1 Introduction

Local Scouring is a complex phenomenon and involves a spectrum of parameters which affect its development. Local Scour is identified as one of the most prominent cause of failure of hydraulic structures (Landers and Mueller 1996; Melville and Coleman 2000; Hong et al. 2012). It is the removal of sediment from around bridge piers or abutments or an obstruction. The sediment removal is caused mainly due to formation of both horse vortex and down flow in front of the piers or obstruction (Moncada-M et al. 2009). Both the obstruction, as well as, the bed material parameters affects the local scouring besides the fluvial properties of the water (Raudkivi and Ettema 1983). The complex nature of the scouring process involves a wide range of theories such as sediment transport, fluvial hydraulics, boundary layer theory, time-dependence, etc. (Fakhri et al. 2014; Mohammadzade Miyab et al. 2017) which hinder the exact quantification of the process. The waterways under consideration, whether it is a natural stream in which the hydraulic structures are laid or the man made channels which also support various structures while dealing with the water resources engineering show a varied scouring pattern. Based on experience from the study of the existing hydraulic structures and the researches available (Laursen and Toch 1956; Chiew and Melville 1987; Melville 1992a, b; Johnson 1995; Heza et al. 2007; Akib and Rahman 2013), it is clearly evident that the bed material of the water way is an important governing feature while computing local scour depths. A complete gradation analysis of the soil or the bed material needs to be done for determining the scouring in that particular setup. The present study was conducted to formulate the detailed effect of the gradation parameters of bed material on the local scour depth.

Most of the local scour studies are accomplished by means of physical modelling (Raudkivi and Ettema 1983; Chiew 1984; Chiew and Melville 1987; Mohammed et al. 2007; Mir et al. 2017; Mir et al. Forthcoming). Physical Modelling along with numerical techniques are an effective means of understanding the local scour mechanism.

The present study was conducted with an aim to study the variation of local scour with bed material characteristics and develop an empirical solution to the identified problem.

2 Dimensional Analysis

Scour depth around the obstruction depends on a number of parameters. These parameters in case of bridge piers can be identified easily as pier size, sediment characteristics, flow conditions and fluid properties (Breusers et al. 1977; Hong et al. 2012). The relationship describing the equilibrium local scour depth for an obstruction may be expressed as follows:

$$d_{s} = \phi \left( {\mu ,\rho ,V,y,d_{5} ,d_{30} ,d_{85} ,f,C_{u} ,D,g,S,b} \right)$$
(1)

where ds = maximum equilibrium scour depth, µ = fluid dynamic viscosity, ρ = water density, V = average velocity of approach flow, y = flow depth, d5 = particle diameter for 5% passing, d30 = particle diameter for 30% passing, d85 = particle diameter for 85% passing, f = silt factor, Cu = uniformity coefficient, D = obstruction size, g = gravitational acceleration, S = bed slope, b = channel width, Fr = Froude number, φ = function of, ψ = function of, ϕ = function of.

Taking repeating variables as ρ, V and D, the functional relationship obtained is given as:

$$\frac{{d_{s} }}{D} = \psi \left( {Fr,\frac{\mu }{\rho VD},\frac{y}{D},\frac{{d_{5} }}{D},\frac{{d_{30} }}{D},\frac{{d_{85} }}{D},f,C_{u} ,S,\frac{b}{D}} \right)$$
(2)

The dependent parameter is ds/D i.e. normalised scour depth and its variation was studied with the normalized gradation parameters. Froude number was varied on account of allowing varying discharges to pass through the flume and slope of the flume adjusted as per the commonly found slope of the rivers in the region.

Also, D, b, µ and ρ for the given experiments were constant. Similarly, for open channel flow, Reynolds number (Re = μ/ρVD) can be ignored. Hence, the functional relationship applicable for this study was reduced to:

$$\frac{{d_{s} }}{D} = \phi \left( {\frac{{d_{5} }}{D},\frac{{d_{30} }}{D},\frac{{d_{85} }}{D},f,C_{u} ,Fr,\frac{y}{D}} \right)$$
(3)

3 Experimental Work

The experimental set-up for the experiments, material used and the procedure adopted is given in the proceeding sections.

3.1 Experimental Setup

The aim of present study was to assess the gradation effect of the bed material on local scour by physical modelling. The experimental set-up used for the purpose consisted of a tilting flume 24 m long having a height of 0.6 m and a width of 1 m, located in the Fluid Mechanics Laboratory of Water Resources Department, National Institute of Technology, Srinagar. Figure 1 shows the glass flume used in the study. The flume had glass side walls and a metallic base with necessary arrangements for water supply regulation and measurements.

