# Case Study of Transient Dynamics in a Bypass Reach

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Governing Physics

#### 2.2. Implementations in Delft3D

#### 2.2.1. Physics

#### 2.2.2. Numerics

#### 2.3. Richardson Extrapolation

## 3. Materials and Methods

#### 3.1. Study Site

^{3}/s for upstream fish migration, during weekends the flow in the reach is increased to 50 m

^{3}/s for aesthetic reasons. In the winter, the reach is mostly dry. The spillways and the fishway spill into the most upstream part of the study reach, the bypass joins the tailrace of the power plant shortly downstream of the study reach, see Figure 1. The reach between the spillways and the confluence is approximately 7-km-long. The entirety of the Ume River is regulated, while the Vindel River—a tributary to the Ume River that merges a couple of kilometers upstream of the study reach—is not regulated. During the spring flood, it is therefore common that spilling occurs in the bypass reach. It would be inaccurate to describe the flow conditions in the study reach as hydropeaked since the discharge in the reach is not necessarily related to the power production of the power plant. The study reach is however subject to rapid changes in discharge, partly during the spring flood and partly during the weekly increase and decrease in discharge.

#### 3.2. Bathymetry and Depth Measurements

#### 3.3. Scenarios

#### 3.3.1. Hysteresis Scenarios

^{3}/s to 21 m

^{3}/s in 5 min. In the second case, the discharge was increased from 21 m

^{3}/s to 50 m

^{3}/s in 5 min. In both simulations, a steady state was ensured both before and after the change in discharge.

#### 3.3.2. Hydropeaking Scenarios

^{3}/s.

#### 3.4. Calibration

^{3}/s. The Manning number was swept from 0.03 s/m

^{1/3}to 0.1 s/m

^{1/3}, which corresponds to the extreme values of the Manning number in natural channels [37]. In each of the simulations, the Manning number was kept constant in the entire reach. This approach has been used with success in other studies [38]. It is assumed that the reach in proximity of the validation points are of the same roughness to the validation point. By comparing the simulated water levels to the measured diver data, it was then possible to find the Manning number that produced the smallest error [20]. The calibrated WSE is plotted against the measured WSE in Figure 4. Relevant statistics can be seen in Table 1. The Pearson correlation of 0.9995 obtained in this study is comparable to the ones obtained in [21,38]. The corresponding Manning number distribution can be seen in Figure 4.

#### 3.5. Model Setup

^{3}/s. At the downstream boundary, the condition was set to “Neumann” with a value of 0.001 for the water surface. The slip condition was set to “free slip”, which for large scale hydrodynamic simulations, is a reasonable assumption [28]. Both boundaries had a reflection parameter of 0. The bathymetry seen in Figure 2 was interpolated on all the meshes using the QUICKIN interpolation tool [39]. The threshold depth was set to 0.1 m and the advection scheme used was “cyclic”, which is the standard advection scheme in Delft3D. The Manning roughness formula was chosen with a roughness file, the distribution can be seen in Figure 4. A timestep of $t=0.005$ min proved to be sufficient to obtain stable solutions.

#### 3.6. Wetted Area Calculation

#### 3.7. Mesh Study

## 4. Results and Discussion

#### 4.1. WSE Hysteresis

#### 4.2. WSE Dynamics with Different Scenarios

^{3}/s and 21 m

^{3}/s in all validation points, see Figure 7 and Figure 9. Similarly for the 20 flow changes per day case, we see that the respective steady state is reached in all validation points except point 8. In this point, there is a state of continuous dynamical change where the WSE never reaches any resemblance of steady state. This effect is noticed for all scenarios except the case of 10 changes per day. For the case of 20 flow changes per day this point of continuous dynamics occurs somewhere between point 7 and point 8. Analogously, this point is between point 5 and point 6 for the 30 flow changes case, see Figure 9. For the case of 40 flow changes per day, the point occurs between point 3 and point 4, see Figure 8. Further, for the 50 and 60 flow changes per day cases, it occurs somewhere upstream of point 1. For the points 5–8 for the 40, 50, and 60 flow changes per day cases (Figure 10), the hysteresis behavior seen in Figure 7 and Figure 9 is no longer observed, rather, the WSE appears to oscillate sinusoidally. One explanation for this behavior could be that the time scales become comparable or smaller than the decrease time and increase time legs seen in Table 4. It is also noticed that in some cases the steady state for the increasing leg will be reached but not for decreasing leg; for instance, see the case with 40 changes per day in Figure 8. This phenomena can also be explained by the timescales of each respective leg. Furthermore, as the number of flow changes per day approaches ∞, the WSE appears to approach the mean of the steady states. This convergence occurs faster for the more-downstream coordinates which can be seen in Figure 10. The cross-section where continuous dynamics is first observed for the five cases where it occurs have been plotted in Figure 11. For the 50 and 60 flow changes per day cases, this point is in close proximity to each other upstream of point 1. Since the width of the river in the lower parts would be reduced given the more frequent scenarios, it would likely affect the erosion and the morphology of the river.

