# Dune Contribution to Flow Resistance in Alluvial Rivers

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{50}/y with d

_{50}being bed material average grain size and y water depth). Julien and Klaassen [21] made a field investigation on the Meuse River and the Rhine River during large floods, and discovered that dune steepness remains relatively constant with discharge, and suggested a linear proportion between wavelength and water depth. The proposed linear coefficient differs from the one empirically proposed by Yalin [22], which either takes into account flume and river data, or theoretically derived [15]. Agarwal et al. [23] stressed that, for specific range of relative depth, dune spacing may greatly differ from what has been suggested by Yalin [15,24], or by Julien and Klaassen [21]. Aberle et al. [25] used a statistical approach to investigate bed forms during different flood conditions in the Elbe River showing how statistical parameters may be used to predict the flow-dependent bed roughness.

## 2. Flow Resistance

_{s}the specific weight of immersed sediment, d

_{s}the representative sediment diameter, Δ the dune height, Λ the dune length, and g the gravity acceleration. Therefore, it must hold a dimensionless general relationship of dependency on the following seven dimensionless parameters:

## 3. Grain Contribution to Flow Resistance

_{s}′ is the equivalent grain roughness. The vertically-averaged velocity U corresponds to the local velocity u at the relative depth z/y = e

^{−1}= 0.368. By integration over the flow depth, it results:

_{s}’. According to different Authors, k

_{s}′ is proportional to a characteristic sediment size d

_{s}, with subscript “s” being equal to 35, 50, 65, 84, 90 (i.e., the percentage of the finer particle size distribution by weight). Typically, k

_{s}′ ranges between 1.25 d

_{35}and 5.10 d

_{84}[1,2,3,4,5,6,7,8,9,16,30,31,32,33], whereas Millar [34] concluded that in gravel streams there is no significant difference between using d

_{35}, d

_{50}, d

_{65}, d

_{84}or d

_{90}. In the present analysis, the mean grain size diameter d

_{50}is assumed as a characteristic sediment size. Different values for k

_{s}were considered and the most appropriate value resulted k

_{s}= 2⋅d

_{50}.

## 4. Sand Dune Contribution to Flow Resistance

**P**represents the pressure vector acting on the boundary of the fixed control between the two consecutive cross sections 1 and 2,

**G**represents the mass force vector acting on the control volume and

**M**represents the momentum flux vector through the boundary. The x-component of the previous equation gives:

_{1}, P

_{2}, P

_{L}the pressure force acting on the upstream cross section, downstream cross section and on the lee side of the dune, respectively. T

_{L}is the shear stress on the lee side of the dune, M

_{1}and M

_{2}the momentum flux across the upstream and downstream cross sections 1 and 2, respectively, ω the angle between the lee dune side and the horizontal, and σ the average bed slope.

_{L}cos(ω)) is already included in the grain roughness component of the total resistance to flow. Introducing the momentum coefficient β (accounting for non-uniform velocity distribution over the cross section), the momentum balance equation per unit width of the channel becomes:

_{i}is water depth at cross section (i = 1,2). The energy balance equation applied between cross sections 1 and 2 results:

_{i}(i = 1,2) is the bed level and α

_{i}the Coriolis coefficient (i = 1,2) referred to cross sections 1 and 2. For the sake of simplicity from now on α

_{1}= α

_{2}= β

_{1}= β

_{2}= 1. The location of the stagnation point is not known a-priori and, taking into account for a mild slope of the toss face of the dune, it may be assumed z

_{1}−Δ = z

_{2}. Moreover, considering a reference dune bed pattern having a vertical dune lee side (i.e., ω = 90°), Equation (15) becomes:

_{Δ/y}is:

^{P}″ indicates the bed form energy loss according to the pressure pipe flow approach. In this case:

