Study on the diameter design method of vortex settling basin for suspended sediment based on the tangential characteristic Reynolds number

Abstract

The vortex settling basin for suspended sediment is a sediment treatment device that enables efficient use of irrigation water, yet the diameter design remains insufficiently defined, leading to large dis crepancies between measured and designed trapping efficiency in practice. This study proposes a diameter design method based on the tangential characteristic Reynolds number. Field observations were first conducted at three representative projects—Donglei, Talan River, and Shangmaxiangdi. Guided by these observations, laboratory physical models were developed, and an Acoustic Doppler Velocimeter was used to measure and analyze the distribution of tangential characteristic Reynolds number in each basin. The results show that when the sediment trapping efficiency is high, the tangential characteristic Reynolds number should be less than 9.5 × 10⁶ for radial positions between 0.80 and 0.92 at the 180-degree section or between 0.60 and 0.92 at the 270-degree section, where the radial position is the distance from the measurement point to the basin center divided by the basin radius. Building on this relationship, a diameter design method for the vortex settling basin for suspended sediment is proposed. Verification using prototype projects indicates that replacing the traditional empirical approach with the proposed method can increase the trapping efficiency of suspended sediment by about 9.4%–21%. These findings provide practical guidance for diameter design.

Introduction

The grain production belt of China has been shifting progressively northward. The mismatch between water supply and demand is therefore intensifying, especially in the arid and semi-arid northwest where the uneven spatial and temporal distribution of water resources places agriculture under increasing pressure^1,2. Efficient use of water resources is essential for safeguarding food security and maintaining a stable supply of agricultural products³.

The rivers of the Tarim River Basin in Xinjiang China are characterized by fine sediment particles and high sediment concentration during the flood season. During the dry season and the normal flow period, streamflow is small and sediment concentration is relatively low. In the flood season, concentrated rainfall increases river discharge and sediment concentration reaches the annual maximum. The flood season usually occurs from July to September, coinciding with the peak of agricultural irrigation. During this period, large volumes of fine suspended sediment enter irrigation systems with the flow, which can easily clog or even damage water saving irrigation equipment, resulting in system failure. To prevent damage to irrigation systems from sediment laden flow, the water is treated before use. Engineering measures for sediment removal include VSBs, settling basins, circular central ring structures, and vortex desilting channels. Among these, the VSB has been applied in rivers such as the Yellow River, the Talan River, and the Muzati River due to its high sediment water separation efficiency, simple structure, low water abstraction ratio, and low construction cost.

A VSB consists of an inlet canal, a deflector, a vortex chamber, a flushing orifice, and an overflow outlet as shown in Fig. 1. Sediment laden flow enters the VSB tangentially through the inlet canal and moves in a spiral pattern. Sediment is carried along the spiral path toward the flushing orifice, while clearer water leaves through the overflow outlet and flows downstream. Because the sediment settling path is much longer than the circumference of the VSB, it occupies less land area than a settling basin when achieving the same sediment removal performance.

Fig. 1

Schematic diagram of VSB operation.

According to the type of sediment it handles, the vortex settling basin (VSB) can be classified into two types: the VSB for bed load sediment and the VSBS for suspended sediment. The VSB is mainly used to treat bed load with a particle size greater than 0.25 mm. To ensure efficient discharge of bed load through the flushing orifice, the flow inside the basin must possess sufficient kinetic energy to transport the sediment toward the outlet. The VSB originated from the annular settling basin proposed by Salakhov⁴, which utilized a central helical flow to intercept sediment. Based on this concept, subsequent researchers have conducted studies focusing on the design methodology of VSBs and on improving and predicting their sediment trapping efficiency.

Paul proposed a design method suggesting that the diameter of the vortex settling basin (VSB) should be five times the bottom width of the inlet canal, and developed a predictive formula for sediment trapping efficiency⁵. Based on the work of Paul, Athar reanalyzed experimental data and field observations to examine the influence of VSB geometric parameters on sediment trapping efficiency and verified the reliability of the design method by paul⁶. Keshavarzi improved the sediment feeding and collection systems and conducted experimental studies, showing that the flow structure is closely related to the water–sediment separation performance. After optimizing the clockwise flow pattern and the arrangement of deflectors, the sediment trapping efficiency increased by 8%⁷. Niknia found that installing deflectors can enhance the sediment trapping efficiency by 5%–15%, reaching up to 88% for sediment with a particle size of 0.35 mm⁸. Ansari developed a predictive model for sediment trapping efficiency using an artificial neural network (ANN), which outperformed traditional regression models and confirmed that the basin diameter is the most critical parameter^9,10. Keshavarzi proposed a method for defining turbulent structures based on three-dimensional bursting events and used quadrant analysis to identify zones sensitive to sediment initiation and deposition on the vortex chamber floor¹¹. Niknia¹² measured the three-dimensional velocity field and flow characteristics around deflectors in the VSB, revealing that the deflector weakens the tangential velocity of the bottom vortex while enhancing radial motion, thereby promoting more effective sediment transport toward the central flushing orifice.

