## Abstract

One of the most common strategies for sewer cleaning is to generate flushing flows using flushing gates to store water in the upstream sewer pipe. Therefore it is important to obtain the flow information on the flushing waves and their eroding effects. In this study, the flow characteristics of the flushing wave and the flushing effect were investigated by a transient flow calculation using a commercial computational fluid dynamics (CFD) code. The values of bottom shear stress were obtained and the effect of several factors are discussed. The water depth and the slope were related to the release rate of the storage volume, while the flushing volume determined the flushing distance at long sewer distances. The initial downstream water level was found to dramatically reduce the flushing effect. Equations based on the storage depth were developed to estimate the flushing effect, and suggestions for the installation and operation of flushing gates are provided.

## INTRODUCTION

Sediment deposits in sewers are responsible for blockages in sewers and negatively affect the hydraulic efficiency of sewer systems (Gent *et al.* 1996). Sediment deposits also promote the formation of hydrogen sulfide, which is commonly related to sewer odor concerns and pipe corrosion (Nielsen & Hvitved-Jacobsen 1988; Ashley *et al.* 2004; Liu *et al.* 2015). Therefore, significant amount of work have been devoted on sediment deposits: for example, Lange & Wichern (2013) conducted successive experiments on the sedimentation process, Ashley & Verbanck (1996) studied the behavior of sediment erosion and transport, and Nalluri & Ab Ghani (1996) proposed several sewer self-cleaning criteria. Shirazi *et al.* (2008) reported that sedimentation in sewers is more likely to happen in locations with adverse hydraulic flow conditions and Seco *et al.* (2014) indicated that the sediment can be consolidated with time under a relatively low shear stress (e.g. 0.15 N/m^{2}). A substantial proportion of the sediments in real sewers is cohesive, from weakly cohesive material to highly cohesive material. The cohesion leads to a greater resistance to erosion due to the presence of organic substances or fine particles (Crabtree 1989; Wotherspoon 1994; Tait *et al.* 2003). When the critical shear stress is exceeded, a collapse happens and the eroded particles can be carried downstream like non-cohesive particles (Nalluri & Alvarez 1992a, 1992b). Efforts have also been devoted to the modeling of sediment transport (Campisano *et al.* 2004; Creaco & Bertrand-Krajewski 2009; Shahsavari 2018).

Flushing is found to be an efficient method for dealing with sediment problems, which is usually based on the storage and subsequent release of water, generating flushing waves to transport the sediments. Several common flush devices have been compared and discussed (e.g. Dettmar *et al.* 2002; Guo *et al.* 2004; Shirazi *et al.* 2010). The flushing gate system is one of the most widely used devices to store the flow in the upstream sewers as it is relatively simple and reliable. They can operate mechanically for a long time without power supply and can use the main sewer to store the needed water volume. To evaluate the eroding effect of the flushing wave, the critical bottom shear stress leading to the initial erosion can be used as a criterion. Several criteria values from previous laboratory and field studies are summarized in Table 1. There are two typical ranges of critical bottom shear stress at which erosion occurred: 2–3 N/m^{2} for weaker materials, and 6–7 N/m^{2} for more consolidated deposits.

τ (N/m_{c}^{2})
. | Source . | Conditions . |
---|---|---|

0.7–3.3 | Ristenpart & Uhl (1993) | Field test in combined sewers |

1.4–3.2 | Campisano et al. (2008) | Laboratory study using different ratios of fine material |

1.8 | Ashley et al. (1992) | Field test in combined sewers |

1.0–3.0 | Ashley et al. (1994) | Field test in combined sewers |

2.0–4.0 | Stotz & Krauth (1986) | Deposits in domestic waste water |

2.5 | Nalluri (1991) | Non-cohesive material in laboratory study |

2.5 | Nalluri & Alvarez (1992a, 1992b) | Weak cohesive material in laboratory study |

4.0–6.0 | Wotherspoon (1994) | Field test in combined sewers |

5.0 | Dettmar & Staufer (2005a, 2005b) | Field test in combined sewers |

5.6 | Ristenpart (1998) | Field test in combined sewers |

6.0–7.0 | Nalluri & Alvarez (1992a, 1992b) | Consolidated material in laboratory study |

6.0–7.0 | Williams et al. (1989) | Synthetic cohesive material in laboratory study |

