1

Numerical prediction of hydrodynamic coefficients for a semi-sub platform by using large eddy simulation

with volume of fluid method and Richardson extrapolation

Jia, PANTakeshi, ISHIHARA

Bridge and Structure Lab, The University of Tokyo2019/01/17

Hydrodynamic coefficients (Ca & Cd)

Target structures1. Heave Plate2. Floater

L.Tao,2004;Lpoez-Pavon, 2015 (CFD)(Shear Stress Transport (SST) model)

Chia-Rong Chen, 2016(CFD)(No free water surface)

Accuracy

• The effects of free water surface and of KC number on hydrodynamic coefficients of asemi-sub model predicted should be systematically investigated by LES with VOF .

• Accuracy of predicted hydrodynamic coefficients by CFD should be improved.

𝐶𝐶𝑎𝑎: Added mass coefficient; 𝐶𝐶𝑑𝑑: Viscous drag coefficient Keulegan-Carpenter (KC) number: 𝐾𝐾𝐶𝐶 = 2𝜋𝜋𝐴𝐴

𝐷𝐷(A: amplitude of motion; D: diameter of typical component)

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Objectives 3

1. To improve accuracy of the predicted hydrodynamiccoefficients by Richardson extrapolation method.

2. To study the effect of KC number and frequency on thehydrodynamic coefficients.

3. To investigate the importance of the free water surface onevaluation of hydrodynamic coefficients by LES with VOF.

Water tank tests

1∇∇

∫∫

∫

TTH0

a HT 02 2 ww 0

F (t)sin(ωt)dtC = = F (t)sin(ωt)dt

πωaρρ a ω sin (ωt)dt

( )∫

∫∫

TTH0

d H2T 02 2 ww 0

F (t)cos(ωt)dt 3C = - = - F (t)cos(ωt)dt1 4ρ Aωaρ A(ωa) cos(ωt) cos(ωt) cos (ωt)dt2

Horizontally forced oscillation Vertically forced oscillation

-40

-20

0

20

40

0 T/4 T/2 3T/4 T

Measured hydrodynamic forceReproduced hydrodynamic force

F H(N

)

Time (s)

H b I KF (t) = F(t) -F -F(t) -F (t)

Forced vibration tests in the horizontal and vertical directions

• KC number

• Definition of hydrodynamic coefficients Ca and Cd

H a d wF (t) = -C Mx(t) - 0.5 C ρ A x(t) x(t)

i i i

maxV ωa 2πaKC = = =D f D f D

4

Large eddy simulation (LES) with volume of fluid (VOF))

Governing equation

Outflow Wall

Symmetry

Outflow

3.4h

1.1h

7.8LComputational domain

S.N.Zhang, T.Ishihara : Numerical study of hydrodynamic coefficients of multiple heave plates by large eddy simulations with volume of fluid method, Ocean Engineering, Vol.163, pp.583-598, 2018.

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i

i

∂u = 0∂x

i i j i j ij

j i j j i j

∂τ∂u ∂uu ∂p ∂ ∂u ∂uρ +ρ =- + μ + -∂t ∂x ∂x ∂x ∂x ∂x ∂x

( ) ( )

w w w w ww

1 ∂ α ρ +∇ α ρ v = 0ρ ∂t

Continuity equation for the volume fraction of water

Numerical simulation by grid refinement

Grid level 1 2 3

Grid size ℎ1 = 8𝑚𝑚𝑚𝑚 ℎ2 = 4𝑚𝑚𝑚𝑚 ℎ3 = 2𝑚𝑚𝑚𝑚Grid number

13.7 million

18.8 million

63.8 million

Grid refinement In the vertical : Refined area in a region of 5cm near Hp, Hp-C, PntnIn the horizontal : Refined area in a region of 5cm near SC, CC

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Exp. Level 1 Level 2 Level 3

Ca3

3 (in

ver

tical

)

0

1

2

3

4

5

6

7

Exp. Level 1 Level 2 Level 3

Cd3

3 (in

ver

tical

)HP1

(Level1)

Predicted Ca & Cd by refined grids

• The accuracy of predicted Cd by using grid refinement is not enough.

5cm HP1(Level 2) SC1

(Level 2)5cm

SC1(Level 1)

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Richardson Extrapolation MethodThe exact solution

where

( )2 1

2 1h h

h h h p

φ φφ ε φ

λ−

Φ = + = +−

1 3 22log(( ) /( ))

,log

h h h hpφ φ φ φ

λ

− −= 1 2 2 3/ /h h h hλ = =( ) ( )

3 2 2 1

3 2

,1 1

h h h hp pp ph hφ φ φ φ

αλ λ

− −= =

− −

0.47p =

• Richardson Extrapolation Method on the finest grid is applied and validated.

Richardson Extrapolation Method

Error:29.9%

0.9%0

0.5

1

1.5

0 2 4 6 8

Cal.

Exp.

