Three-Dimensional Numerical Simulation of Flow Around a Spur Dike in a Meandering Channel Bend

Abstract

This paper presents a three-dimensional (3D) free surface model to predict incompressible flow around a spur dike in a meandering channel bend, which is highly 3D due to the presence of curvature effects. The model solves the Reynolds-averaged Navier–Stokes (RANS) equations using an explicit projection method. The 3D grid system is built from a two-dimensional grid by adding dozens of horizontal layers in the vertical direction. Numerical simulations consider four test cases with different spur dike locations in the same meandering channel bend with the same Froude numbers as 0.22. Four turbulence models, the standard k-ε model, the k-ω model, the RNG k-ε model and a nonlinear k-ε model, are implemented in our three-dimensional free surface model. The performance of these turbulence models within the RANS framework is assessed. Comparisons between the model results and experimental data show that the nonlinear k-ε model behaves better than the three other models in general. Based on the results obtained by the nonlinear k-ε model, the highly 3D flow field downstream of the spur dike was revealed by presenting velocity vectors at representative cross-sections and streamlines at the surface and bottom layers. Meanwhile, the 3D characteristics of the downstream separation zone were also investigated. In addition, to highlight the advantage of the nonlinear turbulence model, comparisons of velocity vectors at representative cross-sections between the results obtained by the linear and nonlinear $k$-$\epsilon$ models are also presented.

1. Introduction

In inland rivers, spur dikes are widely used to deepen the navigational channel, to protect river banks from erosion and even to improve aquatic habitat because of their low cost and simple construction. Over the past few decades, a great deal of effort has been paid investigations of flow past a spur dike in a straight channel using laboratory experiments [1,2,3,4] or numerical simulations [3,5,6,7,8,9,10,11]. However, spur dikes are usually constructed in channel bends in revetment engineering or navigation channel training works, and studies of flow in curved channels with one or more spur dikes are more meaningful from an engineering application perspective. The presence of a channel bend will complicate the flow filed and flow separation zone downstream of the spur dike because the flow passing through meandering channels is obviously of a three-dimensional (3D) nature because of the secondary flow, which is a transverse circulation induced by the centrifugal force, which is characterized by not only the maximum streamwise velocity occurring near the inner bank and shifting towards the opposite bank at the exit of the bend but also transverse velocity towards the inner bank near the bed and towards the outer bank near the water surface [12,13,14].

To the best of our knowledge, some attempts involving laboratory experiments [15,16,17] and numerical simulations [16,18,19,20] have been made to study the flow field in a channel bend with a spur dike. Ghodsian and Vaghefi [15] conducted experiments on scour and flow field around a T-shape spur dike in a 90-degree channel bend. A 3D flow field in a scour hole was presented. Giri et al. [16] carried out an experimental study for flow induced by non-submerged spur dikes protruding from the bank of a meandering laboratory flume with a smooth rigid bed. They measured the mean velocity field near the spur dikes for various combinations and locations of spur dikes and used the measured data to validate their two-dimensional model. However, a three-dimensional (3D) flow field was not presented by the experiment or simulation. Sharma and Mohapatra [17] conducted a series of laboratory experiments for flow past a spur dike on a meandering channel with a trapezoidal cross-section. They mainly studied the effects of spur dike location, contraction ratio of spur dike length to channel width and inflow Froude number on the separation zone parameters. However, there are few measured data concerning the flow field downstream of the spur dike. Tripathi and Pandey [18] simulated the flow characteristics around a T-shaped spur dike in a reverse meandering channel with an ANSYS 2018 Fluent software. The effect of Froude number on flow pattern and several other characteristics, including velocity distribution, flow separation and bed shear stress distribution, was presented. Vaghefi et al. [20] and Vaghefi et al. [19] investigated the flow field around a T-shaped spur dike in a 90-degree bend using SSIIM CFD software.

In this paper, a 3D free surface model is employed to investigate flow in a meandering channel bend with a spur dike. The configuration of the meandering channel is identical to that presented in [16]. The employed model is developed based upon previous models for water waves [21,22,23,24] and has been successfully applied to predict flow and bed evolution in a meandering channel bend [25]. The model solves the Reynolds-averaged Navier–Stokes (RANS) equations by the use of an explicit projection method. The extension of the previous model is carried out by implementing four turbulence models to close the RANS. The performance of these turbulence models has been assessed in the test cases with available experimental data. Numerical results are presented to reveal the highly 3D flow field downstream of the spur dike. Separation zone parameters are also provided to show its 3D characteristics.

