To investigate the influences of river bend geometry on flow structures, three-dimensional simulations were carried out. The influence of aspect ratio, channel sinuousness and curvature ratio were investigated. Simulation results show that with an increasing of channel aspect ratios, the streamwise variation of flow velocities decreases significantly, which is not the case for the spanwise direction. Simulation results also reveal the significant influence of the curvature ratio on the redistribution of primary flows. The streamwise variation of free-surface and bottom-wall velocities was compared and the phase differences between them was identified, which was described as anti-phase and out-of-phase patterns. Deep-water channels with an aspect ratio of 0.15 show significant phase-lag of flow velocities compared with channel geometry, this newly observed phenomenon implies the shift of the peak scouring point in fluvial river meanders. For the first time to our knowledge, the streamwise and spanwise variation of primary and secondary flow strength, as well as their spatial derivatives, are quantitatively provided, which are valuable for theoretical studies. It is also found that the widely adopted assumption that the streamwise velocity component is one order magnitude larger than the spanwise, is only valid for river bends with small curvatures and aspect ratios.

## INTRODUCTION

As one of the most common river patterns in nature, meandering rivers have flow structures much more complicated than those observed in straight rivers. They are characterized by a traverse-inclined free-surface, secondary currents and spiral flows (Johannesson & Parker 1989a, 1989b). These complicated flow structures contribute to sediment transport, bed deformation, bank erosion and river meander development (Kikkawa *et al.* 1976; Olsen 2003; Hooke & Yorke 2010, 2011; Xu *et al.* 2011; Kasvi *et al.* 2013).

A deep understanding on the flow structures in meandering rivers is of great importance in theoretical studies by using the perturbation method (Ikeda *et al.* 1981; Liu *et al.* 2009). For the perturbation method, it is crucial to correctly estimate the magnitude of each variable in the governing equations of flow motion, and choose small parameters rationally. Different estimations may lead to completely different theoretical results (Smith & McLean 1984). To the authors’ knowledge, however, a comprehensive understanding on the flow structures in river bends with diverse geometric shapes is still lacking.

*et al.*2008), as shown in Figure 1. Well verified numerical simulations can be a convenient extension to the experiments for investigation of flow in meandering rivers (Nguyen

*et al.*2007; De Marchis & Napoli 2008).

Two-dimensional (2D) numerical modeling of river bend flows also have some limitations. To investigate the flow and the resultant landform development in river bends, many 2D numerical models have been developed based on the shallow-water equations (Liu *et al.* 2009; Hardy *et al.* 2011; Parker *et al.* 2011). However, it should be noted that most of these models are based on certain assumptions, such as hydrostatic pressure, small curvature, large aspect ratio (wide-to-depth ratio) etc., which limit their applications.

Three-dimensional (3D) numerical simulations of flows in open channels using computational fluid dynamics (CFD) is nowadays becoming more and more powerful and efficacious owing to the significant development of high-performance computers and numerical methods. Using modern CFD techniques like direct numerical simulation and large eddy simulation (LES), even the most complex turbulent structures in straight open channels can be resolved (Ji *et al.* 2012, 2013; San 2014; Li *et al.* 2015). Recently, LES of turbulent flows in curved open channels were carried out and the results agreed well with experimental measurement (Booij 2003; Van Balen *et al.* 2009; Xu *et al.* 2013). However, as indicated by Xu *et al.* (2013), LES of turbulent flows requires extremely high-resolution grids especially near the channel walls and even the most powerful parallel computing facility cannot afford the simulation of turbulent channel flows with high Reynolds numbers. This greatly limits its application in systematic investigations of flow structures in river bends with a variety of geometric shapes. With regards to this, traditional CFD models based on the Reynolds averaged Navier–Stokes (RANS) equations and certain turbulent models are effective options for simulation of turbulent flows in channel bends due to their high computing efficiency and reasonably good results, especially where the time-averaged flow velocities are concerned (Nguyen *et al.* 2007; De Marchis & Napoli 2008; Kashyap *et al.* 2012; Ramamurthy *et al.* 2013). Kashyap *et al*. (2012) investigated the influences of the curvature ratio (bend radius/channel width, R/B) and aspect ratio (channel width/flow depth, B/H) using the RANS method. Ramamurthy *et al.* (2013) also achieved good simulation results on flow in a 90 degree bend using both k-*ε* renormalization group (RNG) and Reynolds stress model in a RANS framework. However, the available studies only focus on a specified channel shape, for example circular shape in Kashyap *et al*. (2012) and 90 degree bend in Ramamurthy *et al.* (2013). Considering the diversity of river bend shapes in nature, more effort is needed for a comprehensive understanding on the complicated flow structures in river bends.

