# Three-Dimensional Flow Characteristics in Slit-Type Permeable Spur Dike Fields: Efficacy in Riverbank Protection

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Laboratory Experiment

^{3}/s was supplied from the upstream region. The five spur dikes were installed 7 m from the upstream end of the right bank. Each spur dike consisted of cylindrical brass piles with diameters of 0.004 m. The spacing between two individual piles was 0.001 m, and the longitudinal interval of each was 0.30 m. A total of five spur dikes were installed, each with a permeability of 71%. Two types of installations were considered: a squared grid-type pile setting and a staggered grid-type pile setting (Figure 3e). The approach flow depth was 0.032 m, and uniform flow was confirmed by adjusting the flume-end weir height. The details of the hydraulic conditions of the flume are described in Table 2.

## 3. Numerical Simulation

#### 3.1. Governing Equations

#### 3.2. Model Schematization

^{3}/s supply of discharge upstream was given at the inlet (variableHeightFlowRateInletVelocity), enabling free water level oscillation (variableHeightFlowRate) and a constant total pressure at the outlet $\left({p}_{0}\right)$. A no-slip condition was applied to the walls as a velocity boundary condition. The upper surfaces of the mesh were considered to correspond to the atmosphere; therefore, flow was allowed to enter and leave the domain freely by applying the pressureInletOutletVelocity condition for velocity and the totalPressure condition for pressure. The wall function approach was used to link the turbulent flow domains near the wall-like structures. kqRWallFunction, omegaWallFunction, and nutUSpaldingWallFunction were applied to the walls as the turbulent boundary conditions. A mesh convergence analysis has been done by the author to some extent (considering the study mesh and 50% refinement of study mesh in all directions) and the author found no significant change in water depth and velocity component.

#### 3.3. Validation of the Flow Hydrodynamics

## 4. Results and Discussion

#### 4.1. Longitudinal Velocity

#### 4.2. Transverse Velocity

#### 4.3. Spatial Velocity

#### 4.4. Flow Depth

#### 4.5. Bed Shear Stress

#### 4.6. Application of Slit-Type Spurs in the Brahmaputra–Jamuna River

^{3}/s, which was used as one of the boundary conditions of the model. The river sediment size, d

_{50,}varied from 150 to 300 $\mu m$

_{,}corresponding to a critical sediment velocity of nearly 0.4 to 0.5 m/s [61]. The total reach length was almost 3290 m long and almost 850 m wide, with bathymetry varying from −0.13 to 9.7 m PDW (meter Public Works Datum, corresponds 0.46 m below mean sea level. The performance of the slit-type spur was compared with that for no spur and conventional impermeable spur dikes. Five spurs aligned 120° with the flow were installed with a permeability of 71%, thereby maintaining similarity with the laboratory experiment. Each spur dike consisted of fifteen individual unsubmerged piles. The pile diameter was 1.25 m, and the gap between the piles was 6.25 m. The spacing of the spurs was 187.46 m, and the length was 113.75 m. In the impermeable case, a similar thickness of the spur was chosen with the same length and spacing.

^{2}) occurred, and due to the installation of the spur dike, this stress decreased. In the case of slit-type spurs, the shear stress varies from 1.45 N/m

^{2}to 4.15 N/m

^{2}. Moreover, in the case of the impermeable spurs, the near-bank shear stress varies between 0.05 and 1.45 N/m

