by
B.S., University of Rochester (2005)
Submitted to the Department of Mechanical Engineering in partial fulfillment of the requirements for the degrees of
Naval Engineer and
Master of Science in Ocean Engineering at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2015
© Massachusetts Institute of Technology 2015. All rights reserved.
Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Mechanical Engineering
May 8, 2015
Certified by. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stefano Brizzolara
Research Scientist & Lecturer
Assistant Director for Research MIT Sea Grant
Thesis Supervisor
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
David E. Hardt
Chairman, Department Committee on Graduate Students

2

by
Jeffrey Kensett White
Submitted to the Department of Mechanical Engineering on May 8, 2015, in partial fulfillment of the requirements for the degrees of
Naval Engineer and
Master of Science in Ocean Engineering
In this paper we investigate the resistance and seakeeping effects of an inverted bow by comparing the motions of an existing combatant hull, the Oliver Hazard Perry class frigate (FFG-7), with a modified version of the same hull with an inverted bow.
The bow of the FFG-7 was redesigned by developing a set of basic curves that define the parametric surface of the new shape. Two 1/80th scale models were built, one of the original and one of the inverted bow frigate, with the same material, machining and finishing standards. Model tests were conducted in the United States Naval
Academy Hydromechanics Laboratory for resistance in calm water and seakeeping in both regular waves and irregular head seas. The differences between the FFG-7 and the inverted bow responses are characterized in terms of pitch, heave, and vertical accelerations. A numerical verification and experimental validation of the seakeeping code Aegir was conducted using the inverted bow.
Thesis Supervisor: Stefano Brizzolara
Title: Research Scientist & Lecturer
Assistant Director for Research MIT Sea Grant
3

4

First and foremost thank you to Professor Brizzolara for your guidance and encouragement. Working with you in the MIT iShip Lab has truly been a pleasure and
I am appreciative for such a great opportunity. I am also extremely grateful to John
Zseleczky for his support of this project and allowing me access to the USNA Hydromechanics Laboratory facilities and staff. Learning the art and nuance of model testing from Bill Beaver was certainly one of the highlights of the project. This project would not have been possible without the expertise of the Hydro Lab technical staff
Don Bunker, Dan Rhodes, Dale Boyer, and Dave Majerowitz.
I owe a special thank you to my family for their love, patience and support over the last three years. To Marissa and Spencer, I love you so very much. It was not easy for two graduate student parents to raise such a wonderful boy and get all of our homework done, but we made it through together. I am very proud of both of you and your accomplishments as well. And to my Dad, my life long teacher - working on this project with you has been so special to me. This was something I wanted to do from day one and I am so fortunate that we were able to make it a reality. Being able to share your passion and to walk a mile in your shoes is something that I will cherish for the rest of my life!
5

6

1 Introduction 19
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
1.1.1 Ulstein X-Bow ® . . . . . . . . . . . . . . . . . . . . . . . . .
20
1.1.2 The Axe-Bow Concept . . . . . . . . . . . . . . . . . . . . . .
21
1.1.3 THALES Programme Frigate Designs . . . . . . . . . . . . . .
23
1.1.4 Flared vs. Tumblehome Hull Forms . . . . . . . . . . . . . . .
24
1.2 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
1.3 Research Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
1.3.1 Design of an Inverted Bow . . . . . . . . . . . . . . . . . . . .
26
1.3.2 Experimental Test Program . . . . . . . . . . . . . . . . . . .
27
1.3.3 Numerical Verification and Validation . . . . . . . . . . . . . .
27
1.3.4 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . .
28
1.3.5 Theoretical Framework . . . . . . . . . . . . . . . . . . . . . .
28
1.3.6 Assumptions, Limitations and Scope . . . . . . . . . . . . . .
29
2 Parametric Model 31
2.1 Application of Parametric Modeling for a Ship . . . . . . . . . . . . .
31
2.1.1 Bow Re-Design . . . . . . . . . . . . . . . . . . . . . . . . . .
32
2.1.2 Features of CAESES . . . . . . . . . . . . . . . . . . . . . . .
32
2.2 Global Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
2.2.1 User Defined Parameters . . . . . . . . . . . . . . . . . . . . .
33
2.2.2 Derived Parameters . . . . . . . . . . . . . . . . . . . . . . . .
34
2.2.3 Translation Parameters . . . . . . . . . . . . . . . . . . . . . .
34
7

2.3 Basic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
2.4 Sections and Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
2.4.1 Surface One . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
2.4.2 Surface Two and Three . . . . . . . . . . . . . . . . . . . . . .
38
2.4.3 Surface Four . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
2.4.4 Fillet Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
2.5 Final Inverted Bow Hull . . . . . . . . . . . . . . . . . . . . . . . . .
41
2.6 Parameter Variation . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
2.6.1 Short Inverted Bow . . . . . . . . . . . . . . . . . . . . . . . .
44
2.6.2 Long Inverted Bow . . . . . . . . . . . . . . . . . . . . . . . .
44
3 Experimental Investigation 47
3.1 Description of Facilities and Instrumentation . . . . . . . . . . . . . .
47
3.1.1 NAHL 120 Foot Towing Tank . . . . . . . . . . . . . . . . . .
48
3.1.2 NAHL 380 Foot Towing Tank . . . . . . . . . . . . . . . . . .
49
3.2 Description of Models . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
3.2.1 Turbulence Stimulation . . . . . . . . . . . . . . . . . . . . . .
52
3.2.2 Static Ballasting . . . . . . . . . . . . . . . . . . . . . . . . .
54
3.2.3 Dynamic Ballasting . . . . . . . . . . . . . . . . . . . . . . . .
56
3.2.4 Free Decay Tests in Pitch and Heave . . . . . . . . . . . . . .
57
3.3 Description of Test Program and Procedures . . . . . . . . . . . . . .
58
3.3.1 Calm Water Resistance Tests . . . . . . . . . . . . . . . . . .
58
3.3.2 Regular Wave Tests . . . . . . . . . . . . . . . . . . . . . . . .
59
3.3.3 Irregular Wave Tests . . . . . . . . . . . . . . . . . . . . . . .
61
3.4 Damped Spring-Mass System Analogy . . . . . . . . . . . . . . . . .
63
3.5 Analysis of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
3.5.1 Resistance Comparison . . . . . . . . . . . . . . . . . . . . . .
66
3.5.2 Pitch Response . . . . . . . . . . . . . . . . . . . . . . . . . .
69
3.5.3 Heave Response . . . . . . . . . . . . . . . . . . . . . . . . . .
74
3.5.4 Acceleration Response . . . . . . . . . . . . . . . . . . . . . .
78
8

3.5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
4 Numerical Investigation 83
4.1 Aegir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
4.1.1 Geometry Preparation . . . . . . . . . . . . . . . . . . . . . .
84
4.1.2 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . .
85
4.2 Steady State Verification . . . . . . . . . . . . . . . . . . . . . . . . .
85
4.2.1 Domain sensitivity analysis . . . . . . . . . . . . . . . . . . .
86
4.2.2 Spatial Sensitivity Analysis . . . . . . . . . . . . . . . . . . .
86
4.2.3 Simulation Settings . . . . . . . . . . . . . . . . . . . . . . . .
89
4.3 Steady State Validation . . . . . . . . . . . . . . . . . . . . . . . . .
89
4.4 Wave Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
5 Conclusions 95
5.1 Approach and Method . . . . . . . . . . . . . . . . . . . . . . . . . .
95
5.2 Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
5.2.1 Performance of the Inverted Bow . . . . . . . . . . . . . . . .
96
5.2.2 Validation of Aegir . . . . . . . . . . . . . . . . . . . . . . . .
97
5.2.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
5.3 Areas for Further Development . . . . . . . . . . . . . . . . . . . . .
97
A Dynamic Ballasting Calculations 99
B Experimental Data 101
B.1 Test Series 1: Calm Water Tests for Model B . . . . . . . . . . . . . . 101
B.2 Test Series 2: Calm Water Tests for Model A . . . . . . . . . . . . . . 102
B.3 Test Series 3: Regular Wave Tests for Model A . . . . . . . . . . . . . 103
B.4 Test Series 4: Regular Wave Tests for Model B . . . . . . . . . . . . . 104
B.5 Test Series 5 and 6: Irregular Wave Tests . . . . . . . . . . . . . . . . 105
C Regular Wave Tests Results 107
C.1 Pitch Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . 108
9

C.2 Heave Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . 109
C.3 Bow Accelerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
C.4 LCB Accelerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
D Irregular Wave Tests Results 113
D.1 MATLAB Script: TimeHistory.m
. . . . . . . . . . . . . . . . . . . . 114
D.2 Run 15: 20 knots in Sea State 4 . . . . . . . . . . . . . . . . . . . . . 117
D.3 Run 17: 25 knots in Sea State 4 . . . . . . . . . . . . . . . . . . . . . 118
D.4 Run 19: 30 knots in Sea State 4 . . . . . . . . . . . . . . . . . . . . . 119
D.5 Run 23: 20 knots in Sea State 6 . . . . . . . . . . . . . . . . . . . . . 120
D.6 Run 26: 25 knots in Sea State 6 . . . . . . . . . . . . . . . . . . . . . 121
D.7 Run 29: 30 knots in Sea State 6 . . . . . . . . . . . . . . . . . . . . . 122
E Spectral Analysis 123
E.1 Discussion of Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 123
E.1.1 Detailed Procedure Used for this Data Set . . . . . . . . . . . 124
E.1.2 Verification of Wave Energy Spectrum . . . . . . . . . . . . . 125
E.2 Pitch and Heave Energy Density Plots . . . . . . . . . . . . . . . . . 127
E.2.1 Run 15: 20 knots in Sea State 4 . . . . . . . . . . . . . . . . . 127
E.2.2 Run 17: 25 knots in Sea State 4 . . . . . . . . . . . . . . . . . 128
E.2.3 Run 19: 30 knots in Sea State 4 . . . . . . . . . . . . . . . . . 129
E.2.4 Run 23: 20 knots in Sea State 6 . . . . . . . . . . . . . . . . . 130
E.2.5 Run 29: 30 knots in Sea State 6 . . . . . . . . . . . . . . . . . 131
E.3 MATLAB Script: FFT.m
. . . . . . . . . . . . . . . . . . . . . . . . 132
F Wave Profiles 143
10

1-1 Model of the AHTS Bourbon Orca . . . . . . . . . . . . . . . . . . .
20
1-2 Body lines of the ESC-4100 (top) and the AXE-4100 (bottom) . . . .
22
1-3 Bodyplan of the of AXE20 (left) and PHF-TH-WP (right) . . . . . .
23
1-4 Section view of ONRFH (left) and ONRTH (right) . . . . . . . . . .
24
1-5 The ship motion coordinate system used for this investigation . . . .
28
2-1 User defined global parameters for the inverted bow shape . . . . . .
34
2-2 Basic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
2-3 Cross section definition and isometric view of surface one . . . . . . .
39
2-4 Cross section definition and isometric view of surfaces two and three .
39
2-5 Cross section definition and isometric view of surface four . . . . . . .
40
2-6 Isometric view of fillet surfaces . . . . . . . . . . . . . . . . . . . . . .
41
2-7 Perspective view of the final inverted bow hull form . . . . . . . . . .
42
2-8 (a) Body Plans for the FFG-7 in red and the Inverted Bow Hull Series in blue (b) Sheer plans for FFG-7 in red and Inverted Bow Hull Series in blue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
2-9 Inverted bow shape with the minimum
LOA
. . . . . . . . . . . . . .
44
2-10 Inverted bow shape with the maximum
LOA
. . . . . . . . . . . . .
45
3-1 Schematic drawing of the NAHL 120 foot towing tank . . . . . . . . .
49
3-2 Model B attached to the 120 foot towing tank carriage . . . . . . . .
50
3-3 Schematic drawing of the NAHL 380 foot towing tank . . . . . . . . .
50
3-4 Schematic drawing of the side-by-side towing arrangement . . . . . .
51
3-5 Models A and B attached to the 380 foot towing tank carriage . . . .
52
11

3-6 Model B under construction . . . . . . . . . . . . . . . . . . . . . . .
53
3-7 Required Hama strip thickness for turbulent flow . . . . . . . . . . .
54
3-8 Hama strip applied to Model B . . . . . . . . . . . . . . . . . . . . .
55
3-9 Dynamic ballasting Model B using the NAHL Lamboley rig . . . . .
57
3-10 Spectral density ordinates for Sea State 4 in ship and model scale . .
63
3-11 Residuary resistance for Model A and Model B . . . . . . . . . . . .
67
3-12 Sinkage for Model A and Model B . . . . . . . . . . . . . . . . . . . .
68
3-13 Trim for Model A and Model B . . . . . . . . . . . . . . . . . . . . .
68
3-14 Pitch transfer function for ship speed of 25 knots . . . . . . . . . . .
70
3-15 (a) Selected time history of pitch (b) Selected time history of water surface elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
3-16 (a) Measured wave data and idealized wave slope spectrum (b) Linear pitch transfer function for Model B (c) Measured and calculated pitch energy spectrum for Model B . . . . . . . . . . . . . . . . . . . . . .
73
3-17 Heave transfer function for ship speed of 25 knots . . . . . . . . . . .
75
3-18 (a) Selected time history of heave (b) Selected time history of waves .
76
3-19 (a) Measured wave data and idealized wave spectrum (b) Linear heave transfer function for Model B (c) Measured and calculated heave energy spectrum for Model B . . . . . . . . . . . . . . . . . . . . . . . . . .
77
3-20 Acceleration transducer location . . . . . . . . . . . . . . . . . . . . .
78
3-21 Bow accelerations for ship speed of 25 knots . . . . . . . . . . . . . .
79
3-22 LCG accelerations for ship speed of 25 knots . . . . . . . . . . . . . .
80
3-23 (a) Selected time history for bow accelerations (b) Selected time history for LCG accelerations (c) Selected time history of water surface elevation 81
4-1 Aegir coordinate system . . . . . . . . . . . . . . . . . . . . . . . . .
85
4-2 Domain converegnce tests for Hull B . . . . . . . . . . . . . . . . . .
87
4-3 Mesh density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
4-4 Spatial converegnce tests for Hull B . . . . . . . . . . . . . . . . . . .
88
4-5 Domain and spatial discretization vizualization . . . . . . . . . . . . .
89
12

4-6 Resistance validation for Hull B . . . . . . . . . . . . . . . . . . . . .
90
4-7 Sinkage validation for Hull B . . . . . . . . . . . . . . . . . . . . . . .
91
4-8 Trim vlidation for Hull B . . . . . . . . . . . . . . . . . . . . . . . . .
91
4-9 Normalized wave profile for Froude number of .201 . . . . . . . . . .
92
A-1 Schematic of Lamboly pendulum test rig in NAHL . . . . . . . . . .
99
E-1 Wave energy spectra for run 29 . . . . . . . . . . . . . . . . . . . . . 125
13

14

3.1 Summary of water properties for the model and full scale ship . . . .
48
3.2 Summary of principal model characteristics . . . . . . . . . . . . . . .
56
3.3 Experimentally determined decay coefficients and natural frequencies 58
3.4 Test conditions for each test series . . . . . . . . . . . . . . . . . . . .
59
3.5 Summary of the standard test conditions to cover the required range of wavelengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
3.6 Annual sea state occurrence in the North Atlantic [3] . . . . . . . . .
62
3.7 Summary of ship motion parameters . . . . . . . . . . . . . . . . . .
65
3.8 Summary of variance of pitch response in irregular waves . . . . . . .
74
3.9 Summary of variance of heave response in irregular waves . . . . . . .
78
3.10 Summary of variance of acceleration response in irregular waves . . .
82
4.1 Summary of domain convergence tests . . . . . . . . . . . . . . . . .
86
4.2 Summary of spatial convergence tests . . . . . . . . . . . . . . . . . .
88
A.1 Gyradius calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
15