τ (N/m_{c}^{2})
. | Source . | Conditions . |
---|---|---|

0.7–3.3 | Ristenpart & Uhl (1993) | Field test in combined sewers |

1.4–3.2 | Campisano et al. (2008) | Laboratory study using different ratios of fine material |

1.8 | Ashley et al. (1992) | Field test in combined sewers |

1.0–3.0 | Ashley et al. (1994) | Field test in combined sewers |

2.0–4.0 | Stotz & Krauth (1986) | Deposits in domestic waste water |

2.5 | Nalluri (1991) | Non-cohesive material in laboratory study |

2.5 | Nalluri & Alvarez (1992a, 1992b) | Weak cohesive material in laboratory study |

4.0–6.0 | Wotherspoon (1994) | Field test in combined sewers |

5.0 | Dettmar & Staufer (2005a, 2005b) | Field test in combined sewers |

5.6 | Ristenpart (1998) | Field test in combined sewers |

6.0–7.0 | Nalluri & Alvarez (1992a, 1992b) | Consolidated material in laboratory study |

6.0–7.0 | Williams et al. (1989) | Synthetic cohesive material in laboratory study |

A number of experiments were conducted to study the parameters which affect the flushing effect. Skipworth *et al.* (1999) used artificial sediments in a pipe to examine the effects of cohesion on the resistance to erosion in a typical storm event. Knight & Sterling (2000) carried out experiments in a pipe of 244 mm in diameter to investigate the influence of fill ratio of the sediment bed (s/D = 0–0.66, with s and D being the sediment depth and the pipe diameter, respectively) under uniform flow condition. Campisano *et al.* (2004, 2007, 2008) investigated the flushing effect of successive flushes generated by a vertical lifting gate in a rectangular channel. Research by Dettmar & Staufer (2005a, 2005b) was located in a real combined sewer (D = 2,500–3,400 mm) and the results show that the influence of dam height on the flushing length is more dominant than the flushing volume. Schaffner & Steinhardt (2006) carried out field investigations to test a flushing gate with a dam height of 0.6–0.8 m in a pipe (a length of 713 m and diameter of 1,100 mm). Seco *et al.* (2014) investigated the behavior of highly organic sediment under an intense rainfall in a real sewer system. Bong *et al.* (2016) and Safari *et al.* (2017) investigated incipient erosion of sediment through laboratory experiments conducted in channels with different cross-section shapes.

However, detailed measurements of the transient flushing event in field and laboratory studies can be expensive and difficult to obtain. Also, the bottom shear stress may not be measured directly. The ability to gain the transient information makes numerical simulation a good tool to help investigate the flow characteristics of flushing waves complemented by experimental data for calibration and validation (Toorman 2000). Previous numerical studies are summarized in Table 2. As is shown in Table 2, previous investigations were mainly based on one-dimensional (1D) models using Saint-Venant equations or two-dimensional models (2D) using Shallow water equations. Very limited numbers of previous simulations were based on three-dimensional (3D) Reynolds-averaged Navier-Stokes (RANS) models due to the enormous computational resources needed. However, the 1D, 2D cases can cause significant difference compared to the 3D cases (Staufer *et al.* 2007) as the flushing event is strongly dependent on the velocity field and on the geometry (Caviedes *et al.* 2017).

Study . | Dimension . | Purpose . |
---|---|---|

Campisano et al. (2004) | 1-D | Investigation of the influence of number of successive flush waves |

Dettmar & Staufer (2005a, 2005b) | 1-D | Investigation of the behavior of flushing waves considering different dam heights |

Schaffner & Steinhardt (2006) | 3-D | Evaluation of the cleaning results for a self-acting flushing system Hydro Flush GS |

Abderrezzak & Paquier (2007) | 1-D | Comparison between the simulated results and the field measurement |

Staufer et al. (2007) | 1-D, 3-D | Comparison between the 1-D and 3-D results using the case by Schaffner (2003) |

Campisano et al. (2008) | 1-D | Investigation of the influence of two upstream water head and different ratio of fine material content (cohesive sediments) |

Staufer et al. (2008) | 1-D | Evaluation of the performance of a flushing device by calculating shear stress. |

Schaffner (2008) | 3-D | Investigation of the propagation of the flushing wave with calculation of the bottom shear stress. |

Schaffner & Steinhardt (2009) | 1-D | Evaluation of two chosen upstream boundary conditions (with/without a drop shaft) |

