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, resulting in flow dominated by gravity but not
hydraulic pressure.
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.[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
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 equation
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 equation
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.