#### 4.3. ${A}_{wetted}$ Dynamics for the Different Scenarios

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

ACUR | Air Cushion Underground Reservoir |

CFD | Computational Fluid Dynamics |

DEM | Digital Elevation Model |

MASL | Meters Above Sea Level |

SWE | Shallow Water Equations |

WSE | Water Surface Elevation |

## References

- UNFCCC. The Paris Agreement. 2016. Available online: https://unfccc.int/process-and-meetings/the-paris-agreement/the-paris-agreement (accessed on 20 March 2020).
- Regeringen. A Coherent Policy for the Climate. 2019. Available online: https://www.government.se/press-releases/2019/12/a-coherent-policy-for-the-climate/ (accessed on 20 March 2020).
- Regjeringen. Norway Steps up 2030 Climate Goal to at Least 50% towards 55%. 2020. Available online: https://www.regjeringen.no/en/aktuelt/norge-forsterker-klimamalet-for-2030-til-minst-50-prosent-og-opp-mot-55-prosent/id2689679/ (accessed on 20 March 2020).
- Ympäristöministeriö. Towards Climate-Smart Day-to-Day Living—Medium-term Climate Change Plan to 2030. 2019. Available online: https://www.ym.fi/en-US/The_environment/Climate_and_air/Mitigation_of_climate_change/National_climate_policy/Climate_Change_Plan_2030 (accessed on 20 March 2020).
- European Comission. 2030 Climate & Energy Framework. Available online: https://ec.europa.eu/clima/policies/strategies/2030_en (accessed on 20 March 2020).
- European Comission. Renewable Energy Statistics. 2020. Available online: https://ec.europa.eu/eurostat/statistics-explained/index.php/Renewable_energy_statistics (accessed on 23 March 2020).
- Statnet, Fingrid, Energinet, Svenska Kraftnät. Nordic Grid Development Plan 2019. Available online: https://www.statnett.no/contentassets/61e33bec85804310a0feef41387da2c0/nordic-grid-development-plan-2019-for-web.pdf (accessed on 23 March 2020).
- North Sea Link. Available online: http://www.northsealink.com/ (accessed on 23 March 2020).
- HydroFlex. Technology for Mitigation of Highly Fluctuating Discharges into Downstream River. Available online: https://www.h2020hydroflex.eu/work-packages/wp-5/task-5-1/ (accessed on 24 March 2020).
- Bunn, S.E.; Arthington, A.H. Basic principles and ecological consequences of altered flow regimes for aquatic biodiversity. Environ. Manag.
**2002**, 30, 492–507. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Saltveit, S.; Halleraker, J.; Arnekleiv, J.; Harby, A. Field experiments on stranding in juvenile Atlantic salmon (Salmo salar) and brown trout (Salmo trutta) during rapid flow decreases caused by hydropeaking. Regul. Rivers Res. Manag. Int. J. Devoted River Res. Manag.
**2001**, 17, 609–622. [Google Scholar] [CrossRef] - Halleraker, J.; Saltveit, S.; Harby, A.; Arnekleiv, J.; Fjeldstad, H.P.; Kohler, B. Factors influencing stranding of wild juvenile brown trout (Salmo trutta) during rapid and frequent flow decreases in an artificial stream. River Res. Appl.
**2003**, 19, 589–603. [Google Scholar] [CrossRef] - McKinney, T.; Speas, D.W.; Rogers, R.S.; Persons, W.R. Rainbow trout in a regulated river below Glen Canyon Dam, Arizona, following increased minimum flows and reduced discharge variability. N. Am. J. Fish. Manag.
**2001**, 21, 216–222. [Google Scholar] [CrossRef] - Choi, B.; Choi, S.U. Impacts of hydropeaking and thermopeaking on the downstream habitat in the Dal River, Korea. Ecol. Inform.
**2018**, 43, 1–11. [Google Scholar] [CrossRef] - Bejarano, M.D.; Jansson, R.; Nilsson, C. The effects of hydropeaking on riverine plants: A review. Biol. Rev.
**2018**, 93, 658–673. [Google Scholar] [CrossRef] [PubMed] - Pisaturo, G.R.; Righetti, M.; Castellana, C.; Larcher, M.; Menapace, A.; Premstaller, G. A procedure for human safety assessment during hydropeaking events. Sci. Total Environ.
**2019**, 661, 294–305. [Google Scholar] [CrossRef] [PubMed] - Premstaller, G.; Cavedon, V.; Pisaturo, G.R.; Schweizer, S.; Adami, V.; Righetti, M. Hydropeaking mitigation project on a multi-purpose hydro-scheme on Valsura River in South Tyrol/Italy. Sci. Total Environ.
**2017**, 574, 642–653. [Google Scholar] [CrossRef] [PubMed] - Gostner, W.; Lucarelli, C.; Theiner, D.; Kager, A.; Premstaller, G.; Schleiss, A. A holistic approach to reduce negative impacts of hydropeaking. In Proc. of International Symposium on Dams and Reservoirs under Changing Challenges; CRC Press, Taylor & Francis Group: Boca Raton, FL, USA, 2011; pp. 857–866. [Google Scholar] [CrossRef] [Green Version]
- Storli, P.T.; Lundström, T.S. A New Technical Concept for Water Management and Possible Uses in Future Water Systems. Water