2. Two-Equation Turbulence Closure Models

The governing equations and boundary conditions for hydrodynamics are given in Appendix A.

In this study, four turbulence models, the standard $k-\epsilon$ model, the $k-\omega$ model, the RNG $k-\epsilon$ model and a nonlinear $k-\epsilon$ model, are considered, in which the Reynolds stress tensor is evaluated by linear or nonlinear eddy-viscosity schemes. The linear eddy-viscosity scheme used in the standard $k-\epsilon$ model, the $k-\omega$ model and the RNG $k-\epsilon$ model is expressed as

$$\overline{u_i{u}_j}={\displaystyle \frac{2}{3}}k{\delta }_{ij}-{\nu }_t{S}_{ij}$$

(1)

where $k$ is the turbulent kinetic energy; ${\nu }_t$ is the eddy viscosity; ${\delta }_{ij}$ is the Kronecker delta and $S_{ij}=\partial U_i/\partial x_j+\partial U_j/\partial x_i$ is the mean strain rate tensor.

The nonlinear eddy-viscosity scheme proposed by [26] is given as follow:

$$\begin{array}{c}\overline{u_i{u}_j}={\displaystyle \frac{2}{3}}k{\delta }_{ij}-{\nu }_t{S}_{ij}\\ +c_1{\displaystyle \frac{k}{\tilde{\epsilon }}}{\nu }_t\left(S_{ik}{S}_{jk}-{\displaystyle \frac{1}{3}}{S}_{kl}{S}_{kl}{\delta }_{ij}\right)+c_2{\displaystyle \frac{k}{\tilde{\epsilon }}}{\nu }_t\left(W_{ik}{S}_{jk}+W_{jk}{S}_{ik}\right)\\ +c_3{\displaystyle \frac{k}{\tilde{\epsilon }}}{\nu }_t\left(W_{ik}{W}_{jk}-{\displaystyle \frac{1}{3}}{W}_{kl}{W}_{kl}{\delta }_{ij}\right)+c_4{\displaystyle \frac{k^2}{{\tilde{\epsilon }}^2}}{\nu }_t\left(S_{ki}{W}_{lj}+S_{kj}{W}_{li}\right)S_{kl}\\ +c_5{\displaystyle \frac{k^2}{{\tilde{\epsilon }}^2}}{\nu }_t\left(W_{il}{W}_{lm}{S}_{mj}+S_{il}{W}_{lm}{W}_{mj}-{\displaystyle \frac{2}{3}}{S}_{lm}{W}_{mn}{W}_{nl}{\delta }_{ij}\right)\\ +c_6{\displaystyle \frac{k^2}{{\tilde{\epsilon }}^2}}{\nu }_t{S}_{ij}{S}_{kl}{S}_{kl}+c_7{\displaystyle \frac{k^2}{{\tilde{\epsilon }}^2}}{\nu }_t{S}_{ij}{W}_{kl}{W}_{kl}\#\end{array}$$

(2)

where $\tilde{\epsilon }=\epsilon -2\nu {\left(\partial k^{1/2}/\partial x_j\right)}^2$ is the so-called isotropic dissipation rate and $W_{ij}=\partial U_i/\partial x_j-\partial U_j/\partial x_i$ is the vorticity tensor.

2.1. Standard $k-\epsilon$ Model

The standard $k-\epsilon$ model is applied with constants defined by [27] and employs the following transport equations for $k$ and $\epsilon$.

$${\displaystyle \frac{\partial k}{\partial t}}+{\displaystyle \frac{\partial k{U}_j}{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left[\left({\displaystyle \frac{{\nu }_t}{{\sigma }_k}}+\nu \right){\displaystyle \frac{\partial k}{\partial x_j}}\right]-\overline{u_i{u}_j}{\displaystyle \frac{\partial U_i}{\partial x_j}}-\epsilon$$

(3)

$${\displaystyle \frac{\partial \epsilon }{\partial t}}+{\displaystyle \frac{\partial \epsilon U_j}{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left[\left({\displaystyle \frac{{\nu }_t}{{\sigma }_{\epsilon }}}+\nu \right){\displaystyle \frac{\partial \epsilon }{\partial x_j}}\right]-c_{\epsilon 1}{\displaystyle \frac{\epsilon }{k}}\overline{u_i{u}_j}{\displaystyle \frac{\partial U_i}{\partial x_j}}-c_{\epsilon 2}{\displaystyle \frac{{\epsilon }^2}{k}}$$

(4)

where $\epsilon$ is the turbulent dissipation rate; ${\sigma }_k=$ 1.0 is the Schmidt number for the eddy diffusivity of turbulent kinetic energy; ${\sigma }_{\epsilon }=$ 1.3 is the Schmidt number for the eddy diffusivity of dissipation; and the other two constants are taken as $c_{\epsilon 1}=$ 1.44 and $c_{\epsilon 2}=$ 1.92, respectively.