Ca

Grid level

Extrapolated value: 1.16

0

1

2

3

4

5

6

7

8

0 2 4 6 8

Cal.

Exp.

Cd

Grid level

Extrapolated value: 5.36

Error:

6.0%

0%

HP1Level 1

HP1Level 2

Fine grid is required to accurately simulate the vortex shedding.

i

ph h h ih Hφ ε φ αΦ = + = + +

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Effect of grid refinement

0

0.5

1

1.5

2

1.5 2 2.5 3 3.5

Exp.Cal.-Coarse gridCal.-Extrapolation

Ca3

3

Period (s)

0

1

2

3

4

5

6

7

8

1.5 2 2.5 3 3.5

Exp.Cal.-Coarse gridCal.-Extrapolation

Cd3

3

Period (s)

0

0.2

0.4

0.6

0.8

1

1.5 2 2.5 3 3.5

Exp.Cal.-Coarse gridCal.-Extrapolation

Ca1

1

Period (s)

0

0.5

1

1.5

2

1.5 2 2.5 3 3.5

Exp.Cal.-Coarse gridCal.-Extrapolation

Cd1

1

Period (s)

• The predicted hydrodynamic coefficients by using LES with VOF method agree well with the experimental data when Richardson extrapolation is performed.

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Effect of KC number and wave frequency

0

0.2

0.4

0.6

0.8

1

1.5 2 2.5 3 3.5

Exp.-KC=4.62Exp.-KC=9.24Cal.-KC=4.62Cal.-KC=9.24Cal.-Potential theory

Ca1

1

Period (s)

-1

-0.5

0

0.5

1

1.5

2

1.5 2 2.5 3 3.5

Exp.-KC=4.62Exp.-KC=9.24Cal.-KC=4.62Cal.-KC=9.24A. Robertson et al. (2014)

Cd1

1

Period (s)

In the horizontal direction

• Potential theory and database have limited accuracy for Ca and Cd, while LES model withVOF can accurately predict the Ca and Cd for different KC numbers and wave frequencies.

In the vertical direction

0

0.5

1

1.5

2

1.5 2 2.5 3 3.5

Exp.-KC=0.9Exp.-KC=1.8Cal.-KC=0.9Cal.-KC=1.8Cal.-Potential theory

Ca3

3

Period (s)

KC Ca11 KC Cd11

0

1

2

3

4

5

6

7

8

1.5 2 2.5 3 3.5

Exp.-KC=0.9Exp.-KC=1.8Cal.-KC=0.9Cal.-KC=1.8A. Robertson et al. (2014)

Cd3

3

Period (s)

KC Ca33 KC Cd33

KC=4.62 KC=9.24

KC=0.9 KC=1.8

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Effect of free water surface In the horizontal direction

0

0.2

0.4

0.6

0.8

1

1.5 2 2.5 3 3.5

Exp.Cal.-W/O F.S.Cal.-With F.S.

Ca1

1

Period (s)

0

0.5

1

1.5

2

1.5 2 2.5 3 3.5

Exp.Cal.-W/O F.S.Cal.-With F.S.

Cd1

1

Period (s)

Ca11Error:

35.8%

8.0%

Cd11Error:25.2%

3.4%

• The free water surface should be included to accurately predict hydrodynamic coefficientsin the horizontal direction and can be captured by using LES with VOF.

KC=9.24 KC=9.24

With free water surface W/O free water surface

SWL

KC=9.24 KC=9.24

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Effect of free water surface In the vertical direction

0

0.5

1

1.5

2

1.5 2 2.5 3 3.5

Exp.Cal.-W/O F.S.Cal.-With F.S.

Ca3

3

Period (s)

0

1

2

3

4

5

6

7

8

1.5 2 2.5 3 3.5

Exp.Cal.-W/O F.S.Cal.-With F.S.

Cd3

3

Period (s)

• The predicted Ca and Cd with and without free surface in the vertical directioncoincide well with those from the water tank test, because the free surface has alimited effect on Ca and Cd in the vertical direction for the deep draft model.

KC=1.8 KC=1.8

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With free water surface W/O free water surface

Prediction of dynamic response

See the poster No.37

The predicted dynamic responses indifferent wave heights by proposedmodel show good agreement withthose from the water tank tests.

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Conclusions

1. The grid refinement can improve accuracy by capturing thevortex shedding near the model and the predicted dragcoefficients by Richardson extrapolation method show goodagreement with those from the water tank test.

2. LES model with VOF can accurately predict the KC numbereffect on the hydrodynamic coefficients in the horizontal andvertical directions, while potential theory and database havelimited accuracy.

3. The hydrodynamic coefficients in the horizontal direction byLES with VOF show good agreement with the experimentaldata, while those predicted by LES without the free surfaceshow significant differences.

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Thank you for your attention!This research is carried out as a part of the Fukushima floatingoffshore wind farm demonstration project funded by Ministry ofEconomy, Trade and Industry.