# The Free Surface Effect: An Overview Guide

### Key Takeaways

• The free surface effect appears in a vessel carrying a fluid in a partially filled space.

• Gravity causes the free surface effect in a fluid when the vessel heels over.

• In watercraft that are transporting fluids, this can cause the vessel to become unstable and possibly capsize.

Gravity seems so simple, but it creates interesting effects observed on earth and in the cosmos. When it comes to fluid dynamics, we sometimes ignore the effects of gravity to describe simple flows, but it is important when describing important points like buoyancy and slosh dynamics in watercraft. The free surface effect is one phenomenon that is driven by gravity and should be accounted for when designing watercraft. The goal is to ensure craft can remain stable and will not capsize should they begin to roll.

Once you are dealing with fluid flow, the free surface can exhibit its own flow characteristics that are distinct from the bulk. CFD simulation methods have been developed to address these aspects of fluid flow, which might involve sloshing or wave propagation along the free surface. To start, we’ll look at how the free surface forms in the presence of gravity, followed by the numerical techniques that can be used in CFD simulation tools to describe free surface flows, including in the presence of the free surface effect.

## The Free Surface Effect

Whenever a fluid is placed in a vessel, it conforms to the shape of its container. As a result of gravity being distributed evenly across the surface of a fluid, the resulting surface of the fluid is completely flat. Assuming the space is partially filled with the fluid and a gas (like air), this interface between the fluid and the gas is called a “free surface”.

The free surface effect describes what happens when the vessel or a liquid-carrying watercraft experiences an attitude change and tilts. If the craft maintains that orientation and the fluid comes to rest, it will maintain a flat surface that is orthogonal to the direction of gravity, while the bulk of the fluid will experience the same change in attitude. This is illustrated below.

When a watercraft that is carrying a fluid begins to roll, the center of gravity (C.O.G.) moves towards the heel, as shown in this image

Due to the motion of the fluid towards one side of the craft, the center of gravity also moves with the craft. The torque exerted by gravity is countered by the torque exerted by the buoyant force, the latter of which works to right the craft. If an attitude change is significant, and the buoyant force cannot exert sufficient torque on the surface of the craft to overcome gravity, the craft can roll and capsize.

The free surface of a fluid does not need to be stationary, meaning a flowing free surface can create the same problem, either due to free flow, sloshing, or the formation of large waves. In extreme cases, such flow could cause the craft to roll and capsize. The free surface itself can exhibit laminar flow, where it moves with the bulk fluid, or it could be turbulent if the Reynolds number is large. In either case, the flow regime can be estimated from the Reynolds number, which would need to be determined by analyzing the flow behavior with standard numerical techniques used in CFD problems. In watercraft, such as tankers hauling liquids, body forces can cause sloshing of the free surface and shallow water waves can arise, which are more complex types of flows. CFD simulation applications are used to address these problems and determine the behavior of the surface using a set of specific numerical methods.

## Simulation Methods for Describing the Free Surface Flow

Numerical techniques in CFD problems extend back to the 1960s, and several methods have been developed for solving fluid flow problems that involve free surfaces. The goal in these problems is to accurately calculate the fluid flow, the interface between the fluid and gas, and how the free surface changes over time. Some of the prominent numerical methods include:

• Marker-and-cell method - This is one of the earliest numerical techniques for calculating free surface flow. This method looks at the motion of a “marker” particle that follows “cells” of the fluid that make up the free surface. This approach can account for complex behavior like wave breaking, but it operates in 3D and requires high computational power.

• Surface marker method - This is related to marker-and-cell, but the simulation region is confined to the surface of the fluid, creating a 2D problem that can be solved with less computing power. Although this provides accurate results in 2D, the method will fail for compressible flow in 3D, as it cannot account for flow regions where the fluid may be expanding.
• Surface height method - This method uses a kinematic equation to describe the time evolution of the free surface height in 3 dimensions. The governing equation for the free surface is linear and 1st order, so it is easy to implement with finite difference in a CFD simulator.
• Lagrangian grid methods - This class of numerical techniques involves generating a time-dependent grid for the fluid, where the grid points perfectly track the flow of the free surface. This is in contrast to the surface height method, where grid points are static. This simulation approach is best used to describe low-amplitude free surface flows.

The other consideration in these simulations is the boundary conditions. These will be a mixed set of equations that impose momentum conservation at the free surface, a no flow condition, and dynamic equilibrium at the boundary of the free surface. CFD simulators will impose the relevant boundary conditions while calculating a solution to the Navier-Stokes equations in a CFD problem. CFD simulators are built to take data directly from a physical layout of the system under consideration, so you should pick an application that is adaptable to your system when you need a CFD simulation application.

The Cadence Omnis platform is ideal for defining and running CFD simulations and analyzing how the free surface effect arises in complex systems.