1. Chapter 9 Applications of Hybrid Wave Models to Irregular Waves 9.1 Computation of Wave Kinematics 9.2 Prediction of Wave Elevation 9.3 Prediction of Wave Elevation Based on Wave Pressures 9.4 Prediction of Wave Properties of Short-crested Waves. 9.5 Accurate determine Wave Energy Loss Due to Wave Breaking. 9.6 Wave Forces on Slender Bodies

2. 9.1 Computation of Wave Kinematics Problems of computing wave kinematics using Linear wave theory Revisit of HW 3 Modifications to LWT (widely used by the offshore industry) (see Zhang et al. 1991, OTC 6522) Wheeler Stretching Linear Extrapolation Vertical Extrapolation 2 nd MCM (Stokes) perturbation method (not convergent) Equivalent Wave + High-Order Wave Theory

3. Prediction of Horizontal Velocity By HWM & MCM Linear 2nd-order HWM Long wave

4. Errors in Using Equivalent Regular Wave to Compute Irregular Wave Kinematics 3rd-order

5. Heuristic Interpretations for the horizontal velocity under the crest -Steep wave crests are caused by many waves of different frequencies. -Waves of high frequencies are on the top of waves of low frequencies. Hybrid Wave Model (HWM) considers these factors -The still water level to a SW is the LW’s surface - Considering Waves of high frequencies are on the top of waves of low frequencies

6. PMM Solution for a SW modulated a LW

7. Generation of Steep Transient Wave in 2-D Flume Wave Kinematics After Wave-breaking Before Wave -breaking

8. Wave Kinematics Prediction Under crest, Largest Horizontal velocity Largest Vertical velocity Linear Extra. HWMWheeler

9. Input -8.53 m Predicted -5.48 m

10. FULWACK Data Fulmar Platform at North Sea Time: Nov. 24,1982. Hs = 11.2m Input: Significant Wave Breaking During the tests Zhang et al. 1999b

11. Output: Predictions are Compared with Measurements.

12. 9.2 Prediction of Wave Elevation Comparison with the corresponding prediction based on LWT (Linear Wave Theory). See 1996 b Zhang et al.

13. 9.3 Prediction of Wave Elevation Based on Wave Pressures (Meza et al. 1999 J. OMAE Vol. 121 pp242-250) Because of ocean environment and the costs of deploying and maintaining surface-piercing instruments, pressure transducers are commonly used for measuring ocean waves. It does not directly measure wave elevation. To obtain the surface elevation, a transfer function relating wave elevation and wave-induced dynamic pressure is required. Traditionally, the deterministic transfer function is based on LWT. Limitation of pressure measurement: When a transducer is deployed at a depth comparable to the wavelength of a measured wave train, the pressure induced by the wave is so weak that the measured pressure is not reliable (comparable to noises).

14. Limitation of Transfer function based on LWT : The relation between the elevation and pressure of a FW is quite different from that of a BW. When waves are steep, BWs are significant or may be dominant at low- and high-frequency ranges. Predicted elevation based on measured pressure through a transfer function of LWT may result in large errors in these frequency ranges. See the comparison between the prediction and the corresponding measurements.

15. Transfer Function in Deep Water

17. Elevation a steep transient wave train is measured. Pressure is also measured there (-16 to -50 cm below SWL, water depth h = 91.0cm). Elevation is predicted based on LWT (Linear Spectral Method). Elevation is predicted using UHWM. Comparison between predictions and measurements. - LWT under-predicts the crest height & greatly over-predicts the trough height. - LWT over-predicts wave heights.

20. Explanation to why LWT under-predicts the crest heights & over-predicts the trough heights.

21. 9.4 Prediction of Wave Properties of Short-crested Waves. Laboratory Measurements: OTRC data see Zhang et al. 1999b Field Measurements: 1.Wave Kinematics (FULWACK cases, see Zhang et al. 1999b) 2.Wave Pressure (Harvest Platform Data, see Zhang et al. 1999b) 3. WACSIS Project (with/without Currents) (Zhang et al. 2004, Zhang & Zhang 2004)

22. WAve Crest Sensor Inter-comparison Study (WACSIS). Platform in 18-m water depth (southern North Sea, 97-98). Various instruments were used. Inter-comparison of measurements of various instruments. Using Directional HWM for the comparison between predictions and measurements. Consistency; May help to correct measurement mistakes.