16

Symbol Description
𝐴
𝑊 𝑃
𝐵𝑀
∆
𝐿
Area of the waterplane
Longitudinal metacentric radius
Displacement h k 𝛿 ∆1
Change in displacement per inch of trim by the stern
DWL Designed load waterline
FP g
𝐻
1 /
3
Forward perpendicular
Acceleration of gravity
Significant wave height
Wave height
Wave number 𝑘
5
KG 𝜆 𝜆
LOA
LBP
LWL
LCB
LCG
Pitch radius of gyration
Distance from the keel to the center of gravity
Linear scale ratio, ship to model
Wavelength
Length of ship, overall
Length of ship between perpendiculars
Load, or design, waterline
Position of the longitudinal center of buoyancy
Position of the longitudinal the center of gravity
Units 𝑓 𝑡 2 in lbs lb
/ in
Symbol 𝑚
0 𝑚
1
MT1 𝜈
Description
Zeroth spectral moment
First spectral moment
Moment to trim one inch
Kinematic viscosity
– NURBS Non-uniform rational b-spline
– PPI
32.174
ft
/ s
2 𝜌
Pounds per inch imersion
Mass density
1 in in
/ in in
RAO
Re
𝑆 𝑚
Response amplitude operator
Reynolds number
Wetted surface area, model
Average period
𝑇
0
Modal period
– in in in
∇
V 𝜔 𝜔 𝑑
Underwater volume
Model speed
Wave frequency
Damped natural frequency in in 𝜔 𝜔 𝑒
0
Encounter wave frequency
Modal Frequency
Units 𝑓 𝑡 3 ft
/ s rad
/ s rad
/ s rad
/ s rad
/ s lb-in
/ in 𝑓 𝑡
2 / s
– lb
/ in 𝑙𝑏 − 𝑠
2 / 𝑓 𝑡
4
–
– 𝑖𝑛 2 s s in 𝜁 Wave elevation in
17

18

This study investigates the resistance and seakeeping effects of an inverted bow by comparing the motions in head seas of an existing U.S. Navy combatant hull, the Oliver Hazard Perry class frigate (FFG-7), with a modified version of the same hull with an inverted bow. Chapter 2 describes the Inverted Bow Hull Series that was created for this study. The modern concept of a systematic series is a fully parametric hull form definition. The bow of the FFG-7 was redesigned by developing a set of parametric curves that define the surface of the new shape. Chapter 3 describes the experimental investigation of the two hulls. Two 1/80th scale models were tested for resistance in calm water and seakeeping for both regular and irregular head seas. The differences between the FFG-7 and the inverted bow responses are characterized in terms of the pitch and heave motions and vertical accelerations.
Chapter 4 describes the numerical investigation of the inverted bow. Using Aegir, a non-linear boundary-element method developed by Navatek Ltd [10], a numerical verification and validation is conducted for the inverted bow. Chapter 5 presents the conclusions drawn from the experimental and numerical tests. Included is a discussion of recommendations for further work.
19

In recent years there have been several studies published where the merits of inverted bows are tested. Universities and research centers around the world are interested in finding ways to improve ship performance, safety, and fuel economy - in certain applications an inverted bow provides all three. Additionally two prominent shipyards in Europe have and continue to build ships that incorporate variations of an inverted bow. In this section we will review recent publications and designs of inverted bows and look for elements in each that will aid in the present study.
®
Ulstein Group is a marine industry group from Norway known for ship building and ship design activities. In 2005 Ulstein introduced the X-Bow ® hull line design which was first incorporated into the Anchor Handling Tug Supply vessel (AHTS)
Bourbon Orca [15].
Figure 1-1: Model of the AHTS Bourbon Orca
As seen in Figure 1-1 the X-Bow ® is characterized by a backward sloping bow that starts at the extreme front of the vessel, a sharper bow entrance, and a smoother volume distribution in the foreship. The shape is optimized for high speed, low
20

resistance and reduced fuel consumption. As a result the X-Bow ® smoothly divides waves and calm water and demonstrates improved fuel efficiency and lower speed loss in waves. Safety is another benefit of the X-Bow ® shape. The redistributed foreship volume helps to eliminate slamming and bow impact, gives a softer entry into waves, has less spray on deck, lower accelerations, and reduced vibrations. Comparative model testing shows the benefits of the X-Bow ® over a conventional bow with a forward sloping blunt bow shape.
The X-Bow ® Hull Line Design is patented in several countries and will only serve to influence the design of the inverted bow for the present study. The bow redesign discussed in Chapter 2 will incorporate elements of the X-Bow ® concept adapted to the FFG-7. Additionally the methodology of comparative model testing will be adapted for use with the FFG-7 and the Inverted Bow Hull Series.
In 1995 Delft University and Damen Shipyards collaborated on a study concerning the influence of hull lengthening on the "practical characteristics" of the ship; what would become known as the Enlarged Ship Concept (ESC). The optimized ESC hull demonstrated a superior resistance and seakeeping performance over the original hull.
In a follow-on research project the authors collaborated with the Royal Netherlands
Navy, U.S. Coast Guard, and the Marine Research Institute Netherlands (MARIN) to decrease the vertical accelerations of the ESC hull. Out of this study the concept of the Axe-Bow was developed [8].
The unique characteristics of the Axe-Bow are a vertical stem line and a bottom centerline that slopes downward to a deeper forefoot. Like the X-Bow ® , the Axe-
Bow also has increased deadrise in the bow and very narrow bow sections. In his paper "Further Investigation into the Hydrodynamic Performance of the Axe-Bow
Concept" [6] Professor Keuning of Delft University presents an extensive experimental and numerical analysis of the performance of the ESC and AXE hulls. The lines plans of the two hulls are shown in Figure 1-2.
For his experimental analysis Keuning ran the models so that they were free to
21

Figure 1-2: Body lines of the ESC-4100 (top) and the AXE-4100 (bottom) pitch and heave in head seas. Each model was instrumented with accelerometers inside the hull at a distance of ten percent LOA aft of the bow and at five percent
LOA forward of the LCG. The second accelerometer was placed in the longitudinal position of the notional pilot house. As they used hollow models constructed of glass fiber reinforced plastic they were able to position the instruments as they pleased inside the hull. For the numerical investigation the models sailed "side-by-side" for irregular seas, a technique that simplified the analysis. However, this side-by-side configuration was not feasible in the towing tank due to the model size and limited width.
For the present study we will make use of elements of both the ESC and Axe-Bow concepts to design the Inverted Bow Hull Series. By increasing the underwater length, and thus the LOA, we expect to also see a decrease in resistance. Also, the vertical stem of the Axe-Bow will serve as the upper limit on entrance angle in the Inverted
Bow Hull Series. Similar instrumentation will be used in the present study as will be discussed in Chapter 3, however, the models weree constructed of solid high density closed cell foam. This limited the placement of acceleration transducers to the main deck only. Finally, the towing tank facilities for the present study are large enough to accommodate a side-by-side towing configuration for the planned model size.
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Following the ESC and Axe-Bow concepts, a group of researchers was assembled under the THALES Programme to explore the hydrodynamic performance on a series of concepts based on a parent monohull frigate [4]. The THALES Programme was an interdisciplinary research and innovation program co-funded by the European Union and Greece that ran from 2000-2004. In their paper "Development of Frigate Designs with good Seakeeping Characteristics" Eefsen et al present the results of resistance and seakeeping experiments on seven design alternatives that have the same deadweight and internal volume. Of note their design alternatives include an Axe-Bow
(AXE20) hull and a wave piercing bow with 10 degree tumblehome from the waterline to the deck (PHF-TH-WP). To meet the similitude requirement the beam of the PHF-TH-WP was widened, to the detriment of the still water performance. The body plans of both hulls are shown below in Figure 1-3.
Figure 1-3: Bodyplan of the of AXE20 (left) and PHF-TH-WP (right)
Their evaluation was based on model tests combined with linear numerical calculations, despite the use of irregular sea states. From the calm water resistance tests the AXE20 shows a noticeable decrease in effective power over the baseline hull.
However the PHF-TH-WP actually exhibits a larger effective power, an effect that is
23

primarily attributable to the widened beam. From the seakeeping analysis it is noted that the PHF-TH-WP suffers from green water on deck and thus requires an increase freeboard in the forward section. For the present study the Inverted Bow Hull Series represents a hybrid of the AXE20 and PHF-TH-WP hull forms. The Inverted Bow
Hull Series also has a similitude requirement, but this is met by increasing the length, not the beam.
In 2005 Dr. Christopher Bassler from the Naval Surface Warfare Center, Carderock Division (NSWCCD) published a study comparing the dynamic stability of a flared hull and a tumblehome hull [5]. The study made use of the Office of Naval
Research (ONR) Topside series hull forms. This series was designed to provide a publicly available hull form which could be used to examine the performance effects from varying topside geometry. As seen in Figure 1-4 the ONR Flared Hull (ONRFH) and the ONR Tumblehome Hull (ONRTH) have the exact same geometry below the waterline.
Figure 1-4: Section view of ONRFH (left) and ONRTH (right)
The purpose of the investigation was to study the effects of increasing KG on the dynamic stability of both hulls. Numerical simulations in regular and irregular waves
24

were used to characterize performance of both hulls in roll, yaw, and pitch. While the study primarily considers transverse motions, the method of analysis presented is useful to the present study. As we are interested in longitudinal motions, the Inverted
Bow Hull Series will be similar to the ONR Topside series except that the forward 30 percent of the hull varies and the remainder of the hull is unchanged.
The U.S. Navy is building a new class of destroyer that has an inverted bow.
Little technical data has been published on the seakeeping effects of inverted bows for military applications. Additionally, in the available literature the inverted bow is often accompanied by negative flare, or tumblehome, over the length of the hull.
In this study we isolate the effect of the inverted bow by comparing the motions of the FFG-7 with a modified version of the same hull with an inverted bow. The gap in knowledge from the reviewed literature is whether the resistance and seakeeping performance of a surface combatant can be improved by changing only the bow shape, from a conventional bow to an inverted bow.
The primary research questions that this study will seek to answer are:
1. Does the inverted bow demonstrate an improved resistance and seakeeping performance over a conventional bow?
Following the analogy of a ship as a classical damped spring-mass system, it is proposed that the inverted bow shape acts to soften the spring constant of the system. The inverted bow should increase the amplitude of pitch and heave motions but reduce vertical accelerations [8]. This improvement in seakeeping performance would be valuable to a U.S Navy surface combatant where the effectiveness of sailors, machinery, and weapons systems are limited by vertical accelerations. Additionally, as a consequence of the underwater shape, the inverted bow should also exhibit a lower resistance.
2. Is Aegir able to accurately predict the motions of the inverted bow hull form?
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Aegir, named for the Norse god of the sea, is a medium-fidelity time-domain seakeeping code that uses an advanced high order boundary element method
(BEM) to solve three-dimensional, potential flow problems emphasizing those involving ocean free surface boundary conditions. Aegir also includes a fully nonlinear steady-state solver for sinkage, trim, and wave resistance predictions.
We believe that Aegir will be able to accurately represent the non-linear response of the Inverted Bow Hull Series. Aegir is advertised as a "panel-less" BEM because it uses the OpenNURBS 4.0 library to represent the hull and the free surface.
To investigate the research questions we first designed a hull from the lines of the
FFG-7 that incorporates an inverted bow. Then a rigorous and systematic testing program was accomplished to collect sufficient data to analyze the motions of the two hulls. Once the analysis of the experimental data was complete it was used for numerical validation of the seakeeping code Aegir, a tool that we believe will allow us to continue the theoretical study of an inverted bow.
Inverted bow shaped hulls have unique characteristics that distinguish them from traditional monohulls. The hull is elongated with a very fine entry, extremely narrow
V-shaped sections below the waterline, and inverted flare and stem profile above the waterline. Additionally, the height of the bow is increased to forestall deck wetness.
In some applications the depth is increased to reduce effects of slamming, but not for this study.
The FFG-7 bare hull was selected as the baseline hull for this study. To truly isolate the effect of the inverted bow we need a baseline ship that does not have a bulbous bow. The FFG-7 is one of the few U.S. Navy surface combatants that meets this requirement. Additionally, as the FFG-7 has limited flare above the waterline
26

the transition to tumblehome in the bow region is more easily accomplished. The forward thirty percent of the baseline hull is modified to an inverted bow shape. The main challenge in converting from the baseline hull to the inverted bow hull was to preserve enough characteristics for a meaningful comparison, but also to make enough changes to see an improved performance. As such we established a firm requirement that the displacement, waterline length, and draft be held constant between hulls.
The Inverted Bow Hull Series is created using a parametric modeling software as described in Chapter 2.
To compare the performance of the two hulls we subjected them to identical testing programs. This was easily accomplished for calm water tests and regular waves for a single model as these test are easily repeated. However in order to more closely represent ocean conditions we also tested the hulls in irregular waves. As such a sideby-side testing configuration, a novel model testing technique discussed in Chapter
3, was used for the irregular wave tests. This allowed for a direct comparison of the motions as each ship sees the exact same wave and the exact same time.
A numerical verification and validation of Aegir for the steady state condition was the next step of the study. Give the hull geometry a numerical verification is required to converge on a valid solution. To do so the domain size is systematically increased and the panel size is systematically decreased until a consistent solution is achieved. Using experimental data to validate Aegir will allow research to continue as we develop a numerical solution for ship motions. Further discussion about the application of Aegir to the present study is described in Chapter 4.
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The motions of a ship at sea are best represented by six degrees of freedom with axes fixed to the ship; three translatory motions and three rotational motions. The three translatory motions of surge, sway, and heave are planar and occur in the x, y, and z planes. The three rotational motions of roll, pitch, and yaw are angular and occur about the x,y, and z axes. Displacement in any direction, either linear or angular, is given as 𝑥 𝑛 where n is the index of the direction as shown in Figure 1-5.
Figure 1-5: The ship motion coordinate system used for this investigation
For a ship it is also convenient to consider the motions grouped together as longitudinal and transverse motions. The longitudinal motions (surge, heave and pitch) occur in the vertical plane and the transverse motions (sway, roll and yaw) occur in the horizontal plane. For the case of a ship with port/starboard symmetry, as are both hulls in the present study, the longitudinal motions are uncoupled from the transverse motions. In head seas we can consider the 2-D vertical plane motions only.
As a further simplification we can assume that the forward speed is constant and thus ignore surge motions.
The motions described above can be modeled by the oscillatory response of a classical damped spring-mass system. While actual ocean waves are non-linear, it has been shown that the behavior of a second-order linear system closely approximates ship motions [13]. A varying force F, which in the case of a ship represents the wave
28

