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Chapter 3 - Theoretical considerations
where Q 0
is the part of the discharge acting on the bed. The term ( Q 0
⁄ Q) ⋅ h accounts for the
influence of the side walls. This term can be replaced by R h0 . The coefficient ( K 0
⁄ K S
) 1.5 introduces
the bedforms in the equation. MEYER-PETER & MÜLLER did not give a formula to determine
K 0 . If the water depth h and the discharge Q are known, the influence of the bedforms can
be computed with
K
K S
⋅ K w
⋅ B 0.67
0
= ---------------------------------------------------------------------------------
(3.47)
1.5
1.5 1.5
[ B ⋅ K w
+ 2 ⋅h ⋅( K w
– K S
)] 0.67
is the roughness coefficient computed with Strickler’s formula (eq. 3.11)
K S
4) Smart & Jäggi (1983)
SMART & JÄGGI (1983) extended the method of MEYER-PETER & MÜLLER (1948) to channels
and rivers with steeper slopes. They based their study on an additional set of tests in a 10 and
20 cm wide flume with bed slopes between 3 and 20 %. Based on their results and on the tests of
MEYER-PETER & MÜLLER, they proposed the following relation to compute the sediment transport
capacity:
Q b
B 4 ⋅ ρ s
----------- ⎛
d
------ 90 ⎞ 0.2 (3.48)
s – 1 ⎝ ⎠
R 0
V S 1.6 θ
1 cr
⋅ ( s – 1) ⋅ d
= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⎛ – -------------------------------------- m ⎞
⎝ ⋅ S ⎠
d 30
R 0
which is equivalent to
d 90
V
Φ = 4 ⋅ ------ ⋅ ⎛------⎞ 0.2 ⋅ S 0.6 ⋅ θ
V∗ ⎝ ⎠
0.5 ⋅ ( θ–
θ cr
)
d 30
(3.49)
With θ cr
= 0.050 , the authors used a slightly higher value compared to MEYER-PETER &
MÜLLER (1948). The sediment transport rate depends only weakly on the coefficient
( d 90
⁄ d 30
) 0.2 . Neglecting this coefficient (put to 1.05) and assuming a relative sediment density of
s = 2.68 , the sediment transport rate can be computed with the following simplified formula:
Q b
2.5 B q S 0.6 d
= ⋅ ⋅ ⋅ ⋅⎛S
– -------------------- m ⎞
(3.50)
⎝ 12.1 ⋅ h ⎠ m
The correlations of equations 3.48 and 3.50 are of r 2 = 0.97 respectively r 2 = 0.95 and standard
errors of s E
= 66 % respectively s E
= 22 % compared with the tests of MEYER-PETER &
MÜLLER.
HUNZIKER (1995) performed another extension of MEYER-PETER & MÜLLER’S formula, allowing
the computation of the sediment transport rate for each sediment fraction of the bed material (see
3.4.3/5)).
5) Hunziker (1995)
In the frame set of his study on sediment transport by grain size fractions, HUNZIKER (1995) analyzed
the results of ZARN (1997) which were performed in a 3 m wide and 26 m long channel. He
further analyzed the results of MEYER-PETER & MÜLLER (1948) and observed a systematic over-
page 38 / November 9, 2002
Wall roughness effects on flow and scouring