Numerical simulations of surf zone wave dynamics using Smoothed Particle Hydrodynamics
Introduction
The accurate prediction of wave transformation and wave breaking in the nearshore zone, including how waves impact coastal structures, remains one of the great challenges in the fields of nearshore oceanography and coastal engineering. Much of this uncertainty stems from how to most accurately simulate the breaking process (i.e., the overturning of the free surface), and in turn, how wave transformation within the surf zone region triggers a range of additional nearshore hydrodynamic processes, including additional sources of water level variations (e.g., low frequency waves and wave setup) and wave-driven mean flows. The accurate description of the full range of nearshore water motions is critical to develop robust predictions of wave-driven coastal impacts, including coastal flooding, erosion and storm damage to coastal infrastructure.
The nonlinear physics that govern nearshore wave transformation (e.g., the cross-shore evolution of wave shape, nonlinear energy transfers, and ultimately dissipation) are especially challenging to predict in practical coastal-scale applications due to the wide range of spatial and temporal scales of the processes involved. For example, both breaking and non-breaking waves drive mass transport over relatively large-scales (i.e., order 10 s to 100 s of metres); whereas, incident wave energy becomes dissipated as heat within turbulent flow fields at much finer-scales (i.e., order centimetres or less). Historically, it has been impractical to directly predict (both analytically and numerically) the full spectrum of hydrodynamic processes in the nearshore zone. To fill this gap, experimental observations (both within the lab and field) have been critical to advance process understanding of nearshore hydrodynamics by supporting the development of empirical formulations to parameterize surf zone processes that occur at scales finer than can be resolved within a coastal model. However, as a general rule, a reliance on the parameterization of physical processes within coastal models can risk undermining their predictive benefits; for example, this can require case-specific (non-physical) tuning of empirical parameters to datasets or may cause models to entirely fail when extended to coastal applications and/or complex study sites beyond the parameter space for which they were initially developed and validated.
Nearshore wave models can be broadly placed into two main categories: phase-averaged (spectral) and phase-resolving models. Phase-averaged wave models attempt to simulate the stochastic properties of waves, usually based on linear wave theory, with empirical formulations to parameterize the nonlinear physics (e.g., wave breaking dissipation, wave–wave interactions, etc.). These models are also commonly coupled to flow models to simulate slowly-varying flow properties (i.e., time-scales greater than the wave group envelope period) such as wave setup and mean wave-driven currents. Given that phase-averaged models can only provide crude representations of the true complex surf zone physics, they often require some degree of parameter tuning to match experimental observations and thus may be incapable of simulating the full range of hydrodynamic processes that are important in a nearshore application. For example, it is relatively common to find that models that have been tuned to optimally reproduce surf zone wave transformation underpredict the magnitude of wave setup, especially when waves break on steep slopes (e.g., Lashley et al., 2018, Skotner and Apelt, 1999). This indicates that these models may not correctly predict the cross-shore distribution of wave forces (radiation stress gradients), which has been attributed to the poor accounting of the conversion of potential to kinetic energy under breaking waves and subsequent dissipation in the inner surf zone (e.g., Buckley et al., 2015). The inclusion of additional empirical formulations (e.g., roller models) have been proposed to account for these unresolved processes during breaking (Svendsen, 1984); however, these approaches have had varying degrees of success and introduce additional model parameters that are not universally applicable across nearshore applications.
Phase-resolving wave models, loosely defined here as any wave-flow model that deterministically resolves motions at time-scales shorter than individual sea-swell waves, are designed to include a more complete representation of the nonlinear physics of nearshore waves. Depth-averaged (2DH, two-dimensional in the horizontal) versions of these models (or multi-layer versions employed with coarse vertical resolution) are commonly either Boussinesq-type or based on the nonlinear shallow water equations with non-hydrostatic pressure corrections (commonly referred to as non-hydrostatic wave-flow models). While 2DH phase-resolving models may more accurately simulate the nonlinear behaviour of non-breaking waves in shallow water, they are still incapable of directly resolving the wave breaking process and thus may suffer from some of the same shortcomings as phase-averaged models (i.e., empirical parameterization of the breaking process). Due to their (quasi) depth-averaged description, 2DH models also do not capture the vertical structure of the (mean) flow dynamics (e.g., the undertow profile). Three-dimensional (3D) phase-resolving wave-flow models (summarized below) provide the most rigorous representation of nearshore hydrodynamics, including the capability to directly resolve at least some of the wave breaking process, but are computationally expensive. These models can be classified as either mesh-based or mesh-free. The former class are based on various solutions of Eulerian forms of the (Reynolds-Averaged) Navier–Stokes equations on numerical grids (meshes). The latter class are based on Lagrangian solution of the Navier–Stokes equations, which include those based on particle methods that attempt to describe the motion of a fluid continuum using discrete particles (Liu and Liu, 2003).
