Download 30.2 Pre entrained hydraulic jump (PHJ)

Hydraulics
Prof. B.S. Thandaveswara
30.2 Pre entrained hydraulic jump (PHJ)
The details so far presented are for the normal hydraulic jump where the approaching
supercritical flow is free from air entrainment. In the hydraulic jumps formed at the foot
of high head structures the approaching supercritical flow is self aerated and such types
of jumps with pre entrainment are called ' pre entrained Hydraulic Jumps ' ( PHJ ).
Earlies studies have been mostly empirical. Douma (1943 ), Gumensky and Yevdjevich
and Levin ( 1953 ) studied the flow characteristics with particular reference to field
observations.
Douma derived a modified formula on the assumption that the air entrained jet has a
lower density but moves with the same velocity as computed by Hall . Hall's test
revealed that the effect of air entrainment is to increase the chute velocity as well as to
the conjugate depth given by the equation
2
V1
y
2
2
y 2 = (1 − C ) y 1 + 2(1 − C ) y 1
[1 − (1 − C ) 1 ]
g
y2
In which C is computed by the equation
C = 10
2
0.2 V
−1
gR
In which V is the mean velocity, R is the hydraulic mean radius.
Gumensky in 1949 derived a similar formula in terms of the equivalent ' solid ' depth and
the actual velocity at entry ( which is not necessarily equal to the values computed on
the assumption that no entrainment occurs ). Yevdjevich and Levin in 1953, presented
an empirical equation for sequent depth
y 2 for hydraulic jump in sloping channels
*
given by
V
r
2
y =
q 1 = y F
2
1 1
g
(1 − C )
*
* *
2
Indian Institute of Technology Madras
2.5 (1 − C )
1
(1 − C )
2
Hydraulics
Prof. B.S. Thandaveswara
In which C
1
and C
2
are the mean air concentrations before and after the jump F1 ,
*
is the initial Froude number . The factor r2 depends on the assumed shape of the
concentration distribution and r2 = 2.0, 2.5, and 3.0 for rectangular, parabolic and
triangular distribution respectively. They concluded that the principal effect of air
entrainment in stilling basins is bulking which leads to considerably greater and more
efficient energy dissipation. In stilling basins the change of slope (to a flatter one)
reduces air concentration and air gets released.
Frankovinc in 1953 concludes that the greater part of energy loss occurs in the channel
itself than through the hydraulic jump in the stilling basin. This loss accounts for 75 %
which means that the depth of the mixture is twice as great as the depth corresponding
to water velocity.
Rajaratnam in 1962, covered a small range of Froude number ( 2.6 to 3.59 ). He derived
a theretical equation for sequent depth ratio for PHJ on a horizontal floor given by
φ13 - [ ( 1 -C ) + 2.52(1 − C )F1' 2 ] φ1 + 2.60(1 − C )2 F1'2 = 0
T
y2
'
In which φ1 = * and F1 =
dT
*
T
V
gdT
, C
T
*
T
*
is transitional mean air concentration of the
approach flow. He computed the sequent depth ratios for different values of CT . He concluded
that φ1 depends with CT for a given value of Froude number.
Rajaratnam formulated an approximate equation for the energy loss for pre entrained jump and
given by
(
El
E1
1−C
T
)
=
Indian Institute of Technology Madras
'2
F1
*
η 12
2
2φ1
⎡
⎤
⎢
⎥
2
⎢⎛ θ Ψ 2 ⎞
⎥
φ1
1
⎜
⎟
1
−
⎢
⎥ −
2
⎜
⎟
2
⎢⎝
⎥
⎠ ⎛ CT ⎞
⎢
⎥
⎜⎜ 1 −
⎟⎟
2
⎝
⎠
⎣⎢
⎦⎥
⎡
⎛ θ Ψ 2 ' 2 ⎞⎤
CT
⎢θ 1 (1 −
) + ⎜ 1
F1 ⎟ ⎥
⎜ 2
2
* ⎟⎥
⎢⎣
⎝
⎠⎦
⎡
CT ⎤
(1
)⎥
φ
−
θ
−
⎢ 1
1
1
⎢⎣
⎥⎦
Hydraulics
Prof. B.S. Thandaveswara
In which η 1 =
d
(1 − CT ) dT
, Ψ is proportionality factor relating the mean velocity, θ 1 is the
percentage of discharge in the lower region and is taken as 0.98. It can be seen from the above
equation that the relative energy loss is a function of C
'
and F1 .
*
T
Rajaratnam has made two assumptions while deriving the above Equation. viz: ( i ) the
approach flow is uniformly aerated and ( ii ) the contribution of momentum to the
supercritical flow by the upper region of the approaching flow is negligible. While
computing he assumed that the mean velocity of the aerated flow is proportinal to the
mean velocity of non -aerated flow and using the data of Straub and Anderson it was
found equal to 1.12.
Rajarathnam defined air pumping capacity as
β=
The air pumping capacity
Qa
Airflow rate
C
=
=
Q w waterflow rate 1 − C
β varies with x. The maximal and β is given by
β max = 0.18 ( F1 − 1)
1.245
Indian Institute of Technology Madras