. Consequences of the axioms of thermostatics 21 1.3.1. Intensive variables 21 1.3.2. Thermodynamic potentials 23 1.4. Out-of-equilibrium states 29 1.4.1. Introduction 29 1.4.2. Discontinuous systems 30 1.4.3. Application to heat engines 45 Chapter 2. Thermodynamics of Continuous Media 47 2.1. Thermostatics of continuous media 47 2.1.1. Reduced extensive quantities 47 2.1.2. Local thermodynamic equilibrium 48 2.1.3. Flux of extensive quantities 50 2.1.4. Balance equations in continuous media 54 2.1.5. Phenomenological laws 57 2.2. Fluid statics 63 2.2.1. General equations of fluid statics 63 2.2.2. Pressure forces on solid boundaries 68 2.3. Heat conduction 72 2.3.1. The heat equation 72 2.3.2. Thermal boundary conditions 72 2.4. Diffusion 73 2.4.1. Introduction 73 2.4.2. Molar and mass fluxes 77 2.4.3. Choice of reference frame 80 2.4.4. Binary isothermal mixture 85 2.4.5. Coupled phenomena with diffusion 97 2.4.6. Boundary conditions 99 Chapter 3. Physics of Energetic Systems in Flow 101 3.1. Dynamics of a material point 101 3.1.1. Galilean reference frames in traditional mechanics 101 3.1.2. Isolated mechanical system and momentum 102 3.1.3. Momentum and velocity 103 3.1.4. Definition of force 104 3.1.5. The fundamental law of dynamics (closed systems) 106 3.1.6. Kinetic energy 106 3.2. Mechanical material system 107 3.2.1. Dynamic properties of a material system 107 3.2.2. Kinetic energy of a material system 109 3.2.3. Mechanical system in thermodynamic equilibrium the rigid solid 111 3.2.4. The open mechanical system 112 3.2.5. Thermodynamics of a system in motion 116 3.3. Kinematics of continuous media 119 3.3.1. Lagrangian and Eulerian variables 119 3.3.2. Trajectories, streamlines, streaklines 121 3.3.3. Material (or Lagrangian) derivative 122 3.3.4. Deformation rate tensors 129 3.4. Phenomenological laws of viscosity 132 3.4.1. Definition of a fluid 132 3.4.2. Viscometric flows 135 3.4.3. The Newtonian fluid 146 Chapter 4. Fluid Dynamics Equations 151 4.1. Local balance equations 151 4.1.1. Balance of an extensive quantity G 151 4.1.2. Interpretation of an equation in terms of the balance equation 153 4.2. Mass balance 154 4.2.1. Conservation of mass and its consequences 154 4.2.2. Volume conservation 160 4.3. Balance of mechanical and thermodynamic quantities 160 4.3.1. Momentum balance 160 4.3.2. Kinetic energy theorem 164 4.3.3. The vorticity equation 171 4.3.4. The energy equation 172 4.3.5. Balance of chemical species 177 4.4. Boundary conditions 178 4.4.1. General considerations 178 4.4.2. Geometric boundary conditions 179 4.4.3. Initial conditions 181 4.5. Global form of the balance equations 182 4.5.1. The interest of the global form of a balance 182 4.5.2. Equation of mass conservation 184 4.5.3. Volume balance 184 4.5.4. The momentum flux theorem 184 4.5.5. Kinetic energy theorem 186 4.5.6. The energy equation 187 4.5.7. The balance equation for chemical species 188 4.6. Similarity and non-dimensional parameters 189 4.6.1. Principles 189 Chapter 5. Transport and Propagation 199 5.1. General considerations 199 5.1.1. Differential equations 199 5.1.2. The Cauchy problem for differential equations 202 5.2. First order quasi-linear partial differential equations 203 5.2.1. Introduction 203 5.2.2. Geometric interpretation of the solutions 204 5.2.3. Comments 206 5.2.4. The Cauchy problem for partial differential equations 206 5.3. Systems of first order partial differential equations 207 5.3.1. The Cauchy problem for n unknowns and two variables 207 5.3.2. Applications in fluid mechanics 210 5.3.3. Cauchy problem with n unknowns and p variables 216 5.3.4. Partial differential equations of order n 218 5.3.5. Applications 220 5.3.6. Physical interpretation of propagation 223 5.4. Second order partial differential equations 225 5.4.1. Introduction 225 5.4.2. Characteristic curves of hyperbolic equations 226 5.4.3. Reduced form of the second order quasi-linear partial differential equation 