1 Introduction

A natural river often consists of a meandering main channel and two floodplains. When a flood occurs,the flow depth will increase,and the floodplains will be submerged,which may result in overbank flows in the meandering channels. The flow characteristics in a meandering compound channel are totally different from those in a straight compound channel. In the straight compound channel,the significant velocity difference between the main channel flow and the floodplain flow leads to a strong lateral shear stress in the mixing region. In the meandering compound channel,the flow under the bankfull level goes downstream in the meandering main channel,and the flood plain flow travels along the valley direction,which generates a strong vertical shear stress around the bankfull level in the main channel. The flow exchanges at the apex sections are weak,while the flow exchanges in the cross-over sections are significant. Besides,the velocities and the secondary flows,which are related to the sediment transports and bed forms,are longitudinally changing in the meandering channel. Therefore,for the engineering purpose,it is necessary to present a simple lateral distribution model (LDM) to accurately predict the depth-averaged velocity in a meandering compound channel.

Shiono and Muto^[1] experimentally investigated the flow structures in the meandering compound channels with fixed beds and different sinuosities. To explore the differences between mobile cases and fixed cases,Shiono et al.^[2] and Spooner^[3] conducted a series of experiments to discuss the flow patterns. Shiono et al.^[4] investigated the meandering compound channels with mobile beds and different floodplain conditions,and showed that the flow characteristics in the meandering compound channels were much more complicated than those in the straight compound channels. The flood plain flow contributed a lot to secondary flows,which resulted in the growth and decay of the secondary currents along the meander. Strong vertical shear stresses were caused due to the different flow directions of the main channel flow and the upper layer flow. The flow contraction and expansion were seen at the cross-over sections,which would affect the internal flow structure in the main channel. All these phenomena are specific characteristics of the meandering channel flow. Therefore,the discharge prediction in meandering compound channels is a challenge. Many methods have been proposed to predict the stagedischarge curve^{[5, 6, 7, 8, 9, 10]}. Morvan et al.^[11],Shao et al.^[12],Zarrati et al.^[13],Guo et al.^[14],and Jing et al.^{[15, 16]} proposed some two-dimensional (2D) and three-dimensional (3D) numerical models. Ervine et al.^[17],McGahey and Samuels^[18],and Huai et al.^[19] proposed 2D analytical models to predict the depth-averaged velocities in meandering main channels based on the method of Shiono and Knight^[20]. However,all these models use vertical coordinates,and calibrate the secondary flow parameter. More importantly,the analytical solutions in these models are very complicated,and are not appliable. Therefore,it is necessary to propose a simple but applicable method for the prediction of the depth-averaged velocity in a meandering compound channel.

This paper presents a simple analytical method with curvilinear coordinates to predict the depth-averaged velocities in the meandering compound channel. The governing equation is derived from the streamwise momentum equation and the flow continuity equation under the condition of quasi-uniform flow. The simple analytical solution is presented with the velocity parameter which contains the longitudinal velocity variation and the lateral velocity variation, and the predictive capability of this model is verified by the experimental data from different sources. Finally,the determination of the velocity parameter and the limitation of the proposed model are discussed.

2 Experimental arrangements

Three groups of experiments (MN1,MN2,and MN3) are conducted in a 35 m long,4 m wide, and 1 m high flume at the State Key Laboratory of Hydraulics and Mountain River Engineering (SKLH),Sichuan University,China. The 3D velocities at the two apex cross sections CS1 and CS7 (see Fig. 1),are recorded by the three-component Sontek Acoustic Doppler Velocimeter (ADV) equipped with up-side and down-side probes. At each point,the measurement period and frequency are set as 30 s and 50 Hz,respectively.

The shape of the meandering main channel is rectangular (see Fig. 2). The sinuosity (s) is designed as 1.381. The total width (B) and the main channel width (b) are 4.0 m and 0.7 m, respectively. The bankfull depth (h) is 0.14 m. Therefore,the aspect ratio (b/h) is 5. The inner radius (r) and the outer radius (R) are 0.9 m and 1.6 m,respectively. Therefore,the medium radius (Rm) is 1.25 m. The bed surface is smoothed and covered by a thin concrete layer. The flume has a fixed bed slope (S) of 0.001. The error in the geometry is controlled to be within 5%.

