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1 SLOSHING IN MARINE- AND LAND-BASED APPLICATIONS 1

1.1 Introduction 1

1.2 Resonant free-surface motions 1

1.3 Ship tanks 5

1.3.1 Oil tankers 10

1.3.2 FPSO ships and shuttle tankers 12

1.3.3 Bulk carriers 12

1.3.4 Liquefied gas carriers 14

1.3.5 LPG carriers 15

1.3.6 LNG carriers 16

1.3.7 Chemical tankers 21

1.3.8 Fish transportation 21

1.3.9 Cruise vessels 21

1.3.10 Antirolling tanks 22

1.4 Tuned liquid dampers 22

1.5 Offshore platforms 24

1.6 Completely filled fabric structure 27

1.7 External sloshing for ships and marine structures 27

1.8 Sloshing in coastal engineering 30

1.9 Land transportation 31

1.10 Onshore tanks 31

1.11 Space applications 32

1.12 Summary of chapters 33

2 GOVERNING EQUATIONS OF LIQUID SLOSHING 35

2.1 Introduction 35

2.2 Navier-Stokes equations 35

2.2.1 Two-dimensional Navier-Stokes formulation for incompressible liquid 35

2.2.1.1 Continuity equation 36

2.2.1.2 Viscous stresses and derivation of the Navier-Stokes equations 36

2.2.2 Three-dimensional Navier-Stokes equations 37

2.2.2.1 Vorticity and potential flow 38

2.2.2.2 Compressibility 39

2.2.3 Turbulent flow 40

2.2.4 Global conservation laws 40

2.2.4.1 Conservation of fluid momentum 40

2.2.4.2 Conservation of kinetic and potential fluid energy 41

2.2.4.3 Examples：two special cases 42

2.3 Tank-fixed coordinate system 43

2.4 Governing equations in a noninertial，tank-fixed coordinate system 45

2.4.1 Navier-Stokes equations 45

2.4.1.1 Illustrative example：application to the Earth as an accelerated coordinate system 46

2.4.2 Potential flow formulation 47

2.4.2.1 Governing equations 47

2.4.2.2 Body boundary conditions 48

2.4.2.3 Free-surface conditions 48

2.4.2.4 Mass （volume） conservation condition 49

2.4.2.5 Free boundary problem of sloshing and initial/periodicity conditions 49

2.5 Lagrange variational formalism for the sloshing problem 51

2.5.1 Eulerian calculus of variations 51

2.5.2 Illustrative examples 53

2.5.2.1 Spring-mass systems 53

2.5.2.2 Euler-Bernoulli beam equation 54

2.5.2.3 Linear sloshing in an upright nonmoving tank 56

2.5.3 Lagrange and Bateman-Luke variational formulations for nonlinear sloshing 57

2.5.3.1 The Lagrange variational formulation 57

2.5.3.2 The Bateman-Luke principle 58

2.6 Summary 59

2.7 Exercises 59

2.7.1 Flow parameters 59

2.7.2 Surface tension 60

2.7.3 Kinematic boundary condition 60

2.7.4 Added mass force for a nonlifting body in infinite fluid 60

2.7.5 Euler-Lagrange equations for finite-dimensional mechanical systems 61

3 WAVE-INDUCED SHIP MOTIONS 63

3.1 Introduction 63

3.2 Long-crested propagating waves 63

3.3 Statistical description of waves in a sea state 67

3.4 Long-term predictions of sea states 70

3.5 Linear wave-induced motions in regular waves 73

3.5.1 Definitions 73

3.5.2 Equations of motion in the frequency domain 76

3.6 Coupled sloshing and ship motions 80

3.6.1 Quasi-steady free-surface effects of a tank 80

3.6.2 Antirolling tanks 82

3.6.3 Free-surface antirolling tanks 83

3.6.4 U-tube roll stabilizer 85

3.6.4.1 Nonlinear liquid motion 88

3.6.4.2 Linear forces and moments due to liquid motion in the U-tube 90

3.6.4.3 Lloyd’s U-tube model 90

3.6.4.4 Controlled U-tank stabilizer 94

3.6.5 Coupled sway motions and sloshing 97

3.6.6 Coupled three-dimensional ship motions and sloshing in beam waves 99

3.7 Sloshing in external flow 103

3.7.1 Piston-mode resonance in a two-dimensional moonpool 103

3.7.2 Piston and sloshing modes in three-dimensional moonpools 108

3.7.3 Resonant wave motion between two hulls 110

3.8 Time-domain response 111

3.9 Response in irregular waves 114

3.9.1 Linear short-term sea state response 114

3.9.2 Linear long-term predictions 115

3.10 Summary 115

3.11 Exercises 117

3.11.1 Wave energy 117

3.11.2 Surface tension 117

