Solutions of transition problems in exponential channels

ABSTRACT

Transition problems in a smooth open-channel flow consist of a solution of the third- or higher-degree algebraic equations to determine the choke-free or post-choking depths at downstream and upstream sections, respectively. Graphical solutions of trapezoidal, circular and exponential channels and analytical solution for rectangular channels have been obtained in the past. However, these solutions are cumbersome and so are difficult for field applications. In the present work, a general transition problem in exponential channels has been formulated in terms of alternate-depth ratio. A governing algebraic equation representing incipient choking condition has been derived for the exponential channel (rectangular, parabolic and triangular). Furthermore, the method of applying the same to calculate the choke free and post-choking depth at downstream and upstream sections has been presented in this paper. Exact solutions for the case of rectangular and parabolic channels have been obtained. After observing, the impossibility of exact solutions for triangular channels and the cumbersome nature of the solutions for rectangular and parabolic channels, empirical solution for the post-choking depth at upstream section has been carried out for the exponential channel. The empirical relation between the shape factor $\sigma$ and upstream Froude number $F_1$ for the incipient condition has been obtained for all channel types. The result shows that for $0<F_1<0.95$, the absolute error in $\sigma$ is less than 1% in all channel types, while for $0.95<F_1<1.0$, this value goes up to 1.1%, 1.6% and 1.8% for rectangular, parabolic and triangular channels, respectively. These empirical solutions are simple for field applications with negligible error. The methodologies presented in this paper have been corroborated using examples from various sources.

KEYWORDS:

• Transition
• alternate-depth ratio
• choking
• incipient choking
• exponential channel

Notation List

$A$	=	= Area of flow
$B$	=	= Bottom width in rectangular channel
$D$	=	= Hydraulic Depth
$D_{C2}$	=	= Critical hydraulic depth at section 2-2
$F$	=	= Froude number
$Q$	=	= Channel discharge
$T$	=	= Top width
$V$	=	= Mean velocity of flow
$a$	=	= Shape parameter of the channel
$a_i$	=	= Regression parameters
$g$	=	= Acceleration due to gravity
$k$	=	= Size parameter of channel
$l$	=	= length of latus rectum of parabolic channel
$m$	=	= Side slow of triangular channel
$y$	=	= epth of flow
$\alpha$	=	= Energy coefficient
$ϵ$	=	= Alternate depth ratio
$\mu$	=	= Elevation-depth ratio
$\sigma$	=	= Shape ratio
$\mathrm{\Delta }z$	=	= Change in channel bed elevation
$\mathrm{\Delta }{z}_m$	=	= Minimum height of hump

Notes: subscripts 1 and 2 denote the variables referred to section 1-1 and 2-2, respectively. Superscript (‘) denotes the new variables corresponding to choking conditions. Subscript M denotes the lower limiting value at which choking at section 2-2 does not occur.

Disclosure statement

No potential conflict of interest was reported by the authors.