Appl. Math. Mech. -Engl. Ed. 3(9), 2 3 (29) DOI:.7/s483-9-96-z c Shanghai University and Springer-Verlag 29 Applied Mathematics and Mechanics (English Edition) Two-dimensional analytical solution or compound channel lows with vegetated loodplains Wen-xin HUAI ( ), Min GAO ( ), Yu-hong ZENG ( ), Dan LI ( ) (State Key Laboratory o Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 4372, P. R. China) (Communicated by Yu-lu LIU) Abstract This paper presents a two-dimensional analytical solution or compound channel lows with vegetated loodplains. The depth-integrated N-S equation is used or analyzing the steady uniorm low. The eects o the vegetation are considered as the drag orce item. The secondary currents arealsotakenintoaccountinthegoverning equations, and the preliminary estimation o the secondary current intensity coeicient K is discussed. The predicted results or the straight channels and the apex cross-section o meandering channels agree well with experimental data, which shows that the analytical model presented here can be applied to predict the low in compound channels with vegetated loodplains. Key words compound channel, vegetation, drag orce, secondary currents Chinese Library Classiication TV33. 2 Mathematics Subject Classiication 76F99 Introduction Vegetated loodplains requently occur in natural rivers. The presence o vegetation plays an essential role in bank stabilization and ecological restoration. However, it changes the internal low structure, enhances low resistance, and reduces lood discharge. These complicate the eorts o low analysis. In recent years, researchers have conducted related numerical simulations to evaluate the eects o vegetation on lows. For example, Shimizu and Tsujimoto [] and Fischer-Antze et al. [2] used the k-ε turbulence model to simulate the behavior o turbulent lows in open channels in vegetated cases. Naot et al. [3] studied the hydrodynamic behavior o lows using a threedimensional algebra stress model (ASM). Nadaoka and Yagi [4] andsuandli [5-6] developed a large eddy simulation (LES) model or turbulent lows in partly vegetated open channels. Li and Yan [7] proposed a 3-D model to simulate the interaction o wave-current-vegetation, which was proved to be a valuable method. In the study o analytical solutions, Rameshwaran and Shiono [8] presented a quasi twodimensional model or straight overbank lows with vegetated loodplains, which was based on Received Sept. 6, 28 / Revised Jun. 29, 29 Project supported by the National Natural Science Foundation o China (Nos. 56796, 57925, and 57493) Corresponding author Wen-xin HUAI, Proessor, E-mail: wxhuai@whu.edu.cn
22 Wen-xin HUAI, Min GAO, Yu-hong ZENG, and Dan LI the Shiono and Knight method (SKM). Huai et al. [9] provided an analytical solution or the depth-averaged velocity or uniorm lows in partially vegetated compound channels, where the theory o eddy viscosity was applied, and the eect o vegetation on lows was considered as the drag orce. The model neglected the second current term and its application was only in straight channels. Ervine et al. [] presented an analytical solution to the depth-averaged velocity or straight and meandering overbank lows without vegetation by introducing a secondary current intensity coeicient. When applied to the apex cross-section o meandering channels, the predicted results do not agree well with the measured data or the cases in which the loodplains are roughed with vertical rods. This is due to the absence o the drag orce caused by vertical rods. Thereore, based on the work o Ervine et al. [], this paper presents a two-dimensional analytical model to predict the lateral variation o the depth-averaged velocity with emergent rigid vegetation on loodplains. In the streamwise direction, the depth-integrated Reynolds averaged Navier-Stokes equation is solved based on the hypothesis o the steady uniorm low, and the drag orce caused by the vegetation and the secondary currents evoked by the lateral depth variation are also considered. The model is validated by the measured data o the UK- Flood Channel Facility (UK-FCF) (see