3.11.3 Added mass and damping 118

3.11.4 Heave damping at small frequencies in finite water depth 118

3.11.5 Coupled roll and sloshing in an antirolling tank of a barge in beam sea 119

3.11.6 Operational analysis of patrol boat with U-tube tank 120

3.11.7 Moonpool and gap resonances 121

4 LINEAR NATURAL SLOSHING MODES 122

4.1 Introduction 122

4.2 Natural frequencies and modes 123

4.3 Exact natural frequencies and modes 125

4.3.1 Two-dimensional case 125

4.3.1.1 Rectangular planar tank 125

4.3.1.2 Wedge cross-section with 45° and 60° semi-apex angles 128

4.3.1.3 Troesch’s analytical solutions 130

4.3.2 Three-dimensional cases 130

4.3.2.1 Rectangular tank 130

4.3.2.2 Upright circular cylindrical tank 133

4.4 Seiching 135

4.4.1 Parabolic basin 136

4.4.2 Triangular basin 136

4.4.3 Harbors 137

4.4.4 Pumping-mode resonance of a harbor 137

4.4.5 Ocean basins 138

4.5 Domain decomposition 138

4.5.1 Two-dimensional sloshing with a shallow-water part 138

4.5.2 Example：swimming pools 140

4.6 Variational statement and comparison theorems 140

4.6.1 Variational formulations 142

4.6.1.1 Rayleigh’s method 142

4.6.1.2 Rayleigh quotient for natural sloshing 144

4.6.1.3 Variational equation 147

4.6.2 Natural frequencies versus tank shape：comparison theorems 150

4.6.3 Asymptotic formulas for the natural frequencies and the variational statement 151

4.6.3.1 Small liquid-domain reductions of rectangular tanks 151

4.6.3.2 Asymptotic formula for a chamfered tank bottom：examples 152

4.6.3.3 Discussion on the analytical continuation and the applicability of formula （4.90） 155

4.7 Asymptotic natural frequencies for tanks with small internal structures 157

4.7.1 Main theoretical background 158

4.7.2 Baffles 161

4.7.2.1 Small-size （horizontal or vertical） thin baffle 161

4.7.2.2 Hydrodynamic interaction between baffles （plates）and free-surface effects 164

4.7.3 Poles 168

4.7.3.1 Horizontal and vertical poles 168

4.7.3.2 Proximity of circular poles 170

4.8 Approximate solutions 171

4.8.1 Two-dimensional circular tanks 171

4.8.2 Axisymmetric tanks 172

4.8.2.1 Spherical tank 173

4.8.2.2 Ellipsoidal （oblate spheroidal） container 175

4.8.3 Horizontal cylindrical container 176

4.8.3.1 Shallow-liquid approximation for arbitrary cross-section 176

4.8.3.2 Shallow-liquid approximation for circular cross-section 177

4.9 Two-layer liquid 179

4.9.1 General statement 179

4.9.2 Two-phase shallow-liquid approximation 182

4.9.2.1 Example：oil-gas separator 183

4.10 Summary 185

4.11 Exercises 186

4.11.1 Irregular frequencies 186

4.11.2 Shallow-liquid approximation for trapezoidal-base tank 186

4.11.3 Annular and sectored upright circular tank 187

4.11.4 Circular swimming pool 187

4.11.5 Effect of pipes on sloshing frequencies for a gravity-based platform 189

4.11.6 Effect of horizontal isolated baffles in a rectangular tank 191

4.11.7 Isolated vertical baffles in a rectangular tank 192

5 LINEAR MODAL THEORY 193

5.1 Introduction 193

5.2 Illustrative example：surge excitations of a rectangular tank 193

5.3 Theory 196

5.3.1 Linear modal equations 196

5.3.1.1 Six generalized coordinates for solid-body，linear dynamics 196

5.3.1.2 Generalized coordinates for liquid sloshing and derivation of linear modal equations 197

5.3.1.3 Linear modal equations for prescribed tank motions 199

5.3.2 Resulting hydrodynamic force and moment in linear approximation 200

5.3.2.1 Force 200

5.3.2.2 Moment 202

5.3.3 Steady-state and transient motions：initial and periodicity conditions 204

5.4 Implementation of linear modal theory 208

5.4.1 Time- and frequency-domain solutions 208

5.4.1.1 Time-domain solution with prescribed tank motion 208

5.4.1.2 Time-domain solution of coupled sloshing and body motion 208

5.4.1.3 Frequency-domain solution of coupled sloshing and body motion 208

