Momentum equation - nptel - Indian Institute of Technology Madras

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