Res. [8] and []). The predictions demonstrate that the two-dimensional analytical model is capable o reproducing the behavior o overbank lows and the secondary currents. For the cases o compound channels with roughed loodplains, better results can be obtained by introducing the drag orce other than the total roughness (Ervine s method [] ). Theoretical analysis For steady uniorm lows, the Reynolds averaged Navier-Stokes equation in the streamwise direction with the eects o emergent vegetation can be simpliied as ρ (u xu y ) = ρgs ρu xu y ρu xu z z ρ u2 x 2 C dλ, () where ρ is the density o water; x, y, and z are the longitudinal, transverse, and vertical axes, respectively (see Fig. ); u x, u y,andu z are the temporal mean velocities in the x, y, and z directions, respectively; g is the gravitation acceleration; S is the longitudinal bed slope; the Reynolds stresses are τ yx = ρu xu y and τ zx = ρu xu z; u x, u y,andu z are the luctuation velocities in the x, y, and z directions, respectively; C d is the drag coeicient o vegetation; λ Temporal mean velocities: u x, u y, u z Fluctuation velocities: u',u',u' x y z z x Secondary currents Point velocity s Side slope y Depth-averaged velocity U Fig. The sketch o the vegetated channel
Two-dimensional analytical solution or compound channel lows with vegetated loodplains 23 is the vegetation coeicient: λ = D N v,whered is the rod diameter, and N v is the vegetation density per unit area. Integrating () over the low depth H, one can obtain the depth-averaged equation ρ H(u xu y ) d = ρghs + Hτ yx τ b + s 2 ρu2 2 C dλh, (2) where τ b is the bed shear stress, s is the bank slope, U is the depth-averaged velocity, τ b + H s 2 = τ zx z dz, and U = H u x dz. H Applying the eddy viscosity theory, i.e., τ yx = ρν t U, ν t = ξhu = ξh ( ) /2 U, 8 we have ( ) /2 τ yx = ρξh U U 8, where ν t is the eddy viscosity, and ξ is the eddy viscosity coeicient. Since the Darcy-Weisbach riction acter = 8τ b ρu, (2) becomes 2 ρ H(u xu y ) d ( = ρghs + ) ρξh 2 8 U U 8 ρu2 + s 2 ρu2 2 C dλh, (3) where ρ H(uxuy) d is the secondary current term. Applying the simpliied ormulation o Ervine et al. [], i.e., u x = K U, u y = K 2 U, u x u y = KU 2, (3) becomes ( ρghs + ) ρξh 2 8 U U ρhku 2 8 ρu2 + s 2 ρu2 2 C dλh =, (4) where K is the secondary current intensity coeicient. Considering the blockage eects on the low by vegetation, Rameshwaran and Shiono [8] introduced the porosity α and noted that α = N v A v, where the cross-section area o a single rod A v = πd 2 /4. In non-vegetated regions, α =. For the constant depth domain, (4) can be rewritten as ( αρghs + α ) ρξh 2 8 U U ρhku 2 8 αρu2 ρ U 2 2 C dλh =. (5) Equation (5) is a common second-order constant-coeicient linear dierential inhomogeneous equation. Its solution equals a special solution plus the general solution o the corresponding homogeneous equation. The analytical solution o (5) is given by U =(σ + C e βy + C 2 e γy ) /2, (6)
24 Wen-xin HUAI, Min GAO, Yu-hong ZENG, and Dan LI where C and C 2 are constants, β = τ 2 + τ2 2 + τ ( 2 αρ +2ρC dλh), 2τ γ = τ 2 τ2 2 + τ ( 2 αρ +2ρC dλh), 2τ τ = 2 αρξh2 8, τ αghs 2 = αρhk, σ = α 8 + 2 C dλh. In the linear side slope region, the low depth linearly changes in the lateral direction (see Fig. 2), and the low depth unction is given by Φ = H y y s. z O x Φ H s y y y Fig. 2 The sketch o the low depth untion Φ Equation (4) becomes ( ρgφs + ) ρξφ 2 8 U U ρφku2 8 ρu2 + =. (7) s2 The analytical solution o (7) is given by where C 3 and C 4 are constants, U =(ωφ+c 3 Φ θ + C 4 Φ θ2 ) /2, (8) L = ξ 2s 2 N = K s 8 2 Experimental data 8, M = ξ s 2 8 + K s, + s 2, ω = gs 8 + s ξ 2 s 2 θ = L M + (L M) 2 4LN, 2L θ 2 = L M (L M) 2 4LN. 2L 8 2K s, 2. Straight channels Rameshwaran and Shiono [8] used experimental data o UK-FCF to study the behavior o straight overbank lows with vegetated loodplains. The experiment was carried on in a channel with the length o 6 m and the width o m. The cross-section is shown in Fig. 3. Two sets o data in Table are used in this study, where Dr is the relative low depth between the main channel and loodplain (Dr = H p /H mc ).
