Hydrodynamic
behavior of inline structures
A Dynamic Fluid Body Interaction study
Technische Universiteit Delft
M.J. Rodermans
Hydrodynamic
behavior of
in-line
structures
A Dynamic Fluid Body Interaction study
by
M.J. Rodermans
to obtain the degree of Master of Science
at the Delft University of Technology,
to be defended publicly on Thursday June 23, 2016 at 11:00 AM.
Student number:
Project duration:
Thesis committee:
1384848
August, 2015 – June, 2016
Prof. dr. A. Metrikine,
Dr. J. M. de Oliveira Barbosa,
Ir. F. W. Renting,
Ir. H. Smienk,
Ir. H. Ottens,
TU Delft
TU Delft
TU Delft
HMC
HMC
This thesis was supported by Heerema Marine Contractors.
This thesis is confidential and cannot be made public until June 23, 2021.
An electronic version of this thesis is available at http://repository.tudelft.nl/.
Abstract
Heerema Marine Contractors (HMC) is, amongst other activities, involved in the installation of subsea pipelines
and subsea structures. These subsea structures are welded into the pipeline instead of installed separately to
increase efficiency during production and installation. The presence of subsea structures in the pipeline increases the stresses in the pipeline during installation and therefore reduces workability.
The stresses in pipelines during installation are analyzed beforehand. These analyses show that with subsea structures becoming bigger, the workability of in-line subsea structure installation becomes unacceptably
low.
Simplified models are used in order to model the hydrodynamic forces acting on subsea structures. The
suspicion is that these simplified models are conservative because of the lack of knowledge about the hydrodynamics around complex shaped structures. A first research [8] on the behavior of in-line structure installation and the effects of alternative hydrodynamic loading models has been performed at HMC. On the basis
of this research questions remain. The most important ones are:
• Is Morison’s equation applicable for sharp edged and asymmetric structures?
• Can forced motion experiments be used to model the dynamic behavior of subsea structures?
• Is the motion of sharp edged and asymmetric structures decoupled?
In order to investigate the hydrodynamics around subsea structures and answer these questions, this thesis is performed. With the use of Dynamic Fluid Body Interaction (DFBI)-simulations the dynamic behavior
of sharp edged and asymmetric structures and the description of this behavior has been investigated.
In this thesis subsea structures are simplified as thin flat plates and simulations in one and two degrees of
freedom have been performed. Over a range of frequencies the behavior in regular oscillating flow has been
investigated.
From the research it can be concluded that over the range of Keulegan-Carpenter (KC) numbers associated with in-line subsea structure installation Morison’s equation can be used to describe forces accurately
and these forces are decoupled. DNV GL prescribed coefficients cause large differences with the simulated
behavior. For thin flat plates KC dependent coefficients model the behavior more accurately. The KC dependent coefficients are determined by forced motion experiments. Further research should be performed on
the hydrodynamic coefficients for subsea structures. Hydrodynamic moments are observed which are not
taken into account during installation analysis. The impact of these moments on the dynamic behavior of
subsea structures should be further investigated.
iii
Contents
List of Figures
vii
List of Tables
xi
List of Abbreviations
xiii
List of Symbols
xv
1 Introduction
1
1.1
1.2
1.3
1.4
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Research objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Research approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Background information
2.1
2.2
2.3
2.4
2.5
5
Hydrodynamic loads modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Installation Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Computational Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
DFBI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-DoF solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Study Case
3.1
3.2
3.3
3.4
3.5
3.6
5
6
7
8
8
9
Load case definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Pipeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Water particle motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Validation
4.1
4.2
4.3
4.4
4.5
4.6
1
1
2
2
15
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Steady Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Oscillating Hydrodynamic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Simulation set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5 1 Degree of Freedom
23
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.3 Decay test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.4 Oscillated simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.5 Added mass and added damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.6 Transmissibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.7 Vortex shedding regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.8 Description with Morison’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.9 KC number based on relative motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.10 Comparison with DNV GL coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.12 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
v
vi
Contents
6 2 Degrees of Freedom
6.1
6.2
6.3
6.4
6.5
6.6
6.7
51
Set up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Resonance in x-direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Drag and added mass in the x-direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Description with Morison’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
7 Conclusions and recommendations
63
7.1 Conclusions 1-DoF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
7.2 Conclusions 2-DoF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
7.3 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
A General Transport Equations
65
B Finite Volume Method
67
B.1
B.2
B.3
B.4
Meshing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Physics Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Computational effort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
C Convergence studies
C.1
C.2
C.3
C.4
73
General mesh convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Prism Layer mesh convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Time-step convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
D Logarithmic decrement
77
E Linearization drag force
79
F Plots description 1 DoF
81
F.1
F.2
F.3
F.4
F.5
F.6
F.7
Simulation #1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Simulation #2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Simulation #3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Simulation #4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Simulation #5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Simulation #6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Simulation #7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
G DNV Comparison
91
H Differential equation solver
95
Bibliography
97
List of Figures
1.1 Typical subsea structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 2D resulting forces and moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Research flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
3
2.1 Variation of drag (left) and inertia (right) coefficients of plates [12] . . . . . . . . . . . . . . . . . .
6
3.1 Load case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Isometric view of 18" production FLET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Equivalent spring stiffnesses Ichthys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
11
13
4.1
4.2
4.3
4.4
4.5
4.6
4.7
Mesh and resulting drag coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hydrodynamic coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Domain size and boundary types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Refinement regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Morison equation fit at K C = 11 and δ = 0.0403 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
In-line force time-traces for a oscillated fluid or plate at KC=8 . . . . . . . . . . . . . . . . . . . . .
Hydrodynamic coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
16
18
19
20
21
21
5.1 Domain and boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Translation of plate during decay test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Damping ratio (ζ) and FFT magnitude plot of decay test . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Simulation #1, Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Simulation #2, Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Displacement FFT magnitude plots of Simulation #1 and #2 . . . . . . . . . . . . . . . . . . . . . .
5.8 Simulation #3, Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9 Simulation #3, Displacement FFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.10 Simulation #4, Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.11 Simulation #5, Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.12 Simulation #6, Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.13 Simulation #7, Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.14 FFT magnitude plots of Simulation #4 and #5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.15 FFT magnitude plots of Simulation #6 and #7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.16 Added mass and added damping from Fourier analysis and Morison’s equation . . . . . . . . . .
5.17 Comparison of forces with Fourier analysis and Morison’s equation . . . . . . . . . . . . . . . . .
5.18 Transmissibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.19 Vortex shedding regime of simulation #2, KC = 9.94 . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.20 Vortex shedding regime of simulation #3, KC = 3.72 . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.21 Comparison between quadratic and linearized drag formulation, Simulation #5 . . . . . . . . . .
5.22 Damping ratios from DFBI and theoretical values . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.23 Simulation #1, Translation comparison at ω/ω0 = 0.23 . . . . . . . . . . . . . . . . . . . . . . . . .
5.24 Simulation #1, Acceleration comparison at ω/ω0 = 0.23 . . . . . . . . . . . . . . . . . . . . . . . .
5.25 Simulation #2, Translation comparison at ω/ω0 = 0.5083 . . . . . . . . . . . . . . . . . . . . . . . .
5.26 Simulation #2, Acceleration comparison at ω/ω0 = 0.5083 . . . . . . . . . . . . . . . . . . . . . . .
5.27 Simulation #1, Force decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.28 Simulation #1, In-line Force comparison at ω/ω0 = 0.23 . . . . . . . . . . . . . . . . . . . . . . . .
5.29 Simulation #2, In-line Force comparison at ω/ω0 = 0.5083 . . . . . . . . . . . . . . . . . . . . . . .
5.30 FFT magnitudes of displacements Simulation #1 and #2 and their description . . . . . . . . . . .
5.31 Improved description simulation #1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
24
25
26
27
27
27
28
28
29
29
29
30
30
30
32
32
33
34
35
36
37
37
38
38
38
39
39
39
40
40
vii
viii
List of Figures
5.32 Improved description simulation #2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.33 Simulation #3, Translation comparison at ω/ω0 = 0.946 . . . . . . . . . . . . . . . . . . . . . . . .
5.34 Simulation #3, Velocity comparison at ω/ω0 = 0.946 . . . . . . . . . . . . . . . . . . . . . . . . . .
5.35 Simulation #3, Acceleration comparison at ω/ω0 = 0.946 . . . . . . . . . . . . . . . . . . . . . . . .
5.36 Simulation #3, In-line Force comparison at ω/ω0 = 0.946 . . . . . . . . . . . . . . . . . . . . . . .
5.37 Simulation #4, Translation comparison at ω/ω0 = 1.568 . . . . . . . . . . . . . . . . . . . . . . . .
5.38 Simulation #4, In-line Force comparison at ω/ω0 = 1.568 . . . . . . . . . . . . . . . . . . . . . . .
5.39 Simulation #5, Translation comparison at ω/ω0 = 6.16 . . . . . . . . . . . . . . . . . . . . . . . . .
5.40 Simulation #5, In-line Force comparison at ω/ω0 = 6.16 . . . . . . . . . . . . . . . . . . . . . . . .
5.41 Simulation #6, Translation comparison at ω/ω0 = 4.17 . . . . . . . . . . . . . . . . . . . . . . . . .
5.42 Simulation #6, In-line Force comparison at ω/ω0 = 4.17 . . . . . . . . . . . . . . . . . . . . . . . .
5.43 Simulation #7, Translation comparison at ω/ω0 = 8.19 . . . . . . . . . . . . . . . . . . . . . . . . .
5.44 Simulation #7, In-line Force comparison at ω/ω0 = 8.19 . . . . . . . . . . . . . . . . . . . . . . . .
5.45 Flow diagram of KC number iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.46 Displacement amplitudes for different modeling methods . . . . . . . . . . . . . . . . . . . . . . .
41
41
42
42
42
43
43
44
44
44
45
45
45
46
48
6.1 Flow direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Hydrodynamic coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Translation comparison y-direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Translation comparison x-direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Translation y-direction at ω = 0.3141r ad /s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6 Translation x-direction at ω = 0.3141r ad /s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7 Translation y-direction at ω = 0.6283r ad /s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.8 Translation x-direction at ω = 0.6283r ad /s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.9 Force x-direction at ω = 0.3141r ad /s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.10 Force x-direction at ω = 0.6283r ad /s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.11 Close-up LSM fit at ω = 0.3141r ad /s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.12 Close-up LSM fit at ω = 0.6283r ad /s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.13 Translation x-direction at ω = 0.3141r ad /s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.14 Translation y-direction at ω = 0.3141r ad /s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.15 Translation x-direction at ω = 0.6283r ad /s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.16 Translation y-direction at ω = 0.6283r ad /s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.17 Translation description x-direction at ω = 0.3141r ad /s . . . . . . . . . . . . . . . . . . . . . . . . .
6.18 Translation description x-direction at ω = 0.6283r ad /s . . . . . . . . . . . . . . . . . . . . . . . . .
6.19 Translation description y-direction at ω = 0.3141r ad /s . . . . . . . . . . . . . . . . . . . . . . . . .
6.20 Force description y-direction at ω = 0.3141r ad /s . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.21 Translation description y-direction at ω = 0.6283r ad /s . . . . . . . . . . . . . . . . . . . . . . . . .
6.22 Force description y-direction at ω = 0.6283r ad /s . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.23 In-line motion of structure and fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.24 Effect of angle factor on force in y-direction at ω = 0.6283r ad /s . . . . . . . . . . . . . . . . . . . .
6.25 Moments acting on plate in 2-DoF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
52
52
53
53
53
54
54
54
55
55
56
56
56
57
57
58
58
59
59
59
60
60
61
62
B.1
B.2
B.3
B.4
2D Finite Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mesh types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Close-up of PLM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Flow regions in a turbulent boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
68
69
71
C.1 Mesh size convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.2 Prism layer mesh convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.3 Time-step convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
74
74
Simulation #1, Translation comparison at ω/ω0 = 0.23 . . . . . . . . . . . . . . . . . . . . . . . . .
Simulation #1, Velocity comparison at ω/ω0 = 0.23 . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simulation #1, Acceleration comparison at ω/ω0 = 0.23 . . . . . . . . . . . . . . . . . . . . . . . .
Simulation #1, In-line Force comparison at ω/ω0 = 0.23 . . . . . . . . . . . . . . . . . . . . . . . .
Simulation #2, Translation comparison at ω/ω0 = 0.5083 . . . . . . . . . . . . . . . . . . . . . . . .
Simulation #2, Velocity comparison at ω/ω0 = 0.5083 . . . . . . . . . . . . . . . . . . . . . . . . . .
81
81
82
82
82
82
F.1
F.2
F.3
F.4
F.5
F.6
List of Figures
ix
F.7 Simulation #2, Acceleration comparison at ω/ω0 = 0.5083 . . . . . . . . . . . . . . . . . . . . . . .
F.8 Simulation #2, In-line Force comparison at ω/ω0 = 0.5083 . . . . . . . . . . . . . . . . . . . . . . .
F.9 Simulation #3, Translation comparison at ω/ω0 = 0.946 . . . . . . . . . . . . . . . . . . . . . . . .
F.10 Simulation #3, Velocity comparison at ω/ω0 = 0.946 . . . . . . . . . . . . . . . . . . . . . . . . . .
F.11 Simulation #3, Acceleration comparison at ω/ω0 = 0.946 . . . . . . . . . . . . . . . . . . . . . . . .
F.12 Simulation #3, In-line Force comparison at ω/ω0 = 0.946 . . . . . . . . . . . . . . . . . . . . . . .
F.13 Simulation #4, Translation comparison at ω/ω0 = 1.568 . . . . . . . . . . . . . . . . . . . . . . . .
F.14 Simulation #4, Velocity comparison at ω/ω0 = 1.568 . . . . . . . . . . . . . . . . . . . . . . . . . .
F.15 Simulation #4, In-line Force comparison at ω/ω0 = 1.568 . . . . . . . . . . . . . . . . . . . . . . .
F.16 Simulation #5, Translation comparison at ω/ω0 = 6.16 . . . . . . . . . . . . . . . . . . . . . . . . .
F.17 Simulation #5, Velocity comparison at ω/ω0 = 6.16 . . . . . . . . . . . . . . . . . . . . . . . . . . .
F.18 Simulation #5, Acceleration comparison at ω/ω0 = 6.16 . . . . . . . . . . . . . . . . . . . . . . . .
F.19 Simulation #5, In-line Force comparison at ω/ω0 = 6.16 . . . . . . . . . . . . . . . . . . . . . . . .
F.20 Simulation #6, Translation comparison at ω/ω0 = 4.17 . . . . . . . . . . . . . . . . . . . . . . . . .
F.21 Simulation #6, Velocity comparison at ω/ω0 = 4.17 . . . . . . . . . . . . . . . . . . . . . . . . . . .
F.22 Simulation #6, Acceleration comparison at ω/ω0 = 4.17 . . . . . . . . . . . . . . . . . . . . . . . .
F.23 Simulation #6, In-line Force comparison at ω/ω0 = 4.17 . . . . . . . . . . . . . . . . . . . . . . . .
F.24 Simulation #7, Translation comparison at ω/ω0 = 8.19 . . . . . . . . . . . . . . . . . . . . . . . . .
F.25 Simulation #7, Velocity comparison at ω/ω0 = 8.19 . . . . . . . . . . . . . . . . . . . . . . . . . . .
F.26 Simulation #7, Acceleration comparison at ω/ω0 = 8.19 . . . . . . . . . . . . . . . . . . . . . . . .
F.27 Simulation #7, In-line Force comparison at ω/ω0 = 8.19 . . . . . . . . . . . . . . . . . . . . . . . .
83
83
83
84
84
84
85
85
85
86
86
86
87
87
87
88
88
88
89
89
89
G.1 Simulation #1, Translation comparison at ω/ω0 = 0.23, KC = 19.23 . . . . . . . . . . . . . . . . . .
G.2 Simulation #2, Translation comparison at ω/ω0 = 0.5083,KC = 9.61 . . . . . . . . . . . . . . . . . .
G.3 Simulation #3, Translation comparison at ω/ω0 = 0.946, KC = 4.2 . . . . . . . . . . . . . . . . . . .
G.4 Simulation #4, Translation comparison at ω/ω0 = 1.568, KC = 9.25 . . . . . . . . . . . . . . . . . .
G.5 Simulation #5, Translation comparison at ω/ω0 = 6.16, KC = 2.1 . . . . . . . . . . . . . . . . . . . .
G.6 Simulation #6, Translation comparison at ω/ω0 = 4.17, KC = 1 . . . . . . . . . . . . . . . . . . . . .
G.7 Simulation #7, Translation comparison at ω/ω0 = 8.19, KC = 0.5 . . . . . . . . . . . . . . . . . . . .
91
91
92
92
92
93
93
List of Tables
3.1
3.2
3.3
3.4
Study case FLET characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Study case model characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Study case pipeline characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fluid motion characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
11
13
14
4.1 Plate geometries used in experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Goodness-of-fit parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
20
5.1 Data decay test (positive peaks) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Data decay test (negative peaks) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Input data simulations below resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Input data simulations near resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Input data simulations above resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Added mass and added damping coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Transmissibilities of simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8 Vortex shedding regime based on KC number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9 Theoretical damped frequencies and damping ratios . . . . . . . . . . . . . . . . . . . . . . . . . .
5.10 Settings simulation #1 and #2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.11 Settings simulation #3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.12 Settings simulation #4, #5, #6 and #7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.13 Drag and inertia coefficients. Based on flow and relative motion . . . . . . . . . . . . . . . . . . .
5.14 KC number iteration, Simulation #1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.15 KC number iteration, Simulation #2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.16 KC number iteration, Simulation #3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.17 KC number iteration, Simulation #4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.18 KC number iteration, Simulation #5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.19 KC number iteration, Simulation #6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.20 KC number iteration, Simulation #7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.21 Comparison of displacement amplitudes for different methods . . . . . . . . . . . . . . . . . . . .
25
25
26
28
29
31
32
34
36
37
41
43
46
46
46
46
46
46
47
47
48
C.1
C.2
C.3
C.4
General mesh settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Prism layer mesh settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Time-step convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Convergence bandwidths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
74
74
75
E.1 Drag force linearization comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
xi
List of Abbreviations
CFD
CFL
Computational Fluid Dynamics.
Courant-Friedrichs-Lewy.
DFBI
DNV GL
DoF
DP
Dynamic Fluid Body Interaction.
Det Norske Veritas Germanischer Lloyd.
Degree of Freedom.
Dynamic Positioning.
FEM
FFT
FLET
FTA
FVM
Finite Element Method.
Fast Fourier Transform.
Flowline End Termination.
Flowline Termination Assembly.
Finite Volume Method.
HMC
Heerema Marine Contractors.
ILS
In-line Structure.
KC
Keulegan-Carpenter.
LSM
Least Squares Method.
PLEM
PLM
Pipeline End Manifold.
Prism Layer Mesh.
RANS
Reynolds-Averaged Navier-Stokes.
xiii
List of Symbols
A
Amplitude [m].
C
c
CA
CD
CM
C max
Courant number [−].
Damping coefficient [N s/m].
Added mass coefficient [−].
Drag coefficient [−].
Inertia coefficient [−].
Maximum Courant number [−].
D
δ
δbound ar y
δl og
Characteristic diameter [m].
Goodness of fit factor [−].
Boundary layer thickness [m].
Logarithmic decrement [−].
E
Modulus of elasticity [N /m 2 ].
F
f
FC F D
Force [N ].
General force vector [ f ].
By CFD caluculated forces [N ].
G
g
Shear modulus [N /m 2 ].
Gravitational constant [m/s 2 ].
h
Hs
Water depth [m].
Significant wave height [m].
I
Moment of inertia [m 4 ].
J
Torsional constant [m 4 ].
k
κ
KC
kw
Spring coefficient [N /m].
Thermal conductivity [W /(mK )].
Keulegan-Carpenter number [−].