5.4.2 Forced sloshing in a two-dimensional rectangular tank 211

5.4.2.1 Hydrodynamic coefficients 211

5.4.2.2 Completely filled two-dimensional rectangular tank 213

5.4.2.3 Transient sloshing during collision of two ships 219

5.4.2.4 Effect of elastic tank wall deflections on sloshing 224

5.4.3 Forced sloshing in a three-dimensional rectangular-base tank 226

5.4.3.1 Hydrodynamic coefficients 226

5.4.3.2 Added mass coefficients in ship applications 229

5.4.3.3 Tank added mass coefficients in a ship motion analysis 233

5.4.4 Hydrodynamic coefficients for an upright circular cylindrical tank 235

5.4.5 Coupling between sloshing and wave-induced vibrations of a monotower 237

5.4.5.1 Theory 237

5.4.5.2 Undamped eigenfrequencies of the coupled motions 240

5.4.5.3 Variational method 240

5.4.5.4 Wave excitation 242

5.4.5.5 Damping 244

5.4.6 Rollover of a tank vehicle 245

5.4.7 Spherical tanks 247

5.4.7.1 Hydroelastic vibrations of a spherical tank 247

5.4.7.2 Simplified two-mode modal system for sloshing in a spherical tank 249

5.4.8 Transient analysis of tanks with asymptotic estimates of natural frequencies 250

5.5 Summary 251

5.6 Exercises 251

5.6.1 Moments by direct pressure integration and the Lukovsky formula 251

5.6.2 Transient sloshing with damping 251

5.6.3 Effect of small structural deflections of the tank bottom on sloshing 252

5.6.4 Effect of elastic deformations of vertical circular tank 252

5.6.5 Spilling of coffee 253

5.6.6 Braking of a tank vehicle 253

5.6.7 Free decay of a ship cross-section in roll 253

6 VISCOUS WAVE LOADS AND DAMPING 254

6.1 Introduction 254

6.2 Boundary-layer flow 254

6.2.1 Oscillatory nonseparated laminar flow 255

6.2.2 Oscillatory nonseparated laminar flow past a circular cylinder 257

6.2.3 Turbulent nonseparated boundary-layer flow 258

6.2.3.1 Turbulent energy dissipation 260

6.2.3.2 Oscillatory nonseparated flow past a circular cylinder 261

6.3 Damping of sloshing in a rectangular tank 262

6.3.1 Damping due to boundary-layer flow （Keulegan’s theory） 262

6.3.2 Incorporation of boundary-layer damping in a potential ow model 264

6.3.3 Bulk damping 265

6.4 Morison’s equation 266

6.4.1 Morison’s equation in a tank-fixed coordinate system 267

6.4.2 Generalizations of Morison’s equation 269

6.4.3 Mass and drag coefficients （CM and CD） 270

6.5 Viscous damping due to baffles 274

6.5.1 Baffle mounted vertically on the tank bottom 275

6.5.2 Baffles mounted horizontally on a tank wall 278

6.6 Forced resonant sloshing in a two-dimensional rectangular tank 280

6.7 Tuned liquid damper （TLD） 280

6.7.1 TLD with vertical poles 282

6.7.2 TLD with vertical plate 283

6.7.3 TLD with wire-mesh screen 283

6.7.4 Scaling of model tests of a TLD 286

6.7.5 Forced longitudinal oscillations of a TLD 286

6.8 Effect of swash bulkheads and screens with high solidity ratio 289

6.9 Vortex-induced vibration （VIV） 294

6.10 Summary 296

6.11 Exercises 297

6.11.1 Damping ratios in a rectangular tank 297

6.11.2 Morison’s equation 297

6.11.3 Scaling of TLD with vertical poles 298

6.11.4 Effect of unsteady laminar boundary-layer flow on potential flow 298

6.11.5 Reduction of natural sloshing frequency due to wire-mesh screen 298

7 MULTIMODAL METHOD 299

7.1 Introduction 299

7.2 Nonlinear modal equations for sloshing 300

7.2.1 Modal representation of the free surface and velocity potential 300

7.2.2 Modal system based on the Bateman-Luke formulation 301

7.2.3 Advantages and limitations of the nonlinear modal method 303

7.3 Modal technique for hydrodynamic forces and moments 304

7.3.1 Hydrodynamic force 305

7.3.1.1 General case 305

7.3.1.2 Completely filled closed tank 306

7.3.2 Moment 306

7.3.2.1 Hydrodynamic moment as a function of the angular momentum 306

7.3.2.2 Potential flow 307

7.3.2.3 Completely filled closed tank 307

7.4 Limitations of the modal theory and Lukovsky’s formulas due to damping 307

