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# Arbitrary and hybrid Lagrangian-Eulerian models

The Arbitrary Lagrangian-Eulerian (ALE) technique is another advantageous approach, that is extremely promising for coastal CFD application. The approach merges the benefits of Eulerian and Lagrangian flow descriptions. The vertices of the computational mesh in ALE models are arbitrary in that they are either allowed to move (in a Lagrangian fashion) or remain fixed in a manner that aids rezoning. In ALE, the computational mesh thus evolves selectively and independently of the fluid motion. There are typically three options for moving the cell vertices:

• • Move with the fluid for Lagrangian computing
• • Stay fixed for Eulerian computing
• • Move in an arbitrarily prescribed way allowing a continuous rezoning capability.

The ALE cycle for hydrodynamic calculations is divided into three phases. Phase I is a standard Lagrangian calculation. Phase II is the rezoning phase, in which rezone velocities are specified to reduce distortions in the mesh. Phase III performs all the advective flux calculations; these are necessary if the mesh is not purely Lagrangian. The purpose of using three phases is to know the Lagrangian motion before a choice is made for rezoning. Through rezoning, a simulation can be run beyond what a pure Lagrangian mesh-based method, such as that described in Chapter 2, will allow. This enables tracking of the interface between fluids, including localised high resolution when it is required and the ability to maintain higher resolution than a Eulerian method for the same number of computational nodes (Donea et ah, 2017).

Moreover, the ALE method can be a particularly powerful tool when combined with techniques for actively controlling mesh quality and mesh refinement . Such methods prevent mesh quality deteriorat ion due to mesh entanglement , the development of skewed cells, and at the same time, they allow for maintaining sufficient refinement around elements such as wall boundaries, fluid interfaces or small scale features. These can be simple techniques, such as the linear elastostatic approach for mesh motion (Dwight, 2009; de Lataillade, 2019), which allows smaller cells to maintain their shape by making them more “rigid” and accommodating mesh deformations by making larger cells more “flexible”. More sophisticated approaches such as mesh monitor functions (Grajewski et ah, 2010) or mesh adaptivity (Bathe and Zhang, 2009) can yield an even higher degree of control to the mesh refinement but these come with a more complicated implementation and higher computational cost. To-date the ALE method has not been widely used for fluid structure interaction (FSI) problems within a coastal engineering setting, but a successful example of application is shown in chapter 8 of this book.

Other hybrid Lagrangian-Eulerian models have been employed within a coastal CFD framework; perhaps the most well-known of which is the Marker and Cell (MAC) method originally developed by Harlow and co-workers (Harlow and Welch, 1965) which uses passive marker particles to track the free surface. In MAC the particles serve the sole purpose of surface tracking. More recently, the Particle in Cell (PIC) approach, also originally due to Harlow and co-workers (Harlow, 1964), has been revived and applied to various Coastal Engineering problems (Kelly, 2012; Chen et al., 2016; Maljaars et al., 2017). The PIC method attempts to get the best from two worlds using a Lagrangian approach efficiently for both the non-linear velocity advection and interface tracking whilst employing an optimized Eulerian approach to enforce solid boundary conditions and also compute the pressure. With recent innovations the PIC approach seems particularly well suited as a CFD method for fluid structure interaction (FSI) problems, including problems that involve floating structures, within a coastal engineering setting (e.g., Chen et al. (2018, 2019)). In common with all particle, or semi-particle, based methods particle clustering, due to numerical error, is a problem with the PIC method. Recent work by Maljaars et al. (2018) uses a hybridized discontinuous Galerkin (HDG) framework for the Eulerian phase of the computations. The HDG approach appears to allow for strict vanishing of the velocity divergence in a local, cell- by-cell, manner. This, combined with the diffusion free advection offered by the Lagrangian phase, retains a uniform particle distribution over time. This removes the need for particle re-distribution schemes which themselves add additional, unwanted, numerical diffusion. The work of Maljaars et al. (2018) appears very promising in this regard; however, the problem of the free surface remains unresolved.

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