Contents preface (VII) introduction 1—37

Fig. 9.8 Determination of type of hydraulic jump on sloping channels
Example 9.1 A discharge of 9.0 m³/s flows in a 6.0 m wide rectangular channel which is inclined at an angle of 3° with the horizontal. Determine the type of jump if h₁ = 0.10 m and h_t = 2.6 m.
Solution:

F1 =	9.0/(6.0 × 0.1)	= 15.15
	9.81 × 0.10

h₂* ( i.e., the sequent depth in a horizontal channel) can be calculated from

h * =

h1

L

1

+ 8F1

2

O

=

0.1

L

1

+ 8(15.15)

2

O

M

− 1

M

− 1

P

2

P

2

2 N

Q

N

Q

= 2.09 m

328 IRRIGATION AND WATER RESOURCES ENGINEERING
Since h_t > h₂*, the depth h₂ should be calculated from Eqs. (9.30)-(9.33).
G₁² = 10^{0.054 (θ)} F₁²

• 
  10^{0.054 (3)} × (15.15)²
  

• 
  333.29
  

∴	h2	=	1	[ 1 + 8 G1^2	− 1]
	(0.10/cos 3° )		2
or	h	=		0.10	[	1 + 8	(333.29) − 1] = 2.54 m
2 cos (3° )
		2

Since h₂ < h_t, the jump is classified as the type D jump. Using Fig. 9.7, the length of the jump, L_j, can be determined by obtaining L_j / h_t for tan 3° = 0.05 and F₁ = 15.15.

	^Lj		= 4.9
	h
		t
∴		^Lj		= 4.9 × 2.6 = 12.74 m
Now		ht	=	2.6		= 26
	h	0.1
	1
and			^h2		=	2.31	= 23.1
	h		0.1
	1

Therefore, relative energy loss can be determined from Eq. (9.36)

∆E	=	(1 − 23.1) + [(15.15)^2 /2] [1 − (1/23.1)^2 ] + tan (3° ) [4.9(26)]	= 0.82
E	1	+		[(15.15)	^2 /2]	+	(4.9) (26) tan (3 )
1					°

9.2.7. Forced Hydraulic Jump

When the tail-water depth h_t is less than the required sequent depth h₂ corresponding to the pre-jump depth h₁, the jump is repelled downstream. However, by introducing devices such as baffle walls or baffle blocks and, thus, increasing the surface friction, the jump can be made to move upstream and form forcibly at the section it would have formed if sufficient tail-water depth (= h₂) were available. Such a jump is called the forced hydraulic jump (Fig. 9.9).

Figure 9.10 shows different types of forced hydraulic jump (8). For small Z and large x_o, type I jump is formed. This is similar to a free jump. When Z is increased and x_o is decreased, the baffle acts like an obstruction placed across the channel having free flow conditions and the jump is of type II*. With increase in tail-water depth, the obstruction is submerged and type II jump forms. On increasing Z and decreasing x_o further, the jump becomes more violent and is of type III. The effect of further increase in Z and decrease in x_o results in jumps of type IV, VI and VI*. A type VI* jump is a type VI jump with low tail-water depth. Between types IV and VI, there occurs an unstable transition phenomenon called type V (not shown in Fig. 9.10).

SURFACE AND SUBSURFACE FLOW CONSIDERATIONS FOR DESIGN OF CANAL STRUCTURES 329

Case 1

h₁

Case 2

h₁

Case 3

h₁

			Case 4
	Z	^hm ht		Z	Z
X	0	h	ht
1

		Case 5
Z	ht	Z	Z	^ht
	h1
		b
	s

^X0 Z	Case 6
Z	ht
ht
	h1

Case 7
Z	^ht
Z
^h1

Fig. 9.9 Devices for producing forced hydraulic jump (8)

330
Type I
^h1

Type II*
h₁

Type II
h₁

Type III
h₁

Type IV
h₁

IRRIGATION AND WATER RESOURCES ENGINEERING

x0	z	ht

6. 
   ^X0
   

^Lrj

x0	ht	0.01 Cd	0.12

x0	ht
		0.4	^X0	< 0.6
			L
			rj
x0	ht	0.05 Cd	0.18
		0.2		^X0	0.4
				L
	ht		rj
^x0
		0.18 Cd	0.5

Type VI
h1	x0		ht
		X0
					00.12
				^Lrj
				0.5 Cd
Type VI*
				Cd = Coefficient
h	1	^x0	h	of Drag
				t

Fig. 9.10 Types of forced hydraulic jump (8)
A large number of experimental studies have been carried out on the forced hydraulic jump as it forms the basic design element of the well-known hydraulic jump type stilling basins. The simplest case of a forced hydraulic jump is the jump forced by a two-dimensional baffle, known as a baffle wall, of height Z kept at a distance x_o from the toe (i.e., the beginning) of the jump (Fig. 9.9, case 1).

