## Abstract

The effect of using permeable spur dikes on the produced maximum scour depth compared to that of solid spur dikes is numerically investigated. The numerical model used for such purpose is the Nays-2DH model of the International River Interface Cooperative (iRIC) software package for bed and bank erosion. The model results are verified using the experimental data collected in this study by conducting experiments on five different models of spur dikes having different opening ratios. Using the statistical performance indices, the root mean square error and the coefficient of determination, the results showed an acceptable agreement between the numerical model results for the relative maximum scour depth defined by the ratio of the maximum scour depth to the flow depth and their corresponding observed values. A new empirical equation using nonlinear regression is developed using the experimental data collected in this study and tested with another existing empirical equation available in the literature for their accuracy in determining the relative maximum scour depth.

## NOTATION

*B*Channel bottom width

*d*_{50}Mean sediment grain size

*d*_{s}Maximum scour depth

*d*_{se}Maximum scour depth, Equation (1)

*F*_{r}Froude number

- g
Gravitational acceleration

*h*Water depth at time t

*H*Total water depth

*C*_{f}Drag coefficient of the bed shear stress

*L*Spur dike length

*n*Number of openings

*n*_{m}Manning's coefficient of roughness

*P*_{r}Opening ratio

*q*_{bx}Bed load transport in x direction

*q*_{by}ed load transport in y direction

*R*Opening ratio, Equation (1)

Hydraulic radius of bed

Grain specific weight

*t*Time

*u*Depth-averaged velocity in

*x*direction*v*Depth-averaged velocity in

*y*direction*V*Average approaching flow velocity

*V*_{c}Critical velocity at incipient flow condition

- x
Width of opening

- y
Water depth

*z*Bed elevation

*ν*_{t}Eddy viscosity coefficient

Porosity of the bed material

Component of the shear stress of river bed in

*x*directionComponents of the shear stress of river bed in

*y*directionNon-dimensional bed shear stress

Critical shear stress

Time step

## INTRODUCTION

Spur dikes are simple structures used to protect river banks from erosion by deviating the flow away from the banks towards the center of the river. However, their construction causes scour activities in their vicinity due to the formation of horseshoe, wake vortices and vertical component of down flow (Ettema & Muste 2004), and which may cause failure to the spur dike itself. Spur dikes can be permeable or impermeable. For permeable spur dikes, the flow partly penetrates the structure which results in a considerable reduction in velocity, vortex strength, and shear force at the nose of the spur dike (Li *et al.* 2005), and hence a remarkable reduction of maximum scour depth is observed (Fukuoka *et al.* 2000). As well, permeable spur dikes have the advantages of being more stable and requiring easier maintenance than impermeable ones (Kang *et al.* 2011). Scour around spur dikes occurs under clear water conditions where there is no transport of sediment by the approaching flow to the region of scour activities around the spur dike or under live bed scour where sediment is transported by the approaching flow as bed load or suspended load to the scour hole at the spur dike (Pandy *et al.* 2015).

*et al.*(2011) experimentally studied the influence of the aspect ratio defined by the interval between a series of permeable spur dikes to the length of the dikes on the transport of suspended sediment in open channels. Nasrollahi

*et al.*(2008) experimentally investigated the local scour at permeable spur dikes under clear water scour condition. The flow intensity defined by the ratio of the average approaching flow velocity

*V*to the critical velocity at incipient flow condition

*V*was assumed equal to unity. They further assumed the approaching flow Froude number is equal to the flow intensity at incipient motion conditions and both are equal to unity. The maximum scour depth,

_{c}*d*was determined using the following relationship: where

_{se}*y*is depth of flow,

*L*is length of spur dike,

*B*is width of main channel and

*R*is the opening ratio. A considerable reduction in maximum scour depth was observed. It is worth mentioning that Equation (1) is independent of the approaching flow Froude number by assuming that the experiments were conducted under incipient motion conditions and that flow intensity is equal to Froude number. This assumption may not be true because critical flow conditions does not necessarily mean incipient motion conditions. Kang

*et al.*(2011) experimentally investigated the flow characteristics for permeable spur dikes of different opening ratios. They observed a decrease in the velocity, the vortex strength, and the scale of the recirculation zone at the tip of the permeable spur dike as the opening ratio increases. Its resistance to the flow is lower than that of the permeable spur dike. By varying the opening ratio, the flow gradient along the bank can be adjusted. However, to maintain a navigation channel, this type of spur dike is less suitable. Zhang

*et al.*(2013) investigated the bed variation around a hybrid type of spur dikes which is a combination of both impermeable and permeable dikes.

