# Validation of a Novel, Shear Reynolds Number Based Bed Load Transport Calculation Method for Mixed Sediments against Field Measurements

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Case Study

_{50}< 70.5 mm, where d

_{50}is the median grain size). Such a wide dispersion of the bed content is a unique feature of the Danube River (~rkm 1600–rkm 1800); at the Lower Austrian Danube, (~rkm 1885, 90 km upstream), the Danube flows through a gravel bed, where d

_{50}is 21.1 mm without any finer fractions [24]. In turn, the middle Hungarian Danube (~200 km downstream) has a typical sandy bed with d

_{50}< 0.05 mm [25]. The complexity of the topography and bed content suggest spatially and temporally varied sediment transport nature [22]. That is in some places the gravel, elsewhere the sand transport dominates [26]. This kind of individual complexity was presented e.g., by the field measurements of Török and Baranya [19], or in [27,28].

## 3. Materials and Methods

#### 3.1. Introducing the Combining Sediment Transport Calculation Method

#### 3.2. Applied 3D Flow Model

#### 3.2.1. Parameterization

_{50}. Thus, applying the fitted function, a transitional and continuous d

_{50}map can be estimated based on the calculated bed shear stress distributions. This method was used based on 33 bed material samples [19,26]. The standard deviation of the calculated d

_{50}to the measured d

_{50}was 3.2 mm.

^{3}/s) [19,24] were simulated. It meant that the inflow boundary condition was set to a continuous flow discharge series, including the highlighted periods in Figure 3a–h, respectively and omitting the intermediate periods. In this range, 66% of the annual bed load amount passed [23]. In turn, it is emphasized that the ignored 34% of the annual bed load yield was still significant. That is, the numerical model neglects the simulation of bed changes that take place during the lower water regime. However, many studies (including papers regarding the investigated river reach) display that the major bed changes such as scouring, bar formations and flushing of the groin fields are expected rather during floods [16,19,24,52]. That is, the mean and lower water stages are less important in the view of bed changes. Based on the above, the bed changes caused by the eight flood waves (Figure 3a–h, a total of 211 days) approach well the real two years changes. Hence, the calculated bed changes were comparable to the measured.

#### 3.2.2. 3D Flow Model Validation

## 4. Results

#### 4.1. Comparison of the Calculation Methods

^{3}/s), but durable (~2 months long) flood wave, while the e was the historical one with a peak higher than 10,000 m

^{3}/s. Thus, the comparative analysis presented the operational characteristics of the sediment transport models for both the durable lower and also for the extreme water regimes. As a benchmark, the measured bed change map indicated the extent of the possible bed changes. However, the measured and calculated maps could not be compared directly, because the measured belonged to the whole two-year-long period (Figure 3).

^{*}[22]. In turn, in the navigation channel, no considerable bed change happened. According to the suspended form of the vR formula, the finer suspended load passes over the calculation domain, while the bed surface remains still, calculated by the W&C formula. This assumption is consistent with the conclusions of the field measurements [19]; the main channel seems to be armored enough to be resistant at the mean water regime.

#### 4.2. Measured Data-Based Verification of the Combined Method

^{3}/s, Figure 3), so the results could not be compared with the measured changes directly. Therefore, to achieve a notionally common scale, the bed change values were normalized. That is both the measured and calculated bed changes got divided by the highest bed level decrease or increase the value of the main channel:

_{50}≈ 0.01 m. Considering the grain-size distributions [19], a still realistic, but considerably finer bed material was presupposed. Therefore, the model was set up by 30% lower d

_{50}, which was d

_{50}≈ 0.007 m. With this only one difference, the model was run for the historical flood wave. Figure 14 presents the bed changes at region A, and at the downstream of it.

