Tag: hydrodynamics

Why an RAO is the dynamic fingerprint of a hull

The Great Blue Hole in Belize looks mysterious from above. It stands out as a perfect dark blue circle – almost black – amid a shallow water atoll. It’s such a dark blue because it is a marine sinkhole. In other words, it’s a large cavern that expands over 100m below the surface. Yet, though it looks mysterious, there’s a lot we know about how it formed.

Thousands of years ago, during an ice age and shallow sea levels, a cavern formed. As it aged, this cavern grew in size and formed many giant stalagmites and stalactites. But eventually, the ice age waned, and sea levels rose. Finally, the weight of seawater collapsed the cavern roof and submerged it. How can we tell all this happened?

The key is the stalagmites and stalactites: they can not form underwater. They tell us the historical characteristics of the cave that became a marine sinkhole and act as a historical fingerprint.

Fingerprints reveal a lot of information. They can even capture a unique identity. When it comes to floating systems, a Response Amplitude Operator (RAO) acts just like a fingerprint. But in this case, the fingerprint gives a hint about how these floating systems respond to ocean waves. All the details of a hull go into making this unique dynamic fingerprint. In this article, we’re going to talk about an RAO and what it tells us about ships and floating structures.

The Great Blue Hole of the coast of Belize looks mysterious, but we know a lot about its history from its geological fingerprints.

What is an RAO?

The RAO shows how much a floating hull responds to ocean waves of different periods. A motion RAO shows how much the hull moves in each degree of freedom. For example, a heave motion RAO will show how much a hull will move up and down across a wide range of ocean wave periods. There will be a different RAO curve for each degree of freedom of the hull – the linear motions surge, sway, heave, and the orientation motions roll, pitch, and yaw.

An RAO gives a hint about motions at sea

The RAO can show you in a single view just how sensitive the hull will be to different ocean wave periods. Depending on the hull type, a few degrees of freedom are often susceptible to a dangerous resonance condition. A resonance condition is when the hull motions or accelerations may get very large. Either of those conditions can lead to injury, damaged equipment, or damage to the vessel itself.

An RAO allows comparison between two different vessels. RAOs can be helpful to understand what might be a more suitable ship for a particular operation. But it can also be beneficial feedback in the design process, where a designer can see how subtle changes to the hull form or load out will affect the overall ship response.

How is an RAO calculated?

A common way to produce RAOs is with a seakeeping analysis based on potential flow theory, such as ShipMo3D. These tools take a wide range of input on the hull shape, hull appendages, system mass and inertia, and then calculate the RAO.

Seakeeping software typically assembles the RAO from calculated steady-state sinusoidal ship movement in sinusoidal ocean waves. The magnitude of the RAO is then the steady-state ship motion amplitude in individual sinusoidal ocean waves. Ultimately, the RAO illustrates the variation in ship motion amplitude across a range of ocean periods.

Since the goal is to show the variation independent of wave height, the RAO may be nondimensionalized by individual sinusoidal wave heights for linear motions and wave slope for rotational movements.

While you get a lot of information from an RAO, it isn’t necessarily representative of motion in an actual sea state.

Ship motion in a sea requires more than the RAO

An actual wave state in the ocean is rarely just a single sinusoidal wave. Typically, there is a combination of many different ocean waves, each with a slightly different direction and period. One way to compute the expected ship motion response in a realistic sea state is with the RAO. But to do this, you need the RAO in combination with the ocean wave state spectrum.

The mathematical combination of the sea state and RAO produces the ship response spectrum in that sea. The ship motion spectrum then tells you precisely the characteristics of the ship’s motion in that sea state.

So the RAO is not in itself the absolute ship motion in a specific sea state. But it is very much a unique dynamic fingerprint. So this fingerprint is then what you can use to determine how the ship will move in a wide range of ocean wave states.

US Navy Ship in a heavy sea state. An RAO tells us a lot about the characteristics of a ship hull. But you need both the RAO and the sea state spectrum to predict ship motion at sea.

Are there other ways to compute ship motion response in the ocean?