A summary of the experimental conditions is shown in Table 1,where Q is the discharge,H is the flow depth,Dr is the relative flow depth,and u∗ is the friction velocity. The parameters

used in the experiments are as follows:

where B m is the width of the meander belt. The friction velocity (u∗) in each case is calculated by ,where g is the local gravitational acceleration,and R_h is the hydraulic radius. The relative flow depth D_r is defined by

The top view of the test meander is shown in Fig. 1. For this channel,the original point of the curvilinear coordinates at C_S7 is set at the left-hand wall of the meandering main channel (see Fig. 2). This figure shows 13 vertical measurement lines denoted by 1 to 13 at C_S1 and C_S7 with a 0.05 m lateral interval from the left (y = 0.0 m) to the right (y = 0.7 m). The measurement intervals between two vertical points are arranged as 0.015 m. All experimental data are recorded under the quasi-uniform flow condition by ensuring that the water surface slope remained to be parallel to the bed slope at each meander by manually adjusting the downstream tailgate. When the deviation of the water surface and the bed slope is less than 5%,the quasi-uniform flow condition is considered to be attained,and the measurements are started.

3 Theoretical background

This model is derived from the momentum equation with the curvilinear coordinates presented by Schlichting^[21]. Under this curvilinear coordinates,the x-axis is parallel to the main channel ridge,and the y-axis is normal to the meandering main channel (see Fig. 1). The original point at CS7 locates at the corner next to the left floodplain (see Fig. 2). For the quasiuniform flow condition in the meandering compound channel,the momentum equation and the flow continuity equation with curvilinear coordinates may be shown as follows.

The momentum equation is

The continuity equation is

In the above equations,r is the radius of the curvature to the inside of the bend,x,y,and z are the streamwise,lateral,and vertical directions,respectively. U,V ,and W are the velocity components corresponding to the x-,y-,and z-directions,respectively. ρ is the flow density. τ xx,τyx,and τzx are the Reynolds stresses.

Combining Eqs.(1) and (2) yields

Since and have the same dimension,it is assumed that

where ϕ is a dimensionless coefficient,and ϕ = 0 when r → ∞.

The ratio of the transverse velocity to the longitudinal velocity (V/U) is used herein to represent the intensity of the secondary flows in the main channel. At the apex sections,the centrifugal force enhances the secondary flow significantly,i.e.,the range of V/U is from −0.43 to 0.37,whereas in the straight compound channel,the range of V/U is from −0.11 to −0.065^[22] (see Fig. 3). This suggests the importance of the centrifugal force to the secondary flow in the meandering main channel. Liu et al.^[23] showed that in the meandering main channel,the measured transverse velocity (V ) consisted of two parts,i.e.,the original secondary current cell (Vo),which was enhanced by the centrifugal force,and the component of the upstream flood plain flow (V1),i.e.,V = (1 + k)Vo + V₁. Moreover,the effect of the centrifugal force on the secondary flows might be considered as ,where k is a dimensionless coefficient to reflect the relation of the centrifugal force and the secondary flow. At the apex sections,the directions of the main channel flow and the flood plain flow seam parallel. Therefore,the angle between them can be considered as zero. This indicates that the contribution of the upstream flood plain flow to the secondary flows may be ignored,i.e.,V₁ = 0.

Only considering the effect of the centrifugal force at the apex,we can rewrite Eq. (3) as follows:

Integrating Eq. (4) over the local flow depth (H) with

yields the depth-averaged equation as follows: where Γ_mc is the velocity parameter representing the longitudinal velocity variation and the lateral secondary flow variation. λ is the dimensionless eddy viscosity. f is the Darcy-Weisbach friction factor. τ _yx is the depth-averaged Reynolds shear stress. ε is the depth-averaged eddy viscosity. Ud is the depth-averaged velocity defined by

At the apex sections (C_S1 and C_S7),the velocity parameter may be presented as follows:

Liu et al.^[23] experimentally investigated the longitudinal distributions of the depth-averaged velocity,and found that at the apex sections,the gradient of the streamwise velocities was small enough to be ignored,i.e., ≈ 0. Therefore,Eq. (7) can be simplified as

Equation (5) is the governing equation with the curvilinear coordinates in the meandering main channel. We understand that its analytical solution is hardly obtained by analyzing its structure. To present the simple analytical solution,Eq. (5) has to be simplified by ignoring some terms. For a specific case,U_d and y are seen as a variable and an independent variable, respectively,and other coefficients are all considered as constants. Hence,Eq. (5) is an inhomogeneous differential equation. For the convenience of comparison,U_d² and are separated, and the five terms from the left to the right are labeled as “1” to “5”,respectively. Therefore, Eq. (5) can be rewritten as follows: where Term 1 is the velocity term,Term 2 is the gravity term,Term 3 and Term 5 are the lateral shear stress terms,and Term 4 is the bed shear stress term.

