Professional baseball pitchers are able to make a baseball move left, right, down, and even up (sort of) to get it by the opposing batter. The physics behind this can be explained by the Magnus effect.

After leaving a pitcher’s hand, a baseball pitch is influenced by three forces: gravity, drag, and the Magnus force. Gravity pulls the ball downwards, drag slows the ball down, and the Magnus force… Well, that depends on the pitch. As the ball spins in its flight path, pressure variations form on it and the Magnus effect generates a force perpendicular to the motion of the ball in the direction of the spin. For a more detailed explanation of *how* the Magnus effect works, check out my colleague Ed Fontes’ great explanation in a related blog post on curving a soccer ball.

In baseball, a pitcher has more control over the spin of the ball than in any other sport. A pitcher can spin the ball to add topspin, backspin, or sidespin depending on the pitch they want to throw. The degree of difficulty in throwing each pitch varies based on the spin. The margin of error is lower as the difficulty rises.

Fastballs are the easiest pitches to throw as they are only slightly affected by the Magnus force. A four-seam fastball is a pitcher’s main pitch, thrown the most often. There is a natural tendency to add backspin to the ball as it is released. The backspin points the Magnus force up, causing the ball to fall slower than other pitches, imparting the illusion that the ball is rising. Other fastballs, such as the two-seam and cutter, are thrown with spin, but are moving too fast for the Magnus effect to change their position drastically.

*Fastball. Thrown by a right-handed pitcher, as seen by the batter. Baseball geometry created using COMSOL Multiphysics.*

Breaking balls are the pitches that most rely on the Magnus force to be effective. The curveball is the most important breaking ball and almost all starting pitchers in Major League Baseball (MLB) must have one in their arsenal. As the pitcher releases the ball, he snaps his wrist over the ball, putting immense amounts of spin on it. This causes the ball to break down and left diagonally (for a right-handed pitcher). If thrown correctly, the curveball can be devastatingly effective, causing the batters to look silly, either by making them swing at pitches in the dirt or even duck out of the way of pitches that end up in the strike zone.

*Curveball. Starts outside and then curves in for a strike.*

A slider is thrown with horizontal spin, causing the ball to break laterally (right to left for a right-handed pitcher). A screwball is thrown with similar spin to a curveball, except it breaks down and *right* instead of left (for a right-handed pitcher). There are other types of breaking balls that pitchers employ, but they are mainly variations of the pitches described here. For example, a 12-6 curveball breaks straight down, without any lateral movement.

*Slider. Looks like a fastball before sliding away out of the strike zone.*

*Screwball. Looks like a fastball before curving down and out of the strike zone.*

The knuckleball is the most majestic pitch of all and the Magnus effect is actually its enemy. A knuckleball is ideally thrown to rotate just once on its way to the catcher. The lack of spin causes a Karman vortex street to be formed behind the ball, as explained by Ed in the World Cup™ blog post. The Karman vortex street causes tiny fluctuations in the ball’s movement, enough to throw off a hitter. In fact, with a great knuckleball, not even the pitcher can know where it will end up. If a knuckleball is thrown with too much spin, it will be easier to hit because the Magnus effect causes both uniformity in the ball’s movement and a straight ball path. When thrown correctly, the ball dances from side-to-side, as seen in this GIF.

The key to effective pitching is mixing the different types of pitches together to keep the batter guessing and off balance. A pitcher also has to mask the pitches as they are thrown, in order to not tip the batter off to where the ball is going.

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The researchers at Argonne National Lab (Argonne) turned to multiphysics simulation and trial-and-error prototyping to optimize the effectiveness of their acoustic levitator. When we want to move an object, sound may not be the tool we would typically reach for. So how does it have the power to float or levitate objects in a lab setting? It’s all about combining forces in just the right way to create lift.

When sound vibrations travel through a medium like air, the resulting compression is measurable and real. By combining acoustophoretic force, gravity, and drag, the pressure is just enough to not only lift a material like liquid medicine, but to also allow the medicine to be positioned, rotated, and moved according to the needs of the operator.

*Pressure pockets created by waves between the transducers of the acoustic levitator do the heavy lifting on a particle scale.*

By keeping the droplets in a steady rotation, researchers are able to work on the chemical reactions while the medicine stays liquid and amorphous. This is key for creating a safe, steady environment where medicine will form correctly.

Every material and measurement in the acoustic levitator will change both whether the device works in its final design and how finely it can be adjusted according to the needs of the scientists who use it.

The geometry of the device includes two small piezoelectric transducers that stand like trumpets above and below the working area where medicine is created, like this:

*The acoustic levitator’s wave patterns are controlled by pieces of Gaussian profile foam located on evenly-spaced transducers.*

Possibly the most important part of the design is the Gaussian profile foam, which consists of polystyrene and coats the ends of each transducer. This foam works to remove acoustic waves that fall outside the required range. It acts as a filter to maintain even, well-defined standing waves.

Using COMSOL Multiphysics together with the Acoustics Module, CFD Module, and Particle Tracing Module, the team at Argonne modeled the acoustic levitator. Working cohesively with simulation, they were able to narrow down the shape of the acoustic field and location of floating droplets.

*The simulation above shows that at T = 0.75 seconds droplets formed from the particles. On the left, the simulation shows the expected particle distribution and on the right, a photograph depicts the actual distribution of the droplets.*

As advances in acoustic levitation expand, the ability to work with finer and finer chemical reactions will allow members of the pharmaceutical science community to expand their reach, perhaps discovering many new medicines with life-saving qualities.

- Learn more about how Argonne improved their acoustic levitation technology.

Almost all of the major scientific breakthroughs in modern society can be traced back to the development of a deeper understanding of materials on the atomic scale. From the advancement of computing and the Internet to developments in modern medicine, knowledge about materials and their physical properties has proven to be vital to the discovery of previously unthought-of inventions. But how do we gain this knowledge of something that takes place on the atomic scale? Neutron scattering is one of the methods that scientists use to investigate the positions and motions of atoms in condensed matter. By aiming a concentrated beam of neutrons at a sample, a lot can be learned about the structure and physical properties of the material. When the neutron beam hits the sample, some particles pass through the material, while others interact with the material and scatter away at an angle.

The deflection patterns and energies of the scattered neutrons can then be interpreted, revealing information about the material’s composition, properties, and behavior. Neutron scattering facilities, such as the High Flux Isotope Reactor or HFIR (pronounced High-FIR) at Oak Ridge National Laboratory (ORNL), are used by scientists and engineers for research into a variety of areas, including materials research, physical chemistry, earth science, and the medical field.

As far back as 1978, there have been conversion efforts in place to redesign research and test reactors currently using highly enriched uranium (HEU) to convert them to low enriched uranium (LEU) fuel. The first initiative to recognize the need for conversion was the Reduced Enrichment for Research and Test Reactors (RERTR) program, which began implementing programs for reducing the use of HEU worldwide.

In 2004, the U.S. Department of Energy began the Global Threat Reduction Initiative (GTRI). The GTRI’s mission is to reduce and protect the use of nuclear and radioactive material at civilian sites located worldwide by contributing to and helping to fund plans to convert reactors using HEU fuel. However, the approach to fuel conversion is different for each reactor depending on the type of research reactor, characteristics of the fuel used, and facility constraints, necessitating that ample research be put into the conversion of each reactor.