Several 3D mesh-based approaches have been developed to simulate waves in the nearshore, which mainly differ in their treatment of the free surface. Multi-layered non-hydrostatic wave-flow models (e.g., Zijlema and Stelling, 2008, Bradford, 2010, Ma et al., 2012) describe the free-surface as a single-valued free-surface (akin to 2DH phase-resolving models). Although this simplification allows them to capture the dissipation of breaking waves and 3D flows more efficiently, it also implies that they cannot resolve all details of the breaking process such as wave overturning and breaking-wave-generated turbulence (e.g., Derakhti et al., 2016b, Rijnsdorp et al., 2017). Alternatively, Computational Fluid Dynamics (CFD) models use more comprehensive techniques that can capture complex details of the free surface, such as under breaking waves. This includes the marker and cell method (Harlow and Welch, 1965), level-set method (Osher and Sethian, 1988), and Volume Of Fluid (VOF) method (Hirt and Nichols, 1981), of which the VOF approach has been most widely adopted to model nearshore processes (e.g., Torres-Freyermuth et al., 2007, Yao et al., 2019). Although these models can capture the breaking process and the turbulent flow field in detail, comparisons between laboratory experiments and mesh-based RANS solvers have highlighted some of the difficulties in accurately predicting turbulent flow fields within the surf zone. In particular, model predictions have been shown to be sensitive to the turbulence model used (e.g., Brown et al., 2016), which can lead to significant discrepancies in the undertow profiles predicted throughout the surf zone (e.g., Brown et al., 2016, Larsen and Fuhrman, 2018).
The most common mesh-free particle methods are Smoothed Particle Hydrodynamics (SPH) and Moving Particle Semi-implicit (MPS) models. The SPH method (which is the specific focus here), was originally developed as a general numerical approach for supporting continuum mechanics applications, and is now used across a range of scientific fields, including astrophysics, fluid mechanics and solid mechanics (Monaghan, 1992). In more recent years (particularly over the past decade), SPH has become an increasingly common technique applied to coastal and ocean engineering problems, due to its ability to deal with complex geometries, account for highly nonlinear flow behaviour, and to simulate large deformations at interfaces (including moving boundaries and at the free surface) (e.g., Monaghan and Kos, 1999, St-Germain et al., 2013, Altomare et al., 2015b, Crespo et al., 2017, González-Cao et al., 2019, Zhang et al., 2018a, Altomare et al., 2014, Domínguez et al., 2019). Nevertheless, SPH is still being continuously developed and improved; for example, there is still concerted international effort to improve performance related to numerical implementations (including enhancing model convergence, consistency, and stability and adaptivity schemes) and improving treatment of boundary conditions (including at solid boundaries, at the free surface and when coupling to other models). In coastal wave applications, SPH techniques are now being increasingly applied to study wave-structure interactions (including loads and overtopping) (e.g., Akbari, 2017, Altomare et al., 2015a, González-Cao et al., 2019) and the dynamics of floating bodies (e.g., Bouscasse et al., 2013, Ren et al., 2017, Crespo et al., 2017). To a lesser degree, SPH approaches have also been used to investigate the physics of surf zone processes; for example, in recent studies of surf zone currents and eddies (Wei et al., 2017, Farahani et al., 2013), nearshore wave breaking (e.g., Issa and Violeau, 2009, Makris et al., 2016, De Padova et al., 2018, Roselli et al., 2019, Shao and Ji, 2006) and surf zone energy balances (Wei and Dalrymple, 2018). Nevertheless, despite the great promise of SPH to nearshore applications, a rigorous assessment of the ability of SPH models to accurately simulate a full range of relevant surf zone hydrodynamic processes is still relatively sparse, certainly in comparison to the wealth of information derived from detailed nearshore wave modelling studies (including mesh-based CFD modelling approaches).
