Large-Eddy Simulation of Hydrodynamic Structure in a Strongly Curved Bank

Abstract

Advances in computer technology have facilitated the use of large-eddy simulation (LES) as a numerical simulation method in fluid mechanics. In this study, LES was used for the numerical simulation of curved water flow with a large curvature. The simulation results show that water separation occurs near the convex bank of the bend transverse section while the separation area increases and then decreases along the water depth direction, where the peak is at half-depth. As the water surface is affected by air shear stress, the velocity decreases. However, as the gradient increases, the vorticity is larger and the vorticity distribution is more even. Reflux occurs in the inlet and outlet sections of the bend but does not occur on the surfaces of z/H = 0.5 and z/H = 0.2, where the vorticity distribution is uneven. In the narrow corridor of the concave and convex banks, notably near the curved top, vorticity distribution is concentrated with a high intensity. At the center of the curve, the vorticity intensity is weak but 10 times higher at the concave and convex banks. A strong correlation exists between circulation structure and vorticity distribution: the vorticity is large where the circulation is intense and the structure is complex.

1. Research Background

Curved flow is the most common flow in plain rivers, and it has complex and dynamic flow patterns. Thus, the hydraulic characteristics of curved flow have been an important subject in river dynamics. Research on open channel bends mainly focuses on water surface line, three-dimensional velocity, pressure distribution, turbulent kinetic energy, and shear stress [1,2,3,4,5].

Currently, research on curved flow is mainly conducted through experiments [6,7] and simulations [8,9]. Ervine presented an exceptional model considering secondary currents to predict depth-averaged velocity and boundary shear stress at the apex of meandering simple and compound channels [10]. Following advancements in computer technology, three-dimensional (3D) numerical simulations of bend flow are being conducted to obtain precise hydraulic characteristics. For example, Xiang [11] used the empirical formula and k-ε numerical simulation to calculate the shear stress distribution of a riverbed and bank slope in 180°-curved river. Wei [12] used the large-eddy simulation (LES) model to simulate the hydraulic flow characteristics of a 90°-curved open channel and studied the secondary strength of the cross section of a curved channel. Zhang [13] conducted a vertical two-dimensional numerical experiment to study the non-uniform flow in an open channel; the vertical distribution results of the Reynolds stress were extended to the bed surface to calculate the velocity in the case of decelerating flow. The author obtained the frictional velocity calculation results of different backwater surfaces and, with the aid of experiment and simulation methods, the flow structures were investigated in a 193 degree channel bend, which included secondary flow, turbulence, boundary shear stress, velocity redistribution, topographic steering, and flow recirculation [14,15,16,17]. Farshi et al. [18] developed an analytical model to evaluate the boundary shear stress in compound open-channel bends, the effectiveness of which was demonstrated by experiments. The results showed that this model could be used to predict the boundary shear stress in simple and compound channels. Russell and Vennell [19] developed a narrow channel shallow water model to calculate the cross-channel distribution of the depth-averaged velocity. The numerical solution was used to find the distribution of secondary flow and vertical velocity on the cross-channel. Bai et al. [20] experimentally investigated the evolution of secondary flows in a U-shaped open channel, with the aid of PIV measurements, in which complex secondary flow patterns were found to be present in the bend and straight inflow/outflow reaches. Experimental and numerical simulation methods were used to investigate the influence of negatively buoyant jets on the flow structures in a curved 135-degree laboratory flume. The results indicated that in the inner bank region, flow separation was strengthened due to the negatively buoyant jet [21]. The simulation of influence of secondary flows on the flow velocity in open channel bends and confluences was carried out using standard k-ε and realizable k-ε models, which was demonstrated by the experiments. The results showed that the standard k-ε model was more suitable for the curved channel, while the realizable k-ε model was better for the confluent channel [22]. Deng et al. [23] investigated the secondary flow in two consecutive sharp bends, which described the ability of reduced-order model of Blanckaert and de Vriend to characterize the flow strength. The parameters of velocity components, streamlines, secondary flow, and wall shear stresses of flow in a 180° mild bend were analyzed by experiments and simulation. The results showed that secondary flow could widely affect the flow pattern in this bend [24]. Vaghefi et al. [25] experimentally studied the effect of secondary flow strength on bed shear stress related to mean and turbulent flow in a 180 degree sharp open channel bend. Detached eddy simulation (DES) method was used to analyze the mean flow three-dimensionality, secondary currents and boundary shear stresses in a strongly curved 193° bend with fixed deformed bed and sedimentation process, and the DES could more accurately predict the mean flow than the Reynolds-averaged Navier-Stokes (RANS) simulation method [26]. Termini carried out research on the evolutionary process of cross-sectional flow along a meandering bend. Their measurements show that a counter-rotating circulation cell takes shape in the upper part near the outer bank, in the case of relative small width-depth ratio [27].