Wave number [r ad /m].
L
λ
Length [m].
Wave length [m].
M
m
ma
µ
Moment [N m].
Mass [kg ].
Added mass [kg ].
Dynamic viscosity [N s/m 2 ].
∇
ν
Vector differential operator.
Kinematic viscosity [m 2 /s].
ω0
ωd
ω
Natural frequency [r ad /s].
Damped frequency [r ad /s].
Frequency [r ad /s].
xv
xvi
List of Symbols
p
Φ
φ
Pressure [P a].
Rayleigh’s Dissipation function.
Arbitrary variable [−].
r
Re
ρ
Stretching factor [−].
Reynolds number [−].
Density [kg /m 3 ].
σ
Variance [−].
T
T
t
t
θ
θF
Period [s].
Temperature [K ].
Structure thickness [m].
Time [s].
Angle [d eg ].
Angle factor [−].
u
u
Um
Fluid velocity [m/s].
Velocity vector [m/s].
Maximum fluid velocity [m/s].
VF
Volume factor [−].
w
Unit weight [kg /m].
X
x
x
Structure displacement amplitude [m].
Structure displacement vector [m].
Displacement structure [m].
Y
∆y 1
∆y n
Fluid displacement amplitude [m].
Thickness first prism layer [m].
Thickness prism layer number n [m].
z
ζ
Depth [m].
Damping ratio [−].
1
Introduction
1.1. Background
Heerema Marine Contractors (HMC) is, amongst other activities, involved in the offshore installation of
pipelines. Included in the scope of some pipelay projects is the installation of subsea structures. In order
to save time and therefore become more competitive, HMC welds these structures into the pipeline on board
and installs the pipeline including the structure as a whole. These structures can either be at the end of a
pipeline or at a point along the pipeline. Figure 1.1 shows a typical subsea structure.
Figure 1.1: Typical subsea structure
The stresses and strains which occur in the pipeline during installation can not exceed certain values in
order to ensure pipeline integrity. Due to environmental conditions the system consisting of the pipeline and
structure will move relative to the water. This motion will cause hydrodynamic forces acting on the structure
and pipeline which cause stresses and strains in the pipeline.
A combination of design and forces acting on the system leads to a workability. This is a percentage of
time in which the environmental conditions are acceptable to allow safe installation.
1.2. Problem description
As subsea structures are large compared to the pipeline, relatively large hydrodynamic forces are acting on
them. Therefore installations of subsea structures have relatively low workability compared to regular pipelay
operations. Other than being large, the lack of knowledge on the hydrodynamic forces acting on these structures causes lower workability. Because of this lack of knowledge, used installation analysis methods are
conservative in the calculation of these forces. It is expected that research on the modeling of hydrodynamic
forces acting on subsea structures will improve workability.
The installation analysis method used at HMC uses empirical formulae to estimate the hydrodynamic
forces acting on a subsea structures (see section 2.2.1). In order to obtain more accurate approximations,
an analysis method was developed [5] using Computational Fluid Dynamics (CFD) (see section 2.2.2). Carel
1
2
1. Introduction
Hoekstra performed a research [8] where the method using CFD was improved with theoretical KeuleganCarpenter number, open-area ratio and angle of incidence dependencies for the hydrodynamic coefficients
(see section 2.2.2).
These hydrodynamic properties analysis methods use Morison’s equation (see section 2.1.1) to calculate
resulting forces. Morison’s equation was developed for in-line forces acting on cylinders. It has been proven
that this equation can also be used to describe in-line forces acting on bodies of other shapes. However, how
accurate this description is for typical subsea structures is not known.
Next to that, the current methods use empirical coefficients found through forced oscillation tests. In a system where vessel motions, waves and currents are present, it is unknown if the current methods are able to
accurately describe the resulting dynamic behavior of the system.
When faced with flow directions not normal to the direction in which the empirical coefficients are defined,
decomposition of motions is used to find resulting forces. This is due to the fact that the current methods
which use Morison’s equation are only able to describe in-line forces. The subsea structures used by HMC are
typified by asymmetry and sharp edges and are therefore likely to create lift and moments (see figure 1.2). If
this is of influence on the response of the system is not yet investigated.
(a) Only in-line force
(b) With lift and moment
Figure 1.2: 2D resulting forces and moments
Summarizing, the applicability and accuracy of Morison’s equation, which is used in installation analysis,
for subsea structures is unknown. This leads to conservatism in force calculation which in turn leads to lower
workability.
1.3. Research objective
Before any research is done on force coefficients, a method for the description of forces acting on subsea
structures needs to be established. This could either be done through validating one of the currently used
methods, or through finding a new method.
The objective of this thesis is therefore:
Investigate the dynamic behavior and the description of this behavior of the installation of in-line subsea structures.
Where the accuracy of the currently used methods is tested on:
• Sharp edged asymmetric bodies
• Unforced body motion
• Decomposition of motions
1.4. Research approach
In order to investigate the installation behavior of subsea in-line structures, there needs to be observable
data. Since no experimental data or real life measurements are available, two methods remain: physical ex-
1.4. Research approach
3
periments or numerical simulation. The latter is chosen. Dynamic Fluid Body Interaction (DFBI) (see section
2.4) is used to simulate the behavior of a subsea structure under certain conditions.
The research approach is divided into several steps:
1. Set up study case. Determine under which conditions the research will be done. Simplifications and
other assumptions are discussed.
2. Validate numerical models.
3. Perform DFBI simulations in one DoF.
4. Analyze and describe the behavior of the one DoF system.
5. Perform DFBI simulations in multiple DoF.
6. Analyze and describe the behavior of the multiple DoF system(s).
Figure 1.3: Research flowchart
These steps are described in this thesis. First in chapter 3 the study case which will be investigated and
the simplifcations are discussed. Next, in chapter 4, the used numerical models are validated. Once the
validation is performed, the study case can be simulated. In chapter 5 results from simulations in 1-DoF
are presented and the description of the behavior using Morison’s equation is investigated. At the end of
this chapter conclusions are drawn based on the findings in 1-DoF and the impact of the simplifications is
discussed. In chapter 6 the same system is analyzed in 2-DoF. Description methods for 2-DoF are investigated
and conclusions specific to 2-DoF are presented at the end of the chapter. Again the impact of simplifications
is discussed.
In chapter 7 the conclusions from chapter 5 and 6 are summarized and general conclusions are drawn
and discussed. Final recommendations are presented in section 7.3.
2
Background information
2.1. Hydrodynamic loads modeling
Any solid which is moving relative to a fluid experiences hydrodynamic loads. Modeling of these loads is
key in modeling its structural behavior. Many methods exist to model the loads acting on a structure. In the
design of offshore structures Morison’s equation is widely used.
2.1.1. Morison’s equation
In 1950 Morison, O’Brien, Johnson and Schaaf published an equation [15] which predicts wave forces on
vertical piles. This equation became known as Morison’s equation. The equation identifies three sources
of hydrodynamic forces. A Froude-Krylov force dependent on fluid acceleration, an inertia force dependent
on relative acceleration and a drag force dependent on relative velocity squared. The diffraction forces are
ignored as the characteristic diameter (D) is assumed to be small relative to the wavelength. As the ratio
between diameter and wavelength in this thesis ranges from 0.38 to 0.002 and short waves (high frequency)
have less impact on the response of the structure than long waves (low frequency), the diffraction forces are
neglected.
Morison’s equation assumes that the inertia and drag forces can be summed to obtain a total force. While
this may be true for very high and very low KC numbers (see section 2.1.2) where one of the contributions is
negligible, this is not the case for flow regimes at intermediate KC numbers. At these intermediate KC numbers the coefficients used in Morison’s equation do describe force amplitude accurately.
The parameter of interest in this thesis is the displacement of the structure as this dictates stresses and strains
in the pipeline. In this case the inaccuracy of Morison’s equation describing force at intermediate KC numbers is insignificant. Displacements are described accurately as total impulse is described accurately.
Morison’s equation is applicable to planar oscillatory flow. When the flow is orbital, Morison’s equation
overestimates the inertial forces acting on structure. In this thesis planar oscillatory flow is used. The effect
this has on inertial forces has to be taken into account when the analysis methods are applied in orbital flow.
Even though this thesis covers complex shaped structures, the formulation of Morison’s equation for
cylinders is used. The experiments with oscillating flat plates presented in section 4.3, which are used for
validation and modeling of forces acting on complex shaped structures, use the cylindrical formulation.
Morison’s equation using relative acceleration and velocity:
F=
π
ρD 2 u̇
4
| {z }
Froude-Krylov force
+
π
1
ρD 2C A (u̇ − ẍ) + ρDC D |u − ẋ|(u − ẋ)
4
|
{z
} |2
{z
}
Inertia force
(2.1)
Drag force
in which ρ is the fluid density, D the structure’s characteristic diameter, u the water particle velocity, x
the structure’s displacement, C A the inertia coefficient and C D the drag coefficient. The Froude-Krylov force
and inertia force are usually combined by replacing C A with C M = 1 +C A . The inertia and drag coefficient are
5
6
2. Background information
dependent on several factors such as but not exclusively shape, porosity and shielding.
These coefficients can be determined with the use of physical experiments or, as a more recent technique,
with numerical simulations. Both these methods for determining inertia and drag coefficients are used at
HMC and will be discussed in sections 2.2.1 and 2.2.2.
Not only the structure’s characteristics influence the inertia and drag coefficients. The flow conditions
itself influence these coefficients as well.
2.1.2. Keulegan-Carpenter number
Keulegan and Carpenter performed experiments with oscillating cylinders and plates in a stationary water
column [12]. They showed that the hydrodynamic coefficients used in Morison’s equation vary with the period parameter.
KC =
Um T
D
(2.2)
Where K C is the Keulegan-Carpenter number, Um the velocity amplitude, T the oscillation period and D
the characteristic diameter.
This period parameter, which became known as the Keulegan-Carpenter number, relates the relative importance of the drag forces over the inertia forces.
Figure 2.1: Variation of drag (left) and inertia (right) coefficients of plates [12]
Since oscillating structures which have characteristic diameters which are small relative to wavelength
are the subject of the research, the Keulegan-Carpenter number is important.
2.2. Installation Analysis
Before pipeline installations are performed, the impact of such an operation on pipeline and equipment integrity needs to be analysed. At HMC such installation analyses are performed using the software package
Flexcom which uses a Finite Element Method (FEM). The impact of an in-line structure can be added by supplying drag and inertia coefficients [11]. These coefficient can be supplied in normal and tangential direction
after which interpolation is used for angles in between. Morison’s equation is than used to calculate forces
acting at the point where the structure is specified by the user.
Another way of modeling the impact of the in-line structure in Flexcom, the FEM software, is to add a
point load. However, this point load needs to be time-dependent since the force acting on a subsea structure
is time-dependent. See [8] for more information.
The forces or hydrodynamic coefficients which serve as an input need to be determined. For subsea
structures this is usually difficult due to the complex shape. Model tests or CFD simulations can be used to
analyze the forces acting on such a structure. However, these are time consuming and expensive methods.
It is common practice to estimate the hydrodynamic coefficients based on characteristic dimensions such as
diameter or volume, although with the developments in computational power CFD is becoming more and
2.3. Computational Fluid Dynamics
7
more a feasible option.
Det Norske Veritas Germanisher Lloyd(DNV GL), a leading classification agency in the maritime and oil &
gas sector, provides guidelines for the analysis method [23] and the calculation of environmental loads [22]
based on characteristic dimensions.
At HMC the predominant method for determining the hydrodynamic coefficients of structures which
serve as input for the installation analyses uses the DNV GL guidelines. Over the past couple of years HMC
has also developed its own method using CFD.
2.2.1. DNV GL based method
The method which is used, is to simplify the structure in a shape, or assembly of shapes, for which empirical
coefficients are known. This simplification can be done in two ways:
1. Consider the structure as one big volume which is determined by its outer dimensions. The coefficients
for this large volume, which are known, are compensated for perforation and aspect ratio.
2. The structure is divided in individual components of which the hydrodynamic coefficients are known.
These are summed to obtain coefficients for the entire structure. This neglects the shielding effect
which is investigated by Cinello et al. [4].
Depending on the estimations and interpretations by the user of these guidelines the results may differ
between methods or even between different users.
2.2.2. CFD based method
Instead of using empirical coefficients, this method uses numerical modeling, also known as CFD (see section 2.3), to simulate the hydrodynamic forces acting on a structure.
The CFD method used at HMC [5] uses forced oscillation simulations to determine added mass and drag
coefficients which are used in Morison’s equation. The fluid is oscillated and the in-line force is measured.
The hydrodynamic coefficients C M and C D are obtained using a Least Squares Method (LSM), fitting Morison’s equation to the in-line force data.
For all three directions the in-line hydrodynamic coefficients are obtained. Flexcom uses interpolation
to find hydrodynamic coefficients in all directions. HMC recognizes that the hydrodynamic coefficients are
dependent on the Keulegan-Carpenter number and applies this dependency in their CFD method. Over a
range of KC numbers simulations are performed and a KC number dependency is interpolated.
Carel Hoekstra presented in his research [8] an alteration to the interpolations which are used in the CFD
method. The same data and LSM to determine C M and C D as in the CFD method are used. But instead of
interpolating between the coefficients, KC dependencies based on research done by Molin [14] and Sandvik
[17] on ventilated structures are fitted to the data. Next to that, an angle of incidence dependency based on
DNV GL guidelines is added to substitute the interpolation done in Flexcom.
2.3. Computational Fluid Dynamics
Computational Fluid Dynamics (CFD) is a collective term for the numerical analysis of problems that involve
fluid flows with the use of computers. The Navier-Stokes equations (see Appendix A), a set of partial differential equations which describe fluid flow, are solved numerically in a domain divided into small volumes,
commonly known as a mesh. The most commonly used CFD method is the Finite Volume Method (FVM)
(see Appendix B).
All CFD methods follow the same basic approach:
1. Pre-Processing
• Defining the problem’s geometry
8
2. Background information
• Discretizing the domain (Meshing)
• Defining the physical models
• Defining boundaries
2. Running the simulation
3. Post-processing
• Visualizing results
• Analysis of results
2.4. DFBI
The STAR-CCM+ User guide [3] defines DFBI as:
"Dynamic Fluid Body Interaction simulates the motion of a rigid body in response to pressure
and shear forces the fluid exerts, and to additional forces you define."
DFBI combines CFD, which computes pressure and shear forces the fluid exerts, with a six Degree of Freedom (6-DoF) solver which computes the rigid body motion. This simulation method allows for freely moving
structures within the CFD environment.
DFBI problems in STAR-CCM+ are solved using a segregated approach. This means that the equation
governing the flow are solved separately from the equations governing the motion of the body. A separate
CFD and 6-DoF solver are employed.
2.5. 6-DoF solver
The 6-DoF solver uses the calculated forces and moments acting on a body and computes the translational
and angular motion of the body. To be more accurate, The 6-DoF solver computes these translational and
angular motions for the center of mass. I.e. the body is assumed to be rigid. This assumption can be made
for subsea structures as any structural deformations of typical subsea structures will be very small compared
to the rigid body motion.
3
Study Case
In order to study the effect different flow conditions have on the behavior of in-line structures, a representative study case is set-up. Just as in the work done by Hoekstra[8], the Ichthys project[10] will be used as the
representative case. HMC is contracted to, amongst other activities, install an 18" production Flowline End
Terminal (FLET).
In this thesis numerical simulations are used to analyze the behavior of in-line structures. The durations
of these simulations are greatly dependent on the complexity of these simulations. This is explained in section B.4. The complexity of the study case is kept as low as possible in order to keep simulation durations low.
How the study case is set up, which assumptions are made and what variables are fixed or varied over
what range will be discussed in this chapter.
3.1. Load case definition
The load case for which the dynamic behavior of subsea structures is investigated is a simplified one. A situation is analyzed where the structure is suspended from the vessel in a vertical direction and is at the end of the
pipeline (see figure 3.1a). This is representative for FLET installation. In this orientation it is the starting point
of the pipeline. Therefore it is called a 1st end FLET. How the subsea structure is simplified will be discussed
in section 3.2.2.
This simplified installation will be analyzed as if it was a mass-spring-damper system. A schematic of the
mass-spring-damper system in one degree of freedom can be found in figure 3.1b. In this figure the forcing
induced by the fluid flow (section 3.4) is present.
The linear equation of motion describing the behavior of this mass-spring-damper system is as follows:
m ẍ + c ẋ + kx = m a (u̇ − ẍ) + c a (u − ẋ)
(3.1)
where x is the motion of the structure and u the fluid velocity. m is the weight of the structure and m a the
added mass. c is used to describe the structural damping, c a the added damping and k is the spring coefficient which models the restoring force due to weight and pipeline stiffness. We are interested in the resulting
structure motions under various fluid motions.
The structure is forced by water motion and vessel motions. Only pitch, roll and heave of the vessel remain as the Dynamic Positioning (DP) system of the vessel will eliminate surge, sway and yaw motions. The
vessel’s motions are neglected.
The fluid motion is due to waves and current. What assumptions are made and how fluid motion is described
is presented in section 3.4.
The mass of the system is made up out of the submerged mass of the structure and its added mass together
with the submerged mass and added mass of the pipeline due to accelerations in the water. The submerged
9
10
3. Study Case
mass and added mass of the pipeline will be neglected as this thesis focuses on the dynamic behavior of the
subsea structure and added mass and damping for pipelines is already accurately described. The submerged
mass of the structure will be discussed in section 3.2.
The damping of the system is due to hydrodynamic damping of the structure and pipeline. Again the
pipeline hydrodynamics will be neglected.
The restoring force is generated by the weight of the pipeline and structure and bending stiffness of the
pipeline. How this is modeled and what assumptions are made is discussed in section 3.3.
In order to reduce simulation durations even further, the system is assessed in a very thin 3D environment.
With the use of symmetry this resembles a 2D analysis while being able to use DFBI simulatoins. This reduces
the amount of cells in the computational domain drastically. On the other side, a 2D assumption is made
which limits the analysis to three degrees of freedom and does not allow flow over two sides of the structure.
This increases the force per reference area or volume.
(b) 1-DoF Load case
simplification
(a) Load case schematic
Figure 3.1: Load case
3.2. Structure
As said in section 3.1, the simplified installation of a 1st end FLET will be used as a study case. One of the
largest FLETs installed by HMC, the 18" production FLET installed at the Ichthys project[10], will be the example on which simplifications are made. These simplifications are necessary to keep simulation durations
as low as possible.
3.2.1. Characteristics
The characteristics concerning geometry, submerged mass and submerged mass distribution of the 18" production FLET of the Ichthys project[6] are shown in table 3.1.
FLET Parameters
Value
Dimensions
Length
Width (wings down)
Width (wings up)
Height
Submerged mass FLET
17.8m
9.5m
6.025m
8.4m
117.39mT
Table 3.1: Study case FLET characteristics
3.2. Structure
11
Figure 3.2: Isometric view of 18" production FLET
3.2.2. Simplification
The entire computational domain is divided into cells. All the equations describing fluid flow need to be
solved for every cell at each iteration. The more cells in the domain, the more computational effort is required per iteration.
In general, the more detail, the more cells. This has to do with the fact that smaller details need more cells
to catch properly and that there are relatively large amounts of cells in the boundary layer mesh compared to
the rest of the mesh. More detail usually means more surface area which in turn means more boundary layer
mesh.
With this in mind, a flat plate with the dimensions of the mud-mat is chosen as a simplification. This
plate is similar to the flat plate normal to the flow used in the validation other than the dimension being a lot
bigger. The same ratios as in the validation will be used. The length of the plate will be modeled as infinite
with the use of symmetry planes.
Molin [14], Sandvik [17], Cinello [4] and others performed research on the hydrodynamic behaviour of
perforated structures. The added mass and damping reduce with perforation ratio as acknowledged by DNV
[23]. In this thesis the plate will be assumed solid. Hoekstra found maximum perforation ratios of 0.25.
