Contents preface (VII) introduction 1—37

9.2.6. Hydraulic Jump on Sloping Channels
On a horizontal floor with little friction, location of hydraulic jump varies considerably with a slight change in the depth or velocity of flow. But, on a sloping floor, location of hydraulic jump is relatively stable and can be closely predicted. However, energy dissipation in the case of jumps on a sloping floor is less owing to the vertical component of velocity remaining intact.
Equation (9.1) is theoretically applicable to hydraulic jumps forming on sloping channels. But the solution of the problem is difficult due to the following reasons :
(i) The length and shapes of the hydraulic jump are not well-defined and, hence, the term W sin θ is poorly computed.
(ii) The specific weight of the liquid in the control volume can change considerably due to air entrainment.
(iii) The pressure terms cannot be determined accurately.
Figure 9.5 shows several cases of hydraulic jumps on sloping channels. In the studies of hydraulic jump on sloping channels, the end of the surface roller is taken as the end of the jump. This means that the length of roller (measured horizontally) is the length of the jump.

	^Lr		Lr
h1			^h1
	h2	= ht		h2 = ht
			Roller
		Type A		Type B
Lr			^Lr
			h1
h1
	h2 = ht	h2	h	t
		Type C		Type D

Fig. 9.5 Different types of hydraulic jump which form in sloping channels

SURFACE AND SUBSURFACE FLOW CONSIDERATIONS FOR DESIGN OF CANAL STRUCTURES

325

When the jump begins at the end of the sloping apron, type A jump occurs and h₂ = h₂^* = h_t. Here, h₂ is the subcritical sequent depth corresponding to h₁, h_t the tail-water depth, and h₂^* is the subcritical sequent depth h₂ given by Eq. (9.9). Type A jump is, obviously, governed by Eqs. (9.9) and (9.10).

When the end of the jump coincides with that of the sloping bed, type C jump occurs. For this case, Kindsvater (6) developed the following equation for the sequent depth h₂ :

h		1	L		2	F	cos^3 θ I	O
2	=		M	1 + 8	G	1 − 2N tan θ ^J	P	(9.29)
h	2 cos θ M	^F1	− 1P
1			N			H	K		Q

in which, θ is the longitudinal slope angle of the channel, and N is an empirical coefficient dependent on the length of the jump. Equation (9.29) can be rewritten as

	^h2 =	^1 [ 1 + 8 G ^2	− 1]
	h1^′	2	1
where,	h′1 = h1/cos θ
and	G ^2 =		cos^3 θ		× F ^2
	1		1 − 2N tan θ	1

Rajaratnam (6) gave the following simple expression :

_cos³ θ

_{1 − 2N tan θ} = 10^{0.054 (θ)}

where, θ is in degrees.

(9.30)
(9.31)
(9.32)

(9.33)

When h_t is greater than the sequent depth h₂ required for type C jump, then type D jump occurs completely on the sloping apron. Bradley and Peterka (7) found that Eqs. (9.30) to (9.33) valid for type C jump can be used for the type D jump also.

If h_t is less than that required for type C jump but greater than h₂^*, the toe of the jump is on the sloping bed, and the end of the jump on the horizontal bed. This jump is classed as type B jump. A graphical solution (Fig. 9.6) has been developed for this type of jump (7).
Bradley and Peterka (7) have developed plots for the estimation of the length of the type D jump (Fig. 9.7). These plots can also be used to determine the lengths of the types B and C jumps.

The energy loss for the type A jump can be estimated from Eq. (9.14). For jumps, one can write

h

u^2

E1

= Lj tan

θ +

1

+

1

cos

2 g

θ

u^2

and

E = h +

2

2

2

2g

Thus,

∆E

=

(1 − h2 /h1 ) + (F1^2 /2) [1 − {1/(h2 /h1)}^2 ] + [(L j /h2 ) (h2 /h1)] tan θ

E1

1 + (F1^2 /2) +

(L j /h2 ) (h2 /h1) tan θ

C and D

(9.34)

(9.35)

(9.36)

326 IRRIGATION AND WATER RESOURCES ENGINEERING

2.8

2.4

2.0

*2 /h t h

1.6

1.2

0.8

0

6

5

4

t /h j L

3

2

0

θ	h*
	2	ht
	L

00		0	50
1		.				.	30			. 25			. 20
	=
.						0				0					0
=		θ			=			=			=	.15
tan
	θ			θ				θ			tan	θ				0
	tan			tan			tan							=
												tan	θ
																		.10
															0
															=
															θ
															tan
																			.05
																θ	=	0
															tan

2 4 6 8 10

L/h*

2
Fig. 9.6 Solution for B jump (2)

tan = 0.05

tan = 0.10

tan = 0.15 tan = 0.20

tan = 0.25

4 8 12 16 20

F ₁ =

^u1

gh₁

Fig. 9.7 Hydraulic jump length for jump types B, C and D (7)

SURFACE AND SUBSURFACE FLOW CONSIDERATIONS FOR DESIGN OF CANAL STRUCTURES

327

Here, the bed level at the end of the jump has been chosen as the datum and the potential energy term h₁/cos θ has been approximated as h₁. Equation (9.36) should not be used when F₁ is less than 4 as in this range very little is known about L_j / h₂ which affects ∆E/E₁. In order to solve a problem of hydraulic jump on a sloping channel, the first step is to determine the type of jump for given slope, the pre-jump supercritical depth, and the tail-water condition, Figure 9.8 illustrates the procedure for the determination of the type of hydraulic jump.

	If		Yes	Type A jump
h *
	> h	t
	2
	No
Find h2 from				If	Yes	Type C jump
Eq. (9.30)				h2 = ht

No
If	No
Type D jump
h2 > ht

Yes
Type B jump