Currently, most studies on flow characteristics in curves are based on the turbulent Reynolds-averaged Navier-Stokes (RANS) equation [28,29], but the description of complex circulation in curves, especially in turbulent structures, is inaccurate [5,30]. LES describes the evolution of unsteady large-scale eddy bodies through the spatial filtering function; thus, the simulation is suitable for the 3D simulation of complex curved flow.

In addition, the geometric shapes of the central angle of the bend, aspect ratio, and width to depth ratio have important influences on the turbulent structure of flow in the bend [21,31]. When R/B < 2, the empirical formula of water surface superelevation has a large error with the measured value [32]. Previous publications have been less involved in detailed research on the critical state where the diameter width ratio is R/B = 2. In our work, LES is further used to simulate the curvature of 180° with the diameter width ratio of 2, which corresponds to the strongly curved bank. Flow turbulence characteristics in bends are analyzed more comprehensively. It can be found that the position of the maximum circulation intensity has a strong correlation with the center angle of the curve. In the test with the center angle of the curve being 180°, the maximum value of the circulation intensity appears near 60°. The turbulent shear stress and the curve circulation influence each other. A strong correlation exists between the circulation structure and vorticity distribution, and the vorticity is larger in the area with a strong circulation intensity and complex structure.

2. Large-Eddy Simulation Turbulence Model and Numerical Method

2.1. Large-Eddy Simulation Turbulence Model

LES decomposes variables into sublattice and solvable scale quantities by filtering operations. The LES filtering is more similar to the spatial average than it is to the RANS average. The filtering is performed by the convolution operation, and the instantaneous variable f is filtered to obtain $\overline{f}$:

$$\overline{f}\left(x\right)={\displaystyle \int }f\left(y\right)\overline{G}\left(x-y,\mathsf{\Delta }\right)dy$$

(1)

$\overline{G}$ is the filtering function, and $\mathsf{\Delta }$ is the filtering scale.

Commonly used filter functions are the box, Gaussian, and spectral truncation filters. If the filtration and derivative processes are exchanged, the RANS equation is filtered to obtain the following [11]:

$$\frac{\partial \overline{u_i}}{\partial t}+\frac{\partial \overline{u_i{u}_j}}{\partial x_j}=-\frac{1}{\rho }\frac{\partial \overline{P}}{\partial x_i}+\mu \frac{{\partial }^2\overline{u_i}}{\partial x_j\partial x_j}$$

(2)

$$\frac{\partial \overline{u_i}}{\partial x_i}=0$$

(3)

Let $u_i{u}_j$ be written in the form of $\overline{u_i}\overline{u_j}+\left(\overline{u_i{u}_j}-\overline{u_i}\overline{u_j}\right)$ in which $-\left(\overline{u_i{u}_j}-\overline{u_i}\overline{u_j}\right)$ is the sub-grid scale model and can be expressed as

$$\frac{\partial \overline{u_i}}{\partial t}+\frac{\partial \overline{u_i{u}_j}}{\partial x_j}=-\frac{1}{\rho }\frac{\partial \overline{P}}{\partial x_i}+\mu \frac{{\partial }^2\overline{u_i}}{\partial x_j\partial x_j}-\frac{\partial \left({\overline{\tau }}_{ij}\right)}{\partial x_j}$$

(4)

$${\overline{\tau }}_{ij}=\left(\overline{u_i}\overline{u_j}-\overline{u_i{u}_j}\right)$$