Absolutely: physical scale model tests predict ship motion response. Commercial Computational Fluid Dynamics software tools resolve fluid physics much more accurately than potential flow methods. But there are advantages and disadvantages to each approach. Calculating an RAO and ship motion response with a potential flow tool like ShipMo3D is typically very fast – on the order of minutes – and generally cost-effective.

On the other hand, physical tank tests may reach tens or even hundreds of thousands of dollars depending on the testing required. Commercial Computational Fluid Dynamics software tools are sophisticated and powerful but incur high computational costs – and that means more time to compute ship motions or higher prices in using a more powerful computational facility to get answers faster.

Example time

In a previous article on seakeeping, we used a Generic Frigate to showcase ship motions in a particular sea state. Part of this process includes calculating the Generic Frigate motion RAO. The RAO for this Generic Frigate configuration at 10kts forward speed in a beam condition is below. The roll motion RAO, the middle plot on the right side, shows a peak around 0.6 rad/s or 10 seconds. The roll RAO peak reaches almost 3 here, which hints that the ship is pretty sensitive to waves around a period of 10 seconds in a beam loading condition.

Generic Frigate motion RAO summary at 10kts in beam condition. Note the roll motion RAO shows a peak around 0.6 rad/s or 10 seconds. This shows the ship is fairly sensitive to roll from ocean wave periods around 10 seconds.

Remember, these values indicate the ship hull response to a single sinusoidal wave. A ship motion time-series requires a combination of the RAO and a specific sea state spectrum. With a particular sea state spectrum, you can compute a ship motion spectrum and time series to predict specific motions and accelerations. A sample time series created from these RAOs in an irregular sea state is below.

A typical sample time series of the Generic Frigate motion in a beam sea condition with short crested, irregular seas.


The RAO shows how a particular hull will respond to a wide range of ocean wave periods. It’s a helpful calculation that helps with a comparison of different ships or how design or configuration changes affect a ship’s response. Most often, it’s a standard computation from commercial seakeeping analysis tools, like ShipMo3D. Calculating the ship motion response in a specific sea state needs the RAO combined with the sea state spectrum, producing the overall ship response spectrum. In this way, the RAO represents a dynamic fingerprint of a specific ship.

A marine sinkhole may look mysterious, but we have many clues about how they formed through their detailed rock formations. Similarly, RAOs provide valuable clues about predicting ship motion in any sea state – so you don’t leave safety at sea as a mystery.

Next step

ShipMo3D is an example of a seakeeping tool that you can use to calculate motion RAOs and better understand all ship motions in various sea conditions. Read more and apply for a free demo of ShipMo3D here.


Read more about the Great Blue Hole near Belize here.

Why drag force is more than just resistance

It’s tough to beat Nature. You don’t have to look in a top secret lab to find one of the most miraculously slippery fluids on the planet. You can just look at your knee! Lining many of your joints is an egg-white coloured substance called Synovial Fluid. Among its many purposes is to keep your joints moving. But not just for a few days – for your entire life. And through all this time, it manages to control and minimize joint resistance.

Certainly, when resistance gets out of control, everything can come to a grinding halt. When it comes to hydrodynamics, when you think of resistance, drag forces may be the first thing that comes to mind. But there’s more to drag than just resistance, and this is what we’re going to cover in this article.

What is the drag force?

Forces appear when there are changes in momentum. Momentum means mass in motion, so for the specific case of fluids, this could mean water or air flowing in a current. The drag force arises when there is a change in momentum in fluids. More specifically, the drag force transfers momentum between a structure and a fluid it is immersed in. This might sound like an abstract idea, but you already have some experience with it every day.

You feel this effect when your hand is in water

A big part of this is the drag force. But if you are in a pool or bathtub, the water isn’t moving around – it’s your hand – and so the drag force in this case will feel like resistance to you. In this case, it’s your hand that has momentum: it is the mass in motion. The drag force is then transferring momentum from your hand into the water.

You can feel drag when moving your hand around in water

What does momentum look like in water?