7.5 Summary 308

7.6 Exercises 309

7.6.1 Modal equations for the beam problem 309

7.6.2 Linear modal equations for sloshing 309

8 NONLINEAR ASYMPTOTIC THEORIES AND EXPERIMENTS FOR A TWO-DIMENSIONAL RECTANGULAR TANK 310

8.1 Introduction 310

8.2 Steady-state resonant solutions and their stability for a Duffing-like mechanical system 315

8.2.1 Nonlinear spring-mass system，resonant solution，and its stability 315

8.2.1.1 Steady-state solution 315

8.2.1.2 Stability 317

8.2.1.3 Damping 319

8.2.2 Steady-state resonant sloshing due to horizontal excitations 319

8.3 Single-dominant asymptotic nonlinear modal theory 323

8.3.1 Asymptotic modal system 323

8.3.1.1 Steady-state resonant waves：frequency-domain solution 325

8.3.1.2 Time-domain solution and comparisons with experiments 327

8.3.2 Nonimpulsive hydrodynamic loads 337

8.3.2.1 Hydrodynamic pressure 337

8.3.2.2 Hydrodynamic force 338

8.3.2.3 Hydrodynamic moment relative to origin O 339

8.3.2.4 Nonimpulsive hydrodynamic loads on internal structures 339

8.3.3 Coupled ship motion and sloshing 340

8.3.4 Applicability：effect of higher modes and secondary resonance 341

8.4 Adaptive asymptotic modal system for finite liquid depth 343

8.4.1 Infinite-dimensional modal system 343

8.4.2 Hydrodynamic force and moment 345

8.4.3 Particular finite-dimensional modal systems 345

8.5 Critical depth 347

8.6 Asymptotic modal theory of Boussinesq-type for lower-intermediate and shallow-liquid depths 352

8.6.1 Intermodal ordering 352

8.6.2 Boussinesq-type multimodal system for intermediate and shallow depths 353

8.6.3 Damping 355

8.7 Intermediate liquid depth 355

8.8 Shallow liquid depth 357

8.8.1 Use of the Boussinesq-type multimodal method for intermediate and shallow depths 357

8.8.1.1 Transients 357

8.8.1.2 Steady-state regimes 358

8.8.2 Steady-state hydraulic jumps 361

8.9 Wave loads on interior structures in shallow liquid depth 371

8.10 Mathieu instability for vertical tank excitation 373

8.11 Summary 375

8.11.1 Nonlinear multimodal method 375

8.11.2 Subharmonics 377

8.11.3 Damping 377

8.11.4 Hydraulic jumps 377

8.11.5 Hydrodynamic loads on interior structures 377

8.12 Exercises 377

8.12.1 Moiseev’s asymptotic solution for a rectangular tank with infinite depth 377

8.12.2 Mean steady-state hydrodynamic loads 378

8.12.3 Simulation by multimodal method 378

8.12.4 Force on a vertical circular cylinder for shallow depth 378

8.12.5 Mathieu-type instability 379

9 NONLINEAR ASYMPTOTIC THEORIES AND EXPERIMENTS FOR THREE-DIMENSIONAL SLOSHING 380

9.1 Introduction 380

9.1.1 Steady-state resonant wave regimes and hydrodynamic instability 380

9.1.1.1 Theoretical treatment by the two lowest natural modes 380

9.1.1.2 Experimental observations and measurements for a nearly square-base tank 381

9.1.2 Bifurcation and stability 385

9.2 Rectangular-base tank with a finite liquid depth 387

9.2.1 Statement and generalization of adaptive modal system （8.95） 387

9.2.2 Moiseev-based modal system for a nearly square-base tank 388

9.2.3 Steady-state resonance solutions for a nearly square-base tank 392

9.2.4 Classification of steady-state regimes for a square-base tank with longitudinal and diagonal excitations 393

9.2.4.1 Longitudinal excitation 394

9.2.4.2 Diagonal excitation 400

9.2.5 Longitudinal excitation of a nearly square-base tank 401

9.2.6 Amplification of higher modes and adaptive modal modeling for transients and swirling 408

9.2.6.1 Adaptive modal modeling and its accuracy 408

9.2.6.2 Transient amplitudes 409

9.2.6.3 Response for diagonal excitations 412

9.2.6.4 Response for longitudinal excitations 414

9.3 Vertical circular cylinder 417

9.3.1 Experiments 419

9.3.2 Modal equations 422

9.3.3 Steady-state solutions 424

9.4 Spherical tank 426

9.4.1 Wave regimes 428

9.4.2 Tower forces 430

