Fractal dimensions of hydraulic parameters in sand mined alluvial channel Open Access

Sand mining in a river is becoming an alarming factor exploiting the river system. Demand for sand has been on the rise due to increasing infrastructural development. The increase in demand for sand stimulates illegal mining activities and leads to the excessive excavation of river bed. Various morphological, hydrological, environmental, and ecological changes occur due to the excavation of the river sand (Rinaldi et al. 2005). The reduction of sediment in a river has a tremendous impact on the river bed dynamic, estuaries, and coastal zone (Yang et al. 2003; Wang et al. 2010; Xia et al. 2016). Various engineering works, such as hydropower generation, bank protection, navigation, irrigation, and soil erosion control, can also cause riverine sediment reduction (Chu et al. 2009; Meade & Moody 2010). Sediment mining can be a potential source for reducing the sediment load in a river (Anthony et al. 2015). The major impact of sand mining on a river bed is the lowering of bed elevation; however, it has a minor impact on reducing suspended sediment load into the sea (Yang et al. 2015; Du et al. 2016). Over excavation of river sand can also cause the collapse of a hydraulic structure (Kondolf 1997). The long-term removal of in-stream bed material creates a large pit on the river bed and changes the river's hydrodynamics. Alfrink & van Rijn (1983) conducted an experimental and numerical study to determine hydrodynamic parameters of flow over a trapezoidal fix trench. Various turbulence parameters of flow, such as time-average flow velocity, turbulence energy, shear stress, eddy viscosity in a trench, were studied by Alfrink & van Rijn (1983). After that, they identified the variations of turbulence parameters in the trench as compared to the plain bed. Disturbance to the alluvial bed in the form of a pit affects the turbulence properties of the flow in a river, which eventually describes the river's sediment transport properties (Barman et al. 2018). Various experimental researches were conducted on erosion due to mining pits (Lee et al. 1993; Neyshabouri et al. 2002). Lee et al. (1993) and Neyshabouri et al. (2002) primarily focused on the physical evolution of the channel bed around the mining pit and described the pit migration phenomenon using empirical relations.

Turbulence characteristics of flow can be used to find the morphological changes caused by the mining pit. Barman et al. (2018, 2019) discussed various important turbulence parameters such as Reynold shear stress, turbulence bursting events, turbulence intensity, higher-order moment analysis, and kinetic energy fluxes in a mining pit region, and their study showed a significant variation of turbulence flow structure in that region. Barman et al. (2019) observed higher values of turbulence kinetic energy fluxes downstream and in the downward direction in the mining pit region. The previous research on the mining pit mostly focused on turbulence flow characteristics; however, a mining pit's flow characteristics using fractal dimension are yet to be defined. This paper aims to define fractal dimensions for velocity fluctuations and Reynolds shear stress (RSS) in the flow region of a mining pit. Completely evolved turbulence comprises a hierarchy of eddies and vortices. These eddies and vortices have different orders of scale (Richardson 1922). The roughness of turbulence flow data is generally scaled invariant and tends to ‘smooth’ the data when using Euclidean methods (Marvasti & Strahle 1995). In that case, to understand the chaotic nature of turbulence structure, fractal analysis provides a technique for analysis (Keshavarzi & Ball 2017). Fractal tools can be the fractal dimension, linear fractal interpolation, and hidden variable fractal interpolation (Marvasti & Strahle 1995). The fractal tool that is used in this study is the fractal dimension. The chaotic nature of turbulence velocity signals can be defined using fractal dimensions. The fractal dimension of the turbulence data set, i.e., time-series data, gives important information regarding the density of the data set in the matric space. Various authors have analyzed the fractal scale of turbulence flow, including fractal properties and spatial fractal dimension of flow dissipation (Sreenivasan & Meneveau 1986; Meneveau & Sreenivasan 1991). Keshavarzi et al. (2005) applied fractal dimension analysis in turbulence bursting events and observed that the RSS mean fractal dimension was less than that of velocity fluctuations. The uniformity of the fractal dimension may differ with direction, as a more constant fractal dimension was found for traversed velocity due to the rigid lateral boundary while there was a variation in fractal dimension in vertical and streamflow directions (Rakhshandehroo et al. 2009). Other researchers, such as Keshavarzi & Gheisi (2007) and Gheisi & Keshavarzi (2008), also observed changes in fractal dimensions due to the boundary conditions. Formation of a pit on the bed creates bed undulation and turbulence flow characteristics changes in the vicinity of the pit. An earlier study showed sediment deposition along the upstream edge of the pit and lowering of bed elevation downstream (Lee et al. 1993; Barman et al. 2018). To better understand the turbulence velocity signals and their variation due to erosion and deposition processes, it is relevant to find the fractal dimension of a turbulence data set for the flow over a pit.

