Free-Surface Flow

1. Basic Concepts

1.1 Introduction

4

1.1.1 Reductionism 4

1.1.2 Free-Surface Flow Models 6

1.2 Macroscopic Theory of Flow

10

1.2.1 Fluid Density 10

1.2.2 The Fluid Continuum Hypothesis 11

1.3 Coordinate Systems

15

1.4 The Laws of Motion

19

1.4.1 Law of Universal Gravitation 19

1.4.2 Prediction by Asymptotic Approximation 22

1.4.3 Pressure Variation in a Static Fluid 23

1.5 Inertial Frames of Reference

27

1.5.1 Spacetime, World Lines, and Wave Surfaces 29

1.5.2 Index Notation – The Summation Convention 31

1.6 Euclidean Space

33

1.6.1 Translation 33

1.6.2 Scaling 33

1.6.3 Reflection 33

1.6.4 Rotation 33

1.6.5 Direction Cosines 34

1.6.6 Rotation in Three-Dimensional Space 35

1.6.7 Matrices 36

1.6.8 Determinants 37

1.6.9 The Levi-Civita Permutation Symbol 38

1.6.10 Index Notation – Range Convention 40

1.7 Simple Harmonic Motion

42

1.7.1 Exponential Representation 44

1.7.2 Fourier Series 45

1.7.3 Probability Density Functions 47

1.7.4 Spectral Analysis 49

1.8 Cartesian Vectors 52

1.8.1 Eigenvalues and Eigenvectors 52

1.8.2 Flux and the Scalar Product 54

1.8.2.1 Equation of a Plane 56

1.8.3 Vector Product 57

1.8.4 Epsilon-Delta Identities 58

1.8.5 Triple Products 60

1.8.6 Orthogonal Decomposition 62

1.9 Cartesian Tensors

63

1.9.1 Physical Meaning of Tensors 64

1.9.2 Quadric Cone 64

1.9.3 Stress at a Point 66

1.9.4 Stress on an Oblique Plane 68

1.9.5 Plane Stress 70

1.9.6 Principal Directions 71

1.9.7 Hydrostatic and Deviatoric Stresses 72

1.9.8 Elements of Tensor Algebra 73

1.9.8.1 Addition 73

1.9.8.2 Multiplication 73

1.9.8.3 Contraction 73

1.9.8.4 Symmetry 74

1.9.8.5 Invariants 74

1.9.9 Isotropic Tensors 75

1.9.10 Tensors in General Coordinates 76

1.9.10.1 Basis for Space Vectors 76

1.9.10.2 The Dual Basis 77

1.9.10.3 Tensor Components 77

1.9.11 Polar and Axial Vectors 78

1.10 Stress in a Moving Fluid

80

1.10.1 Newton’s Law of Viscosity 80

1.10.2 Uniform Shear Flow 81

1.10.3 Coefficient of Dynamic Viscosity 83

1.10.4 No-Slip Condition 84

1.10.5 Other Flux-Gradient Laws 86

Problems

89

References

92

Notes

93

2. Kinematics, Composition, and Thermodynamics

2.1 Introduction

96

2.2 Scalar and Vector Fields

97

2.2.1 Gradient of a Scalar Field 98

2.2.2 Directional Derivative 99

2.2.3 Divergence of a Vector Field 100

2.2.4 Generalization of Gradient and Divergence 102

2.2.5 Curl of a Vector Field 103

2.2.6 Curl of a Vector Product 106

2.2.7 Scalar Potential 107

2.2.8 Vector Potential 109

2.3 Curvilinear Coordinates

110

2.3.1 Cylindrical Coordinates 110

2.3.2 Spherical Coordinates 111

2.4 The Laplacian

113

2.4.1 Curvilinear Coordinates 113

2.4.2 Vector Laplacian 114

2.5 Material Coordinates and Derivatives

115

2.5.1 Fluid Acceleration 118

2.6 Pathlines and Streamlines

120

2.6.1 Stream Surfaces and Streaklines 122

2.6.2 Streamlines in Unsteady Flow 124

2.7 Relative Motion in Eulerian Coordinates

