Numerical investigation of solitary wave attenuation and resistance induced by rigid vegetation based on a 3-D RANS model

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Highlights

  • Solitary wave interaction with emergent/submerged vegetation using a 3-D RANS model.

  • The effects of wave nonlinearity and vegetation configuration on solitary wave attenuation.

  • Forces exerted on the cylinders and velocity pattern inside the vegetation field are explored.

  • Comparison between representative CD and period-averaged CD based on direct force method.

  • Generic CD-Re/KC formulae for solitary wave interaction with vegetation are proposed.

Abstract

Mangrove forests can significantly attenuate tsunami waves and thus play an important role in coastal protection. As a first approximation, the problem is modeled utilizing solitary waves impinging on emergent/submerged rigid cylinders. A three-dimensional (3-D) numerical model using cyclic boundary conditions was developed based on the IHFOAM solver to investigate the effects of wave nonlinearity and vegetation configuration on the solitary wave attenuation. The numerical model was established based on the Reynolds Averaged Navier-Stokes (RANS) equations combined with the standard kω shear stress transport (SST) turbulence model and the volume of fluid (VOF) surface capturing schemes. The results indicate that different patterns are found in terms of flow field characteristics (velocity and turbulent kinetic energy) and forces exerted on the cylinders for various wave nonlinearity and vegetation configuration, which helps to better understand wave dissipation mechanism induced by vegetation. Different from the bulk drag coefficient derived by the conventional wave dissipation models, the direct force method was applied to quantify the time-varying and period-averaged drag coefficients (CD) of individual cylinders. The time-varying CD associated with maximum force and local velocity is defined as the representative CD, for comparison with the period-averaged CD in detail. Besides, by considering the submergence ratio, new generic CD formulas are proposed as functions of the modified Reynolds number (Re) and Keulegan-Carpenter number (KC) for illustrating the CD dynamics under solitary wave conditions. Finally, a preliminary comparison between the proposed CD formulas and existing formulas are given to reveal the intrinsic CD law, which may lead to improved understanding and modeling concerning wave-vegetation interaction.

Introduction

Coastal vegetation (such as seagrasses, salt marshes, kelp forests, and mangroves, etc.) can significantly attenuate incident wave energy and protect coastal habitats and structures, which has long been a topic of interest (Anderson and Smith, 2014; Chen et al., 2018; Huang et al., 2011), and its wave damping efficiency is remarkable even in extreme sea events (Möller et al., 2014; Vuik et al., 2018). For example, mangrove forests effectively attenuated wave energy from the Indian Ocean tsunami to save several villages in 2004 (Huang et al., 2011). Therefore, numerous studies in field-scale (e.g., Bradley and Houser, 2009; Gaylord et al., 2003; Jadhav et al., 2013) and in laboratory-scale (e.g., Augustin et al., 2009; Hu et al., 2014; Huang et al., 2011; Koftis et al., 2013; Ozeren et al., 2014; Wu and Cox, 2015), as well as numerical modeling (e.g., Ma et al, 2013; Mattis et al., 2015, 2019; Maza et al., 2013, 2015; Tang et al., 2017), have been conducted to better understand the protection role of coastal vegetation. However, in addition to wave damping induced by vegetation, merely few studies have focused on the turbulence characteristics in vegetated oscillatory flows (Chen et al., 2020) and the vegetation stem-scale turbulence was found to play a significant role in in the water environment and ecological processes, such as dissolved-nutrients supply (Cornelisen and Thomas, 2004), sediment suspension and transport (Chen et al., 2007). Due to the complicated process and multivariate factors, the mechanisms of vegetation–hydrodynamics interactions have not been fully understood.

In recent years, wave dissipation induced by vegetation (hereafter referred to as WDV) has attracted wide interest, and its relationship was examined with the vegetation characteristics (i.e., the stem diameter, density, stiffness and distribution pattern, etc.) and incident wave conditions (regular or irregular, incident wave height and wave period, etc.) (e.g., Bradley and Houser, 2009; Jadhav et al., 2013; Koftis et al., 2013; Möller, 2006; Yang et al., 2012). Previous studies have mostly focused on vegetation-induced attenuation under either regular or irregular wave conditions; the mechanism of wave attenuation by vegetation during extreme sea events (e.g., tsunami) is still not fully understood. Moreover, few studies consider the effect of wave nonlinearity on WDV (Wu and Cox, 2015), which is of great importance to the understanding of the hydrodynamic, sedimentation, and exchange processes in mangrove forests.

WDV is primarily induced by the drag force (FD) provided by the vegetation acting on the wave motion, which can be quantified by the Morison equation (Dalrymple et al., 1984; Morison et al., 1950); FD is a function of the vegetation drag coefficient, which is a dimensionless quantity used in force equation to define the drag or the resistance of vegetation in a fluid flow, which is generally denoted as CD (Yuce and Kareem, 2016). Thus, to adequately describe WDV, accurate parameterization of CD is essential (Mendez and Losada, 2004; Möller et al., 2014; Ozeren et al., 2014). However, CD parameterization is still challenging when describing vegetation-flow interactions in coastal and freshwater environments, since it is affected by many factors of wave and vegetation parameters (Hu et al., 2014; Henry et al., 2015).