Schaffner & Steinhardt (2011) | 1-D | Investigation of the influence of downstream water levels. |

Schaffner & Steinhardt (2013) | 1-D | Investigation of boundary conditions like bottom roughness and lateral slope |

Yu & Duan (2014) | 1-D, 2-D | Comparison of the dam-break waves between the numerical model and the measurement for the simulation of dam-break flow over a mobile bed. |

Caviedes et al. (2017) | 2-D | Evaluation of the bed-load transport models comparing with experimental results |

Study . | Dimension . | Purpose . |
---|---|---|

Campisano et al. (2004) | 1-D | Investigation of the influence of number of successive flush waves |

Dettmar & Staufer (2005a, 2005b) | 1-D | Investigation of the behavior of flushing waves considering different dam heights |

Schaffner & Steinhardt (2006) | 3-D | Evaluation of the cleaning results for a self-acting flushing system Hydro Flush GS |

Abderrezzak & Paquier (2007) | 1-D | Comparison between the simulated results and the field measurement |

Staufer et al. (2007) | 1-D, 3-D | Comparison between the 1-D and 3-D results using the case by Schaffner (2003) |

Campisano et al. (2008) | 1-D | Investigation of the influence of two upstream water head and different ratio of fine material content (cohesive sediments) |

Staufer et al. (2008) | 1-D | Evaluation of the performance of a flushing device by calculating shear stress. |

Schaffner (2008) | 3-D | Investigation of the propagation of the flushing wave with calculation of the bottom shear stress. |

Schaffner & Steinhardt (2009) | 1-D | Evaluation of two chosen upstream boundary conditions (with/without a drop shaft) |

Schaffner & Steinhardt (2011) | 1-D | Investigation of the influence of downstream water levels. |

Schaffner & Steinhardt (2013) | 1-D | Investigation of boundary conditions like bottom roughness and lateral slope |

Yu & Duan (2014) | 1-D, 2-D | Comparison of the dam-break waves between the numerical model and the measurement for the simulation of dam-break flow over a mobile bed. |

Caviedes et al. (2017) | 2-D | Evaluation of the bed-load transport models comparing with experimental results |

In this study, a commercial software, ANSYS CFX 17.0, was used to solve the Navier-Stokes equations employed with the standard k-*ɛ* turbulence model to investigate the initial erosion in flushing events. The objectives of this study are: (1) to predict the propagation of the flood wave with time and to investigate the flow pattern, (2) to calculate the bottom shear stress of a flushing wave, (3) to evaluate the effect of factors including pipe slope, bottom roughness, storage water level, and downstream water level, (4) to provide suggestions for the installation and operation of the flushing gate systems for practical engineering purposes.

### Model setup

The flushing efficiency of a flushing gate was studied by an upstream sloped pipe (forming the storage region) and a downstream horizontal pipe (forming the flushing region) as shown in Figure 1. Both pipes had a diameter of 0.5 m. The flushing region is assumed as a horizontal pipe because sediments often take place in very flat sewer sections (Dettmar 2007). Besides, the flushing device is located in front of the flushing region with the assumption that the slope of upstream is relatively steep (*i* = 5–10‰). The sediments are immobile as this study focuses on the initial erosion. Influences of several factors were included in the study including the roughness (n), the pipe slope (*i*), the storage water depth (H) and the downstream water depth (H’). Simulated scenarios are listed as follows. As is shown in Table 3, series 1 and series 2 compare the scenarios with different roughness and slope, respectively. And the impact of storage water depth H was investigated by series 3. In addition, in many cases, the sewer is not totally drained before the flushing wave is reaching it (Schaffner & Steinhardt 2011), which may greatly affect the flushing efficiency. In this study, the influence of the downstream water height H’ was investigated as well.

