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{{short description|Type of liquid flow within a conduit}}
In [[fluid mechanics]] and [[hydraulics]], '''open-channel flow''' is a type of [[liquid]] flow within a conduit with a [[free surface]], known as a [[Stream channel|channel]].<ref>{{Cite book|last=Chow|first=Ven Te|url=https://heidarpour.iut.ac.ir/sites/heidarpour.iut.ac.ir/files/u32/open-chow.pdf|title=Open-Channel Hydraulics|publisher=The Blackburn Press|year=2008|isbn=978-1932846188|___location=Caldwell, NJ}}</ref><ref>{{Cite book|last=Battjes|first=Jurjen A.|url=https://www.cambridge.org/core/books/unsteady-flow-in-open-channels/5CCE099F37BCC5AF4E67B35F15666E7B|title=Unsteady Flow in Open Channels|last2=Labeur|first2=Robert Jan|publisher=Cambridge University Press|year=2017|isbn=9781316576878|___location=Cambridge, UK}}</ref> The other type of flow within a conduit is [[pipe flow]]. These two types of flow are similar in many ways but differ in one important respect: open-channel flow has a free surface, whereas pipe flow does not, resulting in flow dominated by gravity but not [[hydraulic pressure]].
[[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:
* '''Time as the criterion'''
** ''Steady flow''
*** The depth of flow does not change over time, or if it can be assumed to be constant during the time interval under consideration.
** ''Unsteady flow''
*** The depth of flow does change with time.
* '''Space as the criterion'''
** ''Uniform flow''
*** The depth of flow is the same at every section of the channel. Uniform flow can be steady or unsteady, depending on whether or not the depth changes with time, (although unsteady uniform flow is rare).
** ''Varied flow''
*** The depth of flow changes along the length of the channel. Varied flow technically may be either steady or unsteady. Varied flow can be further classified as either rapidly or gradually-varied:
**** ''Rapidly-varied flow''
***** The depth changes abruptly over a comparatively short distance. Rapidly varied flow is known as a local phenomenon. Examples are the [[hydraulic jump]] and the [[hydraulic drop]].
**** ''Gradually-varied flow''
***** The depth changes over a long distance.
** ''Continuous flow''
*** The discharge is constant throughout the [[Reach (geography)|reach]] of the channel under consideration. This is often the case with a steady flow. This flow is considered continuous and therefore can be described using the [[continuity equation]] for continuous steady flow.
** ''Spatially-varied flow''
*** 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" />
== Formulation ==
{{further|Computational methods for free surface flow}}
It is possible to formulate equations describing three [[conservation law]]s for quantities that are useful in open-channel flow: mass, momentum, and energy. The governing equations result from considering the dynamics of the [[flow velocity]] [[vector field]] <math>{\bf v}</math> with components <math>{\bf v} = \begin{pmatrix} u & v & w \end{pmatrix}^{T}</math>. In [[Cartesian coordinate system|Cartesian coordinates]], these components correspond to the flow velocity in the x, y, and z axes respectively.
To simplify the final form of the equations, it is acceptable to make several assumptions:
# The flow is [[Incompressible flow|incompressible]] (this is not a good assumption for rapidly-varied flow)
# The Reynolds number is sufficiently large such that viscous diffusion can be neglected
# The flow is one-dimensional across the x-axis
=== Continuity equation ===
The general [[continuity equation]], describing the conservation of mass, takes the form:<math display="block">{\partial \rho\over{\partial t}} + \nabla \cdot (\rho {\bf v}) = 0</math>where <math>\rho</math> is the fluid [[density]] and <math>\nabla \cdot()</math> is the [[divergence]] operator. Under the assumption of incompressible flow, with a constant [[control volume]] <math>V</math>, this equation has the simple expression <math>\nabla \cdot {\bf v} = 0</math>. However, it is possible that the [[Cross section (geometry)|cross-sectional area]] <math>A</math> can change with both time and space in the channel. If we start from the integral form of the continuity equation:<math display="block">{d\over{dt}}\int_{V}\rho \; dV = -\int_{V} \nabla\cdot(\rho {\bf v}) \; dV</math>it is possible to decompose the volume integral into a cross-section and length, which leads to the form:<math display="block">{d\over{dt}}\int_{x}\left(\int_{A}\rho \; dA \right) dx = -\int_{x}\left[\int_{A}\nabla\cdot(\rho {\bf v}) \; dA \right] dx</math>Under the assumption of incompressible, 1D flow, this equation becomes:<math display="block">{d\over{dt}}\int_{x}\left(\int_{A}dA \right) dx = -\int_{x}{\partial\over{\partial x}}\left(\int_{A} u \; dA \right) dx</math>By noting that <math>\int_{A}dA = A</math> and defining the [[volumetric flow rate]] <math>Q = \int_{A}u \; dA</math>, the equation is reduced to:<math display="block">\int_{x}{\partial A\over{\partial t}} \; dx = -\int_{x}{\partial Q\over{\partial x}} dx</math>Finally, this leads to the continuity equation for incompressible, 1D open-channel flow:{{Equation box 1|cellpadding|border|indent=:|equation=<math> {\partial A\over{\partial t}} + {\partial Q\over{\partial x}} = 0 </math>|border colour=#0073CF|background colour=#F5FFFA}}