eral shear stress terms,and Term 4 is the bed shear stress term. To understand which term in Eq. (9) can be reasonably ignored,the experimental data in the case MN2 are taken as an example to show the quantity grade comparison of the five terms. The velocity term is unknown,which may be calculated a posteriori by the results of other four terms. All the parameters are shown in Table 1 except the dimensionless eddy viscosity which is a “catch all” parameter to describe the various 3D effects^[20]. This parameter is 0.07 at the apex section in the meandering main channel,the same as that demonstrated by Liu et al.^[23]. Therefore,the coefficients of Term 2,Term 3,Term 4,and Term 5 can be carried out. In Eq. (9),Term 2 (ρgHS) itself is a constant,i.e.,1.532. The coefficient in Term 3 is

which equals 0.109. In Term 4,the coefficient equals 3.972. Finally,the coefficient in Term 5 is

which equals 0.218. To analyze the coefficients of Term 2,Term 3,Term 4,and Term 5 more clearly,they are linearly mapped to ^{[0, 1]},and Term 2 of Eq. (9) is mapped to 1. Therefore,the coefficients in Term 3,Term 4,and Term 5 are rewritten as 0.071,2.592,and 0.142,respectively. It is noticeable that the coefficients of Term 3 and Term 5 are relatively small when they are compared with those of Term 2 and Term 4. The coefficients in Term 2,Term 3,Term 4,and Term 5 are determined by the hydraulic condition and the river geometry. Therefore,these coefficients are fixed when the boundary and hydraulic conditions are given.

Based on the experimental velocity in the case MN2,it is possible to calculate the values of

at 13 lateral measurement lines. To avoid the offset of the positive value and the negative one, a new parameter (Φ) is defined by where A is the calculated results for each term or variable,N is the number of the results (N = 13 in the case MN2). Φ is used to represent the averaged absolute value for every term or variable.

The values of Φ for

are carried out,which equal 0.050,0.030,and 0.057,respectively. The values of Φ for Term 2,Term 3,Term 4,and Term 5 are 1.532,0.013,0.419,and 0.006,respectively,and the value of Φ for Term 1 is 1.132. We notice that the magnitude of the five terms is quite different. It is apparent that the values of Term 3 and Term 5 are much smaller than those of the other three terms. Therefore,they may be ignored reasonably in practical applications. However, this assumption induces errors.

To verify this finding further,six cases are used herein (see Table 2). For the study of Spooner^[3],the experimental parameters are as follows:

The flow depths (H) of the three cases,i.e.,g4_5,g4_7,and g4_10,are 0.05 m,0.057 2 m,and 0.072 8 m,respectively,and the bankfull depth (h) is 0.04 m. Hence,the values of the relative flow depth D_r in g4_5,g4_7,and g4_10 are 0.2,0.3,and 0.45,respectively.

The results of the six cases are shown in Table 2. From the table,we can see that the magnitude of Term 5 is much smaller than those of other four terms. Therefore,this term is first ignored. When Term 1,Term 2,Term 3,and Term 4 are compared,it is noticeable that in most cases,the magnitude of Term 3 is about ten times smaller than that of Term 4,and is much smaller than those of Term 1 and Term 2. Therefore,Term 3 is also ignored,and Eq. (9) may be rewritten as follows:

where Γ_mc(11) is a velocity parameter,but its value does not equal the value of Γ_mc in Eq. (9). In the following context,the subscript “mc(11)” of Γ indicates the velocity parameter in Eq. (11), while the subscript “mc” means the one defined in Eq. (9). In Eq. (11),the values of Φ of Term 2 and Term 4 are still 1.532 and 0.419,respectively. Though the value of Φ for the new term 1 (Γ_mc(11)) is changed to 1.113,the error reflected in the velocity parameter seems quite small, i.e.,

which can meet the engineering purpose.

The analytical solution of Eq. (11) may be obtained as follows:

The velocity parameter (Γ_mc(11)) and the Darcy-Weisbach friction factor (f) are the only two control factors,which reflect the longitudinal and lateral velocity variations and the bed friction,respectively. The expression f = 8gn2 R1/3 h proposed by Knight et al.^[24] and Huai et al.^[25] was used to predict the friction factor in the main channel,where n is Manning’s coefficient.

The velocity parameters Γ_mc(11) and Γmc are calculated by Eqs. (9) and (11),respectively. Therefore,we can see the slight difference between Γ_mc(11) and Γmc due to the ignored contributions of Term 3 and Term 5. Apparently,Γ_mc(11) and Γmc are lateral variables,and are not easy for practical applications. For the convenience of applications,the averaged values of Γ_mc(11) and Γmc,i.e.,Γ_mc(11a) and Γ_mc(a),are used,and the effect of their differences on the predictions will be discussed later.

4 Applications

Four cases with different flow depths and river conditions are selected here to verify the predictive capability of the proposed model in the meandering compound channel. The experimental data of MN2 and MN3 are recorded in a large-scale channel (B = 4 m) with S = 0.001, and the details of the experimental parameters are shown in Section 2. The relative flow depth (D_r) is 0.35 for MN2,and is 0.25 for MN3. The measurements of g4_7 and g4_10 by Spooner^[3] are taken in a relatively small channel with the width of 2.4 m and S = 0.002. The details of the flow parameters are introduced in the foregoing section. The values of D_r in g4_7 and g4_10 are 0.3 and 0.45,respectively. The values of s in Ref. ^[3] and our experiments are 1.384 and 1.381,respectively,which are almost identical.

