turbulence of gravity waves in laboratory experiments s lukaschuk 1, p denissenko 1, s nazarenko 2 1...
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Turbulence of Gravity Waves in Laboratory
Experiments
S Lukaschuk1, P Denissenko1, S Nazarenko2
1 Fluid Dynamics Laboratory, University of Hull2 Mathematics Institute, University of Warwick
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Plan
• Introduction
• Experimental set-up and methods
• Measurements of the frequency spectra and PDF for wave elevation
• Comparison with numerical results and discussion
• Further experiments
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Theoretical prediction forsurface gravity wave Kolmogorov spectra
• Kinetic equation approach for WT in an ensemble of weakly interacted low amplitude waves(Theory and numerical experiment - Hasselman, Zakharov, Lvov, Falkovich, Newell, Hasselman, Nazarenko … 1962 - 2006)
Assumption: weak nonlinearityrandom phase (or short correlation length)spatial homogeneity stationary energy flow from large to small scales
Kolmogorov spectra for gravity waves in infinite space
42
7
wEkE wk
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Phillips Spectrum
Surface elevation
-space: asymptotic of sharp wave crests or dimensional analysis.
-space: either dimensional analysis or using Dissipation is determined by sharp wave crests (due to wave breaking)
Strong nonlinearity
54 wEkE wk
gkw
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Finite size effects
Most exact wave resonances are lost on discrete k-space (Kartashova’1991) “Frozen turbulence” (Pushkarev, Zakharov’2000) Recent numerics by Pokorni et al & Korotkevich et al (2005).
To restore resonant interaction, their nonlinear brodening δ must be greater than the -grid spacing 2/L
Which in our case means
>1/(kL)1/4 (Nazarenko, 2005),
In numerics, this means 10000x10000 resolution for Intensity ~0.1.
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Discrete scenario (Nazarenko’2005)
• Ineficient cascade at small amplitudes
• Accumulation of spectrum at the forcing scale until δk reaches to the k-grid spacing 2/L
• Excess of energy is released via an avalanche
• Mean spectrum settles at a critical slope determined by δk ~2/L, i.e. E ~ -6.
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Numerical experiments• Convincing claims of numerical confirmation of ZF:
A.I. Dyachenko, A.O. Korotkevich, V.E. Zakharov, (2003,2004)
M. Onorato, V Zakharov et al., (2002).
N. Yokoyama, JFM 501 (2004) 169–178.
Lvov, Nazarenko and Pokorni (2005)
• Results are not 100% satisfying because no greater than 1 decade inertial range
• Phillips spectrum could not be expected in direct numerical simulations because:
1) nonlinearity truncation at cubic terms 2) artificial numerical dissipation at high k to prevent numerical
blowups.
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N. Yokayama (JFM, 2004) direct numerical simulationsWave action spectra
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A.Dyachenko, O.Korotkevich, Zakharov (JETP Lett. 2003)
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Y. Lvov, Nazarenko, Pokorni:numerical experiment: Physica D, 2006
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ExperimentsAirborne Measurements of surface elevation k-spectra
P.A. Hwang, D.W.Wang (2000)
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Advantages of the laboratory experiment:
• Wider inertial interval – two decades in k
• Possibility to study both weakly and strongly nonlinear waves
• No artificial dissipation – natural wavebreaking dissipation mechanism.
Goals:
Long-term: to study transport and mixing generated by wave turbulence
Short-term: to characterize statistical properties of waves in a finite system
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Total Environmental SimulatorThe Deep, Hull
• 6 x 12 x 1.6 m water tank
• 8 panels wave generator
• 1 m3/ s – flow
• rain generator
• PIV & LDV systems
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6 metres12
met
res
90 c
m8 Panel Wave Generator
Laser
Capacity Probes
Rain Generator
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Wave generation and measurements
2 capacitance probes at distance 40 cm\Sampling frequency - 50-200 Hz each channelAcquisition time 2000 s
jk
HztrkatrA iji
ii ]2.15.045.0,4.0[ sin,,
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Small amplitude
100
101
10-6
10-5
10-4
10-3
10-2
10-1
100
Elevation spectrum (file 81)
Frequency, [Hz]
Po
we
r s
pe
ctr
um
: E
lev
ati
on
2
Elevation std1 = 1.8 cm
Elevation std = 1.9 cm
excitation
frequences
Fit range
Probe 1 -7.45 Probe 2 -7.14
400 405 410 415 420 425 430
-5
0
5
Ele
vatio
n, c
m
Elevation as function of time: Ch 1(red), Ch 2(blue), (file 81)
400 405 410 415 420 425 430
-5
0
5
ch 2
time, [s]
Ele
vatio
n, c