*The HFIR Gamma Irradiation Facility at ORNL, an experimental facility designed to irradiate materials with gamma radiation from spent fuel elements for studies into the effects of radiation on materials. Image courtesy of Oak Ridge National Laboratory, U.S. Dept. of Energy.*

So, what is the difference between HEU fuel and LEU fuel? Natural uranium is composed of about 99% ^{238}U isotope and about 0.7% ^{235}U isotope. In general, LEU fuel is defined as a type of enriched uranium that has a percent composition of uranium-235 of 20% or less. HEU fuel, on the other hand, is composed of 20% or more of uranium-235. So far, 20 U.S. reactors have been converted using LEU fuel composition that has already been tested and licensed. Work is currently being done to develop replacement fuels for the remaining five rectors that could not be converted using an existing LEU fuel.

HFIR is one such reactor that has yet to undergo conversion. This is because of the HFIR’s unique core design, high power density, and current fuel characteristics, which make conversion a challenging task. Additionally, since the HFIR provides one of the highest steady-state neutron fluxes of any research reactor in the world, the impact on reactor performance must be minimal in order to preserve the reactor’s current neutron scattering mission, as well as other missions, including isotrope production, irradiation facilities, and neutron activation analysis.

While research into fuel conversion of the HFIR is ongoing, the process is expected to take several years and HFIR is currently scheduled to be the last research reactor operating with HEU fuel in the United States. ORNL researchers are currently using multiphysics simulation to explore and verify new designs for the reactor running on LEU fuel.

In a recent article in *COMSOL News 2014*, I discussed some of the research currently taking place at ORNL for HFIR conversion. Dr. James D. Freels and his graduate student associate, Franklin G. Curtis, two research engineers at ORNL, are using the fluid-structure interaction (FSI) capabilities of COMSOL Multiphysics to explore new designs for the HFIR core. A schematic of the HFIR core, as well as a photo of the core, are shown below:

The fuel plates, depicted above, are one of the main components of the reactor core. They are used to control the distribution of velocity and temperature at which coolant enters and flows through the core. Curtis’ current project is to model the small fuel plate deflections and oscillations that could occur during reactor operation and determine how the deflections will change when the new LEU fuel is used.

Currently, a final fuel design has not been settled upon, but a few alternative fuel design have been proposed — and this fuel is used in Curtis’ analyses. The alternative fuel has a uranium-235 enrichment of 19.75%, as opposed to the reactor’s current fuel, which has an enrichment of 93%.

One of the factors that make modeling the HFIR’s core so complex is the involute shape of the fuel plates, which deform differently than the fuel plates in many other reactors. Here you can see a fuel plate model showing the exaggerated flow-induced deflection of the involute-shaped plate:

*Leading edge deflection of the involute fuel plate. Eigenfrequency analyses predict the “S” shape deflection of the involute fuel plate of the HFIR.*

Curtis’ research included an analysis of a similar proposed reactor, the Advanced Neutron Source Reactor (ANSR), which was used to validate both the COMSOL Multiphysics code and the current safety basis calculations of the HEU fuel.

The rest of their story is very fascinating. You can learn more about Curtis’ work and the research currently being conducted at ORNL in the full-length article “Researching a New Fuel for the HFIR: Advancements at ORNL Require Multiphysics Simulation to Support Safety and Reliability” in the 2014 edition of *COMSOL News*.

Ever since the first offshore wind farm was built off the Danish coast in 1991, offshore wind has been gaining in popularity. Just over two decades later, at the end of 2012, the European Union was producing enough electricity from offshore wind farms to power approximately five million households. In the coming decade, offshore wind farms are expected to generate nearly one fifth of the European Union’s power consumption, jumping from about 6.04 GW in 2013 to over 150 GW by 2030, according to a report by the European Wind Energy Association.

*Windmill park in Oresund between Copenhagen, Denmark and Malmo, Sweden. Photo credit: Ziad, Wikipedia Commons.*

With this huge increase in wind power expected, engineers are being called in to investigate the effect that offshore turbines could have on marine life. In a recent report conducted by Xi Engineering Consultants for the Scottish Government, Brett Marmo, Iain Roberts, and Mark-Paul Buckingham investigated how different types of wind turbine foundations affect the vibrations that propagate from the turbine into the sea, and ultimately how these vibrations could affect surrounding marine life. Also involved in the project were Ian Davies and Kate Brookes of Marine Scotland, who helped define the water depth, turbine size, and foundation types of the turbines modeled in the study based off of the types of turbines submitted to the Scottish Government for licensing permits. Additionally, Davies and Brookes helped identify the marine species most likely to be affected by offshore wind.

I recently interviewed Brett Marmo about the project. “In our research, we explored how different bases affect the noise that is produced by offshore turbines, and whether or not this noise was loud enough to be heard by marine life,” Marmo explained. “We studied three different wind turbine bases and examined the possible effect that noise could have on various types of local whales, porpoise, seals, dolphins, trout, and salmon.”

Vibrations produced by offshore turbines travel from the tower into the turbine foundation and are released as noise into the surrounding marine environment. “Because the noise is emitted at the interface between the foundation and seawater, it’s likely that the intensity and frequency of the noise will vary with the type of foundation used,” described Marmo. “Using finite element analysis, we modeled three identical wind turbines, only altering the structure of the foundation.”

Below, you can see the three most common foundation types: the gravity base, jack foundation, and monopile foundation. Generally, the jacket and gravity base are used in water 50 meters or deeper, while the monopile is generally not used at depths exceeding 30 meters. Due to the different structures, materials, and size of each of these bases, the vibrations that propagate through the base behave very differently, leading to noise produced with different frequencies and sound pressure levels (SPL).

*Three different foundation types are shown: a gravity base structure sitting on the seabed (left), a jacket with pin pile connections to the seabed (middle), and monopile placed onto the seabed with a transition piece (right).*

“Using simulation allowed us to model the noise produced by the foundations under identical operating conditions — something that we wouldn’t have been able to achieve by just taking measurements of in-service wind turbines,” says Marmo. “Without simulation, the different environments and wind loads that these turbines experience would have made it very difficult to determine if it was truly the foundation that was affecting the noise produced and not another unaccounted for variable.”

Before delving into the simulations, let’s first explore where it is that the noise itself comes from. Noise from wind turbines can come from two places; aerodynamic noise is produced by the blades slicing through the air, and mechanical noise is generated by machinery housed in the gearbox. Almost all of the noise produced by the blades themselves gets reflected back from the water’s surface due to the large refractive difference between the air and water, and does not enter the marine environment.

Therefore, the majority of noise is created by mechanical vibrations produced in the turbine’s gearbox and drivetrain by rotational imbalances, gear meshing, blade pass, and by electromagnetic effects between the poles and stators in the generator. Each of these noise sources produce vibrations with a different frequency, which then transmit down the turbine pole and into the foundation. Here is a table of the different frequencies produced and their origin:

Frequency | |
---|---|

Rotational imbalance of rotor | 0.05 to 0.5 Hz |

Rotational imbalance of high-speed shaft between gearbox and generator | 10 to 50 Hz |

Gear teeth meshing | 8 to 1000 Hz |

Electro-magnetic interactions in the generator | 50 to 2000 Hz |

*Frequency bands likely to contain vibration tones produced in the drive train of wind turbines. Table courtesy of Xi Engineering and adapted from their report*.

