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Modelling open-channel flow with rigid vegetation based on two-dimensional shallow water equations using the lattice Boltzmann method
Ecological Engineering 106 (2017) 75–81
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Ecological Engineering journal homepage: www.elsevier.com/locate/ecoleng
Modelling open-channel flow with rigid vegetation based on two-dimensional shallow water equations using the lattice Boltzmann method Zhonghua Yang a , Fengpeng Bai a,∗ , Wenxin Huai a , Ruidong An b , Hanyong Wang a a b
State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065, China
a r t i c l e
i n f o
Article history: Received 14 February 2017 Received in revised form 3 May 2017 Accepted 22 May 2017 Keywords: Lattice Boltzmann Flow-vegetation interactions Open channel Drag force Large eddy simulation
a b s t r a c t A two-dimensional lattice Boltzmann model with a D2Q9 lattice arrangement is developed to simulate the flow-vegetation interactions in an open channel. The rigid vegetation is modelled as vertical cylinders. A formula of drag force induced by the rigid vegetation is included in the momentum equations as a sink term. The large eddy simulation technique is adopted to simulate turbulent flows. External forces, such as bed friction and drag force, are discretised with a centred scheme. The mixed scheme of no-slip and slip boundary conditions is considered to correctly describe the interaction between the fluid and the boundary wall. The proposed lattice Boltzmann model was used to simulate two experimental results in partially vegetated channels. The results show that the presented model can simulate the vegetated channel flows correctly. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Vegetation growing on open channels, floodplains and coasts are responsible for ecological and hydraulic effects and biological processes in river and coastal systems. Vegetation can increase bed roughness, reduce flow velocity and channel conveyance capacity (Nepf, 1999; Wu et al., 1999; Devi and Kumar, 2016). Also, vegetation is an important driver of river ecosystem change, and it is a sensitive and key environmental factor in the river ecosystem. Thus, vegetation plays an important role in stabilising the shoreline, mitigating fluvial flood risk and increasing the ecological value for ecological engineering and restoration. In recent years, a considerable amount of research has been devoted to study flow-vegetation interactions; such research includes laboratory experiments, field measurements, analytical solutions and numerical models. An analytical model for hydraulic roughness with submerged vegetation was presented by Klopstra et al. (1996) by dividing the water body for open channel flow into vegetation layer and surface free water layer. Through a laboratory study, Stone and Shen (2002) showed that flow resistance varies with flow depth, stem concentration, stem length and stem diameter. Huai et al. (2009) proposed a three-layer model for the
∗ Corresponding author. E-mail address: [email protected] (F. Bai). http://dx.doi.org/10.1016/j.ecoleng.2017.05.039 0925-8574/© 2017 Elsevier B.V. All rights reserved.
vertical velocity profile with rigid vegetation in open channel flows, and the flow body was distributed into an upper non-vegetated layer, an outer layer and a bottom layer within vegetation. Complex flow-vegetation interactions have been simulated with various numerical models, in which a formula of drag force induced by the rigid vegetation is included in the momentum equations as a sink term. The commonly used numerical approaches include the finite volume scheme based on the Boussinesq wave equations (Augustin et al., 2009; Huang et al., 2011; Kuiry et al. 2011). Reza and Wu (2014) presented a three-dimensional model for solving Reynolds-averaged Navier–Stokes equations by using the finite volume method. The depth-averaged two-dimensional model is another popular method (Struve et al., 2003; Wu and Marsooli, 2012; Guan and Liang, 2017). Weissteiner et al. (2015) studied the spatial–structural properties of woody riparian vegetation under hydrodynamic loading with 3D tree models. New alternatives are always sought to simulate flows considering the effect of vegetation. The lattice Boltzmann method (LBM) is a relatively new discrete numerical approach that has elicited increasing attention recently. Unlike conventional numerical methods, the LBM describes macroscopical fluid flows from microscopic flow behaviour through particle distribution functions. The advantages of the LBM, such as simplicity, efficiency and easy treatment of boundary conditions, in simulating fluid flows have been demonstrated (Zhou, 2004; Li and Huang, 2008; Fernandino et al. 2009). Without considering the bed slope, Jiménez-Hornero et al. (2007)
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Z. Yang et al. / Ecological Engineering 106 (2017) 75–81
Fig. 1. Sketch of flow over vegetation: (a) flow over submerged vegetation; (b) flow over unsubmerged vegetation.
developed a two-dimensional lattice model to describe the influence of vegetation on the turbulent flow by using a semi-slip boundary condition. Considering the vegetation as solid boundaries in flows, Gac (2014) presented a three-dimensional lattice model and computed the vertical velocity profile in an open channel flow. This paper aims to develop a lattice Boltzmann model on the basis of two-dimensional shallow water equations to simulate the flow-vegetation interactions in open channels with large eddy simulation (LES) technique. The rigid vegetation is treated as vertical cylinders. The drag force caused by the vegetation is considered. A mixed scheme of no-slip and slip boundary conditions is introduced. To validate the proposed model, two laboratory cases are applied by comparing the numerical predictions with experimental results. The rest of this paper is organised as follows: Section 2 presents the governing equations and the LBM. Section 3 presents an evaluation of the scheme’s performance in two laboratory cases. Section 4 provides the conclusions.
Sbi =
gn2 h1/3
ui
uj uj
(3)
where n is the Manning’s coefficient. 2.2. Drag force caused by rigid vegetation For the rigid vegetation, a common approach is to treat it as vertical cylinders (Bennett et al., 2008; Guan and Liang, 2017; Wu and Marsooli, 2012). For the submerged vegetation, the velocity in the plant layer is smaller than that in the upper water layer due to the effect of drag force (Fig. 1a). In case of unsubmerged vegetation, the magnitude of velocity in the entire water-vegetation body is significantly affected (Fig. 1b). The drag force can be expressed as (Morison et al., 1950) Svi =
1 Cd huvi uvj uvj 2
(4)
where Cd is the drag force coefficient and usually in the range of 0.8 and 3.5; uvi denotes the apparent velocity on the vegetation elements in the i direction; and is the projected area of vegetation normal to the streamwise and given by
2. Mathematical formulas 2.1. Governing equations The natural flow through vegetation is usually unsteady and turbulent. In practice, the time- and space-averaged characteristic is mostly considered (Liang and Marche, 2009; Hou et al., 2013). The two dimensional shallow water equations can be derived through an in-depth integration of the Reynolds-averaged Navier–Stokes equations. Considering the flow-vegetation interactions, the nonlinear shallow water equations can be written in a tensor form as follows:
∂h ∂(huj ) =0 + ∂t ∂xj
bed shear stress term in i direction and is expressed as a Manning formula
(1)
2 ∂(hui ) ∂(hui uj ) ∂h ∂ (hui ) ∂z + gh = (v + ve ) − gh b − Sbi − Svi (2) + ∂t ∂xj ∂xi ∂xj xj ∂xi
where the Einstein summation convention over Latin indices is used; h is the water depth; t denotes the time; ui is the depthaveraged velocity; v and ve stand for the kinematic and eddy viscosity, respectively; zb is the bed elevation above datum; Svi is the drag force term caused by the rigid vegetation; and Sbi is the
=
4˛v c dv
(5)
where ␣v is a vegetation element shape factor; c is the vegetation density in vegetation zones; and dv represents the diameter of vegetation element. The drag coefficient Cd can be selected as a constant in the range of 0.7 and 3.5 (García et al., 2004; Wu et al., 2005; López and García, 2001). For unsubmerged vegetation, the apparent flow velocity on the vegetation uvi in Eq. (4) is equal to the depth-averaged velocity ui in Eq. (2). For submerged vegetation, uvi is the averaged velocity in the vegetation layer. Stone and Shen (2002) presented a solution for uvi in case of submerged vegetation 1/2
uvi = v ui (
hv ) h
(6)
where v is a coefficient and is defined as 1.0 in this paper, and hv is the vegetation height. 2.3. Lattice boltzmann method In this paper, we consider the LBM to solve the two-dimensional nonlinear shallow water equations with turbulence considering the effect of rigid vegetation. The D2Q9 (shown in Fig. 2) lattice pattern
Z. Yang et al. / Ecological Engineering 106 (2017) 75–81
77
Fig. 3. Distribution functions at the boundaries.