force, is applied to the system. The force is absorbed in proportion by each of the components of the system such that: 𝑎 ¨ + 𝑏 𝑥 + 𝑐𝑥 = 𝐹
(1.1)
Where a represents the terms associated with mass and is directly proportional to the acceleration of the motion; b represents the terms associated with hydrodynamic damping and is directly proportional to the velocity of the motion; and c represents the spring stiffness, or in the case of ships the hydrostatic terms, which is directly proportional to the displacement of the motion. In Chapter 3 this analogy will be tailored to the specific conditions of the model tests; the coupled motions of pitch and heave in head seas. Understanding the meaning of each component in equation 1.1
will help to characterize the different seakeeping responses of both hulls.
There are inherent limitations in predicting the motions of a full scale ship in the open ocean with a scale model in a laboratory environment. Assumptions are required for model scaling to accept the results of model tests as an accurate representation of a full scale ship. In the present study we have mitigated some of the uncertainty associated with model testing by making a relative comparison of the performance of two hulls. The two models are tested in the same laboratory, using the same test conditions, and in most cases even on the same day.
The scale of the models to be tested is limited by the size of the laboratory facilities. Specifically the towing tank must be wide enough in relation to the model’s beam to minimize interactions with the tank wall. Also, the towing tank must be deep enough relative to the height of the generated waves to eliminate bottom effects.
Additionally, the material from which the model is constructed limits the placement of instruments. In the present study the model is constructed from solid high density foam which requires a certain hull thickness for structural integrity and thus prohibits instruments from being placed at will inside the hull.
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The scope of this study was set by several factors. First this study is narrowly focused on a modified FFG-7 hull with an inverted bow. We maintain consistency between key hull parameters to enable a direct comparison. A full hull design incorporating an inverted bow is a subject for future research. Second, we are in an early stage of research on the subject and the natural starting point is testing in head seas.
These tests are the most straightforward to conduct in the towing tank. Head seas tests are reliable and repeatable and will provide ample data for numerical validation.
Finally, we designed a testing program to make the most efficient use of the available laboratory time. As a result we limited the number of speeds, wave heights and sea states to values that were representative for conditions the ship would likely encounter in service.
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The "ship’s lines" are the set of curves that describe the shape of the hull. They are a two-dimensional representation of a three-dimensional shape. Before the age of computers most naval architects would begin a ship design by carving a model from wood, extracting a set of offsets, and then generating the lines. Today it is common practice to begin with the lines of a parent hull and change key parameters to produce a new shape. This project has elements of both approaches in that we started with a baseline hull, but redesigned the bow section by developing a set of parameter based curves that defined the new shape. This chapter will look at the unique case of re-designing a ship’s bow section parametrically, the tools and process for the bow re-design, and the flexibility of the final parametric model.
A fully parametric model of a ship hull is made by defining the ship’s lines with a set of form parameters, not the position of individual control points or offsets.
Instead of simply replacing numbers by variables and then assigning values to them, parameters feature relationships. The value of a parameter might thus be computed from a formula using a higher ranked parameter. As a result when one curve changes, the hull changes with it. Additionally to maintain the fairness of the hull, practical limits are established for each parameter. A fully parametric model allows the designer
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to rapidly make changes to the complex geometry of the ship and still produce a realistic hull shape. The parametric model for this project was created with Dr.
Guiliano Vernengo following the methods developed in his doctoral thesis [16].
The main challenge in converting from the baseline hull to the inverted bow hull was to preserve enough characteristics for a meaningful comparison, but also to make enough changes to see an improved performance. As such we established a firm requirement that the waterline length (LWL) of both hulls would be the same. It was also important to respect the displacement (
∆
) and longitudinal center of gravity
(LCG) of the original hull as much as possible. At the beginning we decided to modify only the forward 25 percent of the baseline hull, but as the design progressed we found this to be insufficient. As the baseline hull has very little flare, we found it necessary to modify more than 30 percent to achieve a "fair" inverted bow hull. A fair line is smooth and continuous, without bumps or hollows, and no sudden changes in curvature. Mathmatically a fair line has a continuous first and second derivative.
The bow redesign started by importing the baseline hull into the 3-D CAD software
Rhinoceros [14] and re-shaping it to approximate the inverted bow shape. The intent of this step in the design process was to quickly create a "rough sketch" of an inverted bow from which measurements could be taken to build the parametric model. These measurements were used to parametrize the new bow shape in the software CAESES by Friendship Systems as described below.
CAESES is a computational fluid dynamics (CFD) integration platform for simulationdriven design of flow exposed surfaces [7]. The first feature of the software relevent to this project is the ability to create FSpline curves. The FSpline is a fairnessoptimized curve that is defined in a principal 2-D plane by user entered parameters, its starting and terminating positions along with their tangents. It also provides the
32

ability to control the curve’s area and centroid values, if desired. A series of FSplines can easily be used to create the definition of the sections along the ship’s length. The next important feature is the ability to create a parametric surface FMetaSurface .
This allows the designer to define the shape of the surface with a set of curves and still achieve a high degree of fairness. To link the form parameters to each section, the software uses a tool called a FCurveEngine . The FCurveEngine describes the distribution of each section along the length of the hull and serves as the basis for generating a parametric surface.
Global parameters are those that apply to the entire ship, not just a single point, curve or surface. The ten global parameters are grouped according to their utility to the parametric model.
Three user defined global parameters represent the characteristic dimensions of the inverted bow. They are the core that provides the integrity between the baseline hull and the inverted bow shape. Each global parameter is linked in some way to each of the basic curves and surfaces that define the hull shape. The three user defined global parameters, illustrated in Figure 2-1, are:
•
LAftBody is the length from the aft end of the bow section to the forward perpendicular (FP).
•
LOA is the overall length of the bow section, given as a percentage of
LAft-
Body
.
•
Draft is the height of the design waterline (DWL) of the bare hull, determined by the baseline hull in a normal underway condition.
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Figure 2-1: User defined global parameters for the inverted bow shape
Seven global parameters are also derived from the baseline hull that help to make a fair connection across the plane where the hull was cut. The quantities of the sectional area (
SACAft
), breadth at the design waterline (
DWLAftDet
), breadth at the main deck (
DeckBeamAft
), and the depth of the main deck (
D
) are detected from the baseline hull. Additionally, the aft tangent for the sectional area curve, DWL curve, and deck plan curve are detected from the baseline hull. These parameters are used to define the aft end location and aft tangent of several curves listed in the next section.
As the inverted bow section was created directly from the curves described below, it is oriented to the global coordinate system. Therefore it was necessary to translate the inverted bow to the correct position once the baseline hull was imported. These translations were accomplished using a set of three further global parameters.
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The seven basic curves below describe the inverted bow section. The FSpline curves were originally created normalized from 0-1 and, then once the new bow was complete, all curves were scaled to the full size. The four local parameters that define each FSpline are the starting (aft) and ending (forward) position and the aft and forward tangents. For some curves additional parameters are required to define the shape as noted below. The basic curves are illustrated in Figure 2-2.
Figure 2-2: Basic curves
1. The Sectional Area Curve (SAC) describes the area of each section along the ship’s length. The integral of the curve with respect to the x-axis represents the underwater volume. The sectional area at the aft end and the aft tangent are detected from the baseline hull. The curve is further defined by a fullness factor to describe the area under the curve with respect to the x-axis. The forward
35

tangent and the fullness factor are manipulated until the underwater volume is equal to the value calculated from the baseline hull.
2. The DWL Curve describes the breadth of the new bow section at the DWL.
The breadth at the aft and its tangent are detected from the baseline hull. The curve is further defined by a fullness factor to describe the area under the curve with respect to the x-axis. Breadth measurements were taken from the modified
Rhinoceros model at three equally spaced points along the length of the bow section to help shape the curve. The forward tanget and fullness factor were then manipulated to make a fair curve fit to the points.
3. The Profile Curve describes the main deck, keel and the stem line. The curve is broken into three parts, consisting of five individual curves.
• The Lower Profile is the part of the Profile Curve that describes the shape of the keel. It consists of a rectilinear aft curve and a BowLow curve that starts at the point of bow rise and extends to the extreme bow. The x-coordinate of the bow rise is defined as a percentage of the global parameter
LAftBody and the z-coordinate of the extreme bow is defined as a percentage of the global parameter
Draft
.
• The Upper Profile is the part of the Profile Curve that describes the stem line. It consists of a BowUp curve that starts at the extreme bow and ends at the FP and a rectilinear BowUp02 curve that begins at the FP and ends at the entrance, the forward most point of the main deck. The x-coordinate of the entrance is defined by a factor that relates the two global parameters
LOA and
LAftBody to keep a rectilinear bow shape above the DWL. The z-coordinate of the entrance is defined by a local paramters as a percentage of the derived parameter
D
.
• The Deck is the part of the Profile Curve that describes the main deck.
The forward position and tangent were adjusted to ensure faired buttocks lines.
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4. The Deck Plan Curve describes the breadth of the new bow section at the main deck. The length of the main deck is determined by the shape of the
Profile Curve . The Deck Plan Curve is further defined by a fullness factor to describe the area under the curve with respect to the x-axis. Similar to the
DWL curve , the breadth at the aft end and its tangent are detected from the baseline hull. Breadth measurements were taken from the modified Rhinoceros model at three equally spaced points along the length of the bow section to help shape the curve. The forward tangent and fullness factor were then manipulated to make a fair curve fit to the points.
5. The Deadrise Curve describes the angle of deadrise along the keel of the inverted bow section. For ease of viewing the aft and forward deadrise angles are assigned coefficient values that are converted to degrees (multiplied by 100) in the curve engine. As the angles are defined with respect to the z-axis, the values of this curve are higher at the aft of the curve than the bow to create the desired narrow v-shaped sections.
6. The Flare at Deck Curve describes the flare angle along the main deck of the inverted bow section. A unique challenge of the bow redesign was to transition from a flared hull to a tumblehome shape above the waterline. As a result we had to define the flare at the deck with three individual curves. Again, the angles are given as coefficients that are converted by the curve engine to degrees with reference to the x-axis.
• C1 describes the flare from the aft end of the bow section to the entrance.
C1 begins with a flare angle of 82 degrees and ends with a flare angle of
101 degrees, indicating a transition from a flared hull to tumblehome. A value of 90 degrees would indicate zero flare.
• C2 describes the tumblehome from the entrance to the FP.
• C3 describes the tumblehome from the FP to the extreme bow.
7. The Flare at DWL Curve describes the flare angle along the DWL of the in-
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verted bow section. This curve also helps to create a faired surface as the hull transitions from flare to tumblehome.
The new bow section is composed of four surfaces. The bow is then joined to the baseline hull by a set of fillet surfaces. For each bow surface a feature definition was created and a FCurveEngine used to relate to the parameters and curves defined above. A new curve, XPos , is defined to distribute the sections evenly over the length of the hull. As seen in the following paragraphs XPos is the x-coordinate of each point in the cross section definitions.
Surface one extends from the aft end of the bow section to the entrance. As demonstrated in Figure 2-3 the definition of the cross section of this surface is based on three points, the keel, the design waterline and the deck. These points are connected with two FSplines which are then joined into one polycurve. The coordinates of each point, as well as the angles and area coefficient, are linked to their corresponding global parameter or basic curve by the curve engine aft . As such the surface will have a constant topology but a shape that represents the distribution of parameters over its length. Note that the curve in the cross section definition is of generic shape, but represents the distribution of the sections as defined by the curves and global parameters. This is a particularly important surface because at the aft end it must join to the baseline hull surface via fillet surfaces.
Surfaces two and three extend from the entrance to the FP. This section was created using two surfaces to increase the fairness. As the surface ends at a sharp point, the intersection of the DWL and FP, it is best to use a shorter surface to
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Figure 2-3: Cross section definition and isometric view of surface one increase the local resolution of the patch. The aft end of surface two is the entrance and it extends forward to just short of the FP. Surface three is a narrow "sliver" that connects surface two to the FP. As demonstrated in Figure 2-4 the definition of the cross section is the same for both surfaces and only differs from the previous definition by the z-coordinate of the top point. Here the surfaces will only extend up to the BowUp02 curve. Both surface two and three use the same curve engine mid .
Figure 2-4: Cross section definition and isometric view of surfaces two and three
Surface four extends from the FP to the extreme bow. As demonstrated in Figure 2-5 the definition of the cross section of this surface is based on two points, the
39

keel and the upper profile, connected with an FSpline . The coordinates of each point, as well as the angles and area coefficient, are linked to their corresponding global parameter or basic curve by the curve engine forw . As this surface represents a section of the hull that is entirely underwater the top of the surface is now defined by the
BowUp curve from the profile and the C3 curve from the flare at deck. It was a challenge to create a fair connection given that the surface begins at the point where the two flare curves meet.
Figure 2-5: Cross section definition and isometric view of surface four
Three fillet surfaces are used to connect the new bow to the baseline hull as shown in Figure 2-6. While the new bow takes inputs from the baseline hull, the fillet surfaces ensure full tangency across the two surfaces. At the outset the intent was to make the fillet surfaces very short. However we found the need to increase the total length of the modified section to fair the buttock lines. As the bow geometry was already well established fairness was achieved by increasing the length of the fillet surfaces.
40

Figure 2-6: Isometric view of fillet surfaces
The offsets for the baseline hull are imported into CAESES and used to define a b-spline surface. The original surface is then trimmed at the appropriate point to make a connection with the new bow and fillet surfaces. To create the final hull, we have built a polysufrace out of the four bow surfaces and three fillet surfaces. The polysurface and the baseline hull surface are added to the hull group, which can be exported as an .iges file for geometry preparation. The final inverted bow shaped hull is shown in Figure 2-7.
The sheer plans and body plans of the FFG-7 and the Inverted Bow Hull Series are shown for comparison in Figure 2-8. Aft of station 4 the hulls are identical so the lines are only shown for the forward half of the ship.
By changing individual parameters we can generate different hull shapes. The global parameters
Draft and
LAftBody provide continuity between the baseline hull and the inverted bow hull and was therefore set as fixed values. For the
Draft we are limited by the desired to maintain a constant displacement between the two
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Figure 2-7: Perspective view of the final inverted bow hull form hulls and the shape of the baseline hull transom. With only changing the forward 30 percent of the hull we retain the stern geometry of the shaft, strut, and rudder - all of which were designed for a specific draft. The
LAftBody was manipulated during the development of the parametric model and was set to the current value as a means to achieve fairness between the baseline and inverted bow hull. As such, in this section we will discuss the variation in the global parameter
LOA and the necessary local variations to maintain the similitude between hulls.
The
LOA is a factor that multiplies the
LAftBody to define the length of the underwater portion of the bow past the FP. A value of 1 would represent a vertical stem at the FP and a value less than 1 would represent a traditional bow shape.
However, in the context of this parametric model neither of these geometries are realistic because they would require a zero or negative area for surface four. As such the practical lower limit for
LOA is 1.01, meaning the length of surface four is one percent of
LAftBody
. The practical upper limit of
LOA is 1.15 any larger and the inverted bow shape becomes too fine. The parameteric model that is used for the remainder of this project has a
LOA of 1.078 and has been discussed in detail. Here
42

Figure 2-8: (a) Body Plans for the FFG-7 in red and the Inverted Bow Hull Series in blue (b) Sheer plans for FFG-7 in red and Inverted Bow Hull Series in blue
43

we will discuss shapes at the lower and upper limits of the parameter.
The first variation, shown in Figure 2-9, resembles the Axe Bow that was discussed in Chapter 1. By shortening the underwater portion of the inverted bow we drastically reduce the underwater volume. To compensate the fullness factor of the Sectional Area
Curve is increased until the area meets the required value. However this change alone leaves an unfaired hull with wavy buttocks lines. The flare is reduced both of the
DWL and of the deck at the entrance to less tumblehome. This shape also requires more fineness at the FP so the forward tangents of the DWL Curve and Deck Plan curve are adjusted accordingly.
Figure 2-9: Inverted bow shape with the minimum
LOA
The second variation, shown in Figure 2-10, represents the extreme end of practicality for an inverted bow shape. The bow has a very shallow entrance angle and
44

also very narrow v-shaped sections. With this bow shape, the length of the deck is shortened dramatically and thus leaves a short distance over which to transition from flare to tumblehome. Any shorter and there would be a very prominent "shoulder" on the bow. A reverse of the process above was used to fair this hull, starting with the value of the sectional area and following with changes to the flare and shape of the DWL and deck.
Figure 2-10: Inverted bow shape with the maximum
LOA
45

46

The experimental investigation for this study was conducted in the U.S. Naval
Academy Hydromechanics Laboratory (NAHL). Two ship models were built for the testing, one of the baseline frigate discussed in Chapter 1 and one of the inverted bow frigate discussed in Chapter 2, hereafter referred to as Model A and Model B. The goal of the experimental test program described in this report was to compare the performance of Model A and Model B in both resistance and seakeeping tests. The testing program consisted of six test series. Test series one and two were calm water resistance tests for each model conducted at speeds relevant to naval operations.
Test series three and four were seakeeping tests with regular waves in head seas for each model. Three representative speeds were tested in an airy wave program with a constant ratio of wave height to wavelength. Test series five and six were seakeeping tests with irregular waves in head seas. The wave spectra were selected to represent realistic ocean conditions for naval operations. This chapter will discuss the preparations of the facilities and the models, as well as the procedure and results of each test series.
The first four test series were conducted for a single model in the NAHL 120 foot towing tank. Test series five and six were conducted with both models in a
47

side-by-side configuration in the NAHL 380 foot towing tank. Model installation and instrumentation were different for each tank and are discussed below. Water temperature measurements were taken in both tanks for each day of testing and was a constant 62.2 F. The water properties for the model and full scale ships are given in Table 3.1.
Property
Temperature 𝜌 𝜈
Model Ship
Fresh water Sea water
62.2 F 59 F
1.93786
𝑙𝑏 − 𝑠
2 / 𝑓 𝑡
4
1.9905
𝑙𝑏 − 𝑠
2 / 𝑓 𝑡
4
1.171
𝑓 𝑡
2 / s
× 10 5 1.2679
𝑓 𝑡
2 / s
× 10 5
Table 3.1: Summary of water properties for the model and full scale ship
The NAHL 120 foot towing tank has dimensions of 120 feet long, 8 feet wide and 5 feet deep. The tank is outfitted with a medium speed towing carriage and a dual flap, electro-hydraulically activated wavemaker. The wavemaker can be programmed to generate long-crested regular or irregular waves. There is a beach at the opposite end of the tank for absorbing wave energy. Control signals and data are transmitted between the towing carriage and the control station via a wireless PC network. Additionally, a dedicated computer workstation for the tank was used to acquire and analyze the experimental data [17]. A schematic drawing of the NAHL
120 foot towing tank is shown in Figure 3-1.
An individual towing rig was assembled, instrumented and calibrated for each model. Prior to testing, each gage was calibrated using a least squares fit regression.
For each test series the appropriate towing rig and model were attached directly to the towing carriage. The model was fit with a pitch pivot and attached to a heave post on the towing rig, and was thus free to heave and pitch. As this study only considers head seas the model was fixed in the roll direction. Likewise maneuvering was not considered in this study so the model is fixed in the surge, sway, and yaw directions. The model was instrumented as follows:
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Figure 3-1: Schematic drawing of the NAHL 120 foot towing tank
• Resistance was measured using a 5 lb. variable reluctance block gauge located on the model’s centerline and at its LCB. The model was connected to the block gauge at a pivot located approximately 2 inches above the waterline.
• Heave and pitch were measured with a pair of potentiometers located on the towing post located at the model’s LCB.
• Vertical accelerations were measured with Silicon Designs (Model 2210-025) 25g capacitive accelerometers mounted on the bow and at the LCB.
• The encountered water surface elevation was measured with a ultrasonic wave gauge mounted on the carriage in line with the FP and 1 foot to starboard of the model’s centerline.
Data was collected at a sampling rate of 50 Hz. A picture of the instrumented
Model B installed on the towing rig is shown in Figure 3-2.
The NAHL 380 foot towing tank has dimensions of 380 feet long, 26 feet wide and 16 feet deep. The tank is outfitted with two high speed carriages (only the forward carriage was used for this study) and a dual flap, electro-hydraulically activated
49