In this study, we conduct a detailed investigation of the ability of the SPH modelling approach to predict a broad range of nearshore processes relevant to coastal applications where wave breaking is important. Using experimental data of wave breaking over both a plane beach and a fringing reef profile, we demonstrate how the model can accurately reproduce a broad range of relevant hydrodynamic processes, ranging from the nonlinear evolution of wave shapes across the surf zone, wave setup distributions, and mean current profiles. We compare the present surf zone predictions with predictions by other classes of wave models from literature and illustrate some of the advantages of the SPH approach (particularly in resolving hydrodynamics within the crest region above the wave trough).
Section snippets
General features of the SPH method
Smoothed Particle Hydrodynamics (SPH) is a mesh-free numerical method where a continuum is discretized into particles. The approach was originally developed within astrophysics (Lucy, 1977, Gingold and Monaghan, 1977) and since then largely applied across a wide range of Computational Fluid Dynamics (CFD) applications. Within SPH, the particles represent calculation nodal points that are free to move in space according to the governing Lagrangian dynamics, such as in fluid mechanics based on
Wave breaking over a plane beach (Ting and Kirby, 1994)
Although both of the TK94 test cases had breaking wave heights that were approximately the same, the difference in wave period resulted in appreciable differences in how waves broke and transformed within the surf zone (Fig. 2). For the “spilling” case, the model predicts a slight overturning of the free surface (albeit a small volume of water) that initiates at a breaking location m, which evolves into a turbulent bore that propagates shoreward. For the “plunging” case, the waves
Wave transformation and wave setup
The cross-shore variations in wave heights and wave shape were accurately reproduced by the SPH simulations over the range of different wave conditions and bathymetry profiles considered in this study. This included a broad range of different wave breaking types, spanning the extremes of spilling waves breaking on a plane beach in TK94 (Fig. 3) to the strongly plunging waves breaking on a nearly dry, steep reef slope in Y12 (Fig. 8). The excellent agreement with the cross-shore evolution of
Conclusions
This study has demonstrated how the mesh-free SPH approach can provide accurate and robust predictions of complex surf zone hydrodynamic processes generated by wave breaking, with model performance comparable to applications of state-of-the-art mesh-based CFD models such as OpenFOAM. Over the wide range of wave breaking types considered, the SPH approach was able to reproduce many of the detailed hydrodynamic processes that govern the nonlinear evolution of wave shape in the nearshore, the free
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.
Acknowledgements
This work was supported by resources provided by the Pawsey Supercomputing Centre with funding from the Australian Government and the Government of Western Australia. We are grateful for the helpful discussions about this modelling with Marion Tissier, Ad Reniers, Ap van Dongeren, Dano Roelvink, Robert McCall, Niels Jacobsen and Marcel Zijlema, while RJL was on a sabbatical in Delft, the Netherlands. RJL thanks both Deltares and Delft University of Technology for hosting his sabbatical time in
References (98)
Simulation of wave overtopping using an improved SPH method
Coast. Eng.
(2017)- et al.
Applicability of smoothed particle hydrodynamics for estimation of sea wave impact on coastal structures
Coast. Eng.
(2015) - et al.
Numerical modelling of armour block sea breakwater with smoothed particle hydrodynamics
Comput. Struct.
(2014) - et al.
Long-crested wave generation and absorption for SPH-based DualSPHysics model
Coast. Eng.
(2017) - et al.
Improved relaxation zone method in SPH-based model for coastal engineering applications
Appl. Ocean Res.
(2018) - et al.
Nonlinear water wave interaction with floating bodies in SPH
J. Fluids Struct.
(2013) - et al.
Boussinesq evolution equations: Numerical efficiency, breaking and amplitude dispersion
Coast. Eng.
(2004) - et al.
Evaluation of turbulence closure models under spilling and plunging breakers in the surf zone
Coast. Eng.
(2016) A note on estimation of the Jacobian elliptic parameter in cnoidal wave theory
Ocean Eng.