229 5.4.4. Second order partial differential equations in a finite domain 232 5.4.5. Second order partial differential equations and their boundary conditions 233 5.5. Discontinuities: shock waves 239 5.5.1. General considerations 239 5.5.2. Unsteady 1D flow of an inviscid compressible fluid 239 5.5.3. Plane steady supersonic flow 244 5.5.4. Flow in a nozzle 244 5.5.5. Separated shock wave 248 5.5.6. Other discontinuity categories 248 5.5.7. Balance equations across a discontinuity 249 5.6. Some comments on methods of numerical solution 250 5.6.1. Characteristic curves and numerical discretization schemes 250 5.6.2. A complex example 253 5.6.3. Boundary conditions of flow problems 255 Chapter 6. General Properties of Flows 257 6.1. Dynamics of vorticity 257 6.1.1. Kinematic properties of the rotation vector 257 6.1.2. Equation and properties of the rotation vector 261 6.2. Potential flows 269 6.2.1. Introduction 269 6.2.2. Bernoulli’s second theorem 269 6.2.3. Flow of compressible inviscid fluid 270 6.2.4. Nature of equations in inviscid flows 271 6.2.5. Elementary solutions in irrotational flows 273 6.2.6. Surface waves in shallow water 284 6.3. Orders of magnitude 288 6.3.1. Introduction and discussion of a simple example 288 6.3.2. Obtaining approximate values of a solution 291 6.4. Small parameters and perturbation phenomena 296 6.4.1. Introduction 296 6.4.2. Regular perturbation 296 6.4.3. Singular perturbations 305 6.5. Quasi-1D flows 309 6.5.1. General properties 309 6.5.2. Flows in pipes 314 6.5.3. The boundary layer in steady flow 319 6.6. Unsteady flows and steady flows 327 6.6.1. Introduction 327 6.6.2. The existence of steady flows 328 6.6.3. Transitional regime and permanent solution 330 6.6.4. Non-existence of a steady solution 334 Chapter 7. Measurement, Representation and Analysis of Temporal Signals 339 7.1. Introduction and position of the problem 339 7.2. Measurement and experimental data in flows 340 7.2.1. Introduction 340 7.2.2. Measurement of pressure 341 7.2.3. Anemometric measurements 342 7.2.4. Temperature measurements 346 7.2.5. Measurements of concentration 347 7.2.6. Fields of quantities and global measurements 347 7.2.7. Errors and uncertainties of measurements 351 7.3. Representation of signals 357 7.3.1. Objectives of continuous signal representation 357 7.3.2. Analytical representation 360 7.3.3. Signal decomposition on the basis of functions; series and elementary solutions 361 7.3.4. Integral transforms 363 7.3.5. Time-frequency (or timescale) representations 374 7.3.6. Discretized signals 381 7.3.7. Data compression 385 7.4. Choice of representation and obtaining pertinent information 389 7.4.1. Introduction 389 7.4.2. An example: analysis of sound 390 7.4.3. Analysis of musical signals 393 7.4.4. Signal analysis in aero-energetics 402 Chapter 8. Thermal Systems and Models 405 8.1. Overview of models 405 8.1.1. Introduction and definitions 405 8.1.2. Modeling by state representation and choice of variables 408 8.1.3. External representation 410 8.1.4. Command models 411 8.2. Thermodynamics and state representation 412 8.2.1. General principles of modeling 412 8.2.2. Linear time-invariant system (LTIS) 420 8.3. Modeling linear invariant thermal systems 422 8.3.1. Modeling discrete systems 422 8.3.2. Thermal models in continuous media 431 8.4. External representation of linear invariant systems 446 8.4.1. Overview 446 8.4.2. External description of linear invariant systems 446 8.5. Parametric models 451 8.5.1. Definition of model parameters 451 8.5.2. Established regimes of linear invariant systems 453 8.5.3. Established regimes in continuous media 458 8.6. Model reduction 465 8.6.1. Overview 465 8.6.2. Model reduction of discrete systems 466 8.7. Application in fluid mechanics and transfer in flows 474 Appendix 1. Laplace Transform 477 A1.1. Definition 477 A1.2. Properties 477 A1.3. Some Laplace transforms 478 A1.4. Application to the solution of constant coefficient differential equations 479 Appendix 2. Hilbert Transform 481 Appendix 3. Cepstral Analysis 483 A3.1. Introduction 483 A3.2. Definitions 483 A3.3. Example of echo suppression 484 A3.4. General case 485 Appendix 4. Eigenfunctions of an Operator 487 A4.1. Eigenfunctions of an operator 487 A4.2. Self-adjoint operator 487 A4.2.1. Eigenfunctions 487 A4.2.2. Expression of a function of f using an eigenfunction basis-set 488 Bibliography 489 Index 497