**2019**, 11, 2528. [Google Scholar] [CrossRef] [Green Version] - Burman, A. Inherent Damping in a Partially Dry River. In Proceedings of the 38th IAHR World Congress, Panama City, Panama, 1–6 September 2019; pp. 5091–5100. [Google Scholar]
- Juárez, A.; Adeva-Bustos, A.; Alfredsen, K.; Dønnum, B.O. Performance of a two-dimensional hydraulic model for the evaluation of stranding areas and characterization of rapid fluctuations in hydropeaking rivers. Water
**2019**, 11, 201. [Google Scholar] [CrossRef] [Green Version] - Xie, Q.; Yang, J.; Lundström, S.; Dai, W. Understanding morphodynamic changes of a tidal river confluence through field measurements and numerical modeling. Water
**2018**, 10, 1424. [Google Scholar] [CrossRef] [Green Version] - Williams, R.D.; Brasington, J.; Hicks, M.; Measures, R.; Rennie, C.; Vericat, D. Hydraulic validation of two-dimensional simulations of braided river flow with spatially continuous aDcp data. Water Resour. Res.
**2013**, 49, 5183–5205. [Google Scholar] [CrossRef] [Green Version] - Horstman, E.; Dohmen-Janssen, M.; Hulscher, S. Modeling tidal dynamics in a mangrove creek catchment in Delft3D. Coast. Dyn.
**2013**, 2013, 833–844. [Google Scholar] - Yunus, A.C. Fluid Mechanics: Fundamentals And Applications (Si Units); Tata McGraw Hill Education Private Limited: New York, NY, USA, 2014; Volume 3. [Google Scholar]
- Ferziger, J.H.; Perić, M.; Street, R.L. Computational Methods for Fluid Dynamics; Springer: Berlin/Heidelberg, Germany, 2020; Volume 4. [Google Scholar]
- Cushman-Roisin, B.; Beckers, J.M. Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects; Academic Press: Cambridge, MA, USA, 2011. [Google Scholar] [CrossRef]
- Deltares. Delft3D-Flow User Manual; Deltares: Delft, The Netherlands, 2014. [Google Scholar]
- Richardson, L.F. IX. The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam. Philos. Trans. R. Soc. Lond. Ser. A Contain. Pap. A Math. Phys. Character
**1911**, 210, 307–357. [Google Scholar] [CrossRef] [Green Version] - Celik, I.B.; Ghia, U.; Roache, P.J.; Freitas, C.J.; Coleman, H.; Raad, P.E.; Celik, Ì.; Freitas, C.; Coleman, H. Procedure for estimation and reporting of uncertainty due to discretization in {CFD} applications. J. Fluids Eng.
**2008**. [Google Scholar] [CrossRef] [Green Version] - Vattenfall. Stornorrfors. Available online: https://powerplants.vattenfall.com/sv/stornorrfors (accessed on 25 March 2020).
- Angele, K.; Andersson, A. Validation of a HEC-RAS Model of the Stornorrfors Fish Migration Dry Reach against New Field Data. In Proceedings of the 12th International Symposium on Ecohydraulics, Tokyo, Japan, 19–24 August 2018. [Google Scholar]
- Enns, R.H.; McGuire, G.C.; Greene, R.L. Nonlinear physics with Maple for scientists and engineers. Comput. Phys.
**1997**, 11, 451–453. [Google Scholar] [CrossRef] - Kumar, V. Hysteresis. In Encyclopedia of Snow, Ice and Glaciers; Singh, V.P., Singh, P., Haritashya, U.K., Eds.; Springer: Dordrecht, The Netherlands, 2011; pp. 554–555. [Google Scholar] [CrossRef]
- Perumal, M.; Shrestha, K.B.; Chaube, U. Reproduction of hysteresis in rating curves. J. Hydraul. Eng.
**2004**, 130, 870–878. [Google Scholar] [CrossRef] - Länsstyrelsen i Norrbotten. Tappningsschema Gamla Älvfåran Stornorrfors 2017; Länsstyrelsen i Norrbotten: Luleå, Sweden, 2017; diarienummer 532-15339-15.
- Te, C.V. Open-Channel Hydraulics: International Student Ed; McGraw Hill: New York, NY, USA, 1959. [Google Scholar]
- Bakken, T.H.; King, T.; Alfredsen, K. Simulation of river water temperatures during various hydro-peaking regimes. J. Appl. Water Eng. Res.
**2016**, 4, 31–43. [Google Scholar] [CrossRef] - Deltares. QUICKIN User Manual; Deltares: Delft, The Netherlands, 2018. [Google Scholar]
- Scipy.org. SciPy. 2020. Available online: https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.optimize.fsolve.html (accessed on 8 April 2020).
- Pisaturo, G.R.; Righetti, M.; Dumbser, M.; Noack, M.; Schneider, M.; Cavedon, V. The role of 3D-hydraulics in habitat modelling of hydropeaking events. Sci. Total Environ.
**2017**, 575, 219–230. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**(