Two-dimensional analytical solution or compound channel lows with vegetated loodplains 25 Table Parameters o straight channels Dr b/m B/m h/m s mc s p S D/mm N v/m 2.2.75 3.5.5...27 3 25 2.25.75 3.5.5...27 3 25 2 2.2 Apex cross-section o meandering channels The measured data o 6 (B37) and (B45) meandering channels are rom Re. [], which are obtained rom UK-FCF. The lume is with the length o 6 m and the width o m. The cross-section o the meandering channel and the directions o longitudinal and transverse velocities are shown in Figs. 4 and 5, respectively. The parameters o B37 and B45 cases are listed in Table 2, in which the sinuosity is the ratio o the channel length and the valley s straight length. Hmc z x s p H p O b y s mc B h Fig. 3 The cross-section o the straight channel H p H mc s 4 s 3 s 2 B s B 2 B 3 B 4 s 4 B m Fig. 4 The apex cross-section o the meandering channel Flow Apex cross-section u y u x Floodplain Main channel Floodplain Fig. 5 The sketch o longitudinal and transverse velocities at the bend apex cross-section
26 Wen-xin HUAI, Min GAO, Yu-hong ZENG, and Dan LI Table 2 Parameters o B37 and B45 Case Dr Sinuosity H mc/m Q/(m 3 s ) Floodplain side slope s 4 D/mm N v/m 2 B37.4.374.25.264.996 3 25 2 B45.25 2.4.2.5 7.2 3 25 2 3 Boundary conditions 3. Straight channels The trapezoidal compound channel shown in Fig. 3 is divided into three subareas: the main channel domain, the side slope domain between the main channel and loodplain, and the loodplain domain. The boundary conditions are as ollows: ) Symmetry condition: U =. y= 2) Continuity conditions: Every domain junction must satisy the continuity o the velocity and the velocity gradient, i.e., U i U i = U i+, = U i+. 3) The velocity must be zero at the edge o the loodplain, i.e., U =. 3.2 Apex cross-section o meandering channels According to the constant and linear low-depth domains, the channel shown in Fig. 4 is divided into several subareas. The boundary conditions are as ollows: ) Every domain junction must satisy the continuity o velocities and their gradients, i.e., U i = U i+, U i = U i+. 2) The velocity must be zero at the edge o the loodplain, i.e., 4 Estimation o parameters U =. Based on the basic hydraulic ormula, the riction actor in each domain can be determined. By using riction velocity ormulae U = ghs, =8U 2 /U 2, U = n R 2 3 J 2, the ormula o can be derived as = 8gHn2, (9) R4/3 where n is the Manning s coeicient, H is the low depth, R is the hydraulic radius, and g is the gravity acceleration. Equation (9) is validated by comparing the calculated riction actor with the measured data [8]. The parameters o Case B are shown in Table. For Case A, the loodplain is smooth,
Two-dimensional analytical solution or compound channel lows with vegetated loodplains 27 and other parameters are the same as those o Case B. Figure 6 shows that (9) reproduces a reasonable prediction. For the constant coeicient ξ in the dimensionless eddy viscosity model, dierent values are used or the main channel and loodplain. The standard value o the open channel lows ξ = κ/6 =.67 (κ is the von Karman constant) is used in the non-vegetated main channel. In the vegetated loodplain, the model in Re. [] is adopted: ξ p = ξ mc ( 2.+.2Dr.44 ). The value o ξ in the linear side slope domain is evaluated as about twice o that in the main channel...8.6 -Case B Present solution-rods-case B -Case A Present solution-smooth-case A.4.2.5..5 2. 