To our knowledge, numerous studies focused on the bulk drag coefficient (C˜D), which is an empirical parameter to quantify the mean resistance force exerted against the waves by the entire vegetation field (Ozeren et al., 2014). Currently, C˜D is typically calculated based on the measured wave height reduction by the theoretical WDV models, which were derived from energy conservation (Dalrymple et al., 1984; Mendez and Losada, 2004) or momentum conservation (Kobayashi et al. 1993; Mendez et al. 1999). As a conventional method, the WDV models have been widely applied in a large number of flume experiments with not only rigid vegetation (e.g., Hu et al., 2014; Ozeren et al., 2014; Chen et al., 2018) but also artificial/real flexible vegetation (e.g., Anderson and Smith, 2014; Cavallaro et al., 2018; Koftis et al., 2013; Möller et al., 2014) to obtain additional empirical formulas for predicting C˜D, as summarized in Table 1. It can be found that for regular or irregular wave flows, the C˜D empirical formulas usually have been proposed as functions of the Reynolds number, Re = Ucbv/ν, or the Keulegan–Carpenter number, KC = UcT/bv, in which Uc is the characteristic velocity, bv is the characteristic length (usually vegetation diameter), ν is the kinematic viscosity of the fluid and T is the wave period that can be replaced by the spectral peak wave period (Tp) in the case of irregular waves. Note that these formulas can take different forms. This can be explained that significant variations exist in the characteristic velocity and length scales, as well as the WDV models adopted in the literature. Besides, the Cauchy Number (Ca) was used to explore the relationship with C˜D for wave flows interacting with flexible aquatic vegetation (Cavallaro et al., 2018; Luhar and Nepf, 2011, Luhar and Nepf, 2016 ).

Although the aforementioned WDV model method has been widely used in predicting C˜D, the limitations of this method should be pointed out. This method derives C˜D from the perspective of wave energy dissipation, the measured wave height attenuation is often assumed to be solely induced by vegetation drag. Other dissipative processes, such as bed friction and wave breaking, are not explicitly considered but lumped into the vegetation drag, which may lead to an overestimated C˜D(Chen et al., 2018; Hu et al., 2014).

Compared to the WDV model method, the direct force method (Hu et al., 2014; Infantes et al., 2011) is a newer approach to determine CD for individual vegetation, which is expected to be different from C˜D (Mattis et al., 2015). The direct force method is based on the original Morison equation, rather than WDV models, and requires synchronized force and local velocity data to determine CD (Chen et al., 2018; Hu et al., 2014; Yao et al., 2018). The main difference between the WDV model and direct force methods is that the former can only provide C˜D for the entire vegetation field, while the latter provides a way to obtain both time-varying CD and period-averaged CD for individual vegetation, which makes it easier to better understand WDV processes by providing more accurate CD values. However, only few studies have been carried out with the goal of considering time-varying and period-averaged CD (Hu et al., 2014; Yao et al., 2018). Using the direct force method for studying WDV in combined wave-current flows, Hu et al. (2014) and Chen et al. (2018) derived the space-averaged period-averaged CD (termed〈CD〉) by averaging the data from the 4 locations within the vegetation field to represent the bulk drag characteristics of the entire vegetation zone, and respectively proposed two empirical formulas as functions of Re and KC, please refer to Table 1. However, the characteristics associated with drag coefficients of the individual vegetation have not yet been clarified. To the best of our knowledge, the current study is the first study to provide a detailed comparison between the time-varying CD and period-averaged CD for individual vegetation along the canopy under solitary wave conditions.

It needs to mention that, during the laboratory measurements, it is challenging to obtain the desirable aligned force-velocity data since it may be limited by synchronized force-velocity measuring systems and the data processing technique (Yao et al., 2018). A small time lag between the original force data and velocity data may lead to large errors in deriving CD; also the intrinsic time shifts in instrument recordings may further contribute to these errors. Moreover, laboratory wave-attenuation studies are extremely expensive to perform and require many testing facilities (Mattis et al., 2019). For this reason, it is arduous to obtain all CD values for individual vegetation along the entire path during the laboratory tests, hence only a few typical measurements are usually selected to represent the bulk characteristic of the entire vegetation field (Hu et al., 2014), showing a certain degree of limitation. Alternatively, a robust numerical model in which the synchronized force-velocity data are determined by post-processing utilities, which can effectively overcome this limitation; as a result, both the time-varying and period-averaged CD for individual vegetation can be accurately provided, which is the main objective of this paper.

