Ali S. Fayad, Discoverer of The Explicit Alternate Flow Depth in Rectangular Open Channels
Converse Flow Depths
Derivation of the Converse Flow Depths Equation:
Begin with the Fayad Alternate Depth:
yd / yu = 0.25{ Fru [Fru + (Fru^2 + 8 )^0.5 ]}
and reorder by multiplying into the equation within brackets
yd/yu= {(Fru^2)/4 + [(Fru^4)/16 + (Fru^2)/2 ]^0.5 ]}
yd/yu= [(Fru^2)/2]/{[-Fru^2]/4 +[(Fru^4)/16 + (Fru^2)/2 ]^0.5}
yd/yu= [(Fru^2)/2]{[Fru^2]/4] [-1+ (1 + 8(Fru^-2))^0.5]}^(-1)
And the above obfuscated form bears a resemblance to the Bélanger form of the Conjugate Depths equation!The obfuscated forms are mere mathematical variations of the same hydraulic relationship. They share the exact utilitarian function of providing the same output for the same input as is the case with the Fayad Alternate Depth equation. The key difference is that the formulaic utility of the obfuscated forms is rendered more cumbersome to comprehend or linearize. In so doing, the solution becomes more cumbersome than the problem. However, stopping here is not the objective of the author. Instead, the formula is revised a couple more times to yield:
Inverse the previous equation to yield:
yu / yd = {[(Fru^2)/2]^-1}{[Fru^2]/4] [-1+ (1 + 8(Fru^-2))^0.5]}
Remove the common term to obtain:
yu / yd= 0.5{-1+ [1 + 8(Fru^-2)]^0.5}
The above equation is called the conjugate depths equation for the Converse Hydraulic Jump Flow of a given Sluice Gate Flow depths. The next figure illustrates the concept.