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Book SynopsisWall turbulence is encountered in many technological applications as well as in the atmosphere, and a detailed understanding leading to its management would have considerable beneficial consequences in many areas. A lot of inspired work by experimenters, theoreticians, engineers and mathematicians has been accomplished over recent decades on this important topic and Statistical Approach to Wall Turbulence provides an updated and integrated view on the progress made in this area. Wall turbulence is a complex phenomenon that has several industrial applications, such as in aerodynamics, turbomachinery, geophysical flows, internal engines, etc. Several books exist on fluid turbulence, but Statistical Approach to Wall Turbulence is original in the sense that it focuses solely on the turbulent flows bounded by solid boundaries. The book covers the different physical aspects of wall turbulence, beginning with classical phenomenological aspects before advancing to recent research in the effects of the Reynolds numbers, near wall coherent structures, and wall turbulent transport process. This book would be of interest to postgraduate and undergraduate students in mechanical, chemical, and aerospace engineering, as well as researchers in aerodynamics, combustion, and all applications of wall turbulence.Table of ContentsForeword ix Ivan MARUSIC Introduction xi Chapter 1. Basic Concepts 1 1.1. Introduction 1 1.2. Fundamental equations 1 1.3. Notation 4 1.4. Reynolds averaged Navier-Stokes equations 4 1.5. Basic concepts of turbulent transport mechanisms 6 1.6. Correlation tensor dynamics 11 1.7. Homogeneous turbulence 15 1.8. Isotropic homogeneous turbulence 20 1.9. Axisymmetric homogeneous turbulence 33 1.10. Turbulence scales 35 1.11. Taylor hypothesis 39 1.12. Approaches to modeling wall turbulence 40 Chapter 2. Preliminary Concepts: Phenomenology, Closures and Fine Structure 45 2.1. Introduction 45 2.2. Hydrodynamic stability and origins of wall turbulence 46 2.3. Reynolds equations in internal turbulent flows 55 2.4. Scales in turbulent wall flow 55 2.5. Eddy viscosity closures 56 2.6. Exact equations for fully developed channel flow 61 2.7. Algebraic closures for the mixing length in internal flows 65 2.8. Some illustrations using direct numerical simulations at low Reynolds numbers 69 2.9. Transition to turbulence in a boundary layer on a flat plate 76 2.10. Equations for the turbulent boundary layer 77 2.11. Mean vorticity 81 2.12. Integral equations 83 2.13. Scales in a turbulent boundary layer 85 2.14. Power law distributions and simplified integral approach 85 2.15. Outer layer 88 2.16. Izakson-Millikan-von Mises overlap 89 2.17. Integral quantities 91 2.18. Wake region 94 2.19. Drag coefficient in external turbulent flows 96 2.20. Asymptotic behavior close to the wall 98 2.21. Coherent wall structures – a brief introduction 101 Chapter 3. Inner and Outer Scales: Spectral Behavior 105 3.1. Introduction105 3.2. Townsend-Perry analysis in the fully-developed turbulent sublayer 107 3.3. Spectral densities 110 3.4. Clues to the 1x k _ behavior, and discussion 124 3.5. Spectral density vv E and cospectral density uv E 129 3.6. Two-dimensional spectral densities 131 Chapter 4. Reynolds Number-Based Effects 137 4.1. Introduction 137 4.2. The von Karman constant and the renormalization group 140 4.3. Complete and incomplete similarity 146 4.4. Symmetries and their consequences 155 4.5. Principle