Fig. 1
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Glass sided tilting flume used in the experimental study

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3.2 Test Material

The obstructions used were modelled in four different shapes of concrete material. The width of the obstructions were modelled as 1/10th of the channel width i.e. the dimension of each obstruction facing the direction of flow was modelled to be about 10 cm. Chiew and Melville (1987) recommended on the basis of their studies that the obstruction diameter should not exceed 10% of the flume width. In the present study, this aspect of the Chiew and Melville study has been taken into account for more reliability of the results. A pictorial view of these shapes is shown in Fig. 2. The shapes selected for this study have been chosen keeping in view the most commonly encountered obstruction shapes in real life hydraulic structures e.g. Bridge piers, view-points constructed in the water bodies, etc.

Fig. 2
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Obstructions used in the experimental study

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Five types of bed material having varying gradation parameters were used. The bed material used represents the existing natural soil of actual river sites of Kashmir region. The soils were non-cohesive though a negligible amount of cohesive material may have been present. Study of the soils of a large number of river sites of the Kashmir valley was conducted, out of which five were selected on the basis of their gradation parameters variation. Five different sites selected were Khor (Sherabad); Sumbal; Ganderbal, Kunzer Nallah (Tangmarg), and Pampore of the Kashmir valley and were designated as M1–M5 respectively. The gradation parameters of the bed material used are given in Table 1.

Table 1 Gradation parameters of the bed material (M1–M5) used in the study
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3.3 Experimental Procedure

The experiments were carried out in the tilting flume described under the experimental set-up sub-heading. The middle section was chosen for placement of the obstructions so as to minimize the effect of inlet disturbances and tail gates. The flume was then filled with the above described bed material, up to a depth of 18 cm. The obstructions were fully penetrated in the bed material up to the flume floor so that the obstructions aren’t destabilized with the increasing discharges and the local scour depth measurements are done accurately. The material was properly levelled in order to achieve results nearer to natural conditions of the river sites. At the entrance section of the flume boulders were placed on the sand bed to still the flow so as to add stability to the bed material and prevent it from getting washed away due to inlet turbulence. Figure 3 shows the experimental set-up used in the study. Water supply to the flume was regulated with the help of valves located in the supply line fed by a constant head tank. The slope of the flume was fixed at 1 in 200 m which is one of the commonly found slopes in the study area.

Fig. 3
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Experimental set-up of the study

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3.4 Data Collection

The variables of the study include bed material, obstruction shapes and the discharge through the flume, the discharge was varied and scouring was measured for all discharges along the boundaries of the obstructions. Discharge was measured at the downstream end of the flume as shown in Fig. 4 with the help of a Sharp Crested Weir. Table 2 gives the values of the various flow parameters for the given experimental runs. After allowing the discharge through the flume, the scour process starts around the obstructions. Over the period of time, the scour hole gets filled with water along with the sediments and it appears opaque and hinders the scour hole measurement. This shortcoming was overcome by use of a laser meter which gives relatively more accurate values of scour depth with the varying discharges over the period of experimenting time. Scour depth was measured along the periphery of the obstructions and the maximum scour depth was considered. Figures 5, 6, 7 and 8 show a few experimental runs.

Fig. 4
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Discharge measurement at the downstream using Sharp Crested Weir

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Table 2 Values of the various flow parameters of the experimental study
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Fig. 5
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Experimental run for circular obstruction

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Fig. 6
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Experimental run for hexagonal obstruction

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Fig. 7
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Experimental run for rectangular obstruction

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Fig. 8
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Experimental run for rounded obstruction

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4 Results and Discussion

The present study was conducted with an aim to relate the gradation parameters of the bed material with the local scour depth. The type of bed material has a prominent effect on the scouring. The bed material is differentiated on the basis of gradation parameters. Data collected from the experimental runs is given in Table 3.

Table 3 Experimental results of local scour depth for varying shapes and bed material
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4.1 Models Predicted

The variables of the study include bed material, obstruction shapes and the discharge through the flume; the discharge was varied and scouring was measured for all discharges along the boundaries of the obstructions. Table 3 gives the experimental values of scour depths for combinations of five different bed materials (M1–M5), five discharges and four obstruction shapes (circular, hexagonal, rectangular and rounded). The experimental data was regressed to obtain the inter-relationships between the local scour depths and bed material parameters (d5, d30, d85 and Cu). For different shapes, multiple regression using “R” software yielded different models given in the following section. For circular obstruction, the model developed is given as Eq. (4).

$$\frac{{d_{s} }}{D} = 0.261 - 171.166\frac{{d_{5} }}{D} - 27.262\frac{{d_{30} }}{D} + 6.896\frac{{d_{85} }}{D} - 0.115C_{u} + 1.136Fr + 0.134\frac{y}{D}$$
(4)

In the equation, ds is local scour depth in metres, d5, d30 and d85 are the gradation parameters of the bed material in mm, y is flow depth in metres and D is obstruction diameter/width perpendicular to the direction of flow in metres. Similarly, Eqs.  (5), (6) and (7) give the models for hexagonal, rectangular and rounded obstructions respectively.

