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1 SLOSHING IN MARINE- AND LAND-BASED APPLICATIONS1

1.1 Introduction1

1.2 Resonant free-surface motions1

1.3 Ship tanks5

1.3.1 Oil tankers10

1.3.2 FPSO ships and shuttle tankers12

1.3.3 Bulk carriers12

1.3.4 Liquefied gas carriers14

1.3.5 LPG carriers15

1.3.6 LNG carriers16

1.3.7 Chemical tankers21

1.3.8 Fish transportation21

1.3.9 Cruise vessels21

1.3.10 Antirolling tanks22

1.4 Tuned liquid dampers22

1.5 Offshore platforms24

1.6 Completely filled fabric structure27

1.7 External sloshing for ships and marine structures27

1.8 Sloshing in coastal engineering30

1.9 Land transportation31

1.10 Onshore tanks31

1.11 Space applications32

1.12 Summary of chapters33

2 GOVERNING EQUATIONS OF LIQUID SLOSHING35

2.1 Introduction35

2.2 Navier-Stokes equations35

2.2.1 Two-dimensional Navier-Stokes formulation for incompressible liquid35

2.2.1.1 Continuity equation36

2.2.1.2 Viscous stresses and derivation of the Navier-Stokes equations36

2.2.2 Three-dimensional Navier-Stokes equations37

2.2.2.1 Vorticity and potential flow38

2.2.2.2 Compressibility39

2.2.3 Turbulent flow40

2.2.4 Global conservation laws40

2.2.4.1 Conservation of fluid momentum40

2.2.4.2 Conservation of kinetic and potential fluid energy41

2.2.4.3 Examples：two special cases42

2.3 Tank-fixed coordinate system43

2.4 Governing equations in a noninertial，tank-fixed coordinate system45

2.4.1 Navier-Stokes equations45

2.4.1.1 Illustrative example：application to the Earth as an accelerated coordinate system46

2.4.2 Potential flow formulation47

2.4.2.1 Governing equations47

2.4.2.2 Body boundary conditions48

2.4.2.3 Free-surface conditions48

2.4.2.4 Mass （volume） conservation condition49

2.4.2.5 Free boundary problem of sloshing and initial/periodicity conditions49

2.5 Lagrange variational formalism for the sloshing problem51

2.5.1 Eulerian calculus of variations51

2.5.2 Illustrative examples53

2.5.2.1 Spring-mass systems53

2.5.2.2 Euler-Bernoulli beam equation54

2.5.2.3 Linear sloshing in an upright nonmoving tank56

2.5.3 Lagrange and Bateman-Luke variational formulations for nonlinear sloshing57

2.5.3.1 The Lagrange variational formulation57

2.5.3.2 The Bateman-Luke principle58

2.6 Summary59

2.7 Exercises59

2.7.1 Flow parameters59

2.7.2 Surface tension60

2.7.3 Kinematic boundary condition60

2.7.4 Added mass force for a nonlifting body in infinite fluid60

2.7.5 Euler-Lagrange equations for finite-dimensional mechanical systems61

3 WAVE-INDUCED SHIP MOTIONS63

3.1 Introduction63

3.2 Long-crested propagating waves63

3.3 Statistical description of waves in a sea state67

3.4 Long-term predictions of sea states70

3.5 Linear wave-induced motions in regular waves73

3.5.1 Definitions73

3.5.2 Equations of motion in the frequency domain76

3.6 Coupled sloshing and ship motions80

3.6.1 Quasi-steady free-surface effects of a tank80

3.6.2 Antirolling tanks82

3.6.3 Free-surface antirolling tanks83

3.6.4 U-tube roll stabilizer85

3.6.4.1 Nonlinear liquid motion88

3.6.4.2 Linear forces and moments due to liquid motion in the U-tube90

3.6.4.3 Lloyd’s U-tube model90

3.6.4.4 Controlled U-tank stabilizer94

3.6.5 Coupled sway motions and sloshing97

3.6.6 Coupled three-dimensional ship motions and sloshing in beam waves99

3.7 Sloshing in external flow103

3.7.1 Piston-mode resonance in a two-dimensional moonpool103

3.7.2 Piston and sloshing modes in three-dimensional moonpools108

3.7.3 Resonant wave motion between two hulls110

3.8 Time-domain response111

3.9 Response in irregular waves114

3.9.1 Linear short-term sea state response114

3.9.2 Linear long-term predictions115

3.10 Summary115

3.11 Exercises117

