3 Hydrodynamics
223
where C is the scale factor. For the average velocity V , it can be obtained
from the total flow continuity equation
V =
V 1 h 1
h 1 + y
The equation of motion of the particle in X-direction is
dx
dt
=
C V 1 h 1
h 1 +
1
2 ( f − g)t 2
Integrate along the above formula along t, that is
L j = C V 1
√
2h 1
f − g
tg
−1
(
f − g
2h 1
t j )
As shown in Fig. 3.50, when t = t j , y = h 2 -h 1 , we get
t j =
2(h 2 − h 1 )
f − g
Let A =
C
√
2
√
f /g−1
, after finishing, get
L j
h 2
= AFr 1
tg −1 √ η − 1
η
where η is the conjugate depth ratio and the constant A is determined
by experiments. Based on the analysis and comparison of the experimental
curves given by USBR, it is suggested that a be 6.55, get
L j
h 2
= 6.55Fr 1
tg −1 √ η − 1
η
The comparison between the formula and the experimental results of the
Bureau of reclamation of the United States Department of internal affairs is
shown in Fig. 3.51. The results show that the relative error between the semi
theoretical formula based on the principle of projectile and the experimental
results is between 2% and 3% in the range of Fr 1 = 2–13.