_{50}resulted in major erosion at region A. Considering the measured changes in the left figure it is obvious that the decreasing of the d

_{50}led to a better match to the real bed changes. The lower row of Figure 14 represents the bed changes at the downstream. Here, important deposition formations could be measured (left Figure 14) in front of the right bank groin pair (region B) and also in the main stream, between the two gravel bars (between region B and D). These changes could not be represented by the original model setup (middle Figure 14). In turn, in the case of finer bed material (right Figure 14), the model predicted important depositions in these regions. The extension of the deposition in front of the groin pair (region B) was very similar to the measured. Also, the lengthwise extension of the deposition downstream (between regions B and D), in the main stream were also reproduced. However, the location of it was not correct. It seems that the model underestimated the crosswise sediment transport, which is a known limitation of the Reynolds averaged description of the flow field [56,57,58,60]. Concluding, the herein presented investigation suggested that the bed material at region A was finer than the predicted d

_{50}allocation for the original model setup. With this assumption, the deposition nature at the downstream has become also detectable.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Wilcock and Crowe Bed Load Transport Formula

_{m}grain size of the initial bed materials was in the range of 4.1–10.5 mm, whereas d

_{m}was 8.7 mm the measurements of Török et al. [34]. The W&C formula is therefore expected to be capable of describing the gravel/sand bed load transport processes in our experiments. In the case of such non-uniform bed material, the interaction between (hiding and exposure) the particles of different sizes plays an essential role in the stability of the sediments and can result in bed armoring. The W&C formula is reported to be capable of predicting the transient conditions of bed armoring or scouring processes. The preliminary numerical simulations of Török et al. [46] also supported this statement.

_{i}*) as the function of the ratio of the shear stress (τ) and the reference shear stress (τ

_{ri}). The reference shear stress τ

_{ri}is the value of τ at which w

_{i}* is equal to a small but already perceptible value w

_{i}* = 0.002. That is the reference shear stress has a similar physical meaning to the critical shear stress (τ

_{c}), which indicates the initiation of motion of particles. The dimensionless transport rate is obtained according to the following equation:

_{bi}can be expressed from the following formula:

_{s}is the proportion of sand in surface size distribution.

_{ri}, the W&C model suggests that the stability of the sediment particles depends on the sand content. Thus τ

_{ri}can be calculated by the following steps:

^{*}

_{rm}is the dimensionless reference shear stress for the mean size of the bed surface. The reference shear stress for the mean size can be obtained from the particle Froude number equation:

#### Appendix A.2. Van Rijn Bed Load Transport Formula

_{*}= 0.04 m/s) were monitored.

_{L}is the lift force, α

_{L}is the lift coefficient, ν is the kinematic viscosity coefficient, ρ is the density of the fluid, v

_{r}is the relative particle velocity and $\partial \mathrm{u}/\partial \mathrm{z}$ is the velocity gradient. The mathematical method uses the turbulent wall law Equation (A9) for calculation of the vertical flow velocity distribution:

_{*}is the bed shear velocity, κ is constant von Karman (κ = 0.41), and z

_{0}is the zero-velocity level above the bed.

_{b}is the bed load transport per unit width, D

_{50}is the particle size, s is the specific density, g is the acceleration of gravity, T is the transport stage parameter, can be represented as:

_{*,cr}is the critical bed shear velocity according to Shields [35], ${u}_{*}^{\prime}=\left({g}^{0.5}/{C}^{\prime}\right)\overline{u}$ is the bed shear velocity related to grains, where C´ is the Chézy-coefficient related to grains and $\overline{u}$ is the mean flow velocity.

_{*}is the particle parameter defined as:

