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

With Fr representing the Froude number of the subcritical flow regime (as in the upstream flow of an idealized sluice gate), the Fayad Alternate Depth equation: 

(Supercritical Depth)/(Subcritical Depth) = 0.25Fr{Fr+(Fr² +8)^0.5 }

can be quickly transformed into a useful and simple linear form for flow situations with low Froude number values as demonstrated next:

Linearized Alternate Depths Ratio

yd/yu ≈ 0.7Fru                  (for Fru <0.1)

The above formula is simple but is useful for cases where the water behind a sluice gate is almost static and the height of the opening of the gate is minimal in relation to the upstream depth. The following shows its direct derivation from the Fayad Alternate Depth

Recall that:

(SuperDepth)/(SubDepth)=(yd)/(yu)

(yd)/(yu)=0.25Fr{Fr+(Fr² +8)^0.5 }

or        

yd/yu

=[Fr/(2)^0.5]{[Fr/(8)^0.5]+[(Fr²/8)+1]^0.5}

Introduce CL as the coefficient of linearization of the alternate flow depth formula such that

(1/CL)={[Fr/(8)^0.5]+[(Fr²/8)+1]^0.5}

Hence,

(SuperDepth)/(SubDepth)=[Fr/(2)^0.5]/CL

Therefore, for cases where CL can be reasonably approximated as unity, the following linearized form of the alternate depth formula may be used in cases where the subcritical Froude number (upstream of the frictionless sluice gate) is less than 0.1:

yd/yu = [Fru/(2)^0.5]                  (for Fru<0.1)

yd/yu ≈ 0.7071Fru =0.7Fru        (for Fru<0.1)

Here, the Froude number in the above formula is that of the subcritical flow

Linearization  Coefficient

CL was defined as the coefficient of linearization for the Fayad Alternate Flow Depth. The function of this coefficient was shown to permit expressing the alternate depths ratio as a linear relation of the Froude number when CL can be approximated by 1. Here, it is further explored beginning with its original definition:

(1/CL)={[Fru/(8)^0.5]+[(Fru²/8)+1]^0.5}

or

(1/CL)={[-Fru/(8)^0.5]+[(Fru²/8)+1]^0.5}^-1

Hence,

 CL ={[-Fru/(8)^0.5]+[(Fru²/8)+1]^0.5}

The Sluice Gate Discharge Equation:

Expanding the linearized form of the Fayad alternate depth rapidly yields the commonly known sluice gate discharge equation:

Recall,

(yd)/(yu)= [Fru/(2)^0.5]/CL

CL(yd)/(yu)= (Q/b)/(2g.yu³)^0.5

CL(yd)=(Q/b)/(2g.yu)^0.5

(Q/b)=CL(yd)(2g.yu)^0.5

(Q/b)=CL(Cc yG)(2g.yu)^0.5

In the above equation Cc is the coefficient of contraction relating the height of the gate, yG, to the water depth at the Vena Contracta.

Combining the linearization coefficient and the contraction coefficient into the flow coefficient yields:

Q=C.b(yG)(2g.yu)^0.5

In Chapter 17 of his book "Open Channel Hydraulics" (Copyright 1959 by McGraw-Hill Inc), Ve Te Chow provides a treatment for underflow gates. Chow's Eq.17-35 and Eq. 17-36 provide an estimate of the flow in terms of upstream depth of the sluice gate, its width, and a correction coefficient that accommodates the geometry and other factors.

Compare the above equations with those in Ve Te Chow and note that Chow's book did not contain an explicit form for the alternate flow depth.

Inverted Alternate Depths Ratio:

A conjugate form of the Fayad Alternate Depth ratio is developed to mathematically relate the alternate depths ratio for the sluice gate to the well-known conjugate depths ratio for the hydraulic jump.

Using the linearization coefficient, express the alternate depths relation as follows:

yu/yd = CL /[Fru/(2)^0.5]

Or

yu/yd

= {(2)^0.5[-Fru/(8)^0.5]+[(Fru²/8)+1]^0.5}/Fru

Or

yu / yd ={(2)^0.5[-1/(8)^0.5]+[(1/8)+Fru^-2]^0.5}

Or

 yu/ yd ={(1/2)[-1 +[1+8Fru^-2]^0.5}

Author's Note: The above direct derivation of the converse flow depths equation from the Fayad Alternate Flow Depth equation is a copyright of Ali S. Fayad

It is noted that the above equation has a form identical to the Bélanger form of the Conjugate depths equation except for the negative power of the upstream Froude number!

This can be further rendered into the Belanger form by setting (1/Fru)= Frcon as follows:

yu/yd = 0.5{(-1+ [1 + 8(Frcon)^2)]^0.5}

This is called the converse flow depths equation.