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The location of fluid within the grid is determined by a set of marker particles that move with the fluid, but otherwise have no volume, mass or other properties. This scheme is based on a fixed, Eulerian grid of control volumes. The earliest numerical method devised for time-dependent, free-surface, flow problems was the Marker-and-Cell (MAC) method (see Ref. Examples of this technique can be found in References 19. Finally, the application of free-surface boundary conditions is also simplified by the condition on the surface that it remains nearly horizontal. Further, only the height values at a set of horizontal locations must be recorded so the memory requirements for a three-dimensional numerical solution are extremely small. $latex \displaystyle \frac=w$įinite-difference approximations to this equation are easy to implement. This equation is a mathematical expression of the fact that the surface must move with the fluid:
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Time evolution of the height is governed by the kinematic equation, where (u,v,w) are fluid velocities in the (x,y,z) directions. Low amplitude sloshing, shallow water waves, and other free-surface motions in which the surface does not deviate too far from horizontal, can be described by the height, H, of the surface relative to some reference elevation. The remaining free-surface methods discussed here use a fixed, Eulerian grid as the basis for computations so that more complicated surface motions may be treated. References 19 may be consulted for early examples of these approaches. Even large amplitude surface motions can be difficult to track without introducing regridding techniques such as the Arbitrary-Lagrangian-Eulerian (ALE) method. The principal limitation of Lagrangian methods is that they cannot track surfaces that break apart or intersect. If this is not done, asymmetries develop that eventually destroy the accuracy of a simulation.
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Because the grid and fluid move together, the grid automatically tracks free surfaces.Īt a surface it is necessary to modify the approximating equations to include the proper boundary conditions and to account for the fact that fluid exists only on one side of the boundary. Many finite-element methods use this approach. Free-surface boundary conditions must be applied at the surface.Ĭonceptually, the simplest means of defining and tracking a free surface is to construct a Lagrangian grid that is embedded in and moves with the fluid.An algorithm is required to evolve the shape and location with time.A scheme is needed to describe the shape and location of a surface.Regardless of the method employed, there are three essential features needed to properly model free surfaces: In the following discussion we will briefly review the types of numerical approaches that have been used to model free surfaces, indicating the advantages and disadvantages of each method.
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Even then, free surfaces require the introduction of special methods to define their location, their movement, and their influence on a flow. For all but the simplest of problems, it is necessary to resort to numerical solutions. Whatever the name, it should be obvious that the presence of a free or moving boundary introduces serious complications for any type of analysis. In this case, however, the boundaries are phase boundaries, e.g., the boundary between ice and water that changes in response to the heat supplied from convective fluid currents. In heat-transfer texts the term ‘Stephen Problem’ is often used to describe free boundary problems. In other words, the gas-liquid surface is not constrained, but free. The only influence of the gas is the pressure it exerts on the liquid surface. In this sense the liquid moves independently, or freely, with respect to the gas. A low gas density means that its inertia can generally be ignored compared to that of the liquid. The reason for the “free” designation arises from the large difference in the densities of the gas and liquid (e.g., the ratio of density for water to air is 1000). An interface between a gas and liquid is often referred to as a free surface.