^{2}, but in this case, the opposite bank experiences a higher (22%) bed shear stress compared to that for slit-type spurs. As the considered river is braided in nature, this increased bed shear stress may expedite the development of the channel. Another observation is that, due to the channel bathymetry in the impermeable spur case, the main deflected flow impacts the bank further downstream (at approximately X = 2500 m). However, in the case of the slit-type spur, the modified field prevents such a bank-directed flow. In a real field, such a phenomenon was also observed. Figure 22 shows such an example in the study river (at Dhunot, Bogra, Bangladesh). Two impermeable spurs were installed during 2002 (left side figure), and they worked well in subsequent years, but the deflected flow (as well as the sedimentation pattern) caused approximately 650 m of bank erosion approximately 6 km downstream from 2003 to 2005.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

e | Exchange coefficient |

F | Froude number |

h | approach flow depth |

n | number of observations |

Q | discharge |

R | Reynolds number |

S | channel slope |

U | approach flow velocity |

W | channel width |

$k$ | turbulence kinetic energy |

$p$ | pressure |

$U$ | flow velocity vector |

$g$ | gravitational acceleration vector |

${p}_{o}$ | total pressure at outlet |

${U}_{*}$ | friction velocity |

$\stackrel{\xb4}{u},\stackrel{\xb4}{v},\stackrel{\xb4}{w}$ | fluctuation in velocity in the longitudinal, transverse, and vertical directions, respectively |

$\underset{\_}{u},\underset{\_}{v},\underset{\_}{w}$ | time-averaged mean velocity in the longitudinal, transverse, and vertical directions, respectively |

${\alpha}_{k1},{\alpha}_{k2},{\alpha}_{\omega 2},{\beta}^{*},\beta \gamma ,{c}_{1},{a}_{1}$ | turbulent model coefficients |

${S}_{k}$ | effective rate of production $k$ |

${S}_{i},{O}_{i}$ | simulated and observed values, respectively |

${S}_{\omega}$ | effective rate of production $\omega $ |

${\rho}_{a}$ | density of air |

${\rho}_{w}$ | density of water |

${\tau}_{b}$ | total bed shear stress |

${\tau}_{b}^{*}$ | dimensionless bed shear stress |

${\tau}_{b}^{x},{\tau}_{b}^{y}$ | components of bed shear stress in longitudinal and transverse directions |

