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.

Field survey and laboratory experiment are effective methods to obtain flow features in river bends (Hooke & Yorke 2010, 2011; Nanson 2010; Termini & Piraino 2011; Schnauder & Sukhodolov 2012). However, due to natural and experimental limitations, the field surveys which have been carried out so far mainly focus on several special river bends under limited hydrological conditions, while the laboratory experiments performed adopt river bend models with special shapes (Richardson & Thorne 1998; Booij 2003; Du & Miwa 2006; Abad & Garcia 2009). Moreover, their results usually cover quite limited channel geometries. More cases need to be investigated to understand the dynamics of natural meandering rivers varying in curvatures, channel widths, water depths, etc. (Bai 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).
Figure 1

River bends in nature with diverse geometric shapes.

Figure 1

River bends in nature with diverse geometric shapes.

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

Following our previous publication (Bai 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: 
formula
1
 
formula
2
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): 
formula
3
 
formula
4
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.
The widely used values of the constants in the above governing equations are: 
formula
5
where , , and the turbulent Prandtl numbers for 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

The FVM was adopted in the 3D incompressible viscous flow solver. In the solver, to decouple the pressure and the velocity, the SIMPLE (semi-implicit method for pressure-linked equations) algorithm was applied. The flux of each control volume was calculated using a second order Upwind scheme. For time stepping, Adam-Bashforth (AB2) scheme was adopted, which has second order accuracy in time. Figure 2 shows the adopted 3D structured grids in which the vertical and cross-flow grid spacing are uniform, while the streamwise grid spacing varies inversely with the local curvature of the channel bend. More details regarding the finite volume solver can be found in our previous paper (Ji et al. 2012; Bai et al. 2014).
Figure 2

Computational grid (a) structured grid indexing scheme (b) schematic of 3D mesh.

Figure 2

Computational grid (a) structured grid indexing scheme (b) schematic of 3D mesh.

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 106. 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

The Group I experiment was carried out in a laboratory channel with a 180° bend and two straight inlet and outlet reaches. The rectangular cross-section is 0.3 m in height and 0.4 m in width with a mean water depth of 0.15 m, yielding an aspect ratio B/H = 2.67. The curvature radius of the inner and outer banks are 1.0 and 1.4 m, respectively. The length of the inlet and outlet reaches is 2.0 m, as shown in Figure 3, acoustic Doppler velocimetry was used to measure the three components of the instantaneous velocities. This experiment was used to study the 3D flow structure in sharply curved channel bends and also for simulation verifications (Bai 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.
Figure 3

Experimental setup of a 180° channel bend.

Figure 3

Experimental setup of a 180° channel bend.

Figure 4

Verification of vertical profiles of streamwise velocity – u and spanwise velocity – v at the cross-section at θ = 90° (x is the coordinate along the channel width with zero locates at the inner bank).

Figure 4

Verification of vertical profiles of streamwise velocity – u and spanwise velocity – v at the cross-section at θ = 90° (x is the coordinate along the channel width with zero locates at the inner bank).

Group II: mildly curved channel

For verification of flow in a mildly curved channel, high precision experimental data from Booij (2003) was adopted. Generally, river bends in nature are shallow (width/water depth ∼O(50)) and mildly curved (radius/width ∼O(10)). The experiment of Booij (2003) carried out in Delft University of Technology was designed with a geometric shape quite close to the above configurations. Owing to the 3D laser Doppler velocimetry system used, the mean flow velocity and the turbulent quantities were well measured. Therefore, it can be used as an excellent verification case for numerical simulations. The experimental flume is a U-shaped open channel consisting of a straight inflow part of 11 m in length, a curved part of 180° and a straight outflow part of 6.7 m in length, see Figure 5(a). The radius of the curved part is R = 4.10 m. The cross-section is rectangular with a channel width of W = 0.50 m and a water depth of h = 52 mm. The width-to-depth ratio is W/h = 96 and the radius-to-width ratio is R/W = 8.2. In our numerical simulations, the computational domain was set up in accordance with the above geometric parameters, except that the straight inflow part and the outflow part was reduced to 1.0 and 0.5R, respectively, in order to lessen the mesh number, see Figure 5(b).
Figure 5

Setup of channel bend: (a) experimental flume (Booij 2003); (b) computational domain.