The VSBS is designed for the treatment of suspended sediment. To ensure effective settling of suspended particles, the flow velocity inside this type of basin is relatively low. Existing studies mainly focus on reducing sediment deposition around the deflector and within the vortex chamber, as well as on optimizing the arrangement of support columns. Li compared the influence of different perforation positions on the deflector and found that when the openings were located only in the outer region of the deflector, the mass of sediment deposition on the deflector decreased by up to 68.8%, and the VSB exhibited higher sediment trapping efficiency¹³. Li further used numerical simulations to reveal the mechanism by which deflector perforation affects water–sediment separation performance in the VSB¹⁴. Mu conducted model experiments and determined that the optimal perforation ratio in the outer region of the deflector ranges from 8.67%−13%¹⁵. Li investigated the influence of support column arrangement, used to sustain the overflow deflector in the VSBS, on water–sediment separation performance. The results indicated that the column position has little effect on sediment trapping efficiency and water abstraction ratio, but the column arrangement increases the mass of sediment deposition in the vortex chamber¹⁶. Wang proposed installing vanes at the bottom of the support columns to reduce sediment deposition in the chamber and found that the maximum mass of sediment deposition decreased by 22.25% after installing vanes¹⁷. Kiringu and Basson employed ANSYS FLUENT software combined with the VOF and DPM methods to investigate the sediment trapping efficiency of the VSB for suspended sediment in small-scale intake projects (with inflow discharge less than 0.1 m³/s) and optimized the overflow outlet design^18,19.

In recent years, VSBs have been applied to the removal of suspended sediment, referred to as VSBS. Suspended sediment has fine particle size, is easily resuspended, and requires lower flow velocity for effective settling. However, designers have adopted the diameter design method proposed by Zhou²⁰ for bed load sediment removal, expressed as D = (Q_i/0.028) ^(1/2.5), where Q_i is the inflow discharge of the VSBS and D is the diameter of the VSBS. For example, in the VSBS project of the Upper Madi Hydropower Station in Nepal, the design discharge was 48 m³/s and the diameter of the VSBS was 50 m. The design value of sediment trapping efficiency for sediment larger than 0.25 mm was set at more than 80%, whereas field observations showed that the actual sediment trapping efficiency for sediment larger than 0.1 mm was 29.3%. This indicates a large discrepancy between the measured and design values.

To establish a diameter design method better suited for VSBS, this study selected three projects, Donglei (DL), Talan River (TLH), and Shangmaxiangdi (SMXD), for field observations. Measurements of water–sediment conditions were conducted systematically under operating conditions. Combined with laboratory flow field tests, the tangential characteristic Reynolds number (Re_t) distribution inside each VSBS was obtained. Using the tangential characteristic Reynolds number distribution under efficient sediment settling conditions as the target, a diameter design method for VSBS was derived. Verification tests were carried out to assess the proposed method. The results provide guidance for the diameter design of VSBS.

Materials and methods

Field observations

Prototypes

Field observations were conducted at three typical VSBSs—DL, TLH, and SMXD—selected from domestic and international projects with similar design methods but markedly different suspended-sediment trapping efficiency. During operation at each site, inflow and outflow discharges and sediment concentrations were measured, and size-graded sediment trapping efficiencies were calculated. The prototype VSBSs for DL, TLH, and SMXD are shown in Fig. 2(a)–(c).

Fig. 2

Examples of prototype projects of the VSBS. (a) The Donglei VSB, located upstream of the Donglei Irrigation District; (b) the Tailan River VSB, located upstream of the Tailan River Reservoir; (c) the Shangmaxiangdi VSB, located upstream of the Shangmaxiangdi Hydropower Station.

DL Project is located on the main stream of the Xiaobei River in the Yellow River basin, with a diameter of 16.00 m, designed for sediment removal in irrigation projects. The design diversion discharge is 2 m³/s, the average annual sediment concentration is 3.5 kg/m³, and d₅₀=0.02 mm. TLH Project is located on the Tailan River, a tributary of the Tarim River basin, with a diameter of 48.00 m, designed for sediment removal in an irrigation reservoir. Its design diversion discharge is 40.0 m³/s, the average annual sediment concentration of the diverted flow is 4.4 kg/m³, and d₅₀=0.08 mm. SMXD Project is located on the upper Marsyangdi River in western Nepal, with a diameter of 50.00 m, designed for sediment removal at a hydropower station intake. Its design diversion discharge is 50.0 m³/s, the average annual sediment concentration of the diverted flow is 2.5 kg/m³, and d₅₀=0.12 mm.

The geometric parameters of each project are listed in Table 1, where D is the basin diameter, B_i is the inlet canal width, H_i is the inlet canal height, R_f is the flushing orifice radius, B_d is the deflector width, h is the deflector height, S is the bottom slope of the vortex chamber.

Table 1 Prototype data of each VSBS Project.

Observation methods

The field observations comprised two parts: (1) obtaining grain-size distribution curves by collecting samples at the inlet canal and the overflow outlet under typical operating conditions, and (2) calculating trapping efficiency by particle size based on measurements of discharge and sediment concentration at the same sections. The instruments and equipment used are listed in Table 2, where the notations are defined as v = flow velocity (m/s) and n = rotation speed (r/s). For the current meters of the same model (LS25-1), instrument-specific calibration equations were used (v = an + b), where a and b are calibration coefficients obtained from individual laboratory calibrations. The differences in a and b among instruments arise from slight variations in the mechanical characteristics of the propeller, bearing friction, and electronic sensing precision during calibration.