127

2.8 Space Curves

130

2.8.1 Curvature and Principal Normal 130

2.8.2 Natural Coordinates 133

2.8.3 Flow in a Circular Path 135

2.8.4 Estimation of the Radius of Curvature 136

2.9 Integrals of Vector Fields

137

2.9.1 The Rate of Expansion of a Material Volume 137

2.9.2 The Divergence Theorem 138

2.9.3 The Transport Theorem 140

2.9.4 The Leibniz Rule 142

2.9.5 Conservation Laws 142

2.10 Composition

144

2.11 Thermodynamic Relations

146

2.12 Entropy Changes

148

2.12.1 Statistical Definition of Entropy 148

2.13 The Equation of State

150

2.13.1 Ideal Gases 151

2.13.2 Pure and Seawater 151

2.14 First Law of Thermodynamics

154

2.14.1 Heat Capacities 154

2.15 Second Law of Thermodynamics

157

2.15.1 Clausius Inequality 158

2.15.2 The Combined Laws 161

2.15.3 Maxwell Relations 162

2.15.4 Isentropic Processes for Ideal Gases 163

2.15.5 Free Energy 163

2.15.6 Free Enthalpy 164

2.15.7 Equation of State for Internal Energy 166

2.16 Fluid Compressibility

167

Problems

168

References

171

Note

172

3. Diffusive Mass Transfer

3.1 Introduction 176

3.2 Fick’s Law of Diffusion

178

3.2.1 The Unit Impulse Load 180

3.2.2 Thermal Energy of Solute Particles 182

3.2.3 Brownian Motion 186

3.2.4 Langevin’s Equation of Motion 188

3.2.5 A Heuristic Model for Diffusion 191

3.3 Differential Mass Balance

194

3.3.1 Macroscopic Mass Balance 196

3.4 Sources and Sinks

199

3.4.1 Distributed Source or Sink 199

3.4.2 Point, Line, and Plane Source or Sink 200

3.5 Sudden Release of Solute in a Channel

202

3.5.1 Scales of the Diffusion Equation 203

3.5.2 Dimensional Analysis 205

3.5.3 Similarity Solution 206

3.5.4 Properties of the Diffusion Equation 208

3.6 The Unit Impulse Response Function

211

3.6.1 Properties of the Unit Impulse Response Function 214

3.6.2 Gaussian Distribution 215

3.6.3 Non-Gaussian Distributions 216

3.7 Continuous Injection of Mass

219

3.7.1 Evolution of an Initial Concentration Profile 222

3.7.2 Spatially Distributed Maintained Source 224

3.8 The Fourier Transform

225

3.8.1 Differential Properties 226

3.8.2 Transform of the Diffusion Equation 226

3.8.3 Unit Impulse Load 227

3.8.4 Convolution 228

3.9 Specified Concentration History

230

3.9.1 Constant Concentration 233

3.9.2 Linear Increase of Concentration 233

3.9.3 Square-Root Increase of Concentration 233

3.9.4 Exponential Increase of Concentration 233

3.10 Diffusion Coupled With Adsorption

235

3.10.1 Diffusion Coupled With Reaction 236

3.11 Transform Methods for Diffusion-Reaction

237

3.11.1 Local Transform 237

3.11.2 Integral Transform 237

3.11.3 Laplace Transform 238

3.12 Inertia-Moderated Diffusion

242

3.13 Multi-Dimensional Diffusion

245

3.14 Boundary Conditions – Method of Images

248

3.14.1 Two-Dimensional Problems 249

Problems

254

References

257

4. Advective Mass Transfer

4.1 Introduction 260

4.2 Advective Mass Balance

261

4.2.1 One-Dimensional Advection 262