the LBM in the way proposed by Liu et al. (2010). The total relaxation time parameter t is
Fig. 2. D2Q9 lattice pattern.
t =
is frequently used in the fluid field in which D means the number of dimensions and Q denotes the number of particles considered. On the basis of the theory of LBM, a streaming step and a collision step are involved. In the streaming step, the particles move to the neighbouring lattice points in their directions and at their velocities. In the collision step, the Bhatnagar–Gross–Krook model is used due to its simplicity and efficiency. Combining the two steps, the LBM evolution equation is f˛ (x + e␣ t, t + t) = f˛ (x, t) −
t 1 eq f˛ (x, t) − f˛ (x, t) + e˛i Fi t N˛ e2 (7) eq f˛ is
ij
=
2 + 18Cs2 /(e2 h)
+
ij
ij
2 eq
e˛i e˛j (f˛ − f˛ )
(12) (13)
˛
where is the single-relaxation time, and Cs is the Smagorinsky constant. The viscosity of the fluid is calculated by
vt = v + ve =
e2 t (2t − 1) 6
(14)
The external forces Fi in Eq. (7) are the source term in the momentum equation in Eqs. (2) and is expressed as Fi = −gh
∂zb gn2 1 − ui uj uj − Cd huvi uvj uvj 2 ∂xi h1/3
(15)
where f␣ is the distribution function of particles; the local equilibrium distribution function; x is the space vector in Cartesian coordinates; e = x/t; x is the lattice size; t is the time step; t is the total relaxation time parameter; Fi is the external forces; and N␣ is a constant and defined by the lattice pattern
The centred scheme is applied in this study which has a second order in space and time. The external force term is evaluated at the mid-point between the lattice point and its neighbouring lattice point as
N˛ =
F˛i = F˛i (xi +
1 e˛i e˛i e2
(8)
˛
e␣ is the particle velocity along the ␣ direction and is defined for the D2Q9 lattice pattern as
e˛ =
⎧ (0, 0), ˛=0 ⎪
(˛ − 1) ⎪ ⎪ (˛ − 1) ⎨ e cos
, sin
4
4
,
˛ = 1, 3, 5, 7
⎪
⎪ ⎪ ⎩ √2e cos (˛ − 1) , sin (˛ − 1) , ˛ = 2, 4, 5, 6 4
(9)
4
For the nonlinear shallow water equations [Eqs. (1) and (2)], the eq local equilibrium distribution function f˛ is expressed as (Zhou, 2004)
⎧ 5gh2 h− ⎪ ⎪ 6e2 ⎪ ⎨ gh
eq
f˛ =
6e2 ⎪ ⎪ ⎪ ⎩ gh
+
24e2
2h − ui ui , 3e2
h h h e˛i ui + e˛i e˛j ui uj − ui ui , 3e2 2e4 6e2
+
˛=0 ˛ = 1, 3, 5, 7
h h h e˛i ui + e˛i e˛j ui uj − ui ui , ˛ = 2, 4, 6, 8 12e2 8e4 24e2
The macroscopic variables of water depth h and flow velocity ui can be calculated from the distribution function h=
˛
f˛ , hui =
e˛i f˛
(11)
˛
To simulate the turbulent flows, the LES that uses the space-filtered governing equations with a subgrid-scale stress model is adopted to
(16)
2.4. Boundary conditions As shown in Fig. 3, at the boundaries, the distribution functions shown in solid lines can be obtained after the streaming step, while the distribution functions shown in dashed lines cannot be determined from the internal lattice nodes. In this paper, the method proposed by Liu et al. (2012) is applied at the inlet and outlet boundaries. In this section, the no-slip and slip boundary conditions are introduced at the upper and lower boundaries. Finally, a mixed scheme of non-slip and slip boundary conditions is given. 2.4.1. Inflow/outflow conditions At the inlet, if the unit discharge hu is given and the v = 0 is assumed, the unknown distributions can be determined as f1 = f5 +
(10)
1 e t, t) 2 ˛i
2hu 3e
(17)
f2 =
hu f7 − f3 + f6 + 6e 2
(18)
f8 =
hu f3 − f7 + f4 + 6e 2
(19)
Similarly, at the outlet, if a fixed water depth is given and a zerogradient condition is set for u, the unknown distributions f4 , f5 , f6 are expressed as f5 = f1 −
2hu 3e
(20)
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Z. Yang et al. / Ecological Engineering 106 (2017) 75–81
2.4.4. Mixed scheme of no-slip and slip boundary conditions In real cases, the interaction between the fluid and the boundary wall cannot be correctly modelled by using only no-slip or slip boundary conditions. Here, a mixed scheme is presented. For the upper boundary, the unknown distribution functions f6 , f7 and f8 are expressed as f6 = rf4 + (1 − r)f2 , f7 = f3 ,
(27)
f8 = rf2 + (1 − r)f4 in which r is a coefficient(0 ≤ r ≤ 1). At the lower boundary, the unknown distribution functions f2 , f3 and f4 can be determined as Fig. 4. Vegetation arrangement of the non-symmetrical compound cross-section.