The eddy viscosity is calculated from the turbulent kinetic energy $k$ and its dissipation rate $\epsilon$ as follows:

$${\nu }_t=c_{\mu }{\displaystyle \frac{k^2}{\epsilon }}$$

(5)

where $c_{\mu }=0.09$ is used.

2.2. $k-\omega$ Model

The two-equation $k-\omega$ model also incorporates a differential equation for $k$, but it solves a transport equation for the specific dissipation $\omega$ instead of $\epsilon$. The governing equations for $k$ and $\omega$ are given as follows [28]:

$${\displaystyle \frac{\partial k}{\partial t}}+{\displaystyle \frac{\partial k{U}_j}{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left[\left({\displaystyle \frac{{\nu }_t}{{\sigma }_k^{\omega }}}+\nu \right){\displaystyle \frac{\partial k}{\partial x_j}}\right]-\overline{u_i{u}_j}{\displaystyle \frac{\partial U_i}{\partial x_j}}-\epsilon$$

(6)

$${\displaystyle \frac{\partial \omega }{\partial t}}+{\displaystyle \frac{\partial \omega U_j}{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left[\left({\displaystyle \frac{{\nu }_t}{{\sigma }_{\omega }}}+\nu \right){\displaystyle \frac{\partial \omega }{\partial x_j}}\right]-c_{\omega 1}{\displaystyle \frac{\omega }{k}}\overline{u_i{u}_j}{\displaystyle \frac{\partial U_i}{\partial x_j}}-c_{\omega 2}{\displaystyle \frac{\omega }{k}}\epsilon$$

(7)

where ${\sigma }_k^{\omega }=$ 2.0, ${\sigma }_{\omega }=$ 2.0, $c_{\omega 1}=$ 0.555 and $c_{\omega 2}=$ 0.833 are used.

The parameter $\omega$ is related to the dissipation rate $\epsilon$ by

$$\omega ={\displaystyle \frac{\epsilon }{c_{\mu }k}}$$

(8)

Therefore, the eddy viscosity ${\nu }_t$ in the $k-\omega$ model is given by

$${\nu }_t={\displaystyle \frac{k}{\omega }}$$

(9)

2.3. RNG $k-\epsilon$ Model

The RNG $k-\epsilon$ model was derived by [29] using the renormalized group (RNG) theory. The model introduced a new term in the $\epsilon$ equation to take into account the highly anisotropic features of turbulence. However, by redefining coefficient $c_{\epsilon 1}$ in Equation (1), it has a similar form to the standard $k-\epsilon$ model. $c_{\epsilon 1}$ in the RNG $k-\epsilon$ model is given by

$$c_{\epsilon 1}=1.42-{\displaystyle \frac{S\left(1-S/S_0\right)}{\left(1+\beta S^3\right)}}$$

(10)

where $\beta =$ 0.015, ${\eta }_0=$ 4.38 and $S=(k/\epsilon )\sqrt{S_{ij}{S}_{ij}/2}$.

The other coefficients are $c_{\mu }=$ 0.085, $c_{\epsilon 2}=$ 1.68, ${\sigma }_k=$ 0.7179 and ${\sigma }_{\epsilon }=$ 0.7179.

2.4. Nonlinear $k-\epsilon$ Model

All the above turbulence models are often called linear turbulence models because they employ a linear relation between Reynolds stresses and mean strains, as described in Equation (1). By introducing nonlinear expressions in this relation, we can create a nonlinear turbulence model. A cubic nonlinear $k-\epsilon$ model proposed by [26,30] using the relation presented in Equation (2) is employed in this study. The coefficients in Equation (2) are given in Table 1. The model also solves Equation (3) for $k$ but solves Equation (4) for $\tilde{\epsilon }$ instead of $\epsilon$. All the coefficients are the same to those used in the standard $k-\epsilon$ model, except $c_{\epsilon 2}$ is replaced by the following expression:

$$c_{\epsilon 2}=1.92\left[1-0.3 {\mathit{exp}\left(R{e}_t\right)}^2\right]$$

(11)

where $R{e}_t=k^2/\left(\nu \tilde{\epsilon }\right)$ is the local turbulence Reynolds number.

Table 1. The coefficients used in Equation (1).

