Large eddy simulation study of a wave-induced shallow-water monopolar vortex
Graphical abstract
Introduction
Bays and harbours are vital to coastal communities providing recreational, economic, and coastal protection benefits. However, various essential coastal infrastructures in bays and harbours (e.g., breakwater and quay wall) are vulnerable to episodic wave events (e.g., tsunamis). For instance, large horizontal eddies (frequently called whirlpools) have been observed and recorded after tsunamis (e.g. Choi et al., 2008, Wilson et al., 2012, Admire et al., 2014). [5] argued that strong currents driven by large offshore eddies can bring a considerable impact and damage to harbour areas. These tsunami effects can persist for several hours, which may be accompanied with coastal inundation. Thus, the characteristics and evolution of these long-wave-induced vorticial currents interacting with surrounding boundaries require comprehensive investigation.
Significant damage from eddies was observed in many ports and harbours after the 2004 Indian Ocean Tsunami (e.g. Okal et al., 2006a, Okal et al., 2006b, Okal et al., 2006c). Most importantly, a ship moored in the port of Salalah, Oman spun and drifted for a significant distance in the vicinity of the port. The trajectory of the ship followed the eddies which were generated inside the harbour after the arrival of the 2004 Indian Ocean Tsunami and migrated offshore (Okal et al., 2006a). This event clearly shows that the eddies induced by long waves can cause the uncontrolled drift of ships, which in turn can cause critical damage to infrastructure or ships. Lynett et al., 2012, Lynett et al., 2014 suggested that a tsunami-induced horizontally-sheared rotating current, generated by the interaction between vertical coastal structures and topographical fluid jets, was the main cause that the ship broke its mooring in port Salalah instead of free surface fluctuations.
Recently, large-scale laboratory experiments were carried out to investigate the dynamics of a long-wave-induced shallow monopolar vortex generated at the edges of hard boundaries, such as a breakwater (Kalligeris, 2017). In the experiment, the passage of a long wave through a narrow port entrance generated horizontal shear and vertical vorticity that drives two-dimensional (2D) turbulent coherent structures (TCSs) that contain 3D sub-structures. The extensive data set collected during the experiment allows to study the generation, growth, and decay of long-wave-induced shallow TCSs. This study further showed that kinetic energy decay and spatial growth of long-wave-induced shallow TCSs followed the first order vortex model of Seol and Jirka (2010). The primary rotating current of the long-wave-induced shallow TCSs matches well with the theoretical -profile for a monopolar vortex (van Heijst and Clercx, 2009).
TCSs represent turbulent eddies that are instantaneously coherent in space and they can be expressed using velocity or vorticity fluctuations (Hussain, 1983). In general, TCSs are 3D and evolve into sub-structures through energy cascade. In sufficiently shallow water, however, flow and turbulence fields are vertically confined (e.g. Batchelor, 1969, Sou et al., 2010, Kim et al., 2017). TCSs can be dominated by 2D features in nearshore environments when the water depth, , is relatively small compared to the wave length, , (i.e., ) (Jirka, 2001). In such conditions, they are often called shallow TCSs (e.g. Negretti and Socolofsky, 2005). Dolzhanskii et al. (1992) argued that the two-dimensionality of shallow flows mainly depends on the Reynolds number, , and bottom friction. Turbulence in quasi-2D flows can be characterised by the 3 power law in the turbulent kinetic energy spectrum denoted as the enstrophy (integral of vorticity squared) transfer regime (Lindborg and Alvelius, 2000, Uijttewaal and Booij, 2000, Uijttewaal and Jirka, 2003). In this regime, enstrophy is transferred from large to small scales while the vortex stretching is confined. The 3 slope is distinct from the typical 5/3 spectrum slope for isotropic turbulence based on the Kolmogorov hypothesis (Pope, 2000). This 3 slope has been found in wavenumber spectra in the swash zone during uprush in both measured data (Sou et al., 2010) and large eddy simulation (LES) results (Kim et al., 2017).
Depth-integrated and depth-resolving numerical models have been adopted to study shallow TCSs (Kim and Lynett, 2011, Son et al., 2011, Williams and Fuhrman, 2016, Larsen et al., 2017). A fully nonlinear Boussinesq-type model for weakly dispersive and rotational flow using a long wave perturbation approach was derived (Kim and Lynett, 2011) and revealed that the dispersive stress plays an important role for energy transfer in a shallow mixing layer. The nonlinear Boussinesq-type model was coupled the with the shallow-water solver for tsunami generation and propagation in the open ocean (Son et al., 2011), that is capable of reproducing the local dynamics experienced in the Port of Salalah, Oman induced by the 2004 Indian Ocean tsunami (Okal et al., 2006a). To resolve a vertical flow structure, a Reynolds-Averaged Navier–Stokes (so-called RANS) model with a transitional two-equation – turbulence model was also proposed (Williams and Fuhrman, 2016). With additional bedload and suspended load descriptions, Larsen et al. (2017) demonstrated that the tsunami-induced scour processes can be limited by finite wave boundary layer thickness.