Table 2 Main equipment for field Observations.

For each VSBS project, three measurement cross sections were arranged, designated as CS1, CS2, and CS3. CS1 was located upstream of the inlet canal, CS2 was located downstream of the overflow outlet, and CS3 was located in the flushing tunnel. Each cross section contained three measuring verticals, positioned at the midpoint of the diversion canal section and at the midpoints between this centerline and each sidewall. The specific arrangement is shown in Fig. 3.

Fig. 3

Prototype observation cross sections and measuring line arrangement (note: red circles indicate the positions of vertical measuring lines).

Flow measurements were conducted using a current meter. Based on established hydrometric practice, the average of the velocities at 0.2H and 0.8H approximates the depth-averaged velocity, while adding a mid-depth point at 0.6H improves representation of the vertical profile. Therefore, we adopted a three-point scheme (0.2H, 0.6H, 0.8H) to enhance the reliability of discharge and sediment measurements. Three measurement points were set al.ong each vertical at 0.2H, 0.6H, and 0.8H from the channel bed (where H is the water depth)²¹. The discharge of each section was then calculated by the velocity–area method using the velocities measured at these points.

Sediment samples were collected using a horizontal sampler at three verticals on each measurement section. At each vertical, samples were taken at 0.2H, 0.6H, and 0.8H depths. For each sampling point, 2 L of water was collected and sampled three times, resulting in a total volume of 6 L per point. After sampling, water samples were settled, dried, and weighed to determine the sediment mass. Sediment concentration at each point was calculated as the sediment mass divided by the total water volume. The mean sediment concentration of each measurement section was obtained by averaging the values from all sampling points.

The dried sediment samples were sieved using a vibrating shaker, while the fine particles were analyzed by pipette method to obtain the complete grain size distribution.

The total discharge of each measurement section was calculated according to Eq. (1):

$${Q_s} = \frac{1}{n}\mathop \sum\limits_{i = 1}^n {v_s}_i \cdot {A_s}_{}$$

(1)

where Q_s is the discharge of the s-th measurement section, m³/s; v_si is the velocity at the i-th measurement point in the section, m/s; A_s is the area of the s-th measurement section, m²; and n is the total number of measurement points in the section.

The sediment concentration at each sampling point after drying was calculated according to Eq. (2):

$${S_c}=\frac{{{M_c}}}{{{V_c}}}$$

(2)

where S_c is the sediment concentration at a given sampling point, kg/m³; M_c is the mass of sediment after settling and drying, kg; V_c is the total volume of the water sample collected at that point, m³. The mean sediment concentration of each section was calculated by averaging the values from all sampling points within the section.

The sediment trapping efficiency for each grain-size class was calculated using Eq. (3):

$$\eta =\left( {1 - \frac{{f_{o}^{{}} \cdot {{{\Delta}}}d_{o}^{{}}}}{{{f_i} \cdot {{{\Delta}}}{d_i}}}} \right) \times 100{{\% }}$$

(3)

where the ${f_i} \cdot {{{\Delta}}}d$ is the product of the total sediment transport rate at the VSBS inlet and the percentage of the corresponding grain-size class (i.e., the sediment load of that class at the inlet), and $f_{o}^{{}} \cdot {{{\Delta}}}d_{o}^{{}}$ is the corresponding value measured at the downstream cross-section of the overflow outlet. This equation quantifies the fractional reduction of sediment for the same grain-size class after passing through the VSBS, namely the trapping efficiency of that class.

Model experiments

Experimental setup

The experimental setup used in this study is shown in Fig. 4. The model adopted was a normal scale model designed according to the Froude similarity criterion. The formulas for calculating the scale ratios of physical quantities are presented in Eq. (4) to (7).

Geometric scale:

$${\lambda _L}={\lambda _H}=\frac{{{L_p}}}{{{L_m}}}$$

(4)

where, λ_L is the horizontal scaling ratio; λ_H is the vertical scaling ratio; L_p is the prototype length; L_m is the model length.

The velocity scaling ratio is:

$${\lambda _v}=\lambda _{L}^{{\frac{1}{2}}}$$

(5)

The discharge scaling ratio is:

$${\lambda _Q}=\lambda _{L}^{{\frac{5}{2}}}$$

(6)

The roughness scaling ratio is:

$${\lambda _n}=\lambda _{L}^{{1/6}}$$

(7)

In addition to satisfying the aforementioned similarity conditions, the suspended sediment model should also ensure similarity in sediment motion. Given that the primary function of the VSBS is to facilitate sediment settling, it is therefore necessary to ensure similarity in sediment settling processes:

$${\lambda _\omega }={\lambda _v}$$

(8)

where λ_w denotes the scale factor for sediment settling velocity, and λ_v denotes the scale factor for flow velocity.

The sediment settling velocity is calculated using the Stokes formula, as shown in Eq. 9.

$$\omega =\frac{g}{{18}}\frac{1}{\nu }\left( {\frac{{{\rho _s}}}{\rho } - 1} \right){d^2}$$

(9)

By combining Eqs. (9), (8), and (5), the scale factor for sediment particle size λd that simultaneously satisfies both gravity similarity and sediment settling similarity can be obtained.

$${\lambda _d}=\sqrt {\frac{{{\lambda _v}}}{{{\lambda _{\frac{{{\rho _s} - \rho }}{\rho }}}}}}$$

(10)

The model scale factors for each project are shown in Table 3.

