↑ arvau

Wikipedia / 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. 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. 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. The parameter is known as the Froude number is the characteristic length 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.

Formulation

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:

  1. The flow is incompressible (this is not a good assumption for rapidly-varied flow)

  2. The Reynolds number is sufficiently large such that viscous diffusion can be neglected

  3. The flow is one-dimensional across the x-axis

Continuity equation

The general continuity equation is the fluid density is the divergence operator. Under the assumption of incompressible flow, with a constant control volume. However, it is possible that the cross-sectional area and defining the volumetric flow rateFinally, 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 is the pressure is the kinematic viscosity is the Laplace operator is the gravitational potentialThe second equation implies a hydrostatic pressure is the shear stress is the hydraulic radius, a way of quantifying friction losses, leads to the final form of the momentum equation:

Energy equation

To derive an energy is the vorticity is the Euclidean normTaking the dot productThis equation was arrived at using the scalar triple product to be the energy density being a constant; this is equivalent to Bernoulli's principle. Of particular interest in open-channel flow is the specific energy, which is used to compute the hydraulic head that is defined as: being the specific weight. However, realistic systems require the addition of a head loss to account for energy dissipation due to friction and turbulence that was ignored by discounting the external forces term in the momentum equation.

Empirical Computational Methods

In addition to the theoretical computation methods above, empirical methods are available which simplify the calculation and the amount of data required about the flow and the channel. In the United States, the most common method for calculating open-channel flow of water is the Manning's Equation. Manning's Equation, which applies to uniform flow, requires only the hydraulic radius of the flow, the channel slope, and a Manning's coefficient for roughness, n.

See also

Fields of study

Types of fluid flow

Fluid properties

Other related articles

References

Further reading

Caltech lecture notes: