Avalanche dynamics : dynamics of rapid flows of dense granular avalanches

Avalanches, mudflows and landslides are common and natural phenomena that occur in mountainous regions. With an emphasis on snow avalanches, this book provides a survey and discussion about the motion of avalanche-like flows from initiation to run out. An important aspect of this book is the formulation and investigation of a simple but appropriate continuum mechanical model for the realistic prediction of geophysical flows of granular material.

Part I Introduction, Conception and the Importance of Avalanche Research
1 Introduction	p. 3
1.1 Motivation	p. 3
1.2 Goals, Methods and Structure	p. 8
1.2.1 Goals	p. 8
1.2.2 Methodology	p. 9
1.2.3 Structure	p. 10
1.3 Necessities for Avalanche Studies	p. 14
1.3.1 Snow Avalanche Hazards and Fatalities	p. 15
1.3.2 Debris and Mud Flows, Pyroclastic Flows and Lahars	p. 17
1.3.3 International Scientific Activities	p. 24
1.4 A History of Avalanche Research	p. 26
1.4.1 Early History	p. 27
1.4.2 Modern History	p. 27
2 Granular Avalanches: Definition, Related Concepts and a Review	p. 47
2.1 The Complexity of Granular Materials	p. 47
2.2 Applications of Granular Flows	p. 48
2.2.1 Chemical Process Engineering	p. 48
2.2.2 Geophysical Flows	p. 49
2.3 Distinctive Properties of Granular Materials	p. 49
2.3.1 Single-phase and Multi-phase Flows	p. 50
2.3.2 Dilatancy	p. 51
2.3.3 Cohesion	p. 53
2.3.4 Lubrication	p. 54
2.3.5 Fluidisation	p. 55
2.3.6 Unlubricated Sliding	p. 57
2.3.7 Segregation, Inverse Grading and the Brazil Nut Effect	p. 60
2.4 Granular Avalanches	p. 62
2.4.1 Definition	p. 62
2.4.2 Pattern Formation by Granular Avalanches	p. 65
2.5 Snow Avalanche Regions, Formation and Dynamics	p. 72
2.5.1 The Home of Natural Snow Avalanches	p. 72
2.5.2 Topographic Conditions	p. 73
2.5.3 Snowpack and Weather Conditions	p. 74
2.5.4 Size and Speed of Snow Avalanches	p. 76
2.5.5 Avalanche Dynamics	p. 77
2.6 Types of Granular Avalanches	p. 79
2.6.1 Flow Avalanches	p. 79
2.6.2 Powder Avalanches	p. 80
2.6.3 Landslides and Avalanches on other Planets	p. 85
2.7 Fundamentals of Granular Avalanches	p. 88
2.7.1 Some Characteristics of Flow Avalanches	p. 88
2.7.2 Stress Generating Mechanisms	p. 90
2.7.3 Density Variations	p. 91
2.7.4 Constitutive Relations	p. 92
2.7.5 The Size Effect	p. 94
2.8 Survey on Avalanche Modelling	p. 96
2.8.1 A View on Some Classical Avalanche Models	p. 97
2.8.2 Voellmy's Pioneering Work	p. 102
2.8.3 Experimental Data	p. 104
2.8.4 Necessity for a New Model	p. 110
Part II A Continuum Mechanical Theory for Dense Avalanches Sliding Down Non-Trivial Topographies
3 A Continuum Mechanical Theory for Granular Avalanches	p. 115
3.1 General Introduction	p. 115
3.2 The SH-Model, Reduced to its Essentials	p. 117
3.3 Generalisations of the Original Theory	p. 123
3.3.1 Generalisation with Respect to the Coordinate System	p. 123
3.3.2 Generalisation with Respect to the Basal Topography	p. 125
3.4 A Three-Dimensional Granular Avalanche Model	p. 130
3.4.1 Field Equations	p. 131
3.4.2 Curvilinear Coordinate System in a Vertical Plane	p. 133