m
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Medium amplitude
100
101
10-6
10-5
10-4
10-3
10-2
10-1
100
E Elevation Spectrum, file 86
Frequency, [Hz]
Po
we
r s
pe
ctr
um
: |
w|2
Elevation std1 = 3 cm
Elevation std = 3.1 cm
excitation
frequences
Fit range
Probe 1 -5.9 Probe 2 -6.08
400 405 410 415 420 425 430
-10
0
10
Ele
vati
on
, cm
Elevation as function of time: Ch 1(red), Ch 2(blue) (file 86)
400 405 410 415 420 425 430
-10
0
10
ch 2
time, sec
Ele
vati
on
, cm
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Large amplitudes
100
101
10-6
10-5
10-4
10-3
10-2
10-1
100
Elevation Spectrum Ch1, Ch2 File No. 88
Frequency, [Hz]
Po
we
r s
pe
ctr
um
: |
w|2
Elevation std1 = 5.2 cm
Elevation std = 5.3 cm
excitation
frequences
Fit range
Probe 1 -3.74 Probe 2 -3.61
400 405 410 415 420 425 430
-10
0
10
20
Ele
vati
on
, cm
Elevation as finction of time: Ch. 1(red), Ch. 1(blue) (file 88)
400 405 410 415 420 425 430
-10
0
10
20
time, sec
Ele
vati
on
, cm
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-8
-7
-6
-5
-4
-3
1 2 3 4 5 6
Slope Ch1Slope Ch2
Sp
ec
tru
m S
lop
e
Elevation RMS
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Energy (elevation) spectrumsmall amplitude (file 80)
100
101
10-6
10-5
10-4
10-3
10-2
10-1
100
wavesignal_80_090cm_0.4..0.05..1.2Hz_en0.0001_gain0.5_multimode+05-28+50-73_79min_splash0.0_from
Frequency, [Hz]
Pow
er s
pect
rum
: Ele
vatio
n2
Elevation std = 1.3 cm
excitation
frequences
Fit range
Probe 1 -6.18 Probe 2 -6.94
100
101
10-6
10-5
10-4
10-3
10-2
10-1
100
wavesignal_84_090cm_0.4..0.05..1.2Hz_en0.0001_gain2.0_multimode+05-28+50-73_66min_splash
Frequency, [Hz]
Pow
er s
pect
rum
: Ele
vatio
n2
Elevation std = 4.1 cm
excitation
frequences
Fit range
Probe 1 -5.08 Probe 2 -5.19
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PDF of and tt
-4 -2 0 2 410
-4
10-3
10-2
10-1
100
PDF of the second derivative tt
, File No. 84
tt
/
= 3.86 cmS = -0.56K = 7.13
-4 -2 0 2 410
-4
10-3
10-2
10-1
100
Normalized PDF of the elevation , File No. 84
/ , = 3.86 cm
PDF of is close to the Gaussian distribution around the mean value and differsat tail region, s>0, corresponds to the waves with steep tops and flat bottom.
PDF of tt more sensetive to the large wavenumbers and also displays the vertical asymmetry of the wave.
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N. Yokayama (JFM, 2004) direct numerical simulationsPDF of the elevation and
2
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-4 -2 0 2 410
-4
10-3
10-2
10-1
100
Normalized PDF of the elevation , File No. 88
/ , = 5.19 cm
-4 -2 0 2 410
-4
10-3
10-2
10-1
100
PDF of the second derivative tt
, File No. 88
tt
/
= 5.19 cmS = -0.7744K = 727
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Skewness and Kurtosis for PDF of 2nd derivative of elevation
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
1 2 3 4 5 6
PDF Skewness of the 2nd derivative of elevation
RMS
3
4
5
6
7
8
1 1.5 2 2.5 3 3.5 4
PDF Kurtosis of the 2nd derivative elevation
RMS
-8
-7
-6
-5
-4
-3
1 2 3 4 5 6
Slope Ch1Slope Ch2
Sp
ec
tru
m S
lop
e
Elevation RMS
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PDF of filtered envelope: small, medium and high amplitudes
0 0.002 0.004 0.006 0.008 0.0110
0
101
102
103
104
105
Filtered elevation PDF (File 81)
Elevation2, cm2
pd
f, a
.u.
stdev
FilterFreq = 6 Hz
0 0.2 0.4 0.6 0.810
0
101
102
103
104
105
PDF filtered elevation, file 88 - high amplitude
Elevation2, cm2
pd
f, a
.u.
stdev
FilterFreq = 6 Hz
0 0.02 0.04 0.06 0.08 0.1 0.1210
0
101
102
103
104
105
PDF of filtered elevation, file 84
Elevation2, cm2
pd
f, a
.u.
stdev
FilterFreq = 6 Hz
0 1 2 3 4 5-800
-600
-400
-200
0
Frequency (Hz)
Ph
ase
(d
eg
ree
s)
0 1 2 3 4 5-100
-50
0
Frequency (Hz)
Ma
gn
itud
e (
dB
)
filter frequency response
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Squared amplitude of surface elevation at 6 ± 1 Hz, wire probes
Pro
be 1
Pro
be 2
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Numerical results – S.Nazarenko et al (2006)
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ConclusionRandom gravity waves were generated in the laboratory flumewith the inertial interval up to 1m - 1cm.
The spectra slopes increase monotonically from -7 to -4 with the amplitude of forcing. At low forcing level the character of wave spectrais defined by nonlinearity and discreteness effects, at high and intermediate forcing - by wave breaking.
PDFs of surface elevation and its second derivative are non-gaussian at high wave nonlinearity.
PDF of the squared wave elevation filtered in a narrow frequencyinterval (spectral energy density) always has an intermittent tail.
Questions: Which model should be used to describe our spectra?
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Acknowledgement
• The work is supported by Hull Environmental Research Institute