2.5 3. 3.5 Fig. 6 Measured and calculated riction actor across a section (Dr =.2) U/(m s )..8.6.4 K mc =.% K mc =.2% K mc =.3% K mc =.4% K mc =.5% U/(m s )..8.6.4 K mc =.3% K mc =.4% K mc =.5% K mc =.6% K mc =.7%.2.2 U/(m s ).5.5 2.5 3.5 (a) Straight channels (Dr =.2).4.3.2. K mc =5% K mc =6% K mc =7% K mc =8% K mc =9% U/(m s ) 2 3 4 (b) Straight channels (Dr =.25) K mc =7% K mc =8% K mc =9% K mc =% K mc =6% 2 4 6 8.5..5 2. (c) Apex cross-section o meandering channel B37 (d) Apex cross-section o meandering channel B45.35.25.5.5 Fig. 7 Comparison o depth-averaged velocity proiles with dierent K mc values in straight channels and the apex cross-section o meandering channels
28 Wen-xin HUAI, Min GAO, Yu-hong ZENG, and Dan LI In the vegetated loodplains, the low velocities decrease or the resistance o vegetation as well as the secondary currents. So, in this region, the secondary current is ignored by assuming K p =. Reerring to the suggested value range o non-vegetated overbank lows in Re. [], i.e., K <.5% or straight channels, 2% <K<5% or the apex cross-section o meandering channels, the trial and error method is used to estimate the value o K mc. The lateral distribution o the depth-averaged velocities with dierent K mc or dierent cases as listed in Tables and 2 are shown in Fig. 7. It can be seen that the results agree well with the measured data when given an optimum K mc. The optimum values o K mc or our study are.3%,.5%, 7%, and 9%, respectively. 5 Results and analysis The lateral distribution o the depth-averaged velocity can be obtained by using the parameters estimated above. The validity o the model is assessed by applying the model to straight channels and the apex cross-section o meandering channels. 5. Straight channels Figure 8 shows the predictions or cases listed in Table, i.e., D r =.2 and D r =.25. Comparing the results when the secondary currents are ignored (K mc = ), the model considering the second currents with an optimum K mc agrees better with the experimental data... U/(m s ).8.6.4 K mc =.3% K mc = U/(m s ).8.6.4 K mc =.5% K mc =.2.2 2 3 4 (a) Dr =.2 2 3 4 (b) Dr =.25 Fig. 8 Calculated and measured depth-averaged velocity proiles in straight channels 5.2 Apex cross-section o meandering channels For B37 and B45 cases as listed in Table 2, Ervine et al. [] used a higher value o roughness coeicient or the loodplains to calculate. In this paper, the drag orce is introduced to account or the eect o vegetation. A comparison o the results o the present model, the results without consideration o secondary currents (K mc = ), and the predicted results o Ervine et al. with the measured data are shown in Fig. 9, which indicates that introducing the secondary current item and the drag orce item can provide a much better prediction. It can be seen that the value o K o the apex cross-section in meandering channels is much larger than that in straight channels. This may be due to the boundary shape changes o meandering channels when the under layer water o the main channels shits along the streamwise boundary. The internal structure o lows adjusts and the velocity ilaments bend, so that the mixing between water is strengthened. This results in a larger secondary current intensity.