f2 = rf8 + (1 − r)f6 , f3 = f7 ,
hu f7 − f3 f4 = − + f8 + 6e 2
(21)
hu f3 − f7 + f2 + 6e 2
(22)
f6 = −
2.4.2. No-slip boundary condition The basic idea of the no-slip boundary condition, which is also called the bounce-back scheme, is that an incoming particle towards the boundary is bounced back into the fluid. At the upper boundary in Fig. 3, the incoming known distribution functions f2 , f3 and f4 are bounced back by the boundary, then the unknown distribution functions f6 , f7 and f8 can be determined as f6 = f2 , f7 = f3 , f8 = f4
(23)
Similarly, at the lower boundary, the unknown distribution functions f2 , f3 and f4 can be determined as f2 = f6 , f3 = f7 , f4 = f8
(24)
(28)
f4 = rf6 + (1 − r)f8 When r = 0, the mixed scheme becomes a no-slip boundary condition. When r = 1, the mixed scheme becomes a slip boundary condition. The solution of the Lattice Boltzmann model to simulate the flow-vegetation interactions is now summarized as follows: (1) given the initial macroscopic variables (water depth, bed elevation and velocity), (2) impose the inflow/outflow conditions with Eqs. (17)–(22) and the wall boundary conditions with Eqs. (27) and (28), (3) compute the external force Fi with Eqs. (15) and (16), (4) obtain the local equilibrium distribution (Eq. (10)) and compute the distribution function (Eq. (7)) with the total relaxation time parameter, (5) update the depth and the velocity according to Eq. (11), (6) return to the step (2) and repeat the above procedure until a solution is obtained.
2.4.3. Slip boundary condition The slip boundary condition means that no momentum exchange occurs between the fluid and the boundary wall. At the upper boundary, the unknown distribution functions f6 , f7 and f8 are determined as
3. Numerical tests
f6 = f4 , f7 = f3 , f8 = f2
The experiments were conducted by Pasche and Rouvé (1985) to study the flow characteristics of compound channels with vegetation. Circular rigid wood dowels with a diameter of 0.012 m are used to represent the vegetation. The experiments were performed in a recirculating flume with a length of 25.5 m and a width of 1 m. For more conformity with natural rivers, the main channel
(25)
At the lower boundary, the unknown distribution functions f2 , f3 and f4 can are expressed as f2 = f8 , f3 = f7 , f4 = f6
(26)
3.1. Steady flow over a non-symmetrical compound-channel with vegetated flood plains
Fig. 5. Comparison of the simulated results and the measured results with different vegetation densities: (a) Case 1 (c = 0.013); (b) Case 2 (c = 0.025).
Z. Yang et al. / Ecological Engineering 106 (2017) 75–81
and flood plain are connected with a sloping bank. The vegetation arrangement of the cross-section is shown in Fig. 4. The Manning’s coefficient n = 0.01 s/m1/3 in the main channel and bank. The initial values of water depth are 0.2 and 0.076 m in the main channel and flood plain, respectively. The initial velocity is zero in the whole domain. The inflow discharge of 0.0345 m3 /s is given at the upstream boundary to drive the flow. Two experiments are considered in this paper. In Case 1, the vegetation density c is 0.013, and the bed slope is 0.001. In Case 2, the vegetation density c is 0.025, and the bed slope is 0.0005. In the computation, the vegetation element shape factor ␣v = 1.0 and drag coefficient Cd = 1.0. The cell size of x = y = 0.02 m is used, and the time step t = 0.01 s. The single-relaxation time = 0.52, and a moderate Cs = 0.3 is selected. For the upper and lower boundary conditions, the mixed coefficient r = 0.6. Fig. 5 shows the comparison between the simulated cross-section velocity profiles and the laboratory measurements; the comparison indicates satisfactory agreement. The dashed line denotes the edge of the vegetation zone. The velocity in the flood plain with vegetation is significantly smaller than that in the main channel, and a large velocity gradient exists in the area between the vegetation and the non-vegetation zones. Also, with the increase in vegetation density, the velocity decreases in the flood plain.
3.2. Flow over a rectangular channel with a finite patch of vegetation Zong and Nepf (2010) conducted an experiment in a laboratory to study the flow and deposition in and around a finite patch of vegetation located at the wall of a channel. The experimental recirculating horizontal flume is 16 m long and 1.2 m wide with a weir at the downstream end to control the water depth. The bed slope of the flume is 0.00001. A patch of unsubmerged vegetation with a length of 8 m and a width of 0.4 m is constructed on one side of the channel. The distance between the vegetation patch and upstream is 2 m. Fig. 6 shows the experimental arrangement. Circular rigid wood cylinders with a diameter of 0.006 m are used to represent the vegetation. The Manning’s coefficient n is 0.0178 s/m1/3 . The initial value of water depth is 0.138 m, and the inflow discharge of 0.01932 m3 /s is given at the upstream boundary. Two experiments with different vegetation densities are considered. In Case 3, the vegetation density c is 0.02. In Case 4, the vegetation density c is 0.1. During the simulation, a 533 × 40 lattice is used in the entire domain, the mixed coefficient r = 0.8 is used for the upper and lower boundary conditions, x = y = 0.03 m, t = 0.005 s, = 0.51 and
79
Fig. 6. Experimental arrangement with a finite patch of vegetation: top view of the channel.
Cs = 0.32. The vegetation element shape factor ␣v = 1.0, and the drag coefficient Cd = 2.0. Fig. 7 presents the simulated cross-section velocity for case 3 (c = 0.02, x = 8.7 m) and case 4 (c = 0.1, x = 5.1 m), where the modelled results are found to be consistent with the measured results. The dashed line in Fig. 7 denotes the edge of the patch of vegetation. The velocity in the vegetation zone is much smaller than that in the free stream zone. Also, the velocity is uniform in most of the vegetation zone, increasing toward the free stream zone within a few centimetres of the edge. Comparisons of the calculated longitudinal and transverse velocities with experimental data are given in Fig. 8 and show good agreement between the findings. The velocity developments along the centreline of vegetation zone for two cases (y = 0.2 m) are shown in Fig. 8a (c = 0.02) and Fig. 8c (c = 0.1). The dashed line represents the beginning of the patch of vegetation. The longitudinal velocity u decreased until the divergence ended at roughly x = 4 m for high density (c = 0.1) and x = 5 m for sparse density (c = 0.02), and then the velocity in the vegetation zone became uniform. Also, in the non-vegetation zone (x < 2 m), the transverse velocity v increased and reached the peak at x = 2 m. Within the patch (x > 2 m), the transverse velocity v declined until it became uniform. Along the patch edge (y = 0.4 m), the transverse velocity development (Fig. 8b and d) is similar to that along the centreline of the vegetation zone (y = 0.2 m, Fig. 8a and c). In contrast to the longitudinal velocity at y = 0.2 m, which remained constant after the divergence ended, that at y = 0.4 m (Fig. 8b and d) began to increase beyond the region of divergence. 4. Conclusion This paper develops a two-dimensional lattice Boltzmann model with a D2Q9 lattice arrangement to simulate channel flow with
Fig. 7. Comparison of the simulated results and the measured results with different vegetation densities: (a) Case 3 (c = 0.02, x = 8.7 m); (b) Case 4 (c = 0.1, x = 5.1 m).
