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

Pre-Submergence Jump

Consider a rectangular open channel flow where the ensuing jump toe is coincident with the vena contracta of the issuing sluice gate. It is a unique situation where the downstream depth of the sluice gate is one and the same as the initiating depth of the hydraulic jump. This is termed here as the pre-submergence jump. It implies that the initiating depth of the jump did not get the chance to develop its boundary layer.

In the following derivation, let the subscript "f" represent the final depth of the jump (referenced with the subscript "2" in the figure. Let the subscript "i" represent the initiating depth of the jump (referenced with the subscript "1" in the figure).

Hence, the Belanger form of the conjugate depth equation for the Hydraulic Jump can be written as:

yf = 0.5 yi [(8Fri² +1)^1/2 -1]

and is re-expressed as

yf,max = 0.5 yd [(8Frd² +1)^1/2 -1]

where yd is the downstream depth of the sluice at the Vena Contracta and Frd is its corresponding Froude number. Using the Fayad alternate depth ratio:

yd = yu .Rf

and recognizing that:

Frd² = Fru² (yu / yd )³

or, Frd² = Fru² .Rf ^-3

 The previous equation for the pre-submergence hydraulic jump in relation to the sluice gate upstream depth becomes:

yf,max

= 0.5 yu.Rf [(8Fru².Rf ^-3 +1)^1/2 -1]

In a rectangular horizontal open channel, the final depth of the pre-submergence jump is the highest depth that can be attained for the same flow, with the same energy content, emanating from the sluice gate. The hydraulic jump formula becomes directly related to the sluice gate formula. This is useful in practical applications. By entwining the two formulas together, the Fayad alternate depths formula with the traditionally known conjugate depths formula of the hydraulic jump, the designer can predict the necessary height of a box culvert to prevent submergence for a given range of flows.

For situations where the Froude number of the upstream flow of the sluice gate is less than 0.1, the linearized form of the Fayad alternate depth can be used in conjunction with the conjugate depths formula for the hydraulic jump to quickly evaluate many practical applications.

If the sluice gate's upstream flow regime has a Froude number that is less than 0.03, the aforementioned formula can be further approximated by:

yf/yu ≈ [1.68(Fru)^0.5]

(for Fru<0.03)