The submerged mass of the real structure will be used but the centre of gravity will be placed in the centre
of the plate. The fact that asymmetry in weight and inertia is neglected is of no concern as asymmetric behavior will occur due to instabilities or geometric asymmetries.
Model Parameters
Value
Dimensions
Length
Width
Thickness
Weight
∞
9.5m
0.285m
117.39mT
Table 3.2: Study case model characteristics
12
3. Study Case
3.3. Pipeline
As discussed in section 3.1 will the hydrodynamic forces acting on the pipeline be neglected. What remains is
a restoring force due to its self weight and bending stiffness which forces the pipeline in a vertical orientation.
The vessel is assumed to be stationary which makes the restoring force solely dependent on the deviation
of the structure from the vertical.
The restoring effect a deflected pipeline has on the structure will be modeled as a linear spring (see figure
3.1b). There are two main contributors in the restoring force. The bending stiffness and the pipeline’s own
weight. The equivalent spring stiffnesses are approximated by the following formulae where the assumption
of small rotations and deflections is made.
Euler-Bernoulli beam
The pipeline is modeled as a cantilever beam with end load. The end load is rewritten as F = kx where k is
the spring coefficient and x is the deflection at the end of the beam. The deflection of a cantilever beam with
an end load is given as:
x=
F L3
3E I
(3.2)
k=
3E I
L3
(3.3)
rewriting for k gives:
Pendulum
The pipeline and the structure are modeled as a pendulum. The restoring force is due to acting moments.
X
MO = 0 =⇒ M st r uc t ur e + M pi pel i ne = M spr i ng
(3.4)
M st r uc t ur e = m st r uc t g x
(3.5)
1
M pi pel i ne = w pi pel i ne Lg x
2
(3.6)
M spr i ng = kxL
(3.7)
Combining equations 3.5, 3.6 and 3.7 gives the following equation for k:
1
k= 2
w pi pel i ne L + m st r uc t
L
g
(3.8)
The characteristics of the pipeline used during the Ichthys project (see table 3.3) are used to compare both
approximations. As can be seen in figure 3.3, at depths over 100 meters the bending stiffness is negligible
compared to the pendulum restoring force of the system. The bending stiffness is neglected and a spring
stiffness of 5000 N/m is taken as typical spring stiffness.
The pipeline used in the Ichthys project is a large and stiff pipe. For other pipelines with a smaller diameter, lower equivalent bending stiffness compared to the own weight is expected. As EI scales with the outer
diameter squared while the structure’s weight is dominant in the restoring force.
3.4. Water particle motion
13
4
3
x 10
Pendulum
Euler beam
Equivalent striffness [N/m]
2.5
2
1.5
1
0.5
0
0
200
400
600
Pipeline length [m]
800
1000
Figure 3.3: Equivalent spring stiffnesses Ichthys
Pipeline Parameters
Dimensions
Diameter
Wall thickness
Weight
Dry weight
Submerged weight
Modulus of Elasticity, E
Bending Stiffness, EI
Torsional Stiffness, GJ
Axial Stiffness, EA
Value
18"
21.9 mm
364 kg/m
73.6 kg/m
2.0 ∗ 1011 N/m2
1.583 ∗ 108 Nm2
1.218 ∗ 108 Nm2
6.760 ∗ 109 N
Table 3.3: Study case pipeline characteristics
3.4. Water particle motion
The structure will encounter moving water particles which increase or decrease the relative velocity and acceleration between structure and water particle. This relative velocity and acceleration influence the inertia
and damping of the system.
Airy wave theory for regular waves [9] is used to model the water particle motions due to waves. For deep
water (water depth > 1/2 wavelength) the water particle velocity in the horizontal direction is:
u x (z, t ) = A w ave ωe k w z si n(ωt )
(3.9)
where A w ave is the wave elevation amplitude, ω the wave frequency, k w the wave number and z the vertical distance from the waterline.
As z is always negative and k w always positive, the highest velocities are encountered at z = 0. In order
to keep assumptions in this thesis conservative maximum velocities are used. Therefore we will evaluate the
particle velocity at z = 0. This gives e k w z = 1.
The range of amplitudes and frequencies which are encountered by typical structures is limited by the
wave frequencies which are encountered. The response of the vessel to second order wave forces is assumed
to be neutralized by the Dynamic Positioning (DP) system of the vessel. For structure installation, the period
range has been assumed to be 4s - 20s and the wave elevation amplitude is set at the significant wave height
(H s ) as prescribed by [23, sec. 3.4.2.11]. The range of significant wave heights during installation is estimated
at 0.5m - 3m. The realistic ranges of water particle characteristics are summarized in table 3.4.
14
3. Study Case
Parameter
Range
Wave period
Wave frequency
Significant wave height
Velocity amplitude
4 - 20 [s]
1.5708 - 0.31416 [rad/s]
0.5 - 3 [m]
0.16 - 4.71 [m/s]
Table 3.4: Fluid motion characteristics
3.5. Hydrodynamics
The different installation analysis methods all use different methods to model the hydrodynamic forces acting
on the structure. The DFBI method differs from the other methods in the fact that it calculates the hydrodynamic forces by solving the Navier-Stokes equations directly where the other methods use Morison’s equation
to calculate these forces.
Morison’s equation is made up out of an inertia part dependent on acceleration (K M u̇) and a damping
(drag) part dependent on velocity squared (K D |u|u). These parts can directly be inserted in the equation of
motion of the system.
1
K D = ρDC D
2
(3.10)
π
ρD 2C M
4
(3.11)
KM =
Where C D and C M are dependent on the KC number (see section 2.1.2)
3.6. Analysis
After the numerical models are validated in chapter 4, the specific load case can be analyzed. The simulations
are started with one degree of freedom where only motion in-line with the fluid is allowed. After that more
degrees of freedom are added.
The simulations with one degree of freedom will be performed over a range of fluid oscillation frequencies
which are lower than resonance frequency, higher than resonance frequency and around resonance. Also a
decay test will be performed. These are theoretical fluid oscillation frequencies as some of these frequencies in reality are never encountered. The simulations with more than one degree of freedom will only be
performed with realistic fluid oscillation frequencies.
4
Validation
4.1. Introduction
The numerical models used to simulate the behavior of the simplified in-line structure need to be validated.
This is done by comparing simulations of oscillating flat plates normal to the flow with experimental results.
These flat plates are chosen as in-line structures are quite similar in shape and movement. Next to that, an
abundance of experimental data is available.
4.2. Steady Flow
First a flat plate in steady flow is simulated in 2D. 2D is chosen to reduce the simulation durations. Geometry
and flow conditions are based on a numerical research done by Raciti Castelli [2]. A sharp edged plate with a
width of 0.1 m and a thickness of 0.003 m is placed in a domain which is 1.2 m upstream and sideways. The
domain is 2.5 m long downstream. Water flows normal to the plate at 1 m/s. The drag coefficient is measured
which is expected to range between 1.9 and 1.98 [23] for flows with a Reynolds number higher than 104 .
CD = 1
F
2 ρU
Re =
2D
uL 1 · 0.1
=
= 105
ν
10−6
(4.1)
(4.2)
The flow is assumed to be incompressible and turbulent. A segregated solver is used. A steady solver
is chosen even though the flow is not expected to be steady due to unsteady vortex shedding. However,
we are looking for time-averaged values (drag coefficient), the use of a steady solver can be justified. A k −
² turbulence model is used with a boundary layer thickness which is approximated by the boundary layer
thickness of flow past a flat plate [20]. As the flow hits the plate and then flows past the plate up to the sharp
corner, this approximation is used.
0.382x
δbound ar y ≈ p
= 0.00382met er s
5
Re
(4.3)
The used mesh and resulting drag coefficient are presented in figure 4.1. As can be seen, the drag coefficient is averaged over a representative number of iterations. This gives an average drag coefficient of 1.955
which is in agreement with empirical data.
15
16
4. Validation
2.8
2.6
2.4
2.2
CD
2
1.8
1.6
1.4
1.2
(a) Mesh
(b) Mesh close-up
1
2
2.2
2.4
2.6
2.8
Iteration
3
3.2
3.4
3.6
4
x 10
(c) Drag Coefficient
Figure 4.1: Mesh and resulting drag coefficient
4.3. Oscillating Hydrodynamic Forces
The hydrodynamic forces acting on flat plates oscillating normal to the flow have been studied by many. The
main field of applicability is the design of bilge keels which add damping to the roll motion of ships.
Martin[13], Keulegan & Carpenter[12], Sarpkaya & O’Keefe[19] and Ridjanovic[16] have all investigated the
inertia and drag coefficients of flat plates oscillating normal to the flow at different Keulegan-Carpenter numbers.
All four experiments agree on the drag and inertia coefficients over a range of KC numbers from 1 to 100.
These results provide a good basis for the validation of the numerical models.
(a) Drag coefficient
(b) Inertia Coefficient
Figure 4.2: Hydrodynamic coefficients
4.4. Simulation set-up
The conditions under which the plate is simulated resemble the experimental conditions and where possible,
the in-line structure installation conditions are used.
4.4.1. Physical set-up
• Geometry
In table 4.1 the geometries of the plates used in the four experiments are presented.
Based on these geometries a plate with a width of 10 cm and a thickness of 3 mm is used in the simulations. The width of the plate is used as the characteristic length (D) throughout the validation. The
plate is modeled in 2D as all experiments use large enough aspect ratio’s to regard the plate as 2D.
1 Wall bounded at both sides to eliminate 3D effects
2 With a 60 degree bevel
4.4. Simulation set-up
17
Ridjanovic
Keulegan & Carpenter
Sarpkaya & O’Keefe
Martin
Width [cm]
1.59-10.16
1.27 - 7.62
9.1
1.6 - 6.3
Aspect ratio
1-19.4
6.8 - 40.9
∞1
4.9 - 19.4
Thickness [mm]
1.59
62
1.59
Table 4.1: Plate geometries used in experiments
In order to aid mesh generation, all corners are rounded. The radius of these corners is set at 0.34 mm.
Large enough to aid the mesh generation but small enough to maintain a sharp corner.
• Forcing
In all experiments the plate is oscillated through the liquid. In later simulations, where the structure
can move freely, it is more convenient to oscillate the fluid. Therefore the fluid is oscillated instead
of the structure during validation. The displacement, velocity and acceleration are described with a
(co)sine function.
Z
u(t )d t = A ∗ si n(ωt )
(4.4)
u(t ) = Aω ∗ cos(ωt )
(4.5)
u̇(t ) = −Aω2 ∗ si n(ωt )
(4.6)
Where A is the amplitude and ω is the frequency of the oscillation. The mean velocity applied is zero.
In order to validate that oscillating the plate and oscillating the structure produce the same results, one
simulation is performed with both. Results of these simulations will be compared.
• Keulegan-Carpenter number
The KC numbers under which the four experiments have been performed, are in the range 1 to 100.
The validation has to be performed in this range of KC numbers. Hoekstra[8] showed in his research
that the oscillations of in-line structures occur at low KC numbers (0-2). The range of experimental results allows proper validation. In order to keep the thickness of the turbulent boundary layer constant
so the same mesh can be used for all simulations, the Reynolds number needs to be kept constant. As
the Reynolds number is influenced by velocity (see equation 4.2, the period is varied in order to vary
the KC number.
KC =
2πA
Um T
=
D
D
(4.7)
An increase in oscillation period will not lead to longer simulations as the amount of time-steps is fixed
per oscillation and not per second.
• Reynolds number
The Reynolds number is solely dependent on the maximum velocity as the plate width and kinematic
viscosity of the fluid remain constant (see equation 4.2).
Keulegan and Carpenter performed their experiments at Reynolds numbers ranging form 4,500 to 12,800.
Sarpkaya and O’Keefe used the frequency parameter β(= Re/K C ) in their experiments. For their ’free’
plate experiment its value was 1,845. As the KC number ranged from approximately 1 to 100, the
Reynolds number ranged from 1,845 to 184,500. In the articles published by Martin and Ridjanovic
the Reynolds number is not mentioned.
In the research done by Hoekstra a maximum relative velocity of 0.9 m/s was found. Combining this
with a characteristic length of 6.6 meters, the Reynolds number is 6 ∗ 106 . The high Reynolds number
18
4. Validation
is mainly due to the large length scale. As the simulated plate is only 0.1 meters wide, the Reynolds
numbers found in the experiments by Keulegan and Carpenter and Sarpkaya and O’Keefe will be used
as a reference. To stay within the experimental ranges of Reynolds numbers, a Reynolds number of
10,000 is chosen. With the kinematic viscosity at 8.9 · 10−7 [m 2 /s], this leads to a velocity amplitude of:
Um =
νRe
m
= 0.089
D
s
(4.8)
4.4.2. Numerical set-up
In this section the numerical set-up and models used in the simulation are discussed.
• Domain, boundaries
The calculation domain is set at 15 characteristic lengths in all directions(see figure 4.3). This is large
enough to ensure that the boundaries have no effect on the solution. In the flow direction two velocity
inlets are used to oscillate the fluid. Both inlets have a defined velocity profile in the form of Um cos(ωt )
where the oscillation frequency is varied depending on the KC number. The plate has a no-slip wall
boundary and the sides of the domain are modelled with a symmetry plane. A no-slip wall boundary
means that no fluid can pass through this wall and that viscous friction occurs at the wall. A symmetry
plane imposes symmetrical flow conditions.
Figure 4.3: Domain size and boundary types
• Physics
Some assumptions have been made to aid simulation speeds:
– Water is modeled as an incompressible liquid because of the low compressibility and relatively
low pressures.
– The plate is assumed to be two dimensional. The aspect ratio’s used in the experiments are large
enough to model the plate in two dimensions
A segregated solver is chosen as the flow is assumed incompressible.
• Mesh
The mesh is divided into four regions:
1. Far field mesh
2. Large refinement region
3. Small refinement region
4. Prism layer mesh
Regions 1 through 3 are Trimmed cell meshes. Closer to the plate refinement zones are introduced to
simulate details accurately. The Prism layer mesh is there to model the turbulent boundary layer.
4.4. Simulation set-up
19
Figure 4.4: Refinement regions
Prism layer mesh
The Prism layer mesh is controlled by three parameters:
1. Total thickness
2. Stretching factor
3. Number of layers
The total thickness is determined by the boundary layer thickness. The approximation for flow along a
flat plate is used as the fluid needs to flow along the plate before it can separate at the edge. The total
width of the plate is used as the characteristic length scale. This is done because the flow pattern is
unknown and in future cases inclined plates are used. The boundary layer thickness is approximated
by equation 4.3.
The thicknesses of the individual Prism layers vary. Equations B.1 and B.2 describe the build-up of the
PLM. The stretching factor (r ) is set at 1.2 and the total PLM thickness is calculated from equation 4.3.
The coarser the mesh, the less cells it contains. The less cells, the shorter the simulation. To find the
optimal mesh settings, a mesh convergence study has been performed. The point where a finer mesh
doesn’t significantly influence the results is the optimal mesh setting. This convergence study can be
found in Appendix C.
• Time-step
Implicit temporal discretization is used. The implicit method is unconditionally stable as opposed to
the explicit method. The CFL-condition does not have to be satisfied when the implicit method is used.
This does not mean that the solution is accurate.
The physical phenomenon is of oscillatory nature. The time-step is chosen to be dependent on the
oscillation frequency of the plate. A convergence study on the amount of time-steps per oscillation has
been performed and can be found in Appendix C.3.
• Turbulence model
Due to the sharp corners separation is likely to occur. According to the STAR-CCM+ User guide [3],
the best option is to use an k − ² turbulence model. This is a model which uses turbulent viscosity
and turbulent kinetic energy to model the Reynolds stress tensor. Again following the user guide and
opting for the most robust and flexible model, the realizable two-layer k − ² turbulence model is chosen. This model uses a wall function which is called "all-y + ". More information on the mathematical
formulation of this wall function can be found in [3].
20
4. Validation
4.5. Results
Simulations have been performed with KC numbers ranging from 1 to 80. The resulting drag and inertia
coefficients from the simulations are similar to the coefficients found in experiments (see figure 4.7).
4.5.1. Post-processing
The experiments all use a Fourier analysis to obtain the drag (C d ) and inertia (C m ) coefficients in Morison’s
equation. According to Sarpkaya [18, p. 76] Fourier analysis and the Least Squares Method (LSM) yield identical C m values and C d values differ only slightly. A LSM is chosen to fit Morison’s equation to the simulated
in-line force.
Sumer and Fredsøe formulated a goodness-of-fit parameter δ [21, p. 147-148] which states how well the
Morison equation fits experimental data.
RT
δ=
0 (FC F D − F f i t )
RT 2
0 FC F D
2
(4.9)
Where T is the total duration of data sampling and FC F D and F f i t are simulated and fitted curves. The
goodness-of-fit parameters are presented in table 4.2 for every simulation.
KC number
1
2
3
5
8
11
15
20
30
40
80
δ
0.0082
0.0122
0.0191
0.0360
0.0392
0.0403
0.0382
0.0257
0.0150
0.0105
0.0171
Table 4.2: Goodness-of-fit parameters
The highest goodness-of-fit parameter (and therefore the worst fit) is inspected visually under the assumption that all other fits are of better quality. As can be seen in figure 4.5, the fit of Morison’s equation to
the simulated data at KC = 11 is of sufficient quality.
3
2
Force [N]
1
0
−1
−2
−3
60
80
100
120
140
Time [s]
160
180
200
Figure 4.5: Morison equation fit at K C = 11 and δ = 0.0403
220
4.6. Conclusions
21
4.5.2. Oscillation source
At a KC number of 8, two simulations have been performed. One where the plate is oscillated and one where
the fluid is oscillated. The drag and inertia coefficients are respectively 2,3 % and 2,5% higher when the fluid
is oscillated compared to an oscillated plate. A plot of the in-line force time traces is presented in figure 4.6.
This means that the difference between oscillation source is relatively small.
3
Forced fluid
Forced plate
2
Force [N]
1
0
−1
−2
−3
50
100
150
200
Time [s]
Figure 4.6: In-line force time-traces for a oscillated fluid or plate at KC=8
4.5.3. Hydrodynamic coefficients
The resulting hydrodynamic coefficients per KC number are plotted in figure 4.7 together with the data from
experiments.
18
5
Sarpkaya & O’Keefe (1996)
Keulegan & Carpenter (1958)
Ridjanovic (1962)
Martin (1958)
CFD simulation
16
14
Sarpkaya & O’Keefe (1996)
Keulegan & Carpenter (1958)
CFD simulation
4.5
4
12
3.5
3
Cd
Cm
10
8
2.5
6
2
4
1.5
2
1
0
−1
10
0.5
−1
10
0
10
1
10
KC number
(a) Drag coefficient
2
10
3
10
0
10
1
10
KC number
2
10
3
10
(b) Inertia Coefficient
Figure 4.7: Hydrodynamic coefficients
4.6. Conclusions
The drag and inertia coefficients are in good agreement with experimental data. There is proof that a Least
Squares Method provides a similar fit accuracy and that there is a small difference between oscillating plate or
oscillating fluid. The conclusion that the used numerical models are suitable for the simulation of oscillating
flat plates normal to the flow can be drawn.
5
1 Degree of Freedom
5.1. Introduction
In this chapter the system will be analyzed in 1 DoF. How the system behaves and the description of this behavior will be the main subjects of this chapter.
First, the set-up of the simulations will be discussed. After that, the system’s behavior is analyzed. A
decay test and simulations at frequencies below, around and above the resonance frequency of the system
are performed. The simulated behavior will then be described with Morison’s equation with KC dependent
and DNV GL prescribed hydrodynamic coefficients. At the end of this chapter conclusions are drawn on the
1 DoF case and the impact of simplifications is discussed.
5.2. Setup
A thin flat plate with the same width to thickness ratio as in chapter 4 is used. The width of the plate is
increased to 9.5 meters whereby the thickness becomes 0.285 meters. The plate is given a mass of 117.39 mT
so it resembles a typical subsea structure as discussed in section 3.2.2.