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Z. Yang et al. / Ecological Engineering 106 (2017) 75–81
Fig. 8. Comparison of the calculated longitudinal u and transverse velocities v with experimental data: (a) Case 3 (c = 0.02, y = 0.2 m); (b) Case 3 (c = 0.02, y = 0.4 m); (c) Case 4 (c = 0.1, y = 0.2 m); (d) Case 4 (c = 0.1, y = 0.4 m).
vegetation by using the LES technique. The proposed model is based on fully two-dimensional shallow water equations. The drag force is included in the momentum equations to consider the effect of rigid vegetation. A mixed scheme of no-slip and slip boundary conditions is presented. The new model is validated by two experiments in partially vegetated channels. The results show that the model can simulate flow-vegetation interactions in open channels correctly. Acknowledgment This work was supported by the Natural Science Foundation of China (Nos. 51679170, 51379157 and 51439007). References Augustin, L.N., Irish, J.L., Lynett, P., 2009. Laboratory and numerical studies of wave damping by emergent and near-emergent wetland vegetation. Coastal Eng. 56 (3), 332–340. Bennett, S.J., Wu, W., Alonso, C.V., et al., 2008. Modeling fluvial response to in-stream woody vegetation: implications for stream corridor restoration. Earth Surf. Processes Landforms 33 (6), 890–909. Devi, T.B., Kumar, B., 2016. Flow characteristics in an alluvial channel covered partially with submerged vegetation. Ecol. Eng. 94, 478–492. Fernandino, M., Beronov, K., Ytrehus, T., 2009. Large eddy simulation of turbulent open duct flow using a lattice Boltzmann approach. Math. Comput. Simul. 79 (5), 1520–1526. Gac, J.M., 2014. A large eddy based lattice-Boltzmann simulation of velocity distribution in an open channel flow with rigid and flexible vegetation. Acta Geophys. 62 (1), 180–198. García, M.H., López, F., Dunn, C., et al., 2004. Flow, turbulence, and resistance in a flume with simulated vegetation. Riparian Veg. Fluvial Geomorphol., 11–27. Guan, M., Liang, Q., 2017. A two-dimensional hydro-morphological model for river hydraulics and morphology with vegetation. Environ. Modell. Softw. 88, 10–21. Hou, J., Simons, F., Mahgoub, M., et al., 2013. A robust well-balanced model on unstructured grids for shallow water flows with wetting and drying over complex topography. Comput. Methods Appl. Mech. Eng. 257, 126–149. Huai, W.X., Zeng, Y.H., Xu, Z.G., et al., 2009. Three-layer model for vertical velocity distribution in open channel flow with submerged rigid vegetation. Adv. Water Resour. 32 (4), 487–492.
Huang, Z., Yao, Y., Sim, S.Y., et al., 2011. Interaction of solitary waves with emergent, rigid vegetation. Ocean Eng. 38 (10), 1080–1088. Jiménez-Hornero, F.J., Giráldez, J.V., Laguna, A.M., et al., 2007. Modelling the effects of emergent vegetation on an open-channel flow using a lattice model. Int. J. Numer. Methods Fluids 55 (7), 655–672. Klopstra, D., Barneveld, H.J., Van Noortwijk, J.M., et al., 1996. Analytical model for hydraulic roughness of submerged vegetation. Proceedings of the Congress-international Association for Hydraulic Research Local Organizing Committee of the Xxv Congress, 775–780. Kuiry, S.N., Wu, W., Ding, Y., 2011. A hybrid finite-volume finite-difference scheme for one-dimensional Boussinesq equations to simulate wave attenuation due to vegetation. In: Proceedings of the ASCE 2011 World Environmental & Water Resources Congress, Palm Springs, CA : ASCE. López, F., García, M.H., 2001. Mean flow and turbulence structure of open-channel flow through non-emergent vegetation. J. Hydraul. Eng. 127 (5), 392–402. Li, Y., Huang, P., 2008. A coupled lattice Boltzmann model for advection and anisotropic dispersion problem in shallow water. Adv. Water Resour. 31 (12), 1719–1730. Liang, Q., Marche, F., 2009. Numerical resolution of well-balanced shallow water equations with complex source terms. Adv. Water Resour. 32 (6), 873–884. Liu, H., Zhou, J.G., Burrows, R., 2010. Lattice Boltzmann simulations of the transient shallow water flows. Adv. Water Resour. 33 (4), 387–396. Liu, H., Zhou, J.G., Burrows, R., 2012. Inlet and outlet boundary conditions for the Lattice-Boltzmann modelling of shallow water flows. Prog. Comput. Fluid Dyn. Int. J. 12 (1), 11–18. Morison, J.R., Johnson, J.W., Schaaf, S.A., 1950. The force exerted by surface waves on piles. J. Petrol. Technol. 2 (05), 149–154. Nepf, H.M., 1999. Drag, Turbulence, and diffusion in flow through emergent vegetation. Water Resour. Res. 35 (2), 479–489. Pasche, E., Rouvé, G., 1985. Overbank flow with vegetatively roughened flood plains. J. Hydraul. Eng. 111 (9), 1262–1278. Stone, B.M., Shen, H.T., 2002. Hydraulic resistance of flow in channels with cylindrical roughness. J. Hydraulic Eng. 128 (5), 500–506. Struve, J., Falconer, R.A., Wu, Y., 2003. Influence of model mangrove trees on the hydrodynamics in a flume. Estuar. Coast. Mar. Sci. 58 (1), 163–171. Weissteiner, C., Jalonen, J., Järvelä, J., et al., 2015. Spatial–structural properties of woody riparian vegetation with a view to reconfiguration under hydrodynamic loading. Ecol. Eng. 85, 85–94. Wu, W., Marsooli, R., 2012. A depth-averaged 2D shallow water model for breaking and non-breaking long waves affected by rigid vegetation. J. Hydraulic Res. 50 (6), 558–575. Wu, F.C., Shen, H.W., Chou, Y.J., 1999. Variation of roughness coefficients for unsubmerged and submerged vegetation. J. Hydraul. Eng. 125 (9), 934–942.
