220
P. Liu
static pressure of the post jump section in unit time. Because the momentum
flowing into or out of a section in a unit time is equivalent to a force, in
this case, it can be considered that the total thrust of the flow on the section
before the jump is equal to the total resistance of the flow on the section after
the jump. By substituting the continuous equation
q = V h = V 1 h 1 = V 2 h 2 , we get
h 2
1
2
+
β 1
g
q 2
h 1
=
h 2
2
2
+
β 2
g
q 2
h 2
where q is single width flow. The hydraulic jump function J (h) can be
expressed as
J (h) =
h 2
2
+
β
g
q 2
h
Its curve with water depth is shown in Fig. 3.48. In the rapid flow area,
dJ /dh < 0; in the slow flow area, dJ /dh > 0; in the critical flow, dJ /dh = 0,
where h = h c , J = J min .
If approximately β 1 ≈ β 2 ≈ 1 is taken, it is obtained from the above
formula
h 2
h 1
=
1
2
1 + 8
q 2
gh 3
1
− 1
Because of formula
q 2
gh 3
1
=
V 2
1
gh 1
= Fr 2
1 , Fr 1 is the Froude number of the
flow in the section before the jump, which can be obtained by substituting
the above formula
h 2
h 1
=
1
2
1 + 8Fr 2
1 − 1
, η =
1
2
1 + 8Fr 2
1 − 1
This is the conjugate depth equation of free hydraulic jump in horizontal
rectangular channel derived by French hydraulician Blaise in 1860. Among
them, η = h 2 / h 1 is called the conjugate water depth ratio of hydraulic jump,
and the verification of this formula and the experimental results are shown in
Fig. 3.49. The formula shows that the conjugate depth ratio in the rectangular
open channel is a function of Froude number (Fr 1 ) of the section before the
jump. The water depth H 2 after the jump is an important basis for the design
of hydraulic stilling basin depth.