=== Momentum equation ===
The momentum equation for open-channel flow may be found by starting from the [[incompressible Navier–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
\end{aligned}</math>The second equation implies a [[hydrostatic pressure]] <math>p = \rho g \zeta</math>, where the channel depth <math>\eta(t,x) = \zeta(t,x) - z_{b}(x)</math> is the difference between the free surface elevation <math>\zeta</math> and the channel bottom <math>z_{b}</math>. Substitution into the first equation gives:<math display="block">{\partial u\over{\partial t}} + u{\partial u\over{\partial x}} + g{\partial \zeta\over{\partial x}} = F_{x} \implies {\partial u\over{\partial t}} + u{\partial u\over{\partial x}} + g{\partial \eta\over{\partial x}} - gS = F_{x}</math>where the channel bed slope <math>S = -dz_{b}/dx</math>. To account for shear stress along the channel banks, we may define the force term to be:<math display="block">F_{x} = -{1\over{\rho}}{\tau\over{R}}</math>where <math>\tau</math> is the [[shear stress]] and <math>R</math> is the [[hydraulic radius]]. Defining the friction slope <math>S_{f} = \tau/\rho g R</math>, a way of quantifying friction losses, leads to the final form of the momentum equation:{{Equation box 1|cellpadding|border|indent=:|equation=<math> {\partial u\over{\partial t}} + u{\partial u\over{\partial x}} + g{\partial \eta\over{\partial x}} + g(S_{f}- S) = 0 </math>|border colour=#0073CF|background colour=#F5FFFA}}
=== Energy equation ===
To derive an [[energy]] equation, note that the advective acceleration term <math>{\bf v}\cdot\nabla {\bf v}</math> may be decomposed as:<math display="block">{\bf v}\cdot\nabla {\bf v} = \omega \times {\bf v} + {1\over{2}}\nabla\|{\bf v}\|^{2}</math>where <math>\omega</math> is the [[vorticity]] of the flow and <math>\|\cdot\|</math> is the [[Euclidean norm]]. This leads to a form of the momentum equation, ignoring the external forces term, given by:<math display="block">{\partial {\bf v}\over{\partial t}} + \omega \times {\bf v} = -\nabla\left({1\over{2}}\|{\bf v}\|^{2} + {p\over{\rho}} + \Phi \right )</math>Taking the [[dot product]] of <math>{\bf v}</math> with this equation leads to:<math display="block">{\partial\over{\partial t}}\left({1\over{2}}\|{\bf v}\|^{2} \right ) + {\bf v}\cdot \nabla \left({1\over{2}}\|{\bf v}\|^{2} + {p\over{\rho}} + \Phi \right ) = 0</math>This equation was arrived at using the [[scalar triple product]] <math>{\bf v}\cdot (\omega \times {\bf v}) = 0</math>. Define <math>E</math> to be the [[energy density]]:<math display="block">E = \underbrace{{1\over{2}}\rho\|{\bf v} \|^{2} }_{\begin{smallmatrix} \text{Kinetic} \\ \text{Energy} \end{smallmatrix}} + \underbrace{\rho\Phi}_{\begin{smallmatrix} \text{Potential} \\ \text{Energy} \end{smallmatrix}}</math>Noting that <math>\Phi</math> is time-independent, we arrive at the equation:<math display="block">{\partial E\over{\partial t}} + {\bf v}\cdot\nabla (E+p) = 0</math>Assuming that the energy density is time-independent and the flow is one-dimensional leads to the simplification:<math display="block">E + p = C</math>with <math>C</math> being a constant; this is equivalent to [[Bernoulli's principle]]. Of particular interest in open-channel flow is the [[specific energy]] <math>e = E/\rho g</math>, which is used to compute the [[hydraulic head]] <math>h</math> that is defined as:{{Equation box 1|cellpadding|border|indent=:|equation=<math> \begin{aligned}
h &= e + {p\over{\rho g}} \\
&= {u^{2}\over{2g}} + z + {p\over{\gamma}}
\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]]
*[[Streamflow]]
*'''Fields of study'''
**[[Computational fluid dynamics]]
**[[Fluid dynamics]]
**[[Hydraulics]]
**[[Hydrology]]
*'''Types of fluid flow'''
**[[Laminar flow]]
**[[Pipe flow]]
**[[Laminar–turbulent transition|Transitional flow]]
**[[Turbulence|Turbulent flow]]
*'''Fluid properties'''
**[[Froude number]]
**[[Reynolds number]]
**[[Viscosity]]
*'''Other related articles'''
**[[Chézy formula]]
**[[Darcy–Weisbach equation|Darcy-Weisbach equation]]
**[[Hydraulic jump]]
**[[Manning formula]]
**[[Shallow water equations#One-dimensional Saint-Venant equations|Saint-Venant equations]]
**[[Standard step method]]
{{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:
**[https://www.its.caltech.edu/~ce112/chapter2text.pdf Derivation of the Equations of Open Channel Flow]
**[https://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?]
*[https://www.youtube.com/watch?v=RXWknY6zaY4 Open Channel Flow Example]
*[https://web.stanford.edu/class/me469b/handouts/turbulence.pdf Simulation of Turbulent Flows] (p. 26-38)
{{Hydraulics}}
{{Authority control}}
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