27. Plan View of Sensor Layout

29. Fig. 4 Power spectrum density according to pressure head time series

30. Fig. 5 Initial phases of each component of pressure and velocity

31. Case Name (m) (s) Wave Direction at The Peak 98030110202.6487.13.69° 98030510403.7868.3-12.44° 98041311003.06410.5-35.15° Table 3 Wave Characteristics of Selected Cases

32. Fig. 7 Comparison of free-wave directions based on DHWM and Waverider (9803011020)

33. Comparison between prediction & measurement of Marex before & after the orientation of S4 was corrected

34. Comparison between prediction & measurement of SAAB before and after the orientation of S4 was corrected

35. Comparison of free-wave directions based on DHWM and Waverider (9803051040)

36. Comparison between prediction & measurement of SAAB before & after the orientation of S4 was corrected

37. Comparison of free-wave directions based on DHWM and Waverider (9804131100)

38. Comparison between prediction & measurement of Marex before & after the orientation of S4 was corrected

39. Comparison between prediction & measurement of SAAB before & after the orientation of S4 was corrected

40. Governing Eq.s & BCs in the Presence of Currents

41. Assuming the current is uniform and steady, we may simplify the computation using a moving coordinate system. In the moving coordinates, the governing Eqs. and BCs reduce to the same form as those in the absence of current except for the Bernoulli constant C(t). The solutions in the moving coordinates in the presence of the current are the same as those given in the absence of the currents.

42. After transferring the variables back to those in the fixed coordinates, we obtain the truncated solutions up to 2 nd -order in wave steepness for the interaction between two free waves using two different perturbation methods, respectively. That is, in the presence of currents the freqs. involved in the phases are the apparent freqs. instead of intrinsic freqs. as in the absence of currents.

43. Numeric Verification Current velo. 1.5 m/s Initial Guess of wave direction 20-40 degree Test 1: PUV Wave Record and Following Currents Test 2: Elevation Records and Opposing Currents

44. 9.5 Accurately determine Wave Energy Loss Due to Wave Breaking. Energy loss due to wave breaking is a dominant sink in wave energy budget. Field Measurements to determine wave energy loss is extremely difficult due severe ocean environment and other terms such as energy input from wind and nonlinear wave-wave interaction.

45. Laboratory measurements of wave breaking Set-up of Laboratory Measurements See Figure 1 from Meza et al. 2000 Noticing the distance from the wave gages (before breaking) to the ones (after breaking) is short, approximately of the order of typical wavelength. Hence S nl is insignificant. No wind in laboratory, and hence S wind = 0. The energy difference between the measurements at the gage before the breaking and the gage after the breaking should equal to the energy loss due to the breaking.

46. Resultant wave spectrum (based on FFT) may change within short distance even in the absence of wave breaking. This is because the superposition of FWs & BWs at the same frequency. FW spectrum (decoupled from BWs) changes very little in a short distance and in the absence of wave breaking. The comparison of FW spectra before & after an isolated wave breaking may accurately reveal the energy loss as a function of wave frequency.

50. Original wave dissipation model Always lose wave energy regardless of wave freq. Energy loss is proportional to 2 nd power or the 4 th power of wave freq Energy loss is proportional to energy density Our Measurements indicates Free waves at frequency lower than the peak freq. may gain small portion of energy loss by high freq. Free waves. Energy loss of free wave at or near the spectral peak freq. Is not proportional their energy density.

51. 9.6 Wave Forces on Slender Bodies Wave Structure Interactions: Floating offshore structures of a slender body (SPAR) or consisting of several slender bodies (TLP). Wave loads are computed using the Morrison Equation or Modified Morrison Equation.

52. Spar Platform

53. Tension Leg Platform TLP

54. Semi-Submersible

55. –Quasi-static Analysis mooring/tendon/risers modeled as nonlinear springs. mooring line damping estimated or neglected –Coupled Dynamic Analysis structure motion equations and mooring/tendon /riser dynamic equations solved simultaneously. mooring line damping included. Empirical formulation for VIV. Time Domain Analyses Used in Motion and Mooring Analysis of a Moored Floating Structure

56. Numerical Code: COUPLE Cable dynamics: CABLE3D –Finite element method (Garrett,1982, Ma & Webster 1994). –Improvement. Wave/Current/Wind Loads on the Hull: –Morison Equation Using Nonlinear Wave Kinematics (HWM). –Diffraction Theory WAMIT (Initial force) +Morison Equation (Drag). Interface Between Hull and Its Mooring System –Hinge Connection

57. Methodology of Coupled Dynamic Analysis –hinged boundary condition Motion equations of a structure Dynamic equations of Cable Motions at top of the rod Forces applied on the structures

