In
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...
and
hydraulics
Hydraulics (from Greek: Υδραυλική) is a technology and applied science using engineering, chemistry, and other sciences involving the mechanical properties and use of liquids. At a very basic level, hydraulics is the liquid counter ...
, open-channel flow is a type of
liquid
A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas, a ...
flow within a conduit with a
free surface
In physics, a free surface is the surface of a fluid that is subject to zero parallel shear stress,
such as the interface between two homogeneous fluids.
An example of two such homogeneous fluids would be a body of water (liquid) and the air in ...
, known as a
channel
Channel, channels, channeling, etc., may refer to:
Geography
* Channel (geography), in physical geography, a landform consisting of the outline (banks) of the path of a narrow body of water.
Australia
* Channel Country, region of outback Austral ...
. The other type of flow within a conduit is
pipe flow
In fluid mechanics, pipe flow is a type of liquid flow within a closed conduit, such as a pipe or tube. The other type of flow within a conduit is open channel flow.
These two types of flow are similar in many ways, but differ in one important ...
. 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.
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.
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
A hydraulic jump is a phenomenon in the science of hydraulics which is frequently observed in open channel flow such as rivers and spillways. When liquid at high velocity discharges into a zone of lower velocity, a rather abrupt rise occurs in ...
and the
hydraulic drop
A hydraulic drop is a type of local phenomena found in open channel flow. It is a rapid change in the depth of flow from a high stage to a low stage that results in a steep depression in the water surface. It is often caused by an abrupt change in ...
.
**** ''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
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. S ...
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
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
and gravity relative to the
inertia
Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
l forces of the flow.
Surface tension
Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) to f ...
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.
The parameter is known as the
Froude number
In continuum mechanics, the Froude number (, after William Froude, ) is a dimensionless number defined as the ratio of the flow inertia to the external field (the latter in many applications simply due to gravity). The Froude number is based on t ...
, and is defined as:
where
is the mean velocity,
is the
characteristic length scale for a channel's depth, and
is the
gravitational acceleration
In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodies ...
. Depending on the effect of viscosity relative to inertia, as represented by the
Reynolds number
In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...
, the flow can be either
laminar
Laminar means "flat". Laminar may refer to:
Terms in science and engineering:
* Laminar electronics or organic electronics, a branch of material sciences dealing with electrically conductive polymers and small molecules
* Laminar armour or "band ...
,
turbulent
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
, or
transitional. However, it is generally acceptable to assume that the Reynolds number is sufficiently large so that viscous forces may be neglected.
Core equations
It is possible to formulate equations describing three
conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, c ...
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
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
vector 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
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An eq ...
(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
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. S ...
, describing the conservation of mass, takes the form:
where
is the fluid
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
and
is the
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
operator. Under the assumption of incompressible flow, with a constant
control volume
In continuum mechanics and thermodynamics, a control volume (CV) is a mathematical abstraction employed in the process of creating mathematical models of physical processes. In an inertial frame of reference, it is a fictitious region of a given v ...
, 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
In physics and engineering, in particular fluid dynamics, the volumetric flow rate (also known as volume flow rate, or volume velocity) is the volume of fluid which passes per unit time; usually it is represented by the symbol (sometimes ). I ...
, the equation is reduced to:
Finally, this leads to the continuity equation for incompressible, 1D open-channel flow:
Momentum equation
The momentum equation for open-channel flow may be found by starting from the
incompressible Navier-Stokes equations :
where
is the
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
,
is the
kinematic viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inter ...
,
is the
Laplace operator, and
is the
gravitational potential
In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric ...
. By invoking the high Reynolds number and 1D flow assumptions, we have the equations:
The second equation implies a
hydrostatic pressure
Fluid statics or hydrostatics is the branch of fluid mechanics that studies the condition of the equilibrium of a floating body and submerged body "fluids at hydrostatic equilibrium and the pressure in a fluid, or exerted by a fluid, on an imme ...
, 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
Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ot ...
and
is the
hydraulic radius The Manning formula or Manning's equation is an empirical formula estimating the average velocity of a liquid flowing in a conduit that does not completely enclose the liquid, i.e., open channel flow. However, this equation is also used for calculat ...
. Defining the friction slope
, a way of quantifying friction losses, leads to the final form of the momentum equation:
Energy equation
To derive an
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
equation, note that the advective acceleration term
may be decomposed as:
where
is the
vorticity
In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along wi ...
of the flow and
is the
Euclidean norm
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
. This leads to a form of the momentum equation, ignoring the external forces term, given by:
Taking the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
of
with this equation leads to:
This equation was arrived at using the
scalar triple product
In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector- ...
. Define
to be the
energy density
In physics, energy density is the amount of energy stored in a given system or region of space per unit volume. It is sometimes confused with energy per unit mass which is properly called specific energy or .
Often only the ''useful'' or extract ...