4.2.2 Depth-Averaged Advection 264

4.2.3 Cross-Sectional Area-Averaged Advection 267

4.3 Fourier Transform of Advection Equation

268

4.3.1 Discrete Fourier Transform 269

4.3.2 Discontinuous Concentration Profiles 270

4.4 Advection Coupled With Diffusion

273

4.4.1 Specified Upstream Concentration of an Active Solute 274

4.4.2 Transverse Diffusion 276

4.5 Order of Magnitude Analysis

279

4.5.1 Steady State Analysis 280

4.6 Mixing in Unidirectional Flow

283

4.6.1 Fixed Sources in Unidirectional Flow 284

4.6.1.1 Steady Point Source in an Unbounded Stream 284

4.6.1.2 Steady Line Source in a Stream 286

4.6.1.3 Plane Source in a Stream 290

4.6.2 Steady-State Plane Source 291

4.6.3 Advection-Reaction Equation 293

4.7 Distance Required for Complete Mixing

296

Problems

302

References

306

5. Viscous Fluid Flow

5.1 Introduction

310

5.2 Conservation of Mass – Continuity

311

5.2.1 Incompressibility Constraint 313

5.3 The Stream Function

316

5.3.1 Flow Between Streamlines 317

5.3.2 Axisymmetric Flow 318

5.4 Conservation of Momentum

320

5.4.1 Body Forces on a Fluid Element 320

5.4.2 Surface Forces on a Fluid Element 322

5.4.3 Equation of Motion 324

5.4.4 Equation of Momentum 325

5.5 Conservation of Energy

327

5.5.1 Equation of Mechanical Energy 327

5.5.2 Total Energy Equation 328

5.5.3 Equation of Internal Energy 330

5.6 Impact of the Velocity Field on a Fluid Element

332

5.6.1 The Rate of Strain of a Fluid Element 334

5.6.2 Parallel Shear Flow 336

5.6.2.1 Strain Ellipse 336

5.7 Evolution of a Fluid Element

339

5.7.1 Fluid Translation 339

5.7.2 Fluid Dilatation 339

5.7.3 Angular Deformation 340

5.7.4 Rotation of a Rigid Body 341

5.7.5 Fluid Rotation 342

5.8 Stress-Strain Rate Relation – The Stokes Hypothesis

344

5.8.1 Pressure in a Moving Fluid 347

5.9 The Incompressible Navier-Stokes Equations

350

5.9.1 Pressure Poisson Equation 352

5.9.1.1 Boundary Conditions 354

5.9.2 Cylindrical Coordinates 356

5.9.3 Viscous Dissipation of Energy 356

5.9.4 An Alternative Form of the Thermal Energy Equation 358

5.9.5 Vorticity-Stream Function Formulation 359

5.10 Scaling the Navier-Stokes Equations

361

5.10.1 Laminar and Turbulent Flow Regimes 363

5.10.2 Effects of Gravity – The Froude Number 364

5.10.3 Periodic Flow Motion – The Strouhal Number 366

5.11 Fundamental Viscous-Flow Problems

369

5.11.1 Flow Driven by a Moving Boundary 369

5.11.1.1 The Rate of Energy Dissipation 371

5.11.2 Flow Between Parallel Walls 372

5.11.3 Two-Layer Flow Between Parallel Walls 373

5.11.4 Unsteady Flow Problems 376

5.11.4.1 Flow Due to a Suddenly Accelerated Plate 376

5.11.4.2 Flow Near an Oscillating Plate 378

5.12 Integral Equations for Fluid Flow

381

5.12.1 Macroscopic Volume Balance 381

5.12.2 Macroscopic Momentum Balance 382

5.12.3 Macroscopic Energy Balance 384

5.13 Creeping Flow

386

5.13.1 Properties of Creeping Flow 387

5.13.2 The Paint-Scraper Problem 388