of asymptotic invariance. Approach of W.K. George 163 4.6. Mean velocity distribution. Summary 185 4.7. Townsend’s attached eddies 185 4.8. Overlap region in internal flows 228 4.9. Two-point correlations 230 4.10. Active and passive Townsend eddies 239 4.11. Fine structure 249 Chapter 5. Vorticity 259 5.1. Introduction 259 5.2. General characteristics of vorticity 259 5.3. Reynolds shear stress and vorticity transport 261 5.4. Characteristics of the vorticity field close to a wall 264 5.5. Statistics and fine structure 270 5.6. Vorticity transport 277 5.7. Estimating the importance of non-linearity close to the wall 284 5.8. Measurements 287 Notations Used 291 Subscripts and superscripts 293 Greek letters 294 Abbreviations 295 Bibliography 297 Index 309

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Book SynopsisIn fluid mechanics, velocity measurement is fundamental in order to improve the behavior knowledge of the flow. Velocity maps help us to understand the mean flow structure and its fluctuations, in order to further validate codes.Laser velocimetry is an optical technique for velocity measurements; it is based on light scattering by tiny particles assumed to follow the flow, which allows the local fluid flow velocity and its fluctuations to be determined. It is a widely used non-intrusive technique to measure velocities in fluid flows, either locally or in a map.This book presents the various techniques of laser velocimetry, as well as their specific qualities: local measurements or in plane maps, mean or instantaneous values, 3D measurements. Flow seeding with particles is described with currently used products, as well as the appropriate aerosol generators. Post-processing of data allows us to extract synthetic information from measurements and to perform comparisons with results issued from CFD codes. The principles and characteristics of the different available techniques, all based on the scattering of light by tiny particles embedded in the flow, are described in detail; showing how they deliver different information, either locally or in a map, mean values and turbulence characteristics.Table of ContentsPreface xi Alain BOUTIER Intoduction xiii Alain BOUTIER Chapter 1. Measurement Needs in Fluid Mechanics 1 Daniel ARNAL and Pierre MILLAN 1.1. Navier-Stokes equations 2 1.2. Similarity parameters 4 1.3. Scale notion 6 1.4. Equations for turbulent flows and for Reynolds stress tensor 6 1.5. Spatial-temporal correlations 8 1.6. Turbulence models 10 1.6.1. Zero equation model 11 1.6.2. One equation model 11 1.6.3. Two equations model12 1.6.4. Reynolds stress models (RSM, ARSM) 12 1.7. Conclusion 13 1.8. Bibliography . 13 Chapter 2. Classification of Laser Velocimetry Techniques 15 Alain BOUTIER 2.1. Generalities 16 2.2. Definitions and vocabulary 17 2.3. Specificities of LDV 19 2.3.1. Advantages 19 2.3.2. Use limitations 20 2.4. Application domain of laser velocimeters (LDV, PIV, DGV) 21 2.5. Velocity measurements based on interactions with molecules 22 2.5.1. Excitation by electron beams 22 2.5.2. Laser fluorescence 23 2.5.3. Spectroscopy with a tunable laser diode in the infrared 23 2.5.4. Coherent anti-Stokes Raman scattering technique 24 2.5.5. Tagging techniques 24 2.5.6. Summary 25 2.6. Bibliography 28 Chapter 3. Laser Doppler Velocimetry 33 Alain BOUTIER and Jean-Michel MOST 3.1. Introduction 33 3.2. Basic idea: Doppler effect34 3.2.1. Double Doppler effect 34 3.2.2. Four optical set-ups 36 3.2.3. Comments on the four configurations 39 3.3. Fringe velocimetry theory40 3.3.1. Fringe pattern in probe volume 40 3.3.2. Interferometry theory42 3.3.3. Comparison between the three theoretical approaches 44 3.3.4. SNR 44 3.4. Velocity sign measurement 48 3.4.1. Problem origin 48 3.4.2. Solution explanation 49 3.4.3. Various means to shift a laser beam frequency 51 3.5. Emitting and receiving optics 56 3.5.1. Emitting 56 3.5.2. Probe volume characteristics 61 3.5.3. Receiving part 64 3.6. General organigram of a mono-dimensional fringe velocimeter 67 3.7. Necessity for simultaneous measurement of 2 or 3 velocity components 68 3.8. 