# Verification of Three-dimensional Numerical Study of Scour in Channel, Sudden and Gradual Contractions using Experimental Data

## Abstract

This paper presents a verification of a numerical and experimental simulation of scour patterns at channel contractions using a 3-dimensional SSIIM (Sediment Simulation in Intakes with Multiblock option) model and laboratory tests. For this purpose, two states of sudden angle¬-90° and gradual angle -50° contraction were simulated. The numerical model was calibrated and verified using the laboratory data. The accuracy of the model was calculated as 0.936 based on the Nash-Sutcliffe model efficiency coefficient, and 10.18% based on the mean relative error. Results showed that around 80% of scouring occurred during the first 20% of the equilibrium time. Also, it was concluded that the maximum rate of scouring occurred during the first hours of experiments and computations, and decreased with time. The results showed that the average scour rate for the sudden contraction state was 29.5% greater than the gradual state, which indicated a positive impact of gradualness of conversion in reducing maximum scour depth. This is an appropriate performance of the numerical model to simulate the scour pattern in channel contraction.

## 1 Introduction

Scour is a phenomenon occurring because of erosion in the bed induced by water flow and washing of materials following the forces of water flow on the bed. The mechanism of the phenomenon is that prior to the destruction of the structure due to flood forces, it faces the risks induced by the erosion of its foundation. Reduction of the width of torrent bed increases the velocity and shear tension of flow, which will increase erosion energy, capacity to carry solid materials, and sediment transport coefficient in liquids. The conditions required for contraction scour to the ground are provided. Moreover, local scour occurs around barriers and structures present in the flow as well. To get a better understanding of scour, analysis of the complex conditions of physical models is required but is extremely costly and time-consuming. However, recent advancements in computer science have brought numerical simulations based on calculative fluid dynamics into the focus of studies on sediment transport and turbulent flow (Olsen and Melaaen 1993; Huang et al. 2009; Ali and Karim 2002; Minor et al. 2007; Akbari and Barati 2012; Barati et al. 2012; Barati 2013; Shahheydari et al. 2015; Hosseini et al. 2016; Alizadeh et al. 2017; Barati et al. 2018; Tajnesaie et al. 2020; and Atashi et al. 2023 a,b,c). In this regard, Aziziyan and Karbalayi (2013) simulated the effect of changes in discharge on the velocity and pattern of the flow in contracted regions of the channel using MIKE-21 software in 2D and compared the results to laboratory findings. The basis for the solution of this model is the non-overlay finite volume method. Unstructured networking is utilized to model the geometry of the channel; however, in the contraction site, a finer mesh was used. Their findings revealed the effect flow speed and depth changes have on flow properties in contractions. In other words, increased flow speed and depth change increase secondary and rotational flows around contraction. Findings of the studies by Weise (2002) and Marek and Dittrich (2004) demonstrated that 2D models are capable of simulating contraction scours; yet, as the contraction spots expand, their results move away from laboratory findings, which could be attributed to the impact of 3D flows and weaknesses of turbulent models. Lai and Greimann (2010) were among the scholars investigating contraction scours and scour patterns through numerical models and simulated empirical findings with acceptable accuracy. Their findings revealed that scours at the beginning of contraction start from the walls and reach their maximum at these spots as well.

Throughout the world, each year, numerous structures are ruined mainly due to ignoring hydraulic factors, including scouring, in their design process. Therefore, the scour phenomenon in rivers and the parameters affecting it are greatly significant, and should be investigated properly, and in depth. The SSIIM numerical model for analyzing 3D patterns of flows and scour is one of the powerful tools in this area (Asadollahi et al. 2020). In fact, to analyze complex issues such as flow in contractions, and due to the changes in pressure, and the presence of vertical velocities, 3D models are required to predict velocity field, water level, and sediment transport. SSIIM is one of these 3D models; the first version of which was developed by Olsen in 1991 to solve the flow and sediment in a 3D format. Mohammad et al. (2015) utilized this model to study scour and the 3D component of velocity in an arched channel, and revealed that SSIIM powerfully determined the limits, rate of scour, and maximum sedimentation, along with longitudinal velocities and resultant horizontal velocities in arched channels with complex geometries.

In another study, Mirmoazen and Pirestani (2009) studied the efficiency of SSIIM models in concrete channels containing sediments. They used one of the water channel networks in Gilan Province to calibrate the model. Their findings demonstrated that the model had an acceptable accuracy in the presence of water and sediments. Shakibaeinia et al. (2010) used a three-dimensional model to investigate the flow properties in the separation zone of a secondary flow in a confluence area with different junction angles. The model’s predictions were compared with those of an available two-dimensional model. Hosseini-Sargheyn et al. (2012) using an SSIIM model, predicted flow patterns in a river bend with a 90-degree angle in a rigid bed. In this study, velocity profiles and shear tensions of the bed were estimated for each convergence rate, and the results showed that velocity and maximum speed location profiles depend on the rate of convergence.