## References

- Meyer-Peter, E.; Müller, R. Formulas for Bed-Load Transport. In Proceedings of the IAHSR 2nd Meeting, Stockholm, Sweden, 7–9 June 1948. [Google Scholar]
- Einstein, H.A. The Bed-Load Function for Sediment Transportation in Open Channel Flows; United States Department of Agriculture, Economic Research Service: Washington, DC, USA, 1950. [Google Scholar]
- Egiazaroff, I.V. Calculation of Nonuniform Sediment Concentrations. J. Hydraul. Div.
**1965**, 91, 225–247. [Google Scholar] - Ashida, K.; Michiue, M. Study on hydraulic resistance and bedload transport rate in alluvial streams. Trans. Jpn. Soc. Civ. Eng.
**1972**, 206, 59–69. [Google Scholar] [CrossRef] - Parker, G.; Klingeman, P.; McLean, D. Bedload and size distribution in natural paved gravel bed streams. J. Hydraul. Eng.
**1982**, 108, 544–571. [Google Scholar] - Parker, G. Surface-based bedload transport relation for gravel rivers. J. Hydraul. Res.
**1990**, 28, 417–436. [Google Scholar] [CrossRef] - Wilcock, P.R.; Kenworthy, S.T. A two-fraction model for the transport of sand/gravel mixtures. Water Resour. Res.
**2002**, 38, 12-1–12-12. [Google Scholar] [CrossRef] - Wilcock, P.R.; Crowe, J.C. Surface-based transport model for mixed-size sediment. J. Hydraul. Eng. ASCE
**2003**, 129, 120–128. [Google Scholar] [CrossRef] - Wu, W.; Wang, S.S.Y.; Jia, Y. Nonuniform sediment transport in alluvial rivers. J. Hydraul. Res.
**2000**, 38, 427–434. [Google Scholar] [CrossRef] - Powell, D.M.; Reid, I.; Laronne, J.B. Evolution of bed load grain size distribution with increasing flow strength and the effect of flow duration on the caliber of bed load sediment yield in ephemeral gravel bed rivers. Water Resour. Res.
**2001**, 37, 1463–1474. [Google Scholar] [CrossRef] - Parker, G. Transport of Gravel and Sediment Mixtures. In Sedimentation Engineering; Garcia, M., Ed.; American Society of Civil Engineers: Reston, VA, USA, 2008; pp. 165–251. ISBN 978-0-7844-0814-8. [Google Scholar]
- Török, G.T.; Baranya, S.; Rüther, N. 3D CFD Modeling of Local Scouring, Bed Armoring and Sediment Deposition. Water
**2017**, 9, 56. [Google Scholar] [CrossRef] - Rákóczi, L.; Sass, J. A Felső-Duna és a Szigetközi mellékágak mederalakulása a dunacsúni duzzasztómű üzembe helyezése után (Changes of the channel of the Hungarian Upper Danube and of the side river arms of the Szigetköz upon putting the Dunacsúny I. river barrage into operati. Vízügyi Közlemények
**1995**, 77, 46–75. [Google Scholar] - Hankó, Z. Gondolatok a Duna Szap és Szob közötti szakaszának fejlesztéséről (Considerations related to the development of the Danube reach between Szap and Szob). Vízügyi Közlemények
**2000**, 82, 285–299. [Google Scholar] - Holubová, K.; Capeková, Z.; Szolgay, J. Impact of hydropower schemes at bedload regime and channel morphology of the Danube River. In Proceedings of the River Flow 2004: Second International Conference on Fluvial Hydraulics, Napoli, Italy, 23–25 June 2004; CRC Press: Napoli, Italy, 2004; pp. 135–142. [Google Scholar]
- Baranya, S.; Józsa, J.; Török, G.T.; Ficsor, J.; Mohácsiné Simon, G.; Habersack, H.; Haimann, M.; Riegler, A.; Liedermann, M.; Hengl, M. A Duna hordalékvizsgálatai a SEDDON osztrák-magyar együttműködési projekt keretében (Introduction of the joint Austro-Hungarian sediment research under the SEDDON ERFE-project). Hidrológiai Közlöny