2D laser velocimetry 70 3.9. 3D laser velocimetry 71 3.9.1. Exotic 3D laser velocimeters 71 3.9.2. 3D fringe laser velocimetry 72 3.9.3. Five-beam 3D laser velocimeters 73 3.9.4. Six-beam 3D laser velocimeters 74 3.10. Electronic processing of Doppler signal 79 3.10.1. Generalities and main classes of Doppler processors 79 3.10.2. Photon converter: photomultiplier 79 3.10.3. Doppler burst detection 84 3.10.4. First processing units 86 3.10.5. Digital processing units 88 3.10.6. Exotic techniques 102 3.10.7. Optimization of signal processing 103 3.11. Measurement accuracy in laser velocimetry 103 3.11.1. Probe volume influence 104 3.11.2. Calibration 105 3.11.3. Doppler signal quality 112 3.11.4. Velocity domain for measurements 114 3.11.5. Synthesis of various bias and error sources117 3.11.6. Specific problems in 2D and 3D devices 123 3.11.7. Global accuracy 126 3.12. Specific laser velocimeters for specific applications 127 3.12.1. Optical fibers in fringe laser velocimetry 127 3.12.2. Miniature laser velocimeters 132 3.12.3. Doppler image of velocity field 133 3.13. Bibliography 134 Chapter 4. Optical Barrier Velocimetry 139 Alain BOUTIER 4.1. Laser two-focus velocimeter 139 4.2. Mosaic laser velocimeter145 4.3. Bibliography 147 Chapter 5. Doppler Global Velocimetry 149 Alain BOUTIER 5.1. Overview of Doppler global velocimetry 149 5.2. Basic principles of DGV 150 5.3. Measurement uncertainties in DGV 153 5.4. Bibliography 156 Chapter 6. Particle Image Velocimetry 159 Michel RIETHMULLER, Laurent DAVID and Bertrand LECORDIER 6.1. Introduction 159 6.2. Two-component PIV 164 6.2.1. Laser light source 164 6.2.2. Emission optics in PIV 168 6.2.3. Image recording 169 6.2.4. PTV (Particle Tracking Velocimetry) 185 6.2.5. Measurement of velocity using PIV 192 6.2.6. Correlation techniques 201 6.3. Three-component PIV 233 6.3.1. Introduction 233 6.3.2. Acquisition of the signal from the particles 234 6.3.3. Evaluation of the particles’ motion 236 6.3.4. Modeling of sensor 237 6.3.5. Stereoscopy: 2D-3C PIV 252 6.3.6. 2.5D-3C surface PIV259 6.3.7. 3C-3D volumic PIV 261 6.3.8. Conclusion 268 6.4. Bibliography 269 Chapter 7. Seeding in Laser Velocimetry 283 Alain BOUTIER and Max ELENA 7.1. Optical properties of tracers 284 7.2. Particle generators 288 7.3. Particle control 292 7.4. Particle behavior 297 7.5. Bibliography 303 Chapter 8. Post-Processing of LDV Data 305 Jacques HAERTIG and Alain BOUTIER 8.1. The average values 306 8.2. Statistical notions 308 8.3. Estimation of autocorrelations and spectra 314 8.3.1. Continuous signals of limited duration 314 8.3.2. Signals sampled periodically (of limited duration T) 316 8.3.3. Random sampling 318 8.4. Temporal filtering: principle and application to white noise 321 8.4.1. Case of white noise 321 8.4.2. Moving average (MA) 323 8.4.3. Autoregressive (AR) process: Markov 324 8.5. Numerical calculations of FT326 8.6. Summary and essential results329 8.7. Detailed calculation of the FT and of the spectrum of fluctuations in velocity measured by laser velocimetry 330 8.7.1. Notations and overview of results regarding the FT 331 8.7.2. Calculating the FT of a sampled function F(t): periodic sampling 333 8.7.3. Calculating the FT of a sampled function F(t): random sampling 335 8.7.4. FT of the sampled signal reconstructed after periodic sampling 339 8.7.5. FT of the sampled signal, reconstructed after random sampling 341 8.7.6. Spectrum of a random signal sampled in a random manner 345 8.7.7. Application to some signals 352 8.7.8. Main conclusions 356 8.8. Statistical bias 358 8.8.1. Simple example of statistical bias 358 8.8.2. Measurement sampling process 360 8.8.3. The various bias phenomena in laser velocimetry368 8.8.4. Analysis of the bias correction put forward by McLaughlin and Tiederman 369 8.8.5. Method