${\vartheta}_{t}$ | kinematic (turbulent) viscosity |

τ_{o} | approach flow bed shear stress |

$\alpha $ | phase fraction |

$\rho $ | flow density |

$\omega $ | turbulence specific dissipation rate |

$\nabla $ | gradient operator for a three-dimensional region |

$\tau $ | stress tensor |

${f}_{\alpha}$ | the surface tension effects at the free surface |

${\mu}_{m}$ and $R$ | dynamic molecular viscosity and Reynolds stress tensor |

$k$ | the turbulent kinetic energy |

I | unit tensor |

${\vartheta}_{t}$ | turbulent kinematic eddy viscosity |

${S}_{k}$ and ${S}_{\omega}$ | effective rate of production |

${\rho}_{a}$ and ${\rho}_{f}$ | densities of air and flow, respectively |

${k}_{\alpha}$ | curvature of the of the interface |

${\sigma}_{T}$ | coefficient of surface tension, and |

${U}_{r}$ | relative velocity |

## References

- Mutton, D.; Haque, C.E. Human Vulnerability, Dislocation and Resettlement: Adaptation Processes of River-bank Erosion-induced Displacees in Bangladesh. Disasters
**2004**, 28, 41–62. [Google Scholar] [CrossRef] [PubMed] - Bryant, S.; Mosselman, E. Taming the Jamuna: Effects of river training in Bangladesh. In NCR Days 2017, Book of Abstracts; Hoitink, A.J.F., De Ruijsscher, T.V., Geertsema, T.J., Makaske, B., Wallinga, J., Candel, J.H.J., Poelman, J., Eds.; NCR publication: Atlanta, GA, USA, 2017; p. 114. [Google Scholar]
- Piégay, H.; Grant, G.; Nakamura, F.; Trustrum, N. Braided river management: From assessment of river behaviour to improved sustainable development. Braided Rivers
**2006**, 36, 257–275. [Google Scholar] - Nakagawa, H.; Zhang, H.; Baba, Y.; Kawaike, K.; Teraguchi, H. Hydraulic characteristics of typical bank-protection works along the Brahmaputra/Jamuna River, Bangladesh. J. Flood Risk Manag.
**2013**, 6, 345–359. [Google Scholar] [CrossRef] - Best, J.L.; Ashworth, P.J.; Sarker, M.H.; Roden, J.E. The Brahmaputra-Jamuna River, Bangladesh. In Large Rivers: Geomorphology and Management; Gupta, A., Ed.; John Wiley & Sons.: Hoboken, NJ, USA, 2007; pp. 395–430. [Google Scholar]
- Sarker, M.H.; Akter, J.; Ruknul, M. River bank protection measures in the Brahmaputra-Jamuna River: Bangladesh experience. In Proceedings of the International Seminar on’River, Society and Sustainable Development, Assam, India, 26–29 May 2011. [Google Scholar]
- Bhuiyan, A.F.; Hossain, M.M.; Heyl, R. Bank erosion and protection on the Brahmaputra (Jamuna) River. In Hydraulic Information Management; Brebbia, C.A., Blain, W.R., Eds.; WIT Press: Southampton, UK, 2002; p. 10. ISBN 1853129127. [Google Scholar]
- Koken, M.; Constantinescu, G. An investigation of the flow and scour mechanisms around isolated spur dikes in a shallow open channel: Conditions corresponding to the initiation of the erosion and deposition process. Water Resour. Res.
**2008**, 44, 1–19. [Google Scholar] [CrossRef] - Rajaratnam, N.; Nwachukwu, B.A. Flow Near Groin-Like Structures. J. Hydraul. Eng.
**1983**, 109, 463–480. [Google Scholar] [CrossRef] - Uijttewaal, W.S. Effects of Groyne Layout on the Flow in Groyne Fields: Laboratory Experiments. J. Hydraul. Eng.
**2005**, 131, 782–791. [Google Scholar] [CrossRef] - Uijttewaal, W.S.J. The flow in groyne fields: Patterns and exchange processes. In Water Quality Hazards and Dispersion of Pollutants; Czernuszenko, W., Rowinski, P., Eds.; Springer: Boston, MA, USA, 2005; pp. 231–246. ISBN 0387233210. [Google Scholar]
- Li, Y.; Altinakar, M. Effects of a Permeable Hydraulic Flashboard Spur Dike on Scour and Deposition Yujian. In Proceedings of the World Environmental and Water Resources Congress 2016, West Palm Beach, FL, USA, 22–26 May 2016; pp. 399–409. [Google Scholar]