Regarding the numerical modeling of the interaction between coastal vegetation and waves, a common approach is to adopt depth-integrated wave equations, such as Boussinesq-type equations (Huang et al., 2011; Karambas et al., 2015; Yang et al., 2018) and nonlinear shallow water equations (Tang et al., 2017; Wu and Marsooli, 2012); the effects of vegetation on waves are considered by introducing a drag force term in the depth-integrated momentum equations. In recent years, the Reynolds Averaged Navier-Stokes (RANS) models have also been applied in several works to solve the 3-D wave-vegetation issues directly. For example, Maza et al. (2015) proposed a 3-D numerical scheme coupled with the k-ω shear stress transport (SST) turbulence model, for simulating the wave flow field by considering the actual geometry of rigid emergent cylinders. Tsai et al. (2017) presented a 3-D RANS model to simulate the effects of flexible vegetation on tsunami damping. Although the RANS numerical model has been proven to capture flow information in the vicinity of vegetation, however, it requires very fine meshes and extremely high computational cost (Maza et al., 2015, 2016). However, this limitation may be alleviated by applying the cyclic boundary conditions in the numerical flume, as used in this study. The cyclic boundary conditions are mostly applied to represent a large (infinite) system by using a small computational domain to effectively eliminate the undesired effects of wall boundaries and the required cell number was reduced to significantly improve the computing efficiency (Wang et al., 2017). And the cyclic boundary conditions have been successfully used to investigate the dynamics of wave-driven oscillatory flow through emergent canopies (Etminan et al., 2019).

The objective of this study was to use a 3-D numerical approach to investigate solitary wave interactions with the rigid emergent/submerged vegetation based on IHFOAM solver built-in OpenFOAM® (Open Field Operation and Manipulation). The numerical model solves the RANS equations for two incompressible phases using a finite volume discretization, the standard k–ω SST model was considered as the turbulence closure and the volume of fluid (VOF) method was employed to capture the free surface. The work is organized as follows. First, the mathematical description of the 3-D numerical model is presented. Mesh sensitivity analysis and model validation are carried out using laboratory data in Huang et al. (2011). In Section 4, the numerical model setup and the direct force method are introduced in detail. Section 5 shows a detailed and intuitive flow field with streamline and vortex information is provided to understand the small-scale mechanisms responsible for WDV inside the canopy. Besides, the effects of wave nonlinearity and vegetation configuration on solitary wave attenuation, velocity and turbulence fields as well as the wave-induced forces exerted on cylinders are also explored. On the other hand, the time-varying CD and period-averaged CD related to individual cylinders are compared based on the direct force method. Furthermore, some new generic CD-Re and CD-KC relations are proposed to describe the CD dynamics. Finally, conclusions are summarized in Section 6.

Section snippets

Numerical model

To simulate tsunami wave interactions with mangrove forests, the direct simulation approach used by Maza et al. (2015, 2016) was followed in this study. In this approach, individual vegetation elements were included in the numerical domain by considering the actual cylinders’ geometry. Note that this approach is free of the complex flow parameterizations such as the drag coefficient used in the macroscopic approach (Maza et al., 2015; Wang et al., 2019).

The numerical model was developed using

Numerical model validation

Laboratory test results by Huang et al. (2011) were used to validate the numerical model, as they provided solitary wave evolution along an idealized mangrove forest using emergent cylinders. It needs to mention that laboratory information about flow characteristics or wave-induced forces on the cylinders is unfortunately lacking, and only free surface evolution is available to validate the numerical model performance.

The characteristics of the laboratory tests by Huang et al. (2011) are as

Numerical simulations

Since the numerical model was demonstrated to reproduce wave height evolution along the canopy with a high degree of accuracy when the cylinders’ geometry was resolved in the domain, it was used in the following sections to explore the detailed hydrodynamic characteristics of the solitary wave interaction with the emergent/submerged vegetation.

Effect of solitary wave nonlinearity and vegetation configuration on WDV

Fig. 7 shows the normalized free surface elevation (η/Hi) at WGs 1 to 11 for emergent/submerged vegetation with different relative wave heights, Hi/h = 0.17 and 0.67. The results show that arrangements A and B generally share a similar free surface behavior. Free surface measurements at WG1 reveal stronger reflections for arrangement A while this effect is slightly weaker for arrangement B. In addition, it was found that the free surface evolutions with larger Hi/h exhibit a certain degree of

Conclusions

The numerical modeling of the interaction of tsunami waves with mangrove forests is studied employing solitary waves impinging on rigid cylinders. Based on the IHFOAM solver, a 3-D RANS numerical model with cyclic boundary conditions is developed considering the actual geometry of the cylinders in the domain. The cyclic boundary condition is adopted to improve computing efficiency. The numerical model is well-validated using the experimental data reported by Huang et al. (2011). A series of

CRediT authorship contribution statement

Yanxu Wang: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing - original draft, Writing - review & editing. Zegao Yin: Resources, Supervision, Writing - review & editing, Data curation. Yong Liu: Supervision, Writing - review & editing, Data curation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The study is financed by National Natural Science Foundation of China (grant nos. 51879251, 51579229, 51725903), Key Research and Development Plan of Shandong Province, China (grant no. 2017GHY15103) and State Key Laboratory of Ocean Engineering, China (grant no. 1602). We sincerely thank the Editor (G.C. Sander), Associate Editor (R. Ferreira) and the anonymous reviewers for providing insightful comments that have significantly improved our paper.

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