In meanders, flow patterns are extremely complex. In these rivers, bends are of particular concern, especially since these bends are not monotonous, i.e., some are divergent, and others are convergent. A numerical model could effectively be used to predict such flow fields (Ghobadian and Mohammadi 2011). Numerical and experimental analyses were performed by Abhari et al. (2010) for studying flow pattern in a 90-degree bend. Their results indicated that streamlines at the level close to the bed surface orient to the inner wall, while streamlines near the water’s surface orient to the outer wall. Omran (2008) studied the flow hydraulics and sediments in open rectangular and straight channels through a mathematical semi-2D model and reported it had a proper performance. Kang (2013) used a three-dimensional model to simulate turbulent flows and morphological changes in an alternate vegetated zone open channel. It was found that the initial rectangular channel became a compound channel, and the flow pattern changed from a straight stream to a meandering stream when the channel morphological changes were considered for a series of floods.

Motamedi et al. (2014) used the laboratory model, along with an SSIIM numerical model, to examine the effect of dune on the separation of flow, considering various parameters of dune height along with the materials, angle, depth, and velocity of flow. The experiment was conducted using four dunes with different slopes. The numerical model used was the Sediment Simulation in Intakes with Multiblock (SSIIM), which is a dynamic open source calculating fluid model. However, considering the height of the dunes, size of particles, flow velocity, and angle on the leeside, empirical observations, and numerical results on the effects of dune on separating flow were of decent coordination. Hammerling et al. (2019) used SSIIM to simulate the structures built to reduce scour in the Warta River on Jeziorsko Island in Poland and showed that the calculated water velocity profiles do not a show significant difference with the measured samples. Nowroozpour and Ettema (2021) undertook a study on contraction scour with three different contraction rates in transparent water and the living bed induced by subcritical flow along a large rectangular channel. Maximum scour depth in a contraction occurs at the corners of the contraction input and output. The scour depth estimated using the existing HEC-18 equations was larger compared to laboratory observations. The deepest scour in input corners occurred, except for the minimum ratio.

Analyses of deposition and erosion of the Iffezheim Reservoir were performed by Zhang et al. (2022) using SSIIM. Simulation of the morphological changes in the weir channel was considered. They developed a classification of the boundary conditions for discharges (Q) and suspended sediment concentrations (SSC) approach to reduce the computation time of long-term simulation (e.g., several decades). The volume change of the reservoir was calculated by adding up the calculated daily volume changes. Eventually, SSIIM was used in various cases including for bed changes in the joints of natural rivers, and flow in the joints of channels, and proved to be properly simulating the distribution model of rotational flow speed and sedimentation on the banks. Considering the advancements in computer technology and the development of powerful numerical models for 3-D simulation of flow and transportation of sediment in complex geometries, engineers have no other choice but to use these numerical models. However, it should be noted that, prior to using these models, it is essential for their precision to be validated in complex conditions based on current theories.

This study aimed to study contraction scour in both sudden and gradual contractions using a three-dimensional SSIIM model, and to calibrate and validate the findings using the results of a physical model with gradual contraction. Moreover, to analyze the scour phenomena, the scour in sudden contractions in fixed hydraulic conditions were conducted using the numerical simulation.

## 2 Material and methods

### 2.1 Experimental data

To calibrate and validate the SSIIM model, the results of eight experiments conducted at the Hydraulic Laboratory of Shahrekord University were used. The experiments were conducted in a laboratory flume with a length of 20 m, a width of 600 mm, and a height of 600 mm. A general description of tests and an overall contraction plan are seen in Table 1 and Figure 1. The average diameter of the sediment used in the experiments was 0.78 mm. Other characteristics of the bed sediments are listed in Table 2. Where, *cv* is the standard variation, *cu* is the uniformity coefficient, *N* is the porosity and *ϴm* is the static angle of the bed particles (Parsmehr 2014).

Figure 1 Contraction plan of the channel.

Table 1 Overall properties of the conducted tests.

Test No. | Contraction (mm) | Flow Discharge (L/s) | Contraction Ratio (%) |

1 | 200 | 20 | 66.67 |

2 | 200 | 30 | 66.67 |

3 | 300 | 20 | 50 |

4 | 300 | 30 | 50 |

5 | 300 | 40 | 50 |

6 | 400 | 20 | 33.3 |

7 | 400 | 30 | 33.3 |

8 | 400 | 40 | 33.3 |

In Table 1, the ratio of contraction (*R _{c}*) is calculated using Equation 1.