Once the vibrations enter the foundation, the amplitude of the noise emitted is affected by the size of the excitation force, the frequency of structural resonance, and the amount of damping in the structure. Additionally, higher wind speeds lead to increased torque acting on the rotor, likely meaning that higher noise is emitted.

“Understanding the effect of damping — the dissipation of vibration energy from a structure — was one of the key analyses conducted in our project,” described Marmo. “In general, steel structures such as the jacket foundation have less damping than those built from granular materials, such as the gravity base, which is made of concrete.” The amount of internal damping taking place within a structure will therefore affect the noise emitted by different structures. In order to determine how these factors affected the noise produced, Marmo and the team turned to simulation with COMSOL Multiphysics.

Noise is produced at the interface between the wind turbine foundation and seawater, where the vibration of the foundation oscillates water molecules to produce a pressure wave that radiates from the foundation as sound. Geometric spreading and absorption reduce the intensity of the sound as it propagates farther from the foundation, with high frequency sound being absorbed more quickly and low frequency sound absorbing slower and therefore propagating further.

Marmo analyzed each of the three foundations at three different wind speeds (5 m/s, 10 m/s, and 15 m/s) and found that typically, the higher the wind speed the louder the noise produced. A comparison of the average sound pressure level at a wind speed of 15 m/s at different frequencies for each of the three foundation types is shown below.

*At frequencies lower than 180 Hz, the monopile produces the largest amount of noise. Of the three foundation types, the monopile continues to produce larger SPL values up to 500 Hz. Around 600 Hz, all three foundation types become comparable in average 30 m SPL with the trend of the jacket foundation rising to become the noisiest at frequencies greater than 700 Hz.*

As the graph shows, the jacket base demonstrates the lowest sound pressure level of the three at low frequencies (around 200 Hz and lower). However, at high frequencies, the jacket produces the highest sound pressure level. The monopile and gravity base exhibit comparable sound pressure levels at lower frequencies, while at higher frequencies the gravity base produces the lowest sound pressure level of the three bases. The images below illustrate the sound pressure level around each of the three foundation types at the frequency at which the foundation produces the loudest noise.

Marmo and the team also created a far-field model that used a Gaussian beam trace model to analyze the distances at which a wind farm containing 16 turbines could be heard. As mentioned above, sound at lower frequencies tends to propagate farther than sound at higher frequencies. Additionally, ambient noise can mask the sound produced by wind turbines, making them nearly impossible to hear. This was also taken into account in Marmo’s analyses.

“We found that each of the different bases produced the loudest sound in the far-field at different frequencies,” described Marmo. “At a wind speed of 10 and 15 m/s, the monopile and gravity bases are audible at least 18 km away at most frequencies below 800 Hz, while the jacket is audible at 250 Hz 10 km away and 630 Hz at least 18 km away.” Here is a summary of these results:

The next step in the project was to determine the frequencies at which marine species could detect the sound and over what distances. Each of the different foundation types emitted different sound pressure levels at different strengths and frequencies. Since various marine animals have different hearing thresholds, this also had to be taken into account.

Cormac Booth and Stephanie King of SMRU Marine at St. Andrews University were the key marine biologists who analyzed the hearing thresholds of different marine species and determined whether or not the noise produced could affect the animal’s behavior.

*Hearing thresholds for dolphins, minke whales, porpoises, and seals.*

Of the species examined, the minke whale had the most sensitive hearing at low frequencies (less than 2000 Hz) and was able to hear the turbine from the farthest distances. “We predict that minke whales will be able to detect wind farms constructed of either monopile or gravity foundations up to 18 km away at most frequencies below 800 Hz and for all three wind speeds,” says Marmo. “On the other hand, bottlenose dolphins and porpoises are less sensitive to low frequencies. Dolphins can detect a wind farm on a gravity base 4 km away at wind speeds above 10 ms, but can only detect jackets and monopiles at close ranges of less than 1 km.”

You can view an example of the results found in Marmo’s report, showing the hearing threshold of a seal for different wind speeds and frequencies:

Determining behavior responses was harder to predict. Using a sensation parameter, Booth and King estimated the upper and lower ranges around the hearing threshold of each of the species. Then, they determined what percentage of animals could be expected to move away from the turbines within a certain sound pressure range.

Neither seal species nor bottlenose dolphins were predicted to exhibit a behavioral response to the sounds generated under any of the operational wind turbine scenarios. However, between around 4 kilometers and 13 kilometers, 10 percent of minke whales encountering the noise field produced by the monopile foundation were expected to move away. Overall, jacket foundations appear to generate the lowest marine mammal impact ranges when compared to gravity and monopile foundations.

What does this mean for the future of offshore wind power? Marmo and his team’s report found that there were little to no detrimental effects from wind turbine noise on marine species. Although more studies still need to be conducted, these findings demonstrate that the future of offshore wind is looking positive.

- Explore the full report by Xi Engineering “Modelling of Noise Effects of Operational Offshore Wind Turbines including noise transmission through various foundation types“
- “Offshore Wind May Provide One-Fifth of EU Electricity“
- Check out this resource: Cape Wind, a proposed farm that will likely become the first offshore wind farm in America

To optimize the Passive Vaccine Storage Device (PVSD), engineers at Intellectual Ventures, as part of the Global Good Program, turned to thermal and vacuum system modeling with COMSOL Multiphysics together with experimentation.

In the early development stages, they began with a design similar to a cryogenic dewar — a specialized vacuum container commonly used in the field. Typical dewars are able to store ice for a few days before it melts, which is not nearly enough time for long trips to remote destinations. Traveling from a source point to areas where people need vaccinations could take weeks depending on their locations. Long travel times in combination with extreme climates present major challenges for experts working in the medical community. The PVSDs need high-performance insulation to create the temperature-controlled environment required for vaccine storage. Each layer of the device can impact overall performance and is designed to add to its insulative strength.

*The Passive Vaccine Storage Device can hold vaccines in a temperature-controlled, easy-to-transport compartment for longer durations than ever before.*

The shell of the PVSD is made of multilayer insulation, which is similar to the materials used for temperature regulation in spacecraft. This design is especially necessary for areas of the world that get incredibly hot because the vaccines need to stay in a cool and narrow temperature range (between 0°C and 10°C). The multilayer insulation consists of several layers of reflective aluminum, a low conductivity spacer, and nonconducting vacuum space.

*When modeling the PVSD, the Intellectual Ventures team considered physics phenomena and design variables including heat transfer, outgassing, and hold time.*

Vaccines require cold chain storage, which entails proper handling from the moment they are manufactured up until they are administered to a patient. Live virus vaccines can quickly deteriorate as soon as they leave their temperature-controlled space, and inactivated vaccines can lose potency from very short temperature fluctuations. Each year, countless doses of vaccines are thrown away or rendered useless because they were not stored and handled correctly.

For experimental tests, the researchers used an environmental chamber to recreate extreme outdoor conditions. In addition to experimental evaluation, multiphysics models were implemented using the Molecular Flow Module and Heat Transfer Module with COMSOL Multiphysics to optimize the PVSD design with regard to thermal performance and hold time. The outside of the device, composed of metal, prevents air inflow and helps maintain the cool temperature within. Added rubber absorbs shock to protect the contents during bumpy travels. Inside the PVSD is a small insulating shell (pictured below) that contains several compartments where the life-saving vaccines are stored.

