On the use of time coefficients of proper orthogonal decomposition for describing quasi-periodic flow processes in the highly turbulent flow of a fish pass

ABSTRACT

Flow velocity measurements using particle tracking velocimetry carried out in a scale model of a vertical-slot fish pass are analysed using proper orthogonal decomposition. Based on the analysis, the oscillating main stream and related time-varying processes are identified as dominant repeating flow processes. In particular, the time coefficients of the modes are examined in detail. Firstly, the cross-correlation functions of the time coefficients are used to identify the modes best representing this process. Secondly, the time series of coefficients themselves are used to identify the temporal occurrences of the repeating process even when the occurrences are neither identical nor periodical. The presented methodology reduces the examination of the full measured velocity dataset to an analysis of a limited number of coefficient time series, which can be used to detect the occurrences of flow processes repeating at irregular time intervals, and hence to describe their temporal evolution.

Keywords:

• Coherent structures
• fish passages
• non-periodic flow
• particle tracking velocimetry (PTV)
• proper orthogonal decomposition (POD)
• quasi-periodic flow
• turbulent flows

Acknowledgements

The authors are grateful to Dr Giordano Lipari of Watermotion|Waterbeweging for helpful comments on the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors .

Notation

A	=	matrix of POD time coefficients (–)
an	=	vector containing the POD coefficients at the nth discrete time for all modes (–)
ai	=	vector containing the time series of POD coefficients corresponding to mode i (–)
ai	=	value of POD coefficient corresponding to mode i (–)
$a_i^n$	=	value of POD coefficient corresponding to mode i at the nth discrete time (–)
D	=	diagonal matrix of eigenvalues (–)
f	=	frequency (s–1)
kuv	=	turbulent kinetic energy contributions of the velocity components u and v (cm2 s–2)
N	=	number of samples (–)
M	=	number of grid points (–)
pm	=	position vector containing x and y coordinates of point m
qi	=	mode i energy contribution to kuv
ri,j	=	cross correlation function between coefficients time series corresponding to POD modes i and j
Re	=	Reynolds number (–)
R	=	auto-covariance matrix for POD
t	=	time (s)
u	=	x-wise instantaneous velocity (m s–1)
u	=	instantaneous velocity vector (m s–1)
u	=	fluctuation of x-wise velocity (m s–1)
u	=	fluctuation of instantaneous velocity vector (m s–1)
um	=	mean of x-wise velocity (m s–1)
um	=	mean velocity vector (m s–1)
v	=	y-wise instantaneous velocity (m s–1)
v	=	fluctuation of y-wise velocity (m s–1)
X	=	input velocity dataset for POD (m s–1)
x	=	streamwise coordinate (m)
y	=	transversal coordinate (m)
Φ	=	matrix of modes (m s–1)
φ	=	mode arranged as vector (m s–1)
λi	=	eigenvalue corresponding to POD mode i (–)
λ	=	vector containing all eigenvalues λi
σi	=	standard deviation of coefficients time series corresponding to POD mode i (–)
σj	=	standard deviation of coefficients time series corresponding to POD mode j (–)
σu	=	standard deviation of x-wise velocity (cm s–1)
σv	=	standard deviation of y-wise velocity (cm s–1)
τ	=	time lag between coefficients time series (s)