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Glossary Symbol h: Water depth (m) t: Time (s) ui : Depth-averaged velocity in i direction (m/s) v: Kinematic viscosity coefficient (m2 /s) ve : Eddy viscosity coefficient (m2 /s) zb : Bed elevation (m)
n: Manning’s coefficient (s/m1/3 ) Cd : Drag force coefficient : Projected area of vegetation (m−1 ) c: Vegetation density dv : Diameter of vegetation element (m) ˛v : Vegetation element shape factor hv : Vegetation height (m) f˛ : Distribution function eq f˛ : Local equilibrium distribution function x: Gird size in x direction (m) y: Gird size in y direction (m) t: Time step (s) t : Total relaxation time parameter e: Particle velocity (m/s) r: Mixed coefficient
81
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Ecological Engineering journal homepage: www.elsevier.com/locate/ecoleng
Modelling open-channel flow with rigid vegetation based on two-dimensional shallow water equations using the lattice Boltzmann method Zhonghua Yang a , Fengpeng Bai a,∗ , Wenxin Huai a , Ruidong An b , Hanyong Wang a a b
State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065, China
a r t i c l e
i n f o
Article history: Received 14 February 2017 Received in revised form 3 May 2017 Accepted 22 May 2017 Keywords: Lattice Boltzmann Flow-vegetation interactions Open channel Drag force Large eddy simulation
a b s t r a c t A two-dimensional lattice Boltzmann model with a D2Q9 lattice arrangement is developed to simulate the flow-vegetation interactions in an open channel. The rigid vegetation is modelled as vertical cylinders. A formula of drag force induced by the rigid vegetation is included in the momentum equations as a sink term. The large eddy simulation technique is adopted to simulate turbulent flows. External forces, such as bed friction and drag force, are discretised with a centred scheme. The mixed scheme of no-slip and slip boundary conditions is considered to correctly describe the interaction between the fluid and the boundary wall. The proposed lattice Boltzmann model was used to simulate two experimental results in partially vegetated channels. The results show that the presented model can simulate the vegetated channel flows correctly. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Vegetation growing on open channels, floodplains and coasts are responsible for ecological and hydraulic effects and biological processes in river and coastal systems. Vegetation can increase bed roughness, reduce flow velocity and channel conveyance capacity (Nepf, 1999; Wu et al., 1999; Devi and Kumar, 2016). Also, vegetation is an important driver of river ecosystem change, and it is a sensitive and key environmental factor in the river ecosystem. Thus, vegetation plays an important role in stabilising the shoreline, mitigating fluvial flood risk and increasing the ecological value for ecological engineering and restoration. In recent years, a considerable amount of research has been devoted to study flow-vegetation interactions; such research includes laboratory experiments, field measurements, analytical solutions and numerical models. An analytical model for hydraulic roughness with submerged vegetation was presented by Klopstra et al. (1996) by dividing the water body for open channel flow into vegetation layer and surface free water layer. Through a laboratory study, Stone and Shen (2002) showed that flow resistance varies with flow depth, stem concentration, stem length and stem diameter. Huai et al. (2009) proposed a three-layer model for the
∗ Corresponding author. E-mail address: [email protected] (F. Bai). http://dx.doi.org/10.1016/j.ecoleng.2017.05.039 0925-8574/© 2017 Elsevier B.V. All rights reserved.
vertical velocity profile with rigid vegetation in open channel flows, and the flow body was distributed into an upper non-vegetated layer, an outer layer and a bottom layer within vegetation. Complex flow-vegetation interactions have been simulated with various numerical models, in which a formula of drag force induced by the rigid vegetation is included in the momentum equations as a sink term. The commonly used numerical approaches include the finite volume scheme based on the Boussinesq wave equations (Augustin et al., 2009; Huang et al., 2011; Kuiry et al. 2011). Reza and Wu (2014) presented a three-dimensional model for solving Reynolds-averaged Navier–Stokes equations by using the finite volume method. The depth-averaged two-dimensional model is another popular method (Struve et al., 2003; Wu and Marsooli, 2012; Guan and Liang, 2017). Weissteiner et al. (2015) studied the spatial–structural properties of woody riparian vegetation under hydrodynamic loading with 3D tree models. New alternatives are always sought to simulate flows considering the effect of vegetation. The lattice Boltzmann method (LBM) is a relatively new discrete numerical approach that has elicited increasing attention recently. Unlike conventional numerical methods, the LBM describes macroscopical fluid flows from microscopic flow behaviour through particle distribution functions. The advantages of the LBM, such as simplicity, efficiency and easy treatment of boundary conditions, in simulating fluid flows have been demonstrated (Zhou, 2004; Li and Huang, 2008; Fernandino et al. 2009). Without considering the bed slope, Jiménez-Hornero et al. (2007)
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Fig. 1. Sketch of flow over vegetation: (a) flow over submerged vegetation; (b) flow over unsubmerged vegetation.
developed a two-dimensional lattice model to describe the influence of vegetation on the turbulent flow by using a semi-slip boundary condition. Considering the vegetation as solid boundaries in flows, Gac (2014) presented a three-dimensional lattice model and computed the vertical velocity profile in an open channel flow. This paper aims to develop a lattice Boltzmann model on the basis of two-dimensional shallow water equations to simulate the flow-vegetation interactions in open channels with large eddy simulation (LES) technique. The rigid vegetation is treated as vertical cylinders. The drag force caused by the vegetation is considered. A mixed scheme of no-slip and slip boundary conditions is introduced. To validate the proposed model, two laboratory cases are applied by comparing the numerical predictions with experimental results. The rest of this paper is organised as follows: Section 2 presents the governing equations and the LBM. Section 3 presents an evaluation of the scheme’s performance in two laboratory cases. Section 4 provides the conclusions.
Sbi =
gn2 h1/3
ui
uj uj
(3)
where n is the Manning’s coefficient. 2.2. Drag force caused by rigid vegetation For the rigid vegetation, a common approach is to treat it as vertical cylinders (Bennett et al., 2008; Guan and Liang, 2017; Wu and Marsooli, 2012). For the submerged vegetation, the velocity in the plant layer is smaller than that in the upper water layer due to the effect of drag force (Fig. 1a). In case of unsubmerged vegetation, the magnitude of velocity in the entire water-vegetation body is significantly affected (Fig. 1b). The drag force can be expressed as (Morison et al., 1950) Svi =
1 Cd huvi uvj uvj 2
(4)
where Cd is the drag force coefficient and usually in the range of 0.8 and 3.5; uvi denotes the apparent velocity on the vegetation elements in the i direction; and is the projected area of vegetation normal to the streamwise and given by
2. Mathematical formulas 2.1. Governing equations The natural flow through vegetation is usually unsteady and turbulent. In practice, the time- and space-averaged characteristic is mostly considered (Liang and Marche, 2009; Hou et al., 2013). The two dimensional shallow water equations can be derived through an in-depth integration of the Reynolds-averaged Navier–Stokes equations. Considering the flow-vegetation interactions, the nonlinear shallow water equations can be written in a tensor form as follows:
∂h ∂(huj ) =0 + ∂t ∂xj
bed shear stress term in i direction and is expressed as a Manning formula
(1)
2 ∂(hui ) ∂(hui uj ) ∂h ∂ (hui ) ∂z + gh = (v + ve ) − gh b − Sbi − Svi (2) + ∂t ∂xj ∂xi ∂xj xj ∂xi
where the Einstein summation convention over Latin indices is used; h is the water depth; t denotes the time; ui is the depthaveraged velocity; v and ve stand for the kinematic and eddy viscosity, respectively; zb is the bed elevation above datum; Svi is the drag force term caused by the rigid vegetation; and Sbi is the
=
4˛v c dv
(5)
where ␣v is a vegetation element shape factor; c is the vegetation density in vegetation zones; and dv represents the diameter of vegetation element. The drag coefficient Cd can be selected as a constant in the range of 0.7 and 3.5 (García et al., 2004; Wu et al., 2005; López and García, 2001). For unsubmerged vegetation, the apparent flow velocity on the vegetation uvi in Eq. (4) is equal to the depth-averaged velocity ui in Eq. (2). For submerged vegetation, uvi is the averaged velocity in the vegetation layer. Stone and Shen (2002) presented a solution for uvi in case of submerged vegetation 1/2
uvi = v ui (
hv ) h
(6)
where v is a coefficient and is defined as 1.0 in this paper, and hv is the vegetation height. 2.3. Lattice boltzmann method In this paper, we consider the LBM to solve the two-dimensional nonlinear shallow water equations with turbulence considering the effect of rigid vegetation. The D2Q9 (shown in Fig. 2) lattice pattern
Z. Yang et al. / Ecological Engineering 106 (2017) 75–81
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Fig. 3. Distribution functions at the boundaries.