$c_1$	$c_2$	$c_3$	$c_4$	$c_5$	$c_6$	$c_7$
−0.1	0.1	0.26	−10$c_{\mu }^2$	0	−5$c_{\mu }^2$	5$c_{\mu }^2$

The eddy viscosity ${\nu }_t$ in the nonlinear $k-\epsilon$ model is given by

$${\nu }_t=c_{\mu }{f}_{\mu }{\displaystyle \frac{k^2}{\tilde{\epsilon }}}$$

(12)

with

$$c_{\mu }=\mathit{min}\left\{0.09,{\displaystyle \frac{1.2}{1+3.5M+0.235\left[\mathit{max}(0,M-3.333\right){]}^2\mathit{exp}\left(-R{e}_t/400\right)}}\right\}$$

(13)

$$f_{\mu }=\left[1-\mathit{exp}\left(-\sqrt{R{e}_t/90}-{\left(e_t/400\right)}^2\right)\right]f_{RS}+\left(1-f_{RS}\right)$$

(14)

$$f_{RS}=\mathit{min}\left[\mathit{max}(P_k/\epsilon ,{\mathit{exp}\left(-R{e}_t/50\right)}^2,0)/\mathrm{0.75,1}\right]$$

(15)

and

$$M=\mathit{max}(S,W)$$

(16)

where $P_k=-\overline{u_i{u}_j}\partial U_i/\partial x_j$ and $W=(k/\epsilon )\sqrt{W_{ij}{W}_{ij}/2}$.

3. Methodology and Study Area

3.1. Methodology

All the governing equations are discretized by means of a finite difference–finite volume method based upon a 3D grid system that is built from a two-dimensional grid by adding dozens of horizontal layers. In the vertical direction, a vertical boundary-fitted coordinate system is employed, in which the physical domain is divided into $N_z$ layers. The interface between two layers is $z_{k+1/2}\left(x,y,t\right)$, which is defined as

$$z_{k+1/2}\left(x,y,t\right)=z_{k-1/2}\left(x,y,t\right)+\left[\eta \left(x,y,t\right)+h\left(x,y\right)\right]/N_z, k=0,\dots ,N_z$$

(17)

Note that $z_{1/2}=-h(x,y)$ and $z_{N_z+1/2}=\eta (x,y,t)$ (see Figure 1).

Figure 1. Vertical boundary-fitted coordinate system.

In the model, the main numerical algorithm consists of the following three steps.

The first step is to solve the RANS by using an explicit projection method, which is subdivided into two stages [24]. The first stage is to project intermediate velocities $u^{n+1/2}$, $v^{n+1/2}$ and $w^{n+1/2}$ by means of solving the momentum equations that contain the non-hydrostatic pressure at the previous time level. In this stage, Perot’s scheme is used to discretize the horizontal advection terms and diffusion terms. More details about this can be found in [22].

In the second stage, the new velocities $u^{n+1}$, $v^{n+1}$ and $w^{n+1}$ are computed by correcting the projected values after including the non-hydrostatic pressure terms, which are obtained by solving the discretized Poisson equation. The Poisson equation is symmetric and positive definite and thus can be solved efficiently by the preconditioned conjugated gradient method.

In the second step, by substituting the resulting velocities into Equation (A4), we can obtain the new free surface elevation ${\eta }^{n+1}$.

In the final step, turbulent quantities are obtained by solving the corresponding two transport equations. Because turbulent quantities are defined in the same location of the vertical velocity $w$, transport equations for these can be solved by using the similar methods employed in the vertical momentum equation.

3.2. Study Area

The present numerical simulations consider four test cases with different spur dike locations in the same meandering channel bend, which was used in the experiments of [16]. Figure 2 shows the channel arrangement. It consists of a 9.8 m long straight entrance reach, three consecutive and opposite bends, and a 10.5 m long straight exit reach. The cross-section is rectangular, and the width of the channel is $B_0=$ 1.0 m. The numerical conditions for the four test cases are summarized in Table 2. The spur dike protruding from left or right bank is non-submerged. The channel bottom and the vertical walls are smooth. The length of the spur dike is $b=$ 0.25 m for all cases. In the former two test cases (Case 2 and Case 3), the spur dike is located at the left bank of the channel, while it is positioned at the corresponding right bank in the latter two test cases (Case 2R and Case 3R). The experimental data in Case 2 and Case 3, reported by [16], are used to assess the performance of different turbulence models on resolving flow field around the spur dike in a meandering channel bend. The numerical results related to cross-stream circulation and separation zone downstream of the spur dike are presented and discussed.

Figure 2. Meandering channel arrangement used in the experiments of [16].

Table 2. Numerical test cases and conditions.

