B-Spline Boundary Element Method for Ships

64 Pages·2013·1.08 MB·English

by  Aditya Mohan Aggarwal

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University of New Orleans Theses and
Dissertations and Theses
Dissertations
8-7-2008
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Aditya Mohan Aggarwal
University of New Orleans
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B-Spline Boundary Element Method for Ships
A Thesis
Submitted to the Graduate Faculty of the
University of New Orleans
in partial fulfillment of the
requirements for the degree of
Master of Science
in
Engineering
Naval Architecture and Marine Engineering
by
Aditya Mohan Aggarwal
B.S. University of New Orleans, 2006
August, 2008
(cid:13)c 2008, Aditya Mohan Aggarwal
ii
Dedication
For their sacrifice, love and support, this thesis is dedicated to Shail, Archana and Yogendra.
Theirfaithinmyabilitieshasbeentheinspirationanddrivingforceforallmyaccomplishments,
past and present.
iii
Acknowledgement
I wish to thank my advisor, Dr. Lothar Birk, Associate Professor of Naval Architecture and
Marine Engineering at the University of New Orleans, who gave me an opportunity to do my
graduate research work under his able and much sought-after mentorship. I have immensely
benefited from his expertise in the field, and his guidance through the course of this thesis.
This thesis would not have been complete without his expert advice and unfailing patience.
I am very grateful to Dr. William Vorus, Professor of Naval Architecture and Marine
Engineering, and Dr. George Ioup, University Research Professor of Department of Physics,
for their participation in my thesis. I would like to take this opportunity to also thank
Dr. Jeffery Falzarano, now a professor at Texas A&M University, who motivated me towards
research work during my undergraduate studies at the University of New Orleans.
This work was supported by the Louisiana Board of Regents Support Fund Research &
Development Program.
And, last but not least, I would also like to express a special word of thanks to the
present faculty and staff of the University of New Orleans. Their unselfish dedication, in the
aftermath of Hurricane Katrina, has been invaluable to the UNO community as we recover
and rebuild.
iv
Table of Contents
List of Figures vii
Nomenclature ix
Abstract x
1 Introduction 1
1.1 Panel Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Research Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 The Method 9
2.1 Surfaces and the Source Strength . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Body Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Free-Surface Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Wave-making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 The Quasi-Singular Integral 27
3.1 The Integrand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Division and Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 Effect of Division and Transformation . . . . . . . . . . . . . . . . . . . . . . 37
4 Results 42
4.1 Double Body Flow Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1.1 Double Body Flow for a Sphere . . . . . . . . . . . . . . . . . . . . . 43
4.1.2 Double Body Flow for a Wigley Hull . . . . . . . . . . . . . . . . . . 48
4.2 Linearized Free-Surface Problem . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2.1 Linearized Free-Surface For a Sphere . . . . . . . . . . . . . . . . . . 51
5 Conclusion and Future work 53
Bibliography 55
Vita 56
v
List of Figures
1.1 Coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1 Source surface transformation from global to parametric domain. And re-
transformationofthesourcestrengthdistributionoverthesourcesurfacefrom
parametric to global domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Body surface and the source surface (offset has been exaggerated) . . . . . . 14
2.3 Free-surface source surface set-up . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1 Transformation of a quarter of a sphere into the parametric domain. . . . . . 28
3.2 Transformation of a half Wigley hull into the parametric domain. . . . . . . 28
3.3 Original X component of the integral for double body flow around a sphere. . 30
3.4 Original Y component of the integral for double body flow around a sphere. . 31
3.5 Original Z component of the integral for double body flow around a sphere. . 31
3.6 Element sub-division to eliminate the singularity [21]. . . . . . . . . . . . . . 34
3.7 Surface sub-division of Y velocity component of a source surface for double
body flow of a sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.8 Transformation Jacobian for ∆(1) . . . . . . . . . . . . . . . . . . . . . . . . 36
3.9 Original Y velocity component for ∆(1) . . . . . . . . . . . . . . . . . . . . . 38
3.10 Transformed Y velocity component for ∆(1) . . . . . . . . . . . . . . . . . . 38
3.11 Original Y velocity component for ∆(2) . . . . . . . . . . . . . . . . . . . . . 39
3.12 Transformed Y velocity component for ∆(2) . . . . . . . . . . . . . . . . . . 39
3.13 Original Y velocity component for ∆(3) . . . . . . . . . . . . . . . . . . . . . 40
3.14 Transformed Y velocity component for ∆(3) . . . . . . . . . . . . . . . . . . 40
3.15 Original Y velocity component for ∆(4) . . . . . . . . . . . . . . . . . . . . . 41
3.16 Transformed Y velocity component for ∆(4) . . . . . . . . . . . . . . . . . . 41
4.1 Source distribution for a quarter of a sphere with double body flow (α = 0.9). 44
4.2 Cp for a sphere with double body flow (α = 0.9). . . . . . . . . . . . . . . . 45
4.3 Velocity field on a sphere with double body flow (α = 0.9). . . . . . . . . . . 45
4.4 Normal velocity on a sphere with double body flow (α = 0.9). . . . . . . . . 46
4.5 Comparison of C for different values of α with the analytical result for a
p
sphere with double body flow. (values are at z=0 for a sphere of unit radius) 47
4.6 Cp for a Wigley hull with double body flow (3×3 vertices). . . . . . . . . . 48
4.7 Velocity field on a Wigley hull with double body flow (3×3 vertices). . . . . 48
4.8 Cp for a Wigley hull with double body flow (4×3 vertices). . . . . . . . . . 48
vi
4.9 Velocity field on a Wigley hull with double body flow (4×3 vertices). . . . . 48
4.10 Cp for a Wigley hull with double body flow (4×4 vertices). . . . . . . . . . 49
4.11 Velocity field on a Wigley hull with double body flow (4×4 vertices). . . . . 49
4.12 Cp for Wigley hull with different number of vertices for double body flow (at
z = -1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.13 Cp for Wigley hull with varying α with 3 × 3 vertices for double body flow
(at z = -1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.14 Free surface elevation for a sphere . . . . . . . . . . . . . . . . . . . . . . . . 52
vii
Nomenclature
µ doublet strength
Φ total potential
φ perturbation potential
φ(0) zero order wave potential
φ(1) first order wave potential
φ potential for the uniform onset flow
∞
σ source strength on the source surface
A vector A (underscore represents a vector)
n normal at point P
p
p point on the hull surface
P hull surface vertices
ij
q point on the source surface
Q source surface vertices
ij
U fluid onflow velocity (opposite of body velocity)
∞
v velocity field
ζ free-surface elevation
ζ(0) zero order wave elevation
M total number of vertices for the surface in v
N total number of vertices for the surface in u
S hull surface
b
S source strength vertices
ij
viii
U body velocity
b
 
n
x
n normal vector,  ny 
n
z
S free surface
F
ix