Table 3 Model scales.

The experimental setup for this study is illustrated in Fig. 4. The VSBS was constructed from stainless steel with a roughness coefficient (i.e., Manning coefficient) ranging from 0.007 to 0.008, satisfying roughness similarity. At the beginning of the experiment, Pump delivers water from reservoir to elevated tank, which has an overflow outlet to maintain a constant water level. The water then flows through supply pipeline and valve into steady water tank, with the flow rate controlled by the valve. After the flow stabilizes, the water enters diversion canal, and flows through inlet canal into the VSBS chamber. During operation, the clear water treated by the VSBS flows out through deflector, while the sediment-laden water is discharged through flushing orifice. The outflow then passes through the pre-calibrated triangular thin plate weir and rectangular thin plate weir, where the corresponding discharge is measured, and finally returns to the storage tank.

Fig. 4

Experimental setup. 1.Water pump; 2. Storage tank; 3. Elevated tank; 4. Pipeline; 5. Valve; 6. Regulating tank; 7. Diversion canal; 8. Sediment feeding device; 9. Inlet canal; 10. Vortex chamber; 11. Deflector; 12. Flushing orifice; 13. Triangular thin plate weir; 14. Rectangular thin plate weir; 15. Backwater canal.

Experimental design

The experiments consisted of clear-water and sediment-laden flow tests. The clear-water tests were conducted at the design discharges of the DL, TLH, and SMXD prototype projects, which are 2.0 m³/s, 40.0 m³/s, and 50.0 m³/s, respectively. The corresponding model inflow discharges were 1.9 L/s, 2.4 L/s, and 2.8 L/s, under which tangential velocity measurements were performed.

To comprehensively capture the distribution characteristics of the tangential characteristic Reynolds number within the VSBS, four typical measurement sections were established at 0°, 90°, 180°, and 270°. Along each section, measurement lines were arranged at 20 mm intervals in the radial direction, and measurement points were set at 1 mm intervals along the axial direction. Details are provided in Fig. 5.

Fig. 5

Measurement-Point Layout: (a) Plan view (b) Section AA view (Note: red circles indicate measurement points).

Flow velocity was measured using a three-dimensional Acoustic Doppler Velocimeter (ADV) with a downward-looking probe (Qingdao Nortek Instruments Co., Ltd.). The system comprised a connector, signal conditioning module, signal receiver, and signal transmitter. The sampling volume was positioned 40–70 mm below the probe, with a vertical measurement interval of 1 mm. The ADV sampling frequency was set to 60 Hz, and the velocity range and accuracy were ± 0.6 m/s and ± 0.5%, respectively. A comparison of records sampled over 10 s and 20 s showed that the difference in mean velocity was < 2%; therefore, a 10 s duration was adopted at each measurement point, yielding 600 samples per point.

After obtaining the characteristics of the tangential characteristic Reynolds number associated with efficient sediment settling in the VSBS, a method for determining the diameter of the VSBS can be proposed. To verify the reliability of this method, the TLH project was selected for validation through sediment-laden flow experiments (the detailed experimental procedure is described in Sect. “Validation of the design method” following the introduction of the diameter design method). During the experiment, water samples were collected every 5 min from the inlet canal, overflow outlet, and flushing orifice using 250 ml rigid transparent glass volumetric flasks that had been strictly calibrated. The mass of each sample was measured using a Mettler-Toledo electronic balance with a precision of 0.001 g. Each sampling point was measured three times, and the average value was used for subsequent analysis. The sediment concentrations in the inlet canal, overflow outlet, and flushing orifice (denoted as S_i, S_o, and S_f, respectively) were determined using the displacement method¹⁷:

$$S=\frac{{\left( {{\rho _{\text{m}}}{\text{-}}\rho } \right) \times {\rho _{\text{s}}}}}{{{\rho _{\text{s}}}{\text{-}}\rho }}=\frac{{\left( {\frac{M}{V} - \rho } \right) \times {\rho _S}}}{{{\rho _S} - \rho }}=\frac{{\left( {\frac{{{M_{\text{m}}}{\text{-}}{M_{\text{b}}}}}{V} - \rho } \right)}}{{{\rho _S} - \rho }}$$

(11)

where ρ is the density of water, M is the mass of the conical flask, and V is the volume of the conical flask. The subscripts s, m, and b represent sediment, sediment-water mixture, and empty flask, respectively.

To verify the applicability of the model experiment results to the prototype project, tangential velocities were measured in the DL prototype project at sections of 180° and 270° with r/R = 0.4. Measurements were taken at positions 0.2, 0.6, and 0.8 of the water depth starting from the VSB bottom. The measured velocities were compared with those obtained from the model experiment, as shown in Table 4. The results show that the velocity differences between the prototype and the model are within ± 0.03 m/s, and the velocity distribution patterns from the bottom to the water surface are consistent. These findings indicate that the model experiment data can, to some extent, represent the flow patterns of the prototype.