:
Noting that
is time-independent, we arrive at the equation:
Assuming that the energy density is time-independent and the flow is one-dimensional leads to the simplification:
with
being a constant; this is equivalent to
Bernoulli's principle
In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy. The principle is named after the Swiss mathematici ...
. Of particular interest in open-channel flow is the
specific energy
Specific energy or massic energy is energy per unit mass. It is also sometimes called gravimetric energy density, which is not to be confused with energy density, which is defined as energy per unit volume. It is used to quantify, for example, sto ...
, which is used to compute the
hydraulic head
Hydraulic head or piezometric head is a specific measurement of liquid pressure above a vertical datum., 410 pages. See pp. 43–44., 650 pages. See p. 22.
It is usually measured as a liquid surface elevation, expressed in units of length, ...
that is defined as:
with
being the
specific weight
The specific weight, also known as the unit weight, is the weight per unit volume of a material.
A commonly used value is the specific weight of water on Earth at , which is .National Council of Examiners for Engineering and Surveying (2005). ''Fu ...
. However, realistic systems require the addition of a
head loss
Hydraulic head or piezometric head is a specific measurement of liquid pressure above a vertical datum., 410 pages. See pp. 43–44., 650 pages. See p. 22.
It is usually measured as a liquid surface elevation, expressed in units of length, ...
term
to account for energy
dissipation
In thermodynamics, dissipation is the result of an irreversible process that takes place in homogeneous thermodynamic systems. In a dissipative process, energy ( internal, bulk flow kinetic, or system potential) transforms from an initial form to ...
due to
friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of t ...
and
turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
that was ignored by discounting the external forces term in the momentum equation.
See also
*
HEC-RAS
HEC-RAS is a computer program that models the hydraulics of water flow through natural rivers and other channels. Prior to the 2016 update to Version 5.0, the program was one-dimensional, meaning that there is no direct modeling of the hydraulic ...
*
Streamflow
Streamflow, or channel runoff, is the flow of water in streams and other channels, and is a major element of the water cycle. It is one component of the movement of water from the land to waterbodies, the other component being surface runoff. Wate ...
*Fields of study
**
Computational fluid dynamics
Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate th ...
**
Fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
**
Hydraulics
Hydraulics (from Greek: Υδραυλική) is a technology and applied science using engineering, chemistry, and other sciences involving the mechanical properties and use of liquids. At a very basic level, hydraulics is the liquid counter ...
**
Hydrology
Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and environmental watershed sustainability. A practitioner of hydrology is calle ...
*Types of fluid flow
**
Laminar flow
**
Pipe flow
In fluid mechanics, pipe flow is a type of liquid flow within a closed conduit, such as a pipe or tube. The other type of flow within a conduit is open channel flow.
These two types of flow are similar in many ways, but differ in one important ...
**
Transitional flow
**
Turbulent flow
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
*Fluid properties
**
Froude number
In continuum mechanics, the Froude number (, after William Froude, ) is a dimensionless number defined as the ratio of the flow inertia to the external field (the latter in many applications simply due to gravity). The Froude number is based on t ...
**
Reynolds number
In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...
**
Viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
*Other related articles
**
Chézy formula
The Chézy formula is an semi-empirical resistance equation which estimates mean flow velocity in Open-channel flow, open channel conduits. The relationship was realized and developed in 1768 by French physicist and engineer Antoine de Chézy (1718 ...
**
Darcy-Weisbach equation
**
Hydraulic jump
A hydraulic jump is a phenomenon in the science of hydraulics which is frequently observed in open channel flow such as rivers and spillways. When liquid at high velocity discharges into a zone of lower velocity, a rather abrupt rise occurs in ...
**
Manning formula The Manning formula or Manning's equation is an empirical formula estimating the average velocity of a liquid flowing in a conduit that does not completely enclose the liquid, i.e., open channel flow. However, this equation is also used for calculat ...
**
Saint-Venant equations
**
Standard step method
The standard step method (STM) is a computational technique utilized to estimate one-dimensional surface water profiles in open channels with gradually varied flow under steady state conditions. It uses a combination of the energy, momentum, and c ...
References
Further reading
* Nezu, Iehisa; Nakagawa, Hiroji (1993).
Turbulence in Open-Channel Flows'. IAHR Monograph. Rotterdam, NL: A.A. Balkema. .
*Syzmkiewicz, Romuald (2010).
Numerical Modeling in Open Channel Hydraulics'. Water Science and Technology Library. New York, NY: Springer. .
External links
*
Caltech
The California Institute of Technology (branded as Caltech or CIT)The university itself only spells its short form as "Caltech"; the institution considers other spellings such a"Cal Tech" and "CalTech" incorrect. The institute is also occasional ...
lecture notes:
Derivation of the Equations of Open Channel FlowSurface Profiles for Steady Channel FlowOpen-Channel FlowOpen Channel Flow ConceptsWhat is a Hydraulic Jump?Open Channel Flow ExampleSimulation of Turbulent Flows(p. 26-38)
{{Hydraulics
Civil engineering
Fluid dynamics
Hydraulics
Hydraulic engineering