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Chapter 3 - Theoretical considerations
a) Wall influence
The wall influence is taken into account based on EINSTEIN (1934) and the friction coefficient of
the bed can be computed according to Table 3.3 for the straight inlet reach.
Measured data
Calculated data for subareas
Target cells Shear
A 0
A w
A tot based on subsections avg stress
Test Q S e h A U P 0 f 0 R h0 P w d w,90 f w R hw f m R h0 /f 0 R hw /f w R hm /f m R hm /f m τ 0
- m3/s % m m2 m/s m - m m mm - m - m m m m N/m2
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19)
B01b 0.15 0.82% 0.15 0.15 0.98 1.00 0.10 0.15 0.15 0.1 0.02 0.03 0.08 1.48 1.48 1.48 1.48 11.8
B01c 0.18 0.96% 0.17 0.17 1.07 1.00 0.11 0.16 0.17 0.1 0.02 0.03 0.08 1.51 1.51 1.51 1.51 15.2
B01d 0.21 0.82% 0.19 0.19 1.09 1.00 0.10 0.18 0.19 0.1 0.02 0.03 0.08 1.84 1.84 1.84 1.84 14.6
C01b 0.15 0.97% 0.15 0.15 1.00 1.00 0.11 0.14 0.15 0.1 0.02 0.03 0.09 1.32 1.32 1.32 1.32 13.6
C01c 0.18 0.90% 0.17 0.17 1.05 1.00 0.10 0.16 0.17 0.1 0.02 0.03 0.08 1.57 1.57 1.57 1.57 14.3
C01d 0.21 0.94% 0.19 0.19 1.11 1.00 0.11 0.18 0.19 0.1 0.02 0.03 0.08 1.68 1.68 1.68 1.68 16.4
D01b 0.15 0.59% 0.16 0.16 0.93 1.00 0.08 0.15 0.16 0.1 0.02 0.03 0.07 1.84 1.84 1.84 1.84 8.8
D01c 0.18 0.61% 0.19 0.19 0.99 1.00 0.08 0.17 0.19 0.1 0.02 0.04 0.07 2.06 2.06 2.06 2.06 10.2
D01d 0.21 0.76% 0.20 0.20 1.08 1.00 0.09 0.18 0.20 0.1 0.02 0.04 0.07 1.94 1.94 1.94 1.94 13.5
Table 3.3: Computation of the friction factor solving simultaneously for all hydraulic radii (Tests without macroroughness)
The discharge Q, the energy slope S e
as well as the characteristic wall roughness size d w,
90
are
supposed to be known. The wall roughness was chosen d w, 90 = 0.1 mm to obtain the roughness
of the plastic walls. The cross-section A = B⋅
h, the velocity V = Q⁄
A, and the wetted
perimeters = B , P w
= h were computed first.
P 0
GESSLER proposed to compute the wall friction with KEULEGAN’S (1938) relationship and to
admit a hydraulically smooth bed. Since the walls in the used channel are much smoother than the
bed, the bed friction factor was determined in the present study with KEULEGAN’S (1938) relationship:
0.0251
f 0
= 2.21 + 2.03 ⋅ log --------------
⎝
⎛ ⎠
⎞
S e
–2
(3.26)
The friction factor for smooth walls is computed with:
--------
1
2.51
= – 2 ⋅ log -----------------------
⎝
⎛ Re w
⋅ f ⎠
⎞
w
f w
(3.27)
For rough walls or banks the following friction factor can be used:
f w
= 2.21 + 2.03 ⋅ log ------------
⎝
⎛ ⎠
⎞
(3.28)
The average bed friction based on the previously computed bed and wall friction can be obtained
with:
R w
d w,
90
P
f 0
⋅ f 0
+ 2 ⋅P w
⋅ f w
m = --------------------------------------------
P 0
+ 2 ⋅ P w
–2
(3.29)
page 32 / November 9, 2002
Wall roughness effects on flow and scouring