In this paper, 3D RANS-based numerical simulations of turbulent flow in a curved open channel are carried out. The RNG k-*ɛ* turbulent model is adopted for the closure of the governing equations and the finite volume method (FVM) is applied for spatial discretization. The rest of the paper is organized as follows. In the next, the adopted methodology is introduced. After that, the numerical model is verified by comparing the numerical results with three groups of experimental data. In the next section, the influences of geometric shapes of channel bends on flow structures are discussed for channels with different curvatures, channel widths and water depths. Streamwise and spanwise velocities, together with their spatial gradients, are presented. The conclusions are presented in the final section.

## 3D CFD SOLVER AND VERIFICATION CASES

### Governing equations

*et al.*2014), an in-house CFD solver for incompressible viscous flow was adopted. The time averaged form of the Navier–Stokes equations, i.e. the RANS equations (Bai

*et al.*2014), are as follows:where is the time averaged velocity, is the time averaged pressure, is the density of the fluid, is the kinematic viscosity, is the fluctuating velocity. To close the RANS equations, the RNG k-

*ɛ*turbulent model was adopted. The transportation equations for

*k*and are as follows (Bai

*et al.*2014):where

*k*and are the turbulent kinetic energy and its dissipation rate, respectively. is the Reynolds stress tensor and can be represented by , where represents the kinematic eddy viscosity. The term represents the generation of turbulence kinetic energy due to the mean velocity gradients.

*k*and are (Yakhot

*et al.*1992). Such values are widely adopted for simulation of flow in river bends (Wilson

*et al.*2003; Stoesser

*et al*. 2010; Bai

*et al.*2014).

It should be noticed that the standard k-*ɛ* model is based on the hypothesis of isotropic eddy viscosity and turbulent diffusion. Therefore, it cannot be applied in the simulation of anisotropic turbulent flows in its original form. However, the k-*ɛ* model is still one of the most popular turbulence models, owing to its simplicity and good performance. Fortunately, its modified version, RNG k-*ɛ* model is capable of handling anisotropic turbulent flow structures, which have been widely verified in the field of river hydrodynamics (Olsen 2003; Abad 2005; Stoesser *et al.* 2010; Bai *et al.* 2014). Therefore, the RNG k-*ɛ* model was adopted in the present study.

### Discretization, grids and boundary conditions

*et al.*2012; Bai

*et al.*2014).

The no-slip boundary condition was applied on the channel bottom and side-walls, while the free-slip boundary condition was adopted on the free-surface. Note that the rigid wall approximation of the free-surface adopted in this study suppresses the development of surface waves. However, the introduced errors on the water level variations is negligible (less than 5% of the water depth (Stoesser *et al.* 2010)) due to the small Froude number (smaller than 0.71 for all cases) in this study. A uniform inflow was imposed on the inlet boundary, and the non-reflecting outflow condition was imposed on the outlet boundary. A standard wall function was applied on the no-slip walls. The Reynolds number based on mean velocity and water depth ranges from 888.9 to 8,571.4. A uniform grid was adopted in all three directions with a total element number around 10^{6}. There were 64 grids in the vertical direction, which yields ranging from 26 to 97 for different cases. A constant time step of 0.01 s was adopted, which yields a CFL (Courant–Friedrichs–Lewy) number ranging from 0.01 to 0.18. Each case ran around 10,000 time steps to reach a quasi-steady state. The running time for each case is around 48 hours on a 4-core 2.4 GHz CPU.

### Verification cases

To validate the numerical model, three groups of laboratory experiments with different curvatures were adopted, namely Group I (sharply curved channel), Group II (mildly curved channel) and Group III (sine-generated channel).

#### Group I: sharply curved channel

*et al.*2014). Figure 4 compares the streamwise and spanwise velocity profiles obtained in the numerical simulations and the laboratory experiments. Good agreements have been achieved, which demonstrates the high accuracy of the numerical model in simulating complex 3D flow structures in channel bends and the secondary currents in the cross-section plane.

