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Fully Lagrangian models

Another method to describe water waves is to use equations of fluid motion in the Lagrangian formulation. These equations are written in coordinates moving with the fluid. Each material point of the fluid continuum is labelled with a specific label, and the labels in the fluid- occupied domain create a continuous set of coordinates. These are Lagrangian coordinates or Lagrangian labels. Equations of fluid motion are solved in a fixed Lagrangian domain with the free surface represented by a fixed domain boundary. Some numerical methods use elements of the Lagrangian description. For example MEL free surface treatment by BEM and FEM described above. The smooth particle hydrodynamics (SPH) approach can be considered as fully Lagrangian. In this method, the fluid domain is represented by a set of material particles which serve as physical carriers of fluid properties. An integral operator with a compact smoothing kernel is used to represent the average properties of the fluid at a certain location, which are used to satisfy the equations of fluid motion. Each particle interacts with nearby particles from a domain specified by the smoothing kernel (e.g., Gomez-Gesteira et ah, 2010; Violeau and Rogers, 2016). However, SPH does not directly refer to the equations of fluid motion in Lagrangian coordinates and should be distinguished from the methods where Lagrangian equations are directly applied to solve water wave problems.

The initial works on discrete approximation of equations of fluid motion in Lagrangian formulation with applications to water wave problems appeared in the early 70s. Brennen and Whitney (1970) used kinematic equations of mass and vorticity conservation for internal points of a domain occupied by an ideal fluid. Flow dynamics were determined by a free-surface dynamic condition. According to Fenton (1999) this approach apparently had not been followed and there are just a few works in which it was used (e.g., Nishimura and Takewaka, 1988). An alternative approach was developed by Hirt et al. (1970) who applied the equations of motion of viscous fluid in material coordinates moving together with the fluid. The next step was the development of an Arbitrary Lagrangian-Eulerian (ALE) formulation (Chan, 1975). ALE formulation uses a computational mesh moving arbitrarily within a computational domain to optimise the shape of computational elements and the problem is formulated in moving coordinates connected to the mesh. At certain regions of a computational domain the formulation can be reduced either to Eulerian (fixed mesh) or to fully Lagrangian (mesh moving with fluid) depending on the problem requirements. The Lagrangian models mentioned so far use quadrangular numerical cells. These models are subject to “alternating errors” and “even-odd” instability (Hirt et al., 1970; Chan, 1975), which is similar to the saw-tooth instability of the ALE approach. Moreover, application of fully-Lagrangian models to viscous problems has serious limitations. Boundary layers, wakes, vortices and other viscous effects lead to complicated deformations of fluid elements and large variations of physical coordinates over cells of a Lagrangian computational mesh. To address these problems the method was generalised for irregular triangular meshes (Fritts and Boris, 1979) and used for development of finite element models (e.g., Ramaswamy and Kawahara, 1987). This method however remains out of the mainstream and only occasionally appears in the literature (e.g., Kawahara and Anjvu, 1988; Radovitzky and Ortiz, 1998; Staroszczyk, 2009). Implementation of a finite element approach with irregular triangular meshes for ALE formulation (Braess and Wriggers, 2000) led to the development of a sophisticated method capable of solving complicated problems with interfaces including surface waves and fluid-structure interaction. A detailed description of the ALE method, examples of application and comprehensive bibliography can be found in Souli and Benson (2013). Finite element Lagrangian models and especially ALE models are complicated in both formulation and numerical realisation and are missing the main advantage expected from a Lagrangian method: simplicity of representing computational domains with moving boundaries. For many problems solved within the framework of ideal fluid, the deformation of the fluid domain remains comparatively simple. These problems can be efficiently approached by much simpler Lagrangian models similar to the original model of Brennen and Whitney

(1970). Recent examples of application of such a model include tsunami waves in a wave flume (Buldakov, 2013), violent sloshing in a moving tank (Buldakov, 2014) and evolution of breaking wave groups (Buldakov et ah, 2019).

A particular advantage of the Lagrangian models compared to the FNPT models is the ability to model vortical flows and, therefore, waves over sheared currents. Potential formulation assumes an irrotational flow and can, therefore, only be applied to a uniform current (Ryu et al., 2003; Chen et ah, 2017). Potential flow methods can also be generalised to flows with constant vorticity, which preserves the linearity of the problem. This allows the modelling of currents with linear profiles (e.g., Da Silva and Peregrine, 1988). On the other hand, in the inviscid Lagrangian formulation, vorticity does not change over time and can be generally defined as a function of Lagrangian labels. This allows a simple application to waves on arbitrary sheared currents. An example of such application can be found in Buldakov et al. (2015) and Chen et al. (2019). This feature of the Lagrangian formulation can also be useful for simulation of wave behaviour after breaking, which generates intensive vortical motion beneath the surface.

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