Well, if it’s your bathtub, there will be swirling and churning water – turbulence, and probably some waves. All this moving water is mass in motion, too. But the drag force does work in reverse, too – it can transfer momentum from water into structures.

What happens when there are ocean waves or currents in the water?

Ocean waves and water currents are also examples of mass in motion, too. If a structure, like a mooring buoy or ship gets in the way of moving water, there’s going to be some drag forces. These forces will transfer momentum from the waves and currents into the structure. So in this way, drag force is really about the exchange and transfer of momentum.

Sometimes this means it creates resistance and slows down a structure. But it can create motion in floating structures, too. The key is that drag is proportional to the relative velocity between a structure and the fluid.

The drag force is one of the essential forces in hydrodynamics

It acts like a resisting effect on many structures. If you are trying to understand what will happen to an ocean robot driving around in the water, it will have a big impact on energy consumption as well as how it can maneuver.

But the drag force is also a key element in mooring designs. In oceanographic mooring designs, the aggregate drag on all the components and the mooring line causes the system to deflect in an ocean current. In reality, the water current loses some momentum from this drag force from the mooring – the wake from the float and mooring components reduces the ocean current flow speed by some small amount.

In an ocean current with speed U, the drag force from all mooring components causes deflection.

Drag is also a key element in the excitation forces when ocean waves are around. You need to know these excitation forces to properly design a system to perform the way you want in the ocean environment. But knowing drag forces is one thing. Knowing how big they are is really the million-dollar question.

How do we know what the actual drag force is?

The drag force is measured from an experiment. These experiments might be a physical test in a lab, or in more modern times, they might be virtual tests using fluid dynamics software. These experiments resolve what the drag forces are in certain specific conditions. There have been hundreds of thousands of tests completed in laboratories for many decades measuring the drag force on a vast array of shapes in different flow speeds and fluid mediums. But how do we take all this information on drag force and then use it in a specific application?

The key is the drag coefficient

Similar shapes produce similar and predictable drag forces. The concept of a drag coefficient works well and covers a wide range of fluid types and conditions. Ultimately, if you’re looking at something like a sphere, you can use a drag coefficient associated with a sphere and predict reasonably well what the drag forces will be for a good range of wind or water current speeds. These drag coefficients are often available in look up tables to help the design process.

But what about the drag on mooring lines?

A mooring line is indeed a much more complicated structure than a sphere. Depending on its orientation to the flow, the local drag can change drastically. The drag forces on mooring lines require a calculation that considers the local flow speed and tilt of the line of the whole system. Suffice to say, it’s not a back-of-the-envelope type calculation you can do like that of the drag on a sphere! But this is what we have programs like ProteusDS Oceanographic that help address these complexities.

It’s summary time

The drag force is an effect that arises when momentum transfers between a fluid and a structure. Yes, a structure will feel resistance when it’s moving in a fluid, but the reverse is true, too: a moving fluid, like ocean currents and waves, can also cause a structure to move around, too – and the drag force plays a big role in that. The drag force is a very important effect that affects a range of systems from vehicle dynamics to oceanographic mooring design.

Even super low friction fluids like those in your knee joints have some drag effects, too. While you don’t need to do much over your life to keep your knee joints going, you do need to be mindful of drag when designing structures in the ocean.

Next step

Figuring out a drag coefficient for a particular structure isn’t always obvious. While ProtesuDS Oceanographic includes drag coefficients for a variety of shapes, you can learn more about different ways to resolve drag coefficients here.


When it is sensible to use nonlinear Froude-Krylov forcing

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.

Figure 1: Antonov A-40 flying tank was a bold idea but impractical. Picture credit: Wikipedia

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.

Figure 2: Triton C WEC showing hull, mooring, power umbilical, tendons and ring reaction structure. Picture credit: Oscilla Power

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.

Figure 3: Wetted area of any hull can change a lot in severe wave conditions

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.

Figure 4: Triton C WEC in extreme wave event during physical model tests at 1:10 scale. Picture credit: Oscilla Power

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.