- Wu, B.; Wang, G.; Ma, J.; Zhang, R. Case Study: River Training and Its Effects on Fluvial Processes in the Lower Yellow River, China. J. Hydraul. Eng.
**2005**, 131, 85–96. [Google Scholar] [CrossRef] - Ali, M.S.; Hasan, M.M.; Haque, M. Two-Dimensional Simulation of Flows in an Open Channel with Groin-Like Structures by iRIC Nays2DH. Math. Probl. Eng.
**2017**, 2017. [Google Scholar] [CrossRef] - Cao, X.; Gu, Z.; Tang, H. Study on spacing threshold of nonsubmerged spur dikes with alternate layout. J. Appl. Math.
**2013**, 2013. [Google Scholar] [CrossRef] - Gisonni, C.; Hager, W.H. Spur Failure in River Engineering. J. Hydraul. Eng.
**2008**, 134, 135–145. [Google Scholar] [CrossRef] - Gu, Z.; Akahori, R.; Ikeda, S. Study on the transport of suspended sediment in an open channel flow with permeable spur dikes. Int. J. Sediment Res.
**2011**, 26, 96–111. [Google Scholar] [CrossRef] - Kang, J.; Yeo, H.; Kim, S.; Ji, U. Permeability effects of single groin on flow characteristics. J. Hydraul. Res.
**2011**, 49, 728–735. [Google Scholar] [CrossRef] - Duan, J.G. Three-dimensional Mean Flow and Turbulence around a Spur Dike. J. Hydraul. Eng.
**2009**, 135, 803–811. [Google Scholar] [CrossRef] - Copeland, R.R. Bank Protection Techniques Using Spur Dikes; PN: Cebu, Philippines, 1983. [Google Scholar]
- Ghodsian, M.; Vaghefi, M. Experimental study on scour and flow field in a scour hole around a T-shape spur dike in a 90° bend. Int. J. Sediment Res.
**2009**, 24, 145–158. [Google Scholar] [CrossRef] - Kuhnle, R.; Alonso, C. Flow near a model spur dike with a fixed scoured bed. Int. J. Sediment Res.
**2013**, 28, 349–357. [Google Scholar] [CrossRef] - Kuhnle, R.A.; Alonso, C.V.; Shields, F.D. Local Scour Associated with Angled Spur Dikes. J. Hydraul. Eng.
**2002**, 128, 1087–1093. [Google Scholar] [CrossRef][Green Version] - Raudkivi, A.J. Permeable Pile Groins. J. Waterw. Port Coastal Ocean Eng.
**1996**, 122, 267–272. [Google Scholar] [CrossRef] - Bakker, W.T.; Hulsbergen, C.H.; Roelse, P.; De Smit, C.; Svasek, J.N. Permeable groynes: Experiments and practice in the Netherlands. Coast. Eng.
**1984**, 1985, 649–660. [Google Scholar] [CrossRef] - Koken, M.; Constantinescu, G. Flow and turbulence structure around a spur dike in a channel with a large scour hole. Water Resour. Res.
**2011**, 47, 1–19. [Google Scholar] [CrossRef] - Mansoori, A.R. Study on Flow and Sediment Transport around Series of Spur Dikes with Different Head Shape. Bachelor’s Thesis, Kyoto University, Kyoto, Japan, 2014. [Google Scholar]
- Zang, H.; Nakagawa, H.; Kawaike, K.; Baba, Y. Experiment and simulation of turbulent flow in local scour around a spur dyke. Int. J. Sediment Res.
**2009**, 24, 33–45. [Google Scholar] [CrossRef] - Alauddin, M. Morphological Stabilization of Lowland Rivers by Using a Series of Groynes. Ph.D. Thesis, Department of Civil Engineering, Nagoya University, Nagoya, Japan, 2011. [Google Scholar]
- Shampa; Hasegawa, Y.; Nakagawa, H.; Takebayashi, H.; Kawaike, K. Installation Effects on Three-Dimensional Flow Characteristics in a Slit-Type Permeable Spur Dike Field. In Proceedings of the 37th Annual Meeting of the Natural Disaster Society of Japan, Sendai, Japan, 6–7 October 2017; pp. 3–4. [Google Scholar]
