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

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

3Wavy properties and analytical modeling of free-surface flow

Fluid-structure interaction

► Viscosity effect Flow separation

VIV and VIM

Dissipation in gap

Roll damping

……

4Wavy properties and analytical modeling of free-surface flow

Fluid-structure interaction

► Non-linearity

Slamming

Ringing and springing

Water-on-deck

……

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

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)

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

8Wavy properties and analytical modeling of free-surface flow

Wavy properties of free surface flows

20 25 30 35 40-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

ζ

t

Experiment HPC results

20 25 30 35 40-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

ζ

t

Experiment HPC results

20 25 30 35 40-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

ζ

t

Experiment HPC results

► Nonlinear waves propagating over a submerged bar

9Wavy properties and analytical modeling of free-surface flow

Wavy properties of free surface flows

► Wave-induced motions (Results calculated by HydroSTAR)

10Wavy properties and analytical modeling of free-surface flow

Wavy properties of free surface flows

► Typical steady ship waves

11Wavy properties and analytical modeling of free-surface flow

Wavy properties of free surface flows

► Ship waves and environment

12Wavy properties and analytical modeling of free-surface flow

Wavy properties of free surface flows

► Time-harmonic waves due to heave motions

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

• …

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

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

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

17Wavy properties and analytical modeling of free-surface flow

Analytical modelling of waves

Right open dispersion curve Inner-V waves

18Wavy properties and analytical modeling of free-surface flow

Analytical modelling of waves

Closed dispersion curve Ring waves

19Wavy properties and analytical modeling of free-surface flow

Analytical modelling of waves

Left open dispersion curve Outer-V waves

20Wavy properties and analytical modeling of free-surface flow

Analytical modelling of waves

Left open dispersion curve Ring-fan and fan waves

21Wavy properties and analytical modeling of free-surface flow

Analytical modelling of waves

► Ring wave animation

22Wavy properties and analytical modeling of free-surface flow

Analytical modelling of waves

Ring wave component

Local component

Wave + local components

► Ring wave system

23Wavy properties and analytical modeling of free-surface flow

Analytical modelling of waves

► V-shaped wave animation

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

25Wavy properties and analytical modeling of free-surface flow

Analytical modelling of waves

Cusp angles in time-harmonic ship waves

26Wavy properties and analytical modeling of free-surface flow

Analytical modelling of waves

Crest lines of wave systems at 𝝉 = 𝟐/𝟐𝟕 ≈ 𝟎. 𝟐𝟕𝟐

27Wavy properties and analytical modeling of free-surface flow

Analytical modelling of waves

► Singular and peculiar properties (Chen & Wu, JFM, 2001)

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

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

30Wavy properties and analytical modeling of free-surface flow

Multi-domain method

Externaldomain

Transitionaldomain

Internaldomain

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

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

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

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

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

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

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

Thank you for your attention!