When to avoid a static solver in oceanographic moorings

It was when the traffic light turned green, and my car lurched forward from a stop when it happened. A warning light flickered on the dash, but then quickly turned off again. Fortunately for me, it wasn’t a sign of disaster with the engine – it was just a light indicating more window washer fluid was needed. But the warning light wasn’t on all the time – yet.

Dashboard with warning light

The to-do list just got a bit longer

Now, it’s right at this stage that I turn into Mr. Scrooge with my window washer fluid. The little bit left in the tank is sloshing around, and I know if I use one tiny spritz more to clean my windshield, the light will stay on. And if the light stays on? Well, it’s another problem I need to solve on my long to-do list! So my short term solution? Never clean the windshield. It may seem counterintuitive because that’s exactly what washer fluid is supposed to be for – cleaning my windshield. But instead, I completely avoid it.

Likewise, there’s some things I also completely avoid in mooring analysis. In particular, when we’re looking for a static mooring profile, it may seem counterintuitive to avoid using a static solver. But there’s good reason to avoid them in some circumstances. In this article, we’re going to focus on when to avoid using a static solver and what to do instead.

A static solver can mean many different things depending on the problem you’re trying to solve

In the case of mooring analysis, one problem we’re trying to solve is to find the static deflection of a mooring system to steady loads. Most often, these steady loads are from effects like ocean currents and wind. In the case of mooring design, this is what we mean by a static solver.

A static solver is a great tool to have

Ideally, they work quickly to give you a solution that shows what the mooring deflection looks like to specific steady loads. These static solvers compute the tensions in the mooring lines, too. This feedback is really helpful in the early stages of the mooring system’s design, so you can narrow down the materials and concepts to find something workable.

Often, you may need to screen mooring designs to make sure the deflection isn’t too high and check the lines are strong enough to handle the loads. But timelines are often tight in these projects. So getting to the next step in mooring design in short order is crucial.

Finding a static mooring deflection may sound like a straightforward problem

But it can be tricky depending on the complexity of the mooring system. Because of this complexity, there’s no single static solver algorithm that’s useful for all mooring systems. Ultimately though, there’s a common idea behind these static solvers. The idea is that they look for a mooring configuration in which the external forces, such as those from ocean currents and wind, are in balance with the internal forces, such as the mooring tensions.

As a static solver goes through a solution process, it can make large jumps to test what the mooring profile looks like. These jumps aren’t random. But they certainly do depend on how the forces are balanced between the environmental effects and the mooring loads. These jumps are a static solver’s greatest advantage: when they work well, they jump instantly to the right solution! But then sometimes they also struggle. When they struggle, it looks like they jump around and around a potential solution, but never quite land on one. Or, in some circumstances, there just may not be any solution at all.

It’s when static solvers struggle that they lose their advantage of speed

Suddenly, the static solver process is churning away, but just not getting anywhere. It can take a long time to process, whirring away on calculations, and never end up with anything useful. We call this a failure to converge to a solution. Suddenly, the trusty tool that quickly got you to the next step has betrayed you!

Now what do you do?

One alternative is dynamic relaxation. Dynamic relaxation is a very simple and direct approach, and there’s only a few steps. First, you start with a basic crude guess for the mooring system layout. This crude guess might look like diagonal straight lines from the anchor to the fairlead. Next, you simply run a dynamic analysis solver to let the system respond dynamically to steady forces from ocean currents and winds. The mooring system “relaxes” and deflects naturally to its steady state configuration.

The advantage here is that it’s a typically very robust approach: there’s no guessing about mooring profiles – it evolves a realistic deflection of the mooring over time and eventually to a steady state profile.

Why don’t we use dynamic relaxation all the time?

It all has to do with time. Unfortunately, dynamic relaxation can be quite slow! The longer the mooring lines and the deeper the water, and the lower the forces involved, the longer it takes to relax and settle to a steady state profile. A full ocean depth mooring might take over 24 hours to compute a steady state profile by dynamic relaxation – but only a few minutes for a static solver. So when do you want to use dynamic relaxation?