So, the main focus of this paper is to analyze the characteristics of fractal dimensions in the vicinity of the mining pit. Fractal dimensions of velocity fluctuations and RSS are computed using Marvasti & Strahle (1995) algorithm. This study also describes the distribution of turbulence kinetic energy dissipation, Taylor microscale and turbulence mixing length along the nondimensional flow depth in a mining pit region.

Experiments were carried out in a 17.2 m long, 1 m wide, and 0.72 m deep laboratory flume. The slope of the flume was kept at 0.0017 during the experiment. The flume used for the experiment was a recirculating flume that drained the water into an underground tank and pumped again to the upstream over-head water storage. Experiments were conducted for discharges 0.0442, 0.0472, 0.0503, 0.0535, 0.0567 m³/sec for subcritical flow conditions with initial depth of flow 0.0987 m, 0.1014 m, 0.1038 m, 0.1079 m, and 0.1101 m. The flow in the flume was highly turbulent with Reynolds number 30,740, 32,712, 34,730, 36,674, 38,756, respectively. Flow development length was obtained as 6.5 m from the upstream entry of the flume (Sharma & Kumar 2019). Therefore, in this experimental study, the test section was started after 6.5 m from the channel's entry and ends before 5 m from the downstream. Uniform sand of 1.1 mm median grain diameter and geometric standard deviation 1.03 was utilized for bed sediment. The critical shear velocity for the bed material was 2.62 cm/s using Paphitis (2001) mean threshold curve.

A mining pit with an irregular plane of 10 cm deep was constructed at 7.5 m from downstream of the flume. The maximum length of the pit was 1.5 m with varying width and vertical sides. The pit is symmetrical about its longitudinal axis. The detailed laboratory setup and the dimensions of the pit are presented in Figures 1 and 2. The discharge was applied gradually on the dry bed condition, and a constant discharge was maintained throughout one experimental run. Initial flow depth was maintained using the tailgate at the channel downstream. Discharge 0.0472 m³/sec is considered as a representative discharge for analysis.

A 16 MHz MicroADV developed by Sontek was used to measure the instantaneous flow velocity at five different locations of the mining pit. It has a data collection frequency of maximum 50 Hz with a sampling volume 0.1 cc. Data were collected for 180 sec at every point to ensure statistically significant velocity time-series data (Mohammadi & Keshavarzi 2019). Section A was located at the upstream edge of the pit and sections D and E at the pit downstream. Sections B and C were placed inside the mining pit as shown in the Figure 2. All sections were considered along the longitudinal centre line of the channel. Since Sontek acoustic Doppler velocimeter (ADV) can capture data 5 cm away from the central transducer, all data were collected near the channel bed to 5 cm below the free surface. The vertical distance of data collection points is measured as z from the channel bed. The data obtained from the ADV contain noise and low correlation. So, the data need to be filtered for further analysis. We filtered the data using correlations (65–70)% and signal-to-noise ratio at 15 (Sharma & Kumar 2021). The ADV data also contain spikes that need to be removed using proper filtering. For despiking the unfiltered data, the acceleration threshold technique established by Goring & Nikora (2002) with threshold values (1–1.5) was used.

