Journal ArticleDOI

Accuracy of Equivalent Roughness Height Formulas in Practical Applications

Benoît Camenen
01 Mar 2013-Journal of Hydraulic Engineering (American Society of Civil Engineers (ASCE))-Vol. 139, Iss: 3, pp 331-335
TL;DR: In this paper, the application and accuracy of roughness height formulas from a practical point of view are discussed, and the application of these relationships requires an iterative solution technique, which yields large discrepancies in the results.
Abstract: This paper presents a discussion on the application and accuracy of roughness height formulas from a practical point of view. Such formulas have been proposed to describe the equivalent roughness height for plane bed conditions on the basis of the Shields parameter. The application of these relationships requires an iterative solution technique. However, as this paper demonstrates, the roughness estimates are not always reliable because the application of the formulas yields large discrepancies in the results. DOI: 10.1061/(ASCE)HY.1943-7900.0000670. (C) 2013 American Society of Civil Engineers.

Introduction

• Estimating bedload transport is of great importance for engineering design and morphodynamic modelling; yet bedload transport rate predictions are based on insufficient understanding of the resistance to flow in alluvial channels.
• Most of the formulas were validated for fine sediments and high shear stresses.
• The flow seems to be affected by the bed load transport for these particular cases as much as in the sheet flow transport cases.
• The main objective of the present study is to highlight these problems induced by the iterative method.

Iterative procedure

• The iterative solution can be expressed using the relationship for the friction coefficient (see Eq. 12 in Camenen et al., 2006) and the definition of the Shields parameter (see Eq. 2 in Camenen et al., 2006) .
• Thus, the Shields parameter can be expressed 5 Author-produced version of the article published in J. Hydraul.
• The original publication is available at http://ascelibrary.org.

Graphical interpretation of the iterative procedure

• The iterative method will never induce a better result compared to the fitted curve because the two curves f and F −1 are increasing functions of θ.
• A slight underestimation is observed using the fitted curve.

Fig. 2 here

• In conclusion, it appears that the iterative method is not very accurate from a practical point of view.
• A solution does not always exist, and most of the time, two solutions coexist.
• If there is an error between the data and the fitted curve, it will necessarily be larger after employing the iterative method.

Tab. 1 here

• The iterative solution for a pair (k s , θ) 9 Author-produced version of the article published in J. Hydraul.
• The original publication is available at http://ascelibrary.org.
• As a matter of fact, the accuracy of the results (P 2) drops with 10% to 30% depending on the formula used (cf. table 1 and Fig Shields parameter, which are underestimated.

Conclusions

• Many formulas have been proposed to estimate the roughness height in the case of plane beds.
• All the previous authors proposed equations assuming that the total Shields parameter is already known.
• And reveal the main factors governing roughness, they are not applicable from an engineering point the view.
• The iterative formulas resulting from these equations yields very large scatter in the results.
• Finally, the use of the skin Shields parameter produces nearly as good results, apart for large values of the total Shields parameter, which are underestimated.

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On the accuracy of equivalent roughness height formulas
in practical applications
B. Camenen, M. Larson
To cite this version:
B. Camenen, M. Larson. On the accuracy of equivalent roughness height formulas in practical appli-
cations. Journal of Hydraulic Engineering, American Society of Civil Engineers, 2013, 139 (3), p. 331
- p. 335. �10.1061/(ASCE)HY.1943-7900.0000670�. �hal-00920719�

On the accuracy of equivalent roughness height
formulas in practical applications
Benoˆıt Camenen
∗†
and Magnus Larson
Abstract1
This paper presents a discussion on the application and accuracy of rough-2
ness height formulas from a practical point of view. Such formulas have been3
proposed to describe the equivalent roughness height for plane bed condi-4
tions based on the Shields parameter. The application of these relationships5
requires an iterative solution technique. However, as this note demonstrates,6
the roughness estimates are not always reliable since the application of the7
formulas yield large discrepancies in the results.8
Corresponding author, benoit.camenen@irstea.fr
Irstea, UR HHLY, Hydrology-Hydraulics Research Unit, 3 bis quai Chauveau, CP
220, F-69336 Lyon cedex 09, France
Dept. of Water Resources Eng., Lund University, Box 118, S-221 00 Lund, Sweden,
E-mail: magnus.larson@tvrl.lth.se
1
Author-produced version of the article published in J. Hydraul. Eng. (2013), 139(3), 331–335.
The original publication is available at http://ascelibrary.org DOI : 10.1061/(ASCE)HY.1943-7900.0000670

Keywords9
Roughness height, friction factor, p lane bed, sheet ﬂow, experimental data,10
predictive formula, iterative formula.11
Introduction12
Estimating bedload transport is of great importance for engineering design13
and morphodynamic modelling; yet bedload transport rate predictions are14
based on insuﬃcient understanding of the resistance to ow in alluvial chan-15
nels. Although the roughness height or resistance coeﬃcient are quite dif-16
ﬁcult to estimate, they remain fundamental parameters in bed shear stress17
and sediment transport calculations. Total resistance to the ﬂow in a river18
includes grain resistance, bedform resistance, resistance due to lateral and19
vertical channel irregularities, as well as dissipation due to the solid trans-20
port (Reckin g et al., 2008). Meyer-Peter and M¨uller (1948) introduced a lin-21
ear decomposition of the total resistance coeﬃcient, which may be rewritten22
as a linear decomposition of the total bed roughness (Soulsby, 1997; Came-23
nen et al., 2006).24
Excluding form drag, th e value of the equivalent roughness height k
s
still25
varies considerably depending on the conﬁ gu ration of the grains forming th e26
bed roughness or the dissip ation due to sediment transport. If k
s
is generally27
assumed to be proportional to a characteristic bed grain size (Bathurst,28
2
Author-produced version of the article published in J. Hydraul. Eng. (2013), 139(3), 331–335.
The original publication is available at http://ascelibrary.org DOI : 10.1061/(ASCE)HY.1943-7900.0000670