(1) |

Where:

b |
= | contracted section, and |

B |
= | width of the main channel. |

Table 2 Characteristics of bed sediments.

Standard Variation cv (nm) |
Uniformity Coefficient cu |
Static angle of the bed particles ϴm (Degree) |
Porosity N |
Grain specific density Gs |
D50 (mm) |

0.8 | 1.5 | 30.5 | 51.85 | 2.7 | 0.78 |

### 2.2 SSIIM numerical model

The three-dimensional SSIIM numerical model solves the continuity equation, and the Reynolds averaged Navier-Stokes equations with the *k_*ε model as a turbulent model for solving Reynolds stress term (). The numerical model was presented by Olsen (2009). To discretize governing equations, the power-law scheme, or the second order upwind scheme, were utilized by applying the SIMPLE method for pressure coupling. The velocity field was calculated using an implicit solver using a finite volume approach. The flow field for three-dimensional geometry was calculated by solving the following equation (Olsen 2009):

(2) |

Where:

u_{i} |
= | velocity, |

x_{i, }x_{j} |
= | coordinates in the tensor notation, |

= | water density, | |

P |
= | total pressure, |

= | Reynold's stress, and | |

= | Kronecker delta, which takes the value 1 when i = j, and in other cases, it is equal to 0 |

The SSIIM model computes the turbulent stresses *u _{i }u_{j}*, using the eddy viscosity relation in Equation 3:

(3) | |

(4) |

Where:

c_{μ} |
= | 0.09 constant, |

k |
= | turbulent kinetic energy, and |

ε |
= | energy dissipation velocity. |

The turbulent kinetic energy *k* and its dissipation rate *ε,* determining the eddy viscosity *θ _{t,}* were obtained using Equations 5, 6, and 7:

(5) | |

(6) | |

(7) |

Where:

v |
= | viscosity of water, |

σ_{ε} |
= | 1.3 (experimentally determined constant), |

σ_{k} |
= | 1.0 (experimentally determined constant), |

C_{ε1} |
= | 1.44 (experimentally determined constant), and |

C_{ε2} |
= | 1.92 (experimentally determined constant). |

The SSIIM can model sediment transport with a movable bed in complex geometry, which is the main advantage of this tool compared to other CFD programs. A sediment transport model can simulate multiple sediment sizes, sorting, bed load and suspended load, bed forms, and the effects of sloping beds, where the unstructured grid can consider wetting and drying cells. Some limitations of the SSIIM include that it ignores non-orthogonal diffusive terms, the grid lines in the vertical direction must be exactly vertical, water kinematic viscosity is constant for 20° C, and it is not made for the marine environment, so all effects of density gradient due to salinity differences are not considered.

van Rijn (1987) developed Equation 8 to determine the concentration of sediment in the areas near the riverbed. This equation is used in the numerical SSIIM model.

(8) |

Where:

τ_{c} |
= | critical shear tension, |

τ |
= | bed’s shear tension, |

d |
= | particle diameter, |

α |
= | level of the base surface, |

g |
= | acceleration of gravity, |

ρ_{s} |
= | density of sediment particles, and |

C_{bed} |
= | sediment transport. |

The basis of calculating sediment discharge in this software is to solve the transport-distribution equation as provided in Equations 9 and 10.

For suspended load:

(9) |

Where:

u_{i} |
= | velocity in three directions, |

w |
= | fall velocity of the sediment particles, |

z |
= | height abvove bed, |

Γ_{t} |
= | diffusion coefficient, and |

C |
= | sediment concentration. |

For bed load:

(10) |

Where:

D_{50} |
= | average diameter of particles, and |

q_{b} |
= | discharge of bed load. |

### 2.3 Grid generation

A channel with 4.5 m length and 0.6 m width, having three contraction widths of 0.2, 0.3, and 0.4 m at two states of graduate, and sudden transitions were used for the purpose of simulations. Fine cells with a grid size of 0.01 x 0.03 m were used close to the contraction to provide accurate results, whereas 0.01 x 0.015 m cells were employed away from the contraction to optimize the computational time as depicted in Figure 2.

Figure 2 Mesh configuration for contraction with a width of 0.3 m.

## 3 Model calibration

The most critical state was chosen for calibration from among the tests conducted. At this state, maximum contraction occurred in the width of the channel. To calibrate, at the first step, the coarseness height was considered equal to the average diameter for sedimentary particles. Then the height was increased to a point at which results from the numerical simulation would have an acceptable agreement with laboratory results, according to the evaluation criteria. For this purpose, tests 1 and 2 with the characteristics listed in Table 1 were selected.

