wavy properties and analytical modeling
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Wavy properties and analytical modeling
of free-surface flows in the solution of fluid-structure interaction
Xiaobo Chen and Hui Liang
DTRC, BV, M&O Division, Singapore
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2Wavy properties and analytical modeling of free-surface flow
Analytical modelling3
Multi-domain method4
Conclusions5
Wavy properties of free surface flows2
Fluid-structure interaction1
Outline
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3Wavy properties and analytical modeling of free-surface flow
Fluid-structure interaction
► Viscosity effect Flow separation
VIV and VIM
Dissipation in gap
Roll damping
……
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4Wavy properties and analytical modeling of free-surface flow
Fluid-structure interaction
► Non-linearity
Slamming
Ringing and springing
Water-on-deck
……
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5Wavy properties and analytical modeling of free-surface flow
Fluid-structure interaction
► Wavy properties
Ocean waves
Motion of a floating body
A traveling object
……
x/L
y/L
-1 -0.5 0 0.5 1 1.5
-1
-0.5
0
0.5
1
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6Wavy properties and analytical modeling of free-surface flow
Wavy properties of free surface flows
► Overview of numerical wave tank
Free-surface boundary condition
• Lagrangian description (fixed point)
• Semi-Lagrangian description (fictitious moving point)
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7Wavy properties and analytical modeling of free-surface flow
Wavy properties of free surface flows
► Overview of numerical wave tank
Damping zone (artificial beach, sponge layer)
Time marching
• 4th-order Runge-Kutta (RK) method
• 5th-order Runge-Kutta-Gil (RKG) method
• 4th-order Adams-Bashforth-Moulton (ABM) method
Filter
• 5-point Chebyshev filter
• 13-point 10-th order Savisky-Golay filter
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8Wavy properties and analytical modeling of free-surface flow
Wavy properties of free surface flows
20 25 30 35 40-0.02
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0.00
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ζ
t
Experiment HPC results
20 25 30 35 40-0.02
-0.01
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ζ
t
Experiment HPC results
20 25 30 35 40-0.02
-0.01
0.00
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0.03
0.04
ζ
t
Experiment HPC results
► Nonlinear waves propagating over a submerged bar
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9Wavy properties and analytical modeling of free-surface flow
Wavy properties of free surface flows
► Wave-induced motions (Results calculated by HydroSTAR)
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10Wavy properties and analytical modeling of free-surface flow
Wavy properties of free surface flows
► Typical steady ship waves
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11Wavy properties and analytical modeling of free-surface flow
Wavy properties of free surface flows
► Ship waves and environment
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12Wavy properties and analytical modeling of free-surface flow
Wavy properties of free surface flows
► Time-harmonic waves due to heave motions
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13Wavy properties and analytical modeling of free-surface flow
Analytical modelling of waves
Classical studies
• Kelvin (1887) Kelvin angle
• Lamb (1932) Hydrodynamics
• Peters (1949) a new treatment of ship wave problem
• Eggers (1957) Kelvin’s ship waves pattern
• Lighthill (1978) Waves in fluid
• Ursell (1960, 1988) Asymptotic of Neumann-Kelvin waves
Recent works
• Noblesse & Chen (1995) Decomposition of free-surface effect
• Chen & Noblesse (1997) Dispersion relation and ship waves
• Chen & Wu (2001) Singular and highly-oscillatory properties
• Chen (2002 - 2003) Effect of surface tension
• Chen & Lu (2007) Effect of fluid viscosity (and surface tension)
• Liang & Chen (2015) Capillary-gravity waves
• …
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14Wavy properties and analytical modeling of free-surface flow
Analytical modelling of waves
Fourier representation
Decomposition (Noblesse & Chen 1995)
Wave component
Dispersion relation
𝑫 = 𝑭𝜶 − 𝒇 𝟐 − 𝜶𝟐 + 𝜷𝟐
gives 2 or 3 distinct curves
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15Wavy properties and analytical modeling of free-surface flow
Analytical modelling of waves
Open D.C.Open D.C.
Closed D.C.
𝜏 < 0.25
𝐹𝛼 − 𝑓 2 = 𝛼2 + 𝛽2
𝜏 =𝑈𝑓
𝑔
Brad number
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16Wavy properties and analytical modeling of free-surface flow
Analytical modelling of waves
Open D.C.Open D.C.
