Flow features of a horseshoe tunnel bend

Abstract

Previous research on bend flow has mainly focused on rectangular channels or circular conduits with small bend radii. This study investigates the dynamics of bend flow in an unexplored geometry: the horseshoe cross-section with large bend radii, a typical configuration in hydraulic tunnels. We examine two critical flow features, energy losses and superelevation, employing theoretical equations and dimensional analysis based on experimental data collected on a physical model of a diversion tunnel. The findings reveal an exponential decrease in the bend loss coefficient with increasing Reynolds numbers and shockwaves within the bend for Froude numbers exceeding 1.945. These findings are relevant in a hydraulic tunnel design with similar features and offer insights to minimize energy losses and maintain stable flow conditions.

Keywords:

• Energy bend loss
• horseshoe tunnel
• hydraulic model
• supercritical flow
• superelevation
• surface profile

Acknowledgments

The authors acknowledge ITINERA S.P.A for the support during the development of this project. Special thanks are extended to engineers E. Castellet and M. Deamici, who are responsible for the execution of the project prototype. We also thank engineer D. Curti for his contribution during the experimental campaign, and to the staff of the Hydraulics Laboratory of Politecnico di Milano. Additionally, we express our gratitude to the reviewers and the associate editor for their insightful comments and suggestions.

Disclosure statement

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

Notation

$\mathit{a}$	=	acceleration vector (m s${}^{-2}$)
$d_h$	=	hydraulic depth (m)
E	=	energy (m)
F	=	Froude number (–)
$f_c$	=	bend loss coefficient (–)
g	=	gravity acceleration (m s${}^{-2}$)
$\mathit{k}$	=	vertical unitary vector (–)
$k_s$	=	roughness coefficient (m${}^{1/3}$ s${}^{-1}$)
L	=	bend length (m)
$L^{\prime }$	=	downstream distance affected by the bend effect (m)
l	=	length between sections (m)
P	=	pressure (Pa)
Q	=	flow discharge (m${}^3$ s${}^{-1}$)
Re	=	Reynolds number (–)
$R_h$	=	hydraulic radius (m)
$r_c$	=	central radius of curvature (m)
$s_o$	=	bottom slope (%)
t	=	water surface width (m)
V	=	flow velocity (m s${}^{-1}$)
y	=	flow depth (m)
$y_c$	=	critical depth (m)
$y_i$	=	flow depth at inner side (m)
$y_n$	=	normal depth (m)
$y_o$	=	flow depth at outer side (m)
z	=	potential head (m)
$\mathrm{\Delta E}$	=	total energy loss (m)
$\mathrm{\Delta }{E}_f$	=	frictional energy loss (m)
$\mathrm{\Delta }{E}_l$	=	local energy loss due to the bend (m)
$\mathrm{\Delta y}$	=	superelevation (m)
Π	=	pi term of Vaschy–Buckingham theorem (–)
ζ	=	deflection angle (${}^{\circ }$)
ν	=	kinematic viscosity (m${}^2$ s${}^{-1}$)
ρ	=	fluid density (kg m${}^{-3}$)
ϕ	=	function of Vaschy–Buckingham theorem (–)

Additional information

Funding

This work was partially supported by the Italian Project “AGRITECH” ref. PNRR-CN-AGRITECH-S3.