Open-channel flow

Open-channel flow, a branch of hydraulics and fluid mechanics, is a type of liquid flow within a conduit or in channel 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: the free surface. 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:
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, and is defined as:formula_1where formula_2 is the mean velocity, formula_3 is the characteristic length scale for a channel's depth, and formula_4 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. 
Core equations.
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 formula_5 with components formula_6. 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:
Continuity equation.
The general continuity equation, describing the conservation of mass, takes the form:formula_7where formula_8 is the fluid density and formula_9 is the divergence operator. Under the assumption of incompressible flow, with a constant control volume formula_10, this equation has the simple expression formula_11. However, it is possible that the cross-sectional area formula_12 can change with both time and space in the channel. If we start from the integral form of the continuity equation:formula_13it is possible to decompose the volume integral into a cross-section and length, which leads to the form:formula_14Under the assumption of incompressible, 1D flow, this equation becomes:formula_15By noting that formula_16 and defining the volumetric flow rate formula_17, the equation is reduced to:formula_18Finally, this leads to the continuity equation for incompressible, 1D open-channel flow:formula_19
Momentum equation.
The momentum equation for open-channel flow may be found by starting from the incompressible Navier-Stokes equations :formula_20where formula_21 is the pressure, formula_22 is the kinematic viscosity, formula_23 is the Laplace operator, and formula_24 is the gravitational potential. By invoking the high Reynolds number and 1D flow assumptions, we have the equations:formula_25The second equation implies a hydrostatic pressure formula_26, where the channel depth formula_27 is the difference between the free surface elevation formula_28 and the channel bottom formula_29. Substitution into the first equation gives:formula_30where the channel bed slope formula_31. To account for shear stress along the channel banks, we may define the force term to be:formula_32where formula_33 is the shear stress and formula_34 is the hydraulic radius. Defining the friction slope formula_35, a way of quantifying friction losses, leads to the final form of the momentum equation:formula_36
Energy equation.
To derive an energy equation, note that the advective acceleration term formula_37 may be decomposed as:formula_38where formula_39 is the vorticity of the flow and formula_40 is the Euclidean norm. This leads to a form of the momentum equation, ignoring the external forces term, given by:formula_41Taking the dot product of formula_5 with this equation leads to:formula_43This equation was arrived at using the scalar triple product formula_44. Define formula_45 to be the energy density:formula_46Noting that formula_47 is time-independent, we arrive at the equation:formula_48Assuming that the energy density is time-independent and the flow is one-dimensional leads to the simplification:formula_49with formula_50 being a constant; this is equivalent to Bernoulli's principle. Of particular interest in open-channel flow is the specific energy formula_51, which is used to compute the hydraulic head formula_52 that is defined as:formula_53with formula_54 being the specific weight. However, realistic systems require the addition of a head loss term formula_55 to account for energy dissipation due to friction and turbulence that was ignored by discounting the external forces term in the momentum equation.