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Turbulence under spilling breakers using discrete wavelets

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Abstract

Measurements of the kinetic energy of turbulence under spilling waves have been analysed using orthogonal wavelets. Data have been collected using 2-D laser Doppler velocimetry for pre-breaking regular waves, generated in a wave tank. The contribution of different scale vortices is computed, and also phase resolved. It is found that micro-vortices (2 mm <l<0.10 m for the tested case) and mid-size vortices (0.10 m<l<4.0 m for the tested case) are generally dominant, carrying more than 70% of the total turbulence energy under the wave crest. The phase resolved energy spectra are computed, which allows the computation of the transverse and of the longitudinal correlations.

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Abbreviations

∼:

phase or ensemble average operator

Λ:

phasic average operator

−:

time average operator

〈...〉:

ensemble average

α∼3α1 :

Kolgomorov constant

γ γ jk :

function

ρ :

mass density (kg/m3)

ν :

kinematic fluid viscosity (m2/s)

Λ:

integral length scale of turbulence (m)

ΔT i :

time interval (s)

λ E :

Taylor length micro-scale (m)

ε :

turbulent energy dissipation rate (m2/s3)

κ :

turbulent kinetic energy (m2/s2)

ζ :

translation parameter

τ E :

Eulerian time micro-scale (s)

τ :

shear stress (Pa)

η K :

Kolgomorov length micro-scale (m)

A j (x):

approximation at level j

a :

dilation parameter

C :

wave celerity (m/s)

D j (x):

detail at level j

DWT:

discrete wavelet transform

E 1(k 1), E 2({ik}1):

energy spectrum in the wave-number domain (m3/s2)

E 1(f):

energy spectrum in frequency domain (m2/s)

f :

frequency (Hz), function

f acq :

sampling frequency (Hz)

g :

gravitational acceleration (m/s2)

h :

local water depth (m)

H :

wave height (m)

k, k 1 :

wave number (m−1)

k min, k max :

minimum, maximum wave number

k d :

dissipative wave number (m−1)

k :

wave number (vector) (m−1)

l :

length scale (m)

LDV:

laser Doppler velocimetry

N :

number of samples

N:

number of levels in wavelet decomposition

PIV:

particle image velocimetry

R E(r):

normalised Eulerian space autocorrelation

R E(τ):

normalised Eulerian time autocorrelation

Re λ :

Reynolds number based on Taylor micro-scale

s sj :

fluctuating rate of strain (s−1)

STFT:

short time Fourier transform

T :

wave period (s)

T m :

period of time averaging (s)

TFR:

time frequency representation

t, t k, t′, τ:

time variable (s)

T E :

time macro-scale of turbulence (s)

ū :

mean velocity

u′ ν′ :

fluctuating velocity (m/s)

~u,~v :

organised fluctuating velocity (m/s)

U, V. W :

velocity scales, velocity components (m/s)

u :

turbulence scale (m/s)

u :

velocity vector (m/s)

VITA:

variable interval time average

W jk :

wavelet coefficients

x, y, z, x i :

spatial co-ordinates (m)

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Authors and Affiliations

  1. Dipartimento di Ingegneria Civile, Université degli Studi di Parma, Parco Area délie Scienze, 181/A, 43100, Parma, Italy

    S. Longo

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  1. S. Longo
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Published online: 23 November 2002

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Longo, S. Turbulence under spilling breakers using discrete wavelets. Experiments in Fluids 34, 181–191 (2003). https://doi.org/10.1007/s00348-002-0545-1

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  • DOI: https://doi.org/10.1007/s00348-002-0545-1

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