(1)
Liu Shanjun1 Xu Weilin Wang Wei
The State Key Hydraulics Lab. of High Speed Flows, Sichuan University
24, South Segment No.1, Ring Road No.1, Chengdu City, 610065, China
028-5404031, E-mail: drliushanjun@sina.com
1Doctoral
Student, Associate Prof. & State Certified Supervisal Engineer of China
Abstrsct:
It is important and frequent for engineers and researchers to compute the
critical water depth, the normal water depth, the water depth at contractive
cross section under a sluice gate and the sequent depth of hydraulic jump in
hydraulic engineering. How to efficiently compute these factors remains a
problem for many years. Diagrammatic method and trial method is known well by
almost all staffs but the precision of such computation methods is not assurable
sufficiently with much miscellaneous work. In this paper, the authors present a
set of iterative formula to compute those water depths in trapeziform open
channels and the astringency of these formulas is proved out strictly with
algebra skill. Now one can compute
those water depths easily, especially for program designing.
Keywords: Iterative Formula, Astringency, Critical Water Depth, Normal Water Depth, Water Depth at Contractive Cross Section under Sluice Gate, Sequent Water Depth of Hydraulic Jump
In this paper, the authors present a set of
iterative formula to compute the critical water depth, the normal water depth,
the water depth at contractive cross section under a sluice gate and the sequent
depth of hydraulic jump in trapeziform open channels, which is very useful in
hydraulic engineering. And the most
important thing is that the astringency of these formulas is proved out strictly
with algebra skill, so any initial value is available for the iterative formula.
Now one can compute those water depths easily and confidently.
The critical water depth is governed by the equation
as follows:
(1)
where g the gravity
acceleration constant, Q the
discharge, b the bottom width, m1, m2 the bank slope respectively (see Fig.1).
Now one can rewrite the Eq.(1) into the form as follows:
(2)
Eq.(2) is the iterative formula for critical water depth. It is very important to prove the astringency of iterative formula, or else it may not be convergent that one can’t employ such a formula to compute the critical depth confidently. The astringency about Eq.(2) demands that the simple differential on variable h is less than 1, so let’s differentiate Eq.(2):
One must mention that the variables are governed by Eq.(2), so one should take the place of above expression by Eq.(2) to simplify it, which is the key to prove the astringency. Thereafter:
let
, then
Now it can be concluded that the astringency of the iterative formula for critical depth as Eq.(2) is assurable strictly, where any assumption is not used.
The normal water depth for uniform flow is governed by the following equation:
(3)
where i the slope along the
channel, n the roughness.
Rewrite
Eq.(3) into the iterative formula:
(4)
and differentiate Eq.(4) with variable h:
It can also be concluded that the astringency of the iterative formula for normal depth as Eq.(4) is assurable strictly without any assumption.
The rapidly varied energy equation through a sluice gate can
be established easily
where E the total potential head(approach head before sluice), hc
the water depth at contraction-cross (see Fig.2),
Similarly, rewrite Eq.(5) into its iterative
formula as follows:
and then differentiate Eq.(6) with variable hc
obviously,
and on anther side, it
is no doubt on almost every cases that the condition E>1.5h is expectable for outflow through a sluice gate, thus
It can also be concluded that the astringency of
the iterative formula for water depth at contraction-cross under a sluice gate
as Eq.(6) is assurable. The hydraulic jump takes place when the rapidly
varied flow transform suddenly into the gradually varied flow. Both the initial
depth and sequent depth of hydraulic jump are governed by:
where J the function of hydraulic jump which can be calculated by
employing the initial depth and initial cross section area.
Rewrite it into its iterative formula as follows:
Similarly, differentiate Eq.(8) with variable h:
where
Obviously,
In order to expect the
upper limit of
thereafter,
It can also be concluded that the astringency of
the iterative formula for sequent water depth of hydraulic jump as Eq.(8) is
assurable strictly without any assumption. Iterative formula to compute a complex equation
is very convenient but the certificate of its astringency is not easy, which
is why the iterative formula to compute hydraulics does not find wide
application. The attempt to prove out the astringency may be failed unless one
must mention the intrinsic relation about the factors and pay much algebra skill
to simplify the expression. In this paper, the authors present a set of
iterative formula to compute such water depths as the critical depth, the normal
depth, the contractive cross section depth and the sequent depth of hydraulic
jump in a trapeziform channel and the astringency of these formulas is proved
out strictly without any assumption, any initial value is available for the
iterative formula. Now one can employ the iterative formula to compute such
water depth easily instead of diagrammatic method and trial method, especially
for program designing. 
(5)
the coefficient of discharge.
(6)
5 ITERATIVE FORMULA OF
SEQUENT DEPTH OF HYDRAULIC JUMP
(7)
(8)
(employing
Eq.(6) to simplify the expression)
the surface width.
, one can employ the expression
and mention that Froude Number
after hydraulic jump is less than 1, thus
( A/Bk
< h )
6 CONCLUSIONS