Compared to the aforementioned numerical models, the main advantage of an LES model is that unsteady motions larger than a pre-determined grid scale is directly resolved in the entire water column. Thus, a more complete understanding of turbulent characteristics, such as the evolution of a TCS, cascade of turbulent kinetic energy and dissipation, and the vertical flow structure can be obtained. For instance, many LES studies have successfully revealed the TCSs induced by wave breaking processes in the surf zone (e.g. Christensen and Deigaard, 2001, Watanabe et al., 2005, Lubin et al., 2006, Derakhti and Kirby, 2014). For tsunami-like long waves, LES has also been adopted to provide more detailed information for the instantaneous turbulent flow feature where large-scale unsteadiness is significant. Wu and Liu (2008) introduced a numerical model that can be applied to landslide-generated tsunamis. Under the solitary wave, Zhou et al. (2014) showed the evolution of rollers into 3-D hairpin vortices due to wave breaking and their touch down onto the bed. When the solitary wave interacts with a cuboid structure, it was also revealed that primary vortex tubes generated at the leading and trailing edges of structures were stretched in the wave propagation direction accompanied with the formation of the secondary vortices (Arabi et al., 2019). Yao et al. (2019) showed that wave run-up is sensitive to reef morphology affecting vortex and current evolution.
The purpose of this study is to unravel the complexities involved in the dynamics of the long-wave-induced shallow monopolar vortex using LES. In particular, since only surface velocities were measured in the physical experiment (Kalligeris, 2017), LES can provide the comprehensive 3D flow structure and turbulence characteristics in full depth at different stages of the vortex evolution, albeit being computationally expensive. The large-scale laboratory experiments of this study are described in Section 2. The mathematical formulations of the model and numerical implementations are presented in Section 3. We devote Section 4 to discuss model setup, main flow characteristics, and comprehensive model validation with the data collected from the physical experiment (Kalligeris, 2017). A discussion on unique flow and turbulence characteristics of long-wave-induced TCSs obtained from LES is presented in Section 5. Lastly, Section 6 provides the main conclusions of this study.
Section snippets
Physical experiment
An overview of the laboratory experiments of long-wave induced shallow monopolar vortex is provided. More details can be found in Kalligeris (2017). The physical experiments were carried out in the 3D wave basin at the O. H. Hinsdale Wave Research Laboratory in Oregon State University. The 3D wave basin is 44 m long, 26.5 m wide, and the still water depth, , was set at 0.55 m. A vertically walled breakwater was installed with a angle to establish a gap of 3.1 m between the tip of the
Governing equations
3D large-eddy simulations were carried out to model the physical experiments. In LES, the filter operation decomposes the velocity field into the resolved and the unresolved components using a prescribed cutoff filter length. The flow fields with length scales larger than the filter length are solved directly, in which the filter length denotes the characteristic length of the grid size as: where each component denotes the grid size in , , directions, respectively.
Model setup
The 3D numerical model domain was identical to the physical model described in Kalligeris (2017). In a preliminary numerical experiment, we found that orthogonality of the mesh is critical in minimising spurious free surface fluctuations due to numerical errors, which will be discussed in Section 4.3. Hence, a rectangular mesh was constructed with a uniform grid size of 2.75 cm in the vertical (bed normal) direction () and of 8.23 cm in the two horizontal directions (, ), where the ()
Secondary flow characteristics
Besides the main flow field characterised by the azimuthal velocity component, the offshore vortex flow field also consists of radial and vertical velocity components. These secondary flow components, which are relatively minor compared to the azimuthal velocity, generate 3D recirculation patterns in the TCS flow field. Fig. 11 shows the temporal evolution of free surface elevation at the vortex core () for Run 2. It should be noted that the vortex core location in each ensemble also varies
Conclusions
A numerical investigation of long-wave-induced TCSs was carried out using a 3D LES approach on a dynamic mesh. Standard Smagorinsky SGS closure was used in the present study. The numerical model was validated with experimental data (Kalligeris, 2017) for free surface elevation, mid-depth velocities, azimuthal velocity at the free surface, and vertical vorticity at the free surface. Model results showed good agreements for the velocity fields (). The ensemble-azimuthal-averaging throughout
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
This study is supported by NSF, USA (OCE-1635151; OCE-1756714; CMMI-1135026) and Office of Naval Research, USA (N00014-17-1-2796). Numerical simulations presented in this study were carried out using the Mills cluster at University of Delaware, and the SuperMic cluster at Louisiana State University via XSEDE (TG-OCE100015). The case setup to reproduce the same results is publicly available via Community Surface Dynamics Modeling System (CSDMS) model repository maintained by GitHub: //github.com/sedwavefoam/Wave-induced-shallow-water-monopolar-vortex
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