the LBM in the way proposed by Liu et al. (2010). The total relaxation time parameter t is
Fig. 2. D2Q9 lattice pattern.
t =
is frequently used in the fluid field in which D means the number of dimensions and Q denotes the number of particles considered. On the basis of the theory of LBM, a streaming step and a collision step are involved. In the streaming step, the particles move to the neighbouring lattice points in their directions and at their velocities. In the collision step, the Bhatnagar–Gross–Krook model is used due to its simplicity and efficiency. Combining the two steps, the LBM evolution equation is f˛ (x + e␣ t, t + t) = f˛ (x, t) −
t 1 eq f˛ (x, t) − f˛ (x, t) + e˛i Fi t N˛ e2 (7) eq f˛ is
ij
=
2 + 18Cs2 /(e2 h)
+
ij
ij
2 eq
e˛i e˛j (f˛ − f˛ )
(12) (13)
˛
where is the single-relaxation time, and Cs is the Smagorinsky constant. The viscosity of the fluid is calculated by
vt = v + ve =
e2 t (2t − 1) 6
(14)
The external forces Fi in Eq. (7) are the source term in the momentum equation in Eqs. (2) and is expressed as Fi = −gh
∂zb gn2 1 − ui uj uj − Cd huvi uvj uvj 2 ∂xi h1/3
(15)
where f␣ is the distribution function of particles; the local equilibrium distribution function; x is the space vector in Cartesian coordinates; e = x/t; x is the lattice size; t is the time step; t is the total relaxation time parameter; Fi is the external forces; and N␣ is a constant and defined by the lattice pattern
The centred scheme is applied in this study which has a second order in space and time. The external force term is evaluated at the mid-point between the lattice point and its neighbouring lattice point as
N˛ =
F˛i = F˛i (xi +
1 e˛i e˛i e2
(8)
˛
e␣ is the particle velocity along the ␣ direction and is defined for the D2Q9 lattice pattern as
e˛ =
⎧ (0, 0), ˛=0 ⎪
(˛ − 1) ⎪ ⎪ (˛ − 1) ⎨ e cos
, sin
4
4
,
˛ = 1, 3, 5, 7
⎪
⎪ ⎪ ⎩ √2e cos (˛ − 1) , sin (˛ − 1) , ˛ = 2, 4, 5, 6 4
(9)
4
For the nonlinear shallow water equations [Eqs. (1) and (2)], the eq local equilibrium distribution function f˛ is expressed as (Zhou, 2004)
⎧ 5gh2 h− ⎪ ⎪ 6e2 ⎪ ⎨ gh
eq
f˛ =
6e2 ⎪ ⎪ ⎪ ⎩ gh
+
24e2
2h − ui ui , 3e2
h h h e˛i ui + e˛i e˛j ui uj − ui ui , 3e2 2e4 6e2
+
˛=0 ˛ = 1, 3, 5, 7
h h h e˛i ui + e˛i e˛j ui uj − ui ui , ˛ = 2, 4, 6, 8 12e2 8e4 24e2
The macroscopic variables of water depth h and flow velocity ui can be calculated from the distribution function h=
˛
f˛ , hui =
e˛i f˛
(11)
˛
To simulate the turbulent flows, the LES that uses the space-filtered governing equations with a subgrid-scale stress model is adopted to
(16)
2.4. Boundary conditions As shown in Fig. 3, at the boundaries, the distribution functions shown in solid lines can be obtained after the streaming step, while the distribution functions shown in dashed lines cannot be determined from the internal lattice nodes. In this paper, the method proposed by Liu et al. (2012) is applied at the inlet and outlet boundaries. In this section, the no-slip and slip boundary conditions are introduced at the upper and lower boundaries. Finally, a mixed scheme of non-slip and slip boundary conditions is given. 2.4.1. Inflow/outflow conditions At the inlet, if the unit discharge hu is given and the v = 0 is assumed, the unknown distributions can be determined as f1 = f5 +
(10)
1 e t, t) 2 ˛i
2hu 3e
(17)
f2 =
hu f7 − f3 + f6 + 6e 2
(18)
f8 =
hu f3 − f7 + f4 + 6e 2
(19)
Similarly, at the outlet, if a fixed water depth is given and a zerogradient condition is set for u, the unknown distributions f4 , f5 , f6 are expressed as f5 = f1 −
2hu 3e
(20)
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2.4.4. Mixed scheme of no-slip and slip boundary conditions In real cases, the interaction between the fluid and the boundary wall cannot be correctly modelled by using only no-slip or slip boundary conditions. Here, a mixed scheme is presented. For the upper boundary, the unknown distribution functions f6 , f7 and f8 are expressed as f6 = rf4 + (1 − r)f2 , f7 = f3 ,
(27)
f8 = rf2 + (1 − r)f4 in which r is a coefficient(0 ≤ r ≤ 1). At the lower boundary, the unknown distribution functions f2 , f3 and f4 can be determined as Fig. 4. Vegetation arrangement of the non-symmetrical compound cross-section.