#### Group II: mildly curved channel

*et al.*2013).

#### Group III: sine-generated channel

## RESULTS AND DISCUSSION

In order to investigate flow structures in channel bends with a wide range of geometries, in total 10 simulation cases were performed, including three cases (RUN F30, F60, F110) with different deflection angles (channel sinuousness) ranging from 3 to 110°, four cases (RUN H15, H30, H60, H100) with different aspect ratios (B/H) ranging from 0.15 to 1.0 and three cases (RUN B2, B5, B15) with different curvature ratios ranging from 1.0 to 7.5. The Reynolds numbers ranged from 888.9 to 8,571.4 and Froude numbers ranged from 0.03 to 0.20.

### Influences of channel bend sinuousness on flow structures

*et al.*2008). The sinuousness of a river can be described in many ways, such as the curve length to distance ratio, the fractal dimension, etc. (Bai

*et al.*2008). In this paper, the maximum deflection angle (angle between the tangential direction of the centerline and the x direction) was used and the planar shape of a meandering river was generalized as the sine-generated curves (Langbein & Leopold 1966), i.e.:where is the channel deflection angle,

*s*is the curved streamwise coordinate,

*M*is the curve length of a single bend, is the maximum deflection angle. Other geometric parameters of the channel bend include the channel width

*B*, the wave amplitude

*A*and the wavelength , as shown in Figure 9.

In order to study the influences of channel sinuousness on flow structures, three maximal deflection angles of 30, 60 and 110° were adopted in the numerical simulations. Table 1 lists other geometry and simulation parameters.

RUN . | Geometric parameters . | Reynolds number, Re . | Water depth, H/m . | Mean velocity, U_{0}/m/s
. | ||
---|---|---|---|---|---|---|

Maximal deflection angle,° . | Wave length along centerline, M/m . | River width, B/m . | ||||

F30 | 30 | 2 | 0.15 | 8,571.4 | 0.10 | 0.2 |

F60 | 60 | 2 | 0.15 | 8,571.4 | 0.10 | 0.2 |

F110 | 110 | 2 | 0.15 | 8,571.4 | 0.10 | 0.2 |

RUN . | Geometric parameters . | Reynolds number, Re . | Water depth, H/m . | Mean velocity, U_{0}/m/s
. | ||
---|---|---|---|---|---|---|

Maximal deflection angle,° . | Wave length along centerline, M/m . | River width, B/m . | ||||

F30 | 30 | 2 | 0.15 | 8,571.4 | 0.10 | 0.2 |

F60 | 60 | 2 | 0.15 | 8,571.4 | 0.10 | 0.2 |

F110 | 110 | 2 | 0.15 | 8,571.4 | 0.10 | 0.2 |

*Note:* Reynolds number Re is based a characteristic length scale of hydraulic radius.

*ɛ*turbulent model (Ramamurthy

*et al.*2013) and also the weakness of the vortex cell compared to the major cross-sectional circulation in channel bends (Van Balen

*et al.*2009; Blanckaert

*et al.*2013).

*B*to the left side-wall. Results on two vertical positions, i.e.

*y*

*=*0.1

*H*(near the bottom-wall) and

*y*= 0.9

*H*(near the free-surface), were plotted. The subscript

*b*indicates the velocity components near the bottom-wall (marked with solid symbols), while the subscript

*s*denotes those near the free-surface (marked with open symbols). Note that the velocity profiles near the right side-wall are the same as those near the left side-wall except for a phase lag of 180° and thus are not presented for conciseness.

It can be seen that for all cases, all three velocity components show quasi-periodic variations along the curved channel, while the streamwise velocity is consistently larger than the others. The flow transition length, i.e. the distance from the inlet of the channel to the position where the quasi-periodic flow developed, decreases with an increasing sinuousness. For example, obvious quasi-periodic variations can be found in the cross-flow velocities beyond s >0.8 m for RUN F30, while large variations of the cross-flow velocities develop from the channel inlet for RUN F110. For the case with small curvature (RUN F30), the streamwise velocity near the free-surface is significantly larger than that near the bottom-wall due to the shear drag effects. However, this is not necessarily the truth for the cases with a large sinuousness. The strong secondary flow developed in the channel bends with a large curvature deflects the ‘flow core’ (the area with large streamwise velocity) from the center of the channel to the side-walls, as shown in Figure 8. The vertical velocity is close to zero, which verifies the assumption of adopted in many theoretical studies (Ikeda *et al.* 1981). The streamwise velocity near the side-walls has a larger variation amplitude and a two-time longer wave length than that on the center lines. The spanwise velocities near the bottom-wall and free-surface show similar variation amplitude and wave length but in an anti-phase synchronized pattern due to the presence of the secondary currents.