SURFACE AND SUBSURFACE FLOW CONSIDERATIONS FOR DESIGN OF CANAL STRUCTURES

331

Considering unit width of the channel, the momentum equation can be written as

1

2

1

2

F q

q

I

ρ gh1 −

ρ ght

− Fb

= ρ q G

−

J

(9.37)

2

2

H ht

h1 K

where,

F

b

= C (ρ u^2/2) Z.

d

1

Equation (9.37), on simplifying, gives

(α − 1) [ 2 F^2

− α ( 1 + α)]

Cd =

1

(9.38)

F^2

α β

1

where, α = h_t/h₁ and β = Z /h₁. Rajaratnam (8) found experimentally that C_d obtained from Eq. (9.38) is a function of only x_o which is made dimensionless by the length of roller of the classical jump, L_rj (Figs. 9.2 and 9.11). Figure 9.11 is based on data from only one source and, therefore, needs further verification. A design chart (Fig. 9.12) has been developed (8), using Eq. (9.38),
with Ψ (= h_t/h₂ = (h_t /h₁)(h₁/h₂) = α/φ where, φ = h₂/h₁) versus F₁ for various values of β C_d. Choosing a C_d value of 0.4 (a very competitive design) or smaller (for a conservative design),
obtain the value of C_d β (and, hence, β) from Fig. 9.12, for known Ψ and F₁. Now obtain x_o/L_rj from Fig. 9.11. Using Fig. 9.2, L_rj (and, hence, x_o) can now be determined. From known β and h₁, the depth of baffle wall Z can also be determined.

1.0
0.8

0.6

VI, VI*

d C

IV III
0.4
0.2
0

0 0.2 0.4 0.6

I & II

0.8 1.0 1.2 1.4 1.6 1.8 2.0 ^XO ^{/ L}rj

Fig. 9.11 Variation of the drag coefficient for forced hydraulic jump (8)
Baffle block (also known as baffle pier or friction block) is the case of a three-dimensional baffle wall. Baffle blocks are generally trapezoidal in shape and are placed in a single row or in two rows with staggered pattern (Fig. 9.13). The momentum equation [Eq. (9.37)] is applicable for this case too but with a different expression for F_b (the force exerted by the baffle blocks per unit width of the channel) which can be written as

F	F x	o		I
b	= f G		, β, F1	, ηJ	(9.39)
F2
	H h1		K

Here, F₂ = ₂¹ ρ gh₂² and η (i.e., the blockage ratio) = W/(W + S) (Fig. 9.13). Based on the analysis of data of Basco and Adams (10), Ranga Raju et al. (9) found that F₁ is unimportant in Eq. (9.39)

332 IRRIGATION AND WATER RESOURCES ENGINEERING
and Ψ₁Ψ₂F_b/F₂ is uniquely related to x_o/h₁ as shown in Fig. 9.14. Ψ₁ and Ψ₂ are empirical correction factors and are functions of Z/h₁ and η, respectively, as shown in Figs. 9.15 and 9.16.

ψ

1.0
		.05
	0
		.10
		0
		.20
0.9		0
		.	30
		0			.40
			0
				.50
0.8				0
				.60
				0
					.75
					0
0.7
							.90
					=	0
				β	Cd
0.6
0.5
1.0	3.0	5.0		7.0	9.0	11.0
							F1

Fig. 9.12 Design chart for baffle wall (9)

		1				3
				2
			t	h′2	h2	^h3
Flow					End sill
	h1	Z				Z
		^Xo	r	Section
				Basin wall
		S/2
		W
		S		Baffle
Flow		W
			blocks
			S

W
S
Plan

Fig. 9.13 Arrangement of trapezoidal baffle blocks

SURFACE AND SUBSURFACE FLOW CONSIDERATIONS FOR DESIGN OF CANAL STRUCTURES 333

) 2 F/

b )(F 2 ψ

1 (ψ

0.6

0.5
0.4

0.3
0.2
0.1
0 0 10 20 30 40 50 60 70 80 90 100 110
X_O / h₁

Fig. 9.14 Variation of (Ψ₁Ψ₂ F_b/F₂) with x_o/h₁ for trapezoidal blocks (9)
9

8
7
6

5

1 ψ

4
3
2

1
0 0 1 2 3 4 5 6 7 8 9
z / h₁
Fig. 9.15 Variation of Ψ₁ with Z/h₁ (9)

These data also indicated no change in the value of F_b when baffle blocks were placed in two rows for the range of r/Z from 2.5 to 5.0. Here, r is the spacing between the two rows of blocks.

334 IRRIGATION AND WATER RESOURCES ENGINEERING
1.2

1.0
0.8

2

ψ

0.6
0.4
0.2
0.3 0.4 0.5 0.6 0.7 0.8 0.9
η
Fig. 9.16 Variation of Ψ₂ with η (9)