**a**) Position of Stornorrfors in Sweden. (

**b**) Key locations in the study reach and the extent of the numerical model.

**Figure 2.**Field measurements in the study reach: (

**a**) Digital elevation model of the study reach in meters above sea level (MASL). (

**b**) Water-surface elevation (WSE) in all eight validation points during a typical increase–decrease scenario.

**Figure 3.**Hydrograph for the six different scenarios under consideration: (

**a**) 10 flow changes per day. (

**b**) 20 flow changes per day. (

**c**) 30 flow changes per day. (

**d**) 40 flow changes per day. (

**e**) 50 flow changes per day. (

**f**) 60 flow changes per day.

**Figure 4.**Outcome of calibration in the study reach. (

**a**) Correlation plot between validation data and simulated WSE. (

**b**) Manning number distribution obtained from calibration.

**Figure 5.**The 400-DPI image files generated for ${A}_{wetted}$ calculation in the mesh study. The black borders correspond to ${A}_{total}$. (

**a**) Coarse mesh. (

**b**) Less-fine mesh. (

**c**) Finer mesh. (

**d**) Finest mesh.

**Figure 6.**Hysteresis loop for $WS{E}_{norm}$ in validation points 1, 3, 5, and 7, given the measured scenarios in Figure 2 and the simulated scenarios described in Section 3.3.1. (