In the present study, the maximum scour depth occurring at permeable and impermeable spur dikes is simulated using the iRIC 2DH numerical model and the results are experimentally verified. Also, a nonlinear regression model is developed to predict the relative maximum scour depth using the experimental data. Both simulated and predicted results were tested for their accuracy and compared to that previously developed by Nasrollahi *et al.* (2008).

## EXPERIMENTAL ARRANGEMENT

*d*

_{50}= 0.75 mm size. All experiments were conducted under clear water condition where the flow intensity

*V/Vc*is less than unity. As shown in Figure 2, five constant length spur dike models having different permeability ratios,

*P*, of 0, 0.23, 0.35, 0.45, and 0.55 were used, where

_{r}*P*= 0 represents the case of impermeable spur dike (reference case). It has to be noted that

_{r}*P*is defined as: where

_{r}*L*is length of spur dike,

*n*is number of openings, and

*x*is width of opening.

All spur dike models were made of polished wood of thickness 3 cm and 15 cm length which provides a contraction ratio of 0.2 which was kept constant for all dike models. The spur dikes were placed at right angles with the flow direction. A discharge of 0.02 m^{3}/s was kept constant for each of the five spur dike models with different water depths of 0.12, 0.1, and 0.08 m adjusted by the pivoted tailgate located at the downstream end of the channel and which provided different values of Froude number, *Fr*, ranging from 0.2 to 0.37. The discharge was measured using a calibrated sharp crested rectangular weir located at the entrance of the intake sump. After equilibrium scour condition was reached, scour depths were measured using digital point gauge with an accuracy of ±0.1 mm. Figure 3 shows views of the scour pattern at equilibrium stage for both permeable and impermeable spur dikes.

It has previously been reported that 94% of the scour process almost occurs within the first 3 hours (Masjedi *et al.* 2010). Karami *et al.* (2012) concluded that 70–90% of the scour process occurs within the first 20% of the equilibrium time. Therefore, the time to equilibrium in the present study was considered to be approximately 3 hours.

## NUMERICAL MODEL

Many computational fluid dynamics (CFD) models, namely, SSIIM 2.0, Fluent and flow 3-D have been developed and applied by several researchers for simulating flow and scour around spur dikes. In the present study, the Nays-2DH model of the iRIC software package (Shimizu & Takebayashi 2011), which is available at http://www.i-ric.org is presented to test its capability in simulating flow and sediment transport around spur dikes. The model has previously been used by Allauddin & Tsujimoto (2012), Kaffle (2014), and Shahjahan *et al.* (2017). The results of the numerical model obtained in the present study are verified using the experimental data collected in this study.

The basic two-dimensional hydrodynamic equations in an orthogonal coordinate system (*x*, *y*) for the model are given as follows (Nays-2DH solver manual).

*x-*and

*y-*directions: in which where

*h*is water depth at time

*t*,

*u*and

*v*are depth-averaged velocities in

*x*- and

*y*-directions,

*g*is gravitational acceleration,

*H*is the total water depth, and are the components of the shear stress of river bed in

*x*and

*y*directions,

*C*is the drag coefficient of the bed shear stress,

_{f}*ν*

_{t}is eddy viscosity coefficient, and

*n*is the Manning's coefficient of roughness. The depth-averaged k-ɛ model is used as turbulence model with finite differential advections as upwind scheme.

_{m}### Sediment transport equations

*z*is the bed elevation, and are the bed load transport in

_{b}*x*- and

*y*-directions, and is the porosity of the bed material. The model allows the user to calculate both and using either Ashida & Michiue (1972) or Meyer-Peter & Muller (1948) formulae. In the present study, the Meyer-Peter and Müller bed load formula was used, which is given by: where is non-dimensional bed shear stress, is critical shear stress, is hydraulic radius of bed and is grain specific weight.

## GRID GENERATION AND COMPUTATIONAL SCHEME

A constant computational domain of 1.0 m in the *x-*direction with a total number of 41 grid points and 0.74 m in the *y*-direction with a number of grid points that varies according to the width of the opening such that each opening is represented by two cells as illustrated in Figure 4, which shows that for the case of opening width of 0.017 m corresponding to an opening ratio, *Pr,* of 0.45, a total number of 88 grid points was considered. The governing equations for velocities and shear stresses were discretized by finite difference method and then solved for the unknown values by an iterative procedure at each grid point in the computational domain with the boundary conditions set by considering a constant discharge of 0.02 m^{3}/sec at the upstream end and a constant depth of 0.12, 0.1, and 0.08 m at the downstream end. The sediment transport model was then solved using the bed load formula and used to compute the change in bed elevations using the sediment continuity equation.