5.13.3 Flow Around a Sphere 390

5.13.3.1 Pressure Distribution Around a Sphere 394

5.13.3.2 Stress Distribution Around a Sphere 394

5.13.4 Drag Force on Sphere 396

5.13.5 Fall Velocity 397

5.13.6 Oseen’s Improved Creeping Flow Approximation 397

5.13.7 Hele-Shaw Flow 400

5.13.7.1 Hele-Shaw Flow Around a Cylinder 401

Problems

403

References

406

6. Ideal Fluid Flow

6.1 Introduction

410

6.2 The Velocity Potential

411

6.2.0.1 Curvilinear Coordinates 412

6.2.1 Equipotential Surfaces and Lines 412

6.2.2 Harmonic Flow Fields 413

6.2.3 The Stream Function in Irrotational Flow 414

6.2.4 Green’s Identities 415

6.2.5 Elliptic Boundary-Value Problems 416

6.2.6 The Mean-Value Property 416

6.2.7 The Free Space Function 417

6.2.8 The Influence of a Closed Boundary 419

6.2.9 Solution of the Dirichlet Problem 421

6.3 Euler’s Equations

426

6.3.1 Boundary Conditions for Ideal Fluid Flow 426

6.3.2 The Role of Vorticity 427

6.4 Bernoulli’s Equation for Irrotational Flow

428

6.4.1 Bernoulli’s Equation for Steady Flow 430

6.4.2 Bernoulli’s Equation Along a Single Streamline 430

6.4.3 Pressure Variation Along a Streamline 431

6.4.4 The Coanda Effect 432

6.4.5 Draining of a Soda Straw 433

6.5 Standard Patterns of Flow

435

6.5.1 Uniform Flow Along the x Axis 435

6.5.2 Uniform Flow in Arbitrary Direction 435

6.5.3 Flow From a Line Source 436

6.5.4 Sink With Spherical Symmetry – Collapse of a Bubble 438

6.5.5 The Free Vortex 440

6.5.6 Source in a Uniform Stream 441

6.5.7 Sink and Source of Equal Strength 444

6.5.8 The Doublet 446

6.5.9 Rankine Oval 447

6.5.10 Flow Past a Circular Cylinder 448

6.5.11 The Flow Net 449

6.5.12 Drag on Cylinder 450

6.5.13 Unsteady Flow and Virtual Mass 452

6.5.14 Potential Flow Past a Sphere 454

6.6 Conformal Mapping

458

6.6.1 Complex Variables 458

6.6.2 Cauchy-Riemann Equations 459

6.6.3 Complex Potential 460

6.6.4 Conformal Transformations 461

6.6.5 Power-Law Mapping 463

6.6.5.1 Uniform Stream; n = 1 463

6.6.5.2 Flow Near a Corner; n = π/α 463

6.6.5.3 Doublet; n=−1 465

6.6.6 Logarithmic Mapping 466

6.6.6.1 Line Source 466

6.6.6.2 Free Vortex 466

6.6.6.3 Line Source Near a Corner 466

6.6.7 Force and Moment on a Cylinder 467

6.6.7.1 Blasius Theorem 467

6.6.7.2 Cauchy’s Integral Theorem 469

6.6.7.3 Cauchy’s Integral Formula 469

6.7 Polygonal Boundaries

473

6.7.1 Change of Direction Under Conformal Mapping 473

6.7.2 Mapping of Polygons 475

6.7.3 The Schwarz-Christofell Transformation 476

6.7.3.1 Map of Semi-Infinite Channel 477

6.7.3.2 Map of Infinite Channel 478

6.7.4 Free Streamlines 479

6.7.4.1 Sequential Transforms 480

6.7.4.2 Flow Exiting Through a Sharp Orifice 481

6.7.4.3 Contraction Coefficient 485

Problems

486

References

488

7. Vorticity Dynamics

7.1 Introduction

492

7.1.1 Vortex Lines 492

7.1.2 Visualization of Vorticity 493

7.2 Vorticity in Shear Flow

495

7.2.1 Horse-Shoe Vortex 496

