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When it is sensible to use nonlinear Froude-Krylov forcing

Originally posted on dsaocean.com

There’s nothing about tanks that makes them look like they’re ready to fly. At the best of times, tanks are squat and stocky, mean-looking machines meant for slowly crawling over the worst terrain the ground has to offer. In World War 2, tanks were a critical element. Getting tanks to the field in time could mean the matter of winning or losing a battle. So could there be a way to get them to the battlefield in a hurry? Perhaps with this motivation in mind, it’s no surprise that a flying tank design was tested in World War 2.

The Antonov T-40 flying tank was a complex arrangement. The tank needed a crew inside and ready to go, so it needed a smooth descent. The wings were detachable so that it could be up and running quickly after landing, too. But the only way to make the whole thing work was to use a massive airplane to tow it. The tank would then detach from the towing craft and glide to its landing spot.

With all these modifications, and the most powerful bomber available to tow it through the air, the Antonov T-40 did indeed successfully fly and land. But it was a struggle to control: the tank was just too heavy and had too much drag. Ultimately, the idea was abandoned as it just wasn’t sensible.

Sometimes you need to explore ideas that might not seem sensible at first. In hydrodynamics analysis, nonlinear forces may seem like overkill. Nonlinear calculations are often complex and can be computationally costly. But sometimes, you need this kind of firepower to understand a critical problem. In this article, we’re going to talk about when you need to use nonlinear Froude-Krylov forcing.

In numerical hydrodynamics, the modifier “nonlinear” makes me anxious

It makes me anxious because the term “nonlinear” is really code for “things are about to get way more complicated”. And the problem is, there’s often a vast range of things that can be changed to introduce nonlinear details.

Fortunately, nonlinear Froude-Krylov forcing is straightforward

Because it’s so straightforward, it’s even intuitive. The Froude-Krylov force considers the effect of the dynamic ocean pressure field on a hull. In this context, nonlinear Froude-Krylov forcing is the detailed calculation of this pressure over the hull.

For a small hull in extreme wave conditions, the Froude-Krylov force may be calculated using only a single point to represent the entire effect. Now, when we talk about nonlinear Froude-Krylov forcing, we are just taking a big step up in detail in evaluating this force. We get a lot more detail in the Froude-Krylov force by adding up the effect of the detailed dynamic pressure field over the hull.

But to do this, you need a mesh of the hull

At each instant in time in the simulation, the incident ocean field’s dynamic pressure is resolved at each hull mesh panel. The dynamic pressure acting on each mesh panel area produces an individual force. The nonlinear Froude-Krylov force is the result of adding up all these little forces on all the hull mesh’s wet panels.

It is nonlinear because the wetted hull area can change

The reason it’s called “nonlinear” is because the portion of the hull that is wet is accounted for automatically. There’s no simplifying assumption about the size or shape of the hull. There’s also no assumption that the wetted area of the hull stays constant over time, either.

This nonlinear capability is a significant advantage

It directly addresses the limitations of other simpler methods that do not account for the change in wetted hull area, or that the hull shape is not so simple. In addition, the nature of the problem is easier to manage: you need to focus on making sure you have a decent mesh of your floater hull. This is a much more tractable problem to deal with than trying to evaluate the validity of assumptions on dynamic wetted hull area through the course of a simulation in ocean waves.

Another benefit of evaluating the Froude-Krylov force in this way is that you can also assess the static pressure field over the hull at the same time. This gives the effect of nonlinear buoyancy, which helps resolve stability and restoring forces in more detail.

When do you use nonlinear Froude-Krylov forcing?

There are certain conditions that you want to watch out for. These conditions are when you really should consider using nonlinear Froude-Krylov forcing. Ultimately, the conditions to watch out for come back to those fundamental assumptions made by simpler models. The biggest one to look out for is large changes in the wetted portion of the floater hull.