(2003)- et al.
Towards simulating floating offshore oscillaTing water column converters with smoothed particle hydrodynamics
Coast. Eng.
(2017)
DualSPHysics: Open-source parallel CFD solver based on Smoothed Particle Hydrodynamics (SPH)
Comput. Phys. Comm.
Numerical modeling of water waves with the SPH method
Coast. Eng.
Wave breaking in the surf zone and deep-water in a non-hydrostatic RANS model. Part 1: Organized wave motions
Ocean Model.
Wave breaking in the surf zone and deep-water in a non-hydrostatic RANS model, Part 2: Turbulence and mean circulation
Ocean Model.
Application of a buoyancy-modified k- SST turbulence model to simulate wave run-up around a monopile subjected to regular waves using OpenFOAM®
Coast. Eng.
Performance of a buoyancy-modified k- and k- SST turbulence model for simulating wave breaking under regular waves using OpenFOAM®
Coast. Eng.
Optimization strategies for CPU and GPU implementations of a smoothed particle hydrodynamics method
Comput. Phys. Comm.
Three-dimensional reversed horseshoe vortex structures under broken solitary waves
Coast. Eng.
On the accuracy of DualSPHysics to assess violent collisions with coastal structures
Comput. & Fluids
Volume of fluid (VOF) method for the dynamics of free boundaries
J. Comput. Phys.
Nonhydrostatic and surfbeat model predictions of extreme wave run-up in fringing reef environments
Coast. Eng.
Comparisons of weakly compressible and truly incompressible algorithms for the SPH mesh free particle method
J. Comput. Phys.
Shock-capturing non-hydrostatic model for fully dispersive surface wave processes
Ocean Model.
Surf zone dynamics simulated by a Boussinesq type model. Part I. Model description and cross-shore motion of regular waves
Coast. Eng.
Numerical modeling of surf zone dynamics under weakly plunging breakers with SPH method
Ocean Model.
A simple procedure to improve the pressure evaluation in hydrodynamic context using the SPH
Comput. Phys. Comm.
Simulating free surface flows with SPH
J. Comput. Phys.
Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations
J. Comput. Phys.
Application of smoothed particle hydrodynamics for modeling the wave-moored floating breakwater interaction
Appl. Ocean Res.
Efficient non-hydrostatic modelling of 3D wave-induced currents using a subgrid approach
Ocean Model.
Shock-capturing Boussinesq-type model for nearshore wave processes
Coast. Eng.
SPH simulation of periodic wave breaking in the surf zone-A detailed fluid dynamic validation
Ocean Eng.
On the parameterization of the free-stream non-linear wave orbital motion in nearshore morphodynamic models
Coast. Eng.
A Boussinesq model for waves breaking in shallow water
Coast. Eng.
Application of a Boussinesq model for the computation of breaking waves: Part 2: Wave-induced setdown and setup on a submerged coral reef
Ocean Eng.
Non-hydrostatic modeling of surf zone wave dynamics
Coast. Eng.
Depth-induced wave breaking in a non-hydrostatic, near-shore wave model
Coast. Eng.
Wave heights and set-up in a surf zone
Coast. Eng.
Surf zone wave parameters from experimental data
Coast. Eng.
Observation of undertow and turbulence in a laboratory surf zone
Coast. Eng.
Dynamics of surf-zone turbulence in a strong plunging breaker
Coast. Eng.
Dynamics of surf-zone turbulence in a spilling breaker
Coast. Eng.
A new approach to handle wave breaking in fully non-linear Boussinesq models
Coast. Eng.
Finite volume scheme for the solution of 2D extended Boussinesq equations in the surf zone
Ocean Eng.
Coupling methodology for smoothed particle hydrodynamics modelling of non-linear wave-structure interactions
Coast. Eng.
Chaos in breaking waves
Coast. Eng.
1DH Boussinesq modeling of wave transformation over fringing reefs
Ocean Eng.
Efficient computation of surf zone waves using the nonlinear shallow water equations with non-hydrostatic pressure
Coast. Eng.
SWASH: An operational public domain code for simulaTing wave fields and rapidly varied flows in coastal waters
Coast. Eng.
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