Open-channel flow: Difference between revisions

Revision as of 08:51, 25 September 2021

Revision as of 21:13, 18 January 2022

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{{short description|Branch of hydraulics and fluid mechanics}}

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

'''Open-channel flow''', a branch of [[hydraulics]] and [[fluid mechanics]], 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. Open-channel flow has a [[free surface]], whereas pipe flow does not.

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

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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 equations ==

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 display="block">{\partial A\over{\partial t}} + {\partial Q\over{\partial x}} = 0</math>

=== Momentum equation ===

The momentum equation for open-channel flow may be found by starting from the [[Incompressible navier-stokes equations|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}

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h &= e + {p\over{\rho g}} \\

&= {u^{2}\over{2g}} + z + {p\over{\gamma}}

\end{aligned}</math>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}}

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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}}

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*[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)

{{Hydraulics}}

[[Category:Civil engineering]]

[[Category:Fluid dynamics]]