Figure 4 shows all the experimental data and predictions. From this figure,we can see that the predictions in the meandering main channel with different hydraulic conditions agree well with four groups of experimental data,i.e.,MN2,MN3,g4_7 and g4_10. To show the accuracy of this model,the averaged absolute relative error (Ψ_r) is defined as follows:

where (U_d)calculated is the calculated depth-averaged velocity,(U_d)_measured is the measured depth-averaged velocity,and N is the number of the measured depth-averaged velocity for each case.

Table 3 shows the values of Γ _mc(11a) and Γ_mc(a) and the values of their corresponding Ψ_r in four cases,where percentages in the parentheses are the values of the averaged absolute relative error (Ψ_r). From the table,we can see that the difference between Γ_mc(11a) and Γ_mc(a) is quite small,which suggests the unimportance of two shear stress terms in Eq. (9),and demonstrates that the assumption of ignoring these two terms is reasonable. In all the cases,the predictions with Γ _mc(11a) and Γ_mc(a) seem almost identical due to the nearly same values of Ψ_r. For all cases,the values of Ψ_r are always small,and the largest Ψ_r is only 4.68% in g4 7,indicating the good capability of this model in predicting the depth-averaged velocity in the meandering main channel. Although the values of Γ_mc(11a) and Γ_mc(a) are a little different,the depthaveraged velocities obtained with Γmc(11a) are still good enough to meet the practical purpose in engineering requirements.

5 Discussion

In the foregoing section,we demonstrated that two shear stress terms are small enough to be ignored in Eq.(9),and the back calculations show that the values of Γ_mc(11a) and _mc(a) are approximately identical. In this section,the determination of Γ_mc(11a) will be discussed in detail.

Based on the experimental data from Spooner^[3] and this paper,the values of _mc(11a) are back calculated. The results are shown in Fig. 5. From the figure,we can see that Γ_mc(11a) can be described as a second-order function of D_r. Therefore,in both channels,the relations shown in Fig. 5 may be used to predict _mc(11a) at other flow depths,and thus the predictive depth-averaged velocities can be obtained. If only the lateral distribution of the depth-averaged velocity is known in a new meandering channel,Eq.(11) can be used to back calculate Γ_mc(11), and the corresponding Γ_mc(11a) can be obtained. Then,the relation between Γ_mc(11a) and D_r can be obtained. Once this has been done,this model can predict the lateral distribution of the depth-averaged velocity at other flow depths for the same geometric and hydraulic condition.

It is worth noting the limitation of the proposed model. Since this method is derived from the momentum equation and the flow continuity equation under the quasi-uniform flow condition, it is necessary to investigate its feasibility under other flow conditions. In the meandering compound channel,the growth and decay of the secondary flows in the meandering main channel are quite complicated,particularly at the cross-over sections,where the generation mechanism of the secondary flows is different from that at the apex section^[23]. Therefore,this method is only suitable for predicting the depth-averaged velocity at the apex sections. In addition,in some cases,we cannot obtain the lateral distribution of the depth-averaged velocity. Therefore, the calibration may be a method to determine the value of Γ_mc(11a).

6 Conclusions

Three groups of experiments are conducted in a large scale meandering compound channel with smooth floodplains. The detailed velocity components are obtained at the apex sections. In the meandering main channel,the depth-averaged governing equation with curvilinear coordinates is derived from the momentum equation and the flow continuity equation under the quasi-uniform flow condition. According to the magnitude analysis of the five terms in Eq. (9), two lateral shear stress terms are ignored due to their minor contributions. Then,a simple but applicable model is obtained,and its analytical solution with a simple structure is presented. The predictive depth-averaged velocity is obtained when the lateral location is given. Four groups of experimental data from different sources are selected to verify the capability of this model for predicting the depth-averaged velocity in the meandering main channel. The predictions with the velocity parameters (Γ_mc(11a) and Γ_mc(a)) are almost identical. The averaged absolute relative error (Ψ_r) is presented to evaluate the predictive accuracy of this model,and the obtained values of Ψ_r in all cases are low,which demonstrates that the proposed model has a good predictive capability. The determination of Γ_mc(11a) is crucial for the practical application in a new meandering channel,and the initial calibration of the relation of Γ_mc(11a) and D_r is required. The results show that,this simple model can predict the depth-averaged velocity at other flow depths with the same geometry. However,this model is only suitable at the apex sections in the meandering compound channel under the quasi-uniform flow condition.