Fluid-Structure Interaction for Beginners: From Bridges to Blood Flow
Let’s explore how the forces between fluids and structures shape the world around us, from the sway of skyscrapers in the wind to the flow of blood through arteries. Welcome to the fascinating world of fluid-structure interaction!
When designing modern engineering systems, it’s important to understand how different physical systems, such as fluids, structures, and heat, interact with one another. This interaction, known as multi-physics simulation, is crucial for many industries. One example is how a fluid interacts with a solid structure. This phenomenon, called Fluid-Structure Interaction (FSI), plays a key role in understanding how buildings respond to wind, how airplane wings handle airflow, or how blood moves through arteries.
An example of FSI can be seen when air flows around a bridge. The structure of the bridge affects the air movement, and in turn, the air impacts the bridge structure, which may lead to oscillations or even failures, as seen in historical bridge collapses like the Tacoma Narrows Bridge.
Coupled systems
A coupled system refers to a set of physical components that interact dynamically, meaning that what happens in one part of the system influences what happens in another. Think about how bending a paper clip rapidly can heat it up; this is because the mechanical energy of bending transforms into heat - an example of coupling between mechanical and thermal systems.
In more complex cases, systems may involve mechanical, chemical, or even biological interactions. For example, astronauts in space experience bone mass loss because the lack of gravity changes the way bones are loaded, which alters their internal structure - another form of coupled interaction.
Examples of Fluid-structure interaction
Fluid-Structure Interaction (FSI) is a specific kind of coupled system where a fluid (air, water, etc.) interacts with a solid structure. FSI is crucial in many engineering fields, as it affects the safety and performance of a wide range of designs, from skyscrapers to airplanes. Understanding how it works can help prevent disasters or improve performance in fields like civil engineering, aerospace, and even bioengineering. Let’s look at some fascinating and diverse examples:
1. Wind on Civil Structures
One of the most iconic examples of FSI is how wind interacts with large structures, such as bridges or skyscrapers.
Vortex-Induced Vibrations in Cables: Another important example is the vibration of cables due to wind. When wind flows past a cylindrical object, like a bridge cable, it creates swirling patterns in the wake, called vortices. These vortices alternate from one side of the cylinder to the other, creating alternating lift forces. If the frequency of vortex shedding matches the natural frequency of the cable, it can cause resonance, leading to what is called vortex-induced vibration. If not properly controlled, this can cause serious damage to structures like bridges, towers, or power lines.
Galloping of Slender Structures: Galloping is another FSI phenomenon where wind causes slender, non-circular structures (like icy cables or chimneys) to oscillate. This happens when wind flows over a non-symmetrical shape, creating an uneven pressure distribution, which in turn generates oscillations that can grow larger if not damped. For instance, when cables collect ice in the winter, their shape changes, making them more susceptible to galloping under wind loads.
Tacoma Narrows Bridge Collapse
The Tacoma Narrows Bridge in the U.S. is perhaps the most famous FSI failure. In 1940, strong winds caused the bridge to sway violently, and eventually, the structure collapsed. The cause was a phenomenon known as aeroelastic stall flutter, where the wind forces interacted with the structure in such a way that it created large oscillations. The structure couldn’t dampen these oscillations, and it failed catastrophically. This incident is a classic case of how FSI can lead to disasters if not understood and accounted for.
2. Aeroelasticity in Aeronautics
In the aerospace industry, FSI is crucial for the design and performance of aircraft. The interaction between airflow and airplane wings or other structures can lead to various FSI-related phenomena, such as flutter or buffeting.
Flutter in Aircraft Wings: Aircraft wings are designed to be lightweight and flexible, but this flexibility can sometimes lead to flutter, a dangerous instability caused by the interaction between the air and the wing's structure. As the air flows over the wing, it can cause the wing to oscillate. If these oscillations match the wing's natural frequency, the vibrations can increase in amplitude until the wing fails. This problem is tackled using computational simulations that predict how an aircraft's structure will respond to airflow, allowing engineers to design wings that are stable under different conditions.
Wind Turbine Blades: FSI also plays a critical role in the design of wind turbines, especially as these structures become larger. Modern wind turbine blades are designed using aeroelastic principles to optimize their performance while minimizing the risk of failure. Engineers must consider how the wind will interact with the flexible blades and ensure that the blades can withstand high winds without vibrating excessively or breaking. Siemens, for example, has developed aeroelastically tailored blades, which use advanced designs to reduce the impact of these forces.
3. Bioengineering Applications
FSI is also important in the field of bioengineering, where the interaction between fluids (like blood) and solid structures (like blood vessels or heart valves) is essential to understanding the human body and designing medical devices.
Cardiovascular Fluid-Structure Interaction: The heart and blood vessels are perfect examples of FSI in the body. Blood, a fluid, flows through elastic blood vessels, which can expand and contract. If the vessel walls are too stiff (as in some diseases), the flow of blood is affected, which can lead to serious health issues. On the other hand, if the blood flow is abnormal (like in aneurysms), the vessel walls might experience too much stress and eventually rupture. Engineers use FSI models to study how blood flows through the heart and arteries, which helps in designing medical devices such as stents, artificial valves, or even bypass grafts.