The 6-DoF solver and therefore DFBI is incompatible with 2D meshes. Therefore this plate is modeled in a
3D domain which is 30 times the width of the plate in the x- and y-direction and 1.05 meters in the z-direction
(see figure 5.1). With the use of symmetry planes the plate is modeled as if it was infinite in the z-direction.
This reduces simulation durations as the amount of cells is reduced (see Appendix B.4). The real problem
is in three dimensions where this simulation treats the problem in two dimensions. This simplification in
two dimensions influences the total force calculated but not the way forces can be described. In the DNV GL
guideline for marine operations [23] Appendix A and B added mass and drag coefficients are given for 2D and
3D shapes. For flat plates both added mass and drag coefficients are lower in 3D than in 2D. The aspect ratio
of the plate determines how big the difference is. The amplitude of the force therefore is influenced but the
description method not. Using a 2D simulation is conservative as the found coefficients are higher than in 3D.
Just as in chapter 4 a Trimmed cell mesh is used with a minimum of two cells over the thickness of the
domain. A PLM is used consisting out of 20 layers with a 1.2 stretching factor. The total thickness of the PLM
is based on flow past a flat plate where the Reynolds number is calculated using the flow velocity amplitude.
In order to aid mesh generation the corners are rounded with 32 points per circle. General mesh and a
close-up can be found in figure 5.2.
A segregated solver using an implicit time discretization scheme is used. The turbulence is modeled using
a k − ² turbulence model. 200 time-steps per fluid oscillation period are set with a minimum of 15 inner
iterations in the segregated solver.
The plate is allowed to move in the y-direction (in-line with fluid motion) and is restrained by a spring as
discussed in section 3.3.
23
24
5. 1 Degree of Freedom
Figure 5.1: Domain and boundaries
(a) General Mesh
(b) Mesh close-up
Figure 5.2
5.3. Decay test
In order to determine the damped natural frequency, damping and added mass, a decay test is performed.
This will give a good first insight in the behavior of the system. It further validates the numerical model and
certain analysis methods can be tested. A spring with a coefficient of 600 N /m is attached to the plate and an
initial displacement of -30 meters is set.
The linearized equation governing the motion of the plate during this decay is as follows:
(m + m a )ẍ + c a ẋ + kx = 0, x(0) = −30, ẋ(0) = 0
(5.1)
where m is the structure mass, m a is the added mass, c a the added damping and k the spring coefficient.
This can also be written as:
ẍ + 2ζω0 ẋ + ω20 x = 0
where ω0 =
q
(5.2)
k
p ca
. The first parameter (ω0 )is the undamped natural frequency and
m+m a and ζ =
(m+m )k
a
the second (ζ) is called the damping ratio. The response frequency of an underdamped system is a function
of these two parameters.
ωd = ω0
q
1 − ζ2
For underdamped systems, the solution of the differential equation of motion is:
(5.3)
5.3. Decay test
25
x(t ) = Ae −ζω0 t si n(ωd t )
(5.4)
where A is the initial amplitude of oscillation.
Noticing in equation 5.4 that the amplitude of oscillation decreases exponentially per oscillation, the difference in amplitude between oscillations can be used to determine the damping ratio (ζ). This is called a
logarithmic decrement (see Appendix D).
The added mass and damping can not be measured directly from a decay test. The damped frequency
and damping ratio can be measured. From these values added mass and damping can be calculated.
The resulting translation of the free decay test is presented in figure 5.3. The damping and periods of
oscillation are determined with the logarithmic decrement method using both the positive peaks (table 5.1)
and the negative peaks (table 5.2). The damping ratios per oscillation are plotted in figure 5.4a using both the
positive and the negative peaks. It can be seen that the difference between the two methods is only noticeable
when the amplitude of oscillation is changing fast. This is to be expected as the negative peaks method
lags half a period compared to the positive peaks method. The damping ratio is dependent on the velocity
amplitude through the KC number. With large differences between velocity amplitudes of the oscillations
using the positive or negative peaks, the difference in damping ratio at the first oscillations is explained.
15
10
5
Displacement [m]
0
−5
−10
−15
−20
−25
−30
0
200
400
600
Time [s]
800
1000
1200
Oscillation number
1
2
3
4
5
6
7
ωd [rad/s]
0.0486
0.0549
0.0554
0.0554
0.0555
0.0556
0.0555
Figure 5.3: Translation of plate during decay test
Oscillation number
1
2
3
4
5
6
7
8
ωd [rad/s]
0.0402
0.0545
0.0566
0.0555
0.0556
0.0556
0.0556
0.0555
ζ [-]
0.4690
0.1819
0.0240
0.0227
0.0216
0.0310
0.0238
0.0193
Table 5.1: Data decay test (positive peaks)
ζ [-]
0.3087
0.1678
0.0697
0.0519
0.0310
0.0241
0.0206
Table 5.2: Data decay test (negative peaks)
Using a Fast Fourier Transform (FFT)[24], the oscillation frequencies of the simulation are analyzed. A
FFT of the entire simulation is presented in figure 5.4b.The oscillation frequency increases every oscillation
as can be observed in tables 5.1 and 5.2. This can also be observed in the FFT magnitude plot. The absence
of a clear spike indicates that the oscillation frequency is spread out over a bandwidth. From this it can be
concluded that the damping ratio varies per oscillation (see equation 5.3). This is due to the varying KC
26
5. 1 Degree of Freedom
number as velocity amplitudes vary over the oscillations (see equation 2.2. Drag and inertia, and therefore
added mass and damping, vary with KC number as shown in section 4.5.3.
9000
0.5
Positive peaks
Negative peaks
0.45
8000
0.4
7000
6000
Magnitude [−]
Damping ratio [ζ]
0.35
0.3
0.25
0.2
5000
4000
3000
0.15
0.1
2000
0.05
1000
0
1
2
3
4
5
Oscillation number
6
7
0
8
(a) Damping ratio with positive and negative peaks
0
0.05
0.1
Frequency [rad/s]
0.15
0.2
(b) FFT magnitude plot of displacements decay test
Figure 5.4: Damping ratio (ζ) and FFT magnitude plot of decay test
5.4. Oscillated simulations
The plate with a width of 9.5 meters is subjected to an oscillating flow. The velocity amplitude of fluid oscillation is set at 1 m/s as the PLM is based on the Reynolds number at this velocity and it is in the range of
naturally encountered amplitudes (see table 3.4).
The oscillation frequency of the fluid is varied and simulations are performed at frequencies below, near
and above the resonance frequency of the system. For all simulations the response frequencies and response
amplitudes are measured and analyzed.
The resonance frequency of the system is determined by:
s
ω0 =
k
m + ma
(5.5)
Where the added mass is approximated by Morison’s equation with KC dependent coefficients:
ma =
π
ρD 2C M
4
(5.6)
5.4.1. Oscillation frequency below resonance
Two simulations have been performed where the fluid oscillation frequency is lower than the resonance frequency of the system. In table 5.3 the input data is presented.
Simulation
#1
#2
ω
0.0344
0.0688
ω0
0.1496
0.1353
Frequency ratio
0.23
0.5083
K C f l ow
19.23
9.61
Table 5.3: Input data simulations below resonance
The resulting displacements of the plate are presented in figures
5.5 and 5.6.
It can be seen that multiple frequencies are present in the response. To further analyze this, FFTs have
been performed on the displacements. The signals between the red dotted lines in figures 5.5 and 5.6 are
used for the FFTs. The magnitudes are plotted in figure 5.7.
5.4. Oscillated simulations
27
6
Displacement [m]
4
2
0
−2
−4
−6
0
500
1000
1500
2000
2500
Time [s]
3000
3500
4000
4500
5000
Figure 5.5: Simulation #1, Translation
15
Displacement [m]
10
5
0
−5
−10
−15
0
500
1000
1500
2000
2500
Time [s]
Figure 5.6: Simulation #2, Translation
4000
6000
3500
5000
3000
Magnitude [−]
Magnitude [−]
4000
2500
2000
1500
3000
2000
1000
1000
500
0
0
0.05
0.1
Frequency [rad/s]
0.15
(a) Simulation #1, Displacement FFT
0.2
0
0
0.1
0.2
0.3
Frequency [rad/s]
0.4
0.5
(b) Simulation #2, Displacement FFT
Figure 5.7: Displacement FFT magnitude plots of Simulation #1 and #2
From the FFT magnitude plots we can conclude that multiple frequencies indeed are present. Both simulations have a dominant response at the fluid forcing frequency and a smaller response at three times the fluid
forcing frequency. These higher frequency oscillations do not have a significant effect on the displacement
amplitude or on the capability of Morison’s equation to describe with an accurate amplitude (see section
5.8.2). However, these higher frequency oscillations can be significant when fatigue is analyzed.
These higher frequency oscillations are caused by the vortex shedding regime (see section 5.7). The regular Morison’s equation is unable to describe these higher frequencies accurately. Superposition of multiple
Morison’s equations with higher frequencies can fit this signal better. More on this improved description in
section 5.8.2
28
5. 1 Degree of Freedom
5.4.2. Oscillation frequency near resonance
One simulation has been performed where the oscillation frequency of the fluid is near the resonance frequency of the system. In table 5.4 the input data is presented.
Simulation
#3
ω
0.15
ω0
0.1586
Frequency ratio
0.946
K C f l ow
4.2
Table 5.4: Input data simulations near resonance
The resulting displacements are presented in figure 5.8. It can be seen that stable behavior is rapidly
obtained.
10
8
6
Displacement [m]
4
2
0
−2
−4
−6
−8
−10
0
100
200
300
Time [s]
400
500
600
Figure 5.8: Simulation #3, Translation
Unlike in simulations #1 and #2 no higher frequency response is observed. Only one response at the fluid
oscillation frequency is observed. This is confirmed by the FFT magnitude plot in figure 5.9.
12000
10000
Magnitude [−]
8000
6000
4000
2000
0
0
0.05
0.1
0.15
0.2
Frequency [rad/s]
0.25
0.3
Figure 5.9: Simulation #3, Displacement FFT
5.4.3. Oscillation frequency above resonance
Four simulations have been performed where the oscillation frequency of fluid is higher than the resonance
frequency of the system. In table 5.5 the input data is presented.
5.4. Oscillated simulations
29
Simulation
4
5
6
7
ω
0.0715
0.3142
0.6614
1.3228
ω0
0.0456
0.051
0.1586
0.1615
Frequency ratio
1.568
6.16
4.17
8.19
K C f l ow
9.25
2.10
1
0.5
Table 5.5: Input data simulations above resonance
The resulting displacements of the plate are presented in figures 5.10, 5.11, 5.12 and 5.13.
15
Displacement [m]
10
5
0
−5
−10
−15
400
600
800
1000
Time [s]
1200
1400
1600
Figure 5.10: Simulation #4, Translation
Displacement [m]
2
1
0
−1
−2
360
370
380
390
400
410
420
430
440
450
Time [s]
Figure 5.11: Simulation #5, Translation
1.5
Displacement [m]
1
0.5
0
−0.5
−1
−1.5
0
20
40
60
80
100
120
140
Time [s]
Figure 5.12: Simulation #6, Translation
From the FFT magnitude plots in figures 5.14 and 5.15 it can be seen that the response is only at the fluid
oscillation frequency.
30
5. 1 Degree of Freedom
Displacement [m]
1
0.5
0
−0.5
−1
0
10
20
30
40
50
60
70
Time [s]
Figure 5.13: Simulation #7, Translation
4
x 10
3.5
800
700
3
600
Magnitude [−]
Magnitude [−]
2.5
2
1.5
1
400
300
200
0.5
0
500
100
0
0.05
0.1
0.15
0.2
Frequency [rad/s]
0.25
0
0.3
0
0.2
0.4
0.6
Frequency [rad/s]
0.8
1
(b) Simulation #5, Displacement FFT
(a) Simulation #4, Displacement FFT
Figure 5.14: FFT magnitude plots of Simulation #4 and #5
1200
450
400
1000
Magnitude [−]
Magnitude [−]
350
800
600
400
300
250
200
150
100
200
50
0
0
0
0.5
1
Frequency [rad/s]
1.5
2
(a) Simulation #6, Displacement FFT
0
0.5
1
1.5
Frequency [rad/s]
2
2.5
3
(b) Simulation #7, Displacement FFT
Figure 5.15: FFT magnitude plots of Simulation #6 and #7
5.5. Added mass and added damping
For each simulation the added mass and added damping are calculated. These are the mass and damping
that are added to the system by the interaction of the fluid and structure.
The equation of motion of the system, where relative motions are used, can be written as follows:
m ẍ + c ẋ + kx = F h yd r o
(5.7)
We assume the motion of the structure to be a perfect sine (x = A · si n(ωt )). From the FFT plots we can
see that this assumption is true for simulations with low KC numbers. At higher KC numbers (higher than 8)
higher frequency oscillations are present. These are neglected in this analysis.
5.5. Added mass and added damping
31
It is known that the hydrodynamic force is relative to acceleration and velocity. The hydrodynamic force
has a sine and cosine part.
F h yd r o = F s si n(ωt ) + F c cos(ωt )
(5.8)
Combining equations 5.7 and 5.8, he equation of motion can be rewritten as:
(m +
1
1
F s )ẍ + (c −
F c )ẋ + kx = 0
ω2 A r el
ωA r el
| {z }
| {z }
added mass
(5.9)
added damping
F s and F c are Fourier coefficients where the Fourier series has a single frequency. These coefficients are
found by:
2
nT
Z nT
Fs =
2
nT
Z nT
Fc =
0
0
F h yd r o si n(ωt )d t
(5.10)
F h yd r o cos(ωt )d t
(5.11)
where n is the number of cycles over which is averaged.
The added mass (m a ) and added damping (c a ) are found by:
ma =
ca =
1
Fs
(5.12)
1
Fc
ωA r el
(5.13)
ω2 A
r el
The added mass and added damping can also be approximated with Morison’s equation where:
ma =
π
ρD 2C M
4
(5.14)
1
8
c a = ρDC D Um
2
3π
(5.15)
In the latter equation the drag is linearized. See Appendix E for the derivation and validation of the linearized drag. The average amplitude of the relative velocity is used in the added damping formulation.
The added mass and added damping are calculated for all simulations with both methods and presented
below.
Simulation
1
2
3
4
5
6
7
KC number
19.23
9.61
4.2
9.25
2.1
1
0.5
m a theoretical
106,064
155,560
141,419
155,560
113,135
81,316
74,245
m a measured
33,259
-20,493
176,350
-1,202
120,260
87,358
97,327
c a theoretical
15,284
20,714
28,637
9,251
12,066
30,407
42,715
c a measured
17,608
27,371
15,240
-22
11,895
17,928
14,354
Table 5.6: Added mass and added damping coefficients
It can be seen that the theoretical values from Morison’s equation and the measured values using Fourier
analysis do not show a perfect match. This is to be expected as Morison’s equation is a model of the forces
and not a fitted description. The difference in forces which are described is relatively small compared to the
difference in coefficients. The description of the forces using Morison’s equation and measured values from
Fourier analysis are compared. The result for simulations #2 and #7 are presented as these show large differences in added mass and added damping. Simulation #4 is a simulation where the total net force approaches
32
5. 1 Degree of Freedom
4
4
x 10
5
Theoretical
Measured
15
Added damping coefficient [N]
Added mass coefficient [N]
20
10
5
0
−5
0
5
10
KC number [−]
15
Theoretical
Measured
4
3
2
1
0
−1
20
x 10
0
5
(a) Added mass
10
KC number [−]
15
20
(b) Added damping
Figure 5.16: Added mass and added damping from Fourier analysis and Morison’s equation
4
3
5
x 10
1
x 10
Morison’s equation
Fourier analysis
2
Morison’s Equation
Fourier Analysis
0.5
Force [N]
Force [N]
1
0
0
−1
−0.5
−2
−3
500
600
700
800
Time [s]
900
(a) Force comparison, Simulation #2
1000
−1
10
15
20
Time [s]
25
30
(b) Force comparison, Simulation #7
Figure 5.17: Comparison of forces with Fourier analysis and Morison’s equation
zero (see figure 5.38). The found Fourier coefficients describe the force accurately. The relative error is large
at this simulation.
It can be concluded that at very high and very low KC numbers the measured values and theoretical values
for the dominating component (drag or inertia) are of the same order. At mid-range KC numbers or when the
total net force is very small the Fourier analysis gives erroneous results. Even up to the point where negative
damping is present and the system will start to resonate.
The miscalculation of added mass and added damping could have no significant effect on the net force
while the effect on the displacements is significant. The Fourier analysis is sensitive to phase differences and
therefore is not advised to use in a freely moving system.
5.6. Transmissibility
Transmissibility is the amplitude of the response (X) divided by the amplitude of the fluid motion (Y). Transmissibilities for the performed simulations are presented in table 5.7 and plotted in figure 5.18.
Simulation
1
2
3
4
5
6
7
ω
0.0344
0.0688
0.15
0.0715
0.3142
0.6614
1.3228
ω0
0.1496
0.1353
0.1586
0.0456
0.051
0.1586
0.1615
Frequency ratio
0.23
0.5083
0.946
1.568
6.16
4.17
8.19
X
4.31
5.37
8.58
14.44
1.76
0.77
0.34
Table 5.7: Transmissibilities of simulations
Y
29.07
14.53
6.67
13.98
3.18
1.51
0.76
X /Y
0.148
0.3695
1.2864
1.033
0.55
0.513
0.456
5.6. Transmissibility
33
1.4
Transmissibility [X/Y]
1.2
1
0.8
0.6
0.4
0.2
0
0
2
4
6
Frequency ratio [ω/ω0]
8
10
Figure 5.18: Transmissibility
At the resonance frequency the transmissibility is below 1.3. This means that dynamic amplification is
limited due to significant damping.
34
5. 1 Degree of Freedom
5.7. Vortex shedding regimes
Due to the sharp edges of the plate, vortices are created. These vortices are local low pressure area’s which
cause forces acting on the plate. The way vortices separate from the plate is called the vortex shedding regime.
There is a distinct difference observed in the vortex shedding regime of simulations with a low frequency ratio
(simulations #1 and #2) and simulations with a higher frequency ratio. This does not imply that frequency
ratio is the determining parameter for a certain vortex shedding regime. Probably the KC number is determining for the vortex shedding regime.
Keulegan and Carpenter [12] investigated the vortex shedding regime of flat plates oscillating normal to
the flow. In their research the vortex shedding regimes are ascribed to the KC number. During the longer
time that the flow moves in one direction at higher KC numbers, individual vortex shedding is observed. For
plates Keulegan and Carpenter found the transition between the two vortex shedding regimes to happen at a
KC number between 4 and 15. The drop in inertia coefficient between KC numbers of 8 and 13 (figure 4.2b)
is often attributed to the transition to an individual vortex shedding regime.
The same dependency is found when the KC numbers and the vortex shedding regimes of the performed
simulations are compared. For this analysis iterated KC numbers are used (see section 5.9).
Simulation
1
2
3
4
5
6
7
KC
19.17
9.94
3.72
4.21
0.64
0.54
0.29
Vortex regime
Individual
Individual
Synchronized
Synchronized
Synchronized
Synchronized
Synchronized
Table 5.8: Vortex shedding regime based on KC number
Installation of subsea structures in performed at KC numbers lower than 5 (see [8]). Therefore it can be
assumed that during subsea structure installation the synchronous vortex shedding regime is present. When
KC numbers increase the vortex shedding regime should be taken into account for fatigue analysis. In section
5.8.2 it will be shown that amplitudes are described accurately at higher KC numbers. Therefore for ultimate
strength analysis the vortex shedding regime is not important.
5.7.1. Individual vortex shedding
In simulations #1 and #2, individual, alternating vortex shedding is observed. Per translation from one side
to the other, three vortices are shed. This can be seen in figure 5.19. In this figure the vorticity (the spinning
motion of the fluid) is shown. The vortices are numbered in the figure. This vortex shedding frequency of
three times the fluid frequency explains the higher frequency component in the displacements of simulations
#1 and #2 (see figures 5.7a and 5.7b).
