Ali S. Fayad, Discoverer of The Explicit Alternate Flow Depth in Rectangular Open Channels

Energy of Open Channel Flow

In rectangular open channel hydraulics, the specific energy of the flow is typically denoted by the letter E. By manipulating Bernoulli’s equation for the special case of open channel flow, the specific energy of the flow is expressed as,

E= z + y + (V^2)/2g = z + y + (q^2)/(2g.y^2)

Hence, the specific energy of uni-dimensional flow is considered to be a function of the height of the invert of the channel relative to a certain datum, the flow depth above the invert, and the square of the average velocity of the flow divided by double the gravitational acceleration. An alternative way of expressing the specific energy of the flow is:

E= z + y + y(V^2)/2gy   = z + y(1 + (Fr^2)/2)

The above equation relates the specific energy to the Froude number of the flow in addition to height of the invert above the datum and the height of the column of flow above the invert.  

Energy-Depths Relationship

What is the Energy-Depths relationship in open-channels flow? To answer the above question, one needs to consider a more factual concept of energy. The sun is considered a source of energy. It emits light rays that can be felt by the senses. But what about the energy content of water; how does one evaluate it?

Open channel hydraulics is not concerned with the energy content of the water on an atomic level. It is concerned with its kinetic energy as a flow regime. Also, it is concerned with its potential energy fluctuation relative to a reference datum. Hence, Bernoulli's energy equation applied to the hydraulics of open channels has a limited window within which to observe and classify the behavior of channel flow. The pressure component of Bernoulli's energy equation is useful for pressurized pipe flow but gets disregarded for the surface of open channel flow since atmospheric pressure or gage pressure is considered as zero pressure.

Another component of the energy in open channel that was disregarded by the traditional theory is the spiral flow aspect of the overall flow. This is acceptable to neglect for continuous, long and straight channels but compromises the predictive ability of the uni-dimensional theory within transitions or discontinuity regions of a flow.

 Depth-Energy Interaction Diagram Branches 

The plot of the Depth vs. Specific Energy of a flow in a rectangular open channel allows for readily identifying and classifying the branches of the plot:

 On first glance, one needs to ask about the practicality of including a plot of the negative depths in the graph. After all, and according the variety of the hydraulic texts, the negative depth is supposed to be an imaginary depth that has no bearance on reality.

Energy-Depths Triad

Once the z value is dropped by setting the datum to an elevation of zero, the previous equation simplifies to:

E =  y(1 + (Fr^2)/2)

or

E =  ysub + ysup + yneg  

This reveals that the specific energy of the flow is equal to the algebraic sum of its three alternate depths for the same flow. This further suggests an inextricable hydraulic triad (as in a musical triad) where the three depths combine into the specific energy of the flow and the flow manifests into three regimes:

• the subcritical flow depth which is stable,
• the supercritical flow depth which is less stable,
• and the hypothetical flow for the negative depth which is extremely unstable to the extent it is assumed that it cannot manifest in the physical realm.