for detecting statistical bias 369 8.8.6. Signal reconstruction methods 372 8.8.7. Interpolation methods applied to the reconstructed signal 374 8.9. Spectral analysis on resampled signals 375 8.9.1. Direct transform 376 8.9.2. Slotting technique 377 8.9.3. Kalman interpolating filter 379 8.10. Bibliography 384 Chapter 9. Comparison of Different Techniques 389 Alain BOUTIER 9.1. Introduction 389 9.2. Comparison of signal intensities between DGV, PIV and LDV 390 9.3. Comparison of PIV and DGV capabilities 394 9.4. Conclusion 396 9.5. Bibliography 397 Conclusion 399 Alain BOUTIER Nomenclature 401 List of Authors 407 Index 409

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Book SynopsisThe aim of this book is to relate fluid flows to chemical reactions. It focuses on the establishment of consistent systems of equations with their boundary conditions and interfaces, which allow us to model and deal with complex situations.Chapter 1 is devoted to simple fluids, i.e. to a single chemical constituent. The basic principles of incompressible and compressible fluid mechanics, are presented in the most concise and educational manner possible, for perfect or dissipative fluids. Chapter 2 relates to the flows of fluid mixtures in the presence of chemical reactions. Chapter 3 is concerned with interfaces and lines. Interfaces have been the subject of numerous publications and books for nearly half a century. Lines and curvilinear media are less known Several appendices on mathematical notation, thermodynamics and mechanics methods are grouped together in Chapter 4.This summary presentation of the basic equations of simple fluids, with exercises and their solutions, as well as those of chemically reacting flows, and interfaces and lines will be very useful for graduate students, engineers, teachers and scientific researchers in many domains of science and industry who wish to investigate problems of reactive flows. Portions of the text may be used in courses or seminars on fluid mechanics.Table of ContentsPreface xiii List of the Main Symbols xv Chapter 1. Simple Fluids 1 1.1. Introduction 1 1.2. Key elements in deformation theory – Lagrangian coordinates and Eulerian coordinates 2 1.2.1. Strain rates 2 1.2.2. Lagrangian coordinates and Eulerian coordinates 7 1.2.3. Trajectories, stream lines, emission lines 8 1.3. Key elements in thermodynamics Reversibility, irreversible processes: viscosity, heat conduction 9 1.3.1. Thermodynamic variables, definition of a system, exchanges, differential manifold of equilibrium states, transformation 9 1.3.2. Laws of thermodynamics 11 1.3.3. Properties of simple fluids at equilibrium. 14 1.4. Balance equations in fluid mechanics. Application to incompressible and compressible perfect fluids and viscous fluids 18 1.4.1. Mass balance 18 1.4.2. Concept of a particle in a continuous medium: local state 19 1.4.3. Balance for the property F 20 1.4.4. Application to volume, to momentum and to energy 22 1.4.5. Entropy balance and the expression of the rate of production of entropy 23 1.4.6. Balance laws for discontinuity 25 1.4.7. Application to incompressible perfect fluids 26 1.4.8. Application to dissipative fluids 31 1.5. Examples of problems with 2D and 3D incompressible perfect fluids 32 1.5.1. Planar 2D irrotational flows: description in the complex plane of steady flows 32 1.5.2. 3D irrotational flows of incompressible perfect fluids: source, sink, doublet 36 1.5.3. Rotational flows of incompressible perfect fluids 41 1.6. Examples of problems with a compressible perfect fluid: shockwave, flow in a nozzle, and characteristics theory 44 1.6.1. General theorems 44 1.6.2. Propagation of sound in an ideal gas 44 1.6.3. Discontinuities 46 1.6.4. Unsteady characteristics 47 1.6.5. Steady normal shockwave: Hugoniot and Prandtl relations 48 1.6.6. Flow in a de Laval nozzle 49 1.6.7. Simple wave 53 1.7. Examples of problems with viscous fluids 56 1.7.1. General equations 56 1.7.2. Incompressible viscous fluid 57 1.7.3. Flow of a compressible dissipative fluid: structure of a shockwave 61 1.8. Exercises 64 1.8.1. Exercises in kinematics (section 1.2) 64 1.8.2. Exercises in thermodynamics (section 1.3). 