Series . | i (‰)
. | H (m) . | Storage volume (m^{3})
. | Remarks . |
---|---|---|---|---|

1 | 10 | 0.2 | 0.610 | Comparison of scenarios with different roughness (n = 0.25 mm, 1 mm and 2.5 mm) |

2 | 5 | 0.3 | 3.171 | Comparison of scenarios with different upstream slopes (i = 5‰, 10‰) |

10 | 1.594 | |||

3 | 10 | 0.2 | 0.610 | Comparison of scenarios with different storage water depths (H = 0.2 m, 0.3 m, 0.4 m) |

0.3 | 1.594 | |||

0.4 | 3.042 | |||

4 | 10 | 0.3 | 1.594 | Comparison of scenarios with varying downstream water depth (H’ = 0 m, 0.05 m, 0.1 m) |

Series . | i (‰)
. | H (m) . | Storage volume (m^{3})
. | Remarks . |
---|---|---|---|---|

1 | 10 | 0.2 | 0.610 | Comparison of scenarios with different roughness (n = 0.25 mm, 1 mm and 2.5 mm) |

2 | 5 | 0.3 | 3.171 | Comparison of scenarios with different upstream slopes (i = 5‰, 10‰) |

10 | 1.594 | |||

3 | 10 | 0.2 | 0.610 | Comparison of scenarios with different storage water depths (H = 0.2 m, 0.3 m, 0.4 m) |

0.3 | 1.594 | |||

0.4 | 3.042 | |||

4 | 10 | 0.3 | 1.594 | Comparison of scenarios with varying downstream water depth (H’ = 0 m, 0.05 m, 0.1 m) |

*k*-

*ɛ*turbulence model. The following equation connects the turbulent kinetic energy with the wall shear stress (Rodi 1993):where is the wall shear stress,

*k*is the turbulent kinetic energy,

*C*= 0.09 and

_{μ}*ρ*is the water density

*ρ*= 1,000 kg/m

^{3}.

In this study, the studied structure is a regular shape, so the structured meshes were chosen. The mesh size was carefully selected according to previous research (Staufer *et al.* 2007; Schaffner 2008) and was further refined with 1,163,769 nodes and 1,112,400 elements (with a height of 0.002 m for the bottom grid cells). Three time steps of 0.005 s, 0.01 s and 0.02 s were selected to be compared. For water level, the results show little difference with different step sizes. However, for the bottom shear stress, difference was found of up to 1 N/m^{2} in the first 10 s and then narrowed to less than 0.2 N/m^{2} afterwards. Therefore, a higher temporal resolution of 0.005 s was used for the first 50 s' calculation and then changed to 0.02 s for the consequent calculation, and it took around 80 hours to model one calculation by parallel computation (with a workstation with 32 Intel cores).

### Model validation

To verify the model and the settings, previous measurements and simulated results by Schaffner (2008) were used to evaluate the simulated settings. The sewer is 380 m in length with an inclination of 3‰, the cross-section is composed of a dry weather channel in the center and lateral berms, see Figure 2(a). As is shown, the flushing wave was induced by a storage height H = 1.8 m. The roughness of the bottom was chosen to be 5 mm, corresponding to light mineral sediments in the fields. Results at the position of 143.5 m downstream of the flushing gate were compared, see Figure 2(b). Compared to the measurements, previous simulated waves by Schaffner (2008) show lower maximum water level by around 5 cm. In this study, the modeled wave shows good agreement with the measurements.

In addition, measurements from a laboratory scale study by Campisano *et al.* (2004) were used to further validate the model. As is shown in Figure 3(a), the measurement was conducted in a rectangular channel (3.9 m in length, 0.15 m in width and 0.35 m in deep) with a slope of 1.45‰. The measured water depth h was compared with the simulation results in Figure 3(b). And the simulation is shown to provide sufficient accuracy in predicting flushing waves even at a laboratory scale.

## RESULTS AND DISCUSSION

### General behavior

To investigate the flushing wave, the spatial and temporal variation of water depth, bottom shear stress, and the local velocity were studied and the results (in a scenario with slope *i* = 10‰, H = 0.3 m) are shown in Figure 4. Figure 4(a) and 4(b) shows the distribution of water depth *h* and the bottom shear stress *τ _{w}* (in the center plane on the bottom of the sewer channel) along the flushing distance at different times. There is a gradual decline in the maximum bottom shear stress with time. In addition, by comparing the distribution of local velocity and the bottom shear stress in the front part of the flushing wave, see Figure 4(c) and 4(d), the maximum velocity appears near the front end of the wave corresponding to the position of the maximum bottom shear stress.