(5)

x_i and x_j (i, j = 1, 2, 3) are cartesian coordinates; ${\overline{u}}_i$ and ${\overline{u}}_j$ (i, j = 1, 2, 3) are the upper velocities filtered by the filtering function; and $\overline{P}$ is the pressure filtered by the filtering function.

$${\overline{\tau }}_{ij}-\frac{1}{3}{\delta }_{ij}{\overline{\tau }}_{kk}=-2{\nu }_T\overline{S_{ij}}$$

(6)

$${\nu }_T=\rho L_s^2\left|\overline{S}\right|$$

(7)

$$\left|\overline{S}\right|=\sqrt{2\overline{S_{ij}{S}_{ij}}}$$

(8)

$$L_s=\min (\kappa d,c_s{V}^{\frac{1}{3}})$$

(9)

${\nu }_T$ is the sublattice vortex viscosity coefficient and sublattice scale mixing length; $\rho$ is the fluid density; $L_s$ is the sublattice scale mixing length; $\overline{S_{ij}}$ is the strain rate tensor; $\kappa$ is the Von Karman coefficient; d is the distance between the grid and the nearest wall; $c_s$ is the Smagorinsky coefficient; and V is the volume of the grid cells.

The used turbulence model is the SubGrid-Scale model, which can be used to accurately resolve the boundary layer detachment and the formation of an internal shear layer at the inner bank as well as the outer-bank cell of secondary flow [4]. In addition, the effectiveness of the used turbulence model is further illustrated by the comparison of water level, flow velocity, and Reynolds stress between the test and the simulation.

2.2. Numerical Method

A computational fluid dynamics (CFD) model is a series of partial differential equations, which have real solutions in theory, but it is difficult to get real solutions due to its complexity. Therefore, it is necessary to use numerical methods to treat the dependent variables on discrete points in the solution domain as basic unknowns so as to establish a series of algebraic equations (called discrete equations) about these unknowns and then obtain the approximate values of these nodes by solving the algebraic equations. The finite volume method is used to discretize the equations. The obtained discrete equations are solved by the semi implicit method (SIMPLE algorithm) of pressure coupled equations, which is now widely used. SIMPLE algorithm is a common algorithm for solving pressure in the process of “guess correction” proposed by Patankar and Spalding in 1972.

3. Meshing Grid

This study is carried out in a U-shaped bend model. The plane diagram and section setting of the bend are shown in Figure 1. The bottom slope of the flume is flat. The total flow of the bend is 17.02 m; the width of the flume is B = 0.4 m; the inner diameter is 0.6 m; the outer diameter is 1.0 m; the center curvature radius is 0.8 m; the entrance straight channel is L₁ = 6.0 m; and the exit straight channel is L₂ = 6.0 m. There are 17 sections in the flume among which there are 13 sections in the bend according to the angle. There are two sections in the entrance straight channel and the exit straight channel, which is 0.5 m and 0.8 m away from D0 section and D180 section, respectively. The calculation data are obtained by FLUENT software. In the simulation, the flow is incompressible, the flow density is a constant with a value of 1.0 × 10³ kg/m³, and the temperature of flow is set to 20 °C.

The width of the entire flume is uniformly B = 0.4 m, and the centerline length is 6 m. The radius of curvature at the centerline of the bend is R = 0.8 m. Water depth of the experiment H is 0.2 m. Therefore, in the bend, the aspect ratio R/B is 2, the diametral depth ration R/H is 3, and the aspect ratio is B/H is 2, which makes it a strongly curved open channel. The entrance flow and velocity of the flow were 37.5 L/s and 0.47 m/s. In the simulation, the outlet flow tends to be stable after the 80th second, which is the same as the given inlet flow with the value of 37.5 L/s. In our work, parameters of flow velocity, water level, secondary flow, shearing stress, and vortices corresponding to the time of 100th second are adopted to investigate the turbulence characteristics of water flow, all of which are under convergence conditions. A three-dimensional acoustic Doppler velocity (ADV) meter was used to measure the three components of velocity, i.e., longitudinal, lateral and vertical components of velocity. The flows in all the experiments were sub-critical flows, the Reynolds number of the test is 46,551 (

Table 1. Hydraulic conditions for the main flow.

