[Fluid Dynamics: Potential Flows] Boundary Element Method (BEM)- Principle

Summary

This talk introduces the principle of the Boundary Element Method (BEM) for fluid-structure interactions using potential flow theory. It covers the distribution of sources on boundaries to represent velocity potential functions and their strengths, with an emphasis on the Green's Theorem and de-singularization methods. The speaker explains how simple potential flows, like a source and doublet in a uniform flow, can model complex structures. The presentation highlights the mathematical foundations behind solving potential functions and applying them to real-world fluid dynamics problems, focusing on how to represent structures via boundary sources rather than in the fluid.

Takeaways

• 😀 The Boundary Element Method (BEM) for fluid-structure interaction is based on potential flow theory and helps to distribute sources on boundaries to represent velocity potential functions.
• 😀 Simple potential flows, such as a source and doublet in a uniform flow, are used to introduce basic principles for BEM in fluid-structure interactions.
• 😀 A source in a uniform flow creates a bullet-shaped structure with no flow crossing the boundary, representing a solid body in 3D potential flow.
• 😀 A doublet in a uniform flow can be represented by a circle in the flow, with zero radial velocity at the boundary, analogous to a body boundary condition.
• 😀 Combining multiple sources on the structure can represent complex shapes by distributing sources and sinks, which simplifies fluid flow modeling.
• 😀 The Green's Theorem is key in deriving boundary integral equations that are fundamental to solving fluid-structure interaction problems.
• 😀 Green’s Theorem relies on Laplace's equation for the velocity potential function and helps to derive equations that apply to both irrotational and incompressible flows.
• 😀 To avoid singularities in the potential function, sources should be distributed on the boundary surface of the structure, not within the fluid or structure itself.
• 😀 The de-singularization method is used to exclude singularities, allowing the potential function to be expressed in terms of sources distributed on the boundary.
• 😀 After de-singularization, boundary integral equations can be used to determine the velocity potential function and the strengths of the distributed sources on the boundaries.

Q & A

What is the focus of the talk in the video?

The talk focuses on the boundary element method (BEM) for fluid-structure interactions, based on potential flow theory. It explains how singularities like sources and doublets are distributed on boundaries to represent the velocity potential function in fluid mechanics.

What are the two simple potential flows introduced in the talk?

The two simple potential flows introduced are a source in a uniform flow and a doublet in a uniform flow. These examples help illustrate how basic structures can be represented using these singularities.

Why is the source in a uniform flow considered equivalent to a bullet-shaped structure?

In a uniform flow past a source, the resulting flow patterns show a bullet-shaped structure, with a red boundary representing the separation between the uniform flow and the source flow. This boundary indicates no flow across it, resembling a solid structure.

How is the doublet in a uniform flow represented?

The doublet in a uniform flow is represented by a circle, as its radial velocity is zero at the boundary, which matches the body boundary condition. This behavior suggests that a doublet is equivalent to a circle in the flow field.

What is the significance of the Green's Theorem in the boundary element method?

Green's Theorem is fundamental to the boundary element method as it relates the potential functions and flow characteristics at boundaries. It allows for the simplification of solving the flow by distributing sources on the surface, rather than in the fluid or structure.

Why can singularities be distributed only on the boundaries and not inside the structure?

Singularities can be distributed only on the boundaries because the Green's Theorem and the de-singularisation method show that the velocity potential can be expressed through boundary integrals. This reduces the problem from a 3D volume to a 2.5D surface problem.

What does the de-singularisation process achieve?

The de-singularisation process eliminates the singularity by creating a small sphere or hemisphere around it. This process allows the velocity potential function to be expressed using the sources distributed on the boundary, making it mathematically correct.

What is the role of the Laplace equation in the boundary element method?

The Laplace equation governs the potential flow in an incompressible and irrotational fluid. It is essential in deriving the boundary integral equation and ensuring that the velocity potential satisfies the required conditions in the fluid domain.

How does the combination of multiple sources represent a more complex structure?

By distributing multiple sources or sinks on or around a structure, the resulting combination of these singularities can represent a more complex flow pattern, similar to the streamlines that would result from the actual structure.

What is the boundary integral equation and how is it used in fluid-structure interaction problems?

The boundary integral equation expresses the velocity potential function in terms of sources distributed on the boundaries. This equation is used to solve for the strengths of the distributed sources, which in turn helps in analyzing the fluid-structure interaction.