Open-channel flow: Difference between revisions

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{{short description|BranchType of hydraulicsliquid andflow fluidwithin mechanicsa conduit}}

{{expert needed|1=Engineering|date=May 2018}}

'''Open-channel flow''', a branch ofIn [[hydraulicsfluid mechanics]] and [[fluid mechanicshydraulics]], '''open-channel flow''' is a type of [[liquid]] flow within a conduit or in channel 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: the free surface. Openopen-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:

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***** The depth changes over a long distance.

** ''Continuous flow''

*** The discharge is constant throughout the [[Reach_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|inertial]]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" />

== Core equationsFormulation ==

{{further|Computational methods for free surface flow}}

It is possible to formulate equations describing three [[Conservation law|conservation laws]] 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.

It is possible to formulate equations describing three [[Conservationconservation law|conservation laws]]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:

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# 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:<math{{Equation displaybox 1|cellpadding|border|indent="block":|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|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

\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:<math{{Equation displaybox 1|cellpadding|border|indent="block":|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:<math{{Equation displaybox 1|cellpadding|border|indent="block":|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.<br />

== See also ==

{{colbegin|colwidth=22em}}

*[[HEC-RAS]]

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**[[Hydraulic jump]]

**[[Manning formula]]

**[[Shallow_water_equationsShallow water equations#One-dimensional_Saintdimensional Saint-Venant_equationsVenant equations|Saint-Venant equations]]

**[[Standard step method]]

{{colend}}

== References ==

{{Reflist}}

<references />

== 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|}}.

*Rafik, Absi (2021).''[https://doi.org/10.3390/hydrology8030126 Reinvestigating the Parabolic-Shaped Eddy Viscosity Profile for Free Surface Flows]''. Hydrology 2021, 8(3), 126.

== 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?]

*[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. &nbsp;26-38)

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