Large changes in the wetted hull area can happen in extreme ocean conditions

Large and steep ocean waves may result in the hull “digging” into an ocean wave, creating severe and rapidly changing forces on the hull. But extreme ocean conditions aren’t the only problematic condition to look out for.

The floater’s natural period can also drastically change the wetted hull area

Even in relatively mild ocean conditions, the floater may be moving a lot at a resonant period. If the floater is pitching or rolling significantly, large changes of the wetted area of the hull through time are possible.

Let’s look at an example

Oscilla Power develops Wave Energy Converter (WEC) technologies. WECs are often deliberately placed in areas with extreme wave conditions to help maximize power generation. They also may have resonant modes, in an effort to maximize power generation, in certain environmental conditions.

The Triton C WEC consists of a central floater hull with a three point mooring. A ring reaction structure is suspended from the floater hull by three tendons to facilitate power capture. There are countless design details that go into making something like a WEC. The mooring design, structural loads, and power capture are some of the details that can be resolved in a dynamic analysis software tool like ProteusDS.

In normal conditions, the top portion of the floater hull is dry. In this specific example, the Triton C is in severe wave conditions consisting of 5m significant wave height. In the picture below, the trough of a large individual wave approaching the floater shows how the wetted hull area can change drastically. This is an example of a specific scenario in which nonlinear Froude-Krylov loading can help provide more detailed information on the floater hull response and loading.

Does nonlinear Froude-Krylov loading provide all the answers?

All models are approximations of reality. Ultimately, dynamic analysis tools do not replace the need for physical tests – but they can help reduce the number of tests needed. More designs can be explored with tools like ProteusDS before costly physical tests need to be done. In tandem with numerical research, Oscilla also completes a wide range of physical tests to validate numerical models and confirm performance metrics, like power capture.

Why not use nonlinear Froude-Krylov forcing all the time?

It might seem like a good idea to just use nonlinear Froude-Krylov forcing all the time. It certainly has a few key advantages as we showed earlier. But there are a few challenges with this that you need to keep in mind.

Making a detailed hull mesh is not always easy. You may have a complex hull shape. Often, CAD tools are used to create a hull design, and they aren’t always set up to quickly create a decent mesh for hydrodynamics calculations. It may be a logistical challenge to make the hull mesh itself, and this can take extra time or need costly software to carry out.

The other problem is that there are many extra calculations needed to evaluate the pressure field. The more panels in the hull mesh, the more steps the simulation tool needs to go through in summing up the pressure field over the hull. All these calculations add up, and it results in a slower simulation time. When you’re trying to evaluate a structural or mooring design in a hurry, this can make it a challenge to balance the time needed to do so. It’s often a good idea to work with a simplified floater model in early stage design.

It’s time to review

When people talk about “nonlinear” models, it often means, one way or another, there’s a step up in complexity. That’s certainly the case with nonlinear Froude-Krylov forcing. But in a way, it’s an intuitive evaluation of the effect of the dynamic pressure field. A numerical mesh of a floater hull is the starting point.

The next step is counting up each force acting on every mesh panel from the ocean’s dynamic pressure field. It’s important to keep in mind if you are looking at simulations where the wetted hull area changes a lot – like in a resonant condition, or in extreme steep ocean waves. But the computational cost can be high, so it isn’t something you should use all the time.

The Antonov T-40 flying tank was a beast. A miraculous one at that, which really did fly once. But it just wasn’t practical to put into use. Nonlinear models may be intimidating and feel impractical. But nonlinear Froude-Krylov forcing is a straightforward and often critical aspect to consider in hydrodynamics.

Next step

Oscilla Power is focused on developing technology for harnessing energy from ocean waves. They use ProteusDS as one of their key tools in developing their WEC design. Read more on their technology approach and upcoming deployments on the Oscilla Power website here.

Thanks to Oscilla Power

It’s hard to talk about niche hydrodynamics topics without specific examples. Thanks to Brian Rosenberg and Tim Mundon from Oscilla Power for the collaboration through sharing data and discussion on hydrodynamics.