To compare the magnitude of characteristic variables of flow in channel bends, the maximum values of the following variables are computed in Table 2. In the following variables, subscribe ‘*s*’ refers to the variation along a streamwise curve and ‘*n*’ along a spanwise direction:

Range of streamwise variation of streamwise velocity and spanwise velocity;

Range of spanwise variation of streamwise velocity and spanwise velocity;

Mean gradient of streamwise variation of streamwise velocity and spanwise velocity;

Mean gradient of spanwise variation of streamwise velocity and spanwise velocity.

RUN . | /U . | /U . | /U . | /U . | . | . | . | . |
---|---|---|---|---|---|---|---|---|

F30 | 1.00 | 0.35 | 0.75 | 0.25 | 0.150 | 0.052 | 0.75 | 0.25 |

F60 | 0.88 | 0.40 | 0.60 | 0.25 | 0.131 | 0.060 | 0.60 | 0.25 |

F110 | 1.10 | 0.50 | 1.00 | 0.40 | 0.165 | 0.075 | 1.00 | 0.40 |

RUN . | /U . | /U . | /U . | /U . | . | . | . | . |
---|---|---|---|---|---|---|---|---|

F30 | 1.00 | 0.35 | 0.75 | 0.25 | 0.150 | 0.052 | 0.75 | 0.25 |

F60 | 0.88 | 0.40 | 0.60 | 0.25 | 0.131 | 0.060 | 0.60 | 0.25 |

F110 | 1.10 | 0.50 | 1.00 | 0.40 | 0.165 | 0.075 | 1.00 | 0.40 |

It can be seen from Table 2 that when the maximal reflection angle ranges from 30 to 110°, the sinuousness has trivial effects on the magnitude of flow velocities in channel bends. The streamwise variation ranges of velocities are equivalent to their spanwise counterpart. However, the spanwise variation gradients of velocities are over five times larger than their streamwise counterpart, because the streamwise length of the channel is much larger than the channel width. Considering the feature of vertical profile of lateral velocity distributed over the depth, see Figure 6(b), can be roughly estimated as twice of the maximum lateral velocity , namely . Here, represents the characteristic velocity of secondary current and can be approximated as . ‘U’ refers to the mean flow velocity in the streamwise direction, which can be viewed as the primary flow velocity. Therefore, /2U can be taken as a ratio of the strength of secondary current to the strength of primary flow. RUN F110 has a ratio of /U = 0.4, see Table 2, which means that the ratio of secondary flow to primary flow strength is about 20%. The velocity magnitude of secondary current in highly sinuous channel bends can be as large as 20% of the primary flow, which may invalidate many theoretical assumptions (Bai & Yang 2011; Bai *et al.* 2012).

### Influences of aspect ratio on flow structures

As one of the key features which significantly affect flow structures, the aspect ratio of the cross-section (channel width/flow depth, B/H) diverges drastically from one river to another. For instance, in China, the averaged aspect ratio of the Yellow River is over 40 times larger than that of the Changjiang River (Bai *et al.* 2008). In fact, even for the same river, the aspect ratio of cross-section varies within a quite wide range in different sections and in different seasons. To study the influences of aspect ratio on flow structures, numerical simulations were carried out for sine-generated river bends with a fixed channel width of *B* = 15 cm and a varying water depth *H* from 15 to 60 cm. The corresponding aspect ratio of the channel ranges from 0.15 to 1. The maximum deflection angle of 110° was adopted on the purpose of strengthening the secondary flows. See Table 3 for other geometric and simulation parameters.