Figure 3-2: Model B attached to the 120 foot towing tank carriage wavemaker. The tank control and data acquisition systems are similar to the 120 foot tank [17]. A schematic drawing of the 380 foot tank is show in Figure 3-3.
Figure 3-3: Schematic drawing of the NAHL 380 foot towing tank
In the NAHL 380 foot towing tank tests were conducted with the models in a sideby-side configuration by mounting each model and towing rig to the same carriage with the FP of both models in line. As the same towing rigs were used in both tanks the only changes to instrumentation was the location of the ultrasonic wave probe
50

and addition of video cameras. In the NAHL 380 foot towing tank the ultrasonic wave probe is positioned such that it is in line with the FP of both models. The video cameras are positioned to show the same bow perspective for each model.
The major difficulty with performing side-by-side tests is ensuring that the model waves do not interfere with each other, and can only be accomplished with small models. The models were positioned on the carriage such that the wakes do not meet until a distance of approximately 2.5 times the LBP aft of the models. To accomplish this the models were spaced 12.7 feet apart with 5.65 feet between the model and tank wall. It was assumed that the tank wall will perfectly reflect the wake pattern.
Figure 3-4 shows a schematic of the arrangement for towing the models.
Figure 3-4: Schematic drawing of the side-by-side towing arrangement
Figure 3-5 shows a picture of the models installed on the NAHL 380 foot tank towing carriage. Model A is on the port rig (left side of tank) and Model B is on the
51

starboard rig (right side of tank).
Figure 3-5: Models A and B attached to the 380 foot towing tank carriage
The models were built in the NAHL Wood/Composite shop from a 3-D CAD files provided by the author. Model A is the baseline hull for this project described in
Chapter 1. The CAD file was created from the offsets of the FFG-7. Model B is the inverted bow hull described in Chapter 2. The CAD file was created from the file exported from CAESES. Both models are
1 /
80 𝑡ℎ scale ( 𝜆 = 80
) and do not have any appendages. The models were cut from high density closed cell foam using a CNC
3-axis milling machine as seen in Figure 3-6. The models were sanded with 400 grit sand paper and coated with DuraTex to achieve a smooth water resistant finish. On each model ten stations were marked at 6.166 in apart and six waterlines were marked at .75 in to represent 5 ft waterlines in ship scale.
A precaution that must be taken in model testing is to ensure that the flow over the model is fully turbulent, as the flow around the full scale ship will be. For both models at low speeds the Reynolds number was within the transition region and must
52

Figure 3-6: Model B under construction thus be "tripped" to turbulent. This is a common occurrence with such small scale models and easily fixed by adding turbulence stimulators. The standard practice for turbulence stimulation on small ship models in the NAHL is to use Hama strips.
Hama strips are made from electrical tape cut with pinking shears to an average thickness of 0.25 in. The triangular shape of the Hama strip causes vortex loop formation at the trailing edge [9]. Using a measure of the Reynolds number per foot on free-stream speed,
𝑅 𝑒𝑓 𝑡
, the required thickness of the Hama strips is determined for each model [1]. Each layer of tape has a thickness of 0.005 in, and the layers are built up to the required thickness. A plot of the required Hama strip thickness, in layers of tape, for this study in shown in Figure 3-7.
From experience in the NAHL using Hama strips for turbulence stimulation in small ship models there is a trade off between turbulent flow at lower speeds and parasitic drag at higher speeds. As we were interested in higher speeds for this study we used seven layers of tape to build the Hama strips. We predict this configuration will create turbulent flow for ship speeds above 15 knots and adds little drag at
53

Figure 3-7: Required Hama strip thickness for turbulent flow higher speeds. The strip was placed with the trailing edge at station 1. Typically the direction of the strip will follow the inclination of the bow. However, for this experiment the bow angle is so different between models that a vertical orientation, the average of the two bow angles, was used for both. A picture of the Hama strip applied to Model B is shown in Figure 3-8.
When model tests are conducted in still water for resistance and speed related sinkage and trim attitudinal changes, the geometrically similar model must be ballasted to the scaled waterline of the subject full scale ship. This means that both displacement and LCG must be scaled geometrically [2]. Each model was tested with a scaled displacement equal to a normal underway condition with the draft equal to the DWL and zero static trim. Hydrostatic parameters are listed in Table 3.2.
54

Figure 3-8: Hama strip applied to Model B
Ship A has a displacement of 4139 LT in the normal underway condition. Model
A must be ballasted to the proper displacement, which is proportional to the scale ratio cubed and reflects the different densities of sea water and fresh water.
∆
𝑀
= 𝜌
𝑀 𝜌
𝑆
∆
𝑆 𝜆 3
=
1 .
93786 4139
LT
1 .
9905 80 3
× 2240 lb
LT
= 17 .
63 lb (3.1)
The bare hull of Model A weighs 7 lbs, thus 10.63 lbs must be added for proper displacement. The tow post and force block weigh 3.64 lbs thus the remaining 6.99
lbs is stowed in the hull.
Ship B has the same displacement thus Model B must weigh the same as Model
A. However, the bare hull of Model B weighs 7.3 lbs, so 10.33 lbs must be added for proper displacement. Using the same tow post and force block the remaining 6.69 lbs is stowed in the hull.
The model wetted surface area,
𝑆
𝑀 to be used in the analysis, is calculated from
55

the CAD files to the still waterline. The transom area is not included in the wetted surface area.
LWL
∇
∆
T
S
𝐴
𝑊 𝑃
LCG aft of Midships
𝐵𝑀
𝐿 𝑘
5
(% LBP)
Model A Model B
5.15 ft
0.283
𝑓 𝑡 3
17.63 lb
5.15 ft
0.283
𝑓 𝑡 2
17.63 lb
0.206 ft
3.08
𝑓 𝑡
2
2.159
𝑓 𝑡 2
.42 in
0.206 ft
3.15
𝑓 𝑡
2.134
𝑓 𝑡
.71 in
2
2
135.82 in 129.78 in
15.53 in (25.2%) 14.83 in (24.1%)
Table 3.2: Summary of principal model characteristics
As the planned testing program includes runs in waves the models were also dynamically ballasted in pitch and heave. When ship models are tested in head sea conditions a quantitative measure of the longitudinal distribution of weight about the LCG must be modeled in addition to static ballasting requirements [2]. The measure of longitudinal weight distribution about the center of gravity is the longitudinal gyradius, 𝑘
5
. Traditionally a value of 𝑘
5 equal to about 25% of the LBP is assumed for ships. Ship models are ballasted to the correct displacement, LCG, and 𝑘
5 by the judicious placement of ballast weights within the test model. When the weights are moved symmetrically away from the center of gravity of a ship, the gyradius of that ship will increase without changing displacement or the position of the LCG. Increasing a ship’s longitudinal gyradius should affect the natural pitching period and the magnitude of pitch response for a given wave excitation. Because of the strong coupling between pitch and heave motions for ships, it is assumed that the heave motion will similarly be affected.
For this project I followed a method of dynamic ballasting developed by Gilbert
Lamboley [11]. This techniques involves swinging the model in pitch from pivots at a
56

know distance apart. Figure 3-9 shows Model B swinging on the NAHL Lamboley rig.
The resultant period is a function of the distance from the pivots to the model center of gravity and the pitch gyradius. By employing two pivot heights and measuring periods for each, simultaneous equations can be created to solve for 𝑘
5
. The pitch gyradius for each model is given in Table 3.2. Appendix A provides details of the method used and the calculations for determining 𝑘
5
. For this study a value for 𝑘
5 of
25 ± 1% was considered satisfactory.
Figure 3-9: Dynamic ballasting Model B using the NAHL Lamboley rig
A simple way to determine the damped natural frequency in pitch and heave is by experiment. First the bow was displaced to the limit of the pitch pivot and allowed to pitch freely. Then the heave post was displaced to the limit and allowed to heave
57

freely. From the recorded free decay data the average pitch period and heave period were calculated. Like all ship models, heavy damping is experienced and only a few oscillations are observed. From the free decay experiment we also determined the decay coefficient ( 𝜂
) by measuring subsequent peaks and fitting a logarithmic decay
[13].
𝜂 =
1 𝜋
𝑋
0 𝐽 𝑙𝑛
𝑋
0 𝐽 +1
(3.2)
The experimentally determined decay coefficients for the pure and coupled motions are listed in Table 3.3. The decay coefficients help characterize the seakeeping performance of each hull.
𝜂 𝜂
33
53
Model A Model B
0.28
0.243
0.126
0.187
0.08
0.212
𝜂
35 𝜂
33 𝜔 𝑑
0.246
pitch 11.63
rad
/ s
0.253
11.22
rad
/ 𝜔 𝑑 heave 11.63
rad
/ s
11.22
rad
/ s s
Table 3.3: Experimentally determined decay coefficients and natural frequencies
The testing program was broken down into six test series as shown in Table 3.4.
This testing program was developed to demonstrate the performance of the inverted bow and to collect sufficient data for numerical validation in the limited testing window in the NAHL facilities.
For test series one and two each model was tested at speeds scaled to represent ship speeds from 16 kts to 40 kts in two knot increments. Additional data points were taken at intermediate speeds where required. Speeds below 16 knots were not
58

Test Series Model Tank
1
2
3
4
5
6
B
A
A
B
120’
A & B 380’
A & B 380’
Seaway
Calm water
Ship Speed
120’ Calm water 16-40 kts
120’ Regular waves 20, 25, 30 kts
120’ Regular waves 20, 25, 30 kts
Sea State 4
Sea State 6
16-40 kts
20, 25, 30 kts
20, 25, 30 kts
Table 3.4: Test conditions for each test series considered due to the limitations on turbulence stimulation of such a small scale model. Before each test series it is standard practice in the NAHL to conduct a high speed run to introduce vorticies into the tank. Subsequent runs alternated between high and low speeds to keep vorticity in the tank. To ensure calm water for each test a period of three to five minutes was observed between runs, depending on the speed of the model, to allow any surface waves to settle. On each run the velocity, resistance, trim (as the average pitch angle), and the sinkage (as the average heave) were recorded. The data was truncated to include only data recorded when the model is steady at full speed. The full data for test series one and two is presented in Appendix B.
For test series three and four each model was tested at three speeds in long crested regular head waves. For regular wave tests the International Towing Tank Conference
(ITTC) recommends that the ratio of generated wave height to wavelength should be maintained constant [9]. For this experiment this ratio was held constant at the largest value where we could reasonable assume that linear theory is still valid.
ℎ 𝜆
1
=
50
The ITTC also recommends that a sufficient number of tests be conducted at each speed over the range of wavelengths from one-half LBP to two times LBP. To generate the required waves the wave maker input was set to an Airy wave program
59

using the piston flap articulation mode. The frequency input for the wave maker was determined using the dispersion relationship for deep water: 𝜔
2
= 𝑔𝑘 =
2 𝜋𝑔 𝜆
(3.3)
The deep water assumption is only valid when the depth is more than half of the wavelength. Given the frequency range of the wave maker in the NAHL 120 foot towing tank and a tank depth of 5.23 feet, eight wavelengths were chosen to sufficiently cover the required range. A summary of these test conditions are listed in Table 3.5. Further tests were conducted as required with more closely spaced wavelengths to ensure a good definition in the resonance region.
f 𝜆 𝜆 h
(Hz) (ft) (in) (in)
1.4
2.613
31.351 1.254
1.3
3.030
36.360 1.454
1.2
3.556
42.672 1.707
1.1
4.232
50.783 2.031
1.0
5.121
61.448 2.458
.9
.8
6.322
8.001
75.861 3.034
96.012 3.840
.7
10.399 124.791 4.992
Table 3.5: Summary of the standard test conditions to cover the required range of wavelengths
For each run the time histories of the pitch, heave, vertical accelerations and water surface elevation were measured. The data was truncated to include a period of at least ten cycles when the model was at full speed, in waves, and between up crossings of the encountered wave. The mean and standard deviation of each quantity over this time interval was recorded. The full data for test series three and four is presented in Appendix B.
Tests in regular waves were conducted to experimentally determine the pitch and heave transfer functions and to study vertical accelerations for both models. Each transfer function presents a non-dimensionalized form of the motion plotted against
60

the encounter frequency ( 𝜔 𝑒
). The encounter frequency for head seas is given as: 𝜔 𝑒
= 𝜔 + 𝑘𝑉 = 𝜔 + 𝜔
2
𝑉 𝑔
(3.4)
As the water surface elevation was measured by a carriage mounted encounter wave probe we will work in the encounter frequency domain throughout the experiment.
As we are primarily interested in evaluating comparative performance between
Model A and Model B, we ran the models in a side-by-side configuration for irregular seas. By towing side-by-side each model saw exactly the same wave pattern at exactly the same encounter frequency. This allowed for direct comparison of the time histories of the vessel response to waves.
The ITTC recommends the Joint North Sea Wave Project (JONSWAP) Spectrum for fetch limited seas. The JONSWAP Spectrum is a distortion of the two-parameter
Bretschneider Spectrum. A peak enhancement factor (C) increases the height of the peak and also enhances non-linear interactions. The JONSWAP spectral formulation, specified in terms of the one-third highest significant wave height and the modal period, is given as:
𝑆
𝐽
( 𝜔 ) = 0 .
658 𝐶 × 𝑆
𝐵
( 𝜔 )
𝐴
𝑆
𝐽
( 𝜔 ) = 0 .
658 𝐶 × 𝜔 5 exp
− 𝐵 𝜔 4
(3.5) with constants:
𝐴 = 487 .
3
𝐻 2
1 /
3
𝑇 4
0
𝐵 =
1949
𝑇
0
4
𝐶 = 3 .
3 exp −
( 𝜔 − 𝜔
0)
2 𝜎
2 𝜔
2
0
2
61

where: 𝜎
= 0.07 ( 𝜔 < 𝜔
0
) or 0.05 ( 𝜔 > 𝜔
0
)
Two spectra were developed for this experiment based on the NATO Standard
Sea States 4 and 6, as seen in Table 3.6.
Ship Scale
Significant Wave Height
𝐻
1 /
3
[ft]
Modal Wave Period
𝑇
0
[sec]
Most
Sea State Range Mean Range
4
6
4.1 - 8.2
13.1 - 19.7
6.2
16.4
6.1 - 15.2
9.8 - 16.2
Probable
8.8
12.4
Significant Wave Height
𝐻
1 /
3
[in]
Model Scale
Modal Wave Period
𝑇
0
[sec]
Most
Range Mean Range
0.62 - 1.23
1.97 - 2.99
0.93
2.46
0.68 - 1.70
1.10 - 1.81
Probable
0.98
1.39
Table 3.6: Annual sea state occurrence in the North Atlantic [3]
The characteristic parameters were converted to model scale for input to the wave maker. The spectra were set to coalesce at a position just ahead of the carriage starting position to maximize encounters on each run. Figure 3-10 shows the spectral density ordinates for sea state 4 in ship and model scale. The ship scale spectrum is plotted on the primary axis (left) and the model scale spectrum is plotted on the secondary axis (right).
The range of frequencies included in the idealized spectrum was limited by the range of the wave maker. To verify that the developed spectra are sufficient the spectral moments 𝑚
0 and 𝑚
1 were calculated with the following relationships:
𝐻
1 /
3
= 4
√ 𝑚
0
(3.6)
𝑇
¯
= 2 𝜋 𝑚
0 𝑚
1 𝜔
0
=
4 .
849
𝑇
¯
𝑇
0
=
2 𝜋 𝜔
0
(3.7)
Additionally, the spectra were experimentally verified by measuring the water surface elevation at the position for the fully developed spectra with the following
62