It’s hard to know when to use dynamic relaxation ahead of time

But there are a few guidelines you can keep in mind. Generally, what you need to keep your eyes peeled for are strong forces or abrupt changes in the environmental effects. This might look like a very severe shear profile in the ocean current, or perhaps an abruptly varying sea bottom. But one of the most common effects you’ll see is the effect of shallow water. In shallow water, small changes in mooring deflection can mean big changes in forces acting on the mooring – from buoyancy if the lines come out of the water, or ground contact if they touch the seabed.

This is where those jumps that static solvers make in guessing the mooring profile can run into problems. It can take a long time to settle and often you may find it fails to converge to a solution. But the good news is that dynamic relaxation can really shine in shallow water conditions. After all, often, mooring lines are shorter in shallow water. It doesn’t take much time for a mooring to deflect a short distance and reach a steady state.

Wait a minute. What do you mean by shallow water?

It’s one of those grey areas. There isn’t an exact answer. But fortunately, it’s often easy to try both techniques in parallel – static solver and dynamic relaxation – so really it’s up to you to learn what works. I think you will find that there is a range where one works better, and a range where they both work OK, too.

NOAA COOPS Currents Buoya (CURBY) mooring in 10m water is pretty shallow water. Picture credit: Laura Fiorentino

No really, can you give me a specific water depth?

Ok. Fine. For oceanographic moorings, I would suggest definitely trying dynamic relaxation if your mooring is in less than 100m of water depth. But again, it’s ultimately up to you to judge what works best for your specific problem.

Let’s look at an example

The CURBY (CURents BuoY) is a shallow water mooring deployed in the Delaware River. It’s certainly shallow at just over 10m depth. But the currents can rip through there fairly quick. The mooring is a simple design with a surface buoy and a few different sizes of chain along the span. In this example, the ProteusDS static solver struggles to find a solution in certain environmental conditions. But the shallow water is so short, that a dynamic relaxation finds a solution in only about 20 seconds of simulation in a strong current.

CURBY mooring static deflection calculated by ProteusDS

It’s time for a summary

We covered a fair bit on static mooring solutions, so now it’s time to review. There are many different techniques that may be used in static solver algorithms. It’s common for them to use an iterative approach that jumps to a solution, continually seeking a balance of internal tensions with external environmental loads from ocean currents and winds. But sometimes they just don’t work. When they don’t work, either they take way too long or fail entirely to reach a solution. You may find this when there are big discontinuities in the environment, like a very strong current shear, or a shallow water depth. In these cases, you may consider dynamic relaxation as an alternative.

Dynamic relaxation is a process that uses a dynamic solver that lets the system evolve through time from the effects of constant loads. While it’s often much slower than static solvers, in shallow water conditions, it should be plenty speedy – and at least there are no problems like failing to converge. Think about using dynamic relaxation in 100m water depth or less – but always be sure to judge for yourself what works best.

You may not be as annoyed as me when the window washer fluid light goes on. Of course, it seems counterintuitive to avoid using the window washer in your car. In a similar way, avoiding static solvers in mooring design may seem counterintuitive. But fortunately, there are only a few circumstances in which you need to use a workaround.

Next step

Learn more about how static solvers work with an article using an example with a deep water oceanographic mooring example here.

Thanks to NOAA CO-OPS

Thanks to Laura Fiorentino from NOAA CO-OPS for sharing technical pointers and data for the CURBY mooring for the example.

How to effectively model oceanographic surface buoy dynamics

Flying sheep may help you get to sleep, but they can save your life, too. In 1934, the Italian army had a huge logistics problem: crossing one of the most inhospitable deserts in the world. Resupply for the trip was vital, and the army had to consider their options carefully.

Flying Sheep

One option was for soldiers to carry their supplies. But weighing down soldiers wasn’t possible: they would move too slowly under the oppressive sun and wouldn’t survive heat stroke.

Another option was trucking in supplies. But that wasn’t going to work either, as the stifling desert heat was already spoiling their food.

