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Long-period Harbor Oscillations due to Short

Random Waves

Meng-Yi Chen & Chiang C. Mei

Massachusetts Institute of Technology

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Typhon Tim 1994: Hualien Harbor,Taiwan

(# 00)(# 00)

## 8 8

# 00# 00

# 05# 05

# 22# 22

# 10# 10

outsideoutside

insideinside

outsideoutside

insideinside

T (sec)0 200

H

inside

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Port of Hualien

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2

2

10

22

8

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Typhoon Longwang, Oct. 2nd 2005

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Past Works

• Harbor Oscillations- Linear theory

Miles & Munk (1961), Miles( 1971), Lee(1971),

Unluata & Mei (1973), (1978) , Carrier,Shaw & Miyata(1971)

– Nonlinear approximation -- narrow-bandedBowers(1977), Agnon & Mei (1989), Wu & Liu (1990)

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Standing waves near a cliff-Random sea

• Sclavounos (1992)-Stochastic theory

-Simple progressive and standing wave in deep water

-Incident waves: stationary, Gaussian

-Higher order spectrum depends on

first, second, and third-order

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21212 ),(

Pairs of frequencies

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0)(])()([

)()(

cosh

)(cosh)()(

112

12

12

1

11111

11

1111

1

hVhUCCkCC

deA

dekh

zhkigA

gg

ti

ti

Frequency responses Frequency responses By Mild slope By Mild slope ApproximationApproximation

First-order First-order

Chamberlain & Porter (1995) Chamberlain & Porter (1995)

Far field : analytical solution +radiation Far field : analytical solution +radiation conditioncondition

Near field: FEMNear field: FEM

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Hybrid finite element method (Chen & Mei,1974)(HFEM)

Far Field

Analytical

Near Field

Finite element

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Square harbor, Normal incidence 300m by 300 m, depth h=20m

Effect of entrance(1) 60 m opening without protection

(2) 30 m opening without protection

(3) 30 m opening with protection

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Random sea: TMA Spectrum

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First-order average response

60m, no protection

30m, no protection

30m with protection

|)(| 1

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

5000

10000

15000

20000

25000

30000

(rad/sec)2S

A(

) (

cm2 -s

ec)

Mean Linear spectrum

212 ||)()( ASS

60m, no protection

30m, no protection

30m with protection)(AS

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Second-order Mean: setup/down

60m

30m, with protection

30m, no protection

111212 ,,,, dyxStx I

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Nonlinear correction: long wave )(22 S

60m, no protection

30m, no protection

30m with protection

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

5000

10000

15000

20000

25000

30000

(rad/sec)2S

A(

) (

cm2 -s

ec)

)(AS

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Mean Harbor Spectrum

60m 30m,no

30m, protected

)(2 fS)(22 fS)(22 fS

)(22 fS

)(2 fS

)(2 fS)()()( 222 fSfSfS

)( 2O )( 4O

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Qualitative comparison with field data

outout

ininoutout

inin

30m, protected

)(22 fS )(2 fS

)()()( 222 fSfSfS )( 2O )( 4O

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Numerical Aspects

• For 2-nd order problem must be solved for a many pairs of frequencies by FEM

• Large sparse matrix for each pair -- for variable depth: modes are coupled

--10620 pairs, each pair need around 15 minutes, at least 100 days for ONE single computer,

--20-25 parallel computer (4G ram, 2.8G Hz), weeks

),( 212

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Summary- - Stochastic theory for long-period harbor resonance Stochastic theory for long-period harbor resonance by a broad-banded sea by a broad-banded sea

-Long-wave part of response spectrum is -Long-wave part of response spectrum is dominated by second-order correction, not dominated by second-order correction, not first or third-orderfirst or third-order

-Mild-slope equation for second order in wave -Mild-slope equation for second order in wave steepness is sufficient steepness is sufficient

--High-frequency part of response spectrum is High-frequency part of response spectrum is dominated by first-order wavedominated by first-order wave

-Extendable to Slow drift of floating structures-Extendable to Slow drift of floating structures