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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
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
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
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
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.
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.
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.
N. Yokayama (JFM, 2004) direct numerical simulationsWave action spectra
A.Dyachenko, O.Korotkevich, Zakharov (JETP Lett. 2003)
Y. Lvov, Nazarenko, Pokorni:numerical experiment: Physica D, 2006
ExperimentsAirborne Measurements of surface elevation k-spectra
P.A. Hwang, D.W.Wang (2000)
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
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
6 metres12
met
res
90 c
m8 Panel Wave Generator
Laser
Capacity Probes
Rain Generator
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,,
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
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
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
-8
-7
-6
-5
-4
-3
1 2 3 4 5 6
Slope Ch1Slope Ch2
Sp
ec
tru
m S
lop
e
Elevation RMS
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
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.
N. Yokayama (JFM, 2004) direct numerical simulationsPDF of the elevation and
2
-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
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
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
Squared amplitude of surface elevation at 6 ± 1 Hz, wire probes
Pro
be 1
Pro
be 2
Numerical results – S.Nazarenko et al (2006)
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?
Acknowledgement
• The work is supported by Hull Environmental Research Institute