A Preliminary result on the long time evolution of non-linear waves --as a partial result of

1
A Preliminary result on the long time evolution
of non-linear waves --as a partial result of

• The Characteristics of Nonlinear Wave
  Transformation
• on
• Sloping Bottoms
• A MOE Program for Promoting Academic Excellence
  of Universities,
• Taiwan
• under grant number A-91-E-FA09-7-3

2
Basic Information of the Project (1/2)

• Objectives
• The objectives of the project are to investigate
  the nonlinear wave transformation, modulation,
  and the characteristics of related flow
  structures during the shoaling process of wave
  propagation.

3
Basic Information of the Project (2/2)

• Sub-Projects

4
Previous work done by (1/3)

• Theoretical studies
• Benjamin and Feir(1967)
• weakly nonlinear deep water wavetrains were
  unstable to modulational perturbations
• Whitham(1967), Chu and Mei(1970,1971)
• Zakharzov(1968) Schrödinger Eq.(third-order of
  ka)
• (1) deep water
• (2) small ka
• (3) slow modulation of amplitude(narrow band)
• Dysthe(1979) forth-order equation
• (1) mean flow effect(mean water level variation)

5
Previous work done by (2/3)

• Numerical development
• Fully Nonlinear Eq.
• Louguet-Higgins(1978)
• Wang Pei(1995)
• Wethuis(2001)
• Cubic Schrödinger Eq.
• Yuen and Lake(1978)
• Shemer et al.(1998)
• Forth-order Eq.
• Janssen(1983)
• Lo and Mei(1985)
• Lo and Mei(1987)

6
Previous work done by (3/3)

• Experimental Studies
• Lake et. al. (1977)F-P-U phenomena
• Su(1982) , Kit et al.(2000), Wethuis(2001)
• Melville(1982)
• Shemer et al.(1998) intermediate depth
• Tulin and Waseda(1999) breaking effect
• Waseda and Tulin(1999)

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The approaches and methodology of the Project

• The basic approaches are to utilize the
  facilities of the THL to investigate this topic
  experimentally followed by developing a numerical
  model that is able to describe the transformation
  of waves from a deep water region to shallow
  water region and even breaking taken place.
• The methodology applied on this project is to
  investigate the subject under a relatively simple
  boundary condition first, then gradually proceed
  to complicated ones.

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Overview of Experiments

• Three phases
• Phase I
• In the Super Tank(5m5m300m), THL
• With a slopping bottom(180)
• 39 wave gages
• Bicromatic waves
• Phase II
• In a meso-flume(7m1m60m), THL
• With a slopping bottom(140)
• Seeded three waves system
• Phase III
• In the Super Tank(5m5m300m)
• Constant water depth
• 6624 wave gages, 81 velocimetries
• Seeded three waves system, un-seeded waves..

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EXP. I (1/2)

• Tests in the Super Tank , THL
• To ensure the capabilities of the facilities
• Wave generator
• Two waves system (bichromatic waves)
• Boundary conditions
• Relative dimensions, sidewall effects,
  reflection
• Instrumentation
• using 39 wave sensors
• Data acquisition and analysis

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EXP. I (2/2)

WH01
WH03
WH04
WH05
WH06
WH07
WH08
WH09
WH10
WH11
WH12
WH13
WH14
WH15
WH16
WH17
WH18
WH19
WH20
WH21
WH22
WH23
WH24
WH25
WH39

3.5m
3.15
45

285m
MNDAS
A Schematic Diagram of the Experimental
set-up(phase I, III)
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(1/10)Results-Exp. Phase I
kx/2?3.5

• (a) Wave profiles showing the modulational wave
  train.(b)Spectral evolution.(c)amplitude envelope
  of surface elevation.

kx/2?4.3
kx/2?4.6
kx/2?8.5
12
(2/10)Results-Exp. Phase I
kx/2?12.5
kx/?16.0

• (a) Wave profiles showing the modulational wave
  train.(b)Spectral evolution.(c)amplitude envelope
  of surface elevation.

kx/2?19.8
kx/2?23.6
180 slope
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(3/10)Results-Exp. Phase I
kx/2?27.3
kx/2?31.1

