In fluid mechanics and hydraulics, open-channel flow is a type of liquid flow within a conduit with a free surface, known as a channel.[1][2] 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.
Open-channel flow can be classified and described in various ways based on the change in flow depth with respect to time and space.[3] 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 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 flowEdit
The behavior of open-channel flow is governed by the effects of viscosity and gravity relative to the inertial 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.[4] The parameter is known as the Froude number, and is defined as:
where is the mean velocity, is the characteristic length scale for a channel's depth, and 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, turbulent, or transitional. However, it is generally acceptable to assume that the Reynolds number is sufficiently large so that viscous forces may be neglected.[4]
It is possible to formulate equations describing three 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 velocityvector field with components . In 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 (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 equationEdit
The general continuity equation, describing the conservation of mass, takes the form:
where is the fluid density and is the divergence operator. Under the assumption of incompressible flow, with a constant control volume, this equation has the simple expression . However, it is possible that the cross-sectional area can change with both time and space in the channel. If we start from the integral form of the continuity equation:
it is possible to decompose the volume integral into a cross-section and length, which leads to the form:
Under the assumption of incompressible, 1D flow, this equation becomes:
By noting that and defining the volumetric flow rate, the equation is reduced to:
Finally, this leads to the continuity equation for incompressible, 1D open-channel flow:
The second equation implies a hydrostatic pressure, where the channel depth is the difference between the free surface elevation and the channel bottom . Substitution into the first equation gives:
where the channel bed slope . To account for shear stress along the channel banks, we may define the force term to be:
where is the shear stress and is the hydraulic radius. Defining the friction slope , a way of quantifying friction losses, leads to the final form of the momentum equation:
Energy equationEdit
To derive an energy equation, note that the advective acceleration term may be decomposed as:
where is the vorticity of the flow and is the Euclidean norm. This leads to a form of the momentum equation, ignoring the external forces term, given by:
Taking the dot product of with this equation leads to:
with being the specific weight. However, realistic systems require the addition of a head loss term to account for energy dissipation due to friction and turbulence that was ignored by discounting the external forces term in the momentum equation.
Syzmkiewicz, Romuald (2010). Numerical Modeling in Open Channel Hydraulics. Water Science and Technology Library. New York, NY: Springer. ISBN9789048136735.
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In fluid mechanics and hydraulics open channel flow is a type of liquid flow within a conduit with a free surface known as a channel 1 2 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 Central Arizona Project channel Contents 1 Classifications of flow 2 States of flow 3 Formulation 3 1 Continuity equation 3 2 Momentum equation 3 3 Energy equation 4 See also 5 References 6 Further reading 7 External linksClassifications of flow EditOpen channel flow can be classified and described in various ways based on the change in flow depth with respect to time and space 3 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 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 EditThe behavior of open channel flow is governed by the effects of viscosity and gravity relative to the inertial 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 4 The parameter is known as the Froude number and is defined as Fr U g D displaystyle text Fr U over sqrt gD where U displaystyle U is the mean velocity D displaystyle D is the characteristic length scale for a channel s depth and g displaystyle g 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 turbulent or transitional However it is generally acceptable to assume that the Reynolds number is sufficiently large so that viscous forces may be neglected 4 Formulation EditFurther information Computational methods for free surface flow It is possible to formulate equations describing three 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 v displaystyle bf v with components v u v w T displaystyle bf v begin pmatrix u amp v amp w end pmatrix T In 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 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 axisContinuity equation EditThe general continuity equation describing the conservation of mass takes the form r t r v 0 displaystyle partial rho over partial t nabla cdot rho bf v 0 where r displaystyle rho is the fluid density and displaystyle nabla cdot is the divergence operator Under the assumption of incompressible flow with a constant control volume V displaystyle V this equation has the simple expression v 0 displaystyle nabla cdot bf v 0 However it is possible that the cross sectional area A displaystyle A can change with both time and space in the channel If we start from the integral form of the continuity equation d d t V r d V V r v d V displaystyle d over dt int V rho dV int V nabla cdot rho bf v dV it is possible to decompose the volume integral into a cross section and length which leads to the form d d t x A r d A d x x A r v d A d x