#### Empirical Coefficient for Bed Form Drag

_{s}= 2·d

_{50}), it is possible to determine the empirical coefficient κ. A selection of 132 field data collected on 7 different sand-bed rivers with dune bed forms is considered (see Table 1). It refers to the Savio and Fiumi Uniti rivers [39], Calamus River [40], Missouri (data from Shen 1978, as reported by Brownlie [41]), Jamuna and Parana [42], Bergsche Maas (data from Adriaanse 1986, as reported by Julien [42]), and Meuse [42]. The database reports for each data set the number of measurements N, and the range of the following observed parameters: water discharge Q, mean depth y, mean flow velocity U, measured energy gradient S, Froude number F, mean grain size diameter d

_{50}, dune height Δ and dune length Λ. Despite the relative scatter of field data, Figure 3 shows how the drag coefficient decreases with increased dune steepness. Some outliers are present, corresponding to few data recorded on the Jamuna and Missouri rivers [42], characterized by very low values of dune steepness (i.e., Δ/Λ < 0.01), and filtered in the fitting procedures. The best fitting equation is:

_{s}. Among the selected field data (river A–G, Table 1), Fiumi Uniti, Savio, Missouri and Meuse River reported information about sediment gradation or d

_{90}. Figure 4 shows a comparison between the drag coefficient κ obtained considering equivalent roughness k

_{s}= 2·d

_{50}or k

_{s}= 3·d

_{90}.

## 5. Model Validation and Sensitivity Analysis to the Bed Form Geometry and Skin Roughness

_{s}′ = 2·d

_{50}, and S″ is calculated using Equations (32) and (24):

_{s}’ = 2·d

_{50}(see column 9 in Table 1 and Table 2); S″ is calculated with Equation (33) along with parameters m and n from Equation (37), and eventually bed form steepness using Equations (35) and (36).

_{*}’ = (g S′ y)

^{0.5}in the range of about 0.02–0.06 m/s, and the dimensionless conveyance coefficient C = (g S y)

^{0.5}respectively correspond to approximately 7–14 and 22–27.

_{s}′ = 1.0·d

_{50}[46] and a different approach, reflecting Manning–Strickler Equation (38), are considered:

_{s}′ = 1 d

_{50}), field data included within the ±30% error band reduce from 93.5% to 90.7%, and the data included within the ±20% error band decrease from 68.4% to 65.5%. On the contrary, using the Manning–Strickler resistance formula (Equation (38)) only 88.2% of the predicted energy slope are within the ±30% band error and the data included within the ±20% band error are reduced to 65.5% (see Figure 6).

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

C′ | dimensionless conveyance Chezy coefficient related to the skin roughness |

d_{s} | representative sediment diameter |

F | Froude number |

g | gravity acceleration |

G | mass force vector acting on the control volume of streamflow |

k | Von Karman’s constant |

k_{s}’ | equivalent grain roughness |

L | river reach length |

m | fitting parameter of empirical correction function κ |

M_{1}, M_{2} | momentum flux in stream wise direction |

M | momentum flux vector trough the surface of the control volume of the streamflow |

n | fitting parameter of empirical correction function |

n′ | Manning coefficient related to grain roughness |

u | local flow velocity |

${{u}_{\ast}}^{\prime}$ | shear velocity related to the skin roughness |

U | mean flow velocity |

P_{1}, P_{2,}P_{L} | pressure force acting on the upstream, downstream cross section, and on the lee side of the dune, with respect to the control volume of the streamflow |

P | pressure vector acting on the boundary surface of the control volume of streamflow |

q | water discharge per unit with of the channel |

Re | Reynolds’ number |

S | friction slope |

S′ | friction slope due to the grain resistance |

S″ | friction slope due to the dune drag |

T_{L} | shear stress on the lee side of the dune |

y | mean flow depth |

$\tilde{y}$ | local water depth |

z | vertical elevation above the river bed |

Z | relative submergence |

α_{i} | Coriolis coefficient |

β | momentum coefficient |

γ_{s} | specific weight of immersed sediment |

Γ_{Δ/y} and Γ P_{Δ/y} | dune geometric correction function |

δ | dune steepness |

Δ | dune height |

ΔH | total head loss |

ΔH′ | head loss due to the grain resistance |

ΔH″ | head loss due to the dune drag |

ΔH^{P}″ | head loss due to the dune drag according to pressure pipe flow approach |