f2 = rf8 + (1 − r)f6 , f3 = f7 ,
hu f7 − f3 f4 = − + f8 + 6e 2
(21)
hu f3 − f7 + f2 + 6e 2
(22)
f6 = −
2.4.2. No-slip boundary condition The basic idea of the no-slip boundary condition, which is also called the bounce-back scheme, is that an incoming particle towards the boundary is bounced back into the fluid. At the upper boundary in Fig. 3, the incoming known distribution functions f2 , f3 and f4 are bounced back by the boundary, then the unknown distribution functions f6 , f7 and f8 can be determined as f6 = f2 , f7 = f3 , f8 = f4
(23)
Similarly, at the lower boundary, the unknown distribution functions f2 , f3 and f4 can be determined as f2 = f6 , f3 = f7 , f4 = f8
(24)
(28)
f4 = rf6 + (1 − r)f8 When r = 0, the mixed scheme becomes a no-slip boundary condition. When r = 1, the mixed scheme becomes a slip boundary condition. The solution of the Lattice Boltzmann model to simulate the flow-vegetation interactions is now summarized as follows: (1) given the initial macroscopic variables (water depth, bed elevation and velocity), (2) impose the inflow/outflow conditions with Eqs. (17)–(22) and the wall boundary conditions with Eqs. (27) and (28), (3) compute the external force Fi with Eqs. (15) and (16), (4) obtain the local equilibrium distribution (Eq. (10)) and compute the distribution function (Eq. (7)) with the total relaxation time parameter, (5) update the depth and the velocity according to Eq. (11), (6) return to the step (2) and repeat the above procedure until a solution is obtained.
2.4.3. Slip boundary condition The slip boundary condition means that no momentum exchange occurs between the fluid and the boundary wall. At the upper boundary, the unknown distribution functions f6 , f7 and f8 are determined as
3. Numerical tests
f6 = f4 , f7 = f3 , f8 = f2
The experiments were conducted by Pasche and Rouvé (1985) to study the flow characteristics of compound channels with vegetation. Circular rigid wood dowels with a diameter of 0.012 m are used to represent the vegetation. The experiments were performed in a recirculating flume with a length of 25.5 m and a width of 1 m. For more conformity with natural rivers, the main channel
(25)
At the lower boundary, the unknown distribution functions f2 , f3 and f4 can are expressed as f2 = f8 , f3 = f7 , f4 = f6
(26)
3.1. Steady flow over a non-symmetrical compound-channel with vegetated flood plains
Fig. 5. Comparison of the simulated results and the measured results with different vegetation densities: (a) Case 1 (c = 0.013); (b) Case 2 (c = 0.025).
Z. Yang et al. / Ecological Engineering 106 (2017) 75–81
and flood plain are connected with a sloping bank. The vegetation arrangement of the cross-section is shown in Fig. 4. The Manning’s coefficient n = 0.01 s/m1/3 in the main channel and bank. The initial values of water depth are 0.2 and 0.076 m in the main channel and flood plain, respectively. The initial velocity is zero in the whole domain. The inflow discharge of 0.0345 m3 /s is given at the upstream boundary to drive the flow. Two experiments are considered in this paper. In Case 1, the vegetation density c is 0.013, and the bed slope is 0.001. In Case 2, the vegetation density c is 0.025, and the bed slope is 0.0005. In the computation, the vegetation element shape factor ␣v = 1.0 and drag coefficient Cd = 1.0. The cell size of x = y = 0.02 m is used, and the time step t = 0.01 s. The single-relaxation time = 0.52, and a moderate Cs = 0.3 is selected. For the upper and lower boundary conditions, the mixed coefficient r = 0.6. Fig. 5 shows the comparison between the simulated cross-section velocity profiles and the laboratory measurements; the comparison indicates satisfactory agreement. The dashed line denotes the edge of the vegetation zone. The velocity in the flood plain with vegetation is significantly smaller than that in the main channel, and a large velocity gradient exists in the area between the vegetation and the non-vegetation zones. Also, with the increase in vegetation density, the velocity decreases in the flood plain.
3.2. Flow over a rectangular channel with a finite patch of vegetation Zong and Nepf (2010) conducted an experiment in a laboratory to study the flow and deposition in and around a finite patch of vegetation located at the wall of a channel. The experimental recirculating horizontal flume is 16 m long and 1.2 m wide with a weir at the downstream end to control the water depth. The bed slope of the flume is 0.00001. A patch of unsubmerged vegetation with a length of 8 m and a width of 0.4 m is constructed on one side of the channel. The distance between the vegetation patch and upstream is 2 m. Fig. 6 shows the experimental arrangement. Circular rigid wood cylinders with a diameter of 0.006 m are used to represent the vegetation. The Manning’s coefficient n is 0.0178 s/m1/3 . The initial value of water depth is 0.138 m, and the inflow discharge of 0.01932 m3 /s is given at the upstream boundary. Two experiments with different vegetation densities are considered. In Case 3, the vegetation density c is 0.02. In Case 4, the vegetation density c is 0.1. During the simulation, a 533 × 40 lattice is used in the entire domain, the mixed coefficient r = 0.8 is used for the upper and lower boundary conditions, x = y = 0.03 m, t = 0.005 s, = 0.51 and
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Fig. 6. Experimental arrangement with a finite patch of vegetation: top view of the channel.
Cs = 0.32. The vegetation element shape factor ␣v = 1.0, and the drag coefficient Cd = 2.0. Fig. 7 presents the simulated cross-section velocity for case 3 (c = 0.02, x = 8.7 m) and case 4 (c = 0.1, x = 5.1 m), where the modelled results are found to be consistent with the measured results. The dashed line in Fig. 7 denotes the edge of the patch of vegetation. The velocity in the vegetation zone is much smaller than that in the free stream zone. Also, the velocity is uniform in most of the vegetation zone, increasing toward the free stream zone within a few centimetres of the edge. Comparisons of the calculated longitudinal and transverse velocities with experimental data are given in Fig. 8 and show good agreement between the findings. The velocity developments along the centreline of vegetation zone for two cases (y = 0.2 m) are shown in Fig. 8a (c = 0.02) and Fig. 8c (c = 0.1). The dashed line represents the beginning of the patch of vegetation. The longitudinal velocity u decreased until the divergence ended at roughly x = 4 m for high density (c = 0.1) and x = 5 m for sparse density (c = 0.02), and then the velocity in the vegetation zone became uniform. Also, in the non-vegetation zone (x < 2 m), the transverse velocity v increased and reached the peak at x = 2 m. Within the patch (x > 2 m), the transverse velocity v declined until it became uniform. Along the patch edge (y = 0.4 m), the transverse velocity development (Fig. 8b and d) is similar to that along the centreline of the vegetation zone (y = 0.2 m, Fig. 8a and c). In contrast to the longitudinal velocity at y = 0.2 m, which remained constant after the divergence ended, that at y = 0.4 m (Fig. 8b and d) began to increase beyond the region of divergence. 4. Conclusion This paper develops a two-dimensional lattice Boltzmann model with a D2Q9 lattice arrangement to simulate channel flow with
Fig. 7. Comparison of the simulated results and the measured results with different vegetation densities: (a) Case 3 (c = 0.02, x = 8.7 m); (b) Case 4 (c = 0.1, x = 5.1 m).