Artificial Heart Valves: When designing artificial heart valves, it’s crucial to model how blood will flow through the valve and how the valve's structure will respond to this flow. The deformation of the valve due to blood pressure and the resulting impact on blood flow must be carefully analyzed to ensure the valve functions properly.
4. Sloshing in Tanks
Sloshing refers to the movement of a liquid inside a container. FSI comes into play because the movement of the liquid can exert forces on the container, and vice versa.
Fuel Tanks in Vehicles: In airplanes, rockets, or even cars, the sloshing of fuel inside a tank can affect the stability of the vehicle. For example, when a rocket is in flight, the liquid fuel inside its tanks can start sloshing, which can destabilize the rocket. Engineers use FSI simulations to design tanks that minimize this effect, often using baffles to limit the movement of the fluid.
Cargo Ships and Trucks: Similarly, cargo ships and tankers that transport liquids must account for sloshing, especially in rough seas. Sudden movements of the liquid can lead to instability or even capsizing. Designing tanks that can handle these forces without compromising the structure is a complex FSI problem.
5. Structural Design in Civil Engineering
Tall Buildings: Skyscrapers, especially tall and flexible ones, must be designed to withstand FSI effects from wind. In some cases, tuned mass dampers (huge weights inside the building) are used to counteract the swaying caused by the wind. For example, the John Hancock Tower in Boston has such a damper installed at its top to reduce swaying and prevent FSI-related instabilities.
Dams and Reservoirs
FSI also plays a significant role in the design of dams. Water in the reservoir exerts pressure on the dam structure, which in turn affects how the dam holds up under different conditions. Simulating these interactions allows engineers to predict how the dam will behave during extreme events, such as floods or earthquakes.
General Formulation of FSI Problems
When we try to model fluid-structure interaction, we often use continuum mechanics. This approach treats both the fluid and the solid as continuous materials rather than breaking them down into individual particles. The main goal is to capture how forces and movements affect the system as a whole. Let’s try to take a closer look at the equations!
Equations of Motion: Solid and Fluid Constitutive Equations
Equations of motion describe how things move and respond to forces. In Fluid-Structure Interaction (FSI), we have separate equations for solids and fluids, but they share the same basic structure: they’re all based on Newton’s second law, which you probably know as F = m*a, or force equals mass times acceleration. Here’s a more detailed breakdown of how these equations work and how you can think about them.
1. For solids
The equation for a solid looks like this:
All these terms must balance out to zero, meaning the structure reaches a state of equilibrium where forces (both internal and external) balance each other out. If they don’t, the solid will keep accelerating, bending, or deforming until it does.
2. For fluids
For a fluid, the equation looks like this:
These two sets of equations—one for solids and one for fluids—are key to understanding how FSI works. They describe how both materials move and react to forces and pressures.
Lagrangian, Eulerian, and ALE Descriptions
When we simulate FSI problems, we need a way to track how materials move and deform over time. There are three common ways to describe this motion:
Lagrangian description:
In this approach, we “follow” individual material points as they move through space and time, like attaching a GPS tracker to each point of the material and watching its journey. This is especially useful for solid structures where the material stays mostly intact, but it can become problematic for fluids, which can distort wildly.
Example: Think of watching a leaf floating down a river. You track the leaf as it moves with the flow.
Eulerian description:
Instead of following individual particles, we fix our viewpoint at certain locations in space and watch as materials flow through these fixed points. This is more suitable for fluids, where the motion can be chaotic and change rapidly.
Example: Imagine standing at the bank of a river and observing the water that passes by a fixed point. You’re not following the water, but just seeing how it moves through your position.
Arbitrary Lagrangian-Eulerian (ALE) description:
ALE is a hybrid approach that combines the best of both Lagrangian and Eulerian methods. In ALE, we can follow the material in some regions (Lagrangian) and keep the viewpoint fixed in others (Eulerian). This flexibility is especially useful in FSI simulations because it allows us to track the solid structure and fluid flow in the same model.
Example: Imagine you’re watching a boat on a river. You can follow the boat (Lagrangian) while also keeping track of the water flowing under the boat (Eulerian).
In Arbitrary Lagrangian-Eulerian (ALE) form, we combine elements of the Lagrangian (which follows material particles) and Eulerian (which observes from fixed points in space) descriptions.
In ALE, the governing equations adapt to situations where we need to keep track of a moving solid or fluid interface while allowing for flexibility in how we define the "grid" for calculations.
Here’s how we think about it:
In the Lagrangian frame, the computational grid moves along with the material. This is great for solids because you can follow how a bridge, for example, deforms under wind pressure.
Example: Imagine marking points on a piece of clay and watching how they move as you stretch the clay. This is like following the motion of individual particles in the solid.
In the Eulerian frame, the grid stays fixed, and the material moves through it. This is perfect for fluids, where we care more about how the fluid flows through space than tracking individual particles.
Example: Picture standing by a riverbank and watching the water flow past. You’re not moving with the water, but you can still track how fast it’s moving through your fixed point.