$$\frac{{d_{s} }}{D} = - 0.086 - 58.022\frac{{d_{5} }}{D} + 2.669\frac{{d_{30} }}{D} - 5.387\frac{{d_{85} }}{D} - 0.0213{\text{Cu}} + 1.039Fr + 0.126\frac{y}{D}$$
(5)
$$\frac{{d_{s} }}{D} = 0.280 - 322.8\frac{{d_{5} }}{D} + 48.88\frac{{d_{30} }}{D} - 11.72\frac{{d_{85} }}{D} + 0.00777Cu + 0.210Fr + 0.225\frac{y}{D}$$
(6)
$$\frac{{d_{s} }}{D} = 0.504 - 346.99\frac{{d_{s} }}{D} - 3.808\frac{{d_{30} }}{D} + 7.010\frac{{d_{85} }}{D} - 0.094Cu + 1.034Fr + 0.063\frac{y}{D}$$
(7)

The summary of the models are tabulated in Table 4. The characteristics given in the table are clearly in the acceptable ranges. Figures 9, 10, 11, 12, 13, 14, 15 and 16 give the graphical presentations of the regression results for the four shapes; where, S represents ds/D, D5 is d5/D, D30 is d30/D, D85 is d85/D, f is silt factor, Fr is Froude Number, Cu is Uniformity Coefficient and Y is y/D.

Table 4 Regression results for the models developed
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Fig. 9
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Residuals versus fitted plot for circular obstruction

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Fig. 10
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Normal Q–Q plot for circular obstruction

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Fig. 11
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Residuals versus fitted plot for hexagonal obstruction

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Fig. 12
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Normal Q–Q plot for hexagonal obstruction

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Fig. 13
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Residuals versus fitted plot for rectangular obstruction

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Fig. 14
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Normal Q–Q plot for rectangular obstruction

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Fig. 15
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Residuals versus fitted plot for rounded obstruction

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Fig. 16
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Normal Q–Q plot for rounded obstruction

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The results clearly indicate a substantive effect of d5 on the local scour depth. Lesser the d5 of the bed material for a constant obstruction width, greater is the local scour. Greater the d5 of the bed material, lesser is the local scour. Similarly, an increase in the uniformity coefficient will decrease the local scour. Also, results indicate the effect of flow depth and Froude number; greater the flow depth, greater is the local scour; greater the Froude number, more is the local scour and vice versa.

4.2 Validation

The obtained regression models were tested for a separate set of experimental data collected in the laboratory with a different bed material (Mv) and compared to validate the models put forth. The gradation parameters of the bed material used for validation purpose are given in Table 5.

Table 5 Parameters of the bed material used for validation
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Figures 17, 18, 19 and 20 give the predicted versus observed plots for the models developed in the present study. The results were found in close approximation and hence, the models put forth were inferred to be acceptable.

Fig. 17
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Predicted versus observed plot for scour depths of circular shape

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Fig. 18
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Predicted versus observed plot for scour depths of hexagonal shape

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Fig. 19
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Predicted versus observed plot for scour depths of rectangular shape

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Fig. 20
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Predicted versus observed plot for scour depths of round nosed shape

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4.3 Efficiency

The models obtained in this study were further investigated for their efficiency and are given in Table 6. The efficiency was calculated as per Eq. (8).

$$Efficiency = \frac{{\left( {Variance_{Initial} - Variance_{Final} } \right)}}{{Variance_{Initial} }} \times 100$$
(8)

where, ds(obs) = observed local scour; ds(mean) = mean of observed local scour; ds(pred) = predicted local scour

Table 6 Variance and efficiency of the developed models
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5 Conclusions

On the basis of the study presented in this paper, four models were developed for four different shapes of the obstructions for estimation of their local scour depth with respect to the bed material gradation, Froude number and the flow depth. Exhaustive experimental runs were carried out to study the in-depth effect of gradation parameters on local scour depth while keeping other parameters such as slope (S), obstruction width/diameter (D), and channel width (b) constant for a given range of Froude number.

The experimental findings showed the dependence of local scour depth not only on the widely used d50 size of the bed material but also to its other gradation parameters, like d5, d30, d85, Cu. The flow depth and the Froude number were also found to play an important role in the local scour phenomenon.

Equations (4)–(7) gives the models for four different obstruction shapes used in the study. The models presented were further investigated to check their efficiency which ranged between 80 and 92%. Also, these models were validated for a different bed material in the laboratory. The experimental results for this bed material were compared with those obtained using the above models for the said material and the shapes. It was found that the experimental and the calculated results for the said bed material were quite in tune with each other.

It is highly recommended to increase the scope of present study, by extending it to more variations of bed material and obstruction shapes. Another recommendation that is suggested is to check the effect of the ratio of obstruction length to its width.