Figure 5

Setup of channel bend: (a) experimental flume (Booij 2003); (b) computational domain.

The velocity profiles at the cross-section with = 135° were measured and the simulation results were compared against the experimental data, as shown in Figure 6. It can be seen that both the primary flow (U) and the secondary current (V, W) of the computed results agree well with the experimental data. The maximum deviation of computed secondary current from the measured data is observed near the upper corner of the outer bank, see Figure 6(b). This can be attributed to the counter-rotating vortex cell, usually less well-resolved in numerical simulations using RANS solver (Booij 2003; Blanckaert et al. 2013).
Figure 6

Verification of vertical profiles of streamwise velocity – U, spanwise velocity – V, vertical velocity – W at the cross-section at θ = 135° (y is the coordinate along the channel width with zero locates at the inner bank).

Figure 6

Verification of vertical profiles of streamwise velocity – U, spanwise velocity – V, vertical velocity – W at the cross-section at θ = 135° (y is the coordinate along the channel width with zero locates at the inner bank).

Group III: sine-generated channel

Channels in groups I and II consist of a circular bend with a constant curvature. They are rare for meandering rivers in nature. To verify the solver's capability in simulating flow in channel bends with streamwise-varying curvatures, a series of laboratory tests were carried out on a sanded bed. The side-walls of the channel consisted of two rows of plastic sheets planted into the sand bed, as depicted in Figure 7. The channel centerline follows a so-called sine-generated curve (Leopold & Wolman 1957) with a wavelength of 2.5 m. In total, five wavelengths were arranged along the channel. The channel width was 0.15 m and mean water depth was 0.05 m. The particle tracking velocimetry technique was used to measure the surface velocity of the channel flow. The measured mean flow velocity is 0.2 m/s. Good agreement has been achieved in the comparison of the simulated and measured surface velocity distribution, as shown in Figure 8. It was found that the velocity magnitude near the convex bank is larger than that near the concave bank at upstream side of the channel bend. Meanwhile, with the flow developing downstream, the maximum velocity shifts to the opposite bank gradually. This trend is clearly shown in both the experimental data and the simulation results. The simulation shows velocity higher than experiment near the convex bank (right bank in Figure 7). This is partly owing to the insufficient tracking particles in this region in the experiment. Due to the centrifugal force towards the concave bank, few particles were observed near the convex bank during the experiment, which may lead to an underestimation of velocity when using the interpolation technique. In general, the simulated primary flow velocities agree well with measured data.
Figure 7

Experimental facility for validation on surface flow in channel bends (unit: mm).

Figure 7

Experimental facility for validation on surface flow in channel bends (unit: mm).

Figure 8

Comparison of measured and calculated surface velocity magnitude.

Figure 8

Comparison of measured and calculated surface velocity magnitude.

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

The varying sinuousness of natural meandering rivers has significant influences on flow structures even at different sections of the same river (Bai 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.: 
formula
6
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.
Figure 9

Schematic of a sine-generated river bend.

Figure 9

Schematic of a sine-generated river bend.

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.

Table 1

Geometric parameters and simulation parameters of bends with various sinuousness (Group I)

RUN Geometric parameters
 
Reynolds number, Re Water depth, H/m Mean velocity, U0/m/s 
Maximal deflection angle,° Wave length along centerline, M/m River width, B/m 
F30 30 0.15 8,571.4 0.10 0.2 
F60 60 0.15 8,571.4 0.10 0.2 
F110 110 0.15 8,571.4 0.10 0.2 
RUN Geometric parameters
 
Reynolds number, Re Water depth, H/m Mean velocity, U0/m/s 
Maximal deflection angle,° Wave length along centerline, M/m River width, B/m 
F30 30 0.15 8,571.4 0.10 0.2 
F60 60 0.15 8,571.4 0.10 0.2 
F110 110 0.15 8,571.4 0.10 0.2 

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

The contours of velocity magnitude on different cross-sections along the channel are shown in Figure 10. It can be seen that the spanwise variation of streamwise velocity of RUN F110 with a larger sinuousness is obviously larger than that of RUN F30 with a smaller sinuousness. This is consistent with the conventional understanding that the non-uniformity of the velocity distribution increases with the increasing sinuousness of the channel bend. For both RUN F30 and F110, the velocity magnitude adjacent to the convex bank near the bend apexes is larger than that near the concave bank, which leads to the erosion and thus the retreat of the convex bank – the chute cutoff mechanisms of meandering rivers.
Figure 10

Contour of velocity magnitude in bends with various sinuousness.

