Mitigating harbour storms by enhancing nonlinear wave interactions Paul Chleboun
Mitigating harbour storms by enhancing nonlinear wave interactions
Paul Chleboun
Petr Denissenko, Colm Connaughton, Sergey Nazarenko,
Sergei Lukaschuk.
Complexity Science Doctoral Training Centre
University of Warwick
9 th April 2009
Outline
1
2
3
4
5
Harbour storms
Motivation
Waves of tens of meters pose risks in oceanic harbours.
Waves driven by wind and waves from the ocean.
Energy dissipation occurs at short wavelengths by wave breaking and white-capping.
[www.hydrolance.net]
Aims
Understand energy transfer by nonlinear wave interaction.
Excite wave modes that enhance the energy transfer.
Harbour storms
Motivation
Waves of tens of meters pose risks in oceanic harbours.
Waves driven by wind and waves from the ocean.
Energy dissipation occurs at short wavelengths by wave breaking and white-capping.
[www.hydrolance.net]
Aims
Understand energy transfer by nonlinear wave interaction.
Excite wave modes that enhance the energy transfer.
The model
Irrotational flow
= ⇒ u = ∇ φ .
Incompressible
∇ · u = 0
= ⇒ ∆ φ = 0.
Definition
ψ ( x , t ) = φ ( x , η ( x , t ) , t ) .
The model
Irrotational flow
= ⇒ u = ∇ φ .
Incompressible
∇ · u = 0
= ⇒ ∆ φ = 0.
Definition
ψ ( x , t ) = φ ( x , η ( x , t ) , t ) .
The model
Irrotational flow
= ⇒ u = ∇ φ .
Incompressible
∇ · u = 0
= ⇒ ∆ φ = 0.
Definition
ψ ( x , t ) = φ ( x , η ( x , t ) , t ) .
The model
Irrotational flow
= ⇒ u = ∇ φ .
Incompressible
∇ · u = 0
= ⇒ ∆ φ = 0.
Definition
ψ ( x , t ) = φ ( x , η ( x , t ) , t ) .
Boundary conditions
Dynamic
∂ψ
∂ t
+
1
2
( ∇ φ )
2
| z = η
+ g η = 0
Together with Laplace equation fully specify system.
Kinematic
∂η
∂ t
∂φ
∂ z
=
∂φ
∂ z on z = η ( x
= 0 on z = − h .
, t )
Wave solution
Fourier transform
1 f ( k ) =
2 π
1 f ( x ) =
2 π
Z
Z f ( x ) e
− i ( k · x ) d x f ( k ) e i ( k · x ) d k
Wave steepness; α = k η ( k ) , assumed small.
η ( k , t ) = η
0
( k ) e i ω ( k ) t and ψ ( k , t ) = ψ
0
( k ) e i ω ( k ) t
Dispersion relation
ω ( k ) = p gk tanh kh
ω ( k ) → p gk as h → ∞ .
Wave solution
Fourier transform
1 f ( k ) =
2 π
1 f ( x ) =
2 π
Z
Z f ( x ) e
− i ( k · x ) d x f ( k ) e i ( k · x ) d k
Wave steepness; α = k η ( k ) , assumed small.
η ( k , t ) = η
0
( k ) e i ω ( k ) t and ψ ( k , t ) = ψ
0
( k ) e i ω ( k ) t
Dispersion relation
ω ( k ) = p gk tanh kh
ω ( k ) → p gk as h → ∞ .
Hamiltonian Equations of motion
η ( k , t ) and ψ ( k , t ) are canonical variables satisfying
Hamiltonian equations [Broer 74, Miles 77].
Assume random phase and amplitude and take infinite region then small nonlinearity limit.
⇒ Derive a kinetic equation describing evolution of the spectrum in terms of wave-action (average amplitude).
∂ n k
∂ t
=
Z
W k
3
, k k
1
, k
2 n k
1 n k
2
( n k
3
+ n k
) − n k
3 n k
( n k
1
+ n k
2
)
δ ( k
1
+ k
2
− k
3
− k ) δ ( ω
1
+ ω
2
− ω
3
− ω ) d k
1 , 2 , 3
Resonant interactions
Resonant manifold
ω ( k
1
) + ω ( k
2
) − ω ( k
3
) − ω ( k
4
) = 0 , k
1
+ k
2
− k
3
− k
4
= 0 , k d 1 k d 2
= ( 3 .
810 , 0 )
= ( 5 .
166 , 0 .
818 )
2 k k d 1 k d 1 d 1
= k
3
+ k
4
.