Test Cases	Spur Dike Location	Flow Depth (m)	Approach Velocity (m/s)
Case 2	Left bank, $\alpha$ = 15°	0.096	0.21
Case 3	Left bank, $\alpha$ = 29°	0.096	0.21
Case 2R	Right bank, $\alpha$ = 15°	0.096	0.21
Case 3R	Right bank, $\alpha$ = 29°	0.096	0.21

In all the computations, the two straight channel reaches and all the channel bends were discretized by rectangular and trapezoid grids, respectively. After conducting a grid convergence analysis, the horizontal domain is covered by 55,840 grid cells in total and 20 horizontal layers were employed in the vertical direction. The time step is taken as 0.005 s. All the simulations were run until the flow reached steady state.

4. Results and Discussion

4.1. Flow Around a Spur Dike Located at the Left Bank (Case 2 and Case 3)

Figure 3 and Figure 4 concern depth-averaged streamwise ($u_s$) and transverse ($u_n$) velocity distributions along the channel width, respectively, at different cross-sections for Case 2. Comparisons between numerical results obtained from the four turbulence models and experimental data are provided. Each cross-section in Figure 3 and Figure 4 is in bend 1 and can be identified by an angle of $\alpha$ (see Figure 2). Cross-sections at $\alpha =$ 20°, 30° and 35° are located in the separation zone downstream of the spur dike. It can be seen from Figure 3 that the four turbulence models produce results that agree closely with each other and are in good agreement with the experimental data. However, it is relatively easy to discern the differences between these model results in Figure 4. By comparing the experimental data, the results obtained from the nonlinear $k$-$\epsilon$ model are better than other model results in general.

Figure 3. Comparisons of depth-averaged streamwise ($u_s$) velocity distributions along the channel width between the numerical results obtained from the four turbulence models and experimental data for Case 2 at cross-sections $\alpha$ = 15°, 20°, 30° and 35°.

Figure 4. Comparisons of depth-averaged transverse ($u_n$) velocity distributions along the channel width between the numerical results obtained from the four turbulence models and experimental data for Case 2 at cross-sections $\alpha$ = 15°, 20°, 30° and 35°.

Similar to Figure 3 and Figure 4, Figure 5 and Figure 6 show comparisons of depth-averaged streamwise and transverse velocity distributions between the numerical results obtained from the four turbulence models and experimental data, respectively, for Case 3. In these two figures, the latter two cross-sections are positioned in bend 2 and are identified by an angle of $\beta$ (see Figure 2). Cross-sections at $\alpha =$ 40°, and $\beta =$ 5° and 10° are located in the separation zone downstream of the spur dike. By comparing the experimental data, the nonlinear $k$-$\epsilon$ model predicts better results than the other three turbulence models, whether in Figure 5 or in Figure 6. Particularly, the agreement between the nonlinear $k$-$\epsilon$ model results and experimental data are quite good in Figure 5.

Figure 5. Comparisons of depth-averaged streamwise (u_s) velocity distributions along the channel width between the numerical results obtained from the four turbulence models and experimental data for Case 3 at cross-sections $\alpha =$ 29°, 40° and $\beta =$ 5° and 10°.

Figure 6. Comparisons of depth-averaged transverse (u_n) velocity distributions along the channel width between the numerical results obtained from the four turbulence models and experimental data for Case 3 at cross-sections $\alpha =$ 29°, 40° and $\beta =$ 5° and 10°.

Figure 7 displays velocity vectors predicted by the standard $k-\epsilon$ model and the nonlinear $k-\epsilon$ model at five representative cross-sections downstream from the spur dike for Case 2. Both model results show very similar velocity fields. A cross-stream circulation outside the downstream separation zone is just generated in the cross-section at $\alpha =$ 22.5°. Then, it is further developed over cross-sections at $\alpha =$ 25° and 27.5°. After the cross-section at $\alpha =$ 27.5°, the center of the cross-stream circulation becomes lower. As a result, the cross-stream circulation almost disappears in the cross-section at $\alpha =$ 35°. Notably, the cross-stream circulations in all the cross-sections rotate anticlockwise, which are induced by the curvature of bend 1.

Figure 7. Velocity vectors predicted by the standard k-ε model (left panel) and the nonlinear k-ε model (right panel) at five representative cross-sections downstream from the spur dike for Case 2.