Table 4 Comparison of tangential velocities between the DL prototype and the model (m/s).

Results

Field observation results

Based on analyzing, calculating, and compiling the observation data from each project, the grain size distribution curves of the treated sediment were obtained, as shown in Table 5; Fig. 6. The median sediment particle sizes for the DL, TLH, and SMXD projects were 0.02 mm, 0.08 mm, and 0.12 mm, respectively, indicating that all treated sediments belong to suspended sediment. In addition, the size-graded sediment trapping efficiency for each representative project was obtained, as presented in Table 6. The results show that, although the same diameter design method was applied, the three projects (DL, TLH, and SMXD) exhibited notable differences in suspended-sediment trapping efficiency.

The overall sediment trapping performance of the DL project was superior to that of the TLH and SMXD projects, especially for sediment with particle sizes between 0.05and 0.10 mm, where the trapping efficiency reached 46.8%, compared to 16.0% for TLH and 7.3% for SMXD. For sediment with particle sizes greater than 0.1 mm, all three projects demonstrated some trapping capability, but the DL project showed the highest efficiency at 95.9%, followed by TLH at 61.9% and SMXD at 29.3%. Even for sediment smaller than 0.05 mm, the DL project still exhibited a certain trapping capacity, with efficiencies ranging from 15.4% to 16.4%. In contrast, TLH and SMXD had extremely low trapping efficiencies for this size range and were almost ineffective at intercepting sediment smaller than 0.025 mm.

Table 5 Grain size distribution of sediment in each VSB project (%).

Fig. 6

Sediment grain size distribution curves.

Table 6 Sediment trapping efficiency of prototype VSBS projects.

Characteristics of Re_t with efficient suspended sediment settling

The sediment settling process inside the VSBS is closely related to the flow characteristics, which in turn depend on the Reynolds number. As the Reynolds number increases, the flow becomes more turbulent, making sediment settling less favorable. The tangential velocity, as the dominant flow component within the VSB, defines the tangential characteristic Reynolds number Re_t which serves as an important indicator of the turbulence intensity inside the basin^14,22. Its calculation is shown in Eq. (12). As indicated by Eq. (12), Re_t is determined by the vortex chamber radius and the tangential velocity. Under a constant inflow discharge, the tangential velocity within the chamber is strongly influenced by the basin diameter. Therefore, when the inflow discharge is fixed, adjusting the basin diameter effectively changes the Re_t value. If the characteristic Re_t corresponding to optimal suspended sediment settling can be identified, it can be used as a target criterion to develop a diameter design method for VSBS.

$${\operatorname{Re} _t}=\frac{{\rho \overline {u} R}}{\mu }$$

(12)

where ρ is the density of water (kg/m³), $\overline {u}$ is the time-averaged tangential velocity at each measurement point (m/s), R is the radius of the VSBS, and µ is the dynamic viscosity (N·s/m²).

After measuring the flow velocity in the DL, TLH, and SMXD projects using ADV, the cross-sectional distribution and magnitude of the tangential characteristic Reynolds number (Re_t) were analyzed to identify the Re_t characteristics corresponding to higher suspended sediment settling efficiency. Based on this characteristic, a diameter design method for the VSBS was proposed.

Figure 7 shows the cross-sectional distribution of the Re_t for each project under normal temperature conditions (approximately 20 °C). The horizontal axis represents r/R, where r is the distance from the measurement point to the center of the basin and R is the basin radius, indicating the radial position of the measurement point. The vertical axis represents z/H, where z is the vertical distance from the measurement point to the basin floor and h is the deflector height, indicating the vertical position of the measurement point.

As shown in the Fig. 7, influenced by both the forced and free vortices within the basin, all VSBSs exhibit a distribution pattern of lower Re_t values at the center and higher values near the sides. Among them, the DL project, which achieved higher suspended sediment removal efficiency, has a maximum Re_t of 9.5 × 10⁶; for the TLH and SMXD projects, the maximum Re_t values are 37.4 × 10⁶ and 45.1 × 10⁶, respectively.

In addition, since the 0° and 90° sections are close to the inlet and overflow outlet, the rotational flow inside the basin interacts with the inflow and overflow, leading to highly turbulent conditions in these regions. In contrast, the 180° and 270° sections are farther from both the inlet and the overflow outlet, resulting in relatively stable flow patterns. Therefore, this study selected the 180° and 270° sections—where turbulence is comparatively weaker—to further analyze the distribution of Ret and to identify representative regions that can provide guidance for subsequent design.

Fig. 7

Cross-sectional distributions of Re_t for each project: (a-c): 0°;(d-f): 90°༛(g-i): 180°༛(j-l): 270°.

The radial distribution of the tangential characteristic Reynolds number (Re_t) along different horizontal lines at the 180° and 270° sections of each VSB is shown in Fig. 8. As seen from the figure, the radial distribution pattern of Re_t in each project is consistent with that of the tangential velocity. Specifically, in the outer region (0.40 < |r/R| < 0.92), Ret increases with radius, while in the inner region (0.20 < |r/R| < 0.40), Re_t decreases with increasing radius. When Re_t is relatively low, viscous forces dominate, the flow remains stable, and sediment deposition is promoted. Conversely, when Re_t is high, inertial forces prevail, turbulence intensity increases, and sediment particles tend to remain in suspension or re-suspend after settling, leading to a decrease in sediment trapping efficiency.