3.4.3 The Model Equations	p. 135
3.4.4 Differences Between Geophysical Mass Flows and Shallow Water Equations	p. 140
3.4.5 Features and Limitations of the Extended Model	p. 141
3.5 Avalanches with Coulomb-Type and Viscous-Type Frictional Resistance	p. 145
3.5.1 Model Equations Including Voellmy Drag	p. 145
3.5.2 Equations for the Motion of the Centre of Mass	p. 147
3.5.3 Equations for the Deformation and Motion of Mass	p. 150
3.6 Avalanches with Erosion and Deposition	p. 152
3.6.1 Coordinate System	p. 153
3.6.2 Accumulation and Deposition	p. 154
3.6.3 The Model Equations	p. 156
3.7 Granular Flows in Rotating Drums	p. 158
3.7.1 Solid-Like and Fluid-Like Regions	p. 158
3.7.2 Coordinate System	p. 159
3.7.3 Governing Equations in a Solid Rotating Body	p. 160
3.7.4 Interfacial Conditions and Scalings	p. 161
3.7.5 Governing Equations in the Avalanche Region	p. 163
3.8 Summary	p. 165
4 Avalanches in Arbitrarily Curved and Twisted Channels	p. 167
4.1 Motivation	p. 167
4.2 The Essence of the New Theory	p. 170
4.3 General Orthogonal Coordinate System	p. 171
4.4 Non-Dimensional Equations	p. 177
4.4.1 Components of the Gravitational Acceleration	p. 179
4.4.2 Balance Equations	p. 181
4.4.3 Kinematic Surface Conditions	p. 183
4.4.4 Traction-Free Condition at the Free Surface	p. 183
4.4.5 The Coulomb Sliding Law at the Base	p. 184
4.5 Depth Integration	p. 185
4.6 Ordering	p. 188
4.7 Closure	p. 191
4.8 Flow Profile	p. 196
4.9 The Model Equations in Conservative Form	p. 198
4.9.1 Avalanche Motions Down Curved and Twisted Channels	p. 198
4.9.2 The Importance of the New Theory	p. 199
4.9.3 The Standard Form of the Differential Equations	p. 202
4.9.4 Characteristic Speeds and Critical Flow	p. 203
4.10 Erosion and Deposition for the Full Set of Equations	p. 204
4.10.1 Inclusion of Erosion and Deposition	p. 204
4.10.2 Functional Relation for Erosion and Deposition	p. 205
4.11 Discussion	p. 207
4.11.1 Summary and Embedding of Earlier Models	p. 207
4.11.2 The Orthogonal Complex vs. the Orthogonal General System	p. 209
4.12 Concluding Remarks and Future Outlook	p. 210
5 Exact and Semi-Exact Solutions of the Model Equations	p. 213
5.1 Solutions of the Model Equations	p. 213
5.1.1 A Complete Analytical Solution	p. 213
5.1.2 Particular Solutions	p. 214
5.1.3 Numerical Solutions	p. 214
5.2 One-Dimensional Similarity Solutions	p. 215
5.2.1 One-Dimensional Flow Down Inclined Planes	p. 215
5.2.2 Flow Over an Arbitrarily Curved and Twisted Channel	p. 224
5.2.3 Moderately Curved Beds	p. 226
5.2.4 Variable Bed Friction	p. 230
5.2.5 Variable Bed Friction, Curved Bed and Voellmy Drag	p. 249
5.3 Two-Dimensional Similarity Solutions	p. 253
6 Exact Solutions for Flow Avalanches in Rotating Drums	p. 265
6.1 A Simple Exact Solution for Steady Flow in a Rotating Drum Without Erosion and Deposition	p. 266
6.1.1 Coordinate System, Geometry of the Drum and the Moving Mass	p. 266
6.1.2 Avalanche Depth Determined Without Wall Friction	p. 267
6.1.3 Avalanche Depth Determined by Including Wall Friction	p. 270
