Open-channel flow

Type of liquid flow within a conduit

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.

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, 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.

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 momentum equation for open channel flow may be found by starting from the incompressible Navier Stokes equations :where is the pressure, is the kinematic viscosity, is the Laplace operator, and is the gravitational potential. By invoking the high Reynolds number and 1D flow assumptions, we have the equations: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:

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:This equation was arrived at using the scalar triple product Define to be the energy density: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. Of particular interest in open channel flow is the specific energy , which is used to compute the hydraulic head that is defined as: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.