+ k d 2
+ k
2
= k
3
= k d 2
+ k
4
.
+ k
4
.
Finite region
1 Initially at rest then a finite number of modes are driven.
3
4
2 Cascade arrest due to k-space discreteness leads to accumulation of energy near forcing scales.
This leads to widening of the nonlinear resonance.
Sufficient widening triggers a cascade.
Finite region
1
2
Initially at rest then a finite number of modes are driven.
Cascade arrest due to k-space discreteness leads to accumulation of energy near forcing scales.
3
4
This leads to widening of the nonlinear resonance.
Sufficient widening triggers a cascade.
Finite region
1
2
3
Initially at rest then a finite number of modes are driven.
Cascade arrest due to k-space discreteness leads to accumulation of energy near forcing scales.
This leads to widening of the nonlinear resonance.
4 Sufficient widening triggers a cascade.
Finite region
3
4
1
2
Initially at rest then a finite number of modes are driven.
Cascade arrest due to k-space discreteness leads to accumulation of energy near forcing scales.
This leads to widening of the nonlinear resonance.
Sufficient widening triggers a cascade.
Turbulance cascade
Wave energy spectrum
E f
=
Z e
2 π ft
0 i h η ( x , t ) η ( x , t + t
0
) i dt
0
Zakharov Filonenko (1967) power law solution to wave kinetic equation.
E f
∼ f
− 4
.
Other power law power spectra
Phillips (58), E f
Kuznetsov (04),
∼ f
− 5
E f
∼ f
− ( 3 + D )
Nazarenko (06), E f
∼ f
− 6
Turbulance cascade
Wave energy spectrum
E f
=
Z e
2 π ft
0 i h η ( x , t ) η ( x , t + t
0
) i dt
0
Zakharov Filonenko (1967) power law solution to wave kinetic equation.
E f
∼ f
− 4
.
Other power law power spectra
Phillips (58), E f
Kuznetsov (04),
∼ f
− 5
E f
∼ f
− ( 3 + D )
Nazarenko (06), E f
∼ f
− 6
Experimental Setup
8 Panel Wave Generator
Fluorescent visualization:
Spatial spectra
Capacitance probes:
Temporal spectra
Laser
6 m
[Total Environment Simulator at “The Deep” Geography
Department, University of Hull]
Data
Elevation, channel 1(top), channel 2(bottom)
10
5
0
−5
−10
50
5
0
−5
40
100 150 200 250 ch 2
300 350 400 450
50 60 70 time, sec
80 90 100
Initial maximum
Root mean squared surface elevation
Averaged of 10 second windows
A = q h ( η − ¯ ) 2 i .
RMS of amplitude
4
3.5
3
2.5
2
1.5
1
0.5
0
0 100 200 300 t, [sec]
400 500 600
Spectra
10
2
10
1
10
0
10
−1
10
−2
10
−3
0−66 f
0 f
1
2f
0
2f
1
10
−4
1 2
Frequency [Hz]
3 4 5
Welch algorithm with Hann windows of length 20 .
48s averaged over minimum of 5 spectra.
Spectra
10
2
10
1
10
0
10
−1
10
−2
10
−3
0−66
66−132 f
0 f
1
2f
0
2f
1
10
−4
1 2
Frequency [Hz]
3 4 5
Welch algorithm with Hann windows of length 20 .
48s averaged over minimum of 5 spectra.
Spectra
10
2
10
1
10
0
10
−1
10
−2
10
−3
0−66
66−132
132−198 f
0 f
1
2f
0
2f
1
10
−4
1 2
Frequency [Hz]
3 4 5
Welch algorithm with Hann windows of length 20 .
48s averaged over minimum of 5 spectra.
Spectra
10
2
10
1
10
0
10
−1
10
−2
10
−3
Fit Range
Probe 1 slope − 5.06
Probe 2 slope − 5.23
1 2 3
Frequency [Hz]
4 5 6 7 8 9 10
Welch algorithm with Hann windows of length 10 .
24s averaged over minimum of 500 spectra.
Spectra
10
2
10
0
0−66
2f
0 f
0
10
−2
10
−4
10
0
10
1
Frequency [Hz]
Welch algorithm with Hann windows of length 20 .
48s averaged over minimum of 5 spectra.
Spectra
10
2
10
0
0−66
66−132 f
0
2f
0
10
−2
10
−4
10
0
10
1
Frequency [Hz]
Welch algorithm with Hann windows of length 20 .
48s averaged over minimum of 5 spectra.