κ | empirical correction coefficient |

Λ | dune length |

ν | kinematic viscosity of water |

ρ | density of water |

σ | average river bedslope |

τ | total bed shear stress |

τ’ | grain shear stress contribution to the total bed shear stress |

τ″ | bedform shear stress contribution to the total bed shear stress |

ω | angle between the lee dune side and the horizontal |

## References

- Cao, Z.; Carling, P.A. Mathematical modelling of alluvial rivers: Reality and myth. Part 1: General review. Proceedings of the Institution of Civil Engineers. Water Marit. Eng.
**2002**, 154, 207–219. [Google Scholar] [CrossRef] - Wang, S.S.Y.; Wu, W. River sedimentation and morphology modeling—The state of the art and future development. In Proceedings of the 9th International Symposium on River Sedimentation, Yichang, China, 18–21 October 2004; pp. 71–94. [Google Scholar]
- Amoudry, L.O.; Souza, A.J. Deterministic coastal morphological and sediment transport modeling: A review and discussion. Rev. Geophys.
**2011**, 49. [Google Scholar] [CrossRef] - Maldonado, S.; Borthwick, A.G.L. Quasi-two-layer morphodynamic model for bedload-dominated problems: Bed slope-induced morphological diffusion. R. Soc. Open Sci.
**2018**, 5, 172018. [Google Scholar] [CrossRef] [PubMed] - Li, J.; Cao, Z.; Pender, G.; Liu, Q. A double layer-averaged model for dam-break flows over mobile bed. J. Hydraul. Res.
**2013**, 51, 518–534. [Google Scholar] [CrossRef][Green Version] - Iverson, R.M.; Ouyang, C. Entrainment of bed material by Earth-surface mass flows: Review and reformulation of depth-integrated theory. Rev. Geophys.
**2015**, 53, 27–58. [Google Scholar] [CrossRef] - Abril, J.B.; Altinakar, M.S.; Wu, W. One-dimensional numerical modelling of river morphology processes with non-uniform sediment. In Proceedings of the River Flow 2012, San Jose, Costa Rica, 5–7 September 2012; pp. 529–535. [Google Scholar]
- Einstein, H.A.; Barbarossa, N.L. River channel roughness. Trans. Am. Soc. Civ. Eng.
**1952**, 117, 1121–1146. [Google Scholar] - Engelund, F.; Hansen, E. A Monograph on Sediment Transport in Alluvial Streams; Teknisk Forlag: Copenhagen, Denmark, 1967; pp. 1–62. [Google Scholar]
- Van Rijn, L.C. Sediment Transport, Part III Bed Forms and Alluvial Roughness. J. Hydraul. Eng. ASCE
**1984**, 110, 1733–1755. [Google Scholar] [CrossRef] - White, P.J.; Paris, E.; Bettess, R. The frictional characteristics of alluvial streams: A new approach. Proc. Inst. Civ. Eng.
**1980**, 69, 737–750. [Google Scholar] [CrossRef] - Karim, M.F.; Kennedy, J.F. Menu of Coupled Velocity and Sediment Discharge Relations for Rivers. J. Hydraul. Eng.
**1990**, 116, 978–996. [Google Scholar] [CrossRef] - Yang, S.; Tan, S. Flow resistance over mobile bed in an open channel flow. J. Hydraul. Eng.
**2008**, 134, 937–947. [Google Scholar] [CrossRef] - Billi, P.; Salemi, E.; Preciso, E.; Ciavola, P.; Armaroli, C. Field measurement of bedload in a sand-bed river supplying a sediment starving beach. Z. Geomorphol.
**2017**, 61, 207–223. [Google Scholar] [CrossRef] - Yalin, M.S. Mechanics of Sediment Transport, 2nd ed.; Pergamon Press: Oxford, UK, 1977. [Google Scholar]
- Van Rjin, L.C. Equivalent roughness of alluvial bed. J. Hydraul. Eng. Div. ASCE
**1982**, 118, 1215–1218. [Google Scholar] - Yu, G.L.; Lim, S.Y. Modified Manning Formula for Flow in Alluvial Channels with SandBeds. J. Hydraul. Res.
**2003**, 41, 597–608. [Google Scholar] [CrossRef] - Meyer-Peter, E.; Müller, R. Formulas for bed-load transport. In Proceedings of the 3rd Meeting of IAHR, Stockholm, Sweden, 7 June 1948; pp. 39–64. [Google Scholar]
- Azareh, S.; Afzalimehr, H.; Poorhosein, M.; Singh, V.P. Contribution of Form Friction to Total Friction Factor. Int. J. Hydraul. Eng.