The army was left with only one viable option: using aircraft. Soldiers were able to travel quickly through the desert on foot, and supplies were sent by airdrop. Everything, including live sheep, was sent in by parachute! It was the only effective option.

In contrast, there’s often more than one effective option in modelling surface buoys. Surface buoys are a critical component of oceanographic mooring analysis and need to be considered by mooring designers carefully. Without careful consideration of dynamics caused by harsh ocean waves, the buoy and mooring won’t survive. In this article, we’re going to cover a few useful approaches for modelling surface buoys:

  1. point mass
  2. simple rigid body
  3. complex rigid body

First, we’re going to look into point mass approach.

The point mass model is one of the most straightforward approaches to use

They’re straightforward because they ignore all rotational effects of the buoy: a single point represents the hull. It can still account for the linear buoy motions: heave, surge, and sway. But it also means only the most basic shapes can be used to represent the buoy hull, like a sphere or in some exceptional cases, a cylinder.

It’s natural to underestimate simple models

While simple, point mass models can be very extremely useful. Often, a point mass model is more than capable enough to analyze an oceanographic mooring response in currents and waves. This level of detail may be all that’s needed to finalize a robust mooring design.

There are only a few critical parameters needed to get started with a point mass buoy model. Fundamentals like buoy mass, net flotation, and basic hull shape are required and sufficient for many simple systems. But in certain circumstances, buoy rotational effects can be significant. This takes us to the second point on modelling surface buoys: using a simple rigid body approach.

Like a point mass model, a rigid body model accounts for all linear motions

These are heave, surge, and sway. But the difference is that rigid body models also account for rotational effects: roll, pitch, and yaw. These rotation motions may be vital because they can have a significant impact on the equipment used on the buoy. Of course, this means more information is needed to get the model set up correctly.

SOFS buoy in ocean waves

SOFS buoy riding through ocean waves while on station in 4km deep water near Tasmania. Picture credit: Eric Schulz from Australia Bureau of Meteorology / IMOS


But some of this information is not easy to find. For example, most buoy spec sheets provide mass, but usually don’t contain information about what the buoy rotational inertia is. This lack of information is not a surprise because the rotational inertia can change depending on the specific buoy loadout.

This is where the simple rigid body model really shines

With a few assumptions and approximations, mooring designers can estimate many of the additional missing parameters. The first thing is to approximate the hull shape with something simple like a sphere or cylinder. The rotational inertia is easily calculated from these basic shapes and the mass of the buoy.

This is a significant improvement on the point mass model because there is some basis for the rotational effects. A rigid body model can also handle more detail like an offset mooring connection point below the hull. However, it’s still an approximation to the actual buoy shape.

Regardless, it’s certainly a starting point to understand rotational motion. Like the point mass approach, it doesn’t need a lot of information to set up the model. But the simple rigid body model relies on a lot of assumptions to help fill the gaps in knowledge. So what can you do if you have a lot more details on the buoy on hand and want to make use of it? This brings us to the third and final section on modelling surface buoys: using the complex rigid body models approach.

There are a lot of assumptions used in the simple rigid body approach

Complex rigid body models are about reducing some of those assumptions. Using a complex rigid body approach will undoubtedly require a lot more information. In some circumstances, a detailed CAD software model of the buoy may be available. This CAD model can help provide the specific buoy hull geometry. The CAD software can also offer more accurate details like rotational inertia and the location of the centre of mass.

However, it can take a lot of time to set up a detailed CAD model. On top of this, generating a custom mesh of a buoy hull can be a challenging task in its own right. Nevertheless, it is the most flexible and powerful approach to modelling surface buoys.

What about the computational cost of these approaches?

It’s reasonable to expect the computational cost to increase if the model complexity increases. This is always on the top of my mind because more computational cost means more time needed to compute a mooring design. However, in this particular case, it’s not so obvious.

An oceanographic mooring simulation tends to require a fair number of elements in the mooring line. The extra equations introduced between a rigid body and point mass model does not make much of a difference in the computational time. That said, some buoy hulls can have intricate shapes. These intricate shapes mean a lot of geometric detail needs to be used in hydrodynamics calculations to resolve the hull forces. In practice, a complex rigid body model with a detailed hull might take something like twice as long to compute a mooring solution when compared to the other models. But it all depends on how much detail is in the hull.