RUN . | Geometric parameters . | Reynolds number, Re . | Water depth, H/m . | Mean velocity, U_{0}/m/s
. | |||
---|---|---|---|---|---|---|---|

Maximal deflection angle /° . | Wave length along centerline M/m . | River width B/m . | Aspect ratio (B/H) . | ||||

H15 | 110 | 2 | 0.15 | 1.0 | 2,500.0 | 0.015 | 0.2 |

H30 | 110 | 2 | 0.15 | 0.5 | 4,285.7 | 0.030 | 0.2 |

H60 | 110 | 2 | 0.15 | 0.25 | 6,666.7 | 0.060 | 0.2 |

H100(F110) | 110 | 2 | 0.15 | 0.15 | 8,571.4 | 0.100 | 0.2 |

RUN . | Geometric parameters . | Reynolds number, Re . | Water depth, H/m . | Mean velocity, U_{0}/m/s
. | |||
---|---|---|---|---|---|---|---|

Maximal deflection angle /° . | Wave length along centerline M/m . | River width B/m . | Aspect ratio (B/H) . | ||||

H15 | 110 | 2 | 0.15 | 1.0 | 2,500.0 | 0.015 | 0.2 |

H30 | 110 | 2 | 0.15 | 0.5 | 4,285.7 | 0.030 | 0.2 |

H60 | 110 | 2 | 0.15 | 0.25 | 6,666.7 | 0.060 | 0.2 |

H100(F110) | 110 | 2 | 0.15 | 0.15 | 8,571.4 | 0.100 | 0.2 |

*Note:* RUN H100 is RUN F110 in simulation of Group I.

*D*

_{n}=

*Re**(

*d*/2

*R*)

^{0.5}, where

*d*is the equivalent diameter of the channel and

*R*is the radius of curvature. When the

*D*

_{n}number is small, a single vortex forms on the cross-section of the channel. Meanwhile, after the

*D*

_{n}number exceeds the critical value, two counter-rotating vortices develops (even more vortices can be found when the

*D*

_{n}number is sufficiently high) (Ligrani & Niver 1988). This explains the variations of the spanwise velocities in Figure 14(e) being out-of-phase.

Table 4 further quantitatively shows the variations of flow velocities in channel bends with different aspect ratios. It can be seen that, with the decreasing of the aspect ratio, the variation amplitude of *U* in the streamwise direction decreases monotonously, while the variation amplitude of *V* in the streamwise direction increases monotonously. This can be attributed to the stronger secondary current in the channel bend with a larger water depth. Similar situations can be found in the spanwise variation of velocity components.

RUN . | /U . | /U . | /U . | /U . | . | . | . | . |
---|---|---|---|---|---|---|---|---|

H15 | 1.60 | 0.63 | 1.25 | 0.35 | 0.24 | 0.090 | 1.25 | 0.35 |

H30 | 1.50 | 0.75 | 1.25 | 0.50 | 0.225 | 0.113 | 1.25 | 0.50 |

H60 | 1.25 | 0.80 | 0.75 | 0.60 | 0.188 | 0.120 | 0.75 | 0.60 |

H100(F110) | 1.10 | 0.50 | 1.00 | 0.40 | 0.165 | 0.075 | 1.00 | 0.40 |

RUN . | /U . | /U . | /U . | /U . | . | . | . | . |
---|---|---|---|---|---|---|---|---|

H15 | 1.60 | 0.63 | 1.25 | 0.35 | 0.24 | 0.090 | 1.25 | 0.35 |

H30 | 1.50 | 0.75 | 1.25 | 0.50 | 0.225 | 0.113 | 1.25 | 0.50 |

H60 | 1.25 | 0.80 | 0.75 | 0.60 | 0.188 | 0.120 | 0.75 | 0.60 |

H100(F110) | 1.10 | 0.50 | 1.00 | 0.40 | 0.165 | 0.075 | 1.00 | 0.40 |

### Influences of channel curvature on flow structures

Channel curvature, usually denoted by the ratio of curvature radius to channel width (R/B), is a geometric parameter which describes the channel's non-straight morphology. The difference between channel curvature and channel sinuousness is that channel sinuousness only represents the curve degree of the channel centerline, while channel curvature combines the effects of channel shape and channel width. Emmett & Leopold (1964) measured 50 meandering rivers and found that the value of (R/B)^{2/3} ranged from 1.5 to 4.2 with an average of 2.7. They concluded that for alluvial meandering rivers, although their flow rates and geometric dimensions differed drastically, the value of (R/B)^{2/3} was close to a constant. Therefore, meandering rivers in alluvial plain areas show similar appearances on satellite images. However, for mountain rivers, the value of (R/B)^{2/3} may be far beyond that range due to the geological conditions. To study the influences of the channel curvature on flow structures, three sine-generated channel bends with the same maximum deflection angle of 110° but different channel widths were adopted in the simulations of Group III. The corresponding curvature ratio ranges from 1.0 to 7.5 (with (R/B)^{2/3} ranges from 1.0 to 20.5). The aspect ratios of the channels were set to 0.25. Table 5 shows the geometric and simulation parameters.