**2014**, 2, 77–84. [Google Scholar] - Yalin, M.S.; Karahan, E. Steepness of sedimentary dunes. J. Hydraul. Div.
**1979**, 105, 381–392. [Google Scholar] - Julien, P.Y.; Klaassen, G. Sand dune geometry of large rivers during floods. J. Hydraul. Eng.
**1995**, 121, 657–663. [Google Scholar] [CrossRef] - Yalin, M.S. On the average velocity of flow over a mobile bed. La Houille Blanche
**1964**, 1, 45–53. [Google Scholar] [CrossRef] - Agarwal, V.C.; Ranga Raju, K.G.; Garde, R.J. Bed-form geometry in sand-bed flows; Discussion. J. Hydraul. Eng.
**2001**, 127, 433–434. [Google Scholar] [CrossRef] - Yalin, M.S. Geometrical properties of sand waves. J. Hydraul. Div.
**1964**, 90, 105–119. [Google Scholar] - Aberle, J.; Nikora, V.; Henning, M.; Ettmer, B.; Hentschel, B. Statistical characterization of bed roughness due to bed forms: A field study in the Elbe River at Aken, Germany. Water Resour. Res.
**2010**, 46. [Google Scholar] [CrossRef] - Yang, S.; Tan, S.; Lim, S. Flow resistance and bed form geometry in a wide alluvial channel. Water Resour. Res.
**2005**, 41. [Google Scholar] [CrossRef][Green Version] - Engelund, F. Hydraulic resistance of alluvial streams. J. Hydraul. Div.
**1966**, 922, 315–326. [Google Scholar] - Ferreira Da Silva, A.M.; Yalin, M.S. Fluvial Processes, 2nd ed.; IAHR Monograph; Taylor & Francis Group: London, UK, 2017. [Google Scholar]
- Einstein, H.A. The Bed Load Function for Sediment Transportation in Open Channel Flows; Technical Bulletin 1026; U.S. Dept of Agriculture Soil Conservation Service: Washington, DC, USA, 1950.
- Ackers, P.; White, W. Sediment Transport: New Approach and Analysis. J. Hydraul. Div.
**1973**, 99, 2041–2060. [Google Scholar] - Hey, R.D. Flow Resistance in Gravel-Bed Rivers. J. Hydraul. Div.
**1979**, 105, 365–379. [Google Scholar] - Kamphuis, J.W. Determination of sand roughness for fixed beds. J. Hydraul. Res.
**1974**, 12, 193–203. [Google Scholar] [CrossRef] - Whiting, P.J.; Dietrich, W.E. Boundary shear stress and roughness over mobile bed. J. Hydraul. Eng.
**1990**, 116, 1495–1511. [Google Scholar] [CrossRef] - Millar, R.G. Grain and form resistance in gravel-bed rivers Résistances de grain et de forme dans les rivières à graviers. J. Hydraul. Res.
**1999**, 37, 303–312. [Google Scholar] [CrossRef] - Tokyay, N.D.; Altan-Sakarya, A.B. Local energy losses at positive and negative steps in subcritical open channel flows. Water SA
**2011**, 37, 237–244. [Google Scholar] [CrossRef] - van der Mark, C.F. A Semi-Analytical Model for Form Drag of River Bedforms. Ph.D. Thesis, University of Twente, Twente, The Nederlands, 1978. [Google Scholar]
- Engel, P. Length of flow separation over dunes. J. Hydraul. Div.
**1981**, 107, 1133–1143. [Google Scholar] - Shen, H.W.; Fehlman, H.M.; Mendoza, C. Bed form resistances in open channel flows. J. Hydraul. Eng.
**1990**, 116, 799–815. [Google Scholar] [CrossRef] - Cilli, S.; Ciavola, P.; Billi, P.; Schippa, L. Moving dunes constrain flow hydraulics in mobile sand-bed streams: The Fiumi Uniti and Savio River cases (Italy). Geomorphology
**2019**. submitted for publication. [Google Scholar] - Gabel, S.L. Geometry and kinematics of dunes during steady and unsteady flows in the Calamus River, Nebraska, USA. Sedimentology
**1993**, 40, 237–269. [Google Scholar] [CrossRef] - Brownlie, W.R. Compilation of Alluvial Channel Data: Laboratory and Field; Report No. KH-R-43B; California Institute of Technology, WM Keck Laboratory of Hydraulics and Water Resources Division of Engineering and Applied Science: Pasadena, CA, USA, 1981; pp. 1–209.
- Julien, P. Study of Bedform Geometry in Large Rivers; Report Q 1386 Delft Hydraulics; Delft Hydraulics: Delft, The Netherlands, 1992; pp. 1–79. [Google Scholar]
- Colby, B.R.; Hembxee, C.H. Computation of Total Sediment Discharge, Niobrara River Near Cody, Nebraska; Geological survey water-supply paper 1357; U.S. Government Printing Office: Washington, DC, USA, 1955; pp. 1–187. [CrossRef]
- Karim, F. Bed form geometry in sand dunes. J. Hydraul. Eng.
**1999**, 152, 1253–1261. [Google Scholar] [CrossRef] - Yen, B.C. Open Channel Flow Resistance. J. Hydraul. Eng.
**2002**, 128, 20–39. [Google Scholar] [CrossRef] - Keulegan, G.H. Laws of turbulent flow in open channels. J. Res. Nalt. Bur. Stand.
**1938**, 21, 707–741. [Google Scholar] [CrossRef]