*The inner shell of the PVSD holds individual vaccine vials that aid workers can easily access without disrupting the vacuum space or controlled environment.
*

By breaking down geographical barriers for aid workers, many lives will be impacted by the PVSDs and their important cargo.

- Molecular Flow Module
- Heat Transfer Module
- Read the full-length article “Innovative Thermal Insulation Techniques Bring Vaccines to the Developing World” in the 2014 edition of
*COMSOL News*

Our test set-up consists of a copper coil wound around a laminated iron core with some plastic and aluminum parts for stability. A conventional computer fan is placed one meter away from it. The occurring electromagnetic losses have to be calculated as well as the turbulent non-isothermal fluid flow around the device. The iron core has an air gap, which is intentionally included in order to analyze the influence it has on currents inside the coil and aluminum parts.

*The inductor device.*

*Schematic of the test set-up.*

Engineers — especially those working within project deadlines — are always looking for the right balance between computational (and modeling) efforts and accuracy. Therefore, it is a good idea to start with thinking of a suitable simplification, since the aspect ratio of the model geometry is quite challenging.

The distance between the fan and device is roughly one meter, while the interior gaps between the copper winding are about 0.1 millimeters, resulting in an aspect ratio of 10,000. In order to keep the processing time as low as possible, we choose a submodeling approach. A first model with a simplified transformer geometry is used to calculate the large-scale flow field around the device. Due to symmetry, only half of the geometry is modeled. The results of this model are exported and used as an inlet condition for the following step.

*Streamline plot of the velocity field. This field was used as an inlet boundary condition in the detailed model (at the position of the slice plot).*

The geometry of the detailed electrical device is built in SolidWorks® software and imported into COMSOL Multiphysics® via the CAD Import Module. Only a small part is used for the non-isothermal flow calculation in the detailed submodel (about 400 mm by 900 mm). The electromagnetic part needs to be solved for an even smaller domain (200 mm by 200 mm).

The iron core is laminated in order to reduce eddy currents. We’ll use the same approach as described by TU Dresden & ABB. The material is homogenized and defined with an orthotropic electrical conductivity. This allows us to keep a single domain and a coarser mesh rather than resolving the lamination geometrically with all small plates.

Due to the alternating current at 500 Hz, inductive effects in the coil (skin and proximity effect) have to be resolved. Additionally, eddy currents in the aluminum plates and iron core will heat up the device.

Due to hysteresis, there are also some magnetization losses. These are quite small in comparison to the eddy current losses and are not explicitly solved for. The table below shows the magnetization losses as functions of the magnetic flux density Q_{mag} = f(B). We could use an interpolation function instead of solving hysteresis time-dependently.

Part |
Electromagnetic losses |
---|---|

Copper coil |
37.2 W |

Aluminum, eddy currents |
36.2 W |

Laminated core, eddy currents |
0.02 W |

Laminated core, magnetic losses |
0.004 W |

The device reaches a maximum temperature of 125°C on the backside of the coil.

Today, our task was to find the best solution for computing thermal designs of transformers. In the case of BLOCK Transformatoren, they decided that COMSOL Multiphysics was the most suitable for their application after comparing the handling and results of several simulation tools.

In the end, this model involved simultaneous solving for a maximum of 8 million degrees of freedom (DOFs), using a robust combination of direct and iterative solvers. Memory (RAM) usage peaked at 89 GB of memory.

In order to be able to solve highly complex models, they chose the Ready-to-Go+ (RTG+) package with a benchmarked cluster for optimal performance. With everything being set for advanced simulations at BLOCK, we can expect their products to be pushed even closer to the limit in the future.

*SolidWorks is a registered trademark of Dassault Systèmes SolidWorks Corp.*

The Beckham and Maradona curl obtained with the inside of the soccer cleat (football boot), and the curl by Eder, Nelinho, and Roberto Carlos with the outside of the cleat, is due to the *Magnus effect*. The effect is named after the scientist who first observed it in a laboratory in the 1850s. The Magnus effect explains the side-force on a sphere that is both rotating and moving forward. Here, we use it to analyze the World Cup™ match ball.

Like many kids around the world, my dream was to become a professional soccer player, or football as it’s called outside the U.S., Canada, and Australia. However, as a nerdy kid, I also had two other main interests: cars and science.

I remember being obsessed with the aerodynamics of cars in the early 1980s. In particular, the drag coefficient in competition and revolutionary aerodynamic designs between Audi and Ford. I was also keen to understand and achieve the specific curl of the ball when violently hitting it with the outside of my cleat. Engineering science was the glue that unified these interests later in life. Now, in preparation for the World Cup™, I will share some of my CFD analyses of the match ball with you.

The spinning of the ball has a stabilizing effect on the flow around it and thereby on its trajectory. But, let’s begin with a case where there is little or no spin.

If there is *no* spin, a Karman vortex street is formed behind the ball. The ball will then be subjected to force fluctuations with the detachment of vortices behind it. The wakes formed behind the ball not only increase the drag, they also contribute to the swerves that can be observed by anybody who has kicked a beach ball or been in the flight path of a knuckleball thrown by a baseball player. This semi-chaotic pattern can be partially explained in a transient simulation using the CFD Module.

The figure and animation below show the Karman vortices behind a ball rotating counter-clockwise with the same velocity at the equator as its forward movement, i.e., at the relatively low spin value. The animation is recorded on a corresponding 2D problem of a cylinder, which qualitatively shows the same effect.

As the rotation velocity increases, the stagnation points on the ball move together and eventually outside its surface. At this point, the velocity due to the rotation of the ball is perfectly balanced by the ball’s forward movement [1]. If it were not for the fact that the ball loses some of its momentum due to friction, there would be a steady solution for this problem, which is in contrast to the case of the lower spin value discussed above. At this stage, the flight of the ball is stabilized and is much more predictable — at least for the player shooting the ball or the goalie.

The figures below show the velocity and pressure fields around the rotating forward-moving ball and a rotating cylinder. The velocity at the equator is much higher on the side of the ball that rotates and slides the air past its surface. On the other side of the ball, the ball’s rotation and the air that has to pass the ball work against each other.

Due to the difference in speed and shear between the two sides of the ball, a pressure difference is also built up between the two sides. This causes a force that sucks the ball towards the side where the air velocity is higher, which is the *Magnus force* acting on the ball. This is also reflected in an increased lift coefficient with the spin.

Although the simulations I shared above help explain the Magnus effect, there is more to the flight of a soccer ball than the simulations under ideal laminar conditions. Being the center of attention in the most popular sport in the world, the ball and its design have been the subjects of plenty of research. Since the release of the Adidas® Jabulani for the FIFA World Cup 2010™ in South Africa, this research has become even more intense due to the ball’s revolutionary design.

The high drag coefficients for laminar flow are caused by boundary layer separation forming a low-pressure wake that slows down the ball in this flow regime. For higher flight velocities, the boundary layer becomes turbulent ahead of the separation point and remains attached further downstream on the rear side of the ball before the flow reverses. This leads to a narrower wake and a correspondingly lower drag coefficient. The phenomenon is generally referred to as the *drag crisis*, which is illustrated below.

