Wavelet spectral analysis of the free surface of turbulent flows

ABSTRACT

This work demonstrates the applicability of the wavelet directional method as a means of characterizing the free surface dynamics in shallow turbulent flows using a small number of sensors. The measurements are obtained with three conductance wave probes in a laboratory flume, in a range of subcritical flow conditions where the Froude number was smaller than one, and the bed was homogeneously rough. The characteristic spatial scale of the surface elevation is found to correspond to the wavelength of stationary waves oriented against the flow. The spectrum of the dominant distribution of waves is characterized in terms of an angular spreading function. A new procedure to estimate the mean surface velocity based on measurements of the surface elevation at only two locations is proposed. The results can inform the development of more accurate models of the surface behaviour, with applications for the remote sensing of rivers and open channel flows.

Keywords:

• Air–water interface interactions
• gravity waves
• hydrodynamic waves
• laboratory studies
• velocity measurements

Acknowledgments

The authors are grateful to Prof. Francisco J. Ocampo Torres for fostering the collaboration which resulted in this paper, and to Prof. Kirill V. Horoshenkov for his helpful comments on the first version of the manuscript. The authors are also grateful to three anonymous reviewers for their useful comments on the manuscript.

ORCID

Giulio Dolcetti  http://orcid.org/0000-0002-0992-0319

Notation
b	=	width parameter of the Von Mises distribution (–)
c	=	absolute phase velocity of waves (m s−1)
$c_0$	=	phase speed in still water (m s−1)
d	=	mean depth (m)
E	=	directional frequency spectrum (m2 s rad−2)
$f_s$	=	sampling frequency (s−1)
F	=	Froude number (–)
g	=	gravity acceleration (m s−2)
G	=	Doppler shear correction (–)
$G_0$	=	Doppler shear correction for the waves with wavenumber $k_0$ (–)
k	=	wavenumber (rad m−1)
$k_x$	=	streamwise wavenumber (rad m−1)
$k_y$	=	transverse wavenumber (rad m−1)
$k_0$	=	wavenumber of the stationary waves oriented against the flow (rad m−1)
$L_x$	=	streamwise distance between probes (m)
$L_y$	=	transverse distance between probes (m)
m	=	scale-frequency conversion factor (–)
n	=	exponent of the power-function velocity profile (–)
N	=	number of samples (–)
$N_F$	=	number of degrees of freedom (–)
$\mathit{r}$	=	co-ordinate vector (m)
R	=	Reynolds number (–)
S	=	frequency spectrum (m2 s rad−1)
t	=	time (s)
T	=	measurement duration (s)
$U_0$	=	mean surface velocity (m s−1)
W	=	Morlet wavelet transform (m s${}^{1/2}$ rad${}^{-1/2}$)
X	=	streamwise frequency-wavenumber spectrum (m3 s rad−2)
Y	=	transverse frequency-wavenumber spectrum (m3 s rad−2)
γ	=	surface tension coefficient (kg s2)
Γ	=	characteristic flow shear (s−1)
ζ	=	surface elevation (m)
$\hat{\zeta }$	=	angular amplitude spectrum of the surface (m rad−1)
$\tilde{\zeta }$	=	Fourier transform in time of the elevation (m)
θ	=	angle of propagation (rad)
Λ	=	width of wavelet (–)
ρ	=	water density (kg m−3)
${\sigma }_{\zeta }$	=	standard deviation of the surface elevation (m)
Φ	=	phase of the Morlet wavelet transform (rad)
χ	=	three-dimensional frequency-wavenumber spectrum (m3 s rad−3)
${\chi }_0$	=	spectrum of the waves with wavenumber $k_0$ (m2 rad−1)
ω	=	radian frequency (rad s−1)
Ω	=	dispersion relation (rad s−1)