As shown in Fig. 8, the difference in Re_t among the projects becomes more pronounced toward the outer region (−0.92 < r/R < −0.80), though the Re_t values within this range exhibit minimal fluctuation for each project. The magnitude of Re_t follows the order DL < TLH < SMXD, which is inversely proportional to the order of sediment trapping efficiency (DL > TLH > SMXD). Notably, the DL project exhibits the lowest Re_t in this region, with values all below 9.5 × 10⁶. The influence range of its outer forced vortex accounts for approximately 45.7%, which is about 0.68 times that of the low-efficiency case (SMXD), and its vortex intensity is roughly 21.01% of that in the low-efficiency sediment removal basin.

Fig. 8

Radial distribution of Re_t in each VSB project: (a) 180° section; (b) 270° section.

As discussed earlier, although the Ret values in the outer regions of each VSB differ in magnitude among the projects, the variation of Re_t within each project remains relatively small until r/R exceeds a certain value, beyond which the fluctuation increases significantly. To reasonably identify representative regions within the 180° and 270° sections, an index K_r was introduced to describe the rate of change of Ret along the radial direction, as expressed in Eq. (13):

$${K_r}=\frac{{{{\operatorname{Re} }_t}\left( {x+1} \right) - {{\operatorname{Re} }_t}\left( x \right)}}{{{{\operatorname{Re} }_t}\left( x \right)}} \times 100\%$$

(13)

where K_r is the percentage change of Re_t between two adjacent measurement points arranged along the radial direction at a fixed water depth, and x denotes the measurement point index.

Four measurement lines at z/h = 0, 0.35, 0.52 (cylindrical region), and − 0.1 (conical region) were selected within the 180° and 270° sections to compare and analyze Kr, as shown in Fig. 9. The results indicate that, in the range of − 0.92 < r/R < − 0.80 for the 180° section and − 0.92 < r/R < − 0.60 for the 270° section, K_r does not exceed 10%, suggesting that the variation in Re_t is relatively small in these regions. Beyond these ranges, K_r increases significantly. This observation is consistent with the Re_t distribution patterns shown in Fig. 8, indicating that Re_t values are similar within the selected regions of the 180° and 270° sections, while the magnitude of variation increases outside these regions.

In summary, within the 180° section (− 0.92 < r/R < − 0.80) and the 270° section (− 0.92 < r/R < − 0.60) of the VSBS, Re_t is relatively large with minimal variation. The tangential velocity in these typical regions plays a crucial role in maintaining the swirling flow within the VSBS, and the turbulence intensity is closely related to the Re_t calculated from the tangential velocity in these regions. Therefore, this study proposes using these typical regions in the 180° and 270° sections as reference zones for design. The average tangential velocity in the typical region of the 180° section is denoted as u_p, and that in the 270° section is denoted as u_L. The Re_t values calculated from u_p and u_L are used as design criteria for determining the diameter of the VSBS. Given that the maximum Re_t associated with high suspended sediment settling efficiency is 9.5 × 10⁶, the diameter of the VSBS should be designed to ensure that Re_t in at least one of these typical regions does not exceed 9.5 × 10⁶.

Fig. 9

Radial distributions of K_r at different z/h: (a): 180°;(b): 270°.

Diameter design method based on Re_t

The value of Re_t in the typical regions is determined by the tangential velocity and the radius of the VSBS. Since the tangential velocity cannot be directly measured, previous studies have indicated that the inlet velocity V_i can be used as a proxy for the tangential velocity. To establish the relationships between the average tangential velocity and V_i in the two typical regions, relevant data from domestic and international sources were collected and compiled, as shown in Table 7. Based on the data summarized in Table 7, the correlations between u_p, u_L, and the inlet velocity V_i were obtained, as illustrated in Fig. 10 and given by Eqs. (14) and (15).

Table 7 Data of u and V_i (m/s).

Fig. 10

Relationship between u_p, u_L, and V_i.

$${u_p}=0.6763{V_i}$$

(14)

$${u_L}=0.633{V_i}$$

(15)

The results of the statistical indicators and the relative errors between the measured and predicted values are presented in Tables 8 and 9. As shown in the tables, the evaluation metrics for the fitting and validation groups are similar, with absolute errors all within 15.0%.

Table 8 Comparison between the measured and calculated values of u_p.

Table 9 Comparison between the measured and calculated values of u_L.

After establishing the calculation methods for u_p and u_L, the corresponding values of Re_t in the relevant regions can be determined at the initial design stage based on the specified treatment discharge. The value of Re_t can then be used to evaluate whether the design requirements for the VSBS are met. The specific procedure for determining the diameter of the VSBS is illustrated in Fig. 11.