Figure 3-10: Spectral density ordinates for Sea State 4 in ship and model scale relationship:
𝐻
1 /
3
= 4 𝜎 𝜁
A log of data runs for test series five and six is presented in Appendix B.
(3.8)
For the two-dimensional vertical plane motions described in Chapter 1 linearized equations can be used to approximate the ship motions as a classical damped springmass system [3]. By studying each term we can explicitly see where the differences lie between Model A and Model B. Starting on the right side of the equations, each model was subject to the same waves, so the excitation forces are equivalent between
63

models.
(∆ + 𝐴
33
) ¨
3
+ 𝐵
33 𝑥 ˙
3
+ 𝐶
33 𝑥
3
+ 𝐴
35 𝑥 ¨
5
+ 𝐵
35 𝑥 ˙
5
+ 𝐶
35 𝑥
5
= 𝐹
𝐸𝑋 3 𝑐𝑜𝑠 ( 𝜔 𝑒 𝑡 + 𝜖
3
)
(3.9)
( 𝐼
55
+ 𝐴
55 𝑥
5
+ 𝐵
55 𝑥 ˙
5
+ 𝐶
55 𝑥
5
+ 𝐴
53 𝑥 ¨
3
+ 𝐵
53 𝑥 ˙
3
+ 𝐶
53 𝑥
3
= 𝐹
𝐸𝑋 5 𝑐𝑜𝑠 ( 𝜔 𝑒 𝑡 + 𝜖
5
)
(3.10)
The
𝐴 𝑗𝑘 terms refer to the added mass. The term
𝐴
33 represents the added mass in the heave direction due to heave motion.
𝐴
33 is related to the underwater volume, which is the same for both models. Due to the transverse symmetry and the assumption of a constant forward speed we treat the terms
𝐴
35 and
𝐴
53 as equal. Both terms are related to the 1st moment of the underwater volume, which is also the same for both models. The term
𝐴
55 is related to the 2nd moment of the underwater volume, which is lower for Model B. This confirms the smaller volume distribution about the center of buoyancy.
By experiment we determined 𝑘
5 for each model, which allows us to calculate the pitch moment of inertia
𝐼
55 using equation 3.11. Ideally both models would have the exact same pitch moment of inertia. However, as stated previously, the values for 𝑘
5 were within the precision of the experimental dynamic ballasting technique. The result is that Model B has a value of
𝐼
55 that is approximately 8% lower than Model
A.
𝐼
55
= 𝑚𝑘
2
5
(3.11)
The
𝐵 𝑗𝑘 terms refer to the hydrodynamic damping of the system. Using the experimentally determined decay coefficients in Table 3.3 we can characterize the hydrodynamic damping. Model B has less damping in pure heave and greater damping in pure pitch. Similarly for the coupled motions Model B has less damping in pitch motion due to heave excitement and more damping in heave motion due to pitch excitement.
The
𝐶 𝑗𝑘 terms are the restoring forces and moments, or the spring constants.
They are hydrostatic restoring coefficients and independent of frequency. For pure
64

heave motion
𝐶
33 is the pounds per inch immersion (PPI). Model B has a smaller waterplane area and thus a lower PPI.
PPI
=
𝐴
𝑊 𝑃
* 𝜌 𝑓 𝑤
* 𝑔
12
(3.12)
For pure pitch motion
𝐶
55 is related to the moment to trim one inch (MT1). Given that the two models have the same displacement and length, the MT1 is determined by
𝐵𝑀
𝐿
. Model B has a smaller metacentric radius and thus a lower MT1.
MT1 ≈ 𝜌
𝐹 𝑊
* 𝑔 * ∇ * 𝐵𝑀
𝐿
12 𝐿
(3.13)
For the coupled pitch and heave motion
𝐶
35
= 𝐶
53 and both are related to the change in displacement per inch of trim by the stern ( 𝛿 ∆1
). LCF represents the distance of the center of flotation abaft of midships.
𝛿 ∆1 =
𝑃 𝑃 𝐼 * 𝐿𝐶𝐹
𝐿
(3.14)
Model A has a slightly larger PPI and a slightly smaller LCF. The result is that
Model B has a larger 𝛿 ∆1 indicating a stiffer spring for coupled motion. This is consistent with the notion that the inverted bow will pass through waves rather than riding on top. A table of the calculated values is shown below in Table 3.7.
𝐼
55
Model A
4252 lb-in 2
Model B
3877 lb-in 2
PPI 11.10
lb
/ in
10.95
lb
/ in
MT1 38.78
lb-in
/ in
37.14
lb-in
/ in
LCF 3.32 in 𝛿 ∆1
0.59
lb
/ in
3.66 in
0.65
lb
/ in
Table 3.7: Summary of ship motion parameters
65

The data collected during this experiment allowed us to characterize the performance of the inverted bow against the baseline frigate in terms of calm water resistance and motions in regular and irregular waves. Calm water tests were conducted to determine the resistance of each model. Regular waves were generated according to linearized theory and thus the ship motions can be considered linear. This data is easily collected and repeatable, but only serves as a first order approximation of the ship motions. Irregular waves were generated according to idealized sea spectra based on average ocean statistics and thus the ship motions can be considered non-linear.
In the following section we first make a comparison of the linear and non-linear responses of the inverted bow. With the understanding of the motions of the inverted bow we then assess its performance against the baseline frigate.
The output of the resistance tests for a model without appendages is the bare hull resistance, or total model resistance
𝑅
𝑇 𝑚
(lbs), where the subscript 𝑚 indicates model scale. The total resistance coefficient of the model is calculated as [12]:
𝐶
𝑇 𝑚
=
𝑅
𝑇 𝑚
.
5 * 𝑆 𝑚
* 𝜌 𝑚
* 𝑉 2 𝑚
(3.15)
Next the friction coefficient is calculated according to the ITTC-57 formula:
𝐶
𝐹 𝑚
=
0 .
075
( 𝑙𝑜𝑔𝑅𝑒 𝑚
− 2) 2
(3.16)
Then the residuary resistance coefficient can determined by subtracting the friction resistance coefficient from the total resistance coefficient.
𝐶
𝑅𝑚
= 𝐶
𝑇 𝑚
− 𝐶
𝐹 𝑚
(3.17)
Using Froude’s Law of Comparison at corresponding speed-length ratios, the co-
66

efficient of residuary resistance is considered to be equal in model and ship scale.
𝐶
𝑅𝑚
( 𝐹 𝑟 ) = 𝐶
𝑅𝑠
( 𝐹 𝑟 )
Finally the residuary resistance of the model is calculated as:
𝑅
𝑅𝑚
= .
5 * 𝐶
𝑅𝑚
* 𝑆 𝑚
* 𝜌 𝑚
* 𝑉
2 𝑚
(3.18)
A comparative plot for the residuary resistance vs. Froude number for Model A and Model B is shown in Figure 3-11. The data points are fit with a polynomial trendline for ease of comparison. From this plot it is clear that Model B shows an improved performance over Model A at each Froude number. This confirms the lower resistance for the inverted bow proposed at the beginning of the paper.
Figure 3-11: Residuary resistance for Model A and Model B
Also illustrative of the improved performance, and related to the ship motion parameters discussed in the previous section, are plots of the sinkage (Figure 3-12) and trim (Figure 3-13) during the calm water tests. Again the data points are fit
67

with a polynomial trendline for ease of comparison.
Figure 3-12: Sinkage for Model A and Model B
Figure 3-13: Trim for Model A and Model B
68

From the regular wave tests the standard deviation of the pitch motion ( 𝜎 𝑥
50
) was determined for a specific period as described in section 3.3.2. Thus the amplitude of the pitch motion 𝑥
50 was non-dimensionalized by the measured wave slope amplitude to create the transfer function. In the case of simple harmonic waves the variance of the wave elevation ( 𝜎
2 𝜁
) for a finite number of complete cycles of a sine wave is equal to one half the square of the amplitude. Each response is also assumed to be harmonic and follows this relationship.
Pitch Transfer Function
= pitch amplitude wave slope amplitude
= 𝜎 𝑥
50 𝑘𝜎 𝜁
√
2
√
2
= 𝑥
50 𝑘𝜁
0
(3.19)
The transfer functions exhibit a speed dependence with the resonant peak generally increasing and moving to a higher encounter frequency as speed increases. However, as both models are heavily damped in pitch the resonant peaks are not very pronounced.
A plot of the pitch transfer functions for a ship speed of 25 knots is shown below in
Figure 3-14. In the body of this paper we will discuss the test case of 25 knots ship speed, but the plots for the remaining speeds are given in Appendix C. An uncertainty analysis was conducted to characterize the randomness of the experimental process
[9]. For a ship speed of 25 knots, the test condition with a wave frequency .7 Hz was repeated ten times. From this sample a population mean and expanded uncertainty were determined for a 95% confidence interval. Due to the relatively low sample size it was assumed that the results follow a t-distribution. Error bars are included as:
¯ ± 𝑡
95 𝜎 𝑥
50 𝑛
(3.20)
In long waves ( 𝜆 > 𝐿𝐵𝑃
) the encounter frequency is very low and excitations experienced by the ship are nearly completely attributable to the buoyance changes as waves pass the hull. The ship will always be aligned with the wave surface so that the pitch amplitude will equal the wave slope amplitude, thus the transfer function
69