**2015**, 95, 41–46. [Google Scholar] - Hankó, Z.; Starosolszky, Ö.; Bakonyi, P. Megvalósíthatósági tanulmány a Duna környezetének és hajózhatóságának fejlesztésére (Danube Environmental and navigation Project, Feasibility Study). Vízügyi Közlemények
**1996**, 78, 291–315. [Google Scholar] - Goda, L. A Duna gázlói Pozsony-Mohács között (Shallows of the River Danube between Pozsony, Bratislava and Mohács. Vízügyi Közlemények
**1995**, 77, 71–102. [Google Scholar] - Török, G.T.; Baranya, S. Morphological Investigation of a Critical Reach of the Upper Hungarian Danube. Period. Polytech. Civ. Eng.
**2017**, 61, 752–761. [Google Scholar] [CrossRef] - Holubová, K.; Comaj, M.; Lukac, M.; Mravcová, K.; Capeková, Z.; Antalová, M. Final Report in DuRe Flood Project—Danube Floodplain Rehabilitation to Improve Flood Protection and Enhance the Ecological Values of the River in the Stretch between Sap and Szob; Danube Transnational Programme: Bratislava, Slovakia, 2015. [Google Scholar]
- Varga-Lehofer, D.T. A Felső-Magyarországi Duna Morfológiai Változásainak Elemzése (Investigation of the Morphological Changes of the Hungarian Upper Danube). Bachelor’s Thesis, Budapest Univerity of Technology and Economics, Budapest, Hungary, 2014. [Google Scholar]
- Török, G.T.; Józsa, J.; Baranya, S. A Shear Reynolds Number-Based Classification Method of the Nonuniform Bed Load Transport. Water
**2019**, 11, 73. [Google Scholar] [CrossRef] - Török, G.T. Methodological Improvement of Morphodynamic Investigation Tools for Rivers with Non-Uniform Bed Material. Ph.D. Thesis, Budapest Univerity of Technology and Economics, Budapest, Hungary, 2018. [Google Scholar]
- Liedermann, M.; Gmeiner, P.; Kreisler, A.; Tritthart, M.; Habersack, H. Insights into bedload transport processes of a large regulated gravel-bed river. Earth Surf. Process. Landf.
**2018**, 43, 514–523. [Google Scholar] [CrossRef] - Baranya, S. Three-Dimensional Analysis of River Hydrodynamics and Morphology. Ph.D. Thesis, Budapest University of Technology and Economics, Budapest, Hungary, 2009. [Google Scholar]
- Török, G.T.; Baranya, S. A shear Reynolds number based sediment transport classification method for complex river beds. In Proceedings of the 8th International Symphosium on Environmental Hydraulics, Notre Dame, IN, USA, 4–7 June 2018. [Google Scholar]
- Rákóczi, L. A Duna Szap-Gönyű Közötti Szakaszának Hajózási Viszonyait Javító Beavatkozások Vizsgálata (Investigation of Interventions for Navigation Improvment at the Danube Channel between Szap and Szob); Technical Report; Budapest University of Technology and Economics: Budapest, Hungary, 2004. [Google Scholar]
- Baranya, S.; Goda, L.; Józsa, J.; Rákóczi, L. Complex hydro- and sediment dynamics survey of two critical reaches on the Hungarian part of river Danube. In Proceedings of the IOP Conference Series: Earth and Environmental Science, Bled, Slovenia, 2–4 June 2008; Volume 4. [Google Scholar]
- Habersack, H.; Haimann, M.; Baranya, S.; Józsa, J.; Riegler, A.; Sindelar, C.; Liedermann, M.; Ficsor, J.; Simon, G.M.; Hengl, M. Gemeinsame österreichisch-ungarische Sedimentforschung im Rahmen des EFRE-Projektes SEDDON. Österreichische Wasser Und Abfallwirtschaft
**2014**, 66, 340–347. [Google Scholar] [CrossRef] - Wu, K.; Yeh, K.-C.; Lai, Y.G. A Combined Field and Numerical Modeling Study to Assess the Longitudinal Channel Slope Evolution in a Mixed Alluvial and Soft Bedrock Stream. Water
**2019**, 11, 735. [Google Scholar] [CrossRef] - Sattar, A.M.A.; Bonakdari, H.; Gharabaghi, B.; Radecki-Pawlik, A. Hydraulic Modeling and Evaluation Equations for the Incipient Motion of Sandbags for Levee Breach Closure Operations. Water
**2019**, 11, 279. [Google Scholar] [CrossRef] - Bogárdi, J. A Hordalékmozgás Elmélete; Vilmos, I., Ed.; Akadémiai Kiadó: Budapest, Hungary, 1955. [Google Scholar]
- Van Rijn, L.C. Sediment Transport, Part I: Bed Load Transport. J. Hydraul. Eng.