Two-dimensional analytical solution or compound channel lows with vegetated loodplains 29 U/(m s ).6.5.4.3.2. 2 4 K mc = Ervine et al. [] K mc =7% 6 8 (a) B37 U/(m s ).45.35.25.5.5 K mc = Ervine et al. [] K mc =9%.5..5 2. (b) B45 Fig. 9 Calculated and measured depth-averaged velocity proiles in the apex cross-section o meandering channels 6 Conclusions (i) This paper presents a 2-D analytical solution or steady uniorm lows with the eects o emergent rigid vegetation. Preliminary estimation o the secondary current intensity coeicient K is discussed. The model is applied to straight channels and the apex cross-section o meandering channels. The comparison between the predicted results and the experimental data shows that the analytical model can provide an eective prediction or the cases studied. Thereore, the model can be applied to quasi-uniorm lows with relatively lat riverbed and regular vegetation on loodplains. (ii) Approximation and empirical ormulae are used to estimate the riction actor and the dimensionless eddy viscosity coeicient ξ, respectively. The results show that the estimation is reasonable. (iii) Adopting the simpliied ormat o Ervine et al. [], the secondary currents are considered by introducing the intensity coeicient K. Reerring to the range o non-vegetated overbank lows studied by Ervine et al., the trial and error method is used to estimate K. About the value o K, preliminary conclusions can be made here: the K value is about.5% or straight channels; about 8% or the apex cross-section o meandering channels. Furthermore, the secondary current item might be negative or positive. It is determined by the direction o low mixing, which is a relection o the depth-averaged low mixing. (iv) Taking the roughness into account, this paper introduces the drag orce term to calculate the loodplains roughed with vertical rods or meandering channels. Compared with the original method in which only the roughness is considered, the results o the present model give much better predictions. Reerences [] Shimizu, Y. and Tsujimoto, T. Numerical analysis o turbulent open-channel low over vegetation layer using a k-ε turbulence model. J. Hydrosci. Hydr. Eng. (2), 57 67 (994) [2] Fischer-Antze, T. et al. 3D numerical modelling o open-channel low with submerged vegetation. J. Hydraulic Research 39(3), 33 3 (2) [3] Naot, D., Nezu, I., and Nakagawa, H. Hydrodynamic behavior o partly vegetated open channels. J. Hydraulic Engineering, ASCE 22(), 625 663 (996) [4] Nadaoka, K. and Yagi, H. Shallow-water turbulence modeling and horizontal large-eddy computation o river low. J. Hydraulic Engineering, ASCE 24(5), 493 5 (998)
3 Wen-xin HUAI, Min GAO, Yu-hong ZENG, and Dan LI [5] Su, Xiaohui and Li, C. W. Large eddy simulation o ree surace turbulent low in partly vegetated open channels. Int. J. Numer. Methods Fluids 39(), 99 937 (22) [6] Su, Xiaohui et al. K-l LES o shallow-water turbulent low in open channels with a vegetated domain (in Chinese). Journal o Dalian University o Technology 43(2), 223 229 (23) [7] Li, C. W. and Yan, K. Numerical investigation o wave-current-vegetation interaction. J. Hydraulic Engineering, ASCE 33(7), 794 83 (27) [8] Rameshwaran, P. and Shiono, K. Quasi two-dimensional model or straight overbank lows through emergent vegetation on loodplains. J. Hydraulic Research 45(3), 32 35 (27) [9] Huai, Wenxin et al. Two dimensional analytical solution or a partially vegetated compound channel low. Appl. Math. Mech. -Engl. Ed. 29(8), 77 84 (28) DOI:.7/s483-8- 8-y [] Ervine, A. D., Babaeyan-Koopaei, K., and Robert, H. J. Sellin. Two-dimensional solution or straight and meandering overbank lows. J. Hydraulic Engineering, ASCE 26(9), 653 669 (2) [] Abril, J. B. and Knight, D. W. Stage-discharge prediction or rivers in lood applying a depthaveraged model. J. Hydraulic Research 42(6), 66 629 (24)