A depth-averaged 2-D shallow water model for breaking and non-breaking long waves affected by rigid vegetation

Authors: WU, WEIMING - University Of Mississippi; MARSOOLI, RESA - University Of Mississippi

Submitted to: Journal of Hydraulic Research IAHR
Publication Type: Peer Reviewed Journal
Publication Acceptance Date: 6/30/2012
Publication Date: 6/30/2012
Citation: Wu, W., Marsooli, R. 2012. A depth-averaged 2-D shallow water model for breaking and non-breaking long waves affected by rigid vegetation. Journal of Hydraulic Research IAHR. 50(6):558-575.

Interpretive Summary: To quantitatively assess the effectiveness and limitations of vegetation in coastal and river protection, a depth-averaged 2D shallow water model was developed to simulate the dynamic deformation processes of long waves such as tsunami and dam-break waves in vegetated water bodies. The effects of vegetation are taken into account by adding the drag and/or inertia forces in the momentum equations. The model adopts an explicit finite-volume method based on rectangular mesh, which is first-order accurate in time and second-order accurate in space. The model uses the Euler scheme to discretize the temporal terms, a stable central difference scheme for the water surface gradient terms, and the central difference scheme for the diffusion fluxes. A novel hybrid approach is established for the convection fluxes at cell faces, which evaluates the streamwise convection fluxes using the HLL approximate Riemann solver with the second-order MUSCL reconstruction and the lateral convection fluxes using the second-order upwind scheme HLPA. This hybrid approach can significantly reduce the numerical diffusion in the lateral direction. The drag force and bed resistance terms are treated semi-implicitly for better numerical stability. The developed model was tested using three sets of laboratory experiments considering vegetation: steady flow around alternate vegetation zones, solitary wave through vegetation zone in a flat-bed flume, and long wave on a partially-vegetated sloping beach, as well as two cases without vegetation effects: solitary wave runup on a sloping beach and dam-break wave over a triangular hump. The drag coefficient was calibrated as a bulk constant and also determined using the formulas of Tanino and Nepf (2008) and Kothyari et al. (2009) as functions of stem Reynolds number, Froude number, and vegetation volume fraction. The calculated flow patterns, velocity distributions, and wave heights were in good agreement with experimental observations when the calibrated bulk constant drag coefficient was used. The model with the drag coefficient determined using the two formulas provided similar reliable results in the first two cases, but less accurate results in the third case. This indicates that in the absence of measured data, one may use these formulas to estimate the drag coefficient, but caution needs to be taken for possible errors. The two cases without vegetation effects demonstrated the developed model is capable of approximately handling breaking waves. The tests also show the wave shape is distorted near the breaking and when the wave is short, due to the limitation of the shallow water model. The model was then applied to assess the effectiveness and limitations of vegetation in coastal and river protection with regard to hydrodynamics. The simulated wave runups on the beach with and without vegetation were compared, showing that it is beneficial to plant vegetation along the coastal shoreline to reduce wave runup. However, a comparison of the simulated dam-break flows in the open channel with and without vegetation on the floodplains shows that vegetation reduces the flood level in the downstream areas, but an increase in the inundation risk in the upstream certain distance. Thus, planting vegetation on floodplains in the open channel may cause conflicting impacts on the upstream and downstream areas.

Technical Abstract: This paper presents a depth-averaged two-dimensional shallow water model for simulating long waves in vegetated water bodies under breaking and non-breaking conditions. The effects of rigid vegetation are modelled in the form of drag and inertia forces as sink terms in the momentum equations. The drag coefficient is treated as a calibrated bulk constant and also determined using two empirical formulas as functions of stem Reynolds number, Froude number, and vegetation volume fraction. The governing equations are solved using an explicit finite-volume method based on rectangular mesh with the Harten, Lax, and van Leer approximate Riemann solver with second-order piecewise linear reconstruction for the streamwise convection fluxes, a second-order upwind scheme for the lateral convection fluxes, and a stable centred difference scheme for the water surface gradient terms. The model was tested using five laboratory experiments, including steady flow in a flume with alternate vegetation zones, solitary wave in a vegetated flatbed flume, long-wave runup on a partially-vegetated sloping beach, the dam-break wave overtopping an obstacle, and breaking the solitary wave on a sloping beach. The computed water levels, flow velocities, wave heights, and runups are in generally good agreement with experimental observations. The model was then applied to assess the hydrodynamic effectiveness and limitations of vegetation in coastal and river protection. It is shown that vegetation along the coastal shoreline has a positive benefit in reducing wave runup on sloping beaches, whereas vegetation in open channels causes conflicting impacts: reducing inundation in the downstream areas, but increasing flood risk in a certain distance upstream.