## 4 Validation and error percentage of the model

To validate the model, a number of tests were not used in the calibration so that the capability of the model for other tests could be evaluated after calibration and execution of the model. In this study, tests 3 to 8 were selected for validation. The Nash–Sutcliffe model efficiency coefficient was used to determine the approximation of calculated and observed results (Equation11). This index varies between minus infinity and positive one. In fact, as its value approaches number one, the similarity is higher, and values below 0.5 for this index demonstrate a lack of coordination and approximation in the two groups. The mean value of the Nash coefficient for laboratory and numerical data was calculated to be 0.936, which is proof of the proper performance of this model. Table 3 demonstrates laboratory and numerical data along with Nash-Sutcliffe coefficient values. These values for all tests showed the accuracy of this model.

(11) |

Where:

RE |
= | Nash-Sutcliffe coefficient, |

x_{0} |
= | modeled value, |

x_{i} |
= | observed value, and |

= | mean observed value. |

Table 3 Results for scour depth for both numerical and laboratory results.

Test No. | Width of Contraction (m) | Water Depth (m) | Discharge (L/s) | Maximum Scour Depth Calculated (mm) | Maximum Scour Depth Observed (mm) | Nash Coefficient | Error Percentage |

1 | 0.2 | 0.3 | 20 | 44 | 42 | 0.92 | 4.76 |

2 | 0.2 | 0.3 | 30 | 120 | 110 | 0.97 | 9.09 |

3 | 0.3 | 0.3 | 20 | 30 | 26 | 0.97 | 15.38 |

4 | 0.3 | 0.3 | 30 | 46 | 43 | 0.76 | 6.97 |

5 | 0.3 | 0.3 | 40 | 93 | 88 | 0.98 | 5.68 |

6 | 0.4 | 0.3 | 20 | 18 | 15 | 0.99 | 20 |

7 | 0.4 | 0.3 | 30 | 31 | 27 | 0.96 | 14.85 |

8 | 0.4 | 0.3 | 40 | 44 | 42 | 0.92 | 4.76 |

To calculate the error percentage of the SSIIM model for calculation of the ultimate scour depth compared to the laboratory mode, Equation 12 was utilized.

(12) |

Where:

Δd |
= | percentage of error, |

d_{m} |
= | scour depth in the model, and |

d_{l} |
= | scour depth in the laboratory. |

## 5 Results and discussions

In this study, results obtained from numerical simulation were compared to those from laboratory tests. The model was also executed for the 30 mm sudden contraction. This section is dedicated to analyzing results from this test. Also, the Tecplot software tool was utilized to demonstrate the numerical results.

Comparing laboratory and simulation results (Table 3 and Figure 3) shows that SSIIM results are of acceptable coordination with those of laboratory tests and determination coefficient. The Nash-Sutcliffe coefficient and acquired error rate for all tests demonstrate proper approximation with the results of the model to determine the rate of scour in the contraction. The results show a strong correlation between observed and simulated maximum scour depth with a correlation coefficient of 0.976.

Figure 3 Comparison of scour depth rate in the laboratory and numerical models (in the gradual state).

### 5.1 Comparison of scouring results for the numerical and laboratory model

According to the simulation results, as the width of the channel decreases and discharge increases, the rate of scour rises. As expected, the maximum scour rate was in the 200 mm contraction with 30 L/s discharge for 120 mm. Figures 4 to 6 and Figures 8 to 10 show the scour pattern in various contractions and discharges. Considering the output model, scour begins from the sides of the channel and at the beginning of the contraction. Considering Figures 4, 5, and 6, at the part of the channel that comes after the contraction, flow velocity decreases with the increase in width channel, and the sediments subside. Results for all models show that as the discharge decreases, the scour rate decreases, and as the width of the contraction decreases, the scour rate increases. The scour pattern in simulated models shows that although the level of contraction and discharge rate influence the rate of scour and its quantity, these two parameters do not have any effect on the beginning point of scour, which is always the beginning of the contraction. In this study, this point is at the 2.2 m distance from the beginning of the channel where erosion begins.

Figure 4 Scour pattern in gradual contraction of 200 mm and discharge of 30 L/s.

Figure 5 Scour pattern in gradual contraction of 300 mm and discharge of 20 L/s.

Figure 6 Scour pattern in gradual contraction of 400 mm and discharge of 40 L/s.