**1984**, 110, 1431–1456. [Google Scholar] [CrossRef][Green Version] - Török, G.T.; Baranya, S.; Rüther, N.; Spiller, S. Laboratory analysis of armor layer development in a local scour around a groin. In Proceedings of the International Conference on Fluvial Hydraulics, RIVER FLOW 2014, Lausanne, Switzerland, 7–10 July 2014; Taylor and Francis Group: Lausanne, Switzerland, 2014; pp. 1455–1462. [Google Scholar]
- Shields, A. Application of Similarity Principles and Turbulence Research to Bed-Load Movement. Mitt. Preuss. Versuchsanst. Wasserbau Schiffbau
**1936**, 26, 47. [Google Scholar] - van Rijn, L.C. Mathematical modelling of morphological processes in the case of suspended sediment transport. Ph.D. Thesis, Civil Engineering and Geoscience, TU Delft, Delft, The Netherlands, 1987. [Google Scholar]
- Reidar, B.; Olsen, N. A Three-Dimensional Numerical Model for Simulation of Sediment Movements in Water Intakes with Moving Option; Trondheim, Norway. 2018. Available online: http://folk.ntnu.no/nilsol/ssiim/manual5.pdf (accessed on 30 September 2019).
- Olsen, N.R.B. Numerical Modelling and Hydraulics; Online Manuscript. 2012. Available online: http://folk.ntnu.no/nilsol/tvm4155/flures6.pdf (accessed on 30 September 2019).
- Pope, S.B. Turbulent Flows; Cambridge University Press: Cambridge, UK, 2000; ISBN 9780521598866. [Google Scholar]
- Patankar, S.V. Numerical Heat Transfer and Fluid Flow; Minkowycz, M.J., Sparrow, E.m., Eds.; McGraw-Hill Book Company: New York, NY, USA, 1980; ISBN 0-07-048740-5. [Google Scholar]
- Glock, K.; Tritthart, M.; Habersack, H.; Hauer, C. Comparison of Hydrodynamics Simulated by 1D, 2D and 3D Models Focusing on Bed Shear Stresses. Water
**2019**, 11, 226. [Google Scholar] [CrossRef] - Schlichting, H. Boundary-Layer Theory; McGraw-Hill: New York, NY, USA, 1979. [Google Scholar]
- Fischer-Antze, T.; Stoesser, T.; Bates, P.; Olsen, N.R.B. 3D numerical modelling of open-channel flow with submerged vegetation 3D numerical modelling of open-channel flow with submerged vegetation Modélisation numérique 3D d’un écoulement en canal avec vegetation submergée. J. Hydraul. Res.
**2013**, 39, 303–310. [Google Scholar] [CrossRef] - Fischer-Antze, T.; Reidar, B.; Olsen, N.; Gutknecht, D. Three-dimensional CFD modeling of morphological bed changes in the Danube River. Water Resour. Res.
**2008**, 44. [Google Scholar] [CrossRef] - Baranya, S.; Józsa, J. Numerical and laboratory investigation of the hydrodynamic complexity of a river confluence. Period. Polytech. Civ. Eng.
**2007**, 51, 3–8. [Google Scholar] [CrossRef] - Török, G.T.; Baranya, S.; Rüther, N. Three-dimensional numerical modeling of non-uniform sediment transport and bed armoring process. In Proceedings of the 18th Congress of the Asia & Pacific Division of the International Association for Hydro-Environment Engineering and Research 2012, Jeju, Korea, 19–23 August 2012. [Google Scholar]
- Bihs, H.; Olsen, N.R.B. Numerical Modeling of Abutment Scour with the Focus on the Incipient Motion on Sloping Beds. J. Hydraul. Eng.