Here, and represent the velocity fluctuations in longitudinal direction and vertical to the flow direction respectively. The water density is ⁠.

The fractal dimension of velocity fluctuations and RSS in rivers and channels indicates the presence of roughness and turbulence. In the present study, fractal dimensions of velocity fluctuations (⁠⁠) and RSS (⁠⁠) are analyzed in different locations of the flume in the presence of a pit. Firstly, we examined the data set for a number of target points per fixed point to observe the variation of fractal dimension.

Figure 4 shows the fractal dimension for a number of target points per fixed point. Fractal dimension varies for different number of target point and this is expected because it indicates different scaling of the data set. It is observed from Figure 4 that the fractal dimension for longitudinal velocity fluctuations, vertical velocity fluctuations and RSS tends to be constant from target points 10 onwards. So, 10 target points per fixed points is considered throughout this study and it is assumed that a close approximation of fractal dimension can be expected at this target points.

Figure 5 shows fractal dimensions of ⁠, and for an irregular pit for ⁠. From the Figure 5(a), it can be seen that the trend of fractal dimension of velocity fluctuation is greater in the upstream (Section A) and far downstream (Section E) section as compared to the pit (Section B and Section C) and downstream edge of the pit (Section D). The reductions of fractal dimension for at Sections B and C are 1.4% and 2.7%, respectively, as compared to the upstream section. There is an increase in fractal dimension at section E by 3.99%. The results indicate that the presence of pit affects the turbulence characteristics of the flow in the pit region. The fractal dimension of velocity fluctuation shows a decrease in flow roughness in the pit and the same increases at the pit downstream. Another observation which can be seen from the longitudinal velocity fluctuation is that the distribution of the fractal dimension is nonuniform throughout the nondimensional flow depth (⁠⁠) and increases as it approaches the channel bed. Figure 5(b) shows vertical distribution of the fractal dimension of the vertical velocity fluctuations and we have observed that it varies significantly along the nondimensional flow depth (⁠⁠), as well as along the flow direction. An average increasing trend of fractal dimension along the nondimensional depth of flow (⁠⁠) is observed for all the sections with an exception at Section A. At Section A, there is a slight decrease in fractal dimension of vertical velocity fluctuations towards the channel bed. The results show that mean fractal dimension for longitudinal velocity fluctuation is lower than that of the vertical velocity fluctuations by 1.9%, 1.3%, 0.14%, 1.6%, and 2.5% respectively for Section A, B, C, D, E. A similar observation was found by Keshavarzi & Ball (2017) at the centreline of a bridge pier. The trend of vertical distribution of fractal dimension of RSS is similar to that of velocity fluctuation. The vertical distribution of fractal dimension for RSS shows that the minimum fractal dimension occurs at Section B. It is also observed from the results that fractal dimensions for RSS is lower than that of the velocity fluctuations. Keshavarzi et al. (2005) also found that the fractal dimension for RSS is lower that for velocity fluctuations. So, the results of the present study are consistent with the previous findings. The average fractal dimension (Figure 6) for longitudinal velocity fluctuation has a larger value at the pit upstream (Section A, F_D = 1.82) that reduces to a minimum value in the mining pit region (Section B, F_D = 1.76; Section C, F_D = 1.77), and then it starts increasing and reaches the maximum value at the far downstream section (Section D, F_D = 1.799; Section E, F_D = 1.83). The values of fractal dimensions are between 1 and 2, which shows that the turbulence data set cannot be taken as linear or planner (Marvasti & Strahle 1995).

The similar trend is observed for the fractal dimension of vertical velocity fluctuations and RSS. The fractal dimensions are lowest in the pit (Section B and Section C). It is reduced by 3.4% and 3.1% for the longitudinal velocity fluctuation, 4.01% and 4.8% for the vertical velocity fluctuation, and 3.4% and 3.4% for RSS. This indicates an overall decrease in roughness of flow in the pit. In the far downstream from the pit (Section E) the average fractal dimension is more compared to the other regions in the channel. The reason for this might be that in the pit the velocity reduces, which ultimately decreases the roughness and then the velocity starts to accelerate and reaches the main flow velocity of the channel. This tells us that the roughness or turbulence increases at the downstream of the pit. The phenomenon of turbulence in a mining pit region is further explained using kinetic energy dissipation and Taylor microscale.