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Fig. 8. Comparison of the calculated longitudinal u and transverse velocities v with experimental data: (a) Case 3 (c = 0.02, y = 0.2 m); (b) Case 3 (c = 0.02, y = 0.4 m); (c) Case 4 (c = 0.1, y = 0.2 m); (d) Case 4 (c = 0.1, y = 0.4 m).
vegetation by using the LES technique. The proposed model is based on fully two-dimensional shallow water equations. The drag force is included in the momentum equations to consider the effect of rigid vegetation. A mixed scheme of no-slip and slip boundary conditions is presented. The new model is validated by two experiments in partially vegetated channels. The results show that the model can simulate flow-vegetation interactions in open channels correctly. Acknowledgment This work was supported by the Natural Science Foundation of China (Nos. 51679170, 51379157 and 51439007). References Augustin, L.N., Irish, J.L., Lynett, P., 2009. Laboratory and numerical studies of wave damping by emergent and near-emergent wetland vegetation. Coastal Eng. 56 (3), 332–340. Bennett, S.J., Wu, W., Alonso, C.V., et al., 2008. Modeling fluvial response to in-stream woody vegetation: implications for stream corridor restoration. Earth Surf. Processes Landforms 33 (6), 890–909. Devi, T.B., Kumar, B., 2016. Flow characteristics in an alluvial channel covered partially with submerged vegetation. Ecol. Eng. 94, 478–492. Fernandino, M., Beronov, K., Ytrehus, T., 2009. Large eddy simulation of turbulent open duct flow using a lattice Boltzmann approach. Math. Comput. Simul. 79 (5), 1520–1526. Gac, J.M., 2014. A large eddy based lattice-Boltzmann simulation of velocity distribution in an open channel flow with rigid and flexible vegetation. Acta Geophys. 62 (1), 180–198. García, M.H., López, F., Dunn, C., et al., 2004. Flow, turbulence, and resistance in a flume with simulated vegetation. Riparian Veg. Fluvial Geomorphol., 11–27. Guan, M., Liang, Q., 2017. A two-dimensional hydro-morphological model for river hydraulics and morphology with vegetation. Environ. Modell. Softw. 88, 10–21. Hou, J., Simons, F., Mahgoub, M., et al., 2013. A robust well-balanced model on unstructured grids for shallow water flows with wetting and drying over complex topography. Comput. Methods Appl. Mech. Eng. 257, 126–149. Huai, W.X., Zeng, Y.H., Xu, Z.G., et al., 2009. Three-layer model for vertical velocity distribution in open channel flow with submerged rigid vegetation. Adv. Water Resour. 32 (4), 487–492.
Huang, Z., Yao, Y., Sim, S.Y., et al., 2011. Interaction of solitary waves with emergent, rigid vegetation. Ocean Eng. 38 (10), 1080–1088. Jiménez-Hornero, F.J., Giráldez, J.V., Laguna, A.M., et al., 2007. Modelling the effects of emergent vegetation on an open-channel flow using a lattice model. Int. J. Numer. Methods Fluids 55 (7), 655–672. Klopstra, D., Barneveld, H.J., Van Noortwijk, J.M., et al., 1996. Analytical model for hydraulic roughness of submerged vegetation. Proceedings of the Congress-international Association for Hydraulic Research Local Organizing Committee of the Xxv Congress, 775–780. Kuiry, S.N., Wu, W., Ding, Y., 2011. A hybrid finite-volume finite-difference scheme for one-dimensional Boussinesq equations to simulate wave attenuation due to vegetation. In: Proceedings of the ASCE 2011 World Environmental & Water Resources Congress, Palm Springs, CA : ASCE. López, F., García, M.H., 2001. Mean flow and turbulence structure of open-channel flow through non-emergent vegetation. J. Hydraul. Eng. 127 (5), 392–402. Li, Y., Huang, P., 2008. A coupled lattice Boltzmann model for advection and anisotropic dispersion problem in shallow water. Adv. Water Resour. 31 (12), 1719–1730. Liang, Q., Marche, F., 2009. Numerical resolution of well-balanced shallow water equations with complex source terms. Adv. Water Resour. 32 (6), 873–884. Liu, H., Zhou, J.G., Burrows, R., 2010. Lattice Boltzmann simulations of the transient shallow water flows. Adv. Water Resour. 33 (4), 387–396. Liu, H., Zhou, J.G., Burrows, R., 2012. Inlet and outlet boundary conditions for the Lattice-Boltzmann modelling of shallow water flows. Prog. Comput. Fluid Dyn. Int. J. 12 (1), 11–18. Morison, J.R., Johnson, J.W., Schaaf, S.A., 1950. The force exerted by surface waves on piles. J. Petrol. Technol. 2 (05), 149–154. Nepf, H.M., 1999. Drag, Turbulence, and diffusion in flow through emergent vegetation. Water Resour. Res. 35 (2), 479–489. Pasche, E., Rouvé, G., 1985. Overbank flow with vegetatively roughened flood plains. J. Hydraul. Eng. 111 (9), 1262–1278. Stone, B.M., Shen, H.T., 2002. Hydraulic resistance of flow in channels with cylindrical roughness. J. Hydraulic Eng. 128 (5), 500–506. Struve, J., Falconer, R.A., Wu, Y., 2003. Influence of model mangrove trees on the hydrodynamics in a flume. Estuar. Coast. Mar. Sci. 58 (1), 163–171. Weissteiner, C., Jalonen, J., Järvelä, J., et al., 2015. Spatial–structural properties of woody riparian vegetation with a view to reconfiguration under hydrodynamic loading. Ecol. Eng. 85, 85–94. Wu, W., Marsooli, R., 2012. A depth-averaged 2D shallow water model for breaking and non-breaking long waves affected by rigid vegetation. J. Hydraulic Res. 50 (6), 558–575. Wu, F.C., Shen, H.W., Chou, Y.J., 1999. Variation of roughness coefficients for unsubmerged and submerged vegetation. J. Hydraul. Eng. 125 (9), 934–942.
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Glossary Symbol h: Water depth (m) t: Time (s) ui : Depth-averaged velocity in i direction (m/s) v: Kinematic viscosity coefficient (m2 /s) ve : Eddy viscosity coefficient (m2 /s) zb : Bed elevation (m)
n: Manning’s coefficient (s/m1/3 ) Cd : Drag force coefficient : Projected area of vegetation (m−1 ) c: Vegetation density dv : Diameter of vegetation element (m) ˛v : Vegetation element shape factor hv : Vegetation height (m) f˛ : Distribution function eq f˛ : Local equilibrium distribution function x: Gird size in x direction (m) y: Gird size in y direction (m) t: Time step (s) t : Total relaxation time parameter e: Particle velocity (m/s) r: Mixed coefficient
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