58. Two Choices of Computation Methods Morison equation (Slender Body Structure) Wave Kinematics using Hybrid Wave Model (HWM). Inertia, drag and lifting (VIV) forces. Ship Type Structures, (2 nd order Wave Diffraction numerical codes, i.e.WAMIT) Drag and lifting force based on Morison Equation Wave Loads on Structures

59. Use of the Hybrid Wave Models Effects of Nonlinear Wave Interactions. Hybrid Wave Model:(Uni-directional & Directional HWMs) Output of HWM: Tine-dependent irregular wave velocity & acceleration along the centerline of a cylinder from its bottom to the free surface, which is the input to the Morrison equation. Considering nonlinear wave effects. Accurate in predicting slow drift motion of slack moored structures.

60. Flowchart of COUPLE 6D

61. Comparison with Measurements JIP Spar Cao & Zhang (1997) IJOPE Vol. 17 p119-126 Deep Star Spar Ding, Kim, Theckum and Zhang (2004,2005) Mini Tension Leg Platform (TLP) Chen et al. (2006) Ocean Engineering, Vol. 33, p93-117.

62. Main Characteristics of JIP Spar (1:55) Diameter 40.54 m Draft 198.12 m Water depth318.50 m Mass (with entrapped water)2.592E8 kg Center of gravity -105.98 m Pitch radius of gyration62.33 m Mooring point-106.62 m TestedPredicted Natural periods (s) Surge331.86330.89 Pitch66.7762.11 Damping ratio Surge0.0526 Pitch0.0086

64. Response Of JIP Spar HWM

65. Response of JIP Spar Linear Extrapolation Wheeler Stretching

66. Environment Conditions –Wave 100-year West Africa storm (JONSWAP Spectrum, H 1/3 =4m & T P =16 s). (Wind & Current). –0 and 45 degree wave heading. Set-up (1:40) –Fixed model tests (no risers and tendons) –Compliant model tests (4 tendons & 4 (12) risers). Measurements –Motions of Hull (TLP) in 6 degrees of freedom. –Tensions at the bottom of each riser and tendon. Tests of Mini TLP

67. Mini TLP Complaint Model Test Scale 1:40

68. Mini TLP Properties (Prototype Scale)

69. Risers and Tendons Properties (Prototype Scale)

70. R1R2 R3R4 T1T2 T4 T3 Wave Probe 3 WAVE (0 deg) Tendon and Riser Locations WAVE (45 deg)

71. Numerical Code: COUPLE Cable dynamics: CABLE3D –Finite element method (Garrett,1982, Ma & Webster 1994). –Improvement. Wave/Current Loads on the Hull: –Morison Equation Using Nonlinear Wave Kinematics (HWM). –Diffraction Theory WAMIT +Morison Equation (Drag). Interface Between Hull and Its Mooring System –Hinge Connection

72. Measured Wave Spectrum (Wave Probe #3) Jonswap Hs=4m Tp=16s  =2.0

73. Measured Natural Periods and Damping Ratios

74. Approach A: Morsion equation+HWM, quasi-static Approach B: WAMIT+viscous, quasi-static Approach C: Morison equation+HWM, coupled dynamic Approach D: WAMIT+viscous, coupled dynamic Time-Domain Simulations (3 hours)

75. Morison Element Model –Four Column diameter: 8.64m, length: 43.5m Cm=1.0 (Guichard,2001) Cd=0.7 ( 0 degree heading) (Teign & Niedzwechi,1998) Cd=1.0 (45 degrees heading) –Four Pontoons diameter: 7.02m, length: 19.935m Cm=1.5 Cd=1.2 ( 0 degree heading) Cd=1.4 (45 degrees heading)

76. Table Division of Frequency Bands Used in Statistic Analysis

77. Comparison of Surge Spectra (0 deg)

78. Comparison of Heave Spectra (0 deg)

79. Comparison of Pitch Spectra (0 deg)

80. Comparison of Surge Spectra (45 deg)

81. Comparison of Heave Spectra (45 deg)

82. Comparison of Pitch Spectra (45 deg)

83. Accurate Boundary Conditions for Numerical Simulation T. Liu & Prof. Chen Accurate Boundary Conditions are crucial to fully nonlinear numerical simulation B. C. based on linear wave theory results in wrong simulation Using finite amplitude wave theory (Coklete 1977), accurate elevation and velocity of a periodic very steep wave train can be computed numerically. B. C. based on FAWT results in accurate predictions.