## EXPERIMENTAL RESULTS, ANALYSIS, AND DISCUSSION

Figure 5 shows the decrease of the relative maximum scour depth with the increase of the opening ratio. The figure also illustrates the increase of the relative scour depth with the increase of the Froude number. A remarkable reduction in the scour depth is observed for permeable spur dikes than impermeable ones especially at higher values of Froude number. For values of opening ratio of 0.23, the reduction reaches about 30% and reaches 80% for opening ratio of 0.55.

Figures 6 and 7 show the scour depth contours and bed topography for both permeable and impermeable spur dikes, respectively.

Figure 8 shows the line of perfect agreement between the predicted relative scour depth using Equation (8) and the observed data at equilibrium stage with R^{2} of 0.968 and RMSE of 0.066. This means that the regression model predicted the relative scour depth as a function of Froude number and opening ratio with the highest possible degree of accuracy for the range of data presented in this study.

## NUMERICAL RESULTS AND ANALYSIS

Figure 9 illustrates the change in flow pattern (streamlines and velocity vectors) as obtained from the results of the numerical model for both impermeable and permeable spur dikes. The penetration of flow through the dike openings for permeable dikes and the deviation of the flow away from the banks for impermeable dikes are clearly detected by the model.

Figure 10 illustrates the accuracy of the numerical model to simulate the relative scour depth as compared to the observed data. An acceptable agreement is obtained as the statistical indices RMSE and R^{2} were determined as 0.125 and 0.859, respectively. The numerical model is very sensitive to the value of Manning's *n _{m}* calculated as a function of the average roughness height

*k*for the given

_{s}*d*of the bed material (Kafle 2014).

_{50}The values of (*ds/y*) determined using Equation (1) by Nasrollahi *et al.* (2008) are tested against the observed values and the results led to a fairly poor agreement. The scatter shown in Figure 11 and the values of the statistical indices RMSE and R^{2} of 0.409 and 0.312, respectively, led to the key role of Froude number as an input variable in determining the scour depth for both permeable and impermeable spur dikes.

A summary of the values of the statistical indices RMSE and R^{2} is shown in Table 1, and which indicates that the nonlinear regression model presented by Equation (7) better predicted the relative maximum scour depth (*ds/y*) than both numerical model and Nasrollahi *et al.* (2008) regression model.

Statistical index | RMSE | R^{2} |
---|---|---|

Relative scour depth (d) _{s}/y | ||

Predicted (nonlinear regression), Equation (8) | 0.066 | 0.968 |

Simulated (Nays-2DH) | 0.125 | 0.859 |

Nasrollahi et al. (2008), Equation (1) | 0.409 | 0.312 |

Statistical index | RMSE | R^{2} |
---|---|---|

Relative scour depth (d) _{s}/y | ||

Predicted (nonlinear regression), Equation (8) | 0.066 | 0.968 |

Simulated (Nays-2DH) | 0.125 | 0.859 |

Nasrollahi et al. (2008), Equation (1) | 0.409 | 0.312 |

## CONCLUSION

Permeable spur dikes as river training structures are very effective in reducing the scour depth produced by impermeable spur dikes. Extensive research work has been conducted for the determination of the maximum scour depth for impermeable spur dikes. However, very few equations are available for the determination of the maximum scour depth for permeable spur dikes. In this research, the relative maximum scour depth was numerically and experimentally investigated for permeable spur dikes with different opening ratios under different flow conditions.

A new empirical equation was developed using the experimental data and showed a better accuracy than the numerical results. The Nays-2DH numerical model of the iRIC software package simulated the scour around permeable and impermeable spur dikes with an acceptable degree of accuracy when verified by the experimental data, and although it is a 2D model, proved to be a promising numerical model for simulating bed and bank erosion in open channels. Furthermore, the accuracy of the numerical model is very sensitive to the estimation of the Manning's roughness coefficient for the given bed material. It was found that the maximum scour depth for permeable spur dikes is a function of both approaching flow Froude number and opening ratio. The reduction in scour depth for permeable spur dikes could reach up to 68% of its respective value for impermeable spur dikes for an opening ratio of 55%. The empirical equation developed by Nasrollahi *et al.* (2008) which is independent of the approaching flow Froude number showed poor agreement when validated through the experimental data, which means that the Froude number is the key role in determining the scour depth for both permeable and impermeable spur dikes.