Figure 10

Contour of velocity magnitude in bends with various sinuousness.

The primary flow (streamwise) and secondary current (cross-sectional) can be obtained by extracting flow velocities on cross-sectional and horizontal planes from the simulation results. Figure 11 shows the secondary current on a cross-section at the bend apex and the primary flow on a horizontal plane for RUN F110. It can be seen that the counter-rotating vortex cell near the outer bank is absent from the results, because of the limitation of the k-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).
Figure 11

Secondary current at bend apex (a) and near bank velocity vector (b).

Figure 11

Secondary current at bend apex (a) and near bank velocity vector (b).

Figure 12 shows the variations of velocity components along the center plane of the channel and a wall-parallel plane 0.01 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.
Figure 12

Streamwise profiles of flow velocities in bends with various sinuousness.

Figure 12

Streamwise profiles of flow velocities in bends with various sinuousness.

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.

Table 2

Magnitude of flow velocities in bends with various sinuousness

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 
Velocities in the above variables are non-dimensionalized by the mean velocity U, while lengths are non-dimensionalized by channel width B.

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.

Table 3

Geometric parameters and simulation parameters of bends with various aspect ratios (Group II)

RUN Geometric parameters
 
Reynolds number, Re Water depth, H/m Mean velocity, U0/m/s 
Maximal deflection angle /° Wave length along centerline M/m River width B/m Aspect ratio (B/H) 
H15 110 0.15 1.0 2,500.0 0.015 0.2 
H30 110 0.15 0.5 4,285.7 0.030 0.2 
H60 110 0.15 0.25 6,666.7 0.060 0.2 
H100(F110) 110 0.15 0.15 8,571.4 0.100 0.2 
RUN Geometric parameters
 
Reynolds number, Re Water depth, H/m Mean velocity, U0/m/s 
Maximal deflection angle /° Wave length along centerline M/m River width B/m Aspect ratio (B/H) 
H15 110 0.15 1.0 2,500.0 0.015 0.2 
H30 110 0.15 0.5 4,285.7 0.030 0.2 
H60 110 0.15 0.25 6,666.7 0.060 0.2 
H100(F110) 110 0.15 0.15 8,571.4 0.100 0.2 

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

Figure 13 shows the contours of streamwise velocity magnitude in the channel bends with various water depths. It can be seen that the maximum streamwise velocity decreases with increasing water depth, which means the primary flow tends to be more uniform in a channel with a smaller aspect ratio. However, in contrast, the strength of secondary currents increases with increasing water depth. This is clearly demonstrated by the highly twisted contour lines in Figure 13(c). Although the secondary currents distort the contours of the streamwise velocity, they also increase the uniformity of the primary flow by deflecting the flow core and mixing fluids with different velocities. It can also be seen that the maximum streamwise velocity is attained near the free-surface at the upstream side of the channel bends in RUN H15, but is convected to the bottom-wall by the strong secondary currents in RUN H60.
Figure 13

Contour of velocity magnitude in bends with various aspect ratios.

Figure 13

Contour of velocity magnitude in bends with various aspect ratios.