7.2.2 Vorticity in Natural Coordinates 496

7.2.3 Circulation 498

7.2.4 Divergence of the Vorticity Field 498

7.3 Vortex Sheets

500

7.3.1 Stream Induced by Vorticity 501

7.4 Concentrated Vortices

503

7.4.1 The Forced Vortex 503

7.4.2 The Free (Irrotational) Vortex 505

7.4.3 The Rankine Vortex 507

7.5 Cellular Flows

510

7.6 The Vorticity Transport Equation

513

7.6.1 Diffusion of Vorticity 517

7.6.2 Vortex Shedding 518

7.6.3 Vortex Lines “Frozen in the Fluid” 519

7.7 Vorticity Theorems

522

7.7.1 Stokes’ Theorem 522

7.7.2 Vortex Strength Theorem 523

7.7.3 Vortex End Theorem 524

7.7.4 Helicity 525

7.7.5 Enstrophy 529

7.7.6 Kelvin’s Circulation Theorem 531

7.7.7 Conservation of Helicity 533

Problems 535

References

536

8. Turbulent Flow

8.1 Introduction

540

8.2 Turbulent Flow

542

8.2.1 Transition to Turbulence 542

8.2.2 Instability of Laminar Flow 543

8.2.3 Orr-Sommerfeld Equation for Stability 543

8.2.4 Inviscid Instability 546

8.2.5 Viscous Instability 547

8.2.6 Squire’s Theorem 548

8.2.7 Stability of Flow in Open-Channel Flow 550

8.3 Averaging of Turbulent Flow Fields

552

8.3.1 Velocity Fluctuations 553

8.3.2 Correlation of Velocity Fluctuations 554

8.3.3 Homogeneous Turbulence 555

8.3.4 Taylor Microscale 557

8.3.5 Isotropic Turbulence 557

8.4 Scales of Turbulent Motion

559

8.4.1 Kolmogorov Microscale 560

8.4.2 Inertial Subrange 561

8.4.3 Energy Spectrum 562

8.4.4 Dissipation Spectrum 564

8.4.5 Universal Equilibrium 565

8.5 Time-Averaged Equations

568

8.6 Transport of Reynolds Stresses

571

8.7 Turbulence Closure Models

574

8.7.1 Eddy Viscosity 574

8.7.2 Mixing-Length Theory 575

8.7.3 von Kármán’s Similarity Hypothesis 578

8.7.4 The Viscous Sublayer 579

8.8 Eddy Viscosity Profile

583

8.9 Unified Model for Channel Flow

585

8.9.1 The Buffer Region 587

8.9.2 The van Driest Model 588

8.9.3 Empirical Velocity Distributions 590

8.10 Kinetic Energy-Dissipation

(kt −ε) Model 593

8.10.1 Transport of TKE 593

8.10.1.1 Homogeneous Turbulence 595

8.10.2 Scaling Considerations 596

8.10.3 Transport Equation for the Dissipation Rate 597

8.10.4 Direct Numerical Simulation 600

8.11 Large-Eddy Simulation

602

8.11.1 Spatial Filtering 603

8.11.1.1 Filter Properties 604

8.11.1.2 Basic Filters 605

8.11.2 Discrete Filtering 606

8.11.3 Filtered Navier-Stokes Equations 608

8.11.4 Smagorinsky Subfilter Model 610

8.11.4.1 Effect of Boundaries 611

8.11.5 Dynamic Smagorinsky Model 612

8.11.6 Turbulence Model Selection 615

Problems

617

References

619

Note

622

9. Boundary-Layer Flow

9.1 Introduction

626

9.2 Boundary-Layer Theory

627

9.2.1 Laminar Boundary Layer Past a Flat Plate 629

9.2.1.1 Scaling the Boundary Layer Equations 631

9.2.1.2 Similarity Solution 632

9.2.1.3 Velocity Distribution 635

9.2.2 Impact of Boundary Layer on Free Stream 637

9.2.3 Wall Suction 639

9.2.4 Skin Friction 640

9.2.5 Integral Relations 642

9.2.5.1 Zero Pressure Gradient 643