The traditional soccer ball (shown above) has 32 panels: 20 regular hexagonal and 12 regular pentagonal panels. The Jabulani, on the other hand, had eight panels, as you can see in this finite element impression of the match ball:

The reduced number of the seams, shown in black, was compensated with grooves that roughened the surface. However, the aerodynamic properties of the Jabulani were quite different from the traditional ball.

The fact that the panels and seams were fewer and shallower on the Jabulani, compared to those on a traditional ball, increased the laminar flow region with high drag coefficients, while it decreased its drag coefficients at higher speeds. The relatively large width of the laminar regime compared to that of a traditional ball gave it the behavior of a beach ball for a wider velocity range, which was the subject of complaints from many goalies. Also, the pattern that the ball presents to the wind has an impact on the sudden changes in direction for these “knuckleball” kind of shots [2].

The new Adidas Brazuca®, the official ball for the FIFA World Cup 2014™ in Brazil, has *only six panels*. However, the total length of the seams is comparable to that of a traditional ball. Additionally, the depth of the seams is larger than for the Jabulani.

The drag coefficient as a function of the Reynolds number for the Brazuca® is, therefore, more similar to that of a traditional ball, as you can see in the graph below. Thus, the ball is expected to have a stable flight over a wider range of velocity due to the turbulence caused by the seams.

In association football, players like Ronaldo can hit a ball hard and consistently without spin, giving it a straight path far from the goal. That is due to turbulence and the swerving chaotic trajectory closer to the goal, when the flow starts to become laminar.

In contrast, the stable curl of the spinning ball, due to the Magnus effect, allowed players like Beckham and Maradona to repeatedly place a 30-meter cross within a radius of half a meter from the target.

The very hard hit together with the spin that Eder, Nelinho, and Roberto Carlos use, in combination with the transition between laminar and turbulent flow, can give the ball an ordered but peculiar trajectory, almost like a guided missile.

Right after impact with the outside of the cleat, when the ball has been accelerated to its top speed, turbulence around the ball and its small drag coefficient gives it a relatively straight trajectory. When the ball slows down, the relative spin becomes stronger and the Magnus effect becomes more pronounced. In other words, the ball first goes straight and then curves suddenly closer to the goal.

This combination of turbulence and the Magnus effect is observed in Roberto Carlos’ famous free kick in the game between Brazil and France in 1997. France’s goalkeeper, Barthez, did not even move until it was too late and a ball-boy standing several meters from the goal ducked his head. Both keeper and ball-boy thought the ball was going far outside the post!

Tip: You can confirm that this goal was not a coincidence in this YouTube video clip.

More incredible goals featuring the Magnus effect can be found in this clip, where players such as Messi, Ronaldo, Ibrahimovic, Ronaldinho, Beckham, Eder, Cruyff, and many others make use of this effect to cheat the goalkeepers. One minute into the clip, you can also see Nelinho’s goal against Italy that, in 1978, got me hooked on hitting the ball with the outside of my cleat. It was so fascinating that I would practice that one shot several hours per week…

In the early 1980s, car ads always featured the car’s drag coefficient. I have always wondered why this relevant measure disappeared from the published specifications for cars. Now, I can instead think of the curves for the drag and lift coefficients for the ball and picture their combination with the Magnus effect. Imagine that while you watch the incredible shots and goals in this year’s World Cup™.

- G. K. Batchelor, “An Introduction to Fluid Dynamics”, Cambridge University Press, ISBN 0 521 09817 3, see plate 12 at page 364 and forward and pages 424-427.
- J. E. Goff, “A Review of Recent Research into Aerodynamic of Sport Projectiles”, Sports Eng (2013), 16:p137-154.
- Download the model: The Magnus Effect

*Adidas and Brazuca are registered trademarks of adidas AG. COMSOL AB and its subsidiaries and products are not affiliated with, endorsed by, sponsored by, or supported by adidas AG.*

*World Cup, 2014 FIFA World Cup, and 2010 FIFA World Cup are trademarks of FIFA.*

Modern building construction requires efficient climate control, preferably using sustainable energy resources. Such resources can be solar collector systems for warm-water support or ambient air or geothermal-based heat pump systems that can be used for cooling in the summer and heating in the winter.

The geothermal applications mostly consist of pipe installations with water or brine as fluid to provide the heat exchange between the subsurface and a heat pump. There are two main types of these so-called closed-loop systems: vertical borehole heat exchangers (BHEs), which are installed in boreholes (see part 1 of this series); and horizontal heat collectors, which are deposited on large areas and in very shallow depths of only one or two meters.

While the vertical BHEs are often simplified as infinite thermal line sources or sinks, the horizontal systems are more complicated from a modeling point of view. This is because they have to be arranged in patterns to cover large surface areas, and the use of this simplification does not match anymore. The total length of the pipes can easily reach 100 meters or more. The flow regime is preferably turbulent because of the resulting lower thermal resistivity between the fluid and the subsurface. You can imagine that a CFD simulation approach of systems with these dimensions is a challenge (despite the computational performance of modern HPC clusters). Fortunately, there is a solution for this kind of task.

Let’s have a look at how the Pipe Flow Module can be used in the growing sector of geothermal heat collector systems.

Thankfully, we do not have to care about the computational difficulties that may arise when simulating long pipe systems. The Pipe Flow Module provides all necessary capabilities to calculate pressure drops, velocities, and foremost — the heat transport between the pipe fluid and the subsurface, using prescribed 3D curved-edge correlation functions. All we need to do is to define the inlet conditions (temperature, velocity) and the pipe, fluid, and subsurface properties. Therefore, we can completely focus on questions like, “What do you think is the best horizontal pipe arrangement underneath your garden?”

Let’s find out.

The heat transfer example, provided in a recent blog post, shows how a prescribed heat extraction rate can be realized in the form of a numerically-calculated inlet temperature boundary condition. By doing so, any desired heat extraction rate can be given and the corresponding inlet temperature can be computed. Assuming that the fluid properties are temperature-independent, a simplified version of the inlet temperature can be written as:

(1)

T_{in}=T_{out}-\frac{P(t)}{\rho_{f} \cdot C_{P,f} \cdot \dot{V}}.

Thus, it is a function of the outlet temperature (T_{out}), density (\rho_f), heat capacity (C_{p,f}), and volumetric flow rate (\dot{V}) of the working fluid in the pipes. The temperature difference between the inlet and outlet temperature is controlled via the heat extracted by the heat pump, P(t), which is a function of time because heat pumps (for house heating purposes) usually do not run all day. A typical heating rate of a one-family house heat pump is 8 kW, where 2 kW of electric power is needed to run it and 6 kW are to be extracted from the subsurface. To reach a daily heat demand of, say, 48 kWh during the winter season in Germany or North America, the heat pump needs to run six hours per day.

*6 kW of heat extraction for six hours per day, in a three-day cycle.*

The number of possibilities of how the heat collector pipes can be arranged in the subsurface is endless. Here, we will have a look at three randomly chosen patterns. Let’s call them snake, snail, and meander designs. The heat collector is embedded in the subsurface with typical thermal properties that you could find in the uppermost soil layer in your garden. The subsurface temperature distribution corresponds to that of the temperature in Germany during the month of January.

*Three different layer designs for a geothermal heat collector system: 1) snake, 2) snail, and 3) meander design.*

The COMSOL Multiphysics feature, *geometry subsequence*, allows us to create all of the three different pipe geometries within the same model. It also allows us to perform a parametric study that solves for each particular sub-geometry. We can compare the different designs by having a look at the resulting outlet temperature as a function of time.