- Zhang, H.; Nakagawa, H.; Kawaike, K.; Nishio, K. Experiment on Suspended sediment Transport around Bank Protection Structures. In Proceedings of the 5th International Conference on Water & Flood Management (ICWFM-2015), Dhaka, Bangladesh, 6–8 March 2015; pp. 127–134. [Google Scholar]
- Ettema, R.; Kirkil, G.; Muste, M. Similitude of Large-Scale Turbulence in Experiments on Local Scour at Cylinders. J. Hydraul. Eng.
**2006**, 132, 33–40. [Google Scholar] [CrossRef] - Ettema, R.; Muste, M. Scale effects in flume experiments on flow around a spur dike in flatbed channel. J. Hydraul. Eng.
**2004**, 130, 635–646. [Google Scholar] [CrossRef] - Safarzadeh, A.; Ali, S.; Salehi, A.; Zarrati, A.R. Experimental Investigation on 3D Turbulent Flow around Straight and T-Shaped Groynes in a Flat Bed Channel. J. Hydraul. Eng.
**2016**, 142, 1–15. [Google Scholar] [CrossRef] - Dey, S.; Barbhuiya, A.K. Turbulent flow field in a scour hole at a semicircular abutment. Can. J. Civ. Eng.
**2005**, 32, 213–232. [Google Scholar] [CrossRef] - Li, H.; Barkdoll, B.D.; Kuhnle, R.; Alonso, C. Parallel Walls as an Abutment Scour Countermeasure. J. Hydraul. Eng.
**2006**, 132, 510–520. [Google Scholar] [CrossRef] - Klingemann, P.C.; Kehe, S.M.; Owusu, Y.A. Streambank Erosion Protection and Channel Scour Manipulation Using Rockfill Dikes and Gabions; Water Resources Research Institute, Oregon State University: Corvallis, OR, USA, 1984; pp. 1–872. [Google Scholar]
- Trampenau, T.; Goricke, F.; Raudkivi, A.J. Permeable Pile Groins. In Coastal Engineering; ASCE: Reston, VA, USA, 1996; Volume 122, pp. 2142–2151. [Google Scholar]
- Fukuoka, S.; Watanbe, A.; Kawaguchi, H.; Yasutake, Y. A study of permeable groins in series installed in a straight channel. Annu. J. Hydraul. Eng.
**2000**, 44, 1047–1052. [Google Scholar] [CrossRef] - Li, Z.; Kohji, M.; Maeno, S.; Ushita, T.; Fujii, A. Hydraulic characteristics of a group of permeable groins constructed in an open channel flow. J. Appl. Mech. JSCE
**2005**, 8, 773–782. [Google Scholar] [CrossRef] - Mioduszewski, T.; Maeno, S. Three Dimensional Around a Porous Analysis Spur of Flow Dike. J. Appl. Mech.
**2005**, 8, 793–801. [Google Scholar] [CrossRef] - Menter, F.R. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J.
**1994**, 32, 1598–1605. [Google Scholar] [CrossRef][Green Version] - Hellsten, A. Some Improvements in Menter’s k-omega SST Turbulence Model. In Proceedings of the 29th AIAA Fluid Dynamics Conference 1998, Albuquerque, NM, USA, 15–18 June 1997; pp. 2–12. [Google Scholar]
- Menter, F.R.; Kuntz, M.; Langtry, R. Ten Years of Industrial Experience with the SST Turbulence Model, Proceedings of the Fourth International Symposium on Turbulence, Heat and Mass Transfer, Antalya, Turkey, 12–17 October 2003; Begell House: Antalya, Turkey, 2003; pp. 625–632. [Google Scholar]
- Hirt, C.W.; Nichols, B.D. Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys.
**1981**, 39, 201–225. [Google Scholar] [CrossRef] - Zhou, L. Numerical Modelling of Scour in Steady Flows. Ph.D. Thesis, University of Lyon, Lyon, France, 2017. [Google Scholar]
- Weller, H.G. Derivation, modelling and solution of the conditionally averaged two-phase flow equations. Tech. Rep.