Figure 5.19: Vortex shedding regime of simulation #2, KC = 9.94
5.8. Description with Morison’s equation
35
5.7.2. Synchronized vortex shedding
In the other simulations synchronized vortex shedding is observed. With the frequency of plate oscillation
two vortices, at both sides of the plate shed at the same time. This causes the vortex induced forces to be in
phase and at the same frequency as the fluid motion. Therefore no higher frequencies are observed in the
FFT magnitude plots of these simulations (figures 5.9, 5.14a and 5.14b). The synchronized vortex shedding
can be observed in figure 5.20.
Figure 5.20: Vortex shedding regime of simulation #3, KC = 3.72
5.8. Description with Morison’s equation
In this section, the simulation results will be compared with predicted results using Morison’s equation.
The prediction of the results is done using the differential equation solver which can be found in Appendix
H. This solver uses the quadratic form of Morison’s equation for cylinders (equation 2.1) to determine the
force acting on the structure.
The experiments performed with plates (see section 4.3) use a reference volume to calculate the inertia
coefficients. This reference volume is that of a cylinder with the diameter equal to the width of the plate
( π4 D 2 ).
The Froude-Krylov force is based on the actual volume. Since the volume of a thin plate is not equal to
that of a cylinder, a volume factor (VF ) needs to be used.
Dt
VF = π 2
4D
(5.16)
Where t is the thickness of the plate
Equation 2.1 is altered to:
F=
π
VF ρD 2 u̇
| 4{z }
Froude-Krylov force
+
π
1
ρD 2C A (u̇ − ẍ) + ρDC D |u − ẋ|(u − ẋ)
4
|
{z
} |2
{z
}
Inertia force
(5.17)
Drag force
With C A = C M − VF .
The equation of motion which is solved is as follows:
π
π
1
(m + ( ρD 2 (C M − VF ))ẍ − ρD 2C M u̇ + ρDC D (ẋ − u)|ẋ − u| + kx = 0
4
4
2
(5.18)
Where x is the motion of the structure and u the velocity of the fluid.
It is essential that the quadratic drag formulation is used as this formulation agrees a lot better with simulated data than the linearized formulation as can be seen in figure 5.21 where the quadratic and linearized
form of drag are compared for simulation #5.
36
5. 1 Degree of Freedom
The hydrodynamic coefficients (C M and C D ) that are used in the predictions are KC number dependent
coefficients for oscillated plates where the KC number is determined by the flow conditions.
4
4
x 10
Simulation
Quadratic drag
Linearized drag
Force [N]
2
0
−2
−4
150
160
170
180
190
200
Time [s]
210
220
230
240
250
Figure 5.21: Comparison between quadratic and linearized drag formulation, Simulation #5
5.8.1. Decay test
The values for damped frequency (ωd ) and damping ratio (ζ) (see equation 5.3) are compared with theoretically calculated values. The values for k and m are known. The values for c a and m a are calculated with the
use of Morison’s equation with linearized drag.
1
8
c a = ρDC D Um
2
3π
ma =
(5.19)
π
ρD 2C M
4
(5.20)
The analysis of this simulation (section 5.3) has been performed with the use of the linearized drag equation (equation E.5). See Appendix E for the derivation and validation of the linearized drag equation.
This was necessary to be able to apply the logarithmic decrement (Appendix D).
In the linearized drag equation a velocity amplitude is needed. The maximum positive and negative velocities over an oscillation are averaged. This value is used as the velocity amplitude (Um ). These values are
also used to calculate a KC number per oscillation.
The DNV GL guidelines prescribe constant inertia and drag coefficients for plates in oscillating flow. The
coefficients for oscillating plates found in the experiments described in section 4.3 do have a KC number
dependency. This causes the the added mass and damping to vary with KC number.
The hydrodynamic coefficients can then be used to calculate theoretical values for ωd and ζ. Either with
constant coefficients prescribed by DNV GL or with KC number dependent coefficients from the experiments.
These values for ωd and ζ can be found in table 5.9.
Oscillation number
1
2
3
4
5
6
7
8
ωd DNV
0.0559
0.0565
0.0565
0.0565
0.0565
0.0565
0.0565
0.0565
ζ DNV
0.1424
0.0158
0.0061
0.0044
0.0035
0.0029
0.0024
0.0022
ωd Exp.
0.0404
0.0555
0.0563
0.0564
0.0564
0.0564
0.0564
0.0564
ζ Exp.
0.5784
0.1832
0.0772
0.0577
0.0454
0.0394
0.0335
0.0295
Table 5.9: Theoretical damped frequencies and damping ratios
The theoretical damping ratios are compared with the damping ratios found using the logarithmic decrement method on the DFBI simulation. In figure 5.22 it can be seen that the DNV GL method underestimates
5.8. Description with Morison’s equation
37
the damping while the KC number dependent coefficients slightly overestimate the damping.
It can be concluded that Morison’s equation with KC dependent coefficients is usable for approximating
added mass and damping during this thesis.
0.7
DFBI simulation
0.6
DNV
Damping ratio
0.5
KC dependent
0.4
0.3
0.2
0.1
0
1
2
3
4
5
Oscillation number
6
7
8
Figure 5.22: Damping ratios from DFBI and theoretical values
5.8.2. Oscillation frequency below resonance
The inputs for simulations #1 and #2 are presented in table 5.10.
Simulation
#1
#2
k
5000
5000
Um
1
1
ω
0.0344
0.0688
KC number
19.23
9.61
CM
1.5
2.2
CD
3.8
5
ω0
0.1496
0.1353
ω/ω0
0.23
0.5083
Table 5.10: Settings simulation #1 and #2
At these lower frequency ratios the KC number is above 8. As we have seen in sections 5.4.1 and 5.7,
there are oscillations present at a frequency higher than the fluid frequency. This is caused by the vortex
shedding regime at KC numbers higher than 8. Morison’s equation does not describe these higher frequency
oscillations accurately. The displacement amplitudes, important for ultimate strength analysis, are described
accurately.
The hydrodynamic force acting on the plate is in balance with the displacement and acceleration of the
plate. The displacement amplitudes are accurately described (see figures 5.23 and 5.25). The accelerations
are described with less accuracy (see figures 5.24 and 5.26).
5
Displacement [m]
Simulation
Morisons Equation
0
−5
1500
2000
2500
Time [s]
3000
Figure 5.23: Simulation #1, Translation comparison at ω/ω0 = 0.23
3500
38
5. 1 Degree of Freedom
0.06
Simulation
Morisons Equation
Acceleration [m/s2]
0.04
0.02
0
−0.02
−0.04
1500
2000
2500
Time [s]
3000
3500
Figure 5.24: Simulation #1, Acceleration comparison at ω/ω0 = 0.23
10
Displacement [m]
Simulation
Morisons Equation
5
0
−5
−10
1000
1100
1200
1300
1400
1500
Time [s]
1600
1700
1800
1900
2000
Figure 5.25: Simulation #2, Translation comparison at ω/ω0 = 0.5083
0.06
Simulation
Morisons Equation
Acceleration [m/s2]
0.04
0.02
0
−0.02
−0.04
−0.06
1000
1100
1200
1300
1400
1500
Time [s]
1600
1700
1800
1900
2000
Figure 5.26: Simulation #2, Acceleration comparison at ω/ω0 = 0.5083
Hydrodynamic forces acting on the plate are equal to the change of the plates momentum (m ẍ) and the
restoring spring force (kx). In the used system the latter is dominant. This can be seen in figure 5.27 where the
total force of simulation #1 is decomposed in the inertial, momentum changing force (m ẍ)and the restoring
spring force (kx).
5.8. Description with Morison’s equation
39
4
2.5
x 10
Spring force
Inertial force
Total force
2
1.5
Force [N]
1
0.5
0
−0.5
−1
−1.5
−2
−2.5
3000
3050
3100
3150
3200
3250
Time [s]
3300
3350
3400
3450
3500
Figure 5.27: Simulation #1, Force decomposition
The accelerations have a relatively small contribution to the total force (12.5 %). The inaccuracy in the
description of the accelerations is therefore insignificant when the total force is described. The displacements
and total forces are accurately described. The description of the total forces is presented in figures 5.28 and
5.29.
4
2
x 10
Simulation
Morisons Equation
Force [N]
1
0
−1
−2
1500
2000
2500
Time [s]
3000
3500
Figure 5.28: Simulation #1, In-line Force comparison at ω/ω0 = 0.23
4
3
x 10
Simulation
Morisons Equation
Force [N]
2
1
0
−1
−2
−3
1000
1100
1200
1300
1400
1500
Time [s]
1600
1700
1800
1900
2000
Figure 5.29: Simulation #2, In-line Force comparison at ω/ω0 = 0.5083
The higher frequency oscillations observed in section 5.4.1 are not described perfectly (figures 5.23 and
5.25). Morison’s equation is able to describe oscillations at uneven multiples of the fluid frequency. In figure 5.30 the magnitudes of the Fourier transform of the displacements and their description is shown. It can
40
5. 1 Degree of Freedom
be seen that Morison’s equation is able to describe some higher frequency oscillations but that the magnitudes of the present frequencies is not accurate. Again, for ultimate strength analysis these higher frequency
oscillations have insignificant impact. For fatigue analysis they may be significant.
3000
7000
Simulation
Morisons Equation
Magnitude [−]
2500
6000
5000
2000
4000
1500
3000
1000
2000
500
0
1000
0
0.05
0.1
Frequency [rad/s]
0.15
0
0.2
0
0.1
(a) Simulation #1
0.2
0.3
0.4
0.5
(b) Simulation #2
Figure 5.30: FFT magnitudes of displacements Simulation #1 and #2 and their description
As stated in section 5.4.1 is it possible to improve the description of the forces acting on flat thin plates
normal to an oscillating flow with an individual vortex shedding regime present. With the use of a superposition of several Morison’s equations at frequencies which are a multiple of the fluid oscillation frequency, a
more accurate fit can be accomplished.
In figures 5.31 and 5.32 you can see the improved descriptions of the force acting on the plate at simulations #1 and #2.
4
2
4
x 10
2
Simulation
Morisons Equation
1
1
0.5
0.5
0
−0.5
0
−0.5
−1
−1
−1.5
−1.5
−2
2000
2200
2400
2600
Time [s]
(a) Regular description
2800
3000
Simulation
Morisons Equation
1.5
Force [N]
Force [N]
1.5
x 10
−2
2000
2200
2400
2600
Time [s]
2800
3000
(b) Description with superposed Morison’s equations
Figure 5.31: Improved description simulation #1
5.8. Description with Morison’s equation
41
4
3
4
x 10
3
Simulation
Morisons Equation
2
Simulation
Morisons Equation
2
1
1
Force [N]
Force [N]
x 10
0
0
−1
−1
−2
−2
−3
1000
1100
1200
1300
Time [s]
1400
−3
1000
1500
1100
1200
1300
Time [s]
1400
1500
(b) Description with superposed Morison’s equations
(a) Regular description
Figure 5.32: Improved description simulation #2
It is clear that a significant improvement in the description of the force can be achieved. However, there
are some practical obstacles which makes this description not usable at the moment.
• The individual vortex shedding regime is present at KC numbers which are not present during subsea
structure installation. Improvements in the description of these higher frequency oscillations is therefore not interesting for HMC.
• The dependency of coefficients is unknown. A wide scale of parameters could influence the forces
acting on the structure.
• No experiments have been performed to determine the coefficients which are to be used in this superposed Morison’s equation.
5.8.3. Oscillation frequency near resonance
The inputs for simulation #3 are presented in table 5.11.
This simulation, which is near resonance, is accurately described by Morison’s equation. This is due to
the fact that the KC number is below 8 and a synchronous vortex shedding regime is present.
k
5000
Um
1
ω
0.15
KC number
4.2
CM
2
CD
8
ω0
0.1586
ω/ω0
0.946
Table 5.11: Settings simulation #3
10
Simulation
Morisons Equation
Displacement [m]
5
0
−5
−10
0
100
200
300
Time [s]
400
Figure 5.33: Simulation #3, Translation comparison at ω/ω0 = 0.946
500
600
42
5. 1 Degree of Freedom
1.5
Simulation
Morisons Equation
Velocity [m/s]
1
0.5
0
−0.5
−1
−1.5
0
100
200
300
Time [s]
400
500
600
Figure 5.34: Simulation #3, Velocity comparison at ω/ω0 = 0.946
0.3
Simulation
Morisons Equation
Acceleration [m/s2]
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
0
100
200
300
Time [s]
400
500
600
Figure 5.35: Simulation #3, Acceleration comparison at ω/ω0 = 0.946
4
3
x 10
Simulation
Morisons Equation
2
Force [N]
1
0
−1
−2
−3
0
100
200
300
Time [s]
400
500
600
Figure 5.36: Simulation #3, In-line Force comparison at ω/ω0 = 0.946
5.8.4. Oscillation frequency above resonance
The inputs for simulations #4, #5, #6 and #7 are presented in table 5.12.
Just as in simulation #3, all parameters are described accurately by the description using Morison’s equation. In simulation #4 the force is very low as the transmissibility, the ratio between structure and fluid dis-
5.8. Description with Morison’s equation
43
placement, is near unity (see table 5.7).
Simulation
#4
#5
#6
#7
k
600
600
5000
5000
Um
1
1
1
1
ω
0.0715
0.3142
0.6614
1.3228
KC number
9.25
2.1
1
0.5
CM
2.2
1.57
1.15
1.05
CD
5
11
14
18
ω0
0.0456
0.051
0.1586
0.1615
ω/ω0
1.568
6.16
4.17
8.19
Table 5.12: Settings simulation #4, #5, #6 and #7
20
Simulation
Morisons Equation
15
Displacement [m]
10
5
0
−5
−10
−15
−20
0
500
1000
1500
Time [s]
Figure 5.37: Simulation #4, Translation comparison at ω/ω0 = 1.568
4
2
x 10
Simulation
Morisons Equation
1.5
Force [N]
1
0.5
0
−0.5
−1
0
500
1000
Time [s]
Figure 5.38: Simulation #4, In-line Force comparison at ω/ω0 = 1.568
1500
44
5. 1 Degree of Freedom
5
Simulation
Morisons Equation
4
Displacement [m]
3
2
1
0
−1
−2
−3
0
50
100
150
200
250
Time [s]
300
350
400
450
Figure 5.39: Simulation #5, Translation comparison at ω/ω0 = 6.16
4
x 10
3
Simulation
Morisons Equation
2
Force [N]
1
0
−1
−2
−3
0
50
100
150
200
250
Time [s]
300
350
400
450
Figure 5.40: Simulation #5, In-line Force comparison at ω/ω0 = 6.16
2
Simulation
Morisons Equation
Displacement [m]
1.5
1
0.5
0
−0.5
−1
−1.5
0
20
40
60
80
100
Time [s]
Figure 5.41: Simulation #6, Translation comparison at ω/ω0 = 4.17
120
140
5.9. KC number based on relative motion
45
4
x 10
6
Simulation
Morisons Equation
Force [N]
4
2
0
−2
−4
0
20
40
60
80
100
120
140
Time [s]
Figure 5.42: Simulation #6, In-line Force comparison at ω/ω0 = 4.17
0.8
Simulation
Morisons Equation
Displacement [m]
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
0
10
20
30
40
50
60
70
Time [s]
Figure 5.43: Simulation #7, Translation comparison at ω/ω0 = 8.19
5
1
x 10
Simulation
Morisons Equation
Force [N]
0.5
0
−0.5
−1
0
10
20
30
40
50
60
70
Time [s]
Figure 5.44: Simulation #7, In-line Force comparison at ω/ω0 = 8.19
5.9. KC number based on relative motion
In the previous sections the KC number is based on flow conditions only. However, the KC number is a parameter of the relative flow. In this section the KC number will be based on the relative motion between fluid
and structure and the difference will be investigated. This is done by using the relative velocity between flow
and structure from the first description and determining a new KC number. The velocity amplitude is the only
varying parameter as the structure and fluid oscillate at the same frequency and the diameter is not changed.
This results in new hydrodynamic coefficients (see table 5.13).
The resulting relative velocity amplitudes, new KC numbers and the goodness of fit parameters are presented in the tables below. The goodness of fit parameters are calculated over the same time trace. The forces
and displacements are dominant and are therefore the only parameters of interest.
46
5. 1 Degree of Freedom
Figure 5.45: Flow diagram of KC number iteration
Simulation
#1
#2
#3
#4
#5
#6
#7
C D f l ow
3.8
5
8
5
14
14
18
C M f l ow
1.5
2.2
2
2.2
1.15
1.15
1.05
C Dr el
3.8
4.8
8.75
8
17.2
18
20
C M r el
1.5
2.3
1.95
2
1.05
1.05
1
Table 5.13: Drag and inertia coefficients. Based on flow and relative motion
Displacement
Force
K C f l ow
19.23
19.23
Umr el
0.9968
0.9968
K C r el at i ve
19.17
19.17
δ f l ow
0.9841
0.9904
δr el at i ve
0.98941
0.9904
Table 5.14: KC number iteration, Simulation #1
Displacement
Force
K C f l ow
9.61
9.61
Umr el
1.034
1.034
K C r el at i ve
9.94
9.94
δ f l ow
0.6639
0.8836
δr el at i ve
0.6403
0.8703
Table 5.15: KC number iteration, Simulation #2
Displacement
Force
K C f l ow
4.2
4.2
Umr el
0.8850
0.8850
K C r el at i ve
3.72
3.72
δ f l ow
0.9760
0.9880
δr el at i ve
0.9805
0.9888
Table 5.16: KC number iteration, Simulation #3
Displacement
Force
K C f l ow
9.25
9.25
Umr el
0.4555
0.4555
K C r el at i ve
4.21
4.21
δ f l ow
0.9982
0.9886
δr el at i ve
0.9997
0.4673
Table 5.17: KC number iteration, Simulation #4
The very low goodness of fit for the force in simulation #4 is due to the fact that the force itself is almost
zero. This can be seen in figure 5.38. The accuracy of describing the forces is therefore, within limits, not
relevant.
Displacement
Force
K C f l ow
2.1
2.1
Umr el
0.3044
0.3044
K C r el at i ve
0.64
0.64
δ f l ow
0.9402
0.7571
Table 5.18: KC number iteration, Simulation #5
δr el at i ve
0.8568
0.3606
5.10. Comparison with DNV GL coefficients
Displacement
Force
K C f l ow
1
1
47
Umr el
0.5392
0.5392
K C r el at i ve
0.54
0.54
δ f l ow
0.9741
0.9960
δr el at i ve
0.9659
0.9871
Table 5.19: KC number iteration, Simulation #6
Displacement
Force
K C f l ow
0.5
0.5
Umr el
0.5894
0.5894
K C r el at i ve
0.3
0.3
δ f l ow
0.9706
0.9959
δr el at i ve
0.9677
0.9922
Table 5.20: KC number iteration, Simulation #7
In all seven simulations the effect of KC number iteration is negligible. This has several causes:
• At simulations #1, #2, #3, #5, #6 and #7 the change in hydrodynamic coefficient is small. With the
exception of C D in simulations #5, #6 and #7, the change in coefficients is 2-10%. The inaccuracy in the
determination of the hydrodynamic coefficients can already be larger than 10% (see figure 4.2)
• Some larger changes in hydrodynamic coefficient at extreme KC numbers, like the change of 23% in
C D in simulation #5 at KC = 0.64 , don’t have a significant effect on the force. This is due to the drag or
inertia dominance.
• When the total hydrodynamic forces, as in simulation #4, are very small, the effect of KC number iteration is negligible.