**2011**, 137, 1287–1292. [Google Scholar] [CrossRef] - Rüther, N.; Olsen, N.R.B. Modelling free-forming meander evolution in a laboratory channel using three-dimensional computational fluid dynamics. Geomorphology
**2007**, 89, 308–319. [Google Scholar] [CrossRef] - Baranya, S. River Bed Material Mapping to Support Habitat Assesment in Large Rivers. In Proceedings of the 12th International Symposium on Ecohydraulics, Tokyo, Japan, 19–24 August 2018. [Google Scholar]
- Krámer, T.; Szilágyi, J.; Józsa, J. Mértékadó árvízszintek: Országos felülvizsgálat után (High Water Levels in Hungary: after the reconsideration). Mérnök Újság
**2015**, 1–2, 22–25. [Google Scholar] - Ficsor, J. Lebegtetett Hordalék Vizsgálata a Felsö-Magyarorszaági Duna-Szakaszon (Study of Suspended Sediment Transport on the Upper Hungarian Reach of the Danube River). Master’s Thesis, Budapest University of Technology and Economics, Budapest, Hungary, 2016. [Google Scholar]
- Luo, P.; Mu, D.; Xue, H.; Ngo-duc, T.; Dang-dinh, K.; Takara, K.; Nover, D.; Schladow, G. Flood inundation assessment for the Hanoi Central Area, Vietnam under historical and extreme rainfall conditions. Sci. Rep.
**2018**, 8, 12623. [Google Scholar] [CrossRef] - Baranya, S.; Józsa, J. Flow analysis in River Danube by field measurement and 3D CFD turbulence modelling. Period. Polytech. Civ. Eng.
**2006**, 50, 57–68. [Google Scholar] - Török, G.T. Vegyes szemcseösszetételű folyómedrek numerikus vizsgálata (Numerical investigation of non-uniform river bed). Hidrológiai Tájékoztató
**2013**, 22–24. Available online: http://www.hidrologia.hu/mht/letoltes/hidrologiai_tajekoztato_2013.pdf (accessed on 30 September 2019). - Guerrero, M.; Lamberti, A. Flow Field and Morphology Mapping Using ADCP and Multibeam Techniques: Flow Field and Morphology Mapping Using ADCP and Multibeam Techniques: Survey in the Po River. J. Hydraul. Eng.
**2011**, 137, 1576–1587. [Google Scholar] [CrossRef] - Koken, M.; Constantinescu, G. An investigation of the flow and scour mechanisms around isolated spur dikes in a shallow open channel: 1. Conditions corresponding to the initiation of the erosion and deposition process. Water Resour. Res.
**2008**, 44, 1–19. [Google Scholar] [CrossRef] - Catalano, P.; Wang, M.; Iaccarino, G.; Moin, P. Numerical simulation of the flow around a circular cylinder at high Reynolds numbers. Int. J. Heat Fluid Flow
**2003**, 24, 463–469. [Google Scholar] [CrossRef] - Roulund, R.; Sumer, B.M.; Fredsøe, J.; Michelsen, J. Numerical and experimental investigation of flow and scour around a circular pile. J. Fluid Mech.
**2005**, 534, 351–401. [Google Scholar] [CrossRef] - Török, G.T.; Baranya, S.; Rüther, N. Validation of a combined sediment transport modelling approach for the morphodynamic simulation of the upper Hungarian Danube River. In Proceedings of the 19th EGU General Assembly, Vienna, Austria, 8–13 April 2018; Volume 19, p. 15749. [Google Scholar]
- Baranya, S.; Olsen, N.R.B.; Stoesser, T.; Sturm, T. Three-dimensional rans modeling of flow around circular piers using nested grids. Eng. Appl. Comput. Fluid Mech.
**2012**, 6, 648–662. [Google Scholar] [CrossRef] - Wilcock, P.R.; Kenworthy, S.T.; Crowe, J.C. Experimental Study of the Transport of Mixed Sand and Gravel. Water Resour. Res.
**2001**, 37, 3349–3358. [Google Scholar] [CrossRef] - Fernandez Luque, R. Erosion and Transport of Bed-load Sediment. Bachelor’s Thesis, Delft Technical University, Delft, The Netherland, 1974. [Google Scholar]
- Fernandez Luque, R.; van Beek, R. Erosion and Transport of Bed-load Sediment. J. Hydraul. Res.
**1976**, 14, 127–144. [Google Scholar] [CrossRef]