**Figure 4.**Empirical drag coefficient κ as a function of dune steepness δ = Δ/Λ. Comparison between results obtained by using different equivalent roughness k

_{s}(Data A–G see Table 1).

**Figure 5.**Comparison of estimated and observed total energy slope (the dashed lines represent the ±30% error band).

**Figure 6.**Comparison of estimated and observed total energy slope (the dashed lines represent the ±30% error band). Note: the total energy slope S is calculated accounting for the grain component S′ obtained by Manning–Strickler Equation (39).

**Table 1.**Summary of field data used to determine empirical coefficient κ (Equation (29)). Each column reports maximum and minimum value.

Code | River | N | Q (m^{3}/s) | Y (m) | U (m/s) | S (m/km) | F (-) | d_{50} (mm) | d_{90} (mm) | Δ (m) | Λ (m) |
---|---|---|---|---|---|---|---|---|---|---|---|

A | Fiumi Uniti | 22 | 358.40–21.17 | 4.72–1.31 | 1.66–0.20 | 0.139–0.002 | 0.24–0.04 | 0.655–0.390 | 2.100–0.630 | 0.28–0.10 | 17.53–13.1 |

B | Savio | 9 | 132.06–7.04 | 3.58–1.77 | 1.50–0.21 | 0.354–0.012 | 0.28–0.05 | 0.548–0.412 | 1.702–0.694 | 0.16–0.12 | 7.16–6.14 |

C | Calamus | 18 | 1.73–0.82 | 0.61–0.34 | 0.77–0.61 | 1.100–0.680 | 0.34–0.29 | 0.410–0.310 | - | 0.20–0.10 | 4.05–2.02 |

D | Missouri | 25 | 1817.20–179.20 | 4.99–2.77 | 1.76–1.28 | 0.185–0.125 | 0.32–0.22 | 0.266–0.190 | 0.311–0.217 | 2.07–0.58 | 735.18–57.91 |