Figure 3-14: Pitch transfer function for ship speed of 25 knots approaches unity. Resonance occurs in the region where
.
75 𝐿𝐵𝑃 < 𝜆 < 𝐿𝐵𝑃 where buoyancy forces begin to alternate along the hull. As both ships exhibit the same natural frequency in pitch from a free decay test, we expect the peaks to closely align.
In short waves ( 𝜆 ≪ 𝐿𝐵𝑃
) the encounter frequency is high and thus responses are almost negligible.
As predicted the inverted bow demonstrates a larger pitch amplitude in the resonance region. However, outside of the resonance region the two transfer functions are nearly identical. Returning to the damped mass-spring analogy the inverted bow has a lower spring constant in pitch, and thus a exhibits a larger pitch amplitude.
Contrary to the results in the regular waves, the inverted bow appears to demonstrate less pitching motion in irregular waves. Figure 3-15 shows a selected time history of the pitch motion for both models at a ship speed of 25 knots from test series 6. Over this twenty second period in the middle of the run you can see that the pitch motions of the two models are different. The full time histories of pitch motions for each test case are given in Annex D. This leads to the conclusion that there are non-linear effects caused by the inverted bow that are not present in regular
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wave tests. In essence, the sequencing of the waves matter for the inverted bow as it pierces waves rather than riding on top of them.
Figure 3-15: (a) Selected time history of pitch (b) Selected time history of water surface elevation
To demonstrate the presence of non-linear pitch motions we must turn to spectral analysis. This will allow us to make a direct comparison of the pitch motion as measured in irregular waves and as calculated using the linear transfer function. A detailed procedure for the spectral analysis is presented in Annex E. First a Fast
Fourier Transform (FFT) is applied to the time history of the irregular water surface elevation and the wave energy spectrum is defined as
𝑆 𝜁
( 𝜔 ) = 𝜁 2 𝑛 0
2 𝛿𝜔
(3.21)
As pitch is an angular motion we will express the wave energy in terms of the wave slope spectrum. The wave slope spectral ordinates are given by
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𝑠 𝛼
( 𝜔 ) = 𝜔 4 𝑛 𝑔 2
× 𝑆 𝜁
( 𝜔 )
(3.22)
The differences between the measured and calculate pitch motion can be explained by Figure 3-16. In Figure 3-16(a) the wave slope spectrum, as measured for the same run from test series 6, is shown. For verification an idealized spectrum based on the measurements taken during the experiment is also plotted.
Figure 3-16(b) shows the pitch transfer function for Model B as determined in the regular wave tests. Again, the pitch motion is normalized by the wave slope so that the transfer function tends to unity at low encounter frequencies. The transfer function is used to determine the Response Amplitude Operator (RAO). The RAO is the square of the transfer function.
Now a FFT is applied to the irregular time history of the pitch motion and the pitch energy spectrum is defined as 𝑠 𝑥
5
( 𝜔 ) = 𝑥 2
50
2 𝛿𝜔
(3.23)
Figure 3-16(c) shows the measured pitch energy spectrum from the irregular wave tests and the calculated pitch energy spectrum using the linear transfer function for
Model B. This spectrum is calculated by multiplying the measured wave slope spectrum by the RAO. In this figure we can see that the measured and the calculated pitch energy spectrum do not match. The linear method over predicts the pitch energy in the resonance region. By using a trapezoidal rule integration of each spectrum it is shown that the zeroth spectral moment 𝑚
0
(as the area under the pitch energy spectrum) for the calculated spectrum is about 10 % larger than the measured spectrum.
Plots for the remaining test conditions are given in Appendix E.
The above analysis serves to demonstrate that the performance of Model B cannot be characterized by regular wave tests alone. However, we would also like to explain that Model B does indeed pitch less than Model A. For this analysis we can return to the time domain to compare the performance of both models as 𝑚
0 is also the variance of the time history. We are able to take this much simpler approach due to
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Figure 3-16: (a) Measured wave data and idealized wave slope spectrum (b) Linear pitch transfer function for Model B (c) Measured and calculated pitch energy spectrum for Model B the side-by-side testing configuration described earlier because both models saw the exact same wave at the exact same time. A summary of 𝑚
0 for the pitch motion for each test condition is given in Table 3.8.
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Test Series 5 (Sea State 4)
Speed (kts) Model A 𝑚
0 𝑃
Model B 𝑚
0 𝑃
20
25
30
0.512
0.428
0.301
0.601
0.369
0.260
Test Series 6 (Sea State 6)
Speed (kts) Model A 𝑚
0 𝑃
Model B 𝑚
0 𝑃
20
25
30
2.928
3.213
2.736
2.452
2.907
2.529
Difference
14.88%
-15.96%
-15.90%
Difference
-19.42%
-10.52%
-8.18%
Table 3.8: Summary of variance of pitch response in irregular waves
The variance is calculated directly from the data, but also by definition is simply the square of the standard deviation. As demonstrated in equation 3.6 the standard deviation is proportional to the significant value of the motion. Model B demonstrates less pitching motion in response to the same waves than Model A in all cases except for the slowest speed in the smaller sea state. It is not immediately apparent why this one result is inconsistent with the rest and it requires further study.
From the regular wave tests the standard deviation of the heave motion ( 𝜎 𝑥
30
) was determined for a specific period as described in section 3.3.2. The amplitude 𝑥
30 was non-dimensionalized by the measured wave amplitude to create the heave transfer function.
Heave Transfer Function
= heave amplitude wave amplitude
= 𝜎 𝑥
30 𝜎 𝜁
√
2
√
2
= 𝑥
30 𝜁
0
(3.24)
A plot of the heave transfer functions for ship speed of 25 knots is shown below in Figure 3-17. The plots for the remaining speeds are listed in Annex C. Error bars are included as described previously.
Again the transfer function tends toward unity at low encounter frequencies, indicating that the heave motion is synchronized with the wave motion. As predicted the inverted bow demonstrates a larger heave amplitude in the resonance region. Return-
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Figure 3-17: Heave transfer function for ship speed of 25 knots ing to the damped mass-spring analogy the inverted bow has a lower spring constant in heave, and thus exhibits a larger heave amplitude.
Consistent with the results in regular waves the inverted bow also exhibits larger heave motions in irregular waves. Figure 3-18 shows a selected time history of heave motion for both models at a ship speed of 25 knots from test series 6. At nearly every single maxima or minima Model B has a larger heave amplitude. The full time histories for each test case are given in Annex D. To evaluate the heave energy spectra we follow the same analysis as for pitch. In Figure 3-19(a) the wave spectrum, as measured by the carriage mounted encounter wave probe, is shown. Again an idealized wave energy spectrum with equivalent energy is also shown. Figure 3-19(b) shows the heave transfer function for Model B as determined in the regular wave tests.
The heave motion is normalized by the wave amplitude so that the transfer function tends toward unity at low encounter frequencies. The transfer function is used to determine the RAO. Figure 3-19(c) shows the measured heave energy spectrum from the irregular wave tests and the calculated heave energy spectrum using the linear transfer function for Model B. The spectrum is calculated by multiplying the measured
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Figure 3-18: (a) Selected time history of heave (b) Selected time history of waves wave slope spectrum by the RAO. In this figure we can see that the measured and the calculated heave energy spectrum also do not match. By using a trapezoidal rule integration of each spectrum it is shown that 𝑚
0 of the calculated spectrum is about
10 % less than the measured spectrum. Plots of the remaining test conditions are given in Appendix E.
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Figure 3-19: (a) Measured wave data and idealized wave spectrum (b) Linear heave transfer function for Model B (c) Measured and calculated heave energy spectrum for
Model B
This again shows that the regular wave tests do not describe well the performance of the inverted bow. However, in the case of heave the inverted bow actually heaves more in irregular waves. Now we compare 𝑚
0 of heave for Model A and Model B
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from the side-by-side tests given in Table 3.9. Model B demonstrates significantly more heave motion in response to the same waves than Model A in all test cases.
Test Series 5 (Sea State 4)
Speed (kts) Model A 𝑚
0 𝐻
Model B 𝑚
0 𝐻
20
25
30
0.018
0.021
0.021
0.033
0.034
0.032
Test Series 6 (Sea State 6)
Speed (kts) Model A 𝑚
0 𝐻
Model B 𝑚
0 𝐻
20
25
30
0.175
0.207
0.200
0.201
0.288
0.313
Difference
44.54%
37.77%
38.84%
Difference
12.78%
28.18%
36.04%
Table 3.9: Summary of variance of heave response in irregular waves
On each model an acceleration transducer was placed at the LCB and on the bow. Due to the hull geometry and solid construction material the bow acceleration transducers were located at different distances from the LCB for each model. On
Model A the transducer was located at station .5 and on Model B the transducer was located at station 1.5 (Figure 3-20).
Figure 3-20: Acceleration transducer location
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Despite the different distances, the physical location of the transducer is still meaningful. Limits are placed on vertical acceleration due to the effect of machinery, combat systems, and human performance. Each transducer is located near the front of the forecastle where sailors are required to work while the ship is at sea. As this is a hydrodynamic study and we are not concerned with general arrangements, we will draw practical conclusions based on the data we were able to collect.
From the regular wave tests plots of the bow accelerations (Figure 3-21) and
LCB accelerations (Figure 3-22) for 25 knots are shown below. The output of the acceleration transducer is given in terms of the acceleration of gravity, thus it is not a transfer function. The plots for the remaining speeds are given in Annex C. Error bars are included but the experimental error was so small that the uncertainty is almost zero.
Figure 3-21: Bow accelerations for ship speed of 25 knots
From the plots the bow accelerations for both models seem very similar while
Model B has a higher LCB acceleration in the resonance region. This data seems to contradict the spring-mass-damper analogy and requires further study to understand.
From the irregular wave tests, selected time histories of the bow and LCB accel-
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Figure 3-22: LCG accelerations for ship speed of 25 knots erations are shown in Figure 3-23. The full time histories of vertical accelerations for each test case are given in Appendix D. From Figure 3-23 the difference in performance between the models is now abundantly clear. Consistent with the damped mass-spring analogy Model B pitches less by the bow, something that was not clear from the regular wave tests. However, the LCB acceleration, also the acceleration in heave ( 𝑥 ¨
3
), are much larger for Model B.
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Figure 3-23: (a) Selected time history for bow accelerations (b) Selected time history for LCG accelerations (c) Selected time history of water surface elevation
These differences are confirmed by a comparison of the variance from the time histories. Here it is shown that Model B has less bow acceleration in all but one test condition and greater LCB acceleration in every test condition. However, the relative magnitude of the two variances are noteworthy. In each case the magnitude of the bow acceleration variance is five times that of the LCB acceleration variance.
So while Model B does exhibit larger LCB accelerations, the impact on performance is likely small.
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Speed (kts) Model A 𝑚
0 𝐴
20
25
30
Test Series 5 (Sea State 4)
Bow/LCB Model B
0.021/0.002
0.023/0.003
0.021/0.004
𝑚
0 𝐴
Bow/LCB
0.022/0.003
0.019/0.004
0.016/0.005
Speed (kts)
20
25
30
Model A
Test Series 6 (Sea State 6) 𝑚
0 𝐻
Model B 𝑚
0 𝐻
0.037/0.004
0.056/0.007
0.062/0.010
0.030/0.005
0.049/0.010
0.056/0.015
Difference
2.54%/38.18%
-21.31%/24.75%
-26.77%/20.44%
Difference
-20.25%/13.33%
-12.52%/27.09%
-10.23%/31.37%
Table 3.10: Summary of variance of acceleration response in irregular waves
The experimental procedure designed for this study has successfully helped to characterize the performance differences between Model A and Model B. It was clearly shown in Figure 3-11 that the inverted bow causes a lower residuary resistance when key parameters are held constant between the models. This result is consistent with the findings in previous research presented in Chapter 1 and serves as a good verification of the design. Then, by conducting the model tests in a side-by-side configuration we were able to compare the motions of the two ships in the time domain. By studying the variance of the time history of the motions we see that Model B generally demonstrates less pitch motion and bow acceleration and more heave motion and
LCB acceleration.
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Computational fluid dynamics (CFD) plays an important role in predicting the hydrodynamic performance of ships. When correctly implemented the use of CFD dovetails nicely with experimental model tests, providing results in a fraction of the time and cost. However, the value of a CFD code greatly depends on the level of confidence in the results. To assess the accuracy and reliability of computational simulation for the inverted bow we conducted a numerical verification and validation for Aegir. Verification is the assessment of numerical accuracy and validation is a comparison with experimental data. To verify that Aegir gives a good solution for the inverted bow, domain and spatial convergence tests were preformed and the solutions were tested for convergence. To validate the solution, the results from Aegir were compared to the experimental data presented in Chapter 3. This chapter will discuss the preparation required to run a simulation in Aegir, the process of numerical validation and the results of the validation.
The numerical analysis for this projected was conducted using the seakeeping code Aegir from Navatek Ltd. Aegir is a "panel-less," high order, boundary-element method for the computation of steady and unsteady forces and motions of systems operating at or near the free surface. The Aegir GUI provides a straightforward guide
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to follow in creating the namelist (.nml) file that Aegir uses to execute the simulations. Numerical simulations in Aegir were run in the steady state, which represents calm water resistance tests. Where ever possible I have tried to replicate the exact conditions that were experienced in the towing tank for a meaningful comparison.
For each hull that was tested during this phase of the project the input file had to be prepared to the specifications of Aegir. Aegir uses the OpenNurbs 4.0 library to read CAD files in the native Rhinoceros format. The AegirGUI, which was used for this project, is only compatible with Rhinoceros version 4.0 files. All geometry preperation was compelted in Rhinoceros version 5.0 and then the file extension changed for compatability. Aegir requries that the surfaces are valid trimmed non-uniform rational B-spline (NURBS) surfaces that are grouped into a single layer. Trimmed
NURBS surfaces are valid if they have are of bi-quadratic or higher degree and the normal vector points out of the fluid.
From CAESES the hull is exported in .igs format and imported into Rhinoceros.
To achieve the best results from Aegir, the following steps were taken for each hull in
Rhinoceros:
• Use only the port side of the hull.
• Loft the hull into as few patches as possible, giving each patch a unique name.
Place all patches into a single layer.
• Translate the hull so that the origin is at midships, on the centerline and at the design water line.
• Scale the model to the correct size. Verify that the LWL and displacement match the baseline hull.
• Orient the normal vectors for each patch. The u vector must point downstream, the v-vector must point from port to starboard, and thus the normal vector points out of the fluid and into the hull.
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• Re-parametrize each patch from 0 to 1 in both the u and v directions
• Remove the transom patch. This allows for the correct free surface type to be selected in Aegir.
The coordinate system in Aegir varies slightly from the coordinate system presented in Chapter 1. This was easily corrected by changing the sign of the sinkage and trim when comparing numerical and experimental data.
Figure 4-1: Aegir coordinate system
Verification is a purely numerical exercise to check that the solution accurately agrees with that of the continuous mathematical model [12]. It is the process of identifying and eliminating sources of numerical error. We followed an iterative approach to first eliminate errors resulting from the domain size. With the appropriate domain, we then iterated on the mesh density assuming that the numerical errors will vanish when the discretization is refined. We made successive grid refinements until we obtained a grid-independent result.
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In CFD it is important to find a domain size that is large enough represent an infinite surface but also small enough to limit computational time. To determine this ideal domain we ran a series of steady state simulations and incrementally increased the domain size, fixing all other paramters. Convergence of the wave resistance solution vs. Froude number was apparent as the domain size increased. The smallest domain size that captured the solution was chosen for testing. A summary of the domain convergence tests, where the domain is listed as multiples of the LWL in the upstream, downstream, and outer directions, is shown below in Table 4.1.
Domain
Panel Length
Sinkage & Trim
Free Surface Setting
Mass
CG (x,y,z)
.8L - 1.6L - .8L
0.1717 ft (LWL/30)
Fixed at 0
Monohull w/symmetry about X-Z, no transom
.5638 slugs
(0.06,0,0)
1L - 2L - 1L
0.1717 ft (LWL/30)
Fixed at 0
Monohull w/symmetry about X-Z, no transom
.5638 slugs
(0.06,0,0)
1.2L - 2.4L - 1.6L
0.1717 ft (LWL/30)
Fixed at 0
Monohull w/symmetry about X-Z, no transom
.5638 slugs
(0.06,0,0)
Table 4.1: Summary of domain convergence tests
As a result of the domain convergence tests, the domain size of L - 2L - L was selected. A plot of the domain sensitivity results is given below in Figure 4-2. For this domain the result is stable and only shows a very slight difference with the next larger domain. To increase from the chosen domain to the next larger domain the computational time doubles. For the purposes of this investigation the chosen domain is the best option.
The discretization of the parameter space imposes a great influence on the accuracy of the calculations and visual outputs. The more the surface is discretized, the more defined it becomes and the more the solution should converge to the accuracy of the actual physical surface. However, fine grid discretization requires greater computation time. When running numerical simulations, realistic accuracy needs to be met without sacrificing too much computational power. This is why spatial, also
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Figure 4-2: Domain converegnce tests for Hull B referred to as grid, convergence tests are necessary. To determine the proper spatial discretization for a specific hull form and free surface, the grid will be broken into more, smaller, panels. For each change in panel size, a simulation will be run and recorded. Domain size is held constant and applied to the parameter space. The adjustable variable can be referred to as the mesh density, depicted in the schematic below (Figure 4-3).
Figure 4-3: Mesh density
The domain size is held constant and the panel length is varied in reference to
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the LWL. All other variables are held constant. A summary of the settings is shown below in Table 4.2.
Domain
Panel Length
Sinkage & Trim
Free Surface Setting
Mass
CG (x,y,z)
L - 2L - L
LWL/20
Fixed at 0
Monohull w/symmetry about X-Z, no transom
.5638 slugs
(0.06,0,0)
L - 2L - L
LWL/25
Fixed at 0
Monohull w/symmetry about X-Z, no transom
.5638 slugs
(0.06,0,0)
L - 2L - L
LWL/30
Fixed at 0
Monohull w/symmetry about X-Z, no transom
.5638 slugs
(0.06,0,0)
Table 4.2: Summary of spatial convergence tests
As a result of the spatial convergence tests, the panel size of 0.1717 ft (LWL/30) was selected. A plot of the spatial sensitivity results is given below in Figure 4-
4. For this domain and panel size the result is stable and only shows a very slight difference with the next smaller panel size. To decrease from the chosen panel size to the next smaller panel size the computational time doubles. For the purposes of this investigation the chosen panel size is the best option.
Figure 4-4: Spatial converegnce tests for Hull B
With numerical verification complete we can now move on to the validation with confidence in the solution from Aegir. A picture of the final discretized domain is
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shown in Figure 4-5.
Figure 4-5: Domain and spatial discretization vizualization
The AEGIR GUI is run by selecting the compiled python file of the latest release of
AEGIR. All of the simulations were conducted with standard English units, consistent with the towing tank tests. It was important to replicate as best as possible the geometric properties of the physical model including the LWL,
𝐴 𝑤𝑝 and underwater volume. Likewise we must also accruately represent the conditions observed in the towing tank. To do so the code is set to an infinite depth, and thus ignoring shallow water effects, as well as the density and kinematic viscosity appropriate for the correct temperature of the water in the towing tank. Finally the numerical model was free to move in pitch and heave only. Motions in the surge, sway, yaw, and roll directions were fixed.
Validation will demonstrate if the solution from Aegir is an accurate representation of the physical reality. Essentially validation is proof of applicability by comparing the numerical results to benchmark data. For this study the benchmark data was the results obtained from our model tests.
Figure 4-6 shows a plot of the residuary resistance from the model tests and the wave making resistance from Aegir. The wave making resistance cannot be calculated directly from the model tests as demonstrated in Chapter 3. Thus the residuary
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Figure 4-6: Resistance validation for Hull B resistance and the wave making resistance differ by the effects of viscous pressure and eddy formation. However, for the purposes of validation we can consider the global shape of these two different quantities. In this regard Aegir provides an accurate prediction of the resistance.
To help explain any differences in the resistance prediction we can also use the sinkage and trim of the model for a direct validation. In Figure 4-7 the experimental and numerical sinkage is plotted and in Figure 4-8 the experimental and numerical trim is plotted. For sinkage Aegir predicts the global trends very well and is almost correct on the magnitude; likely not a source of error in the resistance calculation.
However, for the trim the predicted trend is accurate but the magnitude differs by as much as 30% at high speeds.
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Figure 4-7: Sinkage validation for Hull B
Figure 4-8: Trim vlidation for Hull B
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Using CDF allows you to "step inside the flow" and see things that are not easily observable in towing tank tests. One such result is the profile of the water surface elevation along the hull or at a certain lateral distance from the hull. This is particulary important to the study of the resistance of the hull as it gives insight to the wave that the hull is making as it moves through the water.
Figure 4-9: Normalized wave profile for Froude number of .201
Figure 4-9 shows the normalized wave profiles along the free surface of the two simulations where the FP at X/L = 0. On the y-axis the wave height is normailzed by the length and then multiplied for 100 for ease of viewing. First, we see the effect of the underwater portion of the inverted bow; the peak of the bow wave is at the
FP. This results in a cancellation effect for the bow wave, similar to the function of a bulbous bow. Thus if we consider the height of the bow wave and the area under the bow wave curve both are lower for the inverted bow. Inspection of these two quantites helps to expain why the resistance is lower. The same happens at each speed in the
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simulation, providing concurrance with the resistacnce calculation. As expected the waves then follow generally the same pattern along the rest of the hull. The plot does not show aft of the transom as the simulation is often not stable in this region, a know weakness of CFD. The plots for the remaining Froude numbers are given in
Appendix F.
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The goal of this project was an to be an unbiased hydrodynamic study of the resistance and seakeeping effects of a U.S. Navy surface combatant with an inverted bow.
Through an experimental and numerical testing program this study demonstrated that the inverted bow as designed exhibts less pitching motion and a lower vertical acceleration on the bow in head seas. It is my hope that the results presented in this study will become part of the larger body of knowledge on the subject of inverted bows and be useful to the ship design community.
Motivated by previous academic studies and existing ship designs this project explored the resistance and seakeeping effects of a U.S. Navy surface combatant that was modified to include an inverted bow. To ensure a meaningful comparison we established that the LWL and displacement must be held constant between the baseline and modified hull. Once the similitude requirements were met models were built for testing. The testing program was designed to make best use of the available facilities to compare the performance of the two hulls. The resistance and regular wave tests, which are easily repeatable, were conducted with a single model at a time. The irregular wave tests were instead conducted with the models in a side by side configuration in a much larger towing tank. This approach facilitated a direct comparison between
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the two models because each model saw the exact same waves at the same time. Once the experimental data was processed it was used for validation of the seakeeping code
Aegir.
The findings of this study have been presented in Chapter 3 and Chapter 4 as well as the appendices. With these findings we can now address the research questions proposed at the beginning of the paper.
The first result from the towing tank experiments was the reduced resistance of the inverted bow. This finding was consistent with the reviewed literature and served to validate that the hull shape was sufficient for further study. The theoretical framework for modeling the motions of a ship with a damped spring-mass system was presented in Chapter 1. It was proposed that the inverted bow acts to "soften" the spring constant of the system, increasing the amplitude of the motion but decreasing the acceleration. For the simple case of linear ship motions in regular waves the findings of this study support this hypothesis. From the regular wave tests the pitch and heave transfer functions show a larger amplitude in the area of resonance. However, the results of the irregular wave tests partially contradict the hypothesis. We see that in irregular waves the inverted bow actually demonstrates less pitching motion and less bow acceleration. The linear model of regular waves does not take into account wave height sequencing or the irregular arrangement of wave peaks and troughs. Rather than responding to each wave it encounters the inverted bow instead slices through the waves. This leads to the conclusion that there is a non-linear interaction between the inverted bow and the irregular waves that improves the seakeeping performance.
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As demonstrated in Chapter 4, Aegir does an adequate job of predicting the performance of the inverted bow in steady state. Aegir almost exactly modeled the sinkage of the inverted bow correctly, but struggled a bit with the trim. As shown in the paper the pitching motion is effected by the inverted bow and Aegir was able to produce the same global shape, but not the correct magnitude. However, when comparing the wave resistance from Aegir and the residuary resistance from model tests the results appear as expected given the differences in the two qualtities. Overall, the predictions of Aegir in the steady state are sufficient that work on this project should include validation in the time domain.
A limitation of the findings is the bow acceleration data. I speculate that the results could be criticized as it was not a "like for like" comparison in the truest sense. The models have different values for
𝐼
55 and the distance from the LCB to the bow accelerometer was different. However the shorter deck length of Model B is a consequence of using an inverted bow when the models have the same displacement, waterline length and draft. The two hulls simply have a different shape in the bow region above the waterline. However, from watching the tests, and slow motion video, it is clear that Model B accelerates less by the bow. Only with the addition of specific owner/operator requirements would we be able to draw any further conclusions.
There are two parallel paths that can make use of the findings of this study to continue research into inverted bows. First, more experimental data can be collected as the models are already build. Second, the numerical investigation can be expanded to validate Aegir for the time domain solution and optimize the Inverted Bow Hull
Series.
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Using the two models from this study, further seakeeping tests are required beyond head seas alone to characterize the performance of the inverted bow. Beyond simply collecting head seas data at more speeds and in more sea states, with the side-byside configuration tests could be conducted in following seas as well. However, due to the significantly reduce the numbers of encounters observed per run these tests would require additional time in the towing tank to complete. Additionally, the performance of the Inverted Bow Hull Series in the roll direction is of significant interest. Using a planar motion mechanism (PMM), with the model free to roll and aligned perpendicular to the wave direction, a zero speed roll test for each model could be used to characterize the transverse stability of the inverted bow. It is expected that both models would exhibit similar performance in roll - further decoupling the inverted bow shape from the tumblehome hull that has traditionally accompanied it.
Finally, tests could be conducted for a single model in quartering seas again utilizing the PMM. As with the following seas tests the scope is limited by the tank geometry.
A small range of angles could be tested due to the width of the tank again with a limited number of encounters. This would allow us to identify the conditions of synchronous roll and "surf riding," perceived areas of instability for an inverted bow.
With a validated seakeeping code we can then work to optimize the hull shape for seakeeping performance. Here the benefit of a parametric model will truly be realized. The global parameters discussed earlier in the paper can be varied over a specified range, and in various combinations using an optimization routine to produce the ideal shape for the inverted bow. Additionally, as this study is for a Navy surface combatant it is very likely that the ship would require a bulbous bow to accommodate a sonar array. The parametric model could be updated to include this feature and could be then used in a similar optimization.
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The pitch gyradius 𝑘
5 was found for each model using the Lamboley method.
Developed by Gilbert Lamboley, this technique involves swinging the models in pitch from pivots a known vertical distance apart [11]. The resultant period for each pivot point is a function of the distance from the pivot to the model center of gravity and the pitch gyradius.
Figure A-1: Schematic of Lamboly pendulum test rig in NAHL
The NAHL has built a pendulum test rig that has two pivot heights 1 inch apart, the value x in figure above. By measuring the periods for each pivot height a set of equations can be created to solve for both the distance to the center of gravity and the pitch gyradius of the model.
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𝑇
1
⎯
⎸
= 2 𝜋
⎸
⎷ 𝑑 2 − 𝑘 2
5 𝑔 × 𝑑 and
𝑇
2
= 2 𝜋
⎯
⎸
⎸
⎷
( 𝑑 − 𝑥 ) 2 + 𝑘 2
5 𝑔 × ( 𝑑 − 𝑥 ) where T = swing period in seconds d = vertical distance from the pivot to the model’s center of gravity x = vertical distance between pivots
These two equations can be solved simultaneously for the distance to the center of gravity, d, and the pitch gyradius 𝑘
5
. To facilitate the simultaneous solution an intermediate quantity, c, is found such that: 𝑔 𝑐 =
4 𝜋 2 𝑥
Then the two unknowns can be found from: 𝑑 = 𝑥 ( 𝑐𝑇
2
2 𝑐 ( 𝑇
2
2 − 𝑇
+ 1)
1
2 ) + 2 and 𝑘
5
=
√︁ 𝑑 𝑥 𝑐 𝑇 2
1
− 𝑑 2 x
Time 1
Time 2
Num Cycles
𝑇
1
𝑇
2 c d 𝑘
5
% LBP
Model A Model B
1 in 1 in
39.01 sec 37.76 sec
40.51 sec 39.17 sec
20 20
1.951 sec 1.888 sec
1.026 sec 1.959 sec
9.780
1 / 𝑠𝑒𝑐
2
9.780
1 / 𝑠𝑒𝑐
2
8.365 in
15.53 in
25.2 %
8.279 in
14.83 in
24.1 %
Table A.1: Gyradius calculation
100