Q (L/s)	U (m/s)	B (m)	H (m)	R (m)	Re	R/B	B/H	R/H
37.5	0.47	0.4	0.2	0.8	46551	2	2	3

).

The wall surface was encrypted as it was affected by viscosity. We used a structured grid, which had two encrypted radial boundaries. As the bend flow structure was complex, the grid of the bend section was encrypted to obtain the computational grid. Here, grid size at the bend takes the values of 1, 0.8, and 0.6, respectively, which correspond to the obtained number of grids with 9.65 miilion, 6.78 million, and 4.59 million, respectively. Additionally, the meshing of the test bend is shown in

Table 2. Mesh parameters.

Grid	Entrance and Exit Straight Grid Size (cm)	Grid size at the Bend (cm)	Total Grid Number (million)
1	1	0.5	9.65
2	1	0.7	6.78
3	1	0.9	4.59

in the revised manuscript, and the meshing of the test bend is shown in Figure 2.

4. Model Validation

Figure 3 shows the comparison between the calculated and measured values of the water surface line of the curved section, and the water surface lines of D0, D45, D90 and D135 are obtained. It can be seen from the figures that the calculation results from grid 1 and grid 2 are in good agreement with the measured values, while the calculation results from grid 3 deviate from the measured values. For example, the D45 section, D90 section and D135 section of the water surface line deviation is larger, especially near the convex bend. It shows that the LES Model used in this article could accurately simulate the distribution characteristics of the profile of water surface of the bend flow.

In order to illustrate the accuracy of the numerical calculation more clearly, the simulated values of the longitudinal time-averaged velocity distribution along the vertical line on the concave bank and the center line and the convex bank of four typical sections are analyzed. Figure 4 show the comparison between the calculated results and the measured values of the longitudinal velocity the latter of which is performed utilizing acoustic Doppler velocimetry (ADV). It can be observed that the calculated values from grid 1 and grid 2 are in good agreement with the measured values. The deviation between the calculated value and the measured value from grid 3 is large, especially in the cross section and the surface layer of the bend, such as the concave bank and the convex bank of the D90 section and the convex bank of the D135 section. The reason for the large deviation is that the simulation of the water flow structure near the wall, especially near the boundary, is not accurate enough under the condition of a coarse grid affected by the wall. Therefore, the simulation of the flow velocity near the concave and convex sides has a large deviation from the measured value.

Since the accuracy of numerical simulation depends on the discrete method, solution method and truncation error, there is a certain error between the calculated value and the experimental value. It can be seen from the comparison between the calculated and measured values of water level and longitudinal velocity in grid 1 and grid 2 that the calculated values of the LES model are basically consistent with the measured values, which verifies the credibility of the model.

In order to observe the cross-section mainstream distribution from the three grids more clearly, the cross-section flow velocity along the way is extracted, as shown in Figure 5. After entering the bend, the mainstream of the water flow under the three grid conditions shifts to the convex bank and shifts to the middle position of the curved top section. After the curved top section, the mainstream shifts to the concave bank, and the mainstream remains in the concave bank to flow out of the bend. From the diagram, it can be observed that in the straight section of the entrance, the flow velocity of the cross section in Figure 5a–c is basically the same, and the difference begins to appear in the D45 section. In Figure 5c, the mainstream concentration is not high, but the cross-section velocity is relatively uniform. This is due to the large grid size and to the insufficient information of the bend flow structure.

To sum up, under three grid conditions, the results of the curved flume test on the water surface line and the longitudinal velocity were verified. The results show that the influence of the near wall on the water flow structure reduce the ability of grid 3 with the lowest grid resolution to capture the details of the flow field, which affects the accuracy of the calculation, resulting in a large deviation between the calculated results and the measured values. Under the conditions of grid 1 and grid 2, it can be used for the numerical simulation of the water flow in the curve. In order to save the calculation time and computer memory, this paper uses grid 2 to carry out the numerical simulation of the curve channel.

It is preliminarily verified that the resolution of grid 2 can be used for simulation calculation and analysis. In order to further ensure the accuracy of LES, the Reynolds stress is verified. Figure 6 and Figure 7 in the revised manuscript show the comparison between the test value and the calculated value of Reynolds stress. It can be seen that they are in good agreement, but there is a slight difference on the surface. This is due to the large measurement error of ADV on the surface. In conclusion, the LES model can be used to simulate the three-dimensional flow numerical simulation of the test bend, and the simulation results are used to further analyze the flow structure.

