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