RUN . | Geometric parameters . | Reynolds number, Re . | Water depth, H/m . | Mean velocity, U_{0}/m/s
. | |||
---|---|---|---|---|---|---|---|

Maximal deflection angle /° . | Wave length along centerline M/m . | River width B/m . | Curvature ratio (R/B) . | ||||

B2 | 110 | 2 | 0.02 | 7.5 | 888.9 | 0.008 | 0.2 |

B5 | 110 | 2 | 0.05 | 3.0 | 2,222.2 | 0.02 | 0.2 |

B15(H60) | 110 | 2 | 0.15 | 1.0 | 6,666.7 | 0.06 | 0.2 |

RUN . | Geometric parameters . | Reynolds number, Re . | Water depth, H/m . | Mean velocity, U_{0}/m/s
. | |||
---|---|---|---|---|---|---|---|

Maximal deflection angle /° . | Wave length along centerline M/m . | River width B/m . | Curvature ratio (R/B) . | ||||

B2 | 110 | 2 | 0.02 | 7.5 | 888.9 | 0.008 | 0.2 |

B5 | 110 | 2 | 0.05 | 3.0 | 2,222.2 | 0.02 | 0.2 |

B15(H60) | 110 | 2 | 0.15 | 1.0 | 6,666.7 | 0.06 | 0.2 |

*Note:* RUN B15 is RUN H60 in simulation of the Group II.

*H*below the free-surface in Figure 16. It can be seen that the flow core is close to the concave side-wall in RUNs B2 and B5 but shifts to the convex side-wall in RUN B15. This can be attributed to the higher inertia of the secondary current directing to the convex side-wall near the free-surface in RUN B15. The stronger secondary current pushes the core flow to the convex side-wall and postpones the transition. Similar results were also reported in other numerical and experimental studies (Booij 2003; Abad & Garcia 2009; Kashyap

*et al.*2012; Bai

*et al.*2014).

The strengthening of the secondary current with increasing channel curvature can also be demonstrated by the variations of spanwise velocity in the streamwise and spanwise directions, i.e. and , as shown in Table 6.

RUN . | /U . | /U . | /U . | /U . | . | . | . | . |
---|---|---|---|---|---|---|---|---|

B2 | 0.85 | 0.25 | 0.75 | 0.10 | 0.09 | 0.005 | 5.625 | 0.75 |

B5 | 0.70 | 0.25 | 0.60 | 0.10 | 0.07 | 0.005 | 1.80 | 0.3 |

B15(H60) | 1.25 | 0.80 | 0.75 | 0.60 | 0.188 | 0.120 | 0.75 | 0.60 |

RUN . | /U . | /U . | /U . | /U . | . | . | . | . |
---|---|---|---|---|---|---|---|---|

B2 | 0.85 | 0.25 | 0.75 | 0.10 | 0.09 | 0.005 | 5.625 | 0.75 |

B5 | 0.70 | 0.25 | 0.60 | 0.10 | 0.07 | 0.005 | 1.80 | 0.3 |

B15(H60) | 1.25 | 0.80 | 0.75 | 0.60 | 0.188 | 0.120 | 0.75 | 0.60 |

Notice that the Reynolds number in the present study is relatively low, which ranges from 2,500 to 8,571.4 for most run cases, except RUN B2, which is 888.9. The flow falls in a regime of hydraulic smooth turbulent or laminar-turbulent transition (RUNB2). These conditions are close to laboratory experiments in mini-flumes. Although the Reynolds stress may not play an important role compared to the inertia force and centrifugal force in channel bends, the quantitative influence of the Reynolds number need to be further investigated. Considering the fact that the same uniform inlet flow velocity was adopted for all run cases in this study, the Froude number may change with the variation of aspect ratio of channel. The influence of Froude number on flow structures was also neglected in the study.