101

102

103

104

The data for each run is contained in an individual file runxx.xlsx
where "xx" corresponds to the run number. Data for each run is imported into the included spreadsheet TruncateWaves.xlsx
for analysis to determine the mean, standard deviation, and variation. Run numbers are restarted at 1 for the tests conducted in the
380 foot towing tank. The first eight runs were used for instrument calibration and runs 9-14 were for verification of the sea spectra. Also included are the start and stop times of each run for the two video files included with the report. It should be noted that the videos are in model-scale time and must be slowed by the square root of 𝜆 to reflect true motions.
105

106

The experimental results for the regular wave tests were already given for the test condition with ship speed of 25 knots in Chapter three. In this appendix are the results of the analysis for the remaining test conditions, with ship speed of 20 and
30 knots. The pitch transfer functions are created using equation 3.19 and the heave transfer functions are created using equation 3.24. The vertical accelerations of the bow and the LCB are given in terms of the acceleration of gravity, the direct output of the instrumentation.
107

As previously mentioned the pitch amplitude was divided by the measured wave slope amplitude to form the linear transfer functions in the frequency domain.
108

As previously mentioned the heave amplitude was divided by the measured wave amplitude to form the linear transfer functions in the frequency domain.
109

As previously mentioned the acceleration transducers are located at different distances from the LCB due to the geometry of the hull and construction of the model.
110

111

112

In this appendix the time histories of the wave amplitude, pitch, heave, and accelerations are displayed for each test condition in irregular waves. Multiple runs were conducted for each test case as listed in Appendix B.5, however only one run is displayed. Note that as the speed increases the duration of the run decreases due to the length of the tank. For each motion plot the time history of the wave elevation and the carriage speed is given as a reference. While the entire run is presented in this appendix, the analysis was only conducted for the time period when the carriage was at a steady speed. The plots were created using a MATLAB script TimeHistory.m
included in the following section.
113

1
%Script written by Jeffrey White - April 2015
2
%MIT Innovative Ship Design Lab (iShip)
3
4
%This script reads excel data files (must be in .xlsx format) created by
5
%the NAHL data acq and plots the time history of the motions with a
6
%reference to the incident wave spectrum and the carriage speed
7
8
%% Read Data from Excel File
9
10
%Note: You must manually change the filename and length of all 10 input
11
%vectors for each run
12
13 filename = 'C:\Users\Jeff\Documents\Thesis\Sea Spectrum\Data29.xlsx' ;
14
15
% Incident wave spectra and the carriage velocity
16 u = xlsread(filename, 'B7:B3006' ); % Velocity data
17 x = xlsread(filename, 'C7:C3006' ); % Wave data (in)
18
19
% Model A
20 ya = xlsread(filename, 'H7:H3006' ); % Pitch data (deg)
21 za = xlsread(filename, 'I7:I3006' ); % Heave data (in)
22 aba = xlsread(filename, 'L7:L3006' ); % Bow acceleration data (g)
23 ala = xlsread(filename, 'M7:M3006' ); % LCB acceleration data (g)
24
25
% Model B
26 y = xlsread(filename, 'E7:E3006' ); % Pitch data - degrees(time domain)
27 z = xlsread(filename, 'F7:F3006' ); % Heave data - inches
28 al = xlsread(filename, 'J7:J3006' ); % Bow acceleration data - g's
29 ab = xlsread(filename, 'K7:K3006' ); % LCB acceleration data - g's
30
31
%% Plot Time Histories
32
114

33
%Note: You must manually change the length of the x axis for subplot 3 on
34
%both figures to accomodate the full run.
The axis must be specified to
35
%control the y axis for readability
36
37
% Plot pitch and heave time history
38 figure(1)
39 subplot(3,1,1)
40 plot(t,ya, 'r' )
41 hold on
42 plot(t,y, 'b' )
43 xlabel( 'time (s)' )
44 ylabel( 'Pitch (deg)' )
45 legend( 'Model A' , 'Model B' )
46
47 subplot(3,1,2)
48 plot(t,za, 'r' )
49 hold on
50 plot(t,z, 'b' )
51 xlabel( 'time (s)' )
52 ylabel( 'Heave (in)' )
53 legend( 'Model A' , 'Model B' )
54
55 subplot(3,1,3)
56 plot(t,x)
57 hold on
58 plot(t,u)
59 legend( '\zeta_0 (in)' , 'Speed (ft/s)' )
60 ylabel( 'Reference' )
61 xlabel( 'time (s)' )
62 axis([0 60 -2 6])
63
64
% Plot acceleration time history
65 figure(2)
66 subplot(3,1,1)
67 plot(t,aba, 'r' )
68 hold on
115

69 plot(t,ab, 'b' )
70 xlabel( 'time (s)' )
71 ylabel( 'Bow Acceleration (g)' )
72 legend( 'Model A' , 'Model B' )
73
74 subplot(3,1,2)
75 plot(t,ala, 'r' )
76 hold on
77 plot(t,al, 'b' )
78 xlabel( 'time (s)' )
79 ylabel( 'LCB Acceleration (g)' )
80 legend( 'Model A' , 'Model B' )
81
82 subplot(3,1,3)
83 plot(t,x)
84 hold on
85 plot(t,u)
86 legend( '\zeta_0 (in)' , 'Speed (ft/s)' )
87 ylabel( 'Reference' )
88 xlabel( 'time (s)' )
89 axis([0 60 -2 6])
116

117

118

119

120

121

122

The primary benefit of the side-by-side model test configuration is the ability to conduct all analysis in the time domain. However, as noted early in the paper one of the objectives is to demonstrate that the inverted bow shape generates significant non-linear ship motions that cannot be captured by linear theory alone. Thus the following spectral analysis was conducted to highlight these differences.
During the regular wave tests we were able to experimentally determine the linear transfer functions for pitch and heave in the frequency domain. From the irregular wave tests we produced the time histories of the pitch and heave motion. In order to make a comparison of the calculated linear motions and the measured non-linear motions we had to turn to spectral analysis to create pitch and heave energy spectra.
We make use of the FFT to convert time domain data into the frequency domain.
To ensure a like-for-like comparison we used the measured sea spectrum from the irregular wave tests for both the measured and calculated motions. The calculated motion energy spectra is determined by multiplying the sea spectrum by the linear motion transfer function squared (also called the Response Amplitude Operator). The measured motion energy spectrum is determined from the FFT of the time history of the motion. The following sections describes the procedure in detail.
123

The spectral analysis was conducted using the script FFT.m
which was written for this project. The full script is included in a later section.
• Set the length of the FFT. This must be a multiple of
2 𝑛 where n is an integer.
Depending on the length of the individual data run this was either set to 2048 or 4096. If the length of the data set is less than the length of the FFT you can add additional zeroes to the end of the data set, but if too many are used the results will be affected. Use this technique sparingly!
• The FFT is symmetric, so we only the first half (up to the Nyquist frequency) and simply multiply the spectral ordinate by 2 to capture the energy of the full wave spectrum. Also, we must use the magnitude of the FFT to calculate the spectral ordinate.
• Wave data was collected from a carriage mounted gauge, thus our FFT is actually in the encounter frequency domain. To calculate the spectral ordinate we must first consider the frequency domain, ignoring the effects of the carriage speed. To do so we must solve for the positive roots of Equation 3.4.
• Wave and wave slope spectra can then be calculated using Equations 3.21
and 3.22. Measurements taken from these spectra are used to generate idealized JONSWAP spectra and are plotted for reference.
• Using a cubic spline interpolation of the linear transfer functions we create transfer functions at the same x values as the wave spectrum, required for multiplication. Now we calculate the motion energy spectrum as described above.
• Now we use the FFT on the time history of the motion to create the measured energy spectrum.
• Finally to compare the two spectra, we calculate the zeroth spectral moment as the area under the curve for each.
124

To ensure that the correct spectrum was being used for each calculation, a plot was created for each run. In Figure E.1.2 the measured wave energy spectrum is plotted in the encounter frequency domain for Run 29. Additionally the wave and wave slope spectra are plotted in the frequency domain (dashed line) and the encounter frequency domain (thick line).
Figure E-1: Wave energy spectra for run 29
The starting point for creating the family of idealized spectra is the wave energy spectrum in the frequency domain. To convert from the frequency domain to the encounter frequency domain in head seas is a two step process. First, the wave energy spectrum must be shifted along the frequency axis to cover a different range of frequencies when observed from the ship. This is because the frequency interval 𝛿𝜔 𝑒 now includes the effect of the carriage speed. Second, transforming the spectrum into the moving frame of reference does not change the energy contained within that band of frequencies. To preserve an equivalent area after the spreading along the
125

frequency axis the spectral ordinate must decrease in proportion to 𝛿𝜔 𝑒
. The result is a a spectrum that is shifted to a higher, wider range of frequencies with reduced height [13].
Returning to the wave energy spectrum in the frequency domain we can now determine the wave slope spectra. Based on the assumption that the wave slope of a regular sine wave also varies sinusoidally the wave slope spectrum can be calculated directly from the wave amplitude energy spectrum. The JONSWAP wave slope spectrum is calculated by multiplying the wave amplitude energy spectrum by 𝜔
4 / 𝑔
2
. The wave slope spectrum has the same sharp peak, but now a greater comparative importance is given to high wave frequencies. To then convert the wave slope spectrum to the encounter frequency domain a similar procedure as described above is followed.
126

