Numerical investigation of water wave near-trapping by rigid emergent vegetation
Introduction
Aquatic vegetation plays a key role in providing a wide range of ecosystem services. It provides different habitat for aquatic animals to promote efficiently biodiversity (Kemp et al., 2000). By means of trapping solute mass and releasing oxygen, vegetation is important to improve water quality (Cotton et al., 2006, Widdows et al., 2008). Aquatic vegetation is considered as a source of flow resistance in increasing flooding. Another issue is that vegetation changes the turbulent flow, and furthermore the distribution of deposition and erosion, which controls the morphology (Bouma et al., 2007, Rominger et al., 2010). Besides the unidirectional flows, waves over vegetation will be attenuated due to the resistance offered by the vegetation. It has been noted that coastal forests can serve as barriers against tides, storm surges and tsunami waves (Tanaka et al., 2007, Dasa and Vincent, 2009).
For investigation of flow through vegetation, circular cylinders are often used to model rigid vegetation piercing the water surface because of the good approximation of the stems. By physical model, the achievements were focused on flow and mass exchange between open water and vegetation (Tanino and Nepf, 2009), the investigation of the wake flow structures (Zong and Nepf, 2011), the drag force (Tanino and Nepf, 2008), and local sediment transport (Chen et al., 2012). Besides the physical model, the numerical model is powerful in the investigation of the turbulent flow structures. In recent decades, numerical models have been promoted to investigate the flow through cylinders, or vegetation patches. Cui and Neary (2008) applied Large Eddy Simulation (LES) to investigate flow in partially of fully vegetated channels containing a uniform layer of vegetation by introducing drag force in the momentum equations. Okamoto and Nezu (2010) promoted a Large Eddy Simulation (LES) model to investigate the flow structure and mass transport in fully vegetated channels, in which model the non-slip solid boundary condition was used to model the vegetation with fine grids. Nicolle and Eames (2011) used two-dimensional cylinders to model a finite porous obstruction, which focused on the local and global effect of an isolated group of cylinders. Baranya et al. (2012) used 3D RANS model based on nested grids to simulate flow around circular piers. The small diameter of the vegetation stem requires high grid resolution, meanwhile, the number of the stem is large, which increases the computational cost. In order to save computational cost, a porous model is popular in numerical simulation of vegetation flow (Li and Yan, 2007, Li and Zeng, 2009, Li and Zhang, 2010, Gao et al., 2011, Ma et al., 2013, Maza et al., 2013).
Compared to researches of unidirectional flow through cylinder array, waves interacting with solid bodies are often considered in ocean engineering, for example, the floating bodies supported by columns. Under the simplifying assumptions of potential flow, the nonlinear interaction of waves with floating bodies is investigated. The analytical model is powerful to reveal some critical parameters identifying the characteristic wave motion including wave diffraction, attenuation and near trapping (Linton and Evans, 1990, Evans and Porter, 1997, Maniar and Newman, 1997, Malenica et al., 1999, Ohl et al., 2001, Meylan and Taylor, 2009, Kashiwagi, 2017). The wave trapping commonly occurs when wave passing around structures with a characteristic geometry. The wave height is commonly amplified under trapping waves, which induces the wave force amplification. With the wave trapped, not only the wave energy focuses in a local region, but also the net flow rate change routes. The wave near trapping is concisely considered as a kind of wave resonance. For waves through vegetation, the Keulegan-Carpenter (KC) number based on the vegetation stem is much larger, and the fluid viscous effect is relatively important. The commonly used potential model is invalidated. Considering the fluid viscosity, the flow through a cylinder group is much more complicated. Whether the wave “near trapping” occurring has not been reported. When the wave is trapped by cylinders with a critical geometry, the wave amplitude is commonly amplified. A deeper motivation is to have an insight into the flow structure, which is the basic force for mass transportation. The present work focuses on the investigation of the wave “near trapping” occurring condition when waves passing through an array of cylinders with a smaller stem, which is a good mimic of the vegetation. Another motivation of the work is focused on the flow structure, which is not only used to state the mechanism of the wave trapping phenomenon, but also contributes to investigate the net flow trajectory. For the ecosystem of vegetation, the water flow has an important influence on the mass transportation and the hydrobios.
In the present paper, the wave near trapping is investigated by designing elaborately the study cases referring to the critical parameters provided by researches (Evans and Porter, 1997; Kashiwagi, 2017). A fully three dimensional numerical model was used to simulate the non-linear water wave passing a finite group of cylinders with different SVF (Φ). Within the vegetation patch, the characteristic turbulence scale is limited by the single cylinder diameter (D), and in the wake region, the relative length scales are commonly characterized by the water depth and the cylinder array diameter (DG). The flow pattern in oscillatory motion is also governed by the KC number.
The paper is structured as follows: in §2, the numerical formulation and the study cases are described. The simulation results are analyzed and interpreted in §3. The general conclusions are presented in §4.
Section snippets
Governing equations
As one extending shallow water equation model with the hydrostatic assumption, the total pressure is split into the hydrostatic component and the non-hydrostatic componentin the present model, in which is the free surface elevation. To capture the variant free surface and uneven bottom, the vertical coordinate z is transformed to σ coordinate (Phillips, 1957). In the transformed coordinate, is a relative coordinate with 0 on the free surface, and −1 on the bottom, i.e. (
Model validation
Available experimental data of wave through slender cylinders is not sufficient to validate the turbulence model used in the present study, while more data are available for the simpler situation of unidirectional flow through vegetation. To demonstrate the applicability of the turbulence model the experiments carried out by Zong and Nepf (2011) were replicated by the numerical model. The experiments were conducted in a laboratory flume under uniform flow condition. The case used for validation
Summary
A 3D numerical model was used to investigate nonlinear waves passing through a finite circular array of cylinders. The cylinder array is one ideal model to represent the vegetation or the forest planted in coastal reach to provide a defence for coasts. The hydrodynamic characteristics of waves passing through cylinders refer to wave reflection, transmission, attenuation and wave forces on solid cylinders. The present 3D numerical model is much powerful in capturing the turbulent flow structures
Acknowledgements
This work was jointly sponsored by the National Nature Science Foundation (No. 11572196 and No. 51479111) and the National Basic Research Program of China (973 Program, No. 2014CB046200).
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