**2002**, TR/HGW 2, 9. [Google Scholar] - Berberović, E.; Van Hinsberg, N.P.; Jakirlić, S.; Roisman, I.V.; Tropea, C. Drop impact onto a liquid layer of finite thickness: Dynamics of the cavity evolution. Phys. Rev. E Stat. Nonlinear Soft Matter Phys.
**2009**, 79, 036306. [Google Scholar] [CrossRef] [PubMed] - Weller, H.G.; Tabor, G.; Jasak, H.; Fureby, C. A tensorial approach to computational continuum mechanics using object-oriented techniques. Comput. Phys.
**1998**, 12, 620–631. [Google Scholar] [CrossRef] - OpenCFD Programmer’s Guide. Available online: http://foam.sourceforge.net/docs/Guides-a4/ProgrammersGuide.pdf (accessed on 26 March 2020).
- Next Foam Boundary Conditions- OpenFOAM-4.1. Available online: http://www.nextfoam.co.kr/lib/download.php?idx=135228&sid=235c1d3fc28364657dbb43ccfe025b25 (accessed on 27 March 2020).
- Roache, P.J. Quantification of Uncertainty in Computational Fluid Dynamics. Annu. Rev. Fluid Mech.
**1997**, 29, 123–160. [Google Scholar] [CrossRef][Green Version] - Roache, P.J. Perspective: Validation—What does it mean? J. Fluids Eng. Trans. ASME
**2009**, 131, 0345031–0345034. [Google Scholar] [CrossRef] - Blocken, B.; Gualtieri, C. Ten iterative steps for model development and evaluation applied to Computational Fluid Dynamics for Environmental Fluid Mechanics. Environ. Model. Softw.
**2012**, 33, 1–22. [Google Scholar] [CrossRef] - Santhi, C.; Arnold, J.G.; Williams, J.R.; Dugas, W.A.; Srinivasan, R.; Hauck, L.M. Validation of the SWAT model on a large river basin with point and nonpoint sources. J. Am. Water Resour. Assoc.
**2002**, 37, 1169–1188. [Google Scholar] [CrossRef] - Bracmort, K.S.; Arabi, M.; Frankenberger, J.R.; Engel, B.A.; Arnold, J.G. Modeling Long-Term Water Quality Impact of Structural BMPs. Trans. Am. Soc. Agric. Biol. Eng.
**2006**, 49, 367–374. [Google Scholar] [CrossRef][Green Version] - Gupta, H.V.; Sorooshian, S.; Yapo, P.O. Status of Automatic Calibration for Hydrologic Models: Comparison With Multilevel Expert Calibration. J. Hydrol. Eng.
**2001**, 4, 135–143. [Google Scholar] [CrossRef] - Melville, B.W.; Coleman, S.E. Bridge Scour; Water Resources Publication: Littleton, CO, USA, 2000; ISBN 1887201181. [Google Scholar]
- Zhang, H.; Nakagawa, H.; Ogura, M.; Mizutani, H. Experiment Study on Channel Bed Characteristics around Spur Dykes of Different Shapes. J. Jpn. Soc. Civ. Eng. Ser. B1
**2013**, 69, 489–499. [Google Scholar] [CrossRef][Green Version] - Dubief, Y.; Delcayre, F. On coherent-vortex identification in turbulence. J. Turbul.
**2000**, 1, 11. [Google Scholar] [CrossRef] - Van Rijn, L.C. Principles of Sedimentation and Erosion Engineering in Rivers, Estuaries and Coastl Seas; Aqua Publications: Blokzijl, The Netherlands, 2005; pp. 1–15. [Google Scholar]
- Gualtieri, C. Numerical simulation of flow patterns and mass exchange processes in dead zones. In Proceedings of the 4th International Congress on Environmental Modelling and Software—Barcelona, Catalonia, Spain, 7–10 July 2008; Volume 1, pp. 150–161. [Google Scholar]
- Weitbrecht, V.; Socolofsky, S.A.; Jirka, G.H. Experiments on Mass Exchange between Groin Fields and Main Stream in Rivers Volker. J. Hydraul. Eng.
**2007**, 134, 173–183. [Google Scholar] [CrossRef] - Ouro, P.; Juez, C.; Franca, M. Drivers for mass and momentum exchange between the main channel and river bank lateral cavities. Adv. Water Resour.
**2020**, 137, 103511. [Google Scholar] [CrossRef]

