Open-channel flow: Difference between revisions

Revision as of 16:32, 30 October 2023

Latest revision as of 21:30, 7 May 2025

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[[File:Arizona cap canal.jpg|alt=|thumb|[[Central Arizona Project]] channel.]]

== Classifications of flow ==

Open-channel flow can be classified and described in various ways based on the change in flow depth with respect to time and space.<ref>{{Cite book|last=Jobson|first=Harvey E.|url=https://pubs.usgs.gov/of/1988/0707/report.pdf|title=Basic Hydraulic Principles of Open-Channel Flow|last2=Froehlich|first2=David C.|publisher=U.S. Geological Survey|year=1988|___location=Reston, VA}}</ref> The fundamental types of flow dealt with in open-channel hydraulics are:

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*** The discharge of a steady flow is non-uniform along a channel. This happens when water enters and/or leaves the channel along the course of flow. An example of flow entering a channel would be a road side gutter. An example of flow leaving a channel would be an irrigation channel. This flow can be described using the continuity equation for continuous unsteady flow requires the consideration of the time effect and includes a time element as a variable.

== States of flow ==

The behavior of open-channel flow is governed by the effects of [[viscosity]] and gravity relative to the [[inertia]]l forces of the flow. [[Surface tension]] has a minor contribution, but does not play a significant enough role in most circumstances to be a governing factor. Due to the presence of a free surface, gravity is generally the most significant driver of open-channel flow; therefore, the ratio of inertial to gravity forces is the most important dimensionless parameter.<ref name=":0">{{Cite book|last=Sturm|first=Terry W.|url=http://docshare03.docshare.tips/files/4233/42333266.pdf|title=Open Channel Hydraulics|publisher=McGraw-Hill|year=2001|isbn=9780073397870|___location=New York, NY|pages=2}}</ref> The parameter is known as the [[Froude number]], and is defined as:<math display="block">\text{Fr} = {U\over{\sqrt{gD}}}</math>where <math>U</math> is the mean velocity, <math>D</math> is the [[characteristic length]] scale for a channel's depth, and <math>g</math> is the [[gravitational acceleration]]. Depending on the effect of viscosity relative to inertia, as represented by the [[Reynolds number]], the flow can be either [[laminar flow|laminar]], [[turbulent flow|turbulent]], or [[Laminar–turbulent transition|transitional]]. However, it is generally acceptable to assume that the Reynolds number is sufficiently large so that viscous forces may be neglected.<ref name=":0" />

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=== Momentum equation ===

The momentum equation for open-channel flow may be found by starting from the [[Incompressible navier-stokes equations|incompressible Navier-StokesNavier–Stokes equations]] :<math display="block">\overbrace{\underbrace{{\partial {\bf v}\over{\partial t}}}_{\begin{smallmatrix} \text{Local} \\ \text{Change} \end{smallmatrix}} + \underbrace{{\bf v}\cdot\nabla {\bf v}}_{\text{Advection}}}^{\text{Inertial Acceleration}} = -\underbrace{{1\over{\rho}}\nabla p}_{\begin{smallmatrix} \text{Pressure} \\ \text{Gradient} \end{smallmatrix}} + \underbrace{\nu \Delta {\bf v}}_{\text{Diffusion}} - \underbrace{\nabla \Phi}_{\text{Gravity}} + \underbrace{{\bf F}}_{\begin{smallmatrix} \text{External} \\ \text{Forces} \end{smallmatrix}}</math>where <math>p</math> is the [[pressure]], <math>\nu</math> is the [[kinematic viscosity]], <math>\Delta</math> is the [[Laplace operator]], and <math>\Phi = gz</math> is the [[gravitational potential]]. By invoking the high Reynolds number and 1D flow assumptions, we have the equations:<math display="block">\begin{aligned}

{\partial u\over{\partial t}} + u{\partial u\over{\partial x}} &= -{1\over{\rho}}{\partial p\over{\partial x}} + F_{x} \\

-{1\over{\rho}}{\partial p\over{\partial z}} - g &= 0

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\end{aligned} </math>|border colour=#0073CF|background colour=#F5FFFA}}with <math>\gamma = \rho g</math> being the [[specific weight]]. However, realistic systems require the addition of a [[head loss]] term <math>h_{f}</math> to account for energy [[dissipation]] due to [[friction]] and [[turbulence]] that was ignored by discounting the external forces term in the momentum equation.

== See also ==

{{colbegin|colwidth=22em}}

*[[HEC-RAS]]

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{{colend}}

== References ==

{{Reflist}}

== Further reading ==

* Nezu, Iehisa; Nakagawa, Hiroji (1993). ''[https://www.crcpress.com/Turbulence-in-Open-Channel-Flows/Nakagawa-Nezu/p/book/9789054101185 Turbulence in Open-Channel Flows]''. IAHR Monograph. Rotterdam, NL: A.A. Balkema. {{ISBN|9789054101185|}}.

*Syzmkiewicz, Romuald (2010). ''[https://www.mobt3ath.com/uplode/book/book-46451.pdf Numerical Modeling in Open Channel Hydraulics]''. Water Science and Technology Library. New York, NY: Springer. {{ISBN|9789048136735|}}.

== External links ==

*[[California Institute of Technology|Caltech]] lecture notes:

**[httphttps://www.its.caltech.edu/~ce112/chapter2text.pdf Derivation of the Equations of Open Channel Flow]

**[httphttps://www.its.caltech.edu/~ce112/chapter3text.pdf Surface Profiles for Steady Channel Flow]

* [https://ocw.mit.edu/courses/earth-atmospheric-and-planetary-sciences/12-090-introduction-to-fluid-motions-sediment-transport-and-current-generated-sedimentary-structures-fall-2006/course-textbook/ch5.pdf Open-Channel Flow]

*[https://www.youtube.com/watch?v=8vmTYmt0Y8Q Open Channel Flow Concepts]

*[https://www.youtube.com/watch?v=7tjf8HWiR3Y What is a Hydraulic Jump?]