Numerical and experimental investigation for the effect of permeability of spur dikes on local scour

B

Channel bottom width

d₅₀

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 direction

Components of the shear stress of river bed in y direction

Non-dimensional bed shear stress

Critical shear stress

Time step

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).

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).

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 m³/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.

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).

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 m³/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.

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 R² 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.

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 R² 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_s for the given d₅₀ of the bed material (Kafle 2014).

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 R² 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 R² 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.

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.