Figure 14 shows the variations of the velocity components along the streamwise direction. After the quasi-periodic flows are fully developed, the streamwise velocity on the center plane only shows small spikes near the channel bends for RUN H15. With the increasing water depth, the variations in the streamwise velocities increase substantially, as shown in Figure 14(c) and 14(e). However, the variations of the streamwise velocities near the side-wall are quite large and seem to be independent to the aspect ratio despite the streamwise velocities near the bottom-wall being slightly increased with the increasing aspect ratio. As shown in Figure 12, similar to the observation in Figure 10, the wave lengths of the streamwise velocities near the side-walls are approximately two times larger than those on the center plane. Moreover, it is interesting that the spanwise velocities near the free-surface and the bottom-wall on the center plane are in anti-phase, as shown in Figure 14(a) and 14(c). However, they become out-of-phase when the water depth is large, as shown in Figure 14(e). This is caused by the effects of the Dean number which represents the square root of the ratio between the product of the inertia and centrifugal forces to the viscous force and is defined as Dn = Re*(d/2R)0.5, where d is the equivalent diameter of the channel and R is the radius of curvature. When the Dn number is small, a single vortex forms on the cross-section of the channel. Meanwhile, after the Dn number exceeds the critical value, two counter-rotating vortices develops (even more vortices can be found when the Dn number is sufficiently high) (Ligrani & Niver 1988). This explains the variations of the spanwise velocities in Figure 14(e) being out-of-phase.
Figure 14

Streamwise profiles of flow velocities in bends with various aspect ratios (case F110 is equivalent to H100).

Figure 14

Streamwise profiles of flow velocities in bends with various aspect ratios (case F110 is equivalent to H100).

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.

Table 4

Magnitude of flow velocities in bends with various aspect ratios

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.

Table 5

Geometric parameters and simulation parameters of bends with various curvature ratios (Group III)

RUN Geometric parameters
 
Reynolds number, Re Water depth, H/m Mean velocity, U0/m/s 
Maximal deflection angle /° Wave length along centerline M/m River width B/m Curvature ratio (R/B) 
B2 110 0.02 7.5 888.9 0.008 0.2 
B5 110 0.05 3.0 2,222.2 0.02 0.2 
B15(H60) 110 0.15 1.0 6,666.7 0.06 0.2 
RUN Geometric parameters
 
Reynolds number, Re Water depth, H/m Mean velocity, U0/m/s 
Maximal deflection angle /° Wave length along centerline M/m River width B/m Curvature ratio (R/B) 
B2 110 0.02 7.5 888.9 0.008 0.2 
B5 110 0.05 3.0 2,222.2 0.02 0.2 
B15(H60) 110 0.15 1.0 6,666.7 0.06 0.2 

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

Figure 15 shows streamwise velocity contours on the cross-sections along the channels for RUNs B2 and B5. The results for RUN B15 are the same as those for RUN H60, as shown in Figure 13(c). It was found that, with the decreasing channel curvature, the contour lines of the streamwise velocity are getting more and more distorted due to the stronger secondary currents. A similar situation occurs when the channel sinuousness increases or the aspect ratio decreases, as shown in Figures 10 and 13. Moreover, the transition of the flow core from one side-wall to the other varies with the channel curvature. In RUN B2, the transition occurs just a little bit downstream from the inflection point of the channel, as indicated by points A and B in Figure 15(a). Meanwhile, it moves further downstream in RUNs B5 and B15. This means that with the decreasing curvature, the bank-to-bank flow transition moves downstream. This trend is also shown by the contours of the streamwise velocity on a horizontal plane 0.1H 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).
Figure 15

Contour of velocity magnitude in bends with various curvature ratios.

Figure 15

Contour of velocity magnitude in bends with various curvature ratios.

Figure 16

Contour of velocity magnitude in bends with various curvature ratios (0.1H below free surface).

Figure 16

Contour of velocity magnitude in bends with various curvature ratios (0.1H below free surface).

Figure 17 shows the variations of velocity components in the streamwise direction. In RUN B2 and B5, 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, which differs from RUN B15 in which the out-of-phase pattern is observed. Such patterns were also identified in the well-presented experimental works by Abad & Garcia (2009) for a high-amplitude Kinoshita meandering channel. It was pointed out that because of the consecutive bends with opposite curvature signs, a hydraulic transitional region was observed where a decaying cell (produced in the upstream bend) interacted with a growing cell (produced by local curvature). The rebalancing between the ‘decaying cell’ and the ‘growing cell’ can interpret this out-of-phase pattern.
Figure 17

Streamwise profiles of flow velocities in bends with various curvature ratios.

Figure 17

Streamwise profiles of flow velocities in bends with various curvature ratios.

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.

Table 6

Magnitude of flow velocities in bends with various curvature ratios

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:

  1. 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.

  2. 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.

  3. 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.

  4. 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).

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