Conceptual analogy between the water hammer phenomenon in free-surface and pressurized-pipe flows

ABSTRACT

The water hammer phenomenon, commonly acknowledged in pressurized pipe-flows is relevant to various water distribution systems. This phenomenon constitutes a major concern for hydraulic researchers and designer, given the dramatic consequences of this phenomenon on hydraulic structures and human life. This study aimed at expanding research on the water hammer waves in the free-surface flow framework. The main objective was to address a comprehensive simulation of the free-surface wave behavior caused by the abrupt closure of sluice-gate in prismatic open channel. The one-dimensional Saint-Venant equations embedding the Boussinesq add-on was used to predict the free-surface wave behavior; which was discretized using the (2–4)-dissipative scheme. Results evidenced the analogy between the water hammer wave behavior in free-surface and pressurized pipe flows. Furthermore, results illustrated that the water hammer maneuver produced a sudden depth rise, about three times of the normal depth value in the cross section adjacent to the gate. From a computational point of view, the proposed solver provided simplicity and accuracy attributes in simulating shock-waves in free-surface-flow. This solver also consumed low computational time compared with the conventional or multiple grid technique – based Finite Element Algorithm.

KEYWORDS:

• (2-4)-dissipative scheme
• boussinesq
• free-surface flow
• water hammer
• shock-wave

Nomenclature

A	=	Wetted cross area of the channel (${\mathrm{m}}^2$).
$\mathit{T}$	=	Channel width ($\mathrm{m}$).
$\mathit{c}$	=	Celerity ($\mathrm{m}/\mathrm{s}$).
$\mathrm{Cr}$	=	Courant number (-).
$\mathit{g}$	=	Acceleration of gravity ($\mathrm{m}/{\mathrm{s}}^2$).
$\mathit{L}$	=	Channel length ($\mathrm{m}$).
$\mathit{n}$	=	Manning roughness coefficient ($\mathrm{s}/{\mathrm{m}}^{1/3}$).
$\mathit{q}$	=	Discharge per unit width (${\mathrm{m}}^2/\mathrm{s}$).
$R_H$	=	Hydraulic radius ($\mathrm{m}$).
$s_0$	=	Longitudinal slope of the channel bottom ($\mathrm{m}/\mathrm{m}$).
$s_f=$n2 x n2$/R_H^{4/3}$	=	Friction slope ($\mathrm{m}/\mathrm{m}$).
$T_1$	=	Period ($\mathrm{s}$).
$T_c$	=	Closure time of the downstream gate ($\mathrm{s}$)
$\mathit{t}$	=	Time ($\mathrm{s}$).
$\mathit{u}$	=	Depth-averaged velocity ($\mathrm{m}/\mathrm{s}$).
$\mathit{x}$	=	Abscissa measured along the channel bed ($\mathrm{m}$).
$\mathit{d}$	=	Flow depth measured perpendicular to the channel bed ($\mathrm{m}$).
$V_w^{+}/\mathrm{\_}$	=	Celerity of positive or negative wave
$\mathit{\mu }$	=	Oscillation phase ($\mathrm{s}$).
Subscripts	=
$0$	=	Initial flow condition (-).
$\mathit{i}$	=	Mesh index in the x-direction (-).
$\mathit{k}$	=	Mesh index in the t-direction (-).
$\mathit{N}$	=	Number of mesh (-).

Disclosure statement

No potential conflict of interest was reported by the author(s).