**Figure 1.**Permeable spur dike in Bangladesh. The shaded area indicates the location perpendicular to the structure.(

**a**) a failed semipermeable spur dike in the Brahmaputra–Jamuna River. (

**b**) initial spur constructed. (

**c**) reinforced cement concrete (RCC) part of the dike. (

**d**) acoustic Doppler current profiler (ADCP).

**Figure 6.**Example comparison of experimental and simulated three-dimensional velocities and water depths.

**Figure 7.**Dimensionless longitudinal velocity vector ($\frac{\underset{\_}{u}}{U},\frac{\underset{\_}{w}}{U}$) 0.11 m from the right bank. The contour shows the variation of the longitudinal velocity component ($\frac{\underset{\_}{u}}{U}$ ).

**Figure 8.**Box-and-whisker plot of the magnitude of the dimensionless longitudinal velocity $\left(=\frac{1}{U}\left(\sqrt{\left({\underset{\_}{u}}^{2}+{\underset{\_}{w}}^{2}\right)}\right)\right)$.

**Figure 9.**The ratio of the depth-averaged longitudinal velocity decrement to the approach velocity 0.02 m upstream and 0.02 m downstream of the spur dike.

**Figure 10.**The dimensionless transverse velocity vector $\left(\frac{\underset{\_}{v}}{U},\frac{\underset{\_}{w}}{U}\right)$ 0.02 m upstream of the third spur dike. The contour shows the variation in the transverse velocity component $\left(\frac{\underset{\_}{v}}{U}\right)$.

**Figure 12.**Dimensionless spatial velocity $\left(\frac{\underset{\_}{u}}{U},\frac{\underset{\_}{v}}{U}\right)$ at Z = 0.01 m. The contour shows the variation in the longitudinal velocity component $\left(\frac{\underset{\_}{u}}{U}\right)$.

**Figure 15.**The distribution of the spatial velocity ($\underset{\_}{u},\underset{\_}{v}$) in m/s. The contours show the magnitude of the spatial velocity ($=\sqrt{\left({\underset{\_}{u}}^{2}+{\underset{\_}{v}}^{2}\right)}$ in m/s.

**Figure 16.**The distribution of the spatial velocity ($\underset{\_}{u},\underset{\_}{v}$) in m/s inside the third embayment. The contours show the magnitude of the spatial velocity ($=\sqrt{\left({\underset{\_}{u}}^{2}+{\underset{\_}{v}}^{2}\right)}$ in m/s.

**Figure 17.**The distribution of the transverse velocity ($\underset{\_}{v},\underset{\_}{w}$) in m/s. The contours show the variation in the transverse velocity $\underset{\_}{v}$ in m/s.

**Figure 18.**Visualization of the vortical structure of the mean flow using the Q criterion for slit-type and impermeable spurs. The magnitude of the velocity $(=\left(\sqrt{\left({\underset{\_}{u}}^{2}+{\underset{\_}{v}}^{2}+{\underset{\_}{w}}^{2}\right)}\right)$ is shown in (m/s).

**Figure 20.**Distribution of pressure gradient magnitude around the third spur (near bed): (

**a**) slit type spur, (

**b**) impermeable spur.

**Figure 21.**The distribution of the bed shear stress along the spur zone (the blue line shows the stream trace).

**Figure 22.**Bank erosion downstream of the impermeable spur at Brahmaputra–Jamuna (location: Dhunot, Bogra, Bangladesh).

Cases | Angle to the u/s Flow (Degree) | Installation Arrangement | Number of Spur Dikes | Number of Piles in Each Spur Dike |
---|---|---|---|---|