*Comparison of the outflow temperatures of the three collector designs.*

The three collectors each show a different thermal behavior. This is due to the varying distances between the pipes, their different lengths, and thereby the surface available for heat transfer. Even though the freezing point of the water-glycol mixture inside the pipes is at about -13°C, the outflow temperature is — for environmental reasons — legally limited to stay above -5°C in Germany.

The snake design tends to go below this point only after a few days and would, thus, not be the preferred design here. However, a geothermal engineer could now proceed to optimize a design (modify the pipe length, pipe diameter, or the covered area to reach a maximum outlet temperature, for example.) One could, of course, also easily perform long time runs to study the impact of seasonal temperature variations or even use the pipes for cooling purposes in the summer.

Once the optimal design of the geothermal heat collector in your garden is found, why not go on and model the modern underfloor heating system in the house? The procedure is straight-forward. There is only a change of sign because heat is injected here.

*Combined model of a garden heat collector together with an underfloor house heating system. The collector extracts heat from the subsurface at 6 kW, while the house heating system injects heat into the floor of the house at 30°C.*

- Read part 1 of the Geothermal Energy series: Modeling Geothermal Processes with COMSOL Software
- Read part 2 of the Geothermal Energy series: Coupling Heat Transfer with Subsurface Porous Media Flow
- Read about designing heat exchangers
- Explore the Pipe Flow Module

One of the greatest challenges in geothermal energy production is minimizing the prospecting risk. How can you be sure that the desired production site is appropriate for, let’s say, 30 years of heat extraction? Usually, only very little information is available about the local subsurface properties and it is typically afflicted with large uncertainties.

Over the last decades, numerical models became an important tool to estimate risks by performing parametric studies within reasonable ranges of uncertainty. Today, I will give a brief introduction to the mathematical description of the coupled subsurface flow and heat transport problem that needs to be solved in many geothermal applications. I will also show you how to use COMSOL software as an appropriate tool for studying and forecasting the performance of (hydro-) geothermal systems.

The heat transport in the subsurface is described by the heat transport equation:

(1)

(\rho C_p)_{eq} \frac{\partial T}{\partial t} + \rho C_p {\bf u } \cdot \nabla T = \nabla \cdot (k_{eq} \nabla T ) + Q + Q_{geo}

Heat is balanced by conduction and convection processes and can be generated or lost through defining this in the source term, Q. A special feature of the *Heat Transfer in Porous Media* interface is the implemented *Geothermal Heating* feature, represented as a domain condition: Q_{geo}.

There is also another feature that makes the life of a geothermal energy modeler a little easier. It’s possible to implement an averaged representation of the thermal parameters, composed from the rock matrix and the groundwater using the matrix volume fraction, \theta, as a weighting factor. You may choose between volume and power law averaging for several immobile solids and fluids.

In the case of volume averaging, the volumetric heat capacity in the heat transport equation becomes:

(2)

(\rho C_p )_{eq} = \sum_{i} ( \theta_{pi}\rho_{pi}C_{p,pi})+(1-\sum_{i}\theta_{pi})\rho C_p

and the thermal conductivity becomes:

(3)

k_{eq}=\sum_{i} \theta_{pi} k_{pi} + ( 1-\sum_{i} \theta_{pi} ) \rho C_p

Solving the heat transport properly requires incorporating the flow field. Generally, there can be various situations in the subsurface requiring different approaches to describe the flow mathematically. If the focus is on the micro scale and you want to resolve the flow in the pore space, you need to solve the creeping flow or Stokes flow equations. In partially saturated zones, you would solve Richards’ equation, as it is often done in studies concerning environmental pollution (see our past Simulating Pesticide Runoff, the Effects of Aldicarb blog post, for instance).

However, the fully-saturated and mainly pressure-driven flows in deep geothermal strata are sufficiently described by Darcy’s law:

(4)

{\mathbf u} = -\frac{\kappa}{\mu} \nabla p

where the velocity field, \mathbf{u}, depends on the permeability, \kappa, the fluid’s dynamic viscosity, \mu, and is driven by a pressure gradient, p. Darcy’s law is then combined with the continuity equation:

(5)

\frac{\partial}{\partial t} (\rho \epsilon_p) + \nabla \cdot ( \rho {\bf u} ) = Q_m

If your scenario concerns long geothermal time scales, the time dependence due to storage effects in the flow is negligible. Therefore, the first term on the left-hand side of the equation above vanishes because the density, \rho, and the porosity, \epsilon_p, can be assumed to be constant. Usually, the temperature dependencies of the hydraulic properties are negligible. Thus, the (stationary) flow equations are independent of the (time-dependent) heat transfer equations. In some cases, especially if the number of degrees of freedom is large, it can make sense to utilize the independence by splitting the problem into one stationary and one time-dependent study step.

Fracture flow may locally dominate the flow regime in geothermal systems, such as in karst aquifer systems. The Subsurface Flow Module offers the *Fracture Flow* interface for a 2D representation of the Darcy flow field in fractures and cracks.

Hydrothermal heat extraction systems usually consist of one or more injection and production wells. Those are in many cases realized as separate boreholes, but the modern approach is to create one (or more) multilateral wells. There are even tactics that consist of single boreholes with separate injection and production zones.

Note that artificial pressure changes due to water injection and extraction can influence the structure of the porous medium and produce hydraulic fracturing. To take these effects into account, you can perform poroelastic analyses, but we will not consider these here.

It is easy to set up a COMSOL Multiphysics model that features long time predictions for a hydro-geothermal application.

The model region contains three geologic layers with different thermal and hydraulic properties in a box with a volume V≈500 [m³]. The box represents a section of a geothermal production site that is ranged by a large fault zone. The layer elevations are interpolation functions from an external data set. The concerned aquifer is fully saturated and confined on top and bottom by aquitards (impermeable beds). The temperature distribution is generally a factor of uncertainty, but a good guess is to assume a geothermal gradient of 0.03 [°C/m], leading to an initial temperature distribution T_{0}(z)=10 [°C] – z·0.03 [°C/m].

*Hydrothermal doublet system in a layered subsurface domain, ranged by a fault zone. The edge is about 500 meters long. The left drilling is the injection well, the production well is on the right. The lateral distance between the wells is about 120 meters.*

COMSOL Multiphysics creates a mesh that is perfectly fine for this approach, except for one detail — the mesh on the wells is refined to resolve the expected high gradients in that area.

Now, let’s crank the heat up! Geothermal groundwater is pumped (produced) through the production well on the right at a rate of 50 [l/s]. The well is implemented as a cylinder that was cut out of the geometry to allow inlet and outlet boundary conditions for the flow. The extracted water is, after using it for heat or power generation, re-injected by the left well at the same rate, but with a lower temperature (in this case 5 [°C]).

The resulting flow field and temperature distribution after 30 years of heat production are displayed below:

*Result after 30 years of heat production: Hydraulic connection between the production and injection zones and temperature distribution along the flow paths. Note that only the injection and production zones of the boreholes are considered. The rest of the boreholes are not implemented, in order to reduce the meshing effort.*

The model is a suitable tool for estimating the development of a geothermal site under varied conditions. For example, how is the production temperature affected by the lateral distance of the wells? Is it worthwhile to reach a large spread or is a moderate distance sufficient?