In the numerical simulation calculation of hydraulic engineering, the interface between the atmosphere and the water body is the focus of the fluid, and it is also one of the most difficult problems. The treatment of the free liquid surface is directly related to the accuracy of the simulation calculation. At present, the most commonly used free surface tracking method is the VOF method, which was proposed by Hirt [33] in 1982. It can effectively reduce the calculation time and improve the calculation efficiency during the calculation process.

5. Analysis of Simulation Results

5.1. The Velocity Section

Figure 8 illustrates the flow velocity distribution along the cross section. Two sections are arranged in the straight entrance and exit of the curve, 0.5 m and 0.8 m away from section D0. The sections are marked as Q0.5, Q0.8, H0.5, and H0.8, respectively. Figure 8 shows that the flow velocity is evenly distributed in the straight entrance of the bend. After the water enters the bend, the water flow characteristics change in centrifugal stress and gravity. The water mainstream begins to shift to the convex bank at the entrance of section D0.

At section D60, the upper mainstream of the water remains in the convex bank, but the lower part begins to shift away from the convex bank to the concave bank. The upper mainstream of the water shifts to the center of the bend and to the concave bank until it completely shifts to and flows out along the concave bank. After entering the bend, the separation phenomenon occurs in the concave and convex banks. The separation area expands and contracts along the straight-lined sections. The separation area is the largest at approximately half of the water depth. As the process continues, the separation area becomes approximately 1/4 of section D30, 1/3 of section D120, and 1/2 of section H0.8.

5.2. Plane Flow

Figure 9 shows the plane velocity and streamline distribution at different water depths. The plane velocity mainstream line at different depths of water starts to shift toward the convex bank after entering the bend. After passing section D90, it moves away from the convex bank to the concave bank until it reaches the center of section D165. Afterward, it continues shifting to the concave bank and flows out of the bend along the concave bank. The schematics of three different water depths are shown in Figure 9, which indicates that the flow velocity increases and then decreases from the bottom to the surface level. The result is consistent with that of the measured test. Both results are obtained because the shear stress between water flow and air at the surface is greater than that inside water, which causes the flow velocity to decrease. Water flow is separated near the rear of the convex bank at section D90, where the separation area first enlarges and then reduces along the water depth direction. At the convex bank area, no backflow occurs at the bottom of the water. However, Figure 4c shows that backflow occurs at the water surface near the low-speed zones at the entrance and exit of the bend.

5.3. Transverse Circulation

The distribution of circulation strength along the flow is very complex, and there are many ways to express it. In this paper, the circulation strength proposed by Shukry [34] is used to analyze the hydraulic characteristics of bend flow. The expression is as follows:

$$S_{xy}=\left(\frac{V_{xy}^2}{2g}\right)/\left(\frac{V^2}{2g}\right)$$

(10)

V_xy = (v² + w²)^0.5, V = (u² + v² + w²)^0.5, u, v, w are the component velocities in three directions: streamwise, transverse and vertical, and g is the gravitational acceleration.

The distribution of circulation strength is shown in Figure 10. After the flow enters the bend, the circulation intensity increases rapidly with the increase of centrifugal stress and flow velocity and reaches the maximum near section D60. After crossing the curved top section, the circulation strength gradually weakens. The circulation intensity has a strong correlation with the change of the main streamline of the bend flow. The main streamline of the water flow near section D60 began to shift to the concave bank, and the main streamline shifted slowly after passing the D90 section so that the secondary flow intensity began to weaken. The lateral circulation begins to form a clockwise secondary circulation near the surface of the concave bank on the D60 section, which makes the secondary flow stronger.

From the maximum position of circulation strength corresponding to different parameters, as shown in Table 3, it can be found that the position of the maximum circulation intensity has nothing to do with the parameters, such as the diameter-to-width ratio, the width-to-depth ratio, and the Froude number but has a strong correlation with the center angle of the curve. In the test with the center angle of the curve being 180°, the maximum value of the circulation intensity appears near 60°.

