Open-channel flow

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.

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: 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:

is the characteristic length scale for a channel's depth, and

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 velocity vector field

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 general continuity equation, describing the conservation of mass, takes the form:

Under the assumption of incompressible flow, with a constant control volume

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:

and defining the volumetric flow rate

Finally, this leads to the continuity equation for incompressible, 1D open-channel flow:

By invoking the high Reynolds number and 1D flow assumptions, we have the equations:

To account for shear stress along the channel banks, we may define the force term to be:

, a way of quantifying friction losses, leads to the final form of the momentum equation:

To derive an energy equation, note that the advective acceleration term

This leads to a form of the momentum equation, ignoring the external forces term, given by:

This equation was arrived at using the scalar triple product

Assuming that the energy density is time-independent and the flow is one-dimensional leads to the simplification:

Of particular interest in open-channel flow is the specific energy

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.