One of the factors affecting contraction scour, is the type of transformation at the site of the contraction section, and in this study, tests were conducted in the gradual transformation state. To analyze the effect of transformation type on contraction scour rate, the SSIIM numerical model was utilized, and the sudden contraction with the width of 300 mm was executed in three different discharge rates. The results revealed that the rate of scour in sudden contractions increased by 29.4 % on average, compared to the gradual contraction (with a 50-degree angle). According to Figure 7, this increase was 36.3, 30.4, and 21.5% for discharge rates of 20, 30 and 40 L/s, respectively. Moreover, Figures 8, 9, and 10 demonstrate the scour pattern in sudden contractions. As can be seen in these figures, at the beginning of the contraction section, scour rate is higher than that of the gradual contraction state, which could be attributed to the sudden nature of contraction.

Figure 7 Changes in maximum scour depth at the discharge, comparison between numerical results and experimental data.

Figure 8 Scour pattern in sudden contraction of 300 mm and discharge of 40 L/s.

Figure 9 Scour pattern in sudden contraction of 300 mm and discharge of 30 L/s.

Figure 10 Scour pattern in the sudden contraction of 300 mm and discharge of 20 L/s.

Figure 11 demonstrates the changes in maximum scour depth in various contraction ratios. For a constant discharge, scour depth increases by increasing contraction ratio (*Rc*). In this figure, changes in the ratio of scour to flow depth with increased discharge were observed. Results revealed that an increase in discharge in a fixed contraction ratio increased the scour ratio, so that when the *Rc *= 0.5, with the increase of discharge rate of 33% (from 20 to 30 L/s), the rate of scour ratio to flow depth () increased by 39.5%, and as the discharge increased by 50% (from 20 to 40 L/s), the value of () increased by 70.4%. Considering this figure, it was concluded that at a fixed discharge, the decrease in contraction ratio (*Rc*) increases the scour rate so that at the fixed discharge rate of 30 L/s, a 16.7% decrease in contraction ratio increased the () ratio by 61%, and a 33.4% decrease increased it by 74.2%.

Figure 11 Changes in maximum scour depth to the contraction ratio of channel, comparison between numerical results and experimental data.

### 5.2 Water flow line in the numerical model

Figures 12 and 13 demonstrate the capability of this numerical model to simulate the form of flow lines, and contraction and rotational flows. Figure 12 shows flow lines for sudden contraction, and Figure 13 shows them for the gradual state. Considering these two figures, gradual contraction of the channel width had a significant effect on the input flow to the contracted section, so that these lines moved in parallel with the body in the gradual state and enter the contracted part with less turbulence. This is, in fact, the main reason for the low scour rate in gradual contractions.

Figure 12 Simulation of rotational flows in sudden 300 mm contraction.

Figure 13 Simulation of rotational flows in gradual 300 mm contraction.

## 6 Conclusion

In this study, flow pattern and scour changes were simulated in three gradual contractions: 200, 300 and 400 mm, and with a sudden contraction of 300 mm with discharges of 20, 30 and 40 L/s. In this simulation, the mean Nash-Sutcliffe coefficient was calculated to be 0.936, which is a sign of approximation of simulation data to laboratory data. The highest error of the model was 20% in Test 6, and the lowest one was in Test 1, with a 4.76% error rate. The following main results were concluded in this research:

- When comparing laboratory results with numerical results, it can be said that the SSIIM numerical model can acceptably predict the value and pattern of scour, and flow pattern changes in river channel contractions.
- Scour pattern in various contractions has a monotonous trend, and the beginning of scour is on the side walls; however, since in gradual contraction flow lines enter the contraction parallel to the body of the channel and the contraction angle, they experience little turbulence. Rotational flows decrease the scour rate in the gradual contraction state, as compared to the sudden one. So, this study demonstrates that scour in a 300 mm contraction with a 50-degree transformation angle is 29.4% less than the sudden state.
- Scouring advances rapidly in the beginning of the process. Both experimental and numerical results indicated that around 80% of scouring occurred during the first 20% of the equilibrium time. It was also observed that the maximum rate of scouring happened during the first hours of experiments and numerical computations and decreased with time.
- The results indicated that to numerically simulate scour in channel sudden and gradual contractions hydraulic structures, it is necessary to use a finer mesh inside the contractions, especially the first one, to simulate the primary vortex, which is the main reason of scouring around the structure (it is recommended to use half size of the mesh before contractions).
- The
*k-e*turbulence model with some RNG extensions showed good agreement with measurements for both flow and sediment transport computation. - Comparison of computed and measured bed changes proved that van Rijn’s sediment transport formula for bed load has high agreement with experimental results when using a default F83 0.053 2.1 0.3 1.5 code in the control file in the SSIIM model.

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