Simple criterion for evaluating stability of hydraulic oscillation based on water-hammer reflection coefficients

Abstract

Hydraulic oscillation is a fluctuating phenomenon of pressure and discharge in pipes, which can threaten the safety of hydropower, pumping and water conveyance systems. To analyse the associated problems, new methods with simple form and clear physics are needed. This paper presents a water-hammer reflection coefficient-based criterion for stability evaluation of free-vibration of hydraulic systems. The stability (or attenuation) condition for a single pipe system is that the modulus of the product of the reflection coefficients at the inlet and outlet should be smaller than 1. For a complex pipe system, the condition necessary for stability is that every single pipe system is stable. To apply this new criterion to stabilizing the reservoir-single pipe-constant power turbine system by adding a head loss valve in the pipe, a formula for the critical head loss is proposed and verified. This new method is theoretically consistent with conventional methods, but more convenient in application.

Keywords:

• Complex water-hammer reflection coefficient
• free-vibration
• hydraulic oscillation
• self-excited oscillation
• valve control
• water-hammer wave

Acknowledgments

The authors thank Mrs. Hou Xiaoxia for checking English grammar of the manuscript.

Disclosure statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Notation

$A$	=	cross-sectional area (m2)
$c$	=	pipe wave speed (m s-1)
$D$	=	pipe diameter (m)
$f$	=	Darcy-Weisbach frictional factor (–)
$f_U$	=	reflected wave function for the inlet of pipe (–)
$f_D$	=	reflected wave function for the outlet of pipe (–)
$F_U$	=	incident wave function for the inlet of pipe (–)
$F_D$	=	incident wave function for the outlet of pipe (–)
$g$	=	gravitational acceleration (m s-2)
$h_V$	=	the head difference across the valve (m)
$H$	=	piezometric water head (m)
$H_0$	=	initial head (m)
$H_1$	=	pressure head on the left side of the valve (m)
$H_2$	=	pressure head on the right side of the valve (m)
$i$	=	the imaginary unit (–)
$l$	=	pipe length (m)
$n$	=	total number of pipes in the complex system (–)
$Q$	=	discharge in the pipe (m)
$Q_0$	=	initial discharge (m)
$r_U$	=	reflection coefficient for the inlet of pipe (–)
$r_D$	=	reflection coefficient for the outlet of pipe (–)
$t$	=	time (s)
$V$	=	pipe velocity (m s-1)
$V_0$	=	initial velocity (m s-1)
$Z_C$	=	the pipe characteristic impedance in friction-less condition (–)
$Z_U$	=	hydraulic impedance at the pipe inlet (–)
$Z_D$	=	hydraulic impedance at the pipe outlet (–)
$\Delta H$	=	water-hammer pressure increment (m)
$\alpha$	=	head loss coefficient (–)
$\gamma$	=	complex-valued constant (s m-1)
$\theta$	=	phase angle corresponding to $r_U{r}_D$ (rad)
${\theta }_x$	=	phase angle corresponding to $r_{U1}{r}_{D1}$ (rad)
${\theta }_y$	=	phase angles corresponding to $r_{U2}{r}_{D2}$ (rad)
$\sigma$	=	damping factor of the corresponding frequency (–)
$\tau$	=	valve opening (–)
$\omega$	=	harmonic frequency of free-vibration (rad s-1).

Subscript

i	=	number of reaches during which the wave function propagates along the pipe from the inlet to the outlet or vice versa
U	=	inlet of the pipe
D	=	outlet of the pipe

Additional information

Funding

This work was supported by National Natural Science Foundation of China (NSFC Grant Nos. 51879087 and 51839008).