9.2.6 Wake Downstream of a Flat Plate 645

9.2.7 Boundary-Layer Separation 646

9.2.8 Wake Downstream of a Bluff Body 649

9.3 Turbulent Boundary-Layer Flow

653

9.3.1 Turbulent Boundary-Layer Equations 655

9.3.2 Integral Relations 656

9.3.3 Direct Numerical Simulation 658

9.4 Free Shear Flows

660

9.4.1 Free Shear Layers 660

9.4.1.1 Asymptotic Solution 662

9.4.2 Axisymmetric Turbulent Jets 664

9.4.2.1 Similarity Solution 666

9.4.2.2 Axial Velocity Profile 667

9.4.2.3 Radial Velocity 668

9.4.3 Turbulent Axisymmetric Wake 669

Problems

673

References

675

10. Geophysical Effects

10.1 Introduction

680

10.2 Effects of the Earth’s Rotation

682

10.2.1 Acceleration in a Rotating Coordinate System 682

10.2.2 Centrifugal (Fictitious) Acceleration 685

10.2.3 Local Coordinates on a Spherical Earth 686

10.2.4 Effective Gravity 687

10.2.4.1 Effect of Altitude 689

10.2.5 Coriolis Acceleration 690

10.2.6 Fictitious Force on a Rotating Frame 691

10.3 The Geopotential Field

693

10.3.1 Geopotential Height 694

10.4 Hydrostatic Equilibrium

697

10.4.1 Pressure Variation in the Atmosphere 698

10.4.2 Potential Temperature and Density 701

10.4.3 Virtual Potential Temperature 702

10.4.4 Pressure Variation in the Ocean 703

10.4.5 The Density Scale Height 704

10.4.6 Condition of Incompressibility 707

10.5 The Boussinesq Approximation

709

10.5.1 Almost Incompressible Fluids 712

10.5.2 Thermal Energy Approximation 713

10.6 Scales of Geophysical Flows

716

10.6.1 Horizontal Momentum 717

10.6.2 Vertical Momentum 719

10.7 Simple Geophysical Flows

721

10.7.1 Inertial Oscillations 721

10.7.2 Geostrophic Balance 722

10.7.3 Barotropic Flow 724

10.7.4 Density Currents 727

10.7.5 The Taylor-Proudman Phenomenon 728

10.7.6 The Ekman Layer 729

10.7.6.1 Bottom Resistance 729

10.7.6.2 Uniform Core Flow 731

10.7.6.3 Ekman Transport and Pumping 733

10.7.7 Wind Stress and the Surface Ekman Layer 735

10.7.7.1 Surface Layer Transport 737

10.7.7.2 Surface Layer Pumping 737

10.7.7.3 Coastal Upwelling 739

Problems

741

References

743

11. Stratified Flow

11.1 Introduction

746

11.1.1 The Richardson Number 748

11.2 Discrete Layer Approximation

750

11.2.1 Viscous Flow in an Open Channel 750

11.2.2 Dense Bottom Currents 752

11.2.3 Wind-Driven Circulation 754

11.3 Interfacial Stability

761

11.3.1 Normal Mode Analysis 762

11.3.2 Effects of Surface Tension 764

11.3.3 Rayleigh-Taylor Instability 766

11.3.4 Kelvin-Helmholtz instability 766

11.4 Continuously Stratified Flow

769

11.4.1 Internal Waves 771

11.4.2 Periodic Internal Waves 772

11.4.3 Internal Wave Orientation 774

11.4.4 Uniform Stratification 776

11.4.5 Internal Seiches 778

11.5 Density Currents

780

11.5.1 Arrested Density Current 781

11.5.2 Density Currents in Stratified Flow 785

11.6 Turbulence and Stratification

789

11.6.1 Taylor-Goldstein Equation 789

11.7 Reynolds-Averaged Boussinesq Equations for Stratified Flow