Further, in a natural meandering river, the water elevation near the concave sidewall is usually higher than that near the convex sidewall – the well-documented water super-elevation transverse slope. The super-elevation usually favors the development of the secondary current and produces a higher velocity and lower pressure region near the concave bank (Nguyen *et al.* 2007; De Marchis & Napoli 2008; Abad & Garcia 2009). However, this effect is usually thought to be relatively weak in the flows with a small Froude number (Stoesser *et al.* 2010), the rigid-lid approximation of the free-surface adopted in this study was also used by many other similar numerical studies (Van Balen *et al.* 2009; Stoesser *et al.* 2010; Bai *et al.* 2014) which yield acceptable agreement with experimental data.

## CONCLUSIONS

By using a RANS-based 3D CFD solver for incompressible viscous flows, numerical simulations for open channel flow in curved channel bends were carried out. A series of run cases were performed and the influences of river channel sinuousness, aspect ratio and curvature ratio were studied. Simulation results show the following:

For all of the channel bends in this study (maximum deflection angle ranges from 30 to 110°, aspect ratio ranges from 0.25 to 1.0, channel curvature ratio ranges from 0.42 to 3.14), the streamwise velocity is 4–10 times larger than the spanwise velocity and the spanwise velocity is 2–10 times larger than the vertical velocity. The streamwise variation of velocity magnitude along the banks are around 50% of the mean flow velocity, while it is less than 25% along the centerline of the channel.

Highly sinuous channel bends with a maximum deflection angle of 110° show much larger cross-sectional variation of velocity compared with 60 and 30° bends. The streamwise variation ranges of velocities are of the same order of magnitude of their spanwise counterpart. However, the spanwise variation gradients can be five times higher than those in the streamwise direction. The velocity magnitude of secondary current in highly sinuous channel bends can be as large as 20% of the primary flow, which may invalidate many theoretical assumptions that secondary current is much weaker than the primary flow.

The spanwise gradient of flow velocities decreases when the aspect ratio of channel bends increases from 0.15 to 1.0. Channels with higher aspect ratios tend to present more uniform velocity distributions on cross-sections. Simulation results show that the lower the water depth, the closer the dynamic flow axis to the channel banks. The variation of streamwise velocity along channel centerlines decreases 70% when the aspect ratio increases from 0.15 to 1.0. Deep-water channels with an aspect ratio of 0.15 show significant phase-lag between flow velocities and channel geometry. The spanwise velocities near the free-surface and the bottom-wall on the center plane are found to be out-of-phase when the water depth is large, and can be attributed to the large Dean number.

When the channel curvature ratio decreases from 7.5 to 1.0, the spanwise and streamwise variation amplitude of velocities increases. For channel bends with a curvature ratio 7.5 and 3.0, the spanwise velocities near the free-surface and bottom-wall vary in the anti-phase pattern due to the Dean numbers being smaller than the critical value. However, when the curvature ratio is 1.0, the out-of-phase pattern is observed.

Note that the channel bend shapes presented in this study are controlled by three factors, i.e. sinuousness, aspect ratio and curvature ratio, and may not cover the practical meandering river shapes in nature due to their diversity. A major contribution of the present study is the flow features in channel bends with various geometries. Such knowledge is necessary for understanding the fundamental mechanism of river meandering dynamics in fluvial environment. Moreover, a systematic analysis of flow velocity magnitudes in various channel bends are presented, which is especially informative and meaningful to theoretical studies.

Nowadays, turbulent models remain a huge challenge for RANS-based CFD simulations, including the RNG k-*ɛ* model adopted in the present study. In this regard, further investigations on the behavior of various turbulent models and their influences on secondary current are necessary, especially for sharply curved channel bends with a strong secondary current. A more advanced numerical technique, such as LES, could be a better option. However, considering LES is computationally expensive, investigation can firstly focus on one geometric factor, such as the deflection angle, before extending to more complex channel shapes.

## ACKNOWLEDGEMENTS

The study is financially supported by the Science Fund for Creative Research Groups of the National Natural Science Foundation of China (51621092), the National Natural Science Foundation of China (51279124, 51009105, 51579175).