This can be studied by performing a parametric study by varying the well distance:

*Flow paths and temperature distribution between the wells for different lateral distances. The graph shows the production temperature after reaching stationary conditions as a function of the lateral distance.*

With this model, different borehole systems can easily be realized just by changing the positions of the injection/production cylinders. For example, here are the results of a single-borehole system:

*Results of a single-borehole approach after 30 years of heat production. The vertical distance between the injection (up) and production (down) zones is 130 meters.*

So far, we have only looked at aquifers without ambient groundwater movement. What happens if there is a hydraulic gradient that leads to groundwater flow?

The following figure shows the same situation as the figure above, except that now there is a hydraulic head gradient of \nablaH=0.01 [m/m], leading to a superposed flow field:

*Single borehole after 30 years of heat production and overlapping groundwater flow due to a horizontal pressure gradient.*

- Explore the Subsurface Flow Module
- Part I of the Geothermal Energy series: Modeling Geothermal Processes with COMSOL Software
- Related papers and posters presented at the COMSOL Conference:
- Hydrodynamic and Thermal Modeling in a Deep Geothermal Aquifer, Faulted Sedimentary Basin, France
- Simulation of Deep Geothermal Heat Production
- Full Coupling of Flow, Thermal and Mechanical Effects in COMSOL Multiphysics® for Simulation of Enhanced Geothermal Reservoirs
- Multiphysics Between Deep Geothermal Water Cycle, Surface Heat Exchanger Cycle and Geothermal Power Plant Cycle
- Modelling Reservoir Stimulation in Enhanced Geothermal Systems

The *Wall Distance* interface calculates the reciprocal distance to selected walls. The value will be small when the object is far away from the respective walls and larger when closer. The exact distance, D, to the closest wall can be found by solving the Eikonal equation:

|\nabla D|=1

where D=0 on selected walls and \nabla D\cdot n=0 on other boundaries. COMSOL Multiphysics solves for a modified version of the Eikonal equation, where the dependent variable is changed from D to G=1/D and an additional smoothing parameter, \sigma_w, is used.

This results in the following equation:

\nabla G \cdot \nabla G + \sigma_wG(\nabla\cdot\nabla G) = (1+2\sigma_w)G^4

with G=G_0=2/l_{ref} on selected walls and homogeneous Neumann conditions on the other boundaries. Here, l_{ref} is a parameter that depends on the geometric shape and is calculated automatically. This parameter can also be defined manually, if necessary.

The resulting wall distance, D_w=1/G-1/G_0, and the direction to the nearest wall are available in COMSOL Multiphysics as predefined variables.

Once we add the *Wall Distance* interface to a model, we simply have to add a Wall boundary condition and select the walls from which we want to calculate the distance. Below, you can see an example where we are interested in knowing the distance to the bottom and right-hand walls of a rectangle:

*Boundary selection of the *Wall Distance* interface.*

Next, we plot the reciprocal wall distance, G, and the wall distance, D_w, and the direction toward the nearest wall:

*Reciprocal wall distance (top figure) and wall distance with arrows showing the direction towards the nearest wall (bottom figure).*

The effect of the smoothing parameter, \sigma_w, can be seen by plotting the wall distance from the top wall to the other two:

*Wall distance at the top wall for different values of the smoothing parameter, \sigma_w.*

For the first half of the top wall (*x* between 0 and 1), the distance to the nearest wall is constant, while it decreases linearly for *x* between 1 and 2. By solving the model for different values of \sigma_w, we can see how, for smaller values of \sigma_w, the wall distance is accurately computed. For larger values, there is a loss in accuracy, but a much smoother transition from the constant value to the linear decrease. Depending on the application being modeled, you can choose the value of \sigma_w that provides the desired accuracy, smoothness, and model convergence.

Now we’re ready to combine the *Wall Distance* interface with another interface. Let’s consider a flow channel containing a solid object subjected to the action of pressure and viscous stress due to fluid flow and a spring pushing it down. If the sum of the forces acting on the object is zero, it will stay still. If the force of the spring is larger than the fluid pressure and viscous stress, then the object will move down and either reach a new equilibrium position or close the channel.

*Schematic of the model.*

The fluid load on the object is included using the Fluid-Solid Interface boundary condition that comes with *Fluid-Structure Interaction* interface, which solves for both the fluid and deformable solid domain, as well as their interaction at the boundaries. To reduce the overall model size, we can build a 2D axisymmetric model and represent the spring by a boundary load depending on the position of the object.

The mesh inside the fluid domain is free to move and deform in order to adjust to the motion of the object. The geometric change of the fluid domain is automatically accounted for in COMSOL Multiphysics. In this approach, we’re not interested in the contact forces between the object and the channel walls as we are only investigating how easy it is to couple different interfaces and create custom functions. We will then close the channel by increasing the viscosity in areas where the object is close to the channel walls. Doing so will stop both the flow and movement of the object.

How, then, do we exclusively increase the viscosity in areas where the object is close to the channel walls? You could, of course, leverage user-defined functions. But in this case, you have to know where and when the object will reach the wall and find smooth functions representing this area. To do this, we will use the *Wall Distance* interface, which can detect the area and trigger changes in the material settings to increase the viscosity. This will enable the object to move in any direction, but increase fluid viscosity when it is likely to hit any of the walls at any time.

We will add two instances of the interface: one on the boundaries of the object (with the dependent variable G2) and another on the walls of the channel (with dependent variable G3). Detection is performed through a function that depends on the sum of both variables (G2 + G3).

As we can see in the below screenshots (all using the same color range), the sum of G2 and G3 will be higher if the object is close to the channel wall. The maximum value for each wall distance is 2,667 1/m (left and middle column). Looking at the sum (right column), the maximum depends on the position of the object. In order to allow the increase in fluid viscosity to “stop” the flow and control the mesh deformation, we set a condition that if the channel is open, the maximum is about 4,000 1/m, and if it is closed, the maximum is more than 5,000 1/m.

*Reciprocal wall distance G2 (left column), G3 (middle column), and sum of G2+G3 (right column) in the opened (top row) and closed (bottom row) state.*

The viscosity is now increased in areas where the maximum is more than 5,000 1/m. To improve convergence, the viscosity change that kicks does so through a smooth ramp function. So, we have two parameters: one for the slope of the function and another for the smoothing of the function.

This function is added to the viscosity of the fluid:

*Manipulation of the dynamic viscosity to account for a smooth increase in viscosity due to the wall distance. The equations are entered directly into the interface’s edit field.*

You can view the results in the animation below. The viscosity of water is about 1×10^{-3} Pa*s (blue in color), while areas with a viscosity higher than 1 Pa*s are red in color.

*Fluid viscosity change over time and dependent on the distance between the object and wall.*

We start with a zero pressure at both the inlet and outlet, and place the object in the closed position at the beginning. A time-dependent function is used to vary the inlet pressure. We will increase the pressure for a short period of time to see how the object reacts to this pressure. In the following animation, we can see both the flow in the channel with the moving object and the inlet pressure, which varies between 0 and 2 mbar. In addition, you can see the resulting mass outflow of the outlet (top boundary).