67 1.8.3. Exercises for the balance equations in fluid mechanics (section 1.4) 68 1.8.4. Examples of problems with 2D and 3D incompressible perfect fluids (section 1.5) 70 1.8.5. Examples of problems with a compressible perfect fluid (section 1.6) 74 1.8.6. Examples of problems with viscous fluids (section 1.7) 77 1.9. Solutions to the exercises 79 1.9.1. Solutions to the exercises in kinematics. 79 1.9.2. Solutions to the Exercises in thermodynamics 83 1.9.3. Solutions to the exercises for the balance of equations in fluid mechanics 88 1.9.4. Solutions to the examples of problems with 2D and 3D incompressible perfect fluids 89 1.9.5. Solutions to the examples of problems with a compressible perfect fluid 93 1.9.6. Solutions to the examples of problems with viscous fluids 95 Chapter 2. Reactive Mixtures 101 2.1. Introduction 101 2.2. Equations of state 103 2.2.1. Definition of the variables of state of a mixture 103 2.2.2. Thermodynamic properties of mixtures 108 2.2.3. Reactive mixture 118 2.2.4. Other issues relating to the thermodynamics of mixtures 123 2.3. Balance equations of flows of reactive mixtures 124 2.3.1. Balance of mass of the species j and overall balance of mass 124 2.3.2. General balance equation of a property F. 127 2.3.3. Momentum balance 129 2.3.4. Energy balance 129 2.3.5. Balance relations in a discrete system. 132 2.3.6. Entropy balance in a continuum 137 2.3.7. Balance equations at discontinuities in continuous media 140 2.4. Phenomena of transfer and chemical kinetics 142 2.4.1. Introduction 142 2.4.2. Presentation of the transfer coefficients by linear TIP 143 2.4.3. Other presentations of the transfer coefficients 147 2.4.4. Elements of chemical kinetics 152 2.5. Couplings 155 2.5.1. Heat transfer and diffusion 155 2.5.2. Shvab-Zeldovich approximation 158 Chapter 3. Interfaces and Lines 163 3.1. Introduction 163 3.1.1. Interfaces 163 3.1.2. Lines 165 3.2. Interfacial phenomena 166 3.2.1. General aspects 166 3.2.2. General form of an interfacial balance law 168 3.2.3. Constitutive laws for interfaces whose variables directly satisfy the classical equations in thermostatics and in 2D-TIP 173 3.2.4. Constitutive laws for interfaces deduced from classical thermostatics and 3D-TIP. Stretched flame example 177 3.2.5. Interfaces manifesting resistance to folding 179 3.2.6. Numerical modeling 179 3.2.7. Interfaces and the second gradient theory. 182 3.2.8. Boundary conditions of the interfaces 185 3.2.9. Conclusion 185 3.3. Solid and fluid curvilinear media: pipes, fluid lines and filaments 186 3.3.1. General aspects 186 3.3.2. Establishing the balance equations in a curvilinear medium. 188 3.3.3. Simplified theories 209 3.3.4. Triple line and second gradient theory 216 3.3.5. Conclusion 220 3.4. Exercises 222 3.4.1. Exercises regarding solid curvilinear media 222 3.4.2. Exercises regarding fluid curvilinear media 222 3.5. Solutions to the exercises 223 3.5.1. Solutions to exercises regarding solid curvilinear media. 223 3.5.2. Solutions to the exercises regarding fluid curvilinear media 225 APPENDICES 229 Appendix 1. Tensors, Curvilinear Coordinates, Geometry and Kinematics of Interfaces and Lines 231 A1.1. Tensor notations 231 A1.1.1. Tensors and operations on tensors 231 A1.2. Orthogonal curvilinear coordinates. 