KC number iteration costs time. The entire analysis needs to be performed multiple times. The benefits
of KC number iteration need to weigh up against the costs of extra analysis. KC number iteration should be
performed when the following conditions are satisfied:
1. The change in hydrodynamic coefficient is significant.
2. The KC number indicates that this force has a significant contribution.
3. The total hydrodynamic force is significant.
5.10. Comparison with DNV GL coefficients
The current method used at HMC for modeling hydrodynamic forces acting on subsea structures uses coefficients prescribed by DNV GL (see section 2.2.1). These coefficients do not have a KC dependency while for
flat plates oscillating normal to the flow this dependency is proven (see figure 4.7).
For infinitely long plates oscillating normal to the flow DNV GL prescribes a drag coefficient (C D ) of 2.5
and an inertia coefficient (C M ) of 1. This causes a difference with the force which is simulation with DFBI.
The displacement amplitudes for both the DNV GL method and KC dependent method are compared with
the simulated amplitude since this is the main input for ultimate strength analysis.
This comparison is only indicative since a difference in force is not linearly correlated with a displacement
amplitude. The force is relative to the relative acceleration and relative velocity while the displacement amplitude is also dependent on spring stiffness and absolute flow characteristics. The displacement amplitude
is the most important parameter in installation analysis and therefore is a good indicator.
In table 5.21 and figure 5.46 the displacement amplitudes are plotted as a percentage of the simulated
displacement against KC number. In Appendix G the time traces including the DNV GL description for simulations #1 through #7 are shown.
When we look at the KC dependent description, it can be noted that the description of the displacement
amplitude is accurate to a maximum difference of 12%. The description is more accurate at low KC numbers. This is due to the fact that at these KC numbers the drag force is very small and the inertia coefficient
converges to a value of 1. At the higher KC numbers (>5) the accuracy of the experiments which are used to
determine the hydrodynamic coefficients is lower which explains the decreased accuracy of the description
48
5. 1 Degree of Freedom
Simulation
#1
#2
#3
#4
#5
#6
#7
K C f l ow
19.23
9.61
4.2
9.25
2.1
1
0.5
X
4.31
5.37
8.58
14.44
1.76
0.76
0.33
X Mor
3.82
6.01
9.04
14.44
1.87
0.76
0.33
X D NV
2.60
2.76
8.22
14.44
1.29
0.63
0.30
Normalized displacement amplitude [−]
Table 5.21: Comparison of displacement amplitudes for different methods
1.3
Simulation
KC depedent
DNV GL
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0
5
10
KC number [−]
15
20
Figure 5.46: Displacement amplitudes for different modeling methods
of displacement amplitudes at these KC numbers.
The description which uses the DNV GL coefficients is always underestimating the displacement amplitude. This has to do with the fact that inertia and drag coefficients are used which are a minimum over the
entire KC number range. When the KC number is either very high or very small, so that inertia or drag forces
are relatively small, the effect of this simplification is limited as the coefficient which is underestimated by the
DNV GL method has a relatively low contribution to the total force. At intermediate KC numbers the effects
are more noticeable. These effects include:
• Underestimated hydrodynamic forces
• Underestimated damping
• Underestimated added mass which leads to overestimated natural frequency
Due to the underestimation of the hydrodynamic forces acting on the structure, the displacement amplitudes are always underestimated. Even at very low KC numbers the DNV GL method underestimates the
displacement amplitude with 10%. This implies that there still is a significant drag force present at this KC
number as the inertia coefficients are equal between the DNV GL method and the KC dependent method.
The drag forces are related to the diameter and the inertia forces are related to the volume. For thin flat
plates the ratio of diameter to volume (D/V ) is large. This explains the significant amount of drag present at
low KC numbers. Typical subsea structures have a smaller diameter to volume ratio (D/V ) which reduces the
significance of drag at these KC numbers.
The underestimation of displacement amplitudes by the DNV GL method is therefore expected to be
smaller for more voluminous subsea structures than thin flat plates.
5.11. Conclusion
49
5.11. Conclusion
Based on the 1-DoF case conclusions can be drawn. These are presented below. The impact of the simplifications used in this thesis are discussed in section 5.12.
• Between KC numbers of 8 and 13 the vortex shedding regime alters. At higher KC numbers the vortex
shedding frequency differs from the oscillation frequency and causes forces oscillating at the vortex
shedding frequency. Morison’s equation is unable to describe the forces associated with vortex shedding at a frequency which differs from the fluid oscillation frequency. Installation of subsea structures
occurs at KC numbers where the vortex shedding frequency and the fluid oscillation frequency are the
same. Morison’s equation is therefore applicable.
• At KC numbers higher than 8, where the vortex shedding frequency differs from the oscillation frequency, Morison’s equation is able to describe force amplitudes correctly. The relative contribution of
vortex induced forces is small.
• Forces on flat plates oscillating normal to the flow are underestimated by the DNV GL prescribed hydrodynamic coefficients. This is due to the fact that DNV GL uses minimum values for drag and inertia
coefficients over the entire KC number range. The amount of underestimation is smallest when drag or
inertia forces are relatively small (very high or very low KC numbers).
This underestimation of forces causes the displacements to be underestimated as well. For ultimate
strength analysis, this leads to overestimated workability.
• Hydrodynamic coefficients determined by forced oscillation experiments are accurate for describing
forces on flat plates moving freely in an oscillating flow.
• KC numbers based on flow conditions instead of relative motion are accurate. Iteration in KC number
is proven to yield little improvement in most cases.
5.12. Discussion
The simplifications made in this research and their expected impact are discussed in this section.
• Subsea structures are more voluminous than thin flat plates. The inertia forces are bigger for subsea
structures than for flat thin plates. The underestimation of forces by DNV GL prescribed coefficients at
low KC numbers is therefore smaller than for thin flat plates.
• Due to the small volume of thin flat plates is the Froude-Krylov force negligible. Whether or not this
formulation is accurate for subsea structures in an oscillating flow can therefore not be confirmed.
• Subsea structures and thin flat plates differ in geometry and porosity. This will cause the flow around
the subsea structure to be different. The KC number dependency of hydrodynamic coefficients will
therefore be different. Effects of multiple parameters, such as porosity, angle of attack and geometry
can not be multiplied or superposed as a combination of parameters affect the flow differently than
alone.
• The description uses an instantaneous motions combined with a KC number. This description method
is also applicable in irregular flow as FEM analysis is able to determine instantaneous motion. The KC
number needs to be determined for irregular flow where the flow field may differ.
• In this thesis a regular oscillating flow is used. In reality this flow is irregular. As KC number iteration
does not have a large influence on forces, an average KC number in irregular flow is suspected to yield
an accurate description.
• Vessel motion is neglected in this thesis. Vessel motions can force the structure in a different frequency
and/or direction than the flow frequency and/or direction. Morison’s equation is not able to deal with
two different frequencies. Using relative motion is expected to give good results.
• The simulations are performed with a model which is regarded as 2D. In other words, the flow is only
allowed to flow over two edges instead of four. This will have an effect on the hydrodynamic coefficients
but not on the method of describing the forces. According to [23] will both the added mass coefficient
50
5. 1 Degree of Freedom
(C A ) and the drag coefficient (C D ) reduce. This has the effect that the natural frequency of the system
will increase and the damping reduce. The forcing amplitude will reduce.
• Forces at multiple frequencies, such as in the individual vortex shedding regime (see section 5.7.1) can
be described by a superposed Morison’s equation more accurately. Research on the coefficients and
their dependencies is not available.
• Inaccuracies will always remain. A perfect description of forces acting on structures is impossible.
Some of the inaccuracies are listed below.
– Determination of flow conditions
– System characteristics (pipeline material, structure volume, ...)
– KC number
– Drag and inertia coefficients (see figure 4.2)
6
2 Degrees of Freedom
In this chapter the simulations with two degrees of freedom will be presented. The simulations will be limited
to a realistic range of oscillation frequencies and amplitudes. The second DoF is translation transverse to the
first DoF. Rotation around the z-axis is not allowed.
6.1. Set up
The oscillating flow now enters at a 30 degree angle measured counterclockwise (see figure 6.1). An angle
where both the decomposed x- and y-component of the flow is significant is chosen to be able to investigate
the possibility of using decomposed flow to describe the system.
The frequency of fluid oscillation is set at a minimum of 0.3141 rad/s which is equal to the longest encountered waves (see table 3.4). The exact oscillation frequencies will be discussed later. The velocity amplitude
of the flow is set at 1 m/s, just as in the 1-DoF case. This way no remeshing needs to be performed as the
Reynolds number stays the same.
Figure 6.1: Flow direction
6.1.1. Springs
There are two different methods which can be used to model the restoring pipeline force. One with a coupled
spring (figure 6.2a) and one with decoupled springs (figure 6.2b).
Linear springs are unable to produce a transverse force. The coupled system has freedom in the direction
tangential to the direction in which the spring is acting. The decoupled system does not have this freedom
because the two springs are at a 90 degree angle to each other.
In the decoupled system the springs are of such a length compared to the motions of the structure that
they can be linearized. They can be seen as moving with the plate and perfectly horizontal or vertical. In
order to produce equal forces in the coupled and decoupled system, the spring coefficients need to be equal.
51
52
6. 2 Degrees of Freedom
Two simulations have been performed to investigate the difference in solution between these two systems.
(a) Coupled spring
(b) Decoupled spring
Figure 6.2: Hydrodynamic coefficients
The first simulation has been performed with a coupled spring. A spring with a relaxation length of zero
meters is attached to the center of the plate with a spring coefficient of 5000 N/m (see section 3.3).
The second simulation has been performed with decoupled springs. Two springs with a spring coefficient
of 5000 N/m have been attached in both the x- and y-direction. A relaxation length of 1000 meters is used so
that the tangential components of the spring forces can be neglected.
3
Displacement in the y−direction [m]
Coupled spring
Decoupled spring
2
1
0
−1
−2
−3
0
100
200
300
400
Time [s]
500
600
700
800
Figure 6.3: Translation comparison y-direction
It can be seen in figures 6.3 and 6.4 that the solutions qualitatively do not differ between the coupled and
decoupled system. The amplitude of oscillation does differ.
HMC uses the Cartesian coordinate system in their analysis. Therefore the decoupled system is chosen.
Usage of the coupled system would result in descriptions in the polar coordinate system.
6.2. Resonance in x-direction
53
1
Coupled spring
Decoupled spring
0.8
Displacement in the x−direction [m]
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
100
200
300
400
Time [s]
500
600
700
800
Figure 6.4: Translation comparison x-direction
6.2. Resonance in x-direction
Two simulations have been performed. One at a fluid frequency of 0.3141 rad/s and one at 0.6283 rad/s. The
motion of the plate is decoupled and will be presented in the x- and y-direction.
The displacements in x- and y-direction for the simulation at 0.3141 rad/s are presented in figures 6.5 and
6.6.
4
Displacement in the y−direction
3
2
1
0
−1
−2
−3
−4
0
200
400
600
800
1000
Time [s]
1200
1400
1600
1800
2000
1600
1800
2000
Figure 6.5: Translation y-direction at ω = 0.3141r ad /s
5
Displacement in the x−direction [m]
4
3
2
1
0
−1
−2
−3
−4
−5
0
200
400
600
800
1000
Time [s]
1200
1400
Figure 6.6: Translation x-direction at ω = 0.3141r ad /s
The motion in the y-direction is at an steady amplitude and is oscillating at the fluid frequency. In the
x-direction unexpected behavior occurs. The plate oscillates at a frequency 32 times the fluid oscillation fre-
54
6. 2 Degrees of Freedom
quency. This frequency, 0.2064 rad/s, corresponds with the natural frequency of the system when no or very
little added mass is taken into account. The amplitude of the motion in x-direction increases over time which
indicates damping is very low.
A second simulation is performed at a fluid frequency two times higher (0.6283 rad/s). The resulting
displacements in the x- and y-direction are presented in figures 6.7 and 6.8.
Displacement in the y−direction [m]
1.5
1
0.5
0
−0.5
−1
−1.5
0
100
200
300
400
Time [s]
500
600
700
800
700
800
Figure 6.7: Translation y-direction at ω = 0.6283r ad /s
Displacement in the x−direction [m]
0.5
0.4
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
−0.5
0
100
200
300
400
Time [s]
500
600
Figure 6.8: Translation x-direction at ω = 0.6283r ad /s
Again the motion in the y-direction shows an oscillation with a steady amplitude at the fluid forcing frequency. The motion in the x-direction is again at the resonance frequency, this time at 13 times the fluid
frequency. This again indicates an absence of significant drag and added mass in the x-direction.
For both simulations the force due to pressure and shear is measured in the x-direction. These have an
amplitude of respectively 400 N and 1000 N. The time traces of these forces are presented in figures 6.9 and
6.10.
1000
Force [N[
500
0
−500
−1000
0
100
200
300
400
500
Time [s]
600
700
Figure 6.9: Force x-direction at ω = 0.3141r ad /s
800
900
1000
6.3. Drag and added mass in the x-direction
55
1500
Force [N]
1000
500
0
−500
−1000
−1500
0
100
200
300
400
500
Time [s]
600
700
800
900
Figure 6.10: Force x-direction at ω = 0.6283r ad /s
6.3. Drag and added mass in the x-direction
In the previous section the suspicion was raised that drag and added mass in the x-direction are relatively
low. In section 3.2.2 the FLET (see figure 3.2) is simplified as a flat plate. This causes the projected area in the
x-direction (along the plate) to be unrealistically small.
In order to make an estimation of the relative significance of the measured forces (figures 6.9 and 6.10), a
theoretical force is compared with the measured force.
The projected area of a FLET in x-direction is approximated by a 5 meter wide plate. The resulting KC
number in x-direction at a frequency of 0.3141 rad/s becomes 2. At this KC number the C D and C M values for
oscillating plates are 10 and 1.6. Using a LSM, Morison’s equation for a plate with a 5 meter diameter is fitted
to the simulation at 0.3141 rad/s. See figure 6.11 for a close-up of the fitting. This results in C D and C M values
of 0.1641 and 0.0607. This is respectively 1.64% and 3.79% of the theoretical values. When the same exercise
is performed for the simulation at 0.6283 rad/s a KC number of 1 is obtained. This corresponds to C D and C M
values of 14 and 1.15. The fitting using a LSM (see figure 6.12) results in measured values of 1.0333 and 0.1327
for C D and C M respectively. These amount to 7.36% and 11.54% of the theoretical values.
Froude-Krylov forces and inertia forces are combined in one inertia force. This is done because a Morison’s equation with a larger diameter than the diameter in the simulation is fitted. Using a separated approach
would result in negative values for C A (equation 2.1).
600
Simulation
LSM fit
400
Force [N]
200
0
−200
−400
−600
500
550
600
650
Time [s]
700
750
800
Figure 6.11: Close-up LSM fit at ω = 0.3141r ad /s
In order to obtain more realistic simulations artificial drag and added mass forces are introduced in the
x-direction in the simulations. The used hydrodynamic coefficients are the theoretical KC dependent values
with the measured values from the initial simulations subtracted. In this way the total forces in the simulation
agree with the theoretically calculated forces. The damping that is introduced in the x-direction is a purely
hydrodynamic added damping. No structural damping is added.
In the y-direction no alterations to the simulation have been made. Description of the forces can still be
done in the y-direction.
56
6. 2 Degrees of Freedom
1500
Simulation
LSM fit
1000
Force [N]
500
0
−500
−1000
−1500
200
220
240
260
280
300
Time [s]
320
340
360
380
400
Figure 6.12: Close-up LSM fit at ω = 0.6283r ad /s
The forces in x- and y-direction acting on the plate in the CFD simulations result in:
1
π
F x−d i r ec t i on = FC F D + ρD 2 (C M t heor et i c al −C Mmeasur ed )(u̇ − ẍ)+ ρD(C D t heor et i c al −C D measur ed )|u − ẋ|(u − ẋ) (6.1)
4
2
F y−d i r ec t i on = FC F D
(6.2)
6.4. Results
Simulations are performed at 10 and 20 second periods with the artificial drag and added mass in x-direction
(see equation 6.1). This results in the following translations in x- and y-direction.
1.5
Displacement [m]
1
0.5
0
−0.5
−1
−1.5
0
100
200
300
400
500
600
700
600
700
Time [s]
Figure 6.13: Translation x-direction at ω = 0.3141r ad /s
3
Displacement [m]
2
1
0
−1
−2
−3
0
100
200
300
400
500
Time [s]
Figure 6.14: Translation y-direction at ω = 0.3141r ad /s
In figures 6.13 and 6.15 the translations in x-direction for both simulations is shown. The motion is in
phase and with the same frequency as the fluid. Realistic ILS behavior is obtained.
6.5. Description with Morison’s equation
57
0.6
Displacement [m]
0.4
0.2
0
−0.2
−0.4
0
100
200
300
Time [s]
400
500
600
500
600
Figure 6.15: Translation x-direction at ω = 0.6283r ad /s
1.5
Displacement [m]
1
0.5
0
−0.5
−1
−1.5
0
100
200
300
Time [s]
400
Figure 6.16: Translation y-direction at ω = 0.6283r ad /s
The testing of the description of this motion in x-direction has become impossible as the forces are prescribed instead of solved with the use of CFD. However, motion in the y-direction is still pure due to simulated hydrodynamic forces. With realistic behavior in the x-direction, the motion in the y-direction can be
described (section 6.5).
The motion in y-direction (figures 6.14 and 6.16) is similar to the motion in y-direction without the artificial added mass and damping (figures 6.5 and 6.7).
6.5. Description with Morison’s equation
The same description as in section 5.8 is used. However, Morison’s equation is only able to describe in-line
motion. Therefore the motion of the plate in the x-y-plane is described with two equations of motion. One in
the x-direction and one in the y-direction.
The fluid motion is decomposed in order to model the hydrodynamic forces acting on the structure. The
flow velocities in x- and y-direction become:
u x (t ) = si n(θ)Um si n(ωt )
(6.3)
u y (t ) = cos(θ)Um si n(ωt )
(6.4)
The hydrodynamic coefficients used in the equations of motion are based on the flow characteristics
and the characteristic diameter in each direction. For these 2-DoF, two separate KC-numbers are calculated
which give different hydrodynamic coefficients.
K Cx =
si n(θ)Um T
Dx
(6.5)
58
6. 2 Degrees of Freedom
KCy =
cos(θ)Um T
Dy
(6.6)
The displacements in the x-direction are underestimated and not accurate (see figures 6.17 and 6.18).
However, accurate description is not of interest. The displacements are acceptable to state that this is realistic
behavior.
Displacement in x−direction [m]
1.5
Simulation
Morison Equation
1
0.5
0
−0.5
−1
−1.5
0
100
200
300
400
500
600
700
Time [s]
Figure 6.17: Translation description x-direction at ω = 0.3141r ad /s
0.4
Simulation
Morison Equation
Displacement in x−direction [m]
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
0
100
200
300
Time [s]
400
500
600
Figure 6.18: Translation description x-direction at ω = 0.6283r ad /s
The description of the displacements and force in the y-direction is presented in figures 6.19 and 6.20
for the simulation with a 10 second period. In figures 6.21 and 6.22 the description in y-direction for the 20
second period oscillation is presented.
We can see that the motion and force in the y-direction is accurately described. The description very
slightly underestimates the force and resulting motion. One of the causes could be that in the formulation
of the drag force (see equation 2.1), the decoupled absolute velocity is used instead of the coupled absolute
velocity. This causes an underestimation of the drag force. The actual coupled absolute relative velocity
cannot be calculated when decoupled equations of motion are used but an approximation of the difference
can be made.
|u − ẋ| > u y − x˙y
(6.7)
If we assume the motion of the plate and that of the fluid to be in-line with each other, an angle factor can
be introduced. The assumption that these motions are in-line is based on the x-y plots of the motion during
both simulations. In figures 6.23a and 6.23b these x-y plots are presented. The red line indicates the fluid
motion angle.