Let’s look at an example

We’ve looked at the Southern Ocean Flux Station (SOFS) system in previous articles. In 4km deep water off the coast of Tasmania, it uses a 3m diameter buoy at the surface. We set up a detailed configuration of the entire mooring and surface buoy in ProteusDS. The ProteusDS model allowed us to examine what happens to the mooring loads and buoy motions when using the three different model approaches.

The general environmental conditions used for this example were 0.5m/s surface current dropping to zero after a few hundred meters. The wave conditions were 3m significant wave height and 10 second spectrum peak period.

We set up separate ProteusDS projects using the three different surface buoy models configured as a point mass, simple rigid body, and complex rigid body models.

A ProteusDS ExtMassCylinder was used to represent the simplest approach – the point mass surface buoy. The diameter and length of the hull were set to encompass the flotation volume of the SOFS buoy. The mass and maximum wet weight were set equal to the SOFS buoy mass and maximum reserve buoyancy, respectively.

Surface buoy in waves

SOFS surface buoy represented by a point mass approach


The simple rigid body approach was represented in ProteusDS by the Rigid Body model along with a Cylinder Mesh hull. Just like the simplest model, the hull geometry was set to encompass the flotation volume of the SOFS buoy. The mass was set equal to the SOFS buoy mass. The rigid body inertia was approximated using a solid homogenous cylinder with the centre of mass right in the middle of the flotation volume.

SOFS surface buoy represented by the simple rigid body approach


The complex rigid body model was represented using a ProteusDS Rigid Body model along with a Custom Mesh hull. The hull mesh was formed based on the shape of the SOFS buoy hull. The rigid body inertia and centre of mass location were computed by a detailed CAD model provided by CSIRO mooring engineer Pete Jansen.

SOFS buoy represented by the complex rigid body approach


Each model approach produced 3 hour peak mooring tensions of 23kN, 24kN, and 25kN, respectively, in the set environmental conditions. This shows how well the simplest model did at driving the peak loads in this storm condition. But what about buoy motions?

The simple point mass model tells us nothing about the rotational motion of the buoy. But there are results from the simple and complex rigid body models. In the 3m significant wave height conditions, we compared standard deviation and maximum buoy tilt. Each rigid body model produced 5.8deg and 5.7deg standard deviation. The maximum tilts were 37deg and 39deg, respectively. Since they were so close together, it shows for this kind of buoy and mooring combination, the simple rigid body approximation does a reasonably good job.


The Italian army only had one option when crossing a desert to succeed: pass as quickly as possible and airdrop supplies (live sheep included). Fortunately, you have more than one option when analyzing buoys in your oceanographic mooring. Now it’s time to review them.

The first approach is the most direct and straightforward way using a point mass. This approach is the fastest to set up and easiest to use. It often does the job for mooring design. But it does not account for any buoy rotational effects.

When these rotational effects may be significant, it’s time to look at the second approach: the simple rigid body model. The simple rigid body approach is excellent when there’s limited information on the buoy available. It uses underlying approximations and assumptions to fill in gaps in details like those for rotational inertia.

If you have a lot more information on hand, and likely a CAD model to help out, the third approach can be useful: the complex rigid body model. Not for the faint of heart, it can take a lot more time to set up. But you can use more detailed buoy hull geometry, and the rotational inertia computed from a CAD tool for much more accuracy.

Next step: check out this video tutorial comparing point mass and rigid body models

DSA develops ProteusDS as a software tool to help evaluate oceanographic mooring designs. We post a variety of materials online, including video tutorials on our YouTube channel. Check out this video tutorial below showing more detail on a comparison of point mass and rigid body models in the software.


Thanks to CSIRO

Thanks to Pete Jansen from CSIRO Marine National Facility / IMOS for sharing technical pointers, sharing data, and helping assess the SOFS mooring system and buoy dynamics.