displaystyle 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 Under the assumption of incompressible 1D flow this equation becomes d d t x A d A d x x x A u d A d x displaystyle d over dt int x left int A dA right dx int x partial over partial x left int A u dA right dx By noting that A d A A displaystyle int A dA A and defining the volumetric flow rate Q A u d A displaystyle Q int A u dA the equation is reduced to x A t d x x Q x d x displaystyle int x partial A over partial t dx int x partial Q over partial x dx Finally this leads to the continuity equation for incompressible 1D open channel flow A t Q x 0 displaystyle partial A over partial t partial Q over partial x 0 Momentum equation EditThe momentum equation for open channel flow may be found by starting from the incompressible Navier Stokes equations v t Local Change v v Advection Inertial Acceleration 1 r p Pressure Gradient n D v Diffusion F Gravity F External Forces displaystyle 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 where p displaystyle p is the pressure n displaystyle nu is the kinematic viscosity D displaystyle Delta is the Laplace operator and F g z displaystyle Phi gz is the gravitational potential By invoking the high Reynolds number and 1D flow assumptions we have the equations u t u u x 1 r p x F x 1 r p z g 0 displaystyle begin aligned partial u over partial t u partial u over partial x amp 1 over rho partial p over partial x F x 1 over rho partial p over partial z g amp 0 end aligned The second equation implies a hydrostatic pressure p r g z displaystyle p rho g zeta where the channel depth h t x z t x z b x displaystyle eta t x zeta t x z b x is the difference between the free surface elevation z displaystyle zeta and the channel bottom z b displaystyle z b Substitution into the first equation gives u t u u x g z x F x u t u u x g h x g S F x displaystyle 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 where the channel bed slope S d z b d x displaystyle S dz b dx To account for shear stress along the channel banks we may define the force term to be F x 1 r t R displaystyle F x 1 over rho tau over R where t displaystyle tau is the shear stress and R displaystyle R is the hydraulic radius Defining the friction slope S f t r g R displaystyle S f tau rho gR a way of quantifying friction losses leads to the final form of the momentum equation u t u u x g h x g S f S 0 displaystyle partial u over partial t u partial u over partial x g partial eta over partial x g S f S 0 Energy equation EditTo derive an energy equation note that the advective acceleration term v v displaystyle bf v cdot nabla bf v may be decomposed as v v w v 1 2 v 2 displaystyle bf v cdot nabla bf v omega times bf v 1 over 2 nabla bf v 2 where w displaystyle omega is the vorticity of the flow and displaystyle cdot is the Euclidean norm This leads to a form of the momentum equation ignoring the external forces term given by v t w v 1 2 v 2 p r F displaystyle partial bf v over partial t omega times bf v nabla left 1 over 2 bf v 2 p over rho Phi right Taking the dot product of v displaystyle bf v with this equation leads to t 1 2 v 2 v 1 2 v 2 p r F 0 displaystyle 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 This equation was arrived at using the scalar triple product v w v 0 displaystyle bf v cdot omega times bf v 0 Define E displaystyle E to be the energy density E 1 2 r v 2 Kinetic Energy r F Potential Energy displaystyle 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 Noting that F displaystyle Phi is time independent we arrive at the equation E t v E p 0 displaystyle partial E over partial t bf v cdot nabla E p 0 Assuming that the energy density is time independent and the flow is one dimensional leads to the simplification E p C displaystyle E p C with C displaystyle C being a constant this is equivalent to Bernoulli s principle Of particular interest in open channel flow is the specific energy e E r g displaystyle e E rho g which is used to compute the hydraulic head h displaystyle h that is defined as h e p r g u 2 2 g z p g displaystyle begin aligned h amp e p over rho g amp u 2 over 2g z p over gamma end aligned with g r g displaystyle gamma rho g being the specific weight However realistic systems require the addition of a head loss term h f displaystyle h f 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 EditHEC RAS Streamflow Fields of study Computational fluid dynamics Fluid dynamics Hydraulics Hydrology Types of fluid flow Laminar flow Pipe flow Transitional flow Turbulent flow Fluid properties Froude number Reynolds number Viscosity Other related articles Chezy formula Darcy Weisbach equation Hydraulic jump Manning formula Saint Venant equations Standard step methodReferences Edit Chow Ven Te 2008 Open Channel Hydraulics PDF Caldwell NJ The Blackburn Press ISBN 978 1932846188 Battjes Jurjen A Labeur Robert Jan 2017 Unsteady Flow in Open Channels Cambridge UK Cambridge University Press ISBN 9781316576878 Jobson Harvey E Froehlich David C 1988 Basic Hydraulic Principles of Open Channel Flow PDF Reston VA U S Geological Survey a b Sturm Terry W 2001 Open Channel Hydraulics PDF New York NY McGraw Hill p 2 ISBN 9780073397870 Further reading EditNezu Iehisa Nakagawa Hiroji 1993 Turbulence in Open Channel Flows IAHR Monograph Rotterdam NL A A Balkema ISBN 9789054101185 Syzmkiewicz Romuald 2010 Numerical Modeling in Open Channel Hydraulics Water Science and Technology Library New York NY Springer ISBN 9789048136735 External links EditCaltech lecture notes Derivation of the Equations of Open Channel Flow Surface Profiles for Steady Channel Flow Open Channel Flow Open Channel Flow Concepts What is a Hydraulic Jump Open Channel Flow Example Simulation of Turbulent Flows p 26 38 Retrieved from https en wikipedia org w index php title Open channel flow amp oldid 1144518754, wikipedia, wiki, book, books, library,