6.5. Description with Morison’s equation
59
Displacement in y−direction [m]
3
Simulation
Morison Equation
2
1
0
−1
−2
−3
0
100
200
300
400
500
600
700
Time [s]
Figure 6.19: Translation description y-direction at ω = 0.3141r ad /s
4
1.5
x 10
Simulation
Morison Equation
Force in y−direction [N]
1
0.5
0
−0.5
−1
−1.5
0
100
200
300
400
500
600
700
Time [s]
Figure 6.20: Force description y-direction at ω = 0.3141r ad /s
Displacement in y−direction [m]
1.5
Simulation
Morison Equation
1
0.5
0
−0.5
−1
−1.5
0
100
200
300
Time [s]
400
500
600
Figure 6.21: Translation description y-direction at ω = 0.6283r ad /s
The angle factor is dependent on the angle of the flow in relation to the direction of interest. In our case
the angle factors (θF x and θF y ) become:
θF x =
1
si n(θ)
(6.8)
θF y =
1
cos(θ)
(6.9)
60
6. 2 Degrees of Freedom
4
3
x 10
Simulation
Morison Equation
Force in y−direction [N]
2
1
0
−1
−2
−3
−4
0
100
200
300
Time [s]
400
500
600
3
1.5
2
1
Translation y−direction [m]
Translation y−direction [m]
Figure 6.22: Force description y-direction at ω = 0.6283r ad /s
1
0
−1
−2
−3
−3
0.5
0
−0.5
−1
−2
−1
0
1
Translation x−direction [m]
2
(a) x-y plot at ω = 0.3141r ad /s
3
−1.5
−1.5
−1
−0.5
0
0.5
Translation x−direction [m]
1
1.5
(b) x-y plot at ω = 0.6283r ad /s
Figure 6.23: In-line motion of structure and fluid
Multiplying the drag component of the hydrodynamic force produces more accurate results. A close-up
of the force description in the y-direction at ω = 0.6283r ad /s is presented in figures 6.24a and 6.24b. The
effect is not that large due to the small angle the flow makes to the y-axis (30 degrees). The angle factor in the
y-direction is therefore 1.15. Next to that is the drag force relatively small compared to the inertia force.
6.6. Conclusion
61
4
2.9
4
x 10
2.9
x 10
Simulation
Morison Equation
Simulation
Morison Equation
2.8
Force in y−direction [N]
Force in y−direction [N]
2.8
2.7
2.6
2.5
2.6
2.5
2.4
2.4
2.3
418
2.7
420
422
424
426
Time [s]
428
430
2.3
418
432
420
422
424
426
Time [s]
428
430
432
(b) with angle factor
(a) Without angle factor
Figure 6.24: Effect of angle factor on force in y-direction at ω = 0.6283r ad /s
6.6. Conclusion
Based on the 2-DoF case additional conclusions can be drawn. These are presented below. The impact of the
simplifications used in this thesis are discussed in section 6.7.
• Thin flat plates in oscillating flow in 2-DoF are accurately described using Morison’s equation with KC
dependent coefficients determined by flow conditions. The motions can be decoupled.
• Drag forces are underestimated in a completely decoupled model as the absolute relative velocity is
lower when decoupled motion is used compared to the coupled absolute relative velocity. In this thesis
an angle correction is used. However, the structure and fluid motion are not necessarily in-line. The
FEM analysis used at HMC is able to compute coupled absolute relative velocity.
• Thin flat plates are prone to resonance in the direction along the plate. This is due to negligible damping
in this direction.
6.7. Discussion
The simplifications made in this research and their expected impact are discussed in this section.
• Subsea structures do not have directions in which damping is very small. Therefore resonance and the
associated large dynamic amplification is not expected.
• The simulations are limited to two degrees of freedom. Rotations are not allowed. Moments measured
during the 2D simulations are significant. As can be seen in figures 6.25a and 6.25b are the moment
amplitudes per meter up to 15 kNm. If these were static moments, using the in table 3.3 defined properties for a 200 meter long pipe, 15 meter long structure and the rotation formula in equation 6.10, the
rotation would amount to 21 degrees.
θ=
Mz
L
GJ
(6.10)
This is an approximation with relatively harsh conditions and a stiff, short pipeline. A buoyancy module which affects the motion as well is neglected.
This approximation is done with a static moment. The system is a dynamic system where the acting
moment will change with rotation. Rotation will also influence the forces acting on the structure as the
relative angle of flow and projected area will change as well. How much and in what way rotation will
influence results is therefore unknown.
Morison’s equation is incapable of describing such a moment acting on a structure. The currently used
method at HMC only takes moments caused by geometric asymmetry into account.
62
6. 2 Degrees of Freedom
10000
20000
15000
6000
Moment [Nm]
Moment [Nm]
8000
4000
10000
5000
2000
0
0
−2000
0
100
200
300
400
Time [s]
500
600
700
(a) Moments 2-DoF simulation at ω = 0.3141r ad /s
−5000
0
100
200
300
Time [s]
400
500
600
(b) Moments 2-DoF simulation at ω = 0.6283r ad /s
Figure 6.25: Moments acting on plate in 2-DoF
• The angle that the flow makes relative to the structure is not varied. This might have an effect on the
vortex shedding regime. The vortices have a relatively small effect on the total force. It is unlikely that
the description method is unable to describe the hydrodynamic forces accurately at certain angles as
the structure in reality does not have extreme aspect ratios.
• Only one load case where the subsea structure is hanging perfectly vertical and is attached to the
pipeline at one end is assessed. This is a simplification of a 1st end FLET installation. There are three
typical differences with other subsea structure installations. The orientation, buoyancy module and
how it is attached to the pipeline or crane, otherwise the suspension.
– Orientation. A rotation around the x- or y-axis is not expected to influence the hydrodynamic
forces acting on the structure differently than a rotation around the z-axis (see chapter 6). The
motions can probably be decoupled. The directions in which the forces act and the stiffness in
these directions is different for all three rotations.
– Buoyancy module. The buoyancy module is not expected to influence the hydrodynamic forces
acting on subsea structures significantly. The buoyancy module does not block the flow or shield
the structure and is located at a distance where the influence on the flow around the subsea structure is negligible. The dynamic behavior of the structure is influenced. The buoynacy module will
dampen the motions and provide a hydrostatic upward force.
– Suspension. Depending on what kind of structure is installed, the suspension is different. An ILS
for instance has a pipeline attached at both ends while a 2nd end FLET hangs in the crane and has
a pipeline connected at the lower end. This has an effect on the response of the system, not on the
hydrodynamic force acting on the structure.
• The natural frequency of the system is highly dependent on the stiffness of the system, the mass and
added mass. In this thesis a relatively stiff and heavy system is used. The natural frequency of realistic
systems is lower than the encountered flow frequencies.
7
Conclusions and recommendations
In this section general conclusions are drawn and the conclusions from chapters 5 and 6 are summarized.
Recommendations on the modeling of hydrodynamic forces acting on subsea structures and further research
are presented in section 7.3.
In general the conclusion can be drawn that there is a wide variety of parameters influencing the behavior
of in-line subsea structures during installation. This thesis focused on a limited set of parameters and a
simplified system. The currently used methods to analyze installation behavior are applicable but can be
improved significantly. Introducing dependencies for the amplitudes of inertial and damping forces increases
the accuracy. These dependencies are known for thin flat plates but need to be researched further for subsea
structures.
7.1. Conclusions 1-DoF
• At KC numbers which are not associated with subsea structure installation vortex shedding regimes exist which cause oscillations at a different frequency than the fluid oscillation frequency and are therefore not described by Morison’s equation.
• The impact of these vortices on the displacement amplitude is negligible.
• DNV GL underestimates forces acting on thin flat plates oscillating normal to the flow
• KC number dependent hydrodynamic coefficients give an accurate description of the forces acting on
thin flat plates oscillating normal to the flow
• KC number dependency of hydrodynamic coefficients determined with forced motion experiments is
accurate for freely moving flat plates in oscillating flow.
• Iteration of relative flow characteristics in order to obtain more accurate KC numbers rarely yields a
significantly more accurate description of motion.
7.2. Conclusions 2-DoF
• The motions in 2-DoF are decoupled.
• In a completely decoupled model the drag forces are underestimated. Coupled absolute relative velocity is necessary for an accurate description.
• When damping in one direction is relatively small compared to the tangential direction, the first direction is prone to resonance.
7.3. Recommendations
Based on the conclusions and discussions in chapters 5 and 6 recommendations can be formulated. These
recommendations are split in two categories. One concerning the modeling of hydrodynamic forces acting
on subsea structures and one focused on further research.
63
64
7. Conclusions and recommendations
7.3.1. Modeling of hydrodynamic forces acting on subsea structures
• Use KC dependent coefficients during installation analysis. By DNV GL prescribed coefficients underestimate forces and displacements.
• KC number iteration is costly and seldom increases accuracy significantly. Perform quick hand calculations before doing a KC number iteration.
• Develop a method for determining average KC numbers in irregular flow. Since KC number iteration
often not leads to major differences, an average KC number should be accurate.
• Since the flow changes over water depth, a water depth dependency for the KC number could be introduced.
• Aim for large frequency ratios during the design of subsea structures. Motion amplitudes are smallest
at large frequency ratios (see figure 5.18).
• Avoid structures with large aspect ratios. The relative absence of damping can cause resonance.
7.3.2. Further research
• Investigate the effects and significance of hydrodynamic moments acting on a subsea structure.
• Use forced motion CFD simulations or physical experiments to investigate the hydrodynamic coefficients of more realistic structures.
• Start with measuring motions of real offshore installations to validate analyses.
• Validate the Froude-Krylov force for subsea structures. Investigate the impact of trapped volume.
• Investigate the superposition of multiple Morison’s equations when structure and fluid oscillate at different frequencies.
A
General Transport Equations
The fundamental equations which need to be solved in order to describe fluid dynamics are the conservation
laws, also known as the general transport equations. This set of equations are known as the Navier-Stokes
equations.
Using differential formulations, the conservation laws are presented below.
1. Mass conservation
Physically, this law states that mass can be created nor destroyed inside a control volume. The rate of
change of mass must be equal to the net inflow of fluid.
∂ρ
+ ∇ · (ρu) = 0
∂t
(A.1)
Many flows can be assumed incompressible
Dρ
=0
Dt
(A.2)
This simplifies the mass conservation equation to
∇·u = 0
(A.3)
2. Momentum conservation
Using Newton’s second law of motion1 , the change in momentum inside a control volume is conserved.
ρ
µ
¶
∂u
1
+ u · ∇u = −∇p + µ∇2 u + µ∇(∇ · u) + ρf
∂t
3
(A.4)
Under the assumption of incompressible flow combining equations A.3 and A.4 gives:
ρ
µ
¶
∂u
+ u · ∇u = −∇p + µ∇2 u + ρf
∂t
(A.5)
When the flow is assumed to be inviscid, we obtain the Euler equation
¶
∂u
ρ
+ u · ∇u = −∇p + ρf
∂t
µ
1 F = ∂(mV)
∂t
65
(A.6)
66
A. General Transport Equations
3. Energy conservation
The first law of thermodynamics2 states that the change in internal energy of a closed system is the sum
of heat added to the system and work done by the system. The properties of a control volume therefore
abide by the equation.
ρ
Dh D p
=
+ ∇ · (κ∇T ) + Φ
Dt
Dt
(A.7)
When incompressible flows are studied, the equations for conservation of mass and momentum are
sufficient to describe the flow. For compressible flows however, the energy equation is needed to copmlete the system.
2 dU = δQ − δW
B
Finite Volume Method
One of the most commonly used methods to discretize the partial differential equations which govern flow,
is the Finite Volume Method (FVM). This method divides the domain in a certain number of control volumes
where the variables are located at the geometric center of said control volumes. The values at these centroids
are obtained through integrating the partial differential equations over the control volume. Interpolation is
used to describe the flux of a certain variable between cells.
The resulting solution is conservative. I.e. the conservation laws for mass, momentum and energy (see
Appendix A) are satisfied.
Figure B.1: 2D Finite Volume
Source:
http://arturo.imati.cnr.it/~marco/resources/GIF/cell-centered_grid-2.gif
B.1. Meshing
The division of the computational domain into control volumes is known as meshing. The resulting grid is
known as the mesh. Types of mesh can be classified based upon the connectivity of the mesh. It is either
structured, unstructured or is a hybrid between these two.
• Structured Mesh. A structured mesh in built up in a regular way so that the relations between cells can
be defined beforehand which saves space and computational power. The cells need to be quadrilateral
in 2D or hexahedral in 3D to allow for a structured mesh.
67
68
B. Finite Volume Method
• Unstructured Mesh. The cells in an unstructured mesh can be of any shape. This is practical when
dealing with flow that is not well resolved in a structured mesh.
• Hybrid Mesh. A hybrid mesh has parts which are structured and parts which are unstructured.
In STAR-CCM+ three types meshers (automatic mesh generators) are available for general volume meshing: Polyhedral, Tetrahedral and Trimmed. Examples of all three types are given in figure B.2. Depending on
geometry and flow characteristics one of these meshers is optimal.
(a) Polyhedral
(c) Trimmed
(b) Tetrahedral
Figure B.2: Mesh types
To resolve the flow in a boundary layer (see section B.3.1) accurately, a special type of mesh is used. In
STAR-CCM+ this is called a Prism Layer Mesh (PLM). This mesh has two special characteristics:
• Wall functions. As described in section B.3.1 can the viscous sublayer and buffer layer be modeled with
the use of wall functions. These wall functions are applied to the first cell of the PLM.
• Stretching. In boundary layers more accuracy is needed close to the boundary compared to further
away from the boundary. Therefore the cell thickness is increased over distance from the plate. The
stretching in STAR-CCM+ is done with a geometric series. The thickness of a cell in the PLM is therefore:
∆y n = ∆y 1 r n−1
(B.1)
Where ∆y 1 is the thickness of the first layer, r the Stretching factor and n the number of the Prism layers
counting from the boundary.
The thickness of the first layer is important for the correct use of wall functions. The thickness of this
first layer can be calculated once the total thickness, the stretching factor and the number of layers is
known. The total thickness is found as the sum of a geometric series. From this definition the thickness
of the first layer can be found.
Total thickness =
n−1
X
k=0
∆y 1 r k = ∆y 1
1−rn
1−r
(B.2)
B.1.1. Mesh size importance
The size of the individual cells in a mesh is important in order to simulate to flow accurately. The gradients
of the flow properties must not be too large. If these gradients are too large, the accuracy of the simulation
will suffer or even numerical instabilities may occur. The mesh size should therefore be small enough to ensure accurate and stable simulations. Through mesh size convergence studies the minimal mesh size can be
found. However, as explained in section B.4, a smaller mesh size leads to longer computational durations.
Mesh size doesn’t need to be constant over the entire computational domain. As the flow conditions vary,
the mesh size can vary as well. Regions where gradients are higher can be refined to assure accuracy where
regions with relatively constant flow can have a coarse mesh to save computational effort.
B.2. Physics Modeling
69
Figure B.3: Close-up of PLM
B.2. Physics Modeling
The flow physics need to be modeled numerically. The most important topics and which decisions come into
play are presented in this section.
B.2.1. Flow simplification
Some parts of the equations which describe the flow, the Navier-Stokes equations, can be simplified to save
computational effort. How these simplifications affect the Navier-Stokes equations is included in Appendix
A.
• Compressibility. Flows can be modeled as incompressible (i.e. with constant density). This is valid for
almost any liquid and for low-speed gas flows. This assumption reduces the required computational
effort to solve the equations describing flow.
• Viscosity. Certain physical situations allow for neglecting the viscous effects in the Navier-Stokes equations. These equations are then called the Euler equations [1]. Not having to resolve the viscous effects
saves a lot of computational effort.
• Steady flow. When the flow pattern can be assumed independent of time, a steady simulation can be
used. Only one converged solution is sought instead of a converged solution at every time-step.
• Number of dimensions. If aspect ratios of the investigated problem are high and therefore the impact
of the flow over one dimension can be neglected, a 2D simulation can be used. The simulations take a
lot less computational effort as the amount of cells and interfaces between cells is lower.
B.2.2. Segregated solver
At every iteration step, the momentum, continuity and turbulence equations need to be solved. This is done
in a segregated manner. The segregated method solves the equation for a certain variable for all cells, then
the equation for the next variable is solved for all cells and so on.
Segregated solvers are quicker than coupled solvers but can have difficulty with convergence. They work
best for incompressible flows or compressible flows at low Mach numbers[3]. Since this thesis only covers
incompressible flows, a segregated solver is used.
B.2.3. Temporal Discretization
When an unsteady problem is numerically simulated, not only space needs to be discretized but also time.
The solution of the used set of governing equations changes over time. How the change of the solution over
time is modeled in a discrete way is called temporal discretization. The change of a certain variable φ over
time is a function of said variable.
∂φ
= F (φ)
∂t
When discretized in time the function becomes:
(B.3)
70
B. Finite Volume Method
φn+1 − φn
= F (φ)
∆t
(B.4)
where n is the current time (t ) and n + 1 is the next time step (t + ∆t ).
The function that defines the change of a variable over time, F (φ), is evaluated at the future time step
(t + ∆t ). This is called implicit.
φn+1 − φn
= F (φn+1 )
∆t
(B.5)
The implicit method can handle larger time-steps and is more stable than the explicit method. This at the
cost of computational effort. Next to that, STAR-CCM+ does not allow the use of explicit time discretization
in combination with a segregated solver. Therefore an implicit method is used.
B.2.4. Time-step
Especially when using an explicit temporal discretization method, the time-step is limited to the CourantFriedrichs-Lewy (CFL) condition [7]. This condition states that a fluid particle can translate only a limited
amount of cells per time-step. The Courant number must remain below a certain maximum value.
C = ∆t
n u
X
xi
i =1 ∆x i
≤ C max
(B.6)
For explicit solvers C max is 1 over the entire domain. The time-step is constant for the entire domain so
the highest local velocity/cell size ratio defines the time-step.
Implicit solvers are capable of handling much larger Courant numbers. When an implicit solver is used,
the time-step is usually based on the timescale of the physical phenomenon that is simulated. These timescales
are usually based on the frequency of the phenomenon or based on the time it takes the flow to travel a certain distance. This timescale is divided in a certain amount of time-steps ranging from 10 up to sometimes
1000. The ideal time-step when taking computational effort into account is found through a convergence
study.
B.3. Turbulence
Turbulent flows are present in almost every real situation. Turbulent flows are characterized by their chaotic
nature and are therefore impossible to solve in a deterministic approach. In CFD statistical methods are used
to model turbulence. There are two main categories of computational approaches:
• Simulations. Equations are solved for a time dependent velocity field which represents one realization
of the turbulent flow.
• Models. Mean values of flow characteristics and fluctuations around this mean are used to model the
turbulent flow. These equations give closure to the Reynolds-Averaged Navier-Stokes (RANS) equations.
Simulations are used when high accuracy is necessary. These simulations require high mesh resolutions
and small time and length scales.
When the RANS equations are used, an additional term in the momentum equations occurs. This additional term is the Reynolds stress tensor. There are two basic approaches to modeling this tensor:
• Eddy viscosity models. These models use the concepts of turbulent viscosity and turbulent kinetic energy to model the Reynolds stress tensor. These models are most commonly used because they require
less computational effort compared to other methods while still having acceptable accuracy.
• Reynolds stress transport models. These models solve transport equations for each component of the
Reynolds stress tensor. These models require relatively a lot of computational effort and are therefore
only used when the flow is highly anisotropic.
B.4. Computational effort
71
B.3.1. Boundary layer
Turbulent flow past a wall can be divided into three regions plus the free-stream region. In these three regions
the velocity differs from the free-stream region because of the present wall friction. Fluid velocity is lowest at
the wall and increases linearly with distance from the wall in a first thin layer (the viscous sublayer). Further
away, the flow starts to transition to turbulent and this is called the buffer layer. Even further away, the average flow velocity follows a logarithmic profile. This region is called the log-law region. When we look even
past this log-law region, the flow is not anymore affected by the wall and the free-stream region is reached.
Figure B.4: Flow regions in a turbulent boundary layer
The distance from the wall to the end of the buffer layer is usually described by δ. RANS can be used
to describe the flow in this very thin first layer. However, the flow can also be estimated with an analytical
expression which reduces computational effort significantly. These estimations are called wall functions.