• (a) Wave profiles showing the modulational wave
  train.(b)Spectral evolution.(c)amplitude envelope
  of surface elevation.

kx/2?34.9
kx/2?38.6
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(4/10)Results-Exp. Phase I
kx/2?43.4
kx/2?46.2

• (a) Wave profiles showing the modulational wave
  train.(b)Spectral evolution.(c)amplitude envelope
  of surface elevation.

kx/2?50.0
kx/2?53.7
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(5/10)Results-Exp. Phase I
kx/2?57.5
kx/2?58.8

• (a) Wave profiles showing the modulational wave
  train.(b)Spectral evolution.(c)amplitude envelope
  of surface elevation.

kx/2?59.2
kx/2?59.5
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(6/10)Results-Exp. Phase I
kx/2?59.7
kx/2?60.2

• (a) Wave profiles showing the modulational wave
  train.(b)Spectral evolution.(c)amplitude envelope
  of surface elevation.

kx/2?60.7
kx/2?61.2
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(7/10)Results-Exp. Phase I
kx/2?61.6
kx/2?61.9

• (a) Wave profiles showing the modulational wave
  train.(b)Spectral evolution.(c)amplitude envelope
  of surface elevation.

kx/2?62.3
kx/2?62.7
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(8/10)Results-Exp. Phase I
kx/2?62.8
kx/2?62.9

• (a) Wave profiles showing the modulational wave
  train.(b)Spectral evolution.(c)amplitude envelope
  of surface elevation.

kx/2?63.0
kx/2?63.1
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(9/10)Results-Exp. Phase I
kx/2?63.2
kx/2?63.29

• (a) Wave profiles showing the modulational wave
  train.(b)Spectral evolution.(c)amplitude envelope
  of surface elevation.

kx/2?63.3
kx/2?63.4
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(10/10)Results-Exp. Phase I
kx/2?63.5

• (a) Wave profiles showing the modulational wave
  train.(b)Spectral evolution.(c)amplitude envelope
  of surface elevation.

kx/2?63.6
kx/2?63.7
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Experiments-Phase II
22
Power spectrum of waveboard motion (EXP.II)
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Data Analysis EXP. II

• Bandpass filter 0.01 Hz 5 Hz
• data low passed by Hilbert transform
• power spectrum were calculated under
• Data length3072
• NFFT1024
• WindowHanning(1024)
• Overlap0
• Frequency resolution1.95310-2

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Test conditions EXP. II

• Three waves system

ka gt0.140.16 breaking, ka gt0.29 three
dimensional
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Spatial evolution of wave profiles(EXP. II, case
2)
kx/2?3.2
kx/2?3.7
kx/2?3.8
kx/2?4.5
kx/2?5.8
26
Spatial evolution of wave profiles(EXP. II, case
2)
kx/2?7.0
kx/2?8.3
kx/2?9.6
kx/2?10.9
kx/2?12.2
27
Spatial evolution of wave profiles(EXP. II, case
2)
kx/2?13.5
kx/2?14.0
kx/2?14.5
kx/2?15.0
kx/2?16.1
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Spatial evolution of wave profiles (EXP. II, case
3)
kx/2?3.2
kx/2?3.7
kx/2?3.8
kx/2?4.5
kx/2?5.8
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Spatial evolution of wave profiles (EXP. II, case
3)
kx/2?7.0
kx/2?8.3
kx/2?9.6
kx/2?10.9
kx/2?12.2
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Spatial evolution of wave profiles (EXP. II, case
3)
kx/2?13.5
kx/2?14.0
kx/2?14.5
kx/2?15.0
kx/2?16.1
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Spatial evolution of wave profiles (EXP. II, case
1)
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Spatial evolution of wave profiles(EXP. II, case
1)
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Spatial evolution of wave profiles(EXP. II, case
1)
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Spatial evolution of wave profiles and spectrum
(EXP. II, case 1)
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Spatial evolution of wave profiles and
spectrum(EXP. II, case 1)
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Spatial evolution of wave profiles and
spectrum(EXP. II, case 1)
37
Side band amplitude evolutions (EXP. II, case 1)
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Growth Curve
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Wave steepness effects