𝜏 > 0.25
𝐹𝛼 − 𝑓 2 = 𝛼2 + 𝛽2
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17Wavy properties and analytical modeling of free-surface flow
Analytical modelling of waves
Right open dispersion curve Inner-V waves
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18Wavy properties and analytical modeling of free-surface flow
Analytical modelling of waves
Closed dispersion curve Ring waves
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19Wavy properties and analytical modeling of free-surface flow
Analytical modelling of waves
Left open dispersion curve Outer-V waves
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20Wavy properties and analytical modeling of free-surface flow
Analytical modelling of waves
Left open dispersion curve Ring-fan and fan waves
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21Wavy properties and analytical modeling of free-surface flow
Analytical modelling of waves
► Ring wave animation
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22Wavy properties and analytical modeling of free-surface flow
Analytical modelling of waves
Ring wave component
Local component
Wave + local components
► Ring wave system
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23Wavy properties and analytical modeling of free-surface flow
Analytical modelling of waves
► V-shaped wave animation
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24Wavy properties and analytical modeling of free-surface flow
Analytical modelling of waves
► V-shaped wave system
Inner-V waves
Local component
Wave + local components
Outer-V waves
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25Wavy properties and analytical modeling of free-surface flow
Analytical modelling of waves
Cusp angles in time-harmonic ship waves
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26Wavy properties and analytical modeling of free-surface flow
Analytical modelling of waves
Crest lines of wave systems at 𝝉 = 𝟐/𝟐𝟕 ≈ 𝟎. 𝟐𝟕𝟐
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27Wavy properties and analytical modeling of free-surface flow
Analytical modelling of waves
► Singular and peculiar properties (Chen & Wu, JFM, 2001)
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28Wavy properties and analytical modeling of free-surface flow
Analytical modelling of waves
► Remarks of the analysis of ship waves
More than one wave system
• Much more complicated than zero forward speed scenario
• Difficult to determine the size of the damping zone
Wave-length covers a very large range (from centimeters to hundreds of meters)
• The size of mesh must be of the order of centimeters
• The size of the computational domain must be hundreds even thousands of meters
Singular and highly-oscillatory behaviors
• It is applicable for a submerged body
• Convergence problem occurs in the simulation of a surface piecing body
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29Wavy properties and analytical modeling of free-surface flow
Multi-domain method
► Control surface of analytical form
• Hemi-sphere, hemi-ellipsoid, vertical cylinder, etc
• Expansion of base functions
► Integral equations
• Use of point solution (Green function)
• Finding elementary solutions by integration of PS
► DtN operator
• Relationship between velocity potential and its radial derivatives
• Use of DtN in the solution of internal domain
► Multi-domain method
• CFD method in the internal domain (near-field)
• Analytical solution in the external domain (far-field)
• Coupling
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30Wavy properties and analytical modeling of free-surface flow
Multi-domain method
Externaldomain
Transitionaldomain
Internaldomain
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31Wavy properties and analytical modeling of free-surface flow
Multi-domain method
Ideas of a new multi-domain method (Liang & Chen 2016)
Advantage of the multi-domain method
• Rankine source function and free-surface Green function
• Internal subdomain and external subdomain
• Control surface in a simple form is NOT panelized
Rankine panel method
Free-surface Green function
• Rankine source function
• Well-suited in a finite domain
• Satisfy linear free-surface condition and radiation condition
• Distribute singularities over the control surface and waterline
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32Wavy properties and analytical modeling of free-surface flow
Multi-domain method
Figure: Schematic of the control surface dividing the fluid domain into
internal and external subdomains.
FE
C C
H
FE
F
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33Wavy properties and analytical modeling of free-surface flow
Figure: Mesh of the hull surface and part of the free surface.
Radiation and diffraction problems of a floating hemi-sphere
Multi-domain method
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34Wavy properties and analytical modeling of free-surface flow
Figure: Added mass and damping coefficients of a floating hemi-sphere
varying with k0R0. k0 = wave number; R0 = radius of hemi-sphere.
Comparison is made with analytical results by Hulme (1982).
Surge and heave added mass and damping coefficients of
a floating hemi-sphere.
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
b11
a1
1 a
nd
b1
1
k0R
0
Hulme
Original
Extendeda11
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
b33
a3
3 a
nd
b3
3
k0R
0
Hulme
Original
Extendeda33
Multi-domain method
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35Wavy properties and analytical modeling of free-surface flow
Figure: Removal of irregular frequencies by introducing an internal
free surface (Zhu, 1994; Lee et al., 1996).
FE
C
FE
C
I
𝜕𝜙
𝜕𝑛= 0
𝜙 = 0 𝜙 = 0
Multi-domain method
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36Wavy properties and analytical modeling of free-surface flow
0 1 2 3 4-1.5
-1.0
-0.5
0.0
0.5
Im{F3}
F3
k0R
0
Chen et al.
Original
Extended
Re{F3}
0 1 2 3 4-0.5
0.0
0.5
1.0
1.5
2.0
Im{F1}
F1
k0R
0
Chen et al.
Original
Extended
Re{F1}
Figure: Linear wave force acting on a floating hemi-sphere varying with
k0R0. k0 = wave number; R0 = radius of hemi-sphere. Comparison is
made with numerical calculation by Chen et al. (2003).
Horizontal and vertical wave force exerting on a floating
hemi-sphere calculated.
Multi-domain method
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37Wavy properties and analytical modeling of free-surface flow
Conclusions
► Wavy properties of free-surface flows
• Different from viscosity and nonlinearity
• Dispersion relation
• Far field
► Difficulties of forward speed seakeeping problem
• Several wave systems with largely different wavelengths
• Singular and highly-oscillatory behaviors
• Convergence problem
► Multi-domain method
• Potential flow and potential flow
• Potential flow and viscous flow
• No mesh over the juncture boundary
• DtN relation
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Thank you for your attention!