1985), several authors showed a signiﬁcant increase in k
s
transport occurs, even for coarse sediments (Camp bell et al., 2005; Recking30
et al., 2008). This increase in roughness is often explained for ﬁne sediments31
by the dissipation of ener gy in the sheet ﬂow layer that appears for large32
Shields parameter values (dimensionless bed s hear stress deﬁned such as33
θ = RI/[(s 1)d
50
], with R: hydraulic radius, I: energy slope, s: relative34
density of the sediment to the ﬂuid, and d: grain s ize). Sediments move35
collectively along the bottom mainly in a layer and k
s
δ
s
, where δ
s
is the36
thickness of the sheet ﬂow layer (Wilson, 1989). This movement induces an37
eﬀective roughn ess ratio k
s
/d, where d is the grain size diameter, up to 5038
(Wilson, 1966; Nnadi and Wilson, 1992; Sumer et al., 1996). Based on a39
large data set, Recking et al. (2008) found a signiﬁcant diﬀerence in the ﬁt40
of a logarithmic law for the friction coeﬃcient depending on the p resence of41
sediment movement, which can be described in terms of a roughness, such42
as the k
s
/d-ratio, that varies from appr oximately 1 for a ﬁxed bed to 3 for43
a moderate sediment transport rate.44
Generally, this phenomenon was expressed using a Shields parameter45
value θ over a critical value f or the upper plane bed regime, where k
s
is not46
only a function of the median grain size d
50
but also of θ (Wilson, 1966; Nnadi47
and Wilson, 1992; Yalin, 1992; van Rijn , 1993; S umer et al., 1996; Bayram48
et al., 2003; Camenen et al., 2006) or of bedload transport rate, if the latter49
is assumed to be directly related to the Shields parameter (Meyer-Peter50
3
Author-produced version of the article published in J. Hydraul. Eng. (2013), 139(3), 331–335.
The original publication is available at http://ascelibrary.org DOI : 10.1061/(ASCE)HY.1943-7900.0000670

and M¨uller, 1948). Most of the formulas were validated for ne sediments51
and high shear stresses. Sumer et al. (1996) observed that the roughness52
height may be a function of a d im ensionless settling velocity
s
= W
s
/u
53
(where W
s
is th e settling velocity and u
the shear velocity). Camenen54
et al. (2006) also showed th at large values on the eﬀective roughness ratio55
k
s
/d
50
can be observed for lower values of the Sh ields parameter. They56
proposed a new relationship f or the critical Shields parameter for the upper57
plane regime, which is a function of a dimensionless settling velocity W
s
=58
[(s 1)
2
/()]
1/3
W
s
(where g the acceleration due to gravity, and ν the59
kinematic viscosity of the ﬂuid) and a Froude number F
r
= U
c
/
gh (where60
U
c
the steady current velocity and h the water depth). Then, for a low61
Froude number and coarse sediment, large eﬀective roughness ratios are62
also obs er ved. The ﬂow seems to be aﬀected by the bed load transport for63
these particular cases as much as in the sh eet ow transport cases.64
However, one important asp ect of these previous studies is that all the65
equations presented in Camenen et al. (2006) have been ﬁtted kn owing the66
total shear stress, since it is directly computed from energy slope measure-67
ments. Therefore, in p ractical applications, as the total S hields parameter68
is an unknown, these relationships induce an iterative method for the com-69
putation of th e roughness height and the total Shields parameter. Two70
main problems may occur: (1) a solution does not always exist d epending71
on the proposed equation (or more than one solution exist), and (2) the72
4
Author-produced version of the article published in J. Hydraul. Eng. (2013), 139(3), 331–335.
The original publication is available at http://ascelibrary.org DOI : 10.1061/(ASCE)HY.1943-7900.0000670

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References
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07 Jun 1948
TL;DR: In this article, an attempt is made to derive an empirical law of bed-load transport based on recent experimental data and the results and interpretation of tests already made known in former publications of the Laboratory for Hydraulic Research and Soil Mechanics at the Federal Institute of Technology, Zurich.
Abstract: In the following paper, a brief summary is first of all given of the results and interpretation of tests already made known in former publications of the Laboratory for Hydraulic Research and Soil Mechanics at the Federal Institute of Technology, Zurich. After that, an attempt is made to derive an empirical law of bed-load transport based on recent experimental data. We desire to state expressly that by bed-load transport is meant the movement of the solid material rolling or jumping along the bed of a river; transport of matter in suspension is not included.

2,182 citations

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[...]

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Book
01 Jan 1993

1,962 citations

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[...]

• ...…+ z0)/R ln ((R + z0)/z0)− 1 ]2 Uc 2 (s− 1)gd50 (1) Employing the relationships proposed by different authors (Wilson, 1966; Yalin, 1992; van Rijn, 1993; Sumer et al., 1996; Bayram et al., 2003; Camenen et al., 2006), which can be expressed as ks/d50 = f(θ), equation 1 can easily be…...

[...]

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TL;DR: In this article, a general procedure and errors and sensitivities properties of water and sand Currents Waves Combined waves and currents Threshold of motion Bed features Suspended sediment Bedload transport Total load transport Morphodynamics and scour Handling the wave-current climate case studies
Abstract: Introduction, including sections on a general procedure and errors and sensitivities Properties of water and sand Currents Waves Combined waves and currents Threshold of motion Bed features Suspended sediment Bedload transport Total load transport Morphodynamics and scour Handling the wave-current climate Case studies

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