3.11.1 Wave energy117

3.11.2 Surface tension117

3.11.3 Added mass and damping118

3.11.4 Heave damping at small frequencies in finite water depth118

3.11.5 Coupled roll and sloshing in an antirolling tank of a barge in beam sea119

3.11.6 Operational analysis of patrol boat with U-tube tank120

3.11.7 Moonpool and gap resonances121

4 LINEAR NATURAL SLOSHING MODES122

4.1 Introduction122

4.2 Natural frequencies and modes123

4.3 Exact natural frequencies and modes125

4.3.1 Two-dimensional case125

4.3.1.1 Rectangular planar tank125

4.3.1.2 Wedge cross-section with 45° and 60° semi-apex angles128

4.3.1.3 Troesch’s analytical solutions130

4.3.2 Three-dimensional cases130

4.3.2.1 Rectangular tank130

4.3.2.2 Upright circular cylindrical tank133

4.4 Seiching135

4.4.1 Parabolic basin136

4.4.2 Triangular basin136

4.4.3 Harbors137

4.4.4 Pumping-mode resonance of a harbor137

4.4.5 Ocean basins138

4.5 Domain decomposition138

4.5.1 Two-dimensional sloshing with a shallow-water part138

4.5.2 Example：swimming pools140

4.6 Variational statement and comparison theorems140

4.6.1 Variational formulations142

4.6.1.1 Rayleigh’s method142

4.6.1.2 Rayleigh quotient for natural sloshing144

4.6.1.3 Variational equation147

4.6.2 Natural frequencies versus tank shape：comparison theorems150

4.6.3 Asymptotic formulas for the natural frequencies and the variational statement151

4.6.3.1 Small liquid-domain reductions of rectangular tanks151

4.6.3.2 Asymptotic formula for a chamfered tank bottom：examples152

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 structures157

4.7.1 Main theoretical background158

4.7.2 Baffles161

4.7.2.1 Small-size （horizontal or vertical） thin baffle161

4.7.2.2 Hydrodynamic interaction between baffles （plates）and free-surface effects164

4.7.3 Poles168

4.7.3.1 Horizontal and vertical poles168

4.7.3.2 Proximity of circular poles170

4.8 Approximate solutions171

4.8.1 Two-dimensional circular tanks171

4.8.2 Axisymmetric tanks172

4.8.2.1 Spherical tank173

4.8.2.2 Ellipsoidal （oblate spheroidal） container175

4.8.3 Horizontal cylindrical container176

4.8.3.1 Shallow-liquid approximation for arbitrary cross-section176

4.8.3.2 Shallow-liquid approximation for circular cross-section177

4.9 Two-layer liquid179

4.9.1 General statement179

4.9.2 Two-phase shallow-liquid approximation182

4.9.2.1 Example：oil-gas separator183

4.10 Summary185

4.11 Exercises186

4.11.1 Irregular frequencies186

4.11.2 Shallow-liquid approximation for trapezoidal-base tank186

4.11.3 Annular and sectored upright circular tank187

4.11.4 Circular swimming pool187

4.11.5 Effect of pipes on sloshing frequencies for a gravity-based platform189

4.11.6 Effect of horizontal isolated baffles in a rectangular tank191

4.11.7 Isolated vertical baffles in a rectangular tank192

5 LINEAR MODAL THEORY193

5.1 Introduction193

5.2 Illustrative example：surge excitations of a rectangular tank193

5.3 Theory196

5.3.1 Linear modal equations196

5.3.1.1 Six generalized coordinates for solid-body，linear dynamics196

5.3.1.2 Generalized coordinates for liquid sloshing and derivation of linear modal equations197

5.3.1.3 Linear modal equations for prescribed tank motions199

5.3.2 Resulting hydrodynamic force and moment in linear approximation200

5.3.2.1 Force200

5.3.2.2 Moment202

5.3.3 Steady-state and transient motions：initial and periodicity conditions204

5.4 Implementation of linear modal theory208

5.4.1 Time- and frequency-domain solutions208

5.4.1.1 Time-domain solution with prescribed tank motion208

5.4.1.2 Time-domain solution of coupled sloshing and body motion208

5.4.1.3 Frequency-domain solution of coupled sloshing and body motion208

5.4.2 Forced sloshing in a two-dimensional rectangular tank211