6.2 An Exact Solution for Steady Flow in a Slowly Rotating Drum with Erosion and Deposition	p. 272
6.2.1 A Steady Flow Avalanche	p. 272
6.2.2 An Exact Solution	p. 273
6.3 Mixing in a Rotating Drum	p. 275
6.3.1 Particle Paths	p. 275
6.3.2 Circuit Time	p. 280
6.4 An Alternative Model Describing the Transverse Flow and Mixing of Granular Material in a Rotating Cylinder	p. 282
6.4.1 Model	p. 282
6.4.2 Experiments	p. 287
6.4.3 Results and Discussion	p. 289
6.5 Concluding Remarks	p. 293
Part III Shock Capturing Numerical Methods and Simulations of Free Surface Flows of Shallow Avalanches Sliding Over Curved and Twisted Channels
7 Classical and High Resolution Shock-Capturing Numerical Methods	p. 297
7.1 Classical Eulerian and Lagrangean Approaches	p. 298
7.1.1 Eulerian Approach	p. 300
7.1.2 Lagrangean Approach	p. 303
7.2 Some Traditional Numerical Methods	p. 307
7.2.1 First-Order Schemes	p. 307
7.2.2 Second-Order Schemes	p. 309
7.3 Appropriate Numerical Modelling	p. 310
7.4 Modern Numerical Methods	p. 312
7.4.1 Total Variation Diminishing Method	p. 312
7.4.2 Second-Order TVD Schemes	p. 313
7.4.3 Cell Reconstruction with Slope Limiters	p. 318
7.4.4 Non-Linear Conservation Law and TVD Methods	p. 320
7.4.5 TVD Lax-Friedrichs Method	p. 321
7.4.6 Modified TVDLF Scheme	p. 322
7.5 NOC Schemes	p. 323
7.6 Alternative Numerical Schemes	p. 326
7.7 Summary	p. 328
8 Two-Dimensional Shock-Capturing Schemes for Avalanching Flow	p. 329
8.1 The Two-Dimensional Lagrangean Techniques	p. 329
8.2 The Two-Dimensional NOC Schemes	p. 331
8.2.1 Description	p. 331
8.2.2 Predictor Step	p. 336
8.2.3 Corrector Step	p. 337
8.3 Two-Dimensional Shock-Capturing Methods Applied to the Extended Avalanche Equations	p. 338
8.4 Summary	p. 341
9 Avalanche Simulations over Curved and Twisted Channels	p. 343
9.1 Performance of Various Numerical Schemes	p. 343
9.1.1 Numerical Performances	p. 344
9.2 Effects of Topographic Variations	p. 350
9.2.1 Constant Cross-Slope Curvature	p. 350
9.2.2 Variable Cross-Slope Curvature	p. 356
9.3 Superimposed Basal Topography	p. 360
9.4 Avalanches Sliding Down Curved and Twisted Channels	p. 363
9.4.1 Flows Through Uniformly Curved and Twisted Channels	p. 364
9.4.2 Avalanching Flows Through Non-Uniformly Curved and Twisted Channels	p. 365
9.5 Sensitivity to Phenomenological Parameters	p. 372
9.6 Pressure Dependence of the Friction Angles	p. 375
9.6.1 Mass-Dependent Bed Friction Angle	p. 378
9.6.2 Scale Effects Due to the Pressure Dependence of [delta]	p. 379
9.7 Formation of Shocks	p. 381
9.8 Summary	p. 385
Part IV Experimental Validation of the Theoretical Prediction with Different Measurement Techniques
10 Experimental Findings and a Comparison with the Theory	p. 389
10.1 Why Are Laboratory Experiments Performed? What Can be Inferred from Them?	p. 389
10.2 Chute Flow Experiments	p. 392
10.2.1 Experimental Set-Up	p. 392
10.2.2 Experimental Procedure	p. 395
10.2.3 Measurement of Phenomenological Coefficients	p. 399
10.2.4 Results	p. 403
10.2.5 Variable Bed Friction Angle (Position-Dependent)	p. 409
10.2.6 Chutes with a Convex Curved Bump	p. 411