127

128

129

130

131

1
%Script written by Jeffrey White - April 2015
2
%MIT Innovative Ship Design Lab (iShip)
3
4
%This script reads excel data files (must be in .xlsx format) created by
5
%the NAHL data acq and plots motion energy specta
6
7
%% Read Data from Excel File
8
9 close all
10 clear all
11
12
%Note: You must manually change the filename for each run
13
14
% Input velocity data
15 filename = 'C:\Users\Jeff\Documents\Thesis\Sea Spectrum\Data26.xlsx' ;
16 u = xlsread(filename, 'B7:B4507' ); % Velocity data
17
18
% Create time vector
19
Fs = 50; % Sampling frequency
20
Nsamps = length(u); % Length of data collection (in seconds)
21 t = (1/Fs)*(1:Nsamps); % Time vector of 1 second
22
23
% Use plot of velocity to set truncation limits
24 figure(1)
25 plot(t,u)
26 pts = ginput(2)
27 xstart = round(pts(1,1));
28 indexstart = (xstart*Fs)+7;
29 xend = round(pts(2,1))
30 indexend = (xend*Fs)+7;
31
32
% Truncate velocity vector
132

33 u = u(indexstart:indexend);
34
U = mean(u) % model speed in ft/s
35
36
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
37
%Input the rest of the data using the truncation limits established above
38
39
% Wave Data
40 x = xlsread(filename, 'C7:C4507' ); % Wave data - inches(time domain)
41 x = x(indexstart:indexend);
42
43
% Model A
44 ya = xlsread(filename, 'H7:H4507' ); % Pitch data - degrees(time domain)
45 ya = ya(indexstart:indexend);
46 za = xlsread(filename, 'I7:I4507' ); % Heave data - inches
47 za = za(indexstart:indexend);
48 aba = xlsread(filename, 'L7:L4507' ); % Bow acceleration data - g's
49 aba = aba(indexstart:indexend);
50 ala = xlsread(filename, 'M7:M4507' ); % LCB acceleration data - g's
51 ala = ala(indexstart:indexend);
52
53
% Model B
54 y = xlsread(filename, 'E7:E4507' ); % Pitch data - degrees(time domain)
55 y = y(indexstart:indexend);
56 z = xlsread(filename, 'F7:F4507' ); % Heave data - inches
57 z = z(indexstart:indexend);
58 al = xlsread(filename, 'J7:J4507' ); % Bow acceleration data - g's
59 al = al(indexstart:indexend);
60 ab = xlsread(filename, 'K7:K4507' ); % LCB acceleration data - g's
61 ab = ab(indexstart:indexend);
62
63
%% Use FFT to generate wave and pitch Spectral Density Functions
64 g = 32.174;
65
SR = 80; % Scale Ratio
66
67
% Convert wave and heave data from inches to ft
68 x = x./12;
133

69 z = z./12;
70 za = za./12;
71
72
% Convert pitch data from degrees to rad
73 y = y.*(pi/180);
74 ya = ya.*(pi/180);
75
76
% Take fft, padding with zeros so that length(X) is equal to nfft
77 nfft = 2048;
78
X = fft(x,nfft);
79
Y = fft(y,nfft);
80
Z = fft(z,nfft);
81
YA = fft(ya,nfft);
82
ZA = fft(za,nfft);
83
AB = fft(ab,nfft);
84
AL = fft(al,nfft);
85
ABA = fft(aba,nfft);
86
ALA = fft(ala,nfft);
87
88
% FFT is symmetric, throw away second half
89
X = X(1:nfft/2);
90
Y = Y(1:nfft/2);
91
Z = Z(1:nfft/2);
92
YA = YA(1:nfft/2);
93
ZA = ZA(1:nfft/2);
94
AL = AL(1:nfft/2);
95
AB = AB(1:nfft/2);
96
ALA = ALA(1:nfft/2);
97
ABA = ABA(1:nfft/2);
98
99
% Take the magnitude of fft
100 mx = abs(X);
101 my = abs(Y);
102 mz = abs(Z);
103 mya = abs(YA);
104 mza = abs(ZA);
134

105 mal = abs(AL);
106 mab = abs(AB);
107 mala = abs(ALA);
108 maba = abs(ABA);
109
110
% Frequency vector and interval
111 f = (1:nfft/2)*Fs/nfft;
112 df = Fs/nfft;
113 w = f*2*pi;
114 dw = df*2*pi; %From Lloyd p.98 (eq 4.11)
115
116
% Convert w from encounter domain to freq domain
117 for i=1:length(w)
118
119
120 p1 = [U/g 1 -w(i)]; %polynomical for encounter freq. equation Lloyd p. 146 (eq. 7.3) r1 = roots(p1); %solve quadratic equation wa(i) = r1(r1>0); %keep only the positive root
121 end
122 wa = wa';
123
124
% Calculate the Spectral Ordinate
125
Sw = (1./(2.*dw))*(2*mx./nfft).^2;
126
Sa = Sw.*(wa.^4/g^2);
127
128
%
129
ModelBSx5 = (1./(2.*dw))*(2*my./nfft).^2;
130
ModelASx5 = (1./(2.*dw))*(2*mya./nfft).^2;
131
132
ModelBSx3 = (1./(2.*dw))*(2*mz./nfft).^2;
133
ModelASx3 = (1./(2.*dw))*(2*mza./nfft).^2;
134
135
ModelBbow = (1./(2.*dw))*(2*mab./nfft).^2;
136
ModelBlcb = (1./(2.*dw))*(2*mal./nfft).^2;
137
ModelAbow = (1./(2.*dw))*(2*maba./nfft).^2;
138
ModelAlcb = (1./(2.*dw))*(2*mala./nfft).^2;
139
140
% Zeroth moment is the variance of the time history
135

141 m0 = var(x)
142 m0ya = var(ya)
143 m0y = var(y)
144 m0za = var(za)
145 m0z = var(z)
146 m0aba = var(aba)
147 m0ab = var(ab)
148 m0ala = var(ala)
149 m0al = var(al)
150
151
Hsig = 4*sqrt(m0);
152 m1 = sum(w*Sw)*dw;
153 wmean = m1/m0;
154
Tmean = 2*pi/wmean;
155
156
% Find the modal period of the FFT data
157
[P, I] = max(Sw);
158 wm = w(I);
159
Tm = 2*pi/wm;
160
161
%% Generate the idealized spectrum
162
163 wplot = 1:.005:20;
164 weplot = wplot+(wplot.^2*(U/g));
165
166
% Need to convert the encounter modal frequency to modal frequency
167 p = [U/g 1 -wm]; %polynomical for encounter freq. equation Lloyd p. 146 (eq. 7.3)
168 r = roots(p); %solve quadratic equation
169 w0 = r(r>0); %keep only the positive root
170
T0 = 2*pi/w0;
171
172 for j=1:length(wplot)
173
174
175
176 if wplot(j)<w0 sigma = 0.07; else sigma = 0.09;
136

177 end
178
A = 487.3*Hsig^2/T0^4;
179
B = 1949/T0^4;
180
C(j) = 3.3.^exp(-((wplot(j)-w0).^2/(2*sigma^2*w0^2)));
181
182
SB(j) = A./wplot(j).^5.*exp(-B./wplot(j).^4);
183
SBa(j) = SB(j).*(wplot(j).^4/g^2);
184
SJ(j) = .658.*C(j).*SB(j);
185
SJa(j) = .658.*C(j).*SBa(j);
186
187
SJE(j)= SJ(j)/(1+(2.*wplot(j).*U)/g);
SJEa(j) = SJa(j)/(1+(2.*wplot(j).*U)/g);
188
189 end
190
191 figure (2)
192 plot(w,Sw)
193 hold on
194 plot(wplot,SJ, '--' )
195 plot(wplot,SJa, '--' )
196 plot(weplot,SJE, 'linewidth' ,1.5)
197 plot(weplot,SJEa, 'linewidth' ,1.5)
198 axis([0 20 0 .0015])
199 title( 'Wave Energy Spectrum' );
200 xlabel( '\omega_e (rad/s)' );
201 ylabel( 'Spectral Ordinate' );
202 legend( 'Data' , 'Wave' , 'Wave Slope' , 'Encounter Wave' , 'Encounter Wave Slope' );
203
204
%% Develop RAOs from Regular Wave Tests
205
206
% Import transfer function from regular wave tests
207 if U<4
208
209 weTF = [5 6.66 7.98 9.40 10.9 12.49 14.18 15.96 17.83];
PitchTF = [1 1.096 1.144 1.091 0.737 0.276 0.103 0.023 0.008];
210
212
HeaveTF = [1 0.936 0.940 1.061 0.976 0.399 0.130 0.118 0.058];
211 elseif U>4 && U<5 weTF = [2 4 7.23 8.72 9.51 10.37 11.18 12.06 13.90 15.85 17.92 20.11];
137

213
214
215 else
PitchTF = [1 1 1.167 1.277 1.319 1.179 0.820 0.610 0.268 0.089 0.014 0.010];
HeaveTF = [1 1 1.037 1.167 1.459 1.520 1.330 1.064 0.447 0.145 0.110 0.146];
216
217
218 weTF = [2 4 6 7.79 8.61 9.45 9.89 10.34 10.8 11.26 12.21 13.21 15.29 17.5 19.87 22.37];
PitchTF = [1 1 1.1 1.242 1.288 1.383 1.208 1.164 0.931 0.853 0.630 0.438 0.175 0.056 0.013 0.009];
HeaveTF = [1 1 1 1.108 1.201 1.463 1.622 1.764 1.717 1.607 1.288 0.832 0.274 0.073 0.048 0.059];
219 end
220
221
% Build a 1x1024 vector of transfer functions
222 g1 = csapi(weTF, PitchTF); %Cubic spline interpolation of Pitch TF
223 g2 = csapi(weTF, HeaveTF); %Cubic spline interpolation of Heave TF
224
225 for j=1:length(w)
226
227
PTF(j) = fnval(g1, w(j)); %Pick pitch TF values at each w
HTF(j) = fnval(g2, w(j)); %Pick heave TF values at each w
228 end
229
230
% Build Pitch and Heave RAO vectors 1x1024
231
PRAO = PTF.^2;
232
HRAO = HTF.^2;
233
%% Build Energy Spectrum
234
PEScalc = PRAO'.*Sa; % Calculated pitch energy spectrum
235
HEScalc = HRAO'.*Sw; % Calculated heave energy spectrum
236
237
% Set desired lower and upper bounds of the integral
238 val1 = 4;
239 val2 = 14;
240
241
% Find the index of the closest values, use this for integration
242 tmp1 = abs(w-val1);
243 tmp2 = abs(w-val2);
244
[idx1 idx1] = min(tmp1); % index of closest value
245
[idx2 idx2] = min(tmp2); % index of closest value
246 lower = w(idx1); % closest value
247 upper = w(idx2); % closest value
248
138

249
% Use indicies to make vectors over desired interval
250 desiredX = w(idx1:idx2);
251
252 desiredY5A = ModelASx5(idx1:idx2);
253 desiredY5B = ModelBSx5(idx1:idx2);
254 desiredY5C = PEScalc(idx1:idx2);
255
256 desiredY3A = ModelASx3(idx1:idx2);
257 desiredY3B = ModelBSx3(idx1:idx2);
258 desiredY3C = HEScalc(idx1:idx2);
259
260 desiredYAB = ModelBbow(idx1:idx2);
261 desiredYAL = ModelBlcb(idx1:idx2);
262 desiredYABA = ModelAbow(idx1:idx2);
263 desiredYALA = ModelAlcb(idx1:idx2);
264
265
% Solve Integrals
266
ModelASx5Area = trapz(desiredX,desiredY5A)
267
ModelBSx5Area = trapz(desiredX,desiredY5B)
268
PEScalcArea = trapz(desiredX,desiredY5C)
269
270
ModelASx3Area = trapz(desiredX,desiredY3A)
271
ModelBSx3Area = trapz(desiredX,desiredY3B)
272
HEScalcArea = trapz(desiredX,desiredY3C)
273
274
ModelBbowArea = trapz(desiredX,desiredYAB)
275
ModelBlcbArea = trapz(desiredX,desiredYAL)
276
ModelAbowArea = trapz(desiredX,desiredYABA)
277
ModelAlcbArea = trapz(desiredX,desiredYALA)
278
279
%% Make Plots
280
281
% Build bar graphs
282 j=1;
283 i=1;
284 while i<length(Sa)-4
139

285
286
287
288
289
290
291
292
293
294 end
295
%%
296
%%
SwAvg(j) = (Sw(i) + Sw(i+1) + Sw(i+2) + Sw(i+3))/4;
SaAvg(j) = (Sa(i) + Sa(i+1) + Sa(i+2) + Sa(i+3))/4;
Sx5Avg(j) = (ModelBSx5(i) + ModelBSx5(i+1) + ModelBSx5(i+2) + ModelBSx5(i+3))/4;
PESAvg(j) = (PEScalc(i) + PEScalc(i+1) + PEScalc(i+2) + PEScalc(i+3))/4;
Sx3Avg(j) = (ModelBSx3(i) + ModelBSx3(i+1) + ModelBSx3(i+2) + ModelBSx3(i+3))/4;
HESAvg(j) = (HEScalc(i) + HEScalc(i+1) + HEScalc(i+2) + HEScalc(i+3))/4; wAvg(j) = w(i); j=j+1; i=i+2;
297
298
299
% Plot of the PITCH motion energy spectrum
300 figure(3);
301 set(gcf, 'PaperPositionMode' , 'manual' );
302 set(gcf, 'PaperUnits' , 'inches' );
303 set(gcf, 'PaperPosition' , [2.25 0.5 4 10]);
304
305 subplot(3,1,1),bar(wAvg,SaAvg,1, 'blue' ) , hold on, plot(weplot,SJEa, 'k' , 'linewidth' ,2) , axis([2 20 0 .0005]), ylabel( 'S_\alpha [rad^2/(rad/s)]' ),legend( 'Data' , 'Idealized' )
306 x1 = .5;
307 y1 = .00035;
308 str1 = '(a)' ;
309 text(x1,y1,str1);
310
311 subplot(3,1,2);
312 fnplt(g1);
313 ylabel( 'X_5_0/k\zeta_0' );
314 axis([2 20 0 2])
315 legend( 'Transfer Function' )
316 x2 = .5;
317 y2 = 1.75;
318 str2 = '(b)' ;
319 text(x2,y2,str2);
320
140

321 subplot(3,1,3);
322 bar(wAvg,PESAvg,1, 'blue' )
323 hold on
324 bar(wAvg,Sx5Avg,1, 'red' )
325 xlabel( '\omega_e (rad/s)' );
326 ylabel( 'S_X_5[rad^2/(rad/s)]' );
327 axis([2 20 0 .0008])
328 legend( 'Calculated' , 'Measured' )
329 x3 = .5;
330 y3 = .0007;
331 str3 = '(c)' ;
332 text(x3,y3,str3);
333
334
% Plot of the HEAVE motion energy spectrum
335 figure(4);
336
337 set(gcf, 'PaperPositionMode' , 'manual' );
338 set(gcf, 'PaperUnits' , 'inches' );
339 set(gcf, 'PaperPosition' , [2.25 0.5 4 10]);
340
341 subplot(3,1,1),bar(wAvg,SwAvg,1, 'blue' ) , hold on, plot(weplot,SJE, 'k' , 'linewidth' ,2), axis([2 20 0 .001]), ylabel( 'S_\zeta [ft^2/(rad/s)]' ), legend( 'Data' , 'Idealized' )
342 x1 = .5;
343 y1 = .00035;
344 str1 = '(a)' ;
345 text(x1,y1,str1);
346
347 subplot(3,1,2);
348 fnplt(g2);
349 ylabel( 'X_3_0/\zeta_0' );
350 axis([2 20 0 2])
351 x2 = .5;
352 y2 = 1.75;
353 str2 = '(b)' ;
354 text(x2,y2,str2);
355
141

356 subplot(3,1,3);
357 bar(wAvg,HESAvg,1, 'blue' )
358 hold on
359 bar(wAvg,Sx3Avg,1, 'red' )
360 xlabel( '\omega_e (rad/s)' );
361 ylabel( 'S_X_3[ft^2/(rad/s)]' );
362 axis([2 20 0 .0012])
363 legend( 'Calculated' , 'Measured' )
364 x3 = .5;
365 y3 = .0007;
366 str3 = '(c)' ;
367 text(x3,y3,str3);
142

The wave profiles for the remaining Froude numbers are given in this appendix.
In each case Hull B shows a lower bow wave. Note that the scale of the y-axis changes for each graph to best display the plots.
143

144

145

146

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