**Figure 1.**The sketch of the investigated Danube study reach (

**left**) and grain size distributions taken from the investigated reach (

**right**). The characteristic water discharges are mean flow Q

_{m}= 2000 m

^{3}/s, range of bankfull discharge Q

_{bf}= 4300–4500 m

^{3}/s, 2, 10 and 100-year flood event Q

_{2}= 5950 m

^{3}/s, Q

_{10}= 7950 m

^{3}/s and Q

_{100}= 10,400 m

^{3}/s [23].

**Figure 3.**Discharge time series at rkm 1801 for the period October 2012–October 2014. The different colored periods marked with letters a–h display the flood waves (including the 100-year historical flood wave from 2013, e) which exceed the bed-forming flow discharge (Q > 2100 m

^{3}/s).

**Figure 6.**Calculated bed changes by the van Rijn (vR) formula for a 2.5 month-long period (Figure 3).

**Figure 7.**Calculated bed changes by the Wilcock and Crowe (W&C) formula for a 2.5 month-long period (Figure 3).

**Figure 8.**Calculated bed changes by the combined method for a 2.5 month-long period (Figure 3).

**Figure 11.**Calculated bed changes by the combined method for the historical flood wave (Figure 3).

**Figure 12.**Normalized bed changes of the measured values (Figure 4) regarding the whole period October 2012–October 2014.

**Figure 13.**Normalized bed changes of the calculated values. The calculation was elaborated for the eight flood waves in the period October 2012–October 2014 (Figure 3).

**Figure 14.**Calculated bed changes by the combining method for the historical flood wave. The model in the middle figure was set up with the initial, while in the right figure with finer bed material.

**Table 1.**The root-mean-square deviation (RMSD) between the measured and calculated velocities, the maximum cross-sectional horizontal velocity and the percentage deviation of them for the seven cross-sections.

Cross-Section | I. | II. | III. | IV. | V. | VI. | VII. |
---|---|---|---|---|---|---|---|

RMSD, m/s | 0.35 | 0.19 | 0.17 | 0.19 | 0.47 | 0.42 | 0.27 |

Max v_{hor}, m/s | 2.73 | 2.84 | 3.01 | 3.14 | 3.13 | 2.88 | 2.62 |

Average dif., % | 12.7 | 6.7 | 5.7 | 6.2 | 14.9 | 14.6 | 10.4 |

**Table 2.**The average daily volume changes for calculated (∆V

_{c}) and measured (∆V

_{m}) volume ratio values ∆V

_{c}/∆V

_{m}for region D. A value of 1 would indicate the perfect match.

Sediment Transport Model | ||||
---|---|---|---|---|

van Rijn | Wilcock and Crowe | Re* Based Combined | ||

The rate of the calculated to measured volumes | Deposition | 48.7 | 4.9 | 3.5 |

Erosion | 7.2 | 0 | 0.7 |

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**MDPI and ACS Style**

Török, G.T.; Józsa, J.; Baranya, S. Validation of a Novel, Shear Reynolds Number Based Bed Load Transport Calculation Method for Mixed Sediments against Field Measurements. *Water* **2019**, *11*, 2051.
https://doi.org/10.3390/w11102051

**AMA Style**

Török GT, Józsa J, Baranya S. Validation of a Novel, Shear Reynolds Number Based Bed Load Transport Calculation Method for Mixed Sediments against Field Measurements. *Water*. 2019; 11(10):2051.
https://doi.org/10.3390/w11102051

**Chicago/Turabian Style**

Török, Gergely T., János Józsa, and Sándor Baranya. 2019. "Validation of a Novel, Shear Reynolds Number Based Bed Load Transport Calculation Method for Mixed Sediments against Field Measurements" *Water* 11, no. 10: 2051.
https://doi.org/10.3390/w11102051