10.2.7 Limitation of the Model	p. 416
10.3 Avalanche Flow Without Side Confinement	p. 417
10.3.1 Experimental Set-Up	p. 417
10.3.2 Rolled Surfaces	p. 420
10.4 Channelised Avalanche Flows	p. 425
10.5 Avalanches Across Irregular Three-Dimensional Terrain	p. 436
10.5.1 The Table-Top Experiments	p. 439
10.5.2 Further Verification of the Model Equations	p. 446
11 Particle Image Velocimetry for Free Surface Flow Avalanches	p. 461
11.1 Introduction	p. 461
11.2 Particle Image Velocimetry Technique	p. 462
11.2.1 Image Intensity Field	p. 463
11.2.2 Cross-Correlation Function	p. 464
11.2.3 Spatial Resolution	p. 466
11.2.4 Summary of the PIV System	p. 466
11.3 Experimental Set-Up for Granular Avalanches	p. 466
11.3.1 Transparent Fluids and the Usual PIV Set-Up	p. 467
11.3.2 Set-Up for Granular Avalanches	p. 467
11.3.3 Technical Details	p. 468
11.4 Experimental Peculiarities Arising for Granular Materials	p. 468
11.4.1 General Errors	p. 469
11.4.2 Particular Errors for Granular Flows	p. 469
11.5 Post-Processing and Evaluation	p. 474
11.6 PIV with Multi-Cameras	p. 475
11.7 Particle Tracking Velocimetry (PTV) Measuring Technique	p. 475
12 Avalanche Experiments Using the PIV Measurement Technique	p. 479
12.1 Experimental Details	p. 480
12.2 Measurement of Avalanche Depth Profiles	p. 483
12.3 Validation of the Theory	p. 484
12.3.1 Experiments Using Small-Cap and Quartz Particles	p. 484
12.3.2 The PIV Measurement and Validation of the Theory	p. 486
12.3.3 Evolution of the Avalanche Geometry	p. 490
12.3.4 Multi-CCD Cameras and Velocity Shearing	p. 490
12.4 Is There a Terminal Velocity on Inclined Planes?	p. 493
12.4.1 Background	p. 493
12.4.2 Remarks on Experimental Procedures	p. 495
12.4.3 Results	p. 495
12.4.4 Summary	p. 502
12.5 Concluding Remarks	p. 503
Part V Avalanche Protection and Defence Structures
13 Protection Against Snow Avalanche Hazards	p. 507
13.1 Types of Avalanche Protection	p. 508
13.1.1 Avalanche Initiation and Protective Measures	p. 508
13.1.2 Early Efforts	p. 510
13.1.3 Modern Methods of Avalanche Defence and Protection	p. 510
13.2 Avalanche Protection in Different Countries	p. 514
13.2.1 Avalanche Protection in Switzerland	p. 514
13.2.2 Avalanche Protection in France	p. 515
13.2.3 Avalanche Protection in Iceland	p. 516
13.2.4 Snow Avalanche Protection in Austria	p. 518
13.2.5 Snow Avalanche Barriers in North America	p. 518
13.3 Laboratory Experiments: A Means to Design Defence Structures	p. 519
13.3.1 Laboratory Models and Experiments	p. 520
13.3.2 Simulation of Avalanche Protection	p. 523
13.3.3 A Structural Protection Technique by Deflection	p. 525
13.4 Conclusion	p. 526
14 Summary and Outlook	p. 529
14.1 Knowledge at Present	p. 530
14.1.1 Theory	p. 530
14.1.2 Numerics	p. 531
14.1.3 Experiments	p. 532
14.2 Attempts in Future	p. 534
14.2.1 Application in Nature	p. 534
14.2.2 Application in the Laboratory	p. 535
14.2.3 Advancing the Numerics	p. 536
14.2.4 More Advanced Measurement Techniques and Experiments	p. 536
References	p. 539
Name Index	p. 565
Index	p. 571

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