5.4.2.1 Hydrodynamic coefficients211

5.4.2.2 Completely filled two-dimensional rectangular tank213

5.4.2.3 Transient sloshing during collision of two ships219

5.4.2.4 Effect of elastic tank wall deflections on sloshing224

5.4.3 Forced sloshing in a three-dimensional rectangular-base tank226

5.4.3.1 Hydrodynamic coefficients226

5.4.3.2 Added mass coefficients in ship applications229

5.4.3.3 Tank added mass coefficients in a ship motion analysis233

5.4.4 Hydrodynamic coefficients for an upright circular cylindrical tank235

5.4.5 Coupling between sloshing and wave-induced vibrations of a monotower237

5.4.5.1 Theory237

5.4.5.2 Undamped eigenfrequencies of the coupled motions240

5.4.5.3 Variational method240

5.4.5.4 Wave excitation242

5.4.5.5 Damping244

5.4.6 Rollover of a tank vehicle245

5.4.7 Spherical tanks247

5.4.7.1 Hydroelastic vibrations of a spherical tank247

5.4.7.2 Simplified two-mode modal system for sloshing in a spherical tank249

5.4.8 Transient analysis of tanks with asymptotic estimates of natural frequencies250

5.5 Summary251

5.6 Exercises251

5.6.1 Moments by direct pressure integration and the Lukovsky formula251

5.6.2 Transient sloshing with damping251

5.6.3 Effect of small structural deflections of the tank bottom on sloshing252

5.6.4 Effect of elastic deformations of vertical circular tank252

5.6.5 Spilling of coffee253

5.6.6 Braking of a tank vehicle253

5.6.7 Free decay of a ship cross-section in roll253

6 VISCOUS WAVE LOADS AND DAMPING254

6.1 Introduction254

6.2 Boundary-layer flow254

6.2.1 Oscillatory nonseparated laminar flow255

6.2.2 Oscillatory nonseparated laminar flow past a circular cylinder257

6.2.3 Turbulent nonseparated boundary-layer flow258

6.2.3.1 Turbulent energy dissipation260

6.2.3.2 Oscillatory nonseparated flow past a circular cylinder261

6.3 Damping of sloshing in a rectangular tank262

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 model264

6.3.3 Bulk damping265

6.4 Morison’s equation266

6.4.1 Morison’s equation in a tank-fixed coordinate system267

6.4.2 Generalizations of Morison’s equation269

6.4.3 Mass and drag coefficients （CM and CD）270

6.5 Viscous damping due to baffles274

6.5.1 Baffle mounted vertically on the tank bottom275

6.5.2 Baffles mounted horizontally on a tank wall278

6.6 Forced resonant sloshing in a two-dimensional rectangular tank280

6.7 Tuned liquid damper （TLD）280

6.7.1 TLD with vertical poles282

6.7.2 TLD with vertical plate283

6.7.3 TLD with wire-mesh screen283

6.7.4 Scaling of model tests of a TLD286

6.7.5 Forced longitudinal oscillations of a TLD286

6.8 Effect of swash bulkheads and screens with high solidity ratio289

6.9 Vortex-induced vibration （VIV）294

6.10 Summary296

6.11 Exercises297

6.11.1 Damping ratios in a rectangular tank297

6.11.2 Morison’s equation297

6.11.3 Scaling of TLD with vertical poles298

6.11.4 Effect of unsteady laminar boundary-layer flow on potential flow298

6.11.5 Reduction of natural sloshing frequency due to wire-mesh screen298

7 MULTIMODAL METHOD299

7.1 Introduction299

7.2 Nonlinear modal equations for sloshing300

7.2.1 Modal representation of the free surface and velocity potential300

7.2.2 Modal system based on the Bateman-Luke formulation301

7.2.3 Advantages and limitations of the nonlinear modal method303

7.3 Modal technique for hydrodynamic forces and moments304

7.3.1 Hydrodynamic force305

7.3.1.1 General case305

7.3.1.2 Completely filled closed tank306

7.3.2 Moment306

7.3.2.1 Hydrodynamic moment as a function of the angular momentum306

7.3.2.2 Potential flow307

7.3.2.3 Completely filled closed tank307

7.4 Limitations of the modal theory and Lukovsky’s formulas due to damping307

