Journal of Hydraulic Research Vol. 44, No. 5 (2006), pp. 715–717
© 2006 International Association of Hydraulic Engineering and Research
Discussion
Exact solutions for normal depth problem
By PRABHATA K. SWAMEE and PUSHPA N. RATHIE, Journal of Hydraulic Research,Volume 42, 2004, Issue 5, pp. 541–548
Discussers:
B. ACHOUR, Professor, Research Laboratory in Subterranean and Surface Hydraulics, Biskra University,
B.P. 145, R.P., 07000, Algeria. E-mail: bachir.achour@larhyss.net
A. BEDJAOUI, PhD Student, Research Laboratory in Subterranean and Surface Hydraulics, Biskra University,
B.P. 145, R.P., 07000,Algeria
The dimensionally consistent uniform flow relationship Q = ϕ
(S
o
, ε, A, R, ν) can be established by the only use of the Darcy–
Weisbach formula, thereby, accounting for the friction factor
f according to Colebrook–White equation. Nevertheless, as it
will be shown, the discussers’ relation (1.19) differs from the
authors’ equation (5) in the first term in parenthesis. This may
be explained by the restrictions which led to the establishment of
Hager’s inequality, numbered (2) by the authors, along with 1.5%
of deviation between friction factor values for complete turbulent
state and transitional state flow was assumed. Moreover, the same
inequality (2) is theoretically valid as long as Reynolds number
Re varies within the confined range 1 × 10
4
to 1 × 10
7
. Further-
more, relating Chezy’s constant C to the known characteristics
of a chosen referential rough channel, the relative normal depth
can be determined using authors’ explicit equations.
Basic equations
The Darcy–Weisbach formula and the Colebrook–White
equation are expressed, respectively, by:
S
o
=
f
8g
P
A
3
Q
2
(1.1)
f
−1/2
=−2 log
ε
14.8R
+
2.51
Re
√
f
(1.2)
where Q is the discharge, S
o
is the energy slope, ε is the absolute
roughness, P is the wetted perimeter, A is the water area, R is the
hydraulic radius, ν is the kinematic viscosity, g is the acceleration
due to gravity, and Re is the Reynolds number. The latter is
defined by:
Re =
4Q
Pν
(1.3)
For any shape of channel section, the geometric elements
A and P can be written, respectively, as:
A = L
2
A
∗
(1.4)
P = LP
∗
(1.5)
in which L is the linear dimension such as the bed width b of
a rectangular channel or the diameter D of a circular section,
etc. Both of the non-dimensional parameters A
∗
and P
∗
depend
solely on the relative normal depth. Inserting Eqs (1.4) and (1.5)
into Eq. (1.1) and rearranging, results in:
L = (f/8)
1/5
(Q/
gS
o
)
2/5
(P
∗
/A
∗3
)
1/5
(1.6)
whereas the combination of Eqs (1.3) and (1.5) leads to:
Re =
4Q
LP
∗
ν
(1.7)
Referential rough channel
With the subscript “r” we refer to a referential rough chan-
nel characterized by ε
r
/R
r
= 0.148 as the arbitrarily assigned
relative roughness value. Moreover, assuming a complete tur-
bulent state flow, the friction factor f
r
is given by Eq. (1.2) for
R
r
→∞, implying f
r
= 1/16. Thus, with the aid of Eqs (1.4)
and (1.5), Eq. (1.1) gives the following expression of the linear
dimension L
r
:
L
r
= (128)
−1/5
(
Q
r
/
gS
o,r
)
2/5
(P
∗
r
/A
∗3
r
)
1/5
(1.8)
According to (1.7), the Reynolds number Re
r
can be expressed as:
Re
r
= 4Q
r
/(L
r
P
∗
r
ν) (1.9)
On the other hand, eliminating Q
r
from Eqs (1.8) and (1.9),
leads to:
Re
r
= 32
√
2
gS
o,r
L
3
r
ν
(A
∗
r
/P
∗
r
)
3/2
(1.10)
which can be rewritten, with the aid of Eqs (1.4) and (1.5), as
follows:
Re
r
= 32
√
2
gS
o,r
R
3
r
ν
(1.11)
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