Numerical modelling of bed topography and bank erosion along tree-lined meandering rivers

Numerical modeling of vegetation effects on open channel flow can follow one of three approaches. Each approach allows a specific range of flow features to be simulated. Computational hydraulics models can be constructed to solve one-dimensional (1D) averaged flow momentum and continuity equations. These models can simulate the effects of vegetative resistance on bulk flow velocity and depth (de Saint-Venant equations). Computational fluid dynamics (CFD) models can be constructed to solve the 1D to 3D steady Reynolds-averaged-Navier-Stokes (RANS) equations. These models can resolve local flow and turbulence features of the temporally averaged turbulent flow field. Finally, unsteady RANS (URANS) and Large eddy simulation CFD models can be constructed to solve the unsteady 3D Navier-Stokes equations. These models can provide a complete description of the instantaneous unsteady 3D turbulent flow field, capturing organized large-scale unsteadiness and asymmetries (coherent structures) resulting from flow instabilities. The characterization of vegetative flow resistance in these models has and will continue to command the attention of both researchers and practitioners alike. For flow through vegetation, where the ratio of plant height K to flow depth d is greater than 0.5, resistance is generally due more to form drag of the vegetation than from bed shear. Emergent vegetation can also induce wave resistance from free surface distortion. Plant properties that affect form drag include the ratio K/d, the relative submergence (K ? d), plant density, distribution, and flexibility. Further complicating matters, unsteady nonuniform flow conditions often prevail, wake interference effects can reduce drag, and a variety of different riparian plant species are typically found in combination, which causes the spatial distribution of plant properties to vary greatly. While it is important to consider the various complexities of flow resistance encountered in fluvial channels, most of our current knowledge on vegetative flow resistance is derived from laboratory flume experiments of steady fully developed flow through simulated vegetation of uniform density within rigid boundary rectangular flumes. These investigations have related vegetative resistance parameters, such as drag coefficients, Manning's n values, and friction factors f, to plant properties, including height, density, and flexibility [e.g. Kouwen and Unny, 1973; Kouwen and Fathi-Moghadam, 2000; Wu et al., 2000; Stone and Shen, 2002]. Presently, computational hydraulics and steady RANS models are the most practical approaches for high Reynolds number fluvial hydraulics applications despite the rapid advancements in computational power and numerical algorithm development. Computational hydraulics models, although limited to the computation of bulk flow properties, are usually sufficient for flood studies. For these models, the bulk flow resistance parameter (e.g., Manning's n or the Darcy-Weisbach friction factor, f) can be modified to account for the measurable physical properties of vegetation based on empirical formulas [Darby, 1999]. Although computationally more intensive, steady RANS models allow resolution of the time-averaged turbulent flow field by adding source terms to the RANS and turbulence transport equations to account for vegetative drag effects. Steady RANS models have simulated 1D laboratory flume flows through simulated rigid vegetation corresponding to the laboratory measurements reported by Shimizu and Tsujimoto [1994] and López and García [1997, 1998; see also López and García, 2001; Neary, 2000, 2003; Choi and Kang, 2001]. Tsujimoto and Kitamura [1998] have incorporated a stem deformation model to extend 1D RANS simulations to flexible vegetation. Naot et al. [1996] and Fischer-Antze et al. [2001] have developed 3D RANS models for vegetated flows in compound channels with vegetation zones in riparian areas and flood plains. These models have enabled prediction of the effects of vegetation on sediment transport in fluvial channels [e.g., Okabe et al., 1997; López and García, 1998]. Mean flow features resolved by the steady RANS models include: (1) the suppression of the streamwise velocity profile in the vegetated zone, (2) the inflection of the velocity profile at the top of the vegetation zone, and (3) the vertical distribution of the streamwise Reynolds stress (turbulent shear), with its maximum value at the top of the vegetation zone. However, for some of the experimental test cases, these models have been less successful at predicting the streamwise turbulence intensity. Also, the bulge in the velocity profile that is sometimes present near the bed cannot be resolved. This feature has been observed for some test cases reported by Shimizu and Tsujimoto [1994] and Fairbanks and Diplas [1998] despite a uniform vertical plant density distribution. The present limitations of the RANS models are due mainly to spatial and temporal averaging, and possibly failure to model the effects of turbulence anisotropy. Some of these deficiencies may be offset somewhat through the treatment of the drag and weighting coefficients in the governing equations that account for vegetative drag effects. However, adopting non-universal drag coefficients or non-theoretical based weighting coefficients to make up for model deficiencies is not particularly desirable [see López and García, 1997; Neary, 2003]. The 1D RANS models eliminate streamwise or spanwise gradients in the flow field and vegetation layer by spatial averaging. The 3D models, while not spatial averaging, distribute the drag uniformly throughout the vegetation layer by introducing body force terms in the RANS equations. To date, neither 1D nor 3D models have actually simulated flow around individual stems. Due to this simplification, streamwise vortices (secondary motion), a suspected mechanism for momentum transfer that produces the near bed velocity bulge [Neary, 2000, 2003], cannot be simulated with any of the present RANS models. As a result of time averaging, RANS models also cannot capture the organized large-scale unsteadiness and asymmetries (coherent structures) resulting from turbulent flow instabilities due to unsteady shear and pressure gradients induced by vegetation. These coherent structures include: (1) the transverse and other secondary vortices described by Finnegan [2000], which occur at the top of the vegetation layer as a result of a Kelvin-Helmholtz instability due to the inflection of the streamwise velocity profile, and (2) 3D vortices produced by the complex interaction of the approach flow with the stem (e.g., horseshoe and necklace vortices) and the oblique vortex shedding in the wake of the stem due to spanwise pressure gradients. These unsteady vortices would also contribute, or possibly play a dominant role, in redistributing momentum and producing the near bed velocity bulge. The use of Reynolds stress transport (RST) modeling to account for turbulence anisotropy and its effects has received only limited numerical investigation [Choi and Kang, 2001] and its benefits are not yet apparent. The laboratory experiments by Nezu and Onitzuka [2001] demonstrate that riparian vegetation has significant effects on secondary currents due to turbulence anisotropy, which increases with Froude number. However, coherent structures may account for a significantly larger percentage of the total Reynolds stresses and anisotropy [Ge et al., 2003]. Under such circumstances, RST modeling would only have limited value. Future numerical modeling efforts will focus on advanced CFD modeling techniques-namely statistical turbulence models that directly resolve large scale, organized, unsteady structures in the flow and advanced numerical techniques for simulating flows around multiple flexible bodies. These would include unsteady 3D Reynolds-averaged Navier-Stokes models [URANS; Paik et al., 2003; Ge et al., 2003] and large eddy simulation models [Cui and Neary, 2002]. Such techniques will elucidate the large-scale coherent structures described above, their important role in vegetative resistance, and their interaction and feedback with Reynolds stresses and lift forces that initiate sediment transport and bed form development