Momentum equation - nptel - Indian Institute of Technology Madras

Hydraulics Prof. B.S. Thandaveswara
28.3 Momentum equation
The ratio of sequent depth
Indian Institute of Technology Madras
y
0
= y
y
• hydrostatic pressure distributions
• uniform velocity distributions
• air entrainment is negligible and
• time-averaged quantities
in sections 1 and 2.
2
1
may be computed assuming
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
Q Q
+ ZA = + ZA
gA gA
1 2
1 2
For a rectangular channel A = b y , A = b y , Q = V A = V A ,
VA Vy y y
V = ,V = , Z = ,Z = ,
1 1 1 1 1 2
2 2
1 2
A2 y2 2 2
Q y Q y
by b
gby 2 gby 2
2 2 2
+ 1
1 = + 2
1 2
Q y Q y 1 y
gy b 2 gb y y 2
2 2 2 2
+ 2
1 = 2
1 + 2
1 1 2
V Q b y Q
2 2 2 2 2
2
1
1 =
gy1 =
gy1 1 = 2 3
gb y1
F
divided
by y
2
1
2 2 2
Q y1 Q y1 1 ⎛y ⎞ 2 1
+ − − 0
3 2 2 2 3 ⎜ ⎟ =
gy1b2y1 gb y1 y2 ⎝ y1 ⎠ 2
2 1 2⎛ y ⎞ ⎛ 1 y ⎞ 2 1
F1 + − F1 ⎜ ⎟+ ⎜ ⎟ = 0
2 ⎝y2 ⎠ ⎝ y1 ⎠
2
2
1 1 1 2 1 2 1 1 2 2
2

Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
2 2 2
Q y1 Q y1 1 ⎛y ⎞ 2 1
+ − − 0
3 2 2 2 3 ⎜ ⎟ =
gy1b2y1 gb y1 y2 ⎝ y1 ⎠ 2
2 1 2⎛ y ⎞ ⎛ 1 y ⎞ 2 1
F1 + − F1 ⎜ ⎟+ ⎜ ⎟ = 0
2 ⎝y2 ⎠ ⎝ y1 ⎠ 2
2 2⎛ y ⎞ ⎛ 1 y ⎞ 2
2F1 + 1 −2F1 ⎜ ⎟− ⎜ ⎟ = 0
⎝y2 ⎠ ⎝ y1
⎠
( )
2 ⎛y ⎞ 2 2 ⎛ y ⎞ 2
2F1 + 1 ⎜ ⎟−2F1 − ⎜ ⎟ = 0
⎝ y1 ⎠ ⎝ y1
⎠
( )
3
⎛y ⎞ 2 2 ⎛y ⎞ 2 2
⎜ ⎟ − ( 2F1 + 1) ⎜ ⎟+
2F1 = 0
⎝ y1 ⎠ ⎝ y1
⎠
This can be rewritten
as
2
⎡ y2 y ⎤
2 2 y2
⎢ 1 ⎥
y1 y1 y1
⎛
⎜
⎢⎣⎝ ⎞
⎟
⎠
+
⎡
−2F ⎢
⎥⎦⎣
⎤
− 1⎥= 0
⎦
y
1 0 y y uniform flow.
2 ∴ − = ∴ 2 = 1
y1
2
⎛y ⎞ 2 y2
2
⎜ ⎟ + − 1 =
y1 y1
⎝ ⎠
2
y211 2
Hence = = 1+ 8F1 −1
y12 2
2
2
3
2F 0 a quadratic equation.
⎛ ⎞ − 1+ 1+ 8F
⎡ ⎤
⎜ ⎟
⎝ ⎠
⎣ ⎦
y2 1 ⎡ ⎤
=
⎢
1+ 8F
2
−1
y ⎣
1 ⎥
1 2
⎦
2
(28.1)
⎛ V1
⎞
in which y2, y1 are sequent and initial depths respectively and F= 1 ⎜ ⎟ is the initial
⎜ gy ⎟
⎝ 1 ⎠
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

Hydraulics Prof. B.S. Thandaveswara
1965, proposed the following momentum equation taking into consideration the
integrated shear force.
Indian Institute of Technology Madras
3
⎛y ⎞
⎜
2 y
⎟ − 2 ⎡ − + ⎤
⎢
1 ε 2F
2
⎥
+ 2F
2
=0
⎜ ⎟ ⎣ ⎦
⎝ y
1 1
1 ⎠ y1
In which ε is the non dimensional integrated shear force, given by
γ y
2
function of Froude number. Pf is the integrated shear force.
P
f
2
1
(28.2)
and is a
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.
y2
___
y1
14
12
10
8
6
4
2
0
Eq. 28.1
Eq. 28.3
Eq. 28.2
Belanger
Rajaratnam
Sarma and Newnham
2 4 6 8 10
Fig. 28.5 - Variation of sequent depth ratio
F1

Hydraulics Prof. B.S. Thandaveswara
Sarma and Newnham 1975 introducing the momentum coefficient ( j 1.045 β = ) for the
Indian Institute of Technology Madras
non uniform velocity distribution obtained the following modified momentum equation
y2 1 ⎡ ⎤
=
⎢
1 + 10.4 F
2
−1
y ⎣
1 ⎥
1 2
⎦
(28.3)
In Eqn. 28.3, a value of β j 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.