**1. Introduction**

Simulations of free surface flows have progressed rapidly over the last decade, and it is now possible to simulate the motion of complicated waves and their interactions with structures considering even deformable bubbles in turbulent flows. In the continuum mechanics, there are two methods to express the motion in the environment. The first description is the Eulerian approach. In this method, attention is paid to a special volume in the space. A mesh remains

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fixed in the Eulerian method and fluid regions change in shape and location on the mesh. It uses a fixed grid system which is not transformed during the solution procedure. The fluid is studied while passing this volume and continuously replaced in time. Therefore, this method is not appropriate for formulation of basic equations of fluid movement. The Eulerian method has some limitations. For example, when the portion of the perimeter to the area of a zone of fluid is large, the error of this method is increased. In the Eulerian method, it is not possible to decompose the equation on the boundaries with the same precision of inner region of fluid and accordingly, the finer mesh should be used near the boundaries. Therefore, when the free surface of a discontinuous region is modeled by this method, finer grid should be employed in order to achieve more precise results, specifically if this surface has large deflections. This is crucial when the portion of the area to the perimeter of a zone is low, for example on phase of a multiphase fluid. In this case, using finer mesh could increase the portion of the number of the inner elements to the boundary elements, which in turn, increases the precision of the numerical solution. The main superiority of the Eulerian description is the possibility of modeling of complicated surfaces. For example, the collapse of a column of a fluid could be modeled in the Eulerian grid which is shown in **Figure 1**.

**Figure 1.** Fluid column in Eulerian grid: (a) before collapse and (b) collapsing flow.

**Figure 2.** A sample of Lagrangian grid in vertical direction.

In Lagrangian method, the flow field of the considered fluid is covered by a mesh moving with the fluid. The fluid boundaries always coincide with the grid boundaries and the fluid inside each cell of the grid always remains in that computational cell. Although this method is not applicable to flows undergoing large distortions, where meshes can be twisted into unaccept‐ able shapes, but its advantage is the ease with which it handles free surfaces and interfaces, which makes it applicable to a wide variety of problems. For example, the grid shown in **Figure 2** is Lagrangian in the vertical coordinate. For free surface problems, if the free surface movement or the tangential acceleration gradient in the perpendicular direction to its surface is not large, the Lagrangian method can be used to simulate free surfaces. The grid lines are located on the free surface and move with it. Therefore, there is no need for any special boundary condition in this location [1].