First, the inlet discharge Q_i of the VSBS is determined based on hydrological data. The initial diameter D of the VSBS is preliminarily selected according to the study by²⁰ The width of the inlet canal, B_i, is then calculated using Eq. (16)⁵, and H_i is determined by Eq. (17)⁵. Next, the inlet velocity V_i is calculated using Eq. (18), followed by the calculation of u_p using Eq. (14) or u_L using Eq. (15). Finally, Re_t is computed. If Re_t in the selected reference region is less than or equal to 9.5 × 10⁶, the diameter D is considered appropriate. Otherwise, D and the relevant dimensions should be adjusted, and the process repeated until Re_t meets the requirement.

$${b_i}=0.2D$$

(16)

$${h_i}=0.07D$$

(17)

$${V_i}=\frac{{{Q_i}}}{{{b_i}{h_i}}}$$

(18)

Fig. 11

Process for determining VSBS diameter based on Re_t.

Validation of the design method

Selection of validation project

A diameter design method for the VSBS was proposed in the previous section. To verify the reliability of this method, this section conducts a redesign based on an existing project. The practical effectiveness and reliability of the proposed method are evaluated by comparing the suspended sediment trapping efficiency of the VSBS before and after the redesign.

The TLH project is located in the Tarim River Basin of Xinjiang. The Tarim River, the longest inland river in China, is formed by the confluence of the Aksu River from the Tianshan Mountains, the Yarkant River from the Karakoram Mountains, and the Hotan River, and ultimately flows into Taitema Lake. This basin is a major water resource region in arid and semi-arid areas, with an extensive coverage and a wide range of sediment particle sizes from 0.002 mm to 10 mm, exhibiting complex sediment characteristics. The basin is a typical region in Northwest China facing significant water and sediment challenges and is highly representative of such issues. Therefore, the TLH VSBS project constructed in this basin was selected as the validation case for the proposed method.

Validation experiment design

The current VSBS design methods used in diversion works are broadly similar, so representativeness of hydro-sedimentary conditions is crucial. TLH is located in the Tarim River Basin—the longest inland river system in China—spanning arid and semi-arid regions and exhibiting a wide sediment size range (0.002–10 mm) with complex characteristics; this basin is a typical hotspot of water–sediment issues in Northwest China. Therefore, TLH provides a representative prototype for validation, and the conclusions are expected to be generalizable to projects designed under the same criteria.

The design diversion discharge for the TLH project is 40.0 m³/s, with a median sediment particle size of 0.08 mm. The redesign of the TLH VSB followed the procedure illustrated in Fig. 11, with an initial diameter set at 50.00 m. Based on this diameter, the width and height of the inlet canal were calculated using Eqs. (16) and (17), resulting in values of 10.00 m and 3.50 m, respectively. With these dimensions, the inlet velocity V_i was obtained as 1.24 m/s using Eq. (18). According to the relationship between inlet velocity and tangential velocity in the measurement region (Eq. 14), up was calculated as 0.84 m/s. The resulting Re_t was 16.1 × 10⁶, which does not meet the design requirement, indicating the need for further adjustment of the diameter. The detailed design process is shown in Table 10. The final selected diameter was D = 100.00 m, with the corresponding dimensions listed in Table 11.

Table 10 Diameter redesign process for TLH project.

Table 11 Engineering parameters of T1 and T2 projects.

The original VSBS project before redesign is referred to as T1, while the redesigned VSBS project (with D = 100 m) is referred to as T2. Sediment-laden flow experiments were conducted for both T1 and T2. The VSBS models were designed according to the gravity similarity criterion, The calculation process of the sediment particle size scale is provided in the Supplementary Materials. In this study, natural sediment with a specific weight of 2650 kg/m³ was selected as the model sediment. According to Eq. (10), when the particle size scale factor λ_d is 2.09 and 2.51, the corresponding median particle sizes d₅₀ are 0.038 mm and 0.032 mm, and the corresponding settling velocity scale factors λ_w are 4.00 and 6.30, respectively, which are approximately equal to the flow velocity scale factor λ_v. The median particle sizes of the model sediment for T1 and T2 are very similar. Therefore, the model sediment used in this experiment (specific weight 2650 kg/m³) was prepared to match the particle size distribution curve of the experimental sand (d₅₀ = 0.035 mm) shown in Fig. 12 for the sediment-laden flow experiments. The remaining similarity ratios are shown in Table 12.

Table 12 Model scale ratio for validation experiments.

Fig. 12

Grain size distribution curve for TLH example verification.

Analysis of validation results

Sediment-laden flow experiments were conducted for both T1 and T2 projects, as shown in Fig. 13, and the sediment trapping efficiencies for different size classes are presented in Table 13. The results indicate that when the diameter of the VSBS was increased from 48.00 m to 100.00 m, the value of Re_t in the reference region decreased from 20.5 × 10⁶ to 8.0 × 10⁶, meeting the design criterion (Re_t≤ 9.5 × 10⁶). Under these conditions, the suspended sediment trapping efficiency of the VSBS improved significantly. The trapping efficiency for sediment with particle sizes of 0.05–0.10 mm increased from 27.2% to 48.2%, approaching the 46.78% observed in the DL project. Additionally, for sediment particles of 0.025–0.05 mm, the trapping efficiency increased from 3.1% to 12.5%. For sediment greater than 0.10 mm, the efficiency increased from 59.0% to 71.3%.

Table 13 Comparison of sediment trapping efficiency between T1 and T2.

Fig. 13

Schematic of the TLH model experiment process: (a) Sediment at the periphery during initial sediment injection phase; (b) Sediment radially moves toward the air vortex during mid-experiment; (c) Experiment end.