9.5 Summary 432

9.5.1 Square-base tank 432

9.5.2 Nearly square-base tanks 433

9.5.3 Circular base 433

9.5.4 Spherical tank 433

9.6 Exercises 434

9.6.1 Multimodal methods for square- and circular-base tanks 434

9.6.2 Spherical pendulum，planar，and rotary motions 434

9.6.3 Angular Stokes drift for swirling 435

9.6.4 Three-dimensional shallow-liquid equations in a body-fixed accelerated coordinate system 436

9.6.5 Wave loads on a spherical tank with a tower 437

10 COMPUTATIONAL FLUID DYNAMICS 439

10.1 Introduction 439

10.2 Boundary element methods 444

10.2.1 Free-surface conditions 445

10.2.2 Generation of vorticity 447

10.2.3 Example：numerical discretization 447

10.2.4 Linear frequency-domain solutions 449

10.3 Finite difference method 450

10.3.1 Preliminaries 451

10.3.2 Governing equations 451

10.3.3 Interface capturing 452

10.3.3.1 Level-set technique 453

10.3.4 Introduction to numerical solution procedures 454

10.3.5 Time-stepping procedures 455

10.3.6 Spatial discretizations 456

10.3.7 Discretization of the convective and viscous terms 456

10.3.8 Discretization of the Poisson equation for pressure 457

10.3.9 Treatment of immersed boundaries 458

10.3.10 Constrained interpolation profile method 459

10.4 Finite volume method 460

10.4.1 Introduction 460

10.4.2 FVM applied to linear sloshing with potentialflow 462

10.4.2.1 Example 464

10.5 Finite element method 465

10.5.1 Introduction 465

10.5.2 A model problem 465

10.5.2.1 Numerical example 466

10.5.3 One-dimensional acoustic resonance 466

10.5.4 FEM applied to linear sloshing with potential flow 468

10.5.4.1 Matrix system 470

10.5.4.2 Example 472

10.6 Smoothed particle hydrodynamics method 472

10.7 Summary 477

10.8 Exercises 478

10.8.1 One-dimensional acoustic resonance 478

10.8.2 BEM applied to steady flow past a cylinder in infinite fluid 479

10.8.3 BEM applied to linear sloshing with potential flow and viscous damping 480

10.8.4 Application of FEM to the Navier-Stokes equations 480

10.8.5 SPH method 480

11 SLAMMING 481

11.1 Introduction 481

11.2 Scaling laws for model testing 484

11.3 Incompressible liquid impact on rigid tank roof without gas cavities 488

11.3.1 Wagner model 489

11.3.1.1 Prediction of wetted surface 491

11.3.1.2 Spray root solution 492

11.3.2 Damping of sloshing due to tank roof impact 494

11.3.3 Three-dimensional liquid impact 496

11.4 Impact of steep waves against a vertical wall 497

11.4.1 Wagner-type model 500

11.4.2 Pressure-impulse theory 502

11.5 Tank roof impact at high filling ratios 503

11.6 Slamming with gas pocket 506

11.6.1 Natural frequency for a gas cavity 509

11.6.1.1 Simplified analysis 511

11.6.2 Damping of gas cavity oscillations 511

11.6.3 Forced oscillations of a gas cavity 513

11.6.3.1 Prediction of the wetted surface 515

11.6.3.2 Case study 515

11.6.4 Nonlinear gas cavity analysis 516

11.6.5 Scaling 516

11.7 Cavitation and boiling 522

11.8 Acoustic liquid effects 522

11.8.1 Two-dimensional liquid entry of body with horizontal bottom 524

11.8.2 Liquid entry of parabolic contour 526

11.8.3 Hydraulic jump impact 526

11.8.4 Thin-layer approximation of liquid-gas mixture 527

11.9 Hydroelastic slamming 528

11.9.1 Experimental study 532

11.9.2 Theoretical hydroelastic beam model 533

11.9.3 Comparisons between theory and experiments 537

11.9.4 Parameter study for full-scale tank 538

11.9.5 Model test scaling of hydroelasticity 544

11.9.6 Slamming in membrane tanks 545

11.10 Summary 548

11.11 Exercises 550

11.11.1 Impact force on a wedge 550

11.11.2 Prediction of the wetted surface by Wagner’s method 550

11.11.3 Integrated slamming loads on part of the tank roof 551

11.11.4 Impact of a liquid wedge 551

11.11.5 Acoustic impact of a hydraulic jump against a vertical wall 551

APPENDIX：Integral Theorems 553

Bibliography 555

Index 571