Figure 8 shows comparisons of velocity vectors between the standard $k-\epsilon$ model and the nonlinear $k-\epsilon$ model at six representative cross-sections downstream from the spur dike for Case 3. Both model results show very similar velocity fields for all the cross-sections except those at $\beta =$ 5° and 10°. In the cross-section at $\alpha =$ 35°, a cross-stream circulation rotating anticlockwise has been formed outside the downstream separation zone, which is induced by the curvature of bend 1, while the cross-stream circulations in the cross-sections at $\beta =$ 15°, 25° and 40° are induced by the curvature of bend 2 and rotate clockwise. The cross-sections at $\beta =$ 5° and 10° are in the transition field from anticlockwise circulation in bend 1 to clockwise circulation in bend 2. In this transition field, the differences between the two model results become discernable because the nonlinear $k-\epsilon$ model predicts a small circulation in the cross-sections at $\beta =$ 5° and 10°, which cannot be found in the results of the standard $k-\epsilon$ model.

Figure 8. Velocity vectors predicted by the standard k-ε model (left panel) and the nonlinear $k-\epsilon$ model (right panel) at six representative cross-sections downstream from the spur dike for Case 3.

Figure 9 shows streamlines computed by the nonlinear $k-\epsilon$ model at the surface ($z=0.95H$) and bottom ($z=0.05H$) layers for Case 2. It can be found that flow downstream of the spur dike is highly 3D because streamlines at the bottom layer are quite different to those at the surface layer. Outside of the separation zone, streamlines at the bottom layer are seen concentrated near the inner right bank of bend 1 and then deflect toward to the inner left bank of bend 2 due to the curvature effect, while at the surface layer, streamlines change very gently along the meandering channel. Similar features of the streamlines can also be observed in Case 3, so streamlines for Case 3 were not provided for brevity.

Figure 9. Streamlines predicted by the nonlinear $k-\epsilon$ model at the surface (z = 0.95H) and bottom (z = 0.05H) layers for Case 2.

It is quite different from the flow in a straight channel with a spur dike that flow inside the separation zone for Case 2 or Case 3 is also characterized as being 3D because of the presence of curvature effects. Following the work of [17], the separation zone downstream of the spur dike may be characterized by the downstream length $L_1$, the maximum width $W_{max}$ and the distance of maximum width from the spur dike $L_2$. To further reveal the 3D characteristics of the downstream separation zone for Case 2 and Case 3, corresponding nondimensional parameters predicted by the nonlinear $k-\epsilon$ model are shown in Figure 10. The nondimensional parameters are obtained by normalizing $L_1$, $L_2$ and $W_{max}$ by the spur dike length $b$. Because the characteristic parameters of the separation zone vary almost monotonously along the water depth, only the results at the surface ($z=0.95H$) and bottom ($z=0.05H$) layers are presented in Figure 10. It is found that $L_1$ at the bottom layer is very close to that at the surface layer. $W_{max}$ and $L_2$ at the bottom layer exceed those at the surface layer by up to 11.9% and 19.2%, respectively, for Case 2 and 25.3% and 11.0%, respectively, for Case 3. The larger value of the maximum width $W_{max}$ is at the bottom layer for both Case 2 and Case 3, which is mainly caused by the curvature effect of bend 1 because the positions of the maximum width in Case 2 and Case 3 are in bend 1.

Figure 10. Nondimensional parameters predicted by the nonlinear $k-\epsilon$ model at the surface (z = 0.95H) and bottom (z = 0.05H) layers for Case 2 and Case 3.

4.2. Flow Around a Spur Dike Located at the Right Bank (Case 2R and Case 3R)

Figure 11 shows comparisons of velocity vectors between the standard $k-\epsilon$ model and the nonlinear $k-\epsilon$ model at six representative cross-sections downstream from the spur dike for the test of Case 2R. Both model results show a cross-stream circulation rotating anticlockwise in the cross-sections at $\alpha =$ 30° and 40°, which is induced by the curvature of bend 1. A cross-stream circulation rotating clockwise in the cross-sections at $\beta =$ 30° and 50° are also predicted similarly by the two turbulence models, which is induced by the curvature of bend 2. However, in the cross-sections at $\beta =$ 10° and 20°, which are in the transition field from anticlockwise circulation in bend 1 to clockwise circulation in bend 2, velocity vectors predicted by the nonlinear $k-\epsilon$ model are different to those obtained by the standard $k-\epsilon$ model. Both model results show two cross-stream circulations in the cross-section at $\beta =$ 10°. The smaller circulation is in the bottom left corner of the cross-section, while the larger one is in the middle water depth. Nevertheless, the center of the larger circulation predicted by the nonlinear $k-\epsilon$ model is very close to the center of the cross-section. In the cross-section at $\beta =$ 20°, both model results show a cross-stream circulation near the bottom of the cross-section, but the nonlinear $k-\epsilon$ model captures another circulation near the water surface, which cannot be found in the results of the standard $k-\epsilon$ model.