The critical value of the bottom shear stress plays a crucial role in determining the erosion of the deposits. In this study, the critical shear stress is regarded as 3 N/m^{2} for the weaker materials and 6 N/m^{2} for the more consolidated deposits, which were suggested by previous field studies (Stotz & Krauth 1986; Ristenpart 1998) and laboratory studies (Williams *et al.* 1989; Campisano *et al.* 2008). To analyze the flushing effect, the term ‘effective flushing time’ *T*_{e} is introduced (Schaffner 2008; Schaffner & Steinhardt 2011) which represents the time duration when the bottom shear stress exceeds the critical values. Similarly, the term ‘effective flushing distance’ *L*_{e} is introduced, which refers to the maximum distance where the flushing wave can produce the bottom shear stress exceeding the criteria.

### Effect of the bottom roughness

In this study, the roughness of the flushing bottom was set as 0.25 mm as suggested by previous numerical research (Schaffner 2008). However, the roughness may be affected by multiple factors including the pipe material, the sedimentation position and the particle size (Kabir & Torfs 1992; Arthur *et al.* 1996; Sakakibara 1996). To investigate the impact of the roughness, scenarios with different roughness values (0.25 mm, 1 mm and 2.5 mm) are compared in Figure 5. In Figure 5(a), the distribution of water level *h* at 5 s, 50 s, 100 s, 200 s, 300 s is compared. The roughness is found to have more obvious effect with time. The greater roughness further slows down the flushing velocity, which leads to a shorter effective flushing distance. In addition, the difference between scenarios with roughness of 1 mm and 2.5 mm is much less obvious compared to the difference between scenarios with 0.25 mm and 1 mm. Figure 5(b) shows the variation of the maximum bottom shear stress during the flushing event. As is shown, a larger roughness reduces the bottom shear stress by around 1 to 2 N/m^{2} at the beginning, and the gap narrows with time. To erode the weaker material, the effective flushing time *T*_{e} for each case is all over 300 s. And to erode the more consolidated material, the effective flushing time *T*_{e} for each scenario is around 10 s, 20 s, and 25 s, respectively.

### Effect of the upstream pipe slope

The flushing wave is also affected by the slope of the storage region in the upstream, the impact is investigated in Figure 6. In Figure 6(a), the solid lines and the dashed lines respectively represent the variation of water depth *h* in the scenario where *i* = 5‰ and *i* = 10‰ during the flushing process. A steeper slope leads to a reduced upstream storage volume, and then results in a smaller pressure to downstream, and the influence is shown to increase with time. Generally speaking, the results show little difference between the scenarios in the first 10 s. However, a distinct gap can be found after 50 s in both the flushing distance and the water depth. In the scenario with 5‰ slope, the flushing wave has higher water depth along the flushing distance with farther flushing distance and the gap tends to be larger with time due to the much larger flushing volume (around double the volume compared to the scenario of 10 ‰). Figure 6(b) compares the maximum bottom shear stress of the flushing wave during the process. In the scenario with slope *i* = 10‰, the standing water is shown to have a faster release process compared with the scenario with slope *i* = 5‰, leading to greater maximum bottom shear stresses in the first 50 s. For the weaker material, effective scouring lasts for over 300 s for both slopes, with the scenario with *i* = 10‰ having a shorter effective flushing distance *L*_{e} due to the reduced upstream storage volume (in the scenario where i = 10‰, V = 1.594 m^{3} and in the scenario where i = 5‰, V = 3.171 m^{3}). However, a steeper slope is beneficial for eroding the more consolidated material.^{.} For example, the effective eroding time *T*_{e} lasts around 20 s in the scenario with *i* = 5‰ and lasts around 40 s in the scenario with *i* = 10‰, which corresponds to an effective distance of 30 m and 45 m, respectively.

### Effect of the upstream storage depth and the downstream water depth

The water depth is one of the most important factors in eroding the deposits. Scenarios with different upstream storage depth H are compared, see Figure 7(a). In addition, the downstream water may not be totally drained when the flushing wave arrives. Thus scenarios with different downstream retaining water are investigated in Figure 7(b). As shown in Figure 7(a), the upstream storage depth determines the release rate of the storage water, and it has a distinct effect on the maximum bottom shear stress in the beginning, and the gap between scenarios narrows with time. For example, compared to the scenario with H = 0.2 m, the maximum shear stress is initially increased by about 2.5 N/m^{2} with H = 0.4 m, and the difference decreases to around 1 N/m^{2} at 50 s and further decreases to around 0.5 N/m^{2} at 200 s. For the purpose of eroding the weaker material, all the scenarios can produce the effective flushing time for over 300 s. However, a higher storage level leads to an increased flushing volume that will create a flush wave with a longer effective flushing distance. To erode the more consolidated sediment, the effective flushing time *T*_{e} for storage depths of 0.2 m, 0.3 m, and 0.4 m is around 25 s, 40 s, and 60 s, respectively, corresponding to the effective flushing distance *L*_{e} of 20 m, 45 m, 70 m for each scenario.