So, maximum kinetic energy dissipation occurs at the pit, while it starts to decrease away from the pit location. Increase in kinetic energy dissipation in the pit suggests possibilities of sediment deposition in the pit, whereas decrease in energy dissipation at the downstream of the pit causes erosion at downstream as energy is transfer to turbulence fluctuations. Earlier research on pit morphological also reported infilling of the mining pit and downstream erosion (Lee et al. 1993; Barman et al. 2018) and hence, these results are consistent with the previous study.

At Sections A, C, D, and E, nondimensional Taylor microscale, decreases away from the bed, which shows an increase in the Taylor microscale. At Section B, increases away from the bed achieving a peak value at (⁠⁠), then it decreases towards the water surface. The significantly large value at for Section B shows a smaller eddy size in that region. However, a decrease in Taylor microscale in the vicinity of the bed surface is observed in all the sections. This mainly occurs due to an increase in turbulence dissipation near the bed region. In the near bed region ⁠), Taylor microscale decreases significantly in the mining pit (Section B and Section C) as compared to upstream Section A. The values of Taylor microscale (⁠⁠) near the bed (⁠⁠) are 1.15 cm, 0.26 cm, 0.03 cm , 0.85 cm, and 1.17 cm for Sections A to E respectively. So the increase in Taylor microscale downstream of the pit shows the increase in eddy size near the bed at downstream locations, that further justify the erosion at the pit downstream (Lee et al. 1993).

Here, nondimensional turbulence mixing or Prandtl mixing length,⁠, ⁠, ⁠. For measuring velocity gradient ⁠, time-average velocity curve was used. Shear velocities (⁠⁠) at Section A to Section E using the RSS damping method are 0.0273 m/s, 0.1037 m/s, 0.0812 m/s, 0.0506 m/s, and 0.0203 m/s, respectively. Nondimensional RSS. (⁠⁠) were taken directly from RSS distribution. Figure 9 shows distribution of nondimensional turbulence mixing length for inner layer of flow (⁠⁠). Turbulence mixing length depicts the distance covered by a fluid particle by keeping its original characteristics prior to its dispersion into the neighboring fluid environment.

Section A shows a satisfactory linear fit between nondimensional mixing length and nondimensional flow depth. The linear fit can be defined as (where k is the von Kármán constant). The value of the von Kármán constant at Section A is 0.4, which is lower than the universal constant k = 0.41 for clear water flow. This indicates the mobility of bed materials at Section A (Barman et al. 2018). Nezu & Nakagawa (1993) described a theoretical curve for turbulence mixing length by ⁠. Nondimensional mixing length from the present experiment at Section A acceptably fitted with the theoretical curve defined by Nezu & Nakagawa (1993). However, at Sections B and C, the linear relation between does not exist. It also shows deviation from the theoretical curve suggested by Nezu & Nakagawa (1993). As we move downstream (Section D), nondimensional turbulence mixing length starts to collapse on the theoretical curve with a very high k value (k = 0.77). Earlier studies also reported deviation in logarithmic velocity distribution for the inner layer of flow in the pit region. This happens due to the negative flow velocity in the pit (Barman et al. 2018). At Section E we have observed k = 0.346 that indicates an increase in sediment mobility at far downstream of the pit. The decrease in traversing length of the eddy at far downstream of the pit represents the increase in sediment mobility that subsequently causes erosion at the pit downstream.

