Mathematics Gives You Wings

TL;DR

Math models airflow around airplane wings by describing how velocity, pressure, momentum, and energy change across space and time under forces such as pressure gradients, gravity, and boundary friction. The same governing equations also apply to weather, ocean currents, yacht sails, and groundwater. At a wing’s surface, air velocity falls from 500 mph to zero across a thin boundary layer, illustrating why the mathematics matters. Read on to understand each part of the model.

Transcript

Stanford University thanks very much I'm a very tickled that you're here because I know I'm competing with a football game the start of a baseball game uh so I'm keeping a very close eye on the time because I know the first pitch is at 4:57 so I want to be out before then uh and I'm also competing with the rain and then the first thing is I'm talki... Read More

Key Insights

• Fluid flow across wildly different domains shares identical governing equations. Climate models, weather models, yacht sail design, airplane wing optimization, and oil and gas reservoir flow are all governed by the same mathematical equations despite appearing completely different.
• The unknown 'U' in fluid equations represents whatever quantity you want to know. In fluid flow this is typically velocity, pressure, or energy, and the equations describe how these change through space and time.
• Momentum is used instead of velocity when mass stays constant. Momentum is mass times velocity, so if air density is treated as constant, you can simplify and think purely in terms of velocity rather than momentum.
• Fluid processes are driven by very few forces. The main forces acting on fluids are pressure gradients, gravity, and friction at boundaries, and these forces determine how the fluid's properties change.
• A strong pressure gradient produces high velocity. When pressure is high in one location and low nearby, the force pushes air quickly from high to low pressure, creating strong wind or fast flow.
• Air at the surface of an airplane wing is not moving at all. Although a plane travels at 500 miles per hour, the air velocity drops from 500 mph to zero across a very thin boundary layer at the wing surface.
• The boundary layer creates strong force on the wing. Because velocity changes extremely rapidly within that thin layer near the wing, there is substantial force on the wing, comparable to stopping a runaway train.
• Flow behavior depends on the boundaries of the domain. How fluid moves is determined by the configuration of the space, such as the size of an opening or the shape of a room, which define the boundaries of the flow.

Questions & Answers

Q: How does math model fluid flow in airplane design?

Fluid-flow equations track how quantities such as velocity, pressure, momentum, and energy change through space and time. They also represent the forces driving those changes, including pressure gradients, gravity, and friction at the wing boundary.

Q: Why can the same equations model airplane wings, weather, and yacht sails?

Professor Margot Gerritsen explains that all fluid-flow processes are governed by fundamentally the same equations. Climate, weather, coastal oceans, wind turbines, yacht sails, airplane wings, groundwater, and oil and gas reservoirs therefore share the same mathematical foundation.

Q: What does U represent in fluid-flow equations?

U is a name for a quantity researchers want to determine about the flow. It commonly represents velocity, pressure, momentum, or energy and allows the equations to describe how that quantity changes across space and time.

Q: Why is air stationary at the surface of an airplane wing?

Boundary friction makes the air directly at the wing surface have zero velocity, even when the airplane is traveling at 500 miles per hour. The velocity falls from 500 mph to zero within a very thin boundary layer over the wing.

Q: Why does the boundary layer create a strong force on an airplane wing?

Velocity changes extremely rapidly across the wing’s thin boundary layer. That sharp change produces substantial force because the wing is effectively stopping the airflow at its surface, an effect compared in the existing explanation to stopping a runaway train.

Q: What forces drive fluid flow?

The main forces identified are pressure gradients, gravity, and friction at boundaries. Pressure differences push fluid from high pressure toward low pressure, gravity affects vertical motion, and boundary friction slows the flow near a surface.

Q: How do pressure gradients affect airflow speed?

A strong pressure gradient creates high velocity. When high pressure and low pressure occur close together, the resulting force pushes air quickly from the high-pressure region toward the low-pressure region, producing strong wind or fast flow.

Q: Why is momentum sometimes used instead of velocity in fluid equations?

Momentum is mass multiplied by velocity, so it accounts for both motion and mass. When mass or air density is treated as constant, the model can be simplified by reasoning directly about velocity instead.

Summary

In this video, the speaker discusses the field of computational mathematics and its application to fluid flow problems. They explain how equations govern fluid flow processes and how these equations can be simplified to understand the behavior of various flow systems. The speaker also introduces the concept of using grids to approximate solutions and the importance of interpolation between grid points. They provide examples of simulations and how these equations can be applied in real-life scenarios, such as sail design and airplane wing optimization.

Questions & Answers

Q: What is the speaker's background in computational mathematics?

The speaker mentions that they obtained their PhD in computer science and energy resources from Stanford University. They are currently the director of the Institute of Computational Mathematical Engineering at Stanford and have worked on various fluid flow projects throughout their career.

Q: How are equations used to describe fluid flow?

Equations are used to describe various properties of fluid flow, such as velocity, pressure, and energy. These properties can be measured and understood through the equations, which help predict and analyze the behavior of fluid flow in different systems. The speaker emphasizes that while the equations may appear complex, they can be simplified and understood to reveal the underlying connections between different flow processes.

Q: How are the equations translated into solutions for fluid flow problems?

The speaker explains that the equations are solved using numerical methods on a grid. Instead of finding the exact solution at every point in a domain, a grid is created and solutions are computed at grid points. These grid points represent approximations of the solution in the region. Interpolation is used to estimate the solution at points that are not on the grid. This allows for efficient computation of solutions while maintaining accuracy.

Q: What are some examples of fluid flow problems that can be solved using computational mathematics?

The speaker provides examples of fluid flow problems they have worked on, such as optimizing wing designs for pterosaurs, studying fluid flow in oil and gas reservoirs, and analyzing airflow around sailboats. They highlight how the same set of equations can be applied to various scenarios, demonstrating the universality of computational mathematics in fluid flow problems.

Q: How is approximation used to simplify the computational process?

The speaker explains that approximation is used to simplify the computation process. Instead of finding the solution at every point in the domain, calculations are focused on a grid of points. This reduces the complexity and computational load while still providing accurate approximations of the solution. Interpolation is used to estimate the solution at points not covered by the grid, further simplifying the computational process.

Q: How are grid points determined in computational mathematics?

Grid points are determined based on the specific problem and the desired level of accuracy. In some cases, a coarse grid with fewer points may be used in regions with steady flow or less complex behavior. In areas with rapid changes or complex flow patterns, a denser grid with more points is employed. The grid points are strategically chosen to capture the important features and behavior of the flow system.

Q: Can the grid be adjusted to include specific points of interest?

Yes, the grid can be adjusted to include specific points of interest. If a point of interest