Discussion

Applicability and generalization

The VSB model established in this study was designed based on the principle of gravitational similarity. The VSB operates under free-surface flow conditions, with the water levels during operation shown in Fig. 14. In the figure, Z_i represents the water level in the upstream inlet canal, and Z_d denotes the water level inside the vortex chamber. The flow within the VSB is primarily driven by the gravitational potential energy of the upstream canal flow entering through the inlet conduit. Similarly, sediment settling within the VSB is governed by gravity. Once the particles settle onto the bottom plate, they are transported toward the flushing orifice under the influence of the gravitational component along the sloped bottom.

Therefore, both the flow motion and sediment transport processes inside the VSB are dominated by gravity, which plays a decisive role in energy transfer and sediment movement. The physical model developed under the criterion of gravitational similarity can effectively reproduce the flow characteristics and sedimentation behavior of the prototype, ensuring good comparability and general applicability of the results.

Fig. 14

Schematic diagram of the water level inside the VSB.

In addition, the diameter design method of the VSB proposed in this study, based on the tangential characteristic Reynolds number (Ret), provides a unified criterion for engineering design under various conditions. Ret characterizes the balance between inertial and viscous forces in the vortex flow field and comprehensively incorporates tangential velocity, characteristic length scale, and fluid viscosity. In practical engineering applications, an appropriate funnel diameter can be determined by maintaining Ret within a target range. Previous studies have shown that hydraulic models developed under Reynolds and Froude similarity principles exhibit strong transferability and stability in both sediment-laden and open-channel flow experiments, which further supports the applicability of the method proposed in this paper^26,27,28.

When parameters such as sediment grain-size distribution, sediment concentration, and design discharge fall within the range validated in this study, the Ret criterion and the corresponding diameter design results can be directly applied. However, for conditions beyond this range—such as finer or coarser sediment gradations, higher sediment concentrations, or extreme discharge conditions—it is recommended that additional verification experiments or prototype monitoring be conducted before applying this method to ensure design reliability.

Uncertainties and limitations

There remain certain uncertainties and potential errors in both the prototype observations and the model experiments of this study. The range of sediment concentrations and particle sizes that could be simulated in the experiments was limited. Although the experimental conditions covered typical suspended sediment characteristics, they still differ from the more complex and variable sediment composition found in natural rivers. In practical engineering applications, sediment grain-size distributions may be broader, and concentration fluctuations more intense—factors that are difficult to fully reproduce under laboratory conditions, which may restrict the representativeness of the results.

Flow velocity measurements were also inevitably influenced by instrument accuracy and operational conditions. ADV measurements are subject to spatial averaging effects and signal noise, and the probe may slightly disturb the local flow field in high-velocity regions. In addition, small positioning errors of the measurement points and the selection of sampling duration could introduce deviations. In the prototype observations, the sampling points were not perfectly aligned with the VSB inlet and outlet, which may have caused discrepancies between the measured and actual sediment concentrations, thereby affecting the accuracy of sediment trapping efficiency calculations.

In addition, minor fluctuations in discharge and sediment concentration may occur during experiments, and environmental factors such as water temperature can change fluid viscosity, thereby affecting the calculation of the tangential Reynolds number. Although repeated measurements, averaging, and outlier removal were used to reduce errors during data processing, the experimental data still exhibit inherent variability; consequently, the sediment trapping performance inferred from the tests may differ across projects.

Moreover, this study is primarily based on tests and observations under fixed inflow conditions, whereas actual VSBS operations often involve pronounced flow-rate fluctuations and abrupt changes in sediment concentration. Under such unsteady conditions, transient flow structures and sediment transport behavior may differ from those observed under steady conditions.

In summary, the conclusions of this work are mainly applicable to the range of conditions covered by our tests and observations (typical suspended-sediment size and concentration ranges, stable inflow conditions, and comparable temperature/viscosity backgrounds). When engineering conditions fall outside this range, targeted supplementary experiments or prototype monitoring—combined with numerical simulations—are still required to verify and refine the design, ensuring its applicability and reliability.

Conclusion

This study focused on three prototype VSBS projects (DL, TLH, and SMXD) with different sediment trapping efficiencies, combining field observations with laboratory experiments. Field observations were used to characterize the water and sediment conditions in each VSBS, and the sediment trapping efficiency for different sediment size classes was calculated. The results showed that the DL project exhibited superior sediment trapping performance across all size classes compared to the TLH and SMXD projects. Based on the field observation results, physical models corresponding to each project were constructed for further laboratory testing.

The experimental results showed that when the suspended sediment settling efficiency was high, the Tangential characteristic Reynolds number within the VSBS should be less than 9.5 × 10⁶ in the region of − 0.92 < r/R < − 0.80 at the 180° section or − 0.92 < r/R < − 0.60 at the 270° section under ambient temperature conditions. In these regions, the average tangential velocities, u_p (at 180°) and u_L (at 270°), were 0.676 and 0.633 of the inlet velocity of the inlet canal, respectively. Validation using prototype engineering cases indicated that replacing the traditional empirical method with this approach for designing the VSBS can increase the suspended sediment trapping efficiency by 9.4% to 21%.