Figure 7(b) compares the scenarios with different downstream water level H’. The condition with H’ = 0.05 m is shown to reduce the maximum bottom shear stress by around up to 30%, which results in the reduction of the eroding effect for the more consolidated material (*T*_{e} is reduced to 20 s from 40 s while *L*_{e} is reduced to 25 m from 45 m). Meanwhile, the downstream water depth of 0.1 m is shown to reduce the maximum value by around 75%, the flushing wave loses its ability to erode any deposits due to the massive drop in bottom shear stress.

### Equations for general prediction

***and the dimensionless flushing distance

*L*

_{e}(for the consolidated material) are then defined:where D is the diameter (m). The results are shown by H* and

*L*

_{e}*. Additionally, results from several previous researches (Dettmar & Staufer 2005a, 2005b; Campisano

*et al.*2008; Schaffner 2008) are included and compared in Figure 8.

As is shown, at a certain H*, the value of *L*_{e}* fluctuates within certain ranges. It is noteworthy that the value of *L*_{e}* in Schaffner (2008) is greater compared to the results from other studies. The difference is caused by the structural diversity. In Schaffner (2008), there is a dry weather channel in the bottom, which leads to a farther effective flushing distance *L*_{e}* (raised by around 20 to 40 m), and the gap is narrowed with the increase of the depth ratio.

The results in this study can fitted by Curve 1 (a = −85, b = 290, c = −60), and it also gives reasonable predictions compared to the measurements from other studies. Meanwhile, Curve 2 (a = −200, b = 380, c = −45) shows the results in the circumstance with a dry weather channel. For general situations, the flushing effect can be estimated by Curve 3 with a = −110, b = 295, c = −45. It is worth noting that the equation is intended to be used for a quick estimation, factors like the downstream water level, the roughness and the slope are not included.

### Summary and conclusions

In this study, two-phase (water-air) transient flow simulations were carried out to investigate the flushing waves created by the flushing gate. A three-dimensional numerical model was employed with the standard k-*ɛ* turbulence closure method. The simulation results were validated using previous field and laboratory measurements. The storage depth, the bottom roughness, the slope, and the downstream water depth were regarded as variables. The bottom shear stress was calculated and two criteria were chosen to evaluate the eroding effect. The following results are obtained from the current study.

When the storage volume is released, the maximum velocity appears in the front end of the flushing wave where the maximum bottom shear stress occurs. In most cases, the flushing wave can produce sufficient shear stress to erode the weaker material. However, the flushing effect on the more consolidated deposits is highly dependent on multiple variables. Specifically, the higher roughness shows a considerable influence on effective flushing time *T*_{e}, and effective flushing distance *L*_{e} (with the shear stress above the criteria value). The slope is related to the release rate of the storage volume and it also affects the effective flushing distance due to different flushing volumes (e.g. *L*_{e} = 30 m in the scenario with *i* = 5‰ can be increased to *L*_{e} = 45 m in the scenario with *i* = 10‰). Another way to increase the eroding ability is to increase the storage depth. In this study, a depth of 0.4 m can produce a much larger effective flushing distance of 75 m compared to the value with a depth of 0.2 m (around 20 m). Therefore, the operation of the gate should try to avoid conditions with a storage depth lower than half of the pipe diameter to improve the flushing efficiency. Moreover, cases with remaining downstream water were investigated. Downstream water with shallow depth (H’ = 0.05 m) are shown to have some certain influence on the bottom shear stress and may reduce the effective flushing time by about half. However, with a deeper depth (0.1 m), the downstream water can dramatically reduce the bottom shear stress by around 75% and the flushing wave entirely loses its scouring effect. Dimensionless equations were also developed to quickly estimate the effective flushing distance, and results from some other researches were also taken into consideration. However, the results in this study are based on theoretical analyses, further investigations and applications of the obtained results for practical projects should be undertaken in the future.

## ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of the China Scholarship Council, the International Cooperation and Exchange of the National Natural Science Foundation of China (No. 51761145022), and the Natural Sciences and Engineering Research Council (NSERC) of Canada.