1 | 90 | Squared | 5 | 13 |

2 | 90 | Staggered | 5 | 13 |

3 | 120 | Squared | 5 | 15 |

4 | 120 | Staggered | 5 | 15 |

5 | 60 | Squared | 5 | 15 |

6 | 60 | Staggered | 5 | 15 |

Parameter | Unit | Values |
---|---|---|

Discharge, Q | m^{3}/s | 0.01 |

Channel slope, s | - | 1/300 |

Channel width, W | m | 0.80 |

Approach flow depth, h | m | 0.032 |

Approach flow velocity, U | m/s | 0.40 |

Friction velocity, ${U}_{*}$ | m/s | 0.0323 |

Reynolds number, R | - | 34,430 |

Froude number, F | - | 0.71 |

Approach flow bed shear stress, τ_{o} | N/m^{2} | 1.0464 |

Manning’s n | s/m^{1/3} | 0.015 |

${\mathit{\alpha}}_{\mathit{k}1}$ | ${\mathit{\alpha}}_{\mathit{k}2}$ | ${\mathit{\alpha}}_{\mathit{\omega}1}$ | ${\mathit{\alpha}}_{\mathit{\omega}2}$ | ${\mathit{\beta}}^{*}$ | $\mathit{\beta}$ | $\mathit{\gamma}$ | ${\mathit{c}}_{1}$ | ${\mathit{a}}_{1}$ |
---|---|---|---|---|---|---|---|---|

0.85 | 1.0 | 0.5 | 0.856 | 0.9 | 0.075 | 0.5532 | 10 | 0.31 |

Domain Patch | Velocity, U | Pressure, $\mathit{p}$ | Turbulent Kinetic Energy, k | Specific Dissipation Rate, ω | Turbulent Kinematic Eddy Viscosity, ${\mathit{\vartheta}}_{\mathit{t}}$ | Phase Fraction, α |
---|---|---|---|---|---|---|

Inlet | variableHeightFlowRateInletVelocity ^{i} | zero gradient | Uniform fixed value | Uniform fixed value | Uniform fixed value | variableHeightFlowRate |

Outlet | pressureInletOutletVelocity ^{ii} | totalPressure ^{iii} | internalField | Uniform fixed value | Calculated from other patch fields | inletOutlet ^{iv} |

Walls | noSlip | zero gradient | kqRWallFunction | omegaWallFunction | nutUSpaldingWallFunction | zero gradient |

Atmosphere | pressureInletOutletVelocity | totalPressure | uniform fixed value | inletOutlet | Calculated from other patch fields | inletOutlet |

Bed | noSlip | zero gradient | kqRWallFunction | omegaWallFunction | nutUSpaldingWallFunction | zero gradient |

^{i}Velocity was estimated by using ${U}_{avg}=\frac{Q}{{\alpha}_{1}.S}$.

^{ii}When p is known at the inlet,

**U**was evaluated from the flux normal to the patch.

^{iii}The patch pressure was estimated from $p={p}_{0}-\frac{1}{2}\left|U\right|$.

^{iv}The outflow condition was generic (zero gradient), with a specified inflow for the case of a return flow.

Cases | PBIAS (%) | |
---|---|---|

Water Depth | Velocity Magnitude | |

Case 1 | −5.06 | −11.05 |

Case 2 | 3.31 | −18.12 |

Case 3 | 1.33 | −15.92 |

Case 4 | 8.76 | 3.79 |

Case 5 | 9.83 | 8.68 |

Case 6 | 11.82 | −6.27 |

Unit | Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Case 6 |
---|---|---|---|---|---|---|

Percentage (%) | 27.60 | 32.83 | 33.71 | 34.48 | 24.69 | 29.97 |

© 2020 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**

Shampa; Hasegawa, Y.; Nakagawa, H.; Takebayashi, H.; Kawaike, K.
Three-Dimensional Flow Characteristics in Slit-Type Permeable Spur Dike Fields: Efficacy in Riverbank Protection. *Water* **2020**, *12*, 964.
https://doi.org/10.3390/w12040964

**AMA Style**

Shampa, Hasegawa Y, Nakagawa H, Takebayashi H, Kawaike K.
Three-Dimensional Flow Characteristics in Slit-Type Permeable Spur Dike Fields: Efficacy in Riverbank Protection. *Water*. 2020; 12(4):964.
https://doi.org/10.3390/w12040964

**Chicago/Turabian Style**

Shampa, Yuji Hasegawa, Hajime Nakagawa, Hiroshi Takebayashi, and Kenji Kawaike.
2020. "Three-Dimensional Flow Characteristics in Slit-Type Permeable Spur Dike Fields: Efficacy in Riverbank Protection" *Water* 12, no. 4: 964.
https://doi.org/10.3390/w12040964