7.5 Summary308

7.6 Exercises309

7.6.1 Modal equations for the beam problem309

7.6.2 Linear modal equations for sloshing309

8 NONLINEAR ASYMPTOTIC THEORIES AND EXPERIMENTS FOR A TWO-DIMENSIONAL RECTANGULAR TANK310

8.1 Introduction310

8.2 Steady-state resonant solutions and their stability for a Duffing-like mechanical system315

8.2.1 Nonlinear spring-mass system，resonant solution，and its stability315

8.2.1.1 Steady-state solution315

8.2.1.2 Stability317

8.2.1.3 Damping319

8.2.2 Steady-state resonant sloshing due to horizontal excitations319

8.3 Single-dominant asymptotic nonlinear modal theory323

8.3.1 Asymptotic modal system323

8.3.1.1 Steady-state resonant waves：frequency-domain solution325

8.3.1.2 Time-domain solution and comparisons with experiments327

8.3.2 Nonimpulsive hydrodynamic loads337

8.3.2.1 Hydrodynamic pressure337

8.3.2.2 Hydrodynamic force338

8.3.2.3 Hydrodynamic moment relative to origin O339

8.3.2.4 Nonimpulsive hydrodynamic loads on internal structures339

8.3.3 Coupled ship motion and sloshing340

8.3.4 Applicability：effect of higher modes and secondary resonance341

8.4 Adaptive asymptotic modal system for finite liquid depth343

8.4.1 Infinite-dimensional modal system343

8.4.2 Hydrodynamic force and moment345

8.4.3 Particular finite-dimensional modal systems345

8.5 Critical depth347

8.6 Asymptotic modal theory of Boussinesq-type for lower-intermediate and shallow-liquid depths352

8.6.1 Intermodal ordering352

8.6.2 Boussinesq-type multimodal system for intermediate and shallow depths353

8.6.3 Damping355

8.7 Intermediate liquid depth355

8.8 Shallow liquid depth357

8.8.1 Use of the Boussinesq-type multimodal method for intermediate and shallow depths357

8.8.1.1 Transients357

8.8.1.2 Steady-state regimes358

8.8.2 Steady-state hydraulic jumps361

8.9 Wave loads on interior structures in shallow liquid depth371

8.10 Mathieu instability for vertical tank excitation373

8.11 Summary375

8.11.1 Nonlinear multimodal method375

8.11.2 Subharmonics377

8.11.3 Damping377

8.11.4 Hydraulic jumps377

8.11.5 Hydrodynamic loads on interior structures377

8.12 Exercises377

8.12.1 Moiseev’s asymptotic solution for a rectangular tank with infinite depth377

8.12.2 Mean steady-state hydrodynamic loads378

8.12.3 Simulation by multimodal method378

8.12.4 Force on a vertical circular cylinder for shallow depth378

8.12.5 Mathieu-type instability379

9 NONLINEAR ASYMPTOTIC THEORIES AND EXPERIMENTS FOR THREE-DIMENSIONAL SLOSHING380

9.1 Introduction380

9.1.1 Steady-state resonant wave regimes and hydrodynamic instability380

9.1.1.1 Theoretical treatment by the two lowest natural modes380

9.1.1.2 Experimental observations and measurements for a nearly square-base tank381

9.1.2 Bifurcation and stability385

9.2 Rectangular-base tank with a finite liquid depth387

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 tank388

9.2.3 Steady-state resonance solutions for a nearly square-base tank392

9.2.4 Classification of steady-state regimes for a square-base tank with longitudinal and diagonal excitations393

9.2.4.1 Longitudinal excitation394

9.2.4.2 Diagonal excitation400

9.2.5 Longitudinal excitation of a nearly square-base tank401

9.2.6 Amplification of higher modes and adaptive modal modeling for transients and swirling408

9.2.6.1 Adaptive modal modeling and its accuracy408

9.2.6.2 Transient amplitudes409

9.2.6.3 Response for diagonal excitations412

9.2.6.4 Response for longitudinal excitations414

9.3 Vertical circular cylinder417

9.3.1 Experiments419

9.3.2 Modal equations422

9.3.3 Steady-state solutions424

9.4 Spherical tank426

9.4.1 Wave regimes428

9.4.2 Tower forces430

9.5 Summary432

9.5.1 Square-base tank432