In ALE, we get the best of both worlds. The computational grid can move with the material in certain regions (like where the fluid meets the solid), but it can also stay fixed in other regions (like the main body of the fluid). This flexibility helps solve FSI problems where both solid deformation and fluid flow are happening simultaneously.
For example, if you’re modeling how wind affects a building, the grid near the building’s surface can follow the structural deformations, while the grid in the surrounding air can stay fixed as it tracks the airflow.
Boundary Conditions at the Fluid-Structure Interface
Interfaces
The boundary conditions at the interface between the fluid and the structure are essential for ensuring that the two systems interact properly. These conditions ensure that both the fluid and the structure behave consistently at their shared interface. In essence, we need to maintain both the movement (velocity) and the force (stress) balance across the boundary.
1. Velocity Continuity (No-Slip Condition)
At the fluid-structure interface, the velocity of the fluid must match the velocity of the structure. This means that there’s no "slipping" of the fluid past the surface of the structure. For example, if air flows over a car’s surface, the air at the immediate boundary of the car should be moving at the same speed as the car's surface at that point.
Why it's important: If this condition isn’t met, it would create unrealistic results, like the fluid somehow moving through the solid or slipping without friction.
2. Stress Continuity
In addition to matching velocities, the forces (or stresses) applied by the fluid on the structure must be balanced by the forces that the structure applies back on the fluid. In other words, the fluid pushes on the structure, and the structure pushes back equally. This ensures that the system is in equilibrium at the interface.
Example: Consider a dam holding back water. The water exerts a force (hydrostatic pressure) on the dam, and the dam exerts an equal and opposite force to hold the water in place. The balance between these forces is crucial for ensuring that the dam doesn't collapse or let water through.
Space-Related Partitioning and Time-Related Splitting
In practice, solving FSI problems can be computationally expensive because fluids and solids behave differently. One common approach to make simulations easier is to divide the problem into separate domains (fluid and solid) and solve them in parts. This is known as partitioning. There are two main types:
1. Space-Related Partitioning
This refers to dividing the system into distinct spatial regions, typically one for the fluid and one for the structure, and solving them separately. These regions exchange information at the interface (like force and velocity) to ensure that the solution is consistent across both regions.
2. Time-Related Splitting
In this approach, the fluid and solid systems are solved at different time steps. For instance, the fluid might be solved first, and then the information from that solution is used to solve the structure’s response. This allows for more manageable computations, but care must be taken to ensure the time-step coupling is accurate and doesn’t introduce errors.
Classification of Coupled Problems
1. Small vs. Large Deformations/Displacements
Small deformations occur when the structure’s movement is minimal, meaning the shape and size of the structure barely change. In these cases, the math is simpler, and linear models can be used.
Example: A small bridge swaying gently in the wind.
Large deformations, on the other hand, are when the structure experiences significant changes in shape or size. This makes the problem much more complex and requires non-linear models to accurately capture the behavior.
Example: A flexible, inflatable dam deforming significantly due to changes in water pressure.
/Photo source/ - M.Dannhauer
2. Conforming vs. Non-Conforming Mesh Methods
Conforming Mesh Methods: These methods use a mesh (grid) that fits exactly at the interface between the fluid and structure. This makes it easier to ensure that the fluid and structure interact properly because the two domains are directly aligned.
Example: When simulating airflow over a perfectly smooth airplane wing, the mesh fits perfectly along the wing surface.
Non-Conforming Mesh Methods: Here, the mesh doesn’t perfectly align between the fluid and structure, which can happen when using different types of simulations for the fluid and the structure. This is more flexible but requires additional computations to make sure the interaction between fluid and structure is accurate.
Example: In some simulations, the mesh for a fluid might be different in shape and size compared to the mesh for a structure, so extra calculations are needed to ensure they interact correctly.
In a nutshell
Coupled Systems: These are systems where different physical phenomena (like fluid flow and structural deformation) interact. Fluid-Structure Interaction (FSI) is a key example, where fluids (air, water) interact dynamically with solid structures (bridges, airplanes, blood vessels).
Fluid-Structure Interaction (FSI): FSI problems involve two-way interactions where a fluid can deform a structure, and the deformed structure can alter the fluid’s behavior. Examples include wind acting on a bridge or blood flow through elastic arteries.
Equations of Motion: Both fluids and solids are described by their own sets of equations of motion. For solids, this typically comes from Newton’s laws of motion (as expressed by stress-strain relationships like Hooke's Law). For fluids, we rely on the Navier-Stokes equations, which govern how fluid flows and interacts with its surroundings.
Describing Motion: There are three ways to track how materials move in FSI problems: the Lagrangian description (follow the material), the Eulerian description (stay fixed in space), and the ALE description (a mix of both, which is particularly useful for FSI).
Boundary Conditions: At the fluid-structure interface, conditions must be set to ensure the fluid and structure behave properly together (i.e., they must move together and exchange forces correctly).
Classifying Problems: FSI problems can be classified based on whether deformations are small or large and whether the mesh used to model the system conforms or doesn’t conform at the interface.