E | Jamuna | 33 | 10000–5000 | 19.50–8.20 | 1.50–1.30 | 0.070 | 0.17–0.09 | 0.200 | - | 5.10–0.80 | 251.00–8.00 |

F | Parana | 13 | 25000 | 26.00–22.00 | 1.50–1.00 | 0.050 | 0.10–0.07 | 0.370 | - | 7.50–3.00 | 450.00–100.00 |

G | Zaire | 29 | 28490–284 | 17.60–6.80 | 1.69–0.32 | 0.345–0.042 | 0.16–0.03 | 0.545–0.430 | 1.900–0.430 | 1.90–1.20 | 450.00–90.00 |

H | Bergsche Maas | 20 | 2160 | 10.50–5.80 | 1.70–1.30 | 0.125 | 0.20–0.13 | 0.520–0.210 | - | 2.50–0.40 | 50.00–6.00 |

Code | River | N | Q (m^{3}/s) | Y (m) | U (m/s) | S (m/km) | F (-) | d_{50} (mm) | d_{90} (mm) | Δ (m) | Λ (m) |
---|---|---|---|---|---|---|---|---|---|---|---|

I | Meuse | 44 | 1743.0–1731.0 | 9.52–8.22 | 1.57–0.87 | 0.141–0.138 | 0.17–0.09 | 0.650–0.500 | 2.500–1.030 | 0.85–0.58 | 13.42–7.03 |

L | ACP–ACOP | 151 | 528.68–27.50 | 4.30–0.76 | 1.29–0.35 | 0.271–0.016 | 0.23–0.12 | 0.364–0.083 | 0.466–0.105 | - | - |

M | Niobrara | 40 | 16.06–5.86 | 0.59–0.40 | 1.27–0.65 | 1.799–1.136 | 0.54–0.30 | 0.359–0.212 | 0.849–0.326 | - | - |

N | Rio Grande | 33 | 42.19–1.67 | 1.51–0.39 | 1–69–0.10 | 0.800–0.450 | 0.49–0.04 | 0.280–0.160 | 0.417–0.198 | - | - |

O | AMC | 11 | 29.42–1.22 | 2.53–0.80 | 0.79–0.42 | 0.3300.058 | 0.25–0.10 | 7.000–0.096 | 1.440–0.331 | - | - |

P | MID | 38 | 13.62–9.03 | 0.41–0.25 | 1.12–0.59 | 1.572–0.929 | 0.72–0.32 | 0.436–0.215 | 1.264–0.346 | - | - |

Q | ATC | 55 | 14186.31–1449.78 | 14.75–6.92 | 2.03–0.64 | 0.051–0.014 | 0.17–0.06 | 0.303–0.085 | 0.708–0.169 | - | - |

Dataset | Λ/y | S′ | Validated Data within Error Band | |
---|---|---|---|---|

30% | 20% | |||

model | 7.30 | Equation (11) k_{s}’ = 2.0·d_{50} | 93.5% | 68.4% |

test | 6.28 | 93.1% | 69.4% | |

test | 5.00 | 91.3% | 68.6% | |

test | 7.30 | Equation (11) k_{s}’ = 1.0·d_{50} | 90.7% | 62.5% |

test | Equation (38) ${n}^{\u2033}=0.0416\xb7{d}_{50}{}^{0.165}$ | 88.2% | 63.9% |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Schippa, L.; Cilli, S.; Ciavola, P.; Billi, P. Dune Contribution to Flow Resistance in Alluvial Rivers. *Water* **2019**, *11*, 2094.
https://doi.org/10.3390/w11102094

**AMA Style**

Schippa L, Cilli S, Ciavola P, Billi P. Dune Contribution to Flow Resistance in Alluvial Rivers. *Water*. 2019; 11(10):2094.
https://doi.org/10.3390/w11102094

**Chicago/Turabian Style**

Schippa, Leonardo, Silvia Cilli, Paolo Ciavola, and Paolo Billi. 2019. "Dune Contribution to Flow Resistance in Alluvial Rivers" *Water* 11, no. 10: 2094.
https://doi.org/10.3390/w11102094