9.5.2 Nearly square-base tanks433

9.5.3 Circular base433

9.5.4 Spherical tank433

9.6 Exercises434

9.6.1 Multimodal methods for square- and circular-base tanks434

9.6.2 Spherical pendulum，planar，and rotary motions434

9.6.3 Angular Stokes drift for swirling435

9.6.4 Three-dimensional shallow-liquid equations in a body-fixed accelerated coordinate system436

9.6.5 Wave loads on a spherical tank with a tower437

10 COMPUTATIONAL FLUID DYNAMICS439

10.1 Introduction439

10.2 Boundary element methods444

10.2.1 Free-surface conditions445

10.2.2 Generation of vorticity447

10.2.3 Example：numerical discretization447

10.2.4 Linear frequency-domain solutions449

10.3 Finite difference method450

10.3.1 Preliminaries451

10.3.2 Governing equations451

10.3.3 Interface capturing452

10.3.3.1 Level-set technique453

10.3.4 Introduction to numerical solution procedures454

10.3.5 Time-stepping procedures455

10.3.6 Spatial discretizations456

10.3.7 Discretization of the convective and viscous terms456

10.3.8 Discretization of the Poisson equation for pressure457

10.3.9 Treatment of immersed boundaries458

10.3.10 Constrained interpolation profile method459

10.4 Finite volume method460

10.4.1 Introduction460

10.4.2 FVM applied to linear sloshing with potentialflow462

10.4.2.1 Example464

10.5 Finite element method465

10.5.1 Introduction465

10.5.2 A model problem465

10.5.2.1 Numerical example466

10.5.3 One-dimensional acoustic resonance466

10.5.4 FEM applied to linear sloshing with potential flow468

10.5.4.1 Matrix system470

10.5.4.2 Example472

10.6 Smoothed particle hydrodynamics method472

10.7 Summary477

10.8 Exercises478

10.8.1 One-dimensional acoustic resonance478

10.8.2 BEM applied to steady flow past a cylinder in infinite fluid479

10.8.3 BEM applied to linear sloshing with potential flow and viscous damping480

10.8.4 Application of FEM to the Navier-Stokes equations480

10.8.5 SPH method480

11 SLAMMING481

11.1 Introduction481

11.2 Scaling laws for model testing484

11.3 Incompressible liquid impact on rigid tank roof without gas cavities488

11.3.1 Wagner model489

11.3.1.1 Prediction of wetted surface491

11.3.1.2 Spray root solution492

11.3.2 Damping of sloshing due to tank roof impact494

11.3.3 Three-dimensional liquid impact496

11.4 Impact of steep waves against a vertical wall497

11.4.1 Wagner-type model500

11.4.2 Pressure-impulse theory502

11.5 Tank roof impact at high filling ratios503

11.6 Slamming with gas pocket506

11.6.1 Natural frequency for a gas cavity509

11.6.1.1 Simplified analysis511

11.6.2 Damping of gas cavity oscillations511

11.6.3 Forced oscillations of a gas cavity513

11.6.3.1 Prediction of the wetted surface515

11.6.3.2 Case study515

11.6.4 Nonlinear gas cavity analysis516

11.6.5 Scaling516

11.7 Cavitation and boiling522

11.8 Acoustic liquid effects522

11.8.1 Two-dimensional liquid entry of body with horizontal bottom524

11.8.2 Liquid entry of parabolic contour526

11.8.3 Hydraulic jump impact526

11.8.4 Thin-layer approximation of liquid-gas mixture527

11.9 Hydroelastic slamming528

11.9.1 Experimental study532

11.9.2 Theoretical hydroelastic beam model533

11.9.3 Comparisons between theory and experiments537

11.9.4 Parameter study for full-scale tank538

11.9.5 Model test scaling of hydroelasticity544

11.9.6 Slamming in membrane tanks545

11.10 Summary548

11.11 Exercises550

11.11.1 Impact force on a wedge550

11.11.2 Prediction of the wetted surface by Wagner’s method550

11.11.3 Integrated slamming loads on part of the tank roof551

11.11.4 Impact of a liquid wedge551

11.11.5 Acoustic impact of a hydraulic jump against a vertical wall551

APPENDIX：Integral Theorems553

Bibliography555

Index571