Figure 11. Velocity vectors predicted by the standard $k-\epsilon$ model (left panel) and the nonlinear $k-\epsilon$ model (right panel) at six representative cross-sections downstream from the spur dike for Case 2R.

Comparisons of velocity vectors between the standard $k-\epsilon$ model and the nonlinear $k-\epsilon$ model at six representative cross-sections downstream from the spur dike for Case 3R are shown in Figure 12. It can be found that there have always been some discernable differences between the two model results in the former four cross-sections, which are in the transition field. Particularly in the cross-sections at $\beta =$ 10°, 15° and 20°, the nonlinear $k-\epsilon$ model predicts more cross-stream circulations than the standard $k-\epsilon$ model. As shown in Figure 12, three circulations are captured by the nonlinear $k-\epsilon$ model in the cross-sections at $\beta =$ 10° and 15°, while the standard $k-\epsilon$ model only predicts two circulations in corresponding cross-sections. In the cross-section at $\beta =$ 20°, both model results show a cross-stream circulation near the bottom of the cross-section, but the nonlinear $k-\epsilon$ model captures another circulation near the water surface, which is not presented by the standard $k-\epsilon$ model. In addition, the two turbulence models predict identical results in the cross-sections at $\beta =$ 25° and 30°, where a cross-stream circulation rotating clockwise is seen to form, which is induced by the curvature of bend 2.

Figure 12. Velocity vectors predicted by the standard $k-\epsilon$ model (left panel) and the nonlinear $k-\epsilon$ model (right panel) at six representative cross-sections downstream from the spur dike for Case 3R.

Figure 13 shows streamlines computed by the nonlinear $k-\epsilon$ model at the surface ($z=0.95H$) and bottom ($z=0.05H$) layers for Case 2R. It is found that downstream of the spur dike, flow, whether in the separation zone or out of it, is highly 3D. Out of the separation zone, streamlines at the bottom layer concentrate along the separation zone and then deflect toward the inner left bank of bend 2 due to the curvature effect, while at the surface layer, streamlines also change very gently along the meandering channel. For Case 3R, streamlines show very similar features to those presented in Figure 13, which were not supplied for brevity.

Figure 13. Streamlines predicted by the nonlinear $k-\epsilon$ model at the surface (z = 0.95H) and bottom (z = 0.05H) layers for Case 2R.

The 3D characteristic parameters of the downstream separation zone for Case 2R and Case 3R are presented in Figure 14. In these two cases, $L_1$ at the bottom layer is also very close to that at the surface layer, and there is very significant difference of $L_2$ between the bottom layer and the surface layer. The maximum width $W_{max}$ at the bottom layer exceeds that at the surface layer by up to 3.5% for Case 2R and is smaller than it by up to 24.5% for Case 3R. For Case 2R, the larger value of $W_{max}$ at the bottom layer may be due to the effect of the spur dike because the position of the maximum width is very close to it.

Figure 14. Nondimensional parameters predicted by the nonlinear $k-\epsilon$ model at the surface ($z=0.95H$) and bottom ($z=0.05H$) layers for Case 2R and Case 3R.

5. Conclusions

In this paper, a 3D numerical model for free surface flows is employed to study flow in a meandering channel bend with a spur dike. Four test cases with different spur dike locations in the same meandering channel bend were considered. The model implements four two-equation turbulence models, including linear and nonlinear eddy-viscosity schemes, to close the RANS equations. The performances of these turbulence models have been assessed in tests with available experimental data. By comparing the experimental data, the results obtained from the nonlinear $k-\epsilon$ model are better than the other model results in general.

Comparisons of the calculated velocity vectors at some representative cross-sections downstream from the spur dike between the standard $k-\epsilon$ model and the nonlinear $k-\epsilon$ model are presented for the four test cases. It is found that the nonlinear $k-\epsilon$ model predicts a more sophisticated flow field than the standard $k-\epsilon$ model in the transition region of the meandering channel, where it is seen to form single, two or three cross-stream circulations in the nonlinear $k-\epsilon$ model results. Outside the transition region, both turbulence models show similar results, in which it can be found that a cross-stream circulation rotates clockwise or anticlockwise due to the curvature effect of the bend.