Table 3. Locations of the maximum circulation strength corresponding to different parameters.

Author	Diameter-Width Ration	Breadth-Depth Ratio	Froude Number	Central Angle	Position of Maximum Circulation Intensity
Gao, S.X. [35]	3	2.67/2.60/0.58	0.15/0.19/0.195	180°	60°
Vaghefi [25]	2	5	0.34	180°	60°
This paper	2	2	0.28	180°	60°

Table 3. Locations of the maximum circulation strength corresponding to different parameters.

Author	Diameter-Width Ration	Breadth-Depth Ratio	Froude Number	Central Angle	Position of Maximum Circulation Intensity
Gao, S.X. [35]	3	2.67/2.60/0.58	0.15/0.19/0.195	180°	60°
Vaghefi [25]	2	5	0.34	180°	60°
This paper	2	2	0.28	180°	60°

The transverse circulation distribution of the section is shown in Figure 11. The circulation structure of each section shows that the upper part flows to the concave bank and the lower part flows to the convex bank, forming a main circulation the center of which is located below the convex bank. The position is continuously shifted downward toward the ledge. After passing through section D30, the circulation intensity of the convex bank is greater than that of the concave bank. Starting from the D60 section, a secondary circulation with the direction opposite to the main circulation forms on the surface layer of the concave bank. The secondary circulation continues to develop along the way with increasing intensity and increasing scope.

When water enters the bend, the open channel water flow is subjected to the joint action of water pressure and centrifugal stress. At the cross section of the bend, the surface water flow moves to the concave bank under the action of centrifugal stress, and the bottom water flow moves from the concave bank to the convex bank under the action of pressure gradient, forming a counterclockwise bend circulation.

Figure 12 illustrates the transverse circulations of the bend sections. The vortex structure obtained with LES is clearer. No circulation structure is formed in the straight sections of the curve. At section D30, a double-vortex structure is formed. A counterclockwise circulation occurs at the bottom of the water, and a clockwise circulation occurs in the concave bank at the water surface. The counterclockwise circulation near the convex bank at section D45 develops into a double-vortex counterclockwise circulation, resulting in the formation of many small circulations near the convex and concave banks.

When bending degree increases, circulations develop and turbulence intensifies, resulting in greater energy loss. At section D180, the water body at the surface flows smoothly, while the lower part of the water body circulates, especially near the concave bank. At the straight sections of the bend, the flow is still affected by the upstream bend; hence, the circulation continues at a reduced strength. Figure 12i,j illustrate the straight sections of the bend. The circulation remains, but a comparison of the circulation characteristics of each section in the bend shows that the circulation strength weakens. LES simulated the secondary flow structure of the cross section more accurately than the k-ε model. This reflects the benefits of applying the LES to studying curved open channel flows, which is consistent with the investigation of Van Balen [4].

Shear stress is an important parameter to predict the degree of scour of the moving bed, and its distribution directly determines the scour form of the bed surface. Figure 13 shows the distribution of the shear stress on the bed surface at the bend. After the water flow enters the bend, the shear stress it receives begins to increase. The shear stress that the concave bank side receives is larger than the convex counterpart. The maximum shear stress appears before the top section of the curve, and it moves from the concave bank to the convex bank along the longitudinal direction of the curve, and the shear stress in the second half of the curve is smaller than that in the first half. After passing through the curved top section, the shear stress on the bed surface presents an opposite distribution characteristic to that before the curved top section, and its value at the convex bank is larger than that at the concave bank. According to the change of the curve circulation in the whole curve, it can be known that the turbulent shear stress and the curve circulation influence each other.

5.4. Vorticity

The vorticity distribution in a flow field reflects the distribution of vortex cluster occurrence and flow separation intensity in that field. Figure 14 illustrates vorticity along each section. The figure shows that the vorticity of the straight sections of the bend is low but increases in the bend sections. The vorticity at the convex bank is higher at both sides of the bend and at the water surface, especially near section D90. After entering the bend, the centrifugal stress readjusts the water flow structure, making the surface water flow move toward the convex bank, thereby increasing its velocity gradient and vorticity. After crossing the D90 bend section, the water flow moves gradually to the concave bank. Momentarily, water separation occurs near the convex bank, making the vorticity increase rapidly and the vorticity distribution area expand intensely.

