Posted on Feb 08, 2022
Ryan Nicoll

Why the added mass force can be confusing

Originally published on

Being invisible makes a lot of sense if you want to avoid predators. Many animals that live deep in the sea use this strategy. One example is the Glass Squid, which is almost entirely transparent. Almost. The big problem is that it’s impossible to be completely invisible.

Eyes absorb light to work, so they will always cast a shadow. And what about food? Food generally isn’t see-through either, so the Glass Squid’s stomach also makes shadows. Yet the Glass Squid has a fantastic ability to help with these problems: it creates an effect of counter-illumination around its eyes and body using light-generating cells. This counter-illumination creates the illusion of faint sparkling sunlight from the water surface. Predators may be looking right at the Glass Squid but will be confused and see something else entirely.

Similarly, you may see something else than what’s really there when looking at a complex problem. There’s certainly no shortage of complex and confusing problems in the realm of hydrodynamics. One facet that can be confusing is the concept of added mass.

The Glass Squid is almost invisible to predators – but not quite. But it creates an illusion with the use of twinkling lights.

Your hands can teach you a lot about added mass

Things feel a lot different when you wave your hand around in the air than in water: waving your hand around in water takes a lot more work! Hydrodynamically speaking, there’s more than one thing going on when you wave your hand around under the water. But one notable effect is added mass. This effect of added mass is something that directly affects accelerations. Whenever anything accelerates in the water, some of the water accelerates around it, too. You don’t feel this effect much waving your hand around in the air because air is so thin compared to water. So how can the concept of added mass be confusing?

The name itself is a little confusing

Make no mistake: added mass is a force – a force that is proportional to acceleration. The name added mass is common because it creates an intuitive link to the effect on a system. When a structure accelerates in a fluid, it acts as if it has an additional amount of mass – this so-called added mass. But the name itself is not the only confusing aspect of added mass.

Mass and weight are intricately linked

But they are two completely separate things. Weight is another example of a force from a gravitational field (usually from Earth in our ocean engineering work!). Mass is not a force and, instead, is a physical property. So when we talk about how water can “add mass” to a body, another part of the confusion is that this in no way contributes to adding to the structure’s weight. Far from it – it is an effect that is entirely separate from any gravitational field. Added mass does not affect the weight of a body in the water at all.

Ultimately, it is a force proportional to acceleration and it arises from the acceleration of some amount of water around an accelerating structure.

How much is “some amount of water”?

How much water is accelerated is not apparent or intuitive and needs to be measured somehow for every structure. For simple shapes, you might expect an added mass around the same amount of displaced water as the hull. This is very simply because the structure and water can’t occupy the same space at the same time, so water needs to accelerate out of the way. But this is not universally true, and the more complex the system, the more complex the resulting acceleration of the water around it. Nevertheless, there are ways of resolving what the effect of added mass is.

A structure and water can’t occupy the same space at the same time. When a shape like a sphere accelerates to the right, the water around it must accelerate around it to the left. This produces a force on the sphere proportional to acceleration: the added mass force.

Historically, scientists and engineers experimentally measured and recorded the characteristics of added mass for a wide range of structural shapes in the literature. This data is often in lookup tables of the dimensionless added mass coefficient. Engineers and designers can use lookup tables of added mass coefficients to account for this effect fairly easily. But new and complex structures may require new physical experiments or advanced software tools to evaluate the added mass.

Added mass does not always come up first when considering a design problem

Yet it can have a critical influence on dynamic systems. The added mass can significantly change how systems accelerate in the water. Because of this, it can shift natural periods of vibration of specific systems. For example, a subsurface oceanographic mooring may have a natural pendulum period that is significantly affected by the added mass of the top float. When the forces from ocean waves line up or even gets close to this pendulum period, the subsurface mooring can have substantial motions in the water, disrupting instrument measurements. If you don’t understand and account for added mass, you can’t anticipate when this effect will happen in reality.

But where do I find lookup tables for added mass coefficients?

Added mass coefficient lookup tables are harder to find than drag coefficient tables. However, there are still many resources around. Fortunately, common shapes like spheres used in oceanographic moorings are well known and even incorporated into software tools like ProteusDS.

Three dimensional added mass coefficients for several shapes. Added mass is proportional to the displaced volume and the added mass coefficient. Adapted from DNVGL RP-C205: Environmental Conditions and Environmental Loads.

A glass squid relies on invisibility to get by and survive in the wild. The counter-illumination trick helps cover the times it can’t be fully invisible, tricking predators into seeing something that isn’t there. Added mass is one of the more complicated effects in hydrodynamics that, without careful consideration, can make you see something that isn’t there, too.

Next step

For fully submerged hulls, added mass is usually a constant. But when a hull is approaching or floating at the water surface, added mass can change dramatically. Learn more on how added mass can change by the effect of wave radiation here.