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28.3 Momentum equation

The ratio of sequent depth 20

1

= y y

y may be computed assuming

• hydrostatic pressure distributions

• uniform velocity distributions

• air entrainment is negligible and

• time-averaged quantities

in sections 1 and 2.

Belanger's momentum equation for sequent depths of a hydraulic jump on a level floor

in a rectangular channel can be derived by applying momentum equation between

sections 1 and 2 as given below.

2 2

1 21 2

1 1 1 2 1 2 1 1 2 2

1 1 1 1 1 21 2

2 22 2

2 2 221

11 2

2 2 2 21 1 2

2 21 1 2

2 2 2 2 22 1 1

1 2 31 1 1

Q QZA ZA

gA gA

For a rectangular channel A b y , A b y , Q VA V A ,

V A V y y yV ,V , Z , Z ,

A y 2 2Q y Q y

by bgby 2 gby 2

Q y Q y 1 ygy b 2 gb y y 2

V Q b y QF

gy gy gb y

di

+ = +

= = = =

= = = =

+ = +

+ = +

= = =

21

22 2 21 1 2

3 2 2 2 31 1 1 2 1

2

2 2 1 21 1

2 1

vided by y

Q y Q y 1 y 10

gy b 2y gb y y y 2

1 y y 1F F 0

2 y y 2

⎛ ⎞+ − − =⎜ ⎟

⎝ ⎠

⎛ ⎞ ⎛ ⎞+ − + =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

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( )

( )

( )

22 2 21 1 2

3 2 2 2 31 1 1 2 1

2

2 2 1 2

1 12 1

2

2 2 1 21 1

2 1

3

2 22 21 1

1 1

3

2 22 21 1

1 1

Q y Q y 1 y 10

gy b 2y gb y y y 2

1 y y 1F F 02 y y 2

y y2F 1 2F 0

y y

y y2F 1 2F 0

y y

y y2F 1 2F 0

y y

This can be r

⎛ ⎞+ − − =⎜ ⎟

⎝ ⎠

⎛ ⎞ ⎛ ⎞+ − + =

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞+ − − =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞+ − − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞− + + =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

2

22 2 21

1 1 1

22 1

1

2

22 21

1 1

2212

11

ewritten as

y y y2F 1 0

y y y

y1 0 y y uniform flow.

y

y y 2F 0 a quadratic equation.y y

1 1 8Fy 1Hence 1 8F 1

y 2 2

⎡ ⎤⎛ ⎞ ⎡ ⎤⎢ ⎥+ − − =⎜ ⎟ ⎢ ⎥⎢ ⎥⎝ ⎠ ⎣ ⎦⎣ ⎦

∴ − = ∴ =

⎛ ⎞ + − =⎜ ⎟⎝ ⎠

− + +⎛ ⎞⎡ ⎤= = + −⎜ ⎟⎣ ⎦

⎝ ⎠

(28.1)⎡ ⎤= + −⎢ ⎥⎣ ⎦

y 1 22 1 8F 11y 21

in which y 2, y 1 are sequent and initial depths respectively and 11

1

VF =

gy

⎛ ⎞

⎜ ⎟⎜ ⎟⎝ ⎠

is the initial

Froude number. Equation 28.1 has been verified by many investigators experimentally

and often a ratio lower than the one calculated by the equation has been recorded.

Belanger , did not consider the bed shear force while deriving Eq. 28.1. Rajaratnam in

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1965, proposed the following momentum equation taking into consideration the

integrated shear force.

(28.2)⎛ ⎞

⎡ ⎤− − + +⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎣ ⎦⎝ ⎠

ε 3

y y 2 22 2 1 2F 2F = 01 1y y1 1

In which ε is the non dimensional integrated shear force, given byγ

f21

Py2

and is a

function of Froude number. P f is the integrated shear force.

He used the data of Rouse et al. , Harleman, Bakhmeteff ,Safranez , Bradley - Peterka ,

along with his own. Figure 2 shows the effect of shear force on sequent depth ratio.

BelangerRajaratnamSarma and Newnham

Eq. 28.1

Eq. 28.3Eq. 28.2

02 4 6 8 10

2

4

6

8

10

12

14

F1

Fig. 28.5 - Variation of sequent depth ratio

y2y1

___

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Sarma and Newnham 1975 introducing the momentum coefficient ( j 1.045 β = ) for the

non uniform velocity distribution obtained the following modified momentum equation

(28.3)⎡ ⎤= + −⎢ ⎥⎣ ⎦

y 1 22 1 10.4 F 11y 21

In Eqn. 28.3, a value of j β was used by them based on the assumption of a similarity

profile for the velocity distribution. Eq. 28.3 gives a higher value for the sequent depth

ratio, compared to the value computed from Eq.28.1. Their analysis was carried out

upto a Froude number value of 4.