793

11.7.1 Kinetic Energy of the Mean Flow 793

11.7.2 Turbulent Kinetic Energy 794

11.7.3 Available Potential Energy 795

Problems

798

References

800

12. Turbulent Mixing and Dispersion

12.1 Introduction

804

12.1.1 Turbulent Mixing 804

12.1.2 Dispersion 805

12.1.3 Time Scale of Turbulent Diffusion 806

12.2 Time-Averaged Equations for Mass Transport

810

12.2.1 Length Scales of Turbulent Transport 813

12.3 Particle Correlations in Turbulent Flow

815

12.3.1 Lagrangian Correlations 815

12.3.2 Lagrangian Autocorrelation Coefficient 817

12.3.3 Lagrangian Integral Time Scale 819

12.3.4 Eddy Diffusion Coefficient 820

12.3.5 Turbulent Diffusion in Three Dimensions 821

12.4 Relative Diffusion of Fluid Particles

823

12.4.1 Small Times 825

12.4.2 Intermediate Times 828

12.4.3 Large Times 829

12.5 Shear Dispersion

831

12.5.1 Dispersion in Parallel Shear Flow 831

12.5.2 Evolution of the Spatial Variance 832

12.5.2.1 Average Concentration 834

12.5.2.2 Center of Mass 836

12.5.2.3 Spatial Variance 837

12.5.2.4 Infinitely Deep Channel 837

12.6 Dispersion in Shallow Water 839

12.6.1 Vertical Mixing in Open Channels 839

12.6.2 Depth-Averaged Laminar Flow 842

12.6.3 Dispersion in Laminar Channel Flow 844

12.6.4 Advection-Dispersion Equation 848

12.6.5 Dispersion in Turbulent Channel Flow 849

12.6.6 Time Scales for Dispersion 852

12.6.7 Dispersion in Two-Dimensional Flow 853

12.6.8 Dispersion in Section-Averaged Flow 855

12.6.9 The Transverse Mixing Coefficient 856

12.6.10 Section-Averaged Dispersion Coefficient 858

12.6.11 Dispersion in a Tidal Estuary 860

Problems

863

References

865

13. Optimal Design and Flow Control

13.1 Introduction

870

13.1.1 Constrained Optimization 871

13.1.2 Optimization Methods 872

13.2 Gradient-Based Methods

874

13.2.1 Steepest Descent 875

13.2.1.1 The Rosenbrock Function 877

13.2.2 Newton’s Method 878

13.2.3 Quasi-Newton Methods 880

13.2.4 Secant Method (BFGS) 880

13.2.5 Hessian Update 881

13.3 Conjugate Gradient Method

884

13.3.1 Line Search 884

13.3.2 Conjugate Gradient 884

13.4 Adjoint Problem Formulation

887

13.4.1 Optimal Source Placement 888

13.4.2 Adjoint Equation 890

13.4.3 Dual Functional 891

13.4.4 Time Dependent Source 891

13.4.5 Variational Approach 895

13.4.5.1 Main Problem 895

13.4.5.2 Adjoint Problem 896

13.4.6 Sensitivity 897

13.5 Generalized Adjoint Problem

899

13.5.1 Construction of Adjoint Problem 901

13.5.2 Optimal Release Sequence in Shallow Water 902

13.5.3 Uncertainty Analysis 906

13.6 Estimation of Dispersion Coefficients

908

13.7 Source Inversion

910

13.7.1 Importance of the Péclet Number 913

13.7.2 Time Dependent Sources 915

13.8 Active Control of Solute Slugs and Plumes 917

13.8.1 Adjoint Problem Formulation 918

13.8.2 Control of Two-Dimensional Slug 920

13.8.3 Control of Three-Dimensional Slug 921

Problems

925

References

927

Epilogue 929

Note

930

Bibliography 931

Index 935