Nomenclature ................................................. xvii
Preface and Acknowledgment .................................... xxv
Acronyms and Abbreviations .................................. xxvii
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 flow 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 (СM 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 О .............................. 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 potential
flow ........................................... 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 AppUcation 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 Нуdroelastic 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
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