Indiscriminate sand mining causes destruction to the river bed and bank. As we can see from our experimental results, flow characteristics change largely not only in the mining pit but they are extended to 0.5 m downstream to the pit. The morphological changes near the pit show erosion at the pit downstream (Barman et al. 2018). The present study found that the erosion at the pit downstream occurs due to the increase in the sediment mobility. The sediment mobility increases with increase in flow roughness near the bed. Fractal dimensions are useful in identifying the flow roughness in the mining pit region. The smaller values of fractal dimensions inside the pit indicate that infilling of the pit may occur due to the decrease in flow roughness in contrast with the increasing erosion at the pit downstream with larger fractal dimensions. We also illustrated this phenomenon by measuring the size of the eddies. Larger eddies were found towards the downstream of the pit and also the traversing length of eddies decreases at the pit downstream. Depositions occur in the pit that cause infilling of the pit (Barman et al. 2018). The deposition in the pit is due to the formation of smaller eddies and high energy dissipation in the pit. The erosion and deposition phenomenon in the mining pit region can also be explained from the occurrence of the turbulence bursting events. Barman et al. (2018) found that the dominance of sweep events increases at the pit downstream. It represents an increase in momentum exchange from the flow to the bed sediment that subsequently increases the sediment mobility.

Removing the sediment layer in the mining region can cause loss of habitat for the species attached to the stream bed, riparian habitat in that location, and the adverse impact extended away from the mining location. Due to the complex interaction between the flow and bed sediment, the ecological condition of a river indirectly is affected (Padmalal et al. 2008). The breeding habitats of Gavialis gangeticus in the Ganga River have been destroyed by eroded river banks due to sand mining activities (Bendixen et al. 2019). Also, the continuous erosion deposition in the mining location may change the sediment characteristics as well as the water quality of the location that further affects the aquatic habitat. Movement of sand along the river is difficult to quantify. The difficulties enhance for a river with active mining zone due to the nonavailability of sand mining data. So, the experimental approach to explain the complex flow–sediment interaction in the presence of mining gives some important insight into the phenomenon. Sand transported from the upstream is deposited in the mining pit but it continues to erode the channel bed far downstream of the pit. For sustainable management of sand mining operations, a detailed understanding of the flow characteristics is required. This present study contributed to the understanding of flow characteristic and erosion deposition pattern in a mining location that can have many real field applications.

In this study, we conducted an experiment to evaluate the fractal dimension and turbulence scale of flow structure in a mining pit. The experiment was conducted for an irregular-shaped mining pit symmetrical about the longitudinal axis. Instantaneous velocities were taken at five different locations of the flume. Using the velocity data, fractal dimensions for longitudinal and vertical velocity fluctuations and RSS in longitudinal-vertical plane were investigated. The distribution of fractal dimension for velocity fluctuations and RSS showed nonuniformity along the nondimensional flow depth. The general observation is that fractal dimension for longitudinal velocity fluctuation is less than that of vertical velocity fluctuations. Again, the fractal dimension for RSS is comparatively lower than the velocity fluctuations. The average fractal dimension decreases in the mining pit region, while it increases far downstream of the mining pit. It shows an increase in flow roughness at the pit downstream. Turbulence kinetic energy dissipation shows an increasing trend towards the channel bed. The highest dissipation occurs at the pit and it significantly decreases downstream of the pit. Taylor microscale increases away from the bed. The decrease in Taylor microscale at the bottom of the pit corresponds to higher energy dissipation in the pit. The decrease in Taylor microscale also indicates the generation of smaller eddies in the mining pit, whereas the eddy size increases at the downstream of the pit. The increased eddy size corresponds to the increase in roughness concentration at the pit downstream and increased particle mobility. From fractal analysis we observed an increase in fractal dimension for RSS and velocity fluctuations at the pit downstream and hence, it increases the roughness concentration downstream of the pit. Therefore, the concept of fractal dimension is effective to illustrate the impact of the mining pit on turbulence flow structures. The analysis was done only for one mining pit in the study reach. There is the possibility of the existence of multiple mining pits in a particular channel reach. Further, continuous interaction between flow and sediment occurs during the active channel mining operation. In this experimental study, a mining pit was constructed prior to applying water into the channel. Thus, these factors give limitations to the applicability of the present results for such cases and further research is required in this regard.

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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