Geoscience Reference
In-Depth Information
Eq. (3.51) is similar to the method adopted by Meyer-Peter and Mueller (1948). It
should be noted that the grain roughness coefficient
n
can be calculated using several
methods, such as
n
d
1
/
6
90
d
1
/
6
21.5 (Strickler, 1923),
n
=
/
=
/
26 (Meyer-Peter and
d
1
/
6
65
d
1
/
6
50
Mueller, 1948),
n
=
24 (Patel and Ranga Raju, 1996), and
n
=
20 (Li and
Liu, 1963; Wu and Wang, 1999). Here, the units of sediment sizes and
n
are m
and s
/
/
m
−
1
/
3
, respectively.
Inserting Eq. (3.51) into Eq. (3.48) leads to
·
n
3
/
2
n
)
3
/
2
n
)
3
/
2
=
(
+
(
(3.52)
Unlike the above Einstein's method, Engelund (1966) suggested the division of the
bed shear stress according to the energy slope and determined the grain and form shear
stresses as
τ
b
=
γ
R
b
S
f
,
τ
b
R
b
S
f
=
γ
(3.53)
where
S
f
and
S
f
are the parts of the energy slope corresponding to the grain and form
roughnesses, respectively.
Applying the equal velocity assumption and the Manning equations
U
R
2
/
3
b
S
1
/
2
=
f
/
R
2
/
3
b
S
1
/
2
f
R
2
/
3
b
S
1
/
2
f
n
, and
U
n
yields
S
f
n
/
2
and
S
f
n
/
2
.
n
,
U
=
/
=
/
=
S
f
(
n
)
=
S
f
(
n
)
Then substituting these two relations into Eq. (3.53) and using Eq. (3.49) results in
n
n
2
n
n
2
τ
b
=
τ
b
τ
b
,
=
τ
(3.54)
b
Inserting Eq. (3.54) into Eq. (3.48) leads to
n
2
n
)
2
n
)
2
=
(
+
(
(3.55)
Note that the exponents are 3/2 in Eq. (3.52), but 2 in Eq. (3.55). However, both
Einstein's and Engelund's methods give the following relation for the Chezy coefficient:
1
C
h
=
1
C
h
+
1
C
2
h
(3.56)
where
C
h
is the total Chezy coefficient; and
C
h
and
C
h
are the fractional Chezy
coefficients corresponding to the grain and form roughnesses, respectively.
3.3.3 Movable bed roughness formulas
Einstein and Barbarossa (1952), Engelund and Hansen (1967), and Alam and Kennedy
(1969) proposed empirical methods for separately calculating the grain and form resis-
tances to flow. Li and Liu (1963), Richardson and Simons (1967), and Wu and Wang
