# Dune Contribution to Flow Resistance in Alluvial Rivers

^{1}

^{2}

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## 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 |

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**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% |

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**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