*Streamline plot of the velocity field with its magnitude represented as the color expression (left). The imposed inlet pressure (blue line) and mass flow at the outlet (green line) are shown in the right figure.*

We can see that there is a small time delay between the pressure increase and the opening of the channel due to the inertia resulting from the spring. If the pressure is too small, the object will not move and the channel stays closed. After opening the channel and then decreasing the pressure, the channel closes and the outflow returns to zero.

We started with the settings of the *Wall Distance* interface and added a wall boundary condition to select the walls of interest. As we saw in the example, the *Wall Distance* interface can be combined with any other interface. We used it to detect the areas where a moving object is close to the walls.

If you want to learn more about modeling with the *Wall Distance* interface, do not hesitate to contact us.

Very broadly speaking, whenever you are modeling a problem that involves computing the velocity and/or the pressure field in a fluid as well as the stresses and strains in a solid material interacting with that fluid, you are solving a fluid-structure interaction (FSI) model.

When modeling an FSI problem, there are various assumptions that we can make to simplify the modeling complexity and reduce the computational burden. To get us started, let’s look at the most complete FSI model that you can build in COMSOL Multiphysics: the fluid flow around a cylinder.

*Deformation of a flexible object in the wake of a cylinder in crossflow.*

The fluid wake behind the cylinder induces large oscillations in the solid protruding from the back of the cylinder. Solving this type of model requires that we address three problems. First, the Navier-Stokes equations are solved in the fluid flow regions. Next, the displacements in the solids are computed. Finally, the deformation of the mesh in the fluid region is solved to account for the deforming region through which fluid can flow.

This nonlinear multiphysics coupling is handled with the *Fluid-Structure Interaction* interface that is available within the MEMS Module or the Structural Mechanics Module. Such models can be solved in either the time-domain or as a steady-state (time-invariant) problem.

The example above considers a linear relationship between the stress and strain in the solid material. If you would like to model materials with a nonlinear stress-strain relationship, such as a hyperelastic material model commonly used to describe rubbers and polymers, you will also want to use the Nonlinear Structural Materials Module.

*Peristaltic pump: A roller pumps fluid along a flexible tube. Image credit: Veryst Engineering.*

On the other hand, you may know ahead of time that the structural displacements will be relatively small, but the stresses may be significant. You can still use the *Fluid-Structure Interaction* interface in this case, but instead also use a *One-Way Coupled* solver, which will compute the flow solution and apply the fluid loads onto the structural problem. That way, you will avoid computing the deformation of the mesh.

It is also possible to put together this type of one-way coupled FSI problem from scratch, without using the *Fluid-Structure Interaction* interface at all. This is demonstrated in the Fluid-Structure Interaction in Aluminum Extrusion example. Additionally, if you are dealing with a very high-speed flow and are not interested in the short time-span chaotic oscillations in the flow, then you can use a turbulent fluid flow model as part of your FSI model. Both the CFD Module and the Heat Transfer Module include various turbulence models appropriate for different flow regimes.

Solar Panel in Periodic Flow: The turbulent air flow around a solar panel and the resultant structural stresses are computed.

If you know in advance that you are dealing with a vibrating structure in a fluid, then you can usually assume that the structural displacements will be relatively small. As a consequence, any induced bulk motion in surrounding fluid is negligible. However, since the structure is vibrating, a pressure wave will be excited in the fluid and sound will radiate away. Within the COMSOL software, this is handled via the *Acoustic-Structure Interaction* interfaces available in the Acoustics Module.

These interfaces assume that the variations in the displacements of the solids are relatively small and therefore do not induce significant bulk motion of the fluid, but only variations in the fluid pressure field. It is possible to solve such problems in the time-domain, but it can also be assumed that the displacements and pressure will vary sinusoidally in time. This allows us to model in the frequency domain, which is less computationally intensive to solve. Bulk losses due to fluid viscosity and material damping can be included.

*The sound pressure level radiated by a loudspeaker.*

It is further possible to solve a *Thermoacoustic-Solid Interaction* problem, which solves a linearized frequency-domain version of the Navier-Stokes equations and can consider losses in the explicitly modeled thermal and viscous boundary layers. Although this is more computationally expensive than an acoustic-solid interaction problem, it is still much more efficient than solving the full FSI problem.

*Vibrating Micromirror: The stresses and displacements of a vibrating micromirror and the surrounding air velocity.*

The Acoustics Module can also handle wave propagation through poroelastic media, such as wetted soils, biological tissues, and sound-damping foams using the *Poroelastic Waves* interface. This solves for both the structural displacements as well as the pressure of the fluid in the pores of the solid. An example is computing the reflection of a sound wave off an interface between water-sediment interface.

If you are interested in modeling poroelastic media, but in the steady-state or time domain instead of the frequency-domain, then you will need the Subsurface Flow Module. It is meant for modeling the steady-state or transient pressure-driven flow and static stresses in soils and other porous media. This module contains a *Poroelasticity* interface, so you can model poroelastic fluid-structure interaction in the steady-state and transient regime.

*Open Hole Multilateral Well: The stresses in the soil and the fluid velocity in the poroelastic domain are plotted.*

All of the approaches I just described explicitly model the volume of the fluid and solve the velocity and/or the pressure throughout these volumes. In circumstances where we have thin layers of fluid, such as in a hydrodynamic bearing, we can avoid a volumetric model of the fluid entirely and solve the Reynold’s equation, which solves only for the pressure in a thin film of fluid.

In this approach, we only solve for fluid flow along a boundary of the domain. This interface is available in the CFD Module and the MEMS Module. We can even take things further and only solve for the fluid flow along a line. In other words, we can solve for flow along a pipe using the Pipe Flow Module.

For an example of a model that considers both pressure variations along the length of a pipe as well as the effect of the elasticity of the pipe walls, please check out this example of solving the Water Hammer Equation.

*Tilted Pad Thrust Bearing: The pressure field in a lubricating layer and deformation of a tilted pad thrust bearing.*

You can probably see that we started with the most complex case and are looking at ways to simplify the computations, especially of the fluid flow field. Let’s take this to the extreme and consider the case of a fluid that isn’t moving at all, but does provide a hydrostatic pressure load on the structure.

In such situations, we can take advantage of the core features of COMSOL Multiphysics: the User-Defined Equations, Component Coupling Operators, and Global Equations. These allow you to include an arbitrary equation into the model to represent any additional variable, such as fluid pressure. As we discussed in a previous blog post, you can include the effect of both a compressible and incompressible fluid within a deforming enclosed cavity, as well as the hydrostatic pressure.

Now that we’ve introduced every way to simplify the fluid flow problem and compute the stresses, let’s finish up by turning things around and modeling the motion of a fluid for cases where we know the rigid body motion of the solid. Such situations can be handled via the Mixer Module, which is meant for addressing mixers and stirred vessels.

The motion of the solid structure is, in this case, completely defined via the rotation, and the movement of the fluid is computed. We can additionally compute the stresses in the moving solid objects, if we assume linear elastic deformations of the solids. This can be handled by a one-way coupled solution that first solves for the fluid flow field due to the moving mixer, and then computes the stresses under the assumption that the structural deformation is small.

*Flow fields in a stirred mixing vessel.*

As you can see, COMSOL Multiphysics is capable of handling a wide range of fluid-structure interaction modeling problems. If you’ve seen something here that you are interested in, or if there is something you’re curious about in this area that isn’t covered here, please contact us.

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