Spectra
10
2
10
0
0−66
66−132
132−198 f
0
2f
0
10
−2
10
−4
10
0
10
1
Frequency [Hz]
Welch algorithm with Hann windows of length 20 .
48s averaged over minimum of 5 spectra.
Spectra
10
2
Fit Range
10
0
10
−2 Probe 1 slope − 4.18
Probe 2 slope − 4.31
10
−4
10
0
10
1
Frequency [Hz]
Welch algorithm with Hann windows of length 10 .
24s averaged over minimum of 500 spectra.
4-wave resonant interactions
(a) Resonant manifold
2 k d 1 k d 1
+
= k k
3 d 2
+ k
4
= k
3
.
+ k
4
.
k d 1
+ k
2
= k d 2
+ k
4
.
(b) Interaction coefficient
Presence of quasi-resonance in spectra
10
2
10
0
10
−2
10
−4
0.5
1
Frequency [Hz]
2 3
| 2 ω ( k d 1
) − ω ( 2 k d 1
− k
4
) − ω ( k
4
) | = 0 .
0003 black
| ω ( k d 1
) + ω ( k d 2
) − ω ( k d 1
+ k d 2
− k
4
) − ω ( k
4
) | = 0 .
0052 blue
| ω ( k d 1
) − ω ( k d 2
) + ω ( k d 2
− k d 1
+ k
4
) − ω ( k
4
) | = 0 .
0051 red .
Conclusion and further work
Conclusions:
1 Accumulation of energy at driving frequency before the onset of nonlinear interactions supports resent theoretical work.
2
3
Greater accumulation for single driving frequency (or well spaced driving frequencies).
Largest interaction coefficient with two close driving modes.
Predict energy transfer based on closest quasi-resonance with eigenmodes of the Harbour.
Further work:
1 More experiments for various driving rates and modes.
2
3
Supported by numerical simulations.
More accurate estimates of interaction coefficient.
Conclusion and further work
Conclusions:
1 Accumulation of energy at driving frequency before the onset of nonlinear interactions supports resent theoretical work.
2
3
Greater accumulation for single driving frequency (or well spaced driving frequencies).
Largest interaction coefficient with two close driving modes.
Predict energy transfer based on closest quasi-resonance with eigenmodes of the Harbour.
Further work:
1 More experiments for various driving rates and modes.
2
3
Supported by numerical simulations.
More accurate estimates of interaction coefficient.
Acknowledgements
Thanks to
Petr Denissenko for supervision and
Sergey Nazarenko for co-supervision.
Sergyei Lukaschuk for the experimental results.
Colm Connaughton for many useful discussions.
Warwick Complexity Science DTC
Funding from the EPSRC
All data supplied by Denissenko and Lukaschuk from the total environment simulator at “The Deep”, University of
Hull Geography Department.
η ( k , t ) and ψ ( k , t ) are canonical variables satisfying
Hamiltonian equations [Broer 74, Miles 77].
∂η ( x , t )
∂ t
=
δ H
δψ ( x , t )
,
∂ψ ( x , t )
∂ t
δ H
= −
δη ( x , t )
Where H = K + Π ,
K =
Π =
1
2
Z Z
η
− h
( ∇ ψ )
2 d z d x
1
2 g
Z
η
2 d x
Assume small nonlinearity (wave steepness)
⇒ Expand the Hamiltonian as an integral power series in conjugate variables.
Canonical transformations reduce the Hamiltonian.
Assume random phase and amplitude and take infinite region then small nonlinearity limit.
⇒ Derive a kinetic equation describing evolution of the spectrum.
Resonant manifold
ω ( k
1
) ± ω ( k
3
) ± ω ( k
3
) = 0 , k
1
ω ( k
1
) + ω ( k
2
) − ω ( k
3
) − ω ( k
4
) = 0 , k
1
± k
2
+ k
2
± k
3
− k
3
= 0 ,
− k
4
= 0 ,
Assume small nonlinearity (wave steepness)
⇒ Expand the Hamiltonian as an integral power series in conjugate variables.
Canonical transformations reduce the Hamiltonian.
Assume random phase and amplitude and take infinite region then small nonlinearity limit.
⇒ Derive a kinetic equation describing evolution of the spectrum.
Resonant manifold
ω ( k
1
) ± ω ( k
3
) ± ω ( k
3
) = 0 , k
1
ω ( k
1
) + ω ( k
2
) − ω ( k
3
) − ω ( k
4
) = 0 , k
1
± k
2
+ k
2
± k
3
− k
3
= 0
− k
4
,
= 0 ,