B.4. Computational effort
The time needed to run an unsteady simulation is dependent on the available computational power and the
computational effort required by the simulation. This computational effort is directly related to the amount
of calculations needed to be made during the entire simulation.
The total amount of physical time that is to be simulated is divided into a certain amount of time-steps.
Within these time-steps, a certain amount of inner iterations need to be performed in order to reach convergence. At every iteration, the solvers need to perform their calculations at every cell and the more complex
the physical models are, the more calculations need to be performed by the solvers.
Total time = # of time-steps × # of inner iterations × # of cells × complexity of physics
(B.7)
It is clear that the number of time-steps, inner iterations, cells and the complexity of physics is kept as
low as possible to reduce the total time a simulation takes without sacrificing accuracy. In order to find this
optimum between simulation duration and accuracy, convergence studies are performed.
C
Convergence studies
C.1. General mesh convergence
The target sizes for the mesh regions are defined by a percentage of a ’Base size’. The Base size is 0.1m. In
table C.1 the settings per mesh are presented together with the resulting cell count. In figure C.1 in can be
seen that the drag and inertia coefficients are converged at mesh # 3.
Mesh #4
Mesh #3
Mesh #2
Mesh #1
Far field
100%
100%
100%
100%
Large refinement
10%
10%
20%
40%
Small refinement
2.5%
5%
10%
20%
2D cell count
53.940
34.242
12.429
7.449
Table C.1: General mesh settings
Mesh size convergence
Mesh size convergence
12.8
1.38
12.78
1.375
12.76
1.37
12.74
1.365
12.72
Cm
Cd
1.36
12.7
1.355
12.68
1.35
12.66
1.345
12.64
12.62
1.34
12.6
1.335
1
1.5
2
2.5
Mesh no.
3
3.5
4
1
(a) Drag coefficient
1.5
2
2.5
Mesh no.
3
3.5
4
(b) Inertia Coefficient
Figure C.1: Mesh size convergence
C.2. Prism Layer mesh convergence
The only parameter that is varied, is the number of layers in the PLM. Convergence in the hydrodynamic
coefficients is sought depending on the number op Prism layers. Convergence is found at 10 Prism layers as
can be seen in table C.2 and figure C.2.
73
74
C. Convergence studies
CD
5.92
5.96
5.97
5.95
# prism layers
5
10
20
30
CM
0.42
0.42
0.42
0.42
Table C.2: Prism layer mesh settings
Prism layer mesh convergence
5.97
0.4205
0.42
5.96
0.4195
5.95
0.419
Cm
Cd
0.4185
5.94
0.418
5.93
0.4175
0.417
5.92
0.4165
5.91
5
10
15
20
no. of layers
25
30
0.416
5
10
(a) Drag coefficient
15
20
no. of layers
25
30
(b) Inertia Coefficient
Figure C.2: Prism layer mesh convergence
C.3. Time-step convergence
A convergence study has been performed to determine the optimal amount of time-steps per oscillation. See
table C.3 and figure C.3 for the results.
CD
9.27
10.26
10.90
11.10
11.10
Steps per period
50
100
200
400
800
CM
1.15
1.11
1.05
1.03
1.02
Table C.3: Time-step convergence
Time step convergence
Time step convergence
1.39
12.6
1.38
12.4
1.37
12.2
1.36
Cd
Cm
12.8
12
1.35
11.8
1.34
11.6
1.33
11.4
0
100
200
300
400
500
Time steps per period
(a) Drag coefficient
600
700
800
1.32
0
100
200
300
400
500
Time steps per period
600
(b) Inertia Coefficient
Figure C.3: Time-step convergence
700
800
C.4. Conclusion
75
C.4. Conclusion
Of all converged values an average value and a bandwidth of convergence is determined.
General mesh
Prism layer mesh
Time-step
CD
12.62 ± 0.16%
5.955 ± 0.25%
12.575 ± 0.20%
CM
1.3725 ± 0.18%
0.41975 ± 0.06%
1.3675 ± 0.55%
Worst case bandwidth
0.61%
0.79%
Table C.4: Convergence bandwidths
Depending on the relative contribution of inertia and drag on the total force, a maximum error of 0.79%
is expected. This is acceptable for this thesis.
D
Logarithmic decrement
We define
xn
δl og = l n
x n+1
µ
¶
(D.1)
For underdamped, unforced systems the response is given by
x(t ) = Ae −ω0 ζt si n(ωd t + φ)
(D.2)
x n = Ae −ω0 ζtn si n(ωd t n + φ)
(D.3)
x n+1 = Ae −ω0 ζtn+1 si n(ωd t n+1 + φ)
(D.4)
x n and x n+1 become
The difference between t n and t n+1 is exactly one oscillation period and therefore we can say:
t n+1 = t n + Td
(D.5)
si n(ωd t n + φ) = si n(ωd t n+1 + φ)
(D.6)
where Td is the damped oscillation period.
Inserting equations (D.3),(D.4),(D.5) and (D.6) into equation (D.1):
Ã
!
Ã
!
¶
µ
Ae −ω0 ζtn
Ae −ω0 ζtn
1
δl og = l n
= l n(e ω0 ζTd ) = ω0 ζTd
=
l
n
=
l
n
Ae −ω0 ζ(tn +Td )
Ae −ω0 ζtn ∗ e −ω0 ζTd
e −ω0 ζTd
p
With Td = 2π/ωd and ωd = ω0 1 − ζ2 we get
2πζ
δl og = p
1 − ζ2
(D.7)
(D.8)
Rearranging for ζ gives
ζ= r
1
³
1 + l n(xn2π
/x n+1 )
77
´2
(D.9)
E
Linearization drag force
In Morison’s equation, the drag or damping term is nonlinear. The damping is dependent on the velocity
squared. This makes dynamic calculations relatively complex. A linear drag force, dependent on velocity,
would simplify the dynamic calculations.
Wolfram[25] presented an extensive summary of methods to linearize the drag in Morison’s equation. One of
the most widely used methods is equivalent linearization. A least-squares approach is used to find factors for
the equivalent linear form which minimize the error between it and the original nonlinear form.
Morison’s equation may be written in a compact form
F = K D |u|u + K M u̇
(E.1)
For the linearized form, the equation is rewritten in the form:
F l = K¯D u + K M u̇
(E.2)
Where F l is the linearized force and the bar indicates the linear coefficient. Applying a least-squares
approach to minimize the difference between the two formulations yields
∂⟨(F − F l )2 ⟩
= −2⟨K D u 2 |u| − K¯D u 2 ⟩ = 0
∂K¯D
(E.3)
Which in turn, because of the speed independent components in K D and K¯D , yields
K¯D = K D
⟨u 2 |u|⟩
⟨u 2 |u|⟩
¯D = C D
→
C
⟨u 2 ⟩
⟨u 2 ⟩
(E.4)
where ⟨⟩ indicates a time-averaged value. Assuming
the water particle velocity to be a Gaussian process
p
with zero mean and a standard deviation of u a / 2 because we assume a regular wave, the linear drag coefficient becomes:
C¯D = C D
q
(8/π)σ3u
σ2u
= CD
p
(8/π)σu = C D Um
8
3π
(E.5)
where σu is the standard deviation of water particle velocity and Um the water particle velocity amplitude.
The difference this linearization induces compared to the quadratic drag is analyzed. Using the simulation data for a forced plate oscillating normal to the flow over a range of KC numbers (see chapter 4), the drag
coefficients are compared.
The drag coefficients are a maximum of 4% higher with linear drag compared to quadratic drag. The
goodness of fit is in the same order or even lower than the quadratic coefficients (see section 4.5). The linear
drag coefficients fit the force time trace with δ < 0.03 and are a good fit to the experimental data presented in
section 4.1.
1 Goodness of fit factor (see section 4.5)
79
80
E. Linearization drag force
KC number
Quad. drag
Lin. drag
Lin. drag %
δ1 .
1
2
3
5
8
11
15
20
30
40
80
11.0786
10.0038
8.4030
5.7672
5.1707
4.5047
4.0330
3.6040
3.1339
2.8713
2.5143
11.3975
10.2406
8.6629
5.9999
5.3405
4.6513
4.1539
3.6733
3.1912
2.9158
2.5615
102.88 %
102.37 %
103.09 %
104.03 %
103.28 %
103.25 %
103.00 %
101.92 %
101.83 %
101.55 %
101.88 %
0.0024
0.0052
0.0080
0.0139
0.0142
0.0193
0.0241
0.0280
0.0198
0.0192
0.0166
Table E.1: Drag force linearization comparison
F
Plots description 1 DoF
F.1. Simulation #1
5
Displacement [m]
Simulation
Morisons Equation
0
−5
1500
2000
2500
Time [s]
3000
3500
Figure F.1: Simulation #1, Translation comparison at ω/ω0 = 0.23
0.4
Simulation
Morisons Equation
Velocity [m/s]
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
1500
2000
2500
Time [s]
3000
Figure F.2: Simulation #1, Velocity comparison at ω/ω0 = 0.23
81
3500
82
F. Plots description 1 DoF
0.06
Simulation
Morisons Equation
Acceleration [m/s2]
0.04
0.02
0
−0.02
−0.04
1500
2000
2500
Time [s]
3000
3500
Figure F.3: Simulation #1, Acceleration comparison at ω/ω0 = 0.23
4
2
x 10
Simulation
Morisons Equation
Force [N]
1
0
−1
−2
1500
2000
2500
Time [s]
3000
3500
Figure F.4: Simulation #1, In-line Force comparison at ω/ω0 = 0.23
F.2. Simulation #2
10
Displacement [m]
Simulation
Morisons Equation
5
0
−5
−10
1000
1100
1200
1300
1400
1500
Time [s]
1600
1700
1800
1900
2000
Figure F.5: Simulation #2, Translation comparison at ω/ω0 = 0.5083
0.6
Simulation
Morisons Equation
Velocity [m/s]
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
1000
1100
1200
1300
1400
1500
Time [s]
1600
1700
Figure F.6: Simulation #2, Velocity comparison at ω/ω0 = 0.5083
1800
1900
2000
F.3. Simulation #3
83
0.06
Simulation
Morisons Equation
Acceleration [m/s2]
0.04
0.02
0
−0.02
−0.04
−0.06
1000
1100
1200
1300
1400
1500
Time [s]
1600
1700
1800
1900
2000
Figure F.7: Simulation #2, Acceleration comparison at ω/ω0 = 0.5083
4
3
x 10
Simulation
Morisons Equation
Force [N]
2
1
0
−1
−2
−3
1000
1100
1200
1300
1400
1500
Time [s]
1600
1700
1800
1900
2000
Figure F.8: Simulation #2, In-line Force comparison at ω/ω0 = 0.5083
F.3. Simulation #3
10
Simulation
Morisons Equation
Displacement [m]
5
0
−5
−10
0
100
200
300
Time [s]
400
Figure F.9: Simulation #3, Translation comparison at ω/ω0 = 0.946
500
600
84
F. Plots description 1 DoF
1.5
Simulation
Morisons Equation
Velocity [m/s]
1
0.5
0
−0.5
−1
−1.5
0
100
200
300
Time [s]
400
500
600
Figure F.10: Simulation #3, Velocity comparison at ω/ω0 = 0.946
0.3
Simulation
Morisons Equation
Acceleration [m/s2]
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
0
100
200
300
Time [s]
400
500
600
Figure F.11: Simulation #3, Acceleration comparison at ω/ω0 = 0.946
4
3
x 10
Simulation
Morisons Equation
2
Force [N]
1
0
−1
−2
−3
0
100
200
300
Time [s]
400
Figure F.12: Simulation #3, In-line Force comparison at ω/ω0 = 0.946
F.4. Simulation #4
500
600
F.5. Simulation #5
85
20
Simulation
Morisons Equation
15
Displacement [m]
10
5
0
−5
−10
−15
−20
0
500
1000
1500
Time [s]
Figure F.13: Simulation #4, Translation comparison at ω/ω0 = 1.568
1.5
Simulation
Morisons Equation
Velocity [m/s]
1
0.5
0
−0.5
−1
−1.5
0
500
1000
1500
Time [s]
Figure F.14: Simulation #4, Velocity comparison at ω/ω0 = 1.568
4
2
x 10
Simulation
Morisons Equation
1.5
Force [N]
1
0.5
0
−0.5
−1
0
500
1000
Time [s]
Figure F.15: Simulation #4, In-line Force comparison at ω/ω0 = 1.568
F.5. Simulation #5
1500
86
F. Plots description 1 DoF
5
Simulation
Morisons Equation
4
Displacement [m]
3
2
1
0
−1
−2
−3
0
50
100
150
200
250
Time [s]
300
350
400
450
Figure F.16: Simulation #5, Translation comparison at ω/ω0 = 6.16
0.8
Simulation
Morisons Equation
0.6
Velocity [m/s]
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0
50
100
150
200
250
Time [s]
300
350
400
450
Figure F.17: Simulation #5, Velocity comparison at ω/ω0 = 6.16
0.2
Simulation
Morisons Equation
Acceleration [m/s2]
0.15
0.1
0.05
0
−0.05
−0.1
−0.15
−0.2
0
50
100
150
200
250
Time [s]
300
350
Figure F.18: Simulation #5, Acceleration comparison at ω/ω0 = 6.16
400
450
F.6. Simulation #6
87
4
x 10
3
Simulation
Morisons Equation
2
Force [N]
1
0
−1
−2
−3
0
50
100
150
200
250
Time [s]
300
350
400
450
Figure F.19: Simulation #5, In-line Force comparison at ω/ω0 = 6.16
F.6. Simulation #6
2
Simulation
Morisons Equation
Displacement [m]
1.5
1
0.5
0
−0.5
−1
−1.5
0
20
40
60
80
100
120
140
Time [s]
Figure F.20: Simulation #6, Translation comparison at ω/ω0 = 4.17
0.6
Simulation
Morisons Equation
0.4
Velocity [m/s]
0.2
0
−0.2
−0.4
−0.6
−0.8
0
20
40
60
80
100
Time [s]
Figure F.21: Simulation #6, Velocity comparison at ω/ω0 = 4.17
120
140
88
F. Plots description 1 DoF
0.4
Simulation
Morisons Equation
Acceleration [m/s2]
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
0
20
40
60
80
100
120
140
Time [s]
Figure F.22: Simulation #6, Acceleration comparison at ω/ω0 = 4.17
4
x 10
6
Simulation
Morisons Equation
Force [N]
4
2
0
−2
−4
0
20
40
60
80
100
120
140
Time [s]
Figure F.23: Simulation #6, In-line Force comparison at ω/ω0 = 4.17
F.7. Simulation #7
0.8
Simulation
Morisons Equation
Displacement [m]
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
0
10
20
30
40
50
Time [s]
Figure F.24: Simulation #7, Translation comparison at ω/ω0 = 8.19
60
70
F.7. Simulation #7
89
0.5
Velocity [m/s]
Simulation
Morisons Equation
0
−0.5
0
10
20
30
40
50
60
70
Time [s]
Figure F.25: Simulation #7, Velocity comparison at ω/ω0 = 8.19
0.8
Simulation
Morisons Equation
Acceleration [m/s2]
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
0
10
20
30
40
50
60
70
Time [s]
Figure F.26: Simulation #7, Acceleration comparison at ω/ω0 = 8.19
5
1
x 10
Simulation
Morisons Equation
Force [N]
0.5
0
−0.5
−1
0
10
20
30
40
50
Time [s]
Figure F.27: Simulation #7, In-line Force comparison at ω/ω0 = 8.19
60
70
G
DNV Comparison
Displacement [m]
5
Simulation
Morisons Equation
DNV GL
0
−5
2000
2200
2400
2600
2800
3000
Time [s]
3200
3400
3600
3800
4000
Figure G.1: Simulation #1, Translation comparison at ω/ω0 = 0.23, KC = 19.23
Displacement [m]
10
Simulation
Morisons Equation
DNV GL
5
0
−5
−10
1000
1100
1200
1300
1400
1500
Time [s]
1600
1700
1800
Figure G.2: Simulation #2, Translation comparison at ω/ω0 = 0.5083,KC = 9.61
91
1900
2000
92
G. DNV Comparison
10
Simulation
Morisons Equation
DNV GL
Displacement [m]
5
0
−5
−10
100
150
200
250
300
350
Time [s]
400
450
500
550
600
Figure G.3: Simulation #3, Translation comparison at ω/ω0 = 0.946, KC = 4.2
Displacement [m]
20
Simulation
Morisons Equation
DNV GL
10
0
−10
−20
500
600
700
800
900
1000
Time [s]
1100
1200
1300
1400
1500
Figure G.4: Simulation #4, Translation comparison at ω/ω0 = 1.568, KC = 9.25
Displacement [m]
2
Simulation
Morisons Equation
DNV GL
1
0
−1
−2
250
300
350
Time [s]
400
Figure G.5: Simulation #5, Translation comparison at ω/ω0 = 6.16, KC = 2.1
450
93
2
Simulation
Morisons Equation
DNV GL
Displacement [m]
1.5
1
0.5
0
−0.5
−1
−1.5
0
20
40
60
80
100
120
140
Time [s]
Figure G.6: Simulation #6, Translation comparison at ω/ω0 = 4.17, KC = 1
0.8
Simulation
Morisons Equation
DNV GL
Displacement [m]
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
0
10
20
30
40
50
Time [s]
Figure G.7: Simulation #7, Translation comparison at ω/ω0 = 8.19, KC = 0.5
60
70
H
Differential equation solver
1
2
3
%% Copy_of_EoM_solve .m: Solver f o r the in ODEEoM.m defined d i f f e r e n t i a l equation
%% Input data
global Cd Cm D rho m k A omega c a a2
4
5
6
7
%Used hydrodynamic c o e f f i c i e n t s
Cd = 5 ;
Cm = 2 . 2 ;
8
9
10
11
12
13
14
D = 9 . 5 ; %P l a t e diameter
rho = 997.561; %Fluid density
m = 117390; %Structure mass
k = 5000; %Spring c o e f f i c i e n t
A = 1 ; %Fluid o s c i l l a t i o n speed amplitude
omega = 0 . 0 7 1 5 ; %Fluid o s c i l l a t i o n frequency
15
16
17
18
19
%Damping and added mass c o e f f i c i e n t s
c = 0 . 5 * rho *D* Cd ;
a = ( pi /4) * rho *D*D*Cm;
a2 = ( pi /4) * rho *D*D* (Cm−0.0382) ; %Added mass minus the volume f a c t o r
20
21
22
23
24
%% I n i t i a l conditions
y0 ( 1 ) =0; %I n i t i a l displacement
y0 ( 2 ) =0; %I n i t i a l v e l o c i t y
tspan = [0 3 0 0 0 ] ; %Timespan of simulation
25
26
27
1
2
%% ODE s o l v e r
[ t , y ] = ode45 ( ’Copy_of_ODEEoM ’ , tspan , y0 ) ; %D i f f e r e n t i a l equation s o l v e r
function yprime = Copy_of_ODEEoM( t , y )
%Defines a c c e l e r a t i o n and speed as functions of speed and position
3
4
global Cd Cm D rho m k A omega c a a2
5
6
7
8
%V e l o c i t y and a c c e l e r a t i o n of f l u i d
v ( 1 , 1 ) = A * sin (omega * t ) ;
v ( 2 , 1 ) = A * omega * cos (omega * t ) ;
9
10
11
%V e l o c i t y and a c c e l e r a t i o n of the p l a te
yprime ( 1 , 1 ) = y ( 2 ) ;
95
96
12
13
H. Differential equation solver
yprime ( 2 , 1 ) = (−c * ( y ( 2 )−v ( 1 ) ) * abs ( ( y ( 2 )−v ( 1 ) ) )−k * y ( 1 ) +a * v ( 2 ) ) / (m+a2 ) ;
end
Bibliography
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flexcom/index.html?theory.html.
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University of Iowa, 1960.
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and Polar Engineers, 2006.
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