Streamlines predicted by the nonlinear $k-\epsilon$ model at the surface and bottom layers for Case 2 and Case 2R indicate that flow out of the separation zone is 3D, and it is very different from the flow in a straight channel with a spur dike, in which flow inside the separation zone is also characterized as being 3D. For the four test cases, the 3D characteristics of the downstream separation zone are investigated by presenting the downstream length $L_1$, the maximum width $W_{max}$ and the distance of maximum width from the spur dike $L_2$ at the surface and bottom layers. It is found that the characteristic parameters of the separation zone are affected by the curvature effects and may be quite different from the bottom layer to the surface layer. For all the tests, $L_1$ at the bottom layer is very close to that at the surface layer, but there is a significant difference in $L_2$ or $W_{max}$ between the surface and the bottom layers. This demonstrates that there is a highly 3D separation zone downstream of the spur dike in the meandering channel bend.

At present, it is common to arrange spur dikes on the outer bank of curved rivers to protect the bank slope from water scouring in the management of small and medium-sized rivers. This method arranges spur dikes on the outer bank to divert high-speed water away from the bank slope area to achieve the purpose of protecting the river bank. In the future, the model can be used to study this problem, and simulate and analyze the number, location and angle of the spur dike group arrangement, involving a lot of design optimization.

Appendix A.1. Reynolds-Averaged Navier–Stokes (RANS) Equations

The model solves the 3D RANS equations to predict free surface flows, which are expressed in the following form:

Continuity equation:

$${\displaystyle \frac{\partial U_i}{\partial x_i}}=0$$

(A1)

Momentum equation:

$${\displaystyle \frac{\partial U_i}{\partial t}}+{\displaystyle \frac{\partial U_i{U}_j}{\partial x_j}}=g_i-{\displaystyle \frac{\partial p}{\partial x_i}}+{\displaystyle \frac{\partial (-\overline{u_i{u}_j})}{\partial x_j}}+\nu {\displaystyle \frac{{\partial }^2{U}_i}{\partial x_j^2}}$$

(A2)

where $t$ is the time; $x_i=(x,y,z)$ is the spatial coordinate; $U_i=(u,v,w)$ and $u_i$ are the mean and turbulent components of velocity, respectively; $\overline{u_i{u}_j}$ is the Reynolds stress tensor; $g_i$ is acceleration due to gravity; $\nu$ is molecular viscosity; and the normalized pressure $p$ is defined as the pressure divided by a constant reference density and is split into hydrostatic and non-hydrostatic ones as follows:

$$p=g\left(\eta -z\right)+q$$

(A3)

where $\eta$ is the free surface elevation and $q$ is the non-hydrostatic pressure component.

Appendix A.2. Boundary Conditions

To calculate the moving surface, the following free surface equation is used, which is obtained by integrating Equation (A1) over water depth and applying the kinematic free surface condition and bottom condition.

$${\displaystyle \frac{\partial \eta }{\partial t}}+{\displaystyle \frac{\partial }{\partial x}}{\int }_{-h}^{\eta }udz+{\displaystyle \frac{\partial }{\partial y}}{\int }_{-h}^{\eta }vdz=0$$

(A4)

where $-h(x,y)$ is the bottom surface.

At rigid wall boundaries, velocity normal to the wall is zero and the tangential velocity is determined by the wall-function approach. In this study, only smooth rigid walls are considered, so velocity parallel to the wall is described by the following logarithmic law:

$${\displaystyle \frac{U_{\tau }}{U_{*}}}={\displaystyle \frac{1}{\kappa }}\mathit{ln}{\displaystyle \frac{9.05U_{*}{D}_n}{\nu }}$$

(A5)

where $\kappa$ is von Karman’s constant; $U_{\tau }$ is velocity parallel to the smooth wall; $U_{*}$ is the shear velocity; and $D_n$ is the normal distance from the wall.

At the free surface, the boundary conditions for turbulent quantities are

$${\displaystyle \frac{\partial k}{\partial z}}=0, \epsilon ={\displaystyle \frac{(k{c}_{\mu }{)}^{1.5}}{0.07\kappa H}}$$

(A6)

where $H=\eta +h$ is the water depth.

At the rigid walls, the following boundary conditions for turbulent quantities are specified.

$$k={\displaystyle \frac{U_{*}^2}{c_{\mu }^{0.5}}}, \epsilon ={\displaystyle \frac{U_{*}^3}{\kappa D_n}}$$

(A7)

At inflow boundaries, the velocity normal to the boundary and turbulent quantities are specified as follows:

$$U_n=w=0, k=0.03U_n^2, \epsilon =c_{\mu }{\displaystyle \frac{k^{1.5}}{0.09H}}$$

(A8)

Notably, boundary conditions for the specific dissipation $\omega$ can be obtained by substituting Equation (8) into Equations (A6) and (A7).