Contents preface (VII) introduction 1—37

Fig. 9.2 Length characteristics of jump (2)
9.2.4. Profile of Hydraulic Jump
In case of overflow structures located on permeable foundations, the concrete aprons of the stilling basins are subjected to uplift pressures which are partly counterbalanced by the weight of water flowing on the apron. Therefore, in the hydraulic jump type stilling basins, determination of profile of the jump becomes necessary. Rajaratnam and Subramanya (3) have
obtained an empirical relation, (Fig. 9.3), between y/[0.75(h₂ – h₁)] and x / X . Here, X is the distance from the beginning of the jump to the section where the depth measured above the x-

axis is 0.75(h2 – h1). The length X is empirically related to h1 and F1 as
		^X = 5.08 F – 7.82					(9.20)
		h1	1
)]	1.2
					y
1
h
–						x
2
(h
	[0.75	0.6					0.75 (h2 – h1)	h
				y	2
/
y						h1	x			x
	0
	0		1.0	2.0		3.0
			x / X

Fig. 9.3 Profile of the hydraulic jump in a rectangular channel (3)

322 IRRIGATION AND WATER RESOURCES ENGINEERING
9.2.5. Calculations for Hydraulic Jump in Horizontal Rectangular Channels
Equations (9.8), (9.11), (9.12), and (9.14) can be used to obtain direct solution of the jump parameters (i.e., E₂, h₂, ∆E, and E₁) for horizontal rectangular channels, if h₁ and q are known. These equations would also yield direct solution for E₁, h₁, ∆E, and E₂, if h₂ and q are known. One can also use Eq. (9.9) or Eq. (9.10) instead of Eq. (9.8) depending upon whether pre-jump or post-jump conditions are known.
However, in an actual design problem, generally the discharge q and the levels of the upstream and downstream total energy lines are known. Thus, q and ∆E are known. Determination of the remaining four parameters of the jump from Eqs. (9.8), (9.11), (9.12), and (9.14) is rather difficult. This difficulty can be overcome by the use of critical depth h_c (= (q²/g)^1/3) and defining

24. 
    = (h₁/h_c) ; Y = (h₂/h_c) ; Z = (∆E/h_c) ;
    

• 
  = E₁/h_c, and η = E₂/h_c
  

so that Eqs. (9.8), (9.11), (9.12), and (9.14) reduce to the following forms respectively :

	XY(X + Y) = 2
	ξ = X +	1
		2X ^2
	η = Y +	1
			2Y ^2
and	Z =	(Y − X)^3
4 XY

(9.21)
(9.22)

(9.23)
(9.24)


Here, X can vary from 0 to 1 only. Using Eqs. (9.21) to (9.24), one can obtain the sets of values of Y, ξ, η and Z for different values of X and, thus, the curves shown in Fig. 9.4. These curves are known as Crump’s curves (4). The method to use these curves is as follows:
(i) Calculate h_c from h_c = (q²/g)^1/3 (ii) Compute Z, i.e., ∆E/h_c.

(iii) Read h₂/h_c from the curve ∆E/h_c versus h₂/h_c.

(iv) Read E₂/h_c from the curve E₂/h_c versus h₂/h_c. (v) Thus, E₁ = ∆E + E₂.

(vi) For known E₁/h_c, obtain h₁/h_c from the curve E₁/h_c versus h₁/h_c.

Thus, for given q and ∆E, one can determine h₂, E₂, E₁, and h₁ from the relationships for Crump’s coefficients.
Combining Eqs. (9.21) and (9.24), one can obtain (5),

	Z =	− X ^6	+ 20 X ^3 + 8 − ( X ^4 + 8X)^3 /2	(9.25)
		16 X ^2
and also	Z =		− Y ^6	− 20Y ^3 − 8 −	(Y ^4 + 8Y)^3 /2	(9.26)
	16Y ^2

For different values of X and Y, the values of Z can be obtained and the curves Y-Z prepared. Approximate equations for these curves were obtained as (5)

	Y = 1 + 0.93556 Z^0.368	for Z < 1	(9.27)
and	Y = 1 + 0.93556 Z^0.240	for Z > 1	(9.28)

SURFACE AND SUBSURFACE FLOW CONSIDERATIONS FOR DESIGN OF CANAL STRUCTURES

323

c E/h

c /h 1 E

1.4

1.2
1.0

0.8
0.6
0.4
0.2
0

1.0 1.2 1.4 1.6 1.8 2.0 h ₂/h_c

(Subcritical, enlarged scale)

14.0

12.0
10.0

8.0
6.0
4.0
2.0
0.0

1. 
   0.2 0.4 0.6 0.8 1.0 h₁/h_c (Supercritical)
   

	Co-ordinates			160
	h/hc	E 2/hc	E/hc
	0.20 12.70
	0.25	7.67
	0.30	5.87					140
	0.60	1.99
	0.80	1.58
	1.00	1.50	0.0
7.0	1.20	1.55	0.014			120
	1.40	1.66	0.1
	1.60	1.80	0.30
	1.80	1.95	0.77
6.0	2.00	2.13	1.21			100
2.20	2.30	1.80
				c
	2.40		2.49	3.00
					E/h
	2.60		2.67		4.42
5.0	2.80	2.86	6.38			E2/hc		80
3.00	3.06	8.82
c	3.50		3.54	17.28
/h
	4.00		4.03	30.50
	2
	E
		4.50	4.52		50.30			60
4.0
		5.00	5.02	74.40
		6.00	6.01		161.1
	3.0						E/hc	40
2.0						20
1.0	2.0	3.0	4.0	5.0	0
1.0	6.0
			h2/hc (Subcritical)

Fig. 9.4 Relationship for Crump’s coefficients (4)
One of the two equations (Eqs. (9.27) and (9.28)) can be used for obtaining the value of Y for a specified value of Z. Equations (9.21), (9.22), and (9.23) can then be used for the determination of X, ξ, and η respectively.
Equations (9.25) and (9.26), can be solved to prepare a table for computations of hydraulic jump elements in rectangular channels (Table 9.1).