In computational fluid dynamics, free-surface modelling (FSM) refers to the numerical modelling of a free surface—a freely moving interface between immiscible fluids—in order to be able to track and locate it.
Common methods used in free-surface modelling include interface-capturing approaches such as the level-set method and the volume of fluid method, as well as interface-tracking and mesh-based techniques such as Lagrangian prediction–correction schemes and arbitrary Lagrangian–Eulerian formulations, including surface-height (free-surface elevation) methods.
Free-surface modelling Free-surface modelling is a sub-discipline of computational fluid dynamics (CFD) and hydraulics concerned with the numerical simulation of liquids where the interface between the fluid and a gas (usually air) is free to move. Unlike internal flows in pipes, the boundary of a free surface is not fixed by the geometry of the container but is determined by the balance of forces within the fluid and gravity.
Overview In free-surface flow, the position of the interface is one of the unknowns that must be solved. The most common examples include waves in the ocean, the flow of water in a river, or the sloshing of fuel in a tank. The primary challenge in modelling these systems is tracking or capturing the moving boundary accurately while maintaining mass conservation.
Mathematical Formulation Free-surface models are typically governed by the Navier–Stokes equations. For shallow applications, such as coastal engineering or flood routing, the Saint-Venant equations are often used to reduce computational complexity by assuming the vertical pressure distribution is hydrostatic.
The boundary condition at the surface usually requires that:
Kinematic condition: No fluid particles cross the surface. Dynamic condition: The pressure at the surface is equal to the atmospheric pressure (neglecting surface tension in large-scale models).
Numerical Methods There are two primary approaches to handling the moving interface:
Interface Tracking This method uses a moving mesh that deforms to follow the free surface.
Pros: Very accurate representation of the boundary. Cons: Struggles with large deformations, such as breaking waves or splashing, where the mesh can become highly distorted.
Interface Capturing This method uses a fixed grid (Eulerian approach) and calculates the shape of the surface based on the fluid's properties within the cells. Common techniques include:
Volume of Fluid (VOF): Tracks the fraction of fluid in each cell. It is robust and excellent at maintaining mass conservation. Level-Set Method: Uses a distance function to define the interface, providing a smooth surface but often requiring "reinitialization" to prevent mass loss. Smoothed-Particle Hydrodynamics (SPH): A mesh-free method where the fluid is represented by discrete particles, making it ideal for highly fragmented flows like dam breaks.
Applications
Coastal Engineering: Predicting storm surges, tsunamis, and wave-structure interactions. Environmental Fluid Mechanics: Modelling oil spills, sediment transport, and river flooding. Industrial Design: Simulating the pouring of molten metal, ink-jet printing, and ship hull resistance. Aerospace: Analyzing fuel sloshing in rocket tanks during maneuvers.
References