**a**) Simulated hysteresis loop. The arrows indicate the direction of the process. (

**b**) Measured hysteresis loop.

**Figure 7.**WSE for the points 1, 2, 3, and 4 given the flow scenarios with 10, 20, and 30 flow changes per day. (

**a**) Validation point 1. (

**b**) Validation point 2. (

**c**) Validation point 3. (

**d**) Validation point 4.

**Figure 8.**WSE for the points 1, 2, 3, and 4 given the flow scenarios with 40, 50, and 60 flow changes per day. (

**a**) Validation point 1. (

**b**) Validation point 2. (

**c**) Validation point 3. (

**d**) Validation point 4.

**Figure 9.**WSE for the points 5, 6, 7, and 8 given the flow scenarios with 10, 20, and 30 flow changes per day. (

**a**) Validation point 1. (

**b**) Validation point 2. (

**c**) Validation point 3. (

**d**) Validation point 4.

**Figure 10.**WSE for the points 5, 6, 7, and 8 given the flow scenarios with 40, 50, and 60 flow changes per day. (

**a**) Validation point 5. (

**b**) Validation point 6. (

**c**) Validation point 7. (

**d**) Validation point 8.

**Figure 11.**Cross-section corresponding to the most upstream observation of continuous dynamics for the five different scenarios where it occurs. The legend is in flow changes per day.

**Figure 12.**Total wetted area dynamics for the six different scenarios. The legend is in flow changes per day.

Property | Value |
---|---|

Maximum error | 0.74 $\left[m\right]$ |

Minimum error | 0.02 $\left[m\right]$ |

Median error | 0.08 $\left[m\right]$ |

Standard deviation | 0.304 $\left[m\right]$ |

Pearson correlation | 0.9995 |

**Table 2.**Mesh properties for the four different meshes used in the study. Representative size is defined in Equation (9).

Grid | ${\mathbf{N}}_{\mathbf{x}}$ | ${\mathbf{N}}_{\mathbf{y}}$ | Nr. of Elements | Representative Size [1/m] |
---|---|---|---|---|

Coarse | 521 | 26 | 13546 | 0.0086 |

Less-Fine | 1040 | 74 | 76960 | 0.0036 |

Finer | 2078 | 218 | 453004 | 0.0015 |

Finest | 4154 | 218 | 905572 | 0.0011 |

**Table 3.**Wetted area for all meshes and Richardson extrapolated area. Error is given in percentage of the Richardson extrapolated value.

Grid | ${\mathbf{A}}_{\mathbf{wetted}}$ [m^{2}] | Error |
---|---|---|

Coarse | 863897 | +26.10% |

Less-Fine | 724442 | +5.71% |

Finer | 746164 | +8.88% |

Finest | 733532 | +7.03% |

Richardson Extrapolation | 685334 | - |

Standard Deviation | 0.304 |

**Table 4.**Time for each leg of the hysteresis loop for $WS{E}_{norm}$ obtained from Figure 6. The time it takes for the $WSE$ to reach the new steady state is referred to as increase time and decrease time. All units of time are in minutes.

Validation Point | Simulated | Measured | ||
---|---|---|---|---|

$\mathbf{WSE}$ Increase Time | $\mathbf{WSE}$ Decrease Time | $\mathbf{WSE}$ Increase Time | $\mathbf{WSE}$ Decrease Time | |

Point 1 | 29 | 44 | 32 | 69 |

Point 3 | 33 | 56 | 35 | 79 |

Point 5 | 35 | 69 | 36 | 88 |

Point 7 | 62 | 99 | 51 | 109 |

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## Share and Cite

**MDPI and ACS Style**

Burman, A.J.; Andersson, A.G.; Hellström, J.G.I.; Angele, K.
Case Study of Transient Dynamics in a Bypass Reach. *Water* **2020**, *12*, 1585.
https://doi.org/10.3390/w12061585

**AMA Style**

Burman AJ, Andersson AG, Hellström JGI, Angele K.
Case Study of Transient Dynamics in a Bypass Reach. *Water*. 2020; 12(6):1585.
https://doi.org/10.3390/w12061585

**Chicago/Turabian Style**

Burman, Anton J., Anders G. Andersson, J. Gunnar I. Hellström, and Kristian Angele.
2020. "Case Study of Transient Dynamics in a Bypass Reach" *Water* 12, no. 6: 1585.
https://doi.org/10.3390/w12061585