As with Figure 9, the circulation strongly correlates with the vorticity distribution: the circulation intensity is complex, and the vorticity distribution is large. Figure 8 illustrates the location of the complex circulation and the corresponding large vorticity of each section. After the bend, the concave bank vorticity enlarges. After exiting the bend, the convex bank vorticity enlarges, as illustrated in Figure 8. The velocity gradient near the convex bank of the straight exit sections of the bend changes significantly. Figure 12 also shows that the circulation is near the convex bank of the straight exit sections of the bend, which causes large vorticity.

Figure 15 illustrates vorticity distributions at different water depths. The surface vorticity is evenly distributed and large because the shear stress between the surface water flow and air increases the flow velocity gradient. Figure 15a,b show that the vorticity distribution is extremely uneven and concentrated in the narrow area of the convex and concave banks of the bend with a high vorticity intensity. Near the top of the bend, the vorticity intensity is greatest but significantly reduced at the middle of the bend. The vorticity intensity distribution of the concentrated zone is 10 times higher than that in the weak distribution zone. The phenomenon can be explained as follows. First, in the longitudinal flow of the flume, the water flow is separated at the corner of the rectangular sections, and the separation point is the origin of the vortex cluster having strong vorticity. Second, near the top of the bend, the cross-sectional circulation causes a strong water separation at the corner of the top sections of the bend, creating a strong vortex, which is reflected in the concentrated strong-vorticity area. Third, after passing the top of the bend, the cross-sectional circulation causes the water flow mainstream to gradually shift from the convex bank to the concave bank, increasing the size of the water separation area at the convex bank and enlarging the strong-vorticity distribution area. Figure 15b shows that the strong-vorticity distribution area of the concave and convex banks at the water surface is large. This is because as the water flow mainstream gradually moves from the convex bank to the concave bank, the separation area first enlarges and then shrinks along the water depth near the convex bank. The separation area near z/H = 0.5 is larger, as shown in Figure 8 and Figure 9.

6. Conclusions

In this study, large-eddy simulation method is employed to conduct 3D numerical simulation of water flow characteristics in a U-shaped bend, and the structure of water flow is analyzed based on the simulation results. Compared with the previous work that concentrated on the vorticity in the curved channel [4,36,37], our work presents an investigation of the vorticity that exits the curved. The effectiveness of large-eddy simulation is demonstrated by the parameter measurement of 3D flow velocity utilizing ADV.

• The simulation results show that after the flow enters the bend, the velocity of the section is separated near the convex bank, and the separation area increases first and then decreases along the water depth direction, where the maximum is nearly half of the water depth. Backflow occurs at the water surface near the low-speed zones at the entrance and exit of the bend.
• Large-eddy simulation adequately simulates the complex structure of curved circulation: the main circulation is nested with multiple secondary circulations, which are constantly changing and interacting. After the flow enters the bend, the circulation intensity increases rapidly with the increase of centrifugal stress and flow velocity and reaches the maximum near section D60. After crossing the curved top section, the circulation strength gradually weakens. It can be found that the position of the maximum circulation intensity has nothing to do with the parameters, such as the diameter-to-width ratio, the width-to-depth ratio, and the Froude number but has a strong correlation with the center angle of the curve. In the test with the center angle of the curve being 180°, the maximum value of the circulation intensity appears near 60°.
• After the water flow enters the bend, the shear stress it receives begins to increase. The maximum shear stress appears before the top section of the curve. After passing through the curved top section, the shear stress on the bed surface presents an opposite distribution characteristic to that before the curved top section. According to the change of the curve circulation in the whole curve, it can be known that the turbulent shear stress and the curve circulation influence each other. Owing to the action of air shear stress, the flow velocity of the surface decreases; however, the gradient increases, the vorticity is large, and the vorticity distribution is uniform. The vorticity distribution is extremely uneven. The vorticity intensity in the concentrated distribution zone is 10 times greater than in the weak distribution zone. A strong correlation exists between the circulation structure and vorticity distribution, and the vorticity is larger in the area with a strong circulation intensity and complex structure.