Transverse surface waves in steady uniform and non-uniform flows through an array of emergent and slightly submerged square cylinders

Abstract

Steady uniform and non-uniform flows through an array of emergent and slightly submerged square cylinders are experimentally investigated with a specific focus on transverse seiche waves induced by vortex shedding. The study is first and foremost aimed at assessing the effect of streamwise flow non-uniformity on seiche waves. Its secondary purpose is to investigate the change in seiche magnitude, when initially emerged cylinders become slightly submerged. Thirdly and lastly, the effect of seiche waves on mean velocities and velocity fluctuations is quantified. The lock-in process between waves and vortex shedding is unaltered by flow non-uniformity and by a change from cylinder emergence to submergence. For non-uniform flows, this results in the co-existence of two differently oscillating transverse waves close to each other. Relative wave amplitude is found to be mainly influenced by relative submergence in the case of submerged cylinders, and by Froude number and oscillation mode in the case of emergent cylinders. Finally, seiche waves modify the streamwise mean velocity, when cylinders are emergent.

Keywords:

• Non-uniform flow
• open-channel flow
• relative submergence
• seiche
• surface wave
• vortex shedding

Acknowledgments

The authors wish to thank André Paquier and Marc Chatelain for their invaluable advice.

Supplemental data

Supplemental data for this article can be accessed http://doi.org/10.1080/00221686.2019.1647885.

Notation

A	=	wave amplitude at a given x-position (mm)
B	=	channel width (mm)
D	=	time-averaged water depth (mm)
$D\left(t\right)$	=	instantaneous water depth (mm)
D/h	=	relative submergence (–)
$F_D$	=	Froude number based on water depth D and velocity $U_{\lambda }$ (–)
$f_D$	=	predominant frequency of the transverse seiche wave based on measured water depth fluctuation spectrum (Hz)
$f_{tw}$	=	theoretical frequency of the natural transverse waves, see Eq. (3(3) $f_{tw}=\sqrt{\frac{\frac{gn}{4\pi B}\tanh \frac{n\pi D}{B}}{S}}$(3) ) (Hz)
$f_V$	=	predominant frequency of the transverse seiche wave based on the measured transverse velocity fluctuation spectrum (Hz)
h	=	height of a square cylinder (mm)
l	=	width of a square cylinder (mm)
L	=	distance between the centres of two adjacent square cylinders (mm)
N	=	number of square cylinders in a transverse row (–)
n	=	oscillation mode (–)
Q	=	discharge (l s−1)
$R_l$	=	Reynolds number based on width l and velocity $U_{\lambda }$ (–)
${\mathit{S}}_{\mathit{0}}$	=	The longitudinal bed slope of the channel (–)
$S_l$	=	Strouhal number based on width l and velocity $U_{\lambda }$ (–)
$U_{\lambda }$	=	average pore velocity (m s−1)
W	=	downstream weir height (mm)
γ	=	transverse wavelength (mm)
λ	=	planar density of square cylinders (–)
${\lambda }_f$	=	frontal density of square cylinders (–)
σ	=	standard deviation of water depth fluctuations (mm)
${\sigma }_{\mathrm{m}\mathrm{a}\mathrm{x}}$	=	maximum value of σ at a given x-position (mm)

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

Meriem Chetibi conducted her experiments at Irstea within the framework of a research project funded by the French National Research Agency (Flowres project, grant number ANR-14-CE03-0010, https://flowres.irstea.fr/en/).