1. The spatial evolution of wave profiles is not
   only related to the initial wave steepness
2. The experiment (ka0.1) is similar to the
   condition of the evolution experiment on a weak
   wave reported in Lake et. al.(1977) and Tulin
   Waseda(1999).
3. ka gt0.140.16 breaking, ka gt0.29 three
   dimensional

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Instrumentation EXP. III(1/2)
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Instrumentation EXP. III(2/2)
42
test conditions were determined mainly based on
Initial growth rate of the sideband disturbance,
based on Krasitskiis reduced four-wave
interaction (Tulin Washeda 1999).
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Data Analysis EXP. III

• 1. following that of EXP. II
• 2. the bi-spectrum analysis

44
Spatial evolution of wave profiles(EXP. III, case
5)
kx/2?4.6
kx/2?5.4
kx/2?5.6
kx/2?5.9
kx/2?7.8
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Spatial evolution of wave profiles(EXP. III, case
5)
kx/2?9.6
kx/2?11.5
kx/2?13.4
kx/2?15.2
kx/2?17.7
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Spatial evolution of wave profiles(EXP. III, case
5)
kx/2?19
kx/2?20.9
kx/2?22.7
kx/2?24.6
kx/2?26.5
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Spatial evolution of wave profiles(EXP. III, case
5)
kx/2?28.4
kx/2?30.2
kx/2?31.2
kx/2?32.1
kx/2?33
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Spatial evolution of wave profiles(EXP. III, case
5)
kx/2?34
kx/2?34.9
kx/2?35.9
kx/2?36.8
kx/2?37.7
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Spatial evolution of wave profiles(EXP. III, case
5)
kx/2?62.1
kx/2?63
kx/2?64
kx/2?64.9
kx/2?65.8
50
Spatial evolution of wave profiles(EXP. III, case
5)
kx/2?66.8
kx/2?67.7
kx/2?68.6
kx/2?69.6
kx/2?70.5
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Spatial evolution of wave profiles(EXP. III, case
5)
kx/2?71.5
kx/2?72.4
kx/2?73.3
kx/2?73.6
kx/2?74.6
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Side band amplitude evolution (EXP. III, case 5)
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Side band amplitude evolution (EXP. III , case 5)
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Side band amplitude evolution (EXP. III , case 5)
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Side band amplitude evolution (EXP. III , case 5)
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Side band amplitude evolution (EXP. III , case 5)
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Side band amplitude evolution (EXP. III , case 5)
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Side band amplitude evolution (EXP. III , case 5)
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Side band amplitude evolution (EXP. III , case 5)
60
Non-dimensional amplitude vs. kx (EXP. III , case
5)
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• Thank you !

62
Lake et al.(1977)

• When experiments were performed using initially
  uniform, or nearly uniform, wave trains with
  large initial steepness, recurrence cycles were
  observed in which the wave trains became strongly
  modulated and then demodulated until they were
  again nearly uniform.

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Waseda and Tulin(1999)

• Wave train
• It is clear from this figure that Benjamin-Feirs
  theory overestimates the growth rate, but the
  Krasitskii(1994) theory predicts the growth rate
  fairly well.

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Tulin and Waseda(1999)

• Wave train
• The evolution of 1.2m breaking wave
• The energy originally residing largely in the
  carrier ware is now divided roughly between two
  waves, the original carrier and the lower
  sideband. The future evolution of this two-wave
  system, which it was impossible to measure here
  for lack of fetch, can be expected to be
  different from that of the Benjamin-Feir
  three-wave systems studied here.

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MNDAS-the Multi-Nods Data Acquisition System
Data base server
Internet
Internet
Internet
NODE 0
NODE 1
NODE M
ADAM4520
ADAM4520
Internet
Internet
ADAM5510 NODE 1
ADAM5510 NODE 1
ADAM5510 NODE 2
ADAM5510 NODE N

• Up to 120 channels(90 wave gages, 19 velocities 1
  wave board motion and 10 spares) sampling at 20
  Hz are available now

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Wave generator, the Super Tank