234 A1.2.1. General aspects 234 A1.2.2. Curl of a vector field 236 A1.2.3. Divergence of a vector field 237 A1.2.4. Gradient of a scalar 238 A1.2.5. Laplacian of a scalar 238 A1.2.6. Differentiation in a curvilinear basis 238 A1.2.7. Divergence of a second order tensor 239 A1.2.8. Gradient of a vector 239 A1.2.9. Cylindrical coordinates and spherical coordinates 240 A1.3. Interfacial layers 242 A1.3.1. Prevailing directions of an interfacial medium 242 A1.3.2. Operators of projection for interfaces 244 A1.3.3. Surface gradients of a scalar field 245 A1.3.4. Curvature vector of a curve 245 A1.3.5. Normal and tangential divergences of a vector field 246 A1.3.6. Extension of surface per unit length 246 A1.3.7. Average normal curvature of a surface 247 A1.3.8. Breakdown of the divergence of a vector field 248 A1.3.9. Breakdown of the Laplacian of a scalar field 249 A1.3.10. Breakdown of the divergence of a second order tensor 249 A1.3.11. Projection operators with the intrinsic definition of a surface 252 A1.3.12. Comparison between the two descriptions 253 A1.4. Curvilinear zones 254 A1.4.1. Presentation 254 A1.4.2. Geometry of the orthogonal curvilinear coordinates 256 A1.4.3. Projection operators and their consequences 257 A1.5. Kinematics in orthogonal curvilinear coordinates 260 A1.5.1. Kinematics of interfacial layers 260 A1.5.2. Kinematics of curvilinear zones 266 A1.5.3. Description of the center line 269 Appendix 2. Additional Aspects of Thermostatics 277 A2.1. Laws of state for real fluids with a single constituent 277 A2.1.1. Diagram of state for a pure fluid 277 A2.1.2. Approximate method to determine the thermodynamic functions 278 A2.1.3. Van der Waals fluid 279 A2.1.4. Other laws for dense gases and liquids 279 A2.2. Mixtures of real fluids 280 A2.2.1. Mixture laws for a real mixture 280 A2.2.2. Expression of the free energy of a real mixture 281 Appendix 3. Tables for Calculating Flows of Ideal Gas ƒ× ƒ­1.4 283 A3.1. Calculating the parameters in continuous steady flow (section 1.6.6.2) 286 A3.2. Formulae for steady normal shockwaves 288 Appendix 4. Extended Irreversible Thermodynamics. 289 A4.1. Heat balance equations in a non-deformable medium in EIT 290 A4.2. Application to a 1D case of heat transfer 293 A4.3. Application to heat transfer with the evaporation of a droplet 296 A4.3.1. Reminders about evaporating droplets 296 A4.3.2. Evaporating droplet with EIT. 300 A4.4. Application to thermal shock 302 A4.4.1. Presentation of the problem and solution using CIT 302 A4.4.2. Thermal shock and EIT 303 A4.4.3. Application of the second order approximation into two examples of thermal shock 305 A4.5. Outline of EIT 307 A4.6. Applications and perspectives of EIT 310 Appendix 5. Rational Thermodynamics 313 A5.1. Introduction 313 A5.2. Fundamental hypotheses and axioms 314 A5.2.1. Basic hypotheses 314 A5.2.2. Basic axioms 316 A5.3. Constitutive laws 318 A5.4. Case of the reactive mixture 320 A5.4.1. Principle of material frame indifference 320 A5.4.2. Constitutive laws for a reactive mixture 321 A5.5. Critical remarks 324 Appendix 6. Torsors and Distributors in Solid Mechanics 325 A6.1. Introduction 325 A6.1.1. Torsor 325 A6.1.2. Distributor 325 A6.1.3. Power 326 A6.2. Derivatives of torsors and distributors which depend on a single position parameter 326 A6.2.1. Derivative of the velocity distributor 327 A6.2.2. Derivative of the tensor of forces 328 A6.3. Derivatives of torsors and distributors dependent on two positional parameters 328 A6.3.1. Expression of the velocity distributor 329 A6.3.2. Derivative of the velocity distributor 329 Appendix 7. Virtual Powers in a Medium with a Single Constituent 331 A7.1. Introduction 331 A7.2. Virtual powers of a system of n material points 332 A7.3. Virtual power law 333 A7.4. The rigid body and systems of rigid bodies 333 A7.4.1. The rigid body 333 A7.4.2. System of rigid bodies, concept of a link 334 A7.5. 3D deformable continuous medium 335 A7.5.1. First gradient theory 335 A7.5.2. A 3D case of perfect internal linkage: the incompressible perfect fluid 337 A7.5.3. Second gradient theory 337 A7.6. 1D continuous deformable medium 338 A7.6.1. First gradient theory 338 A7.6.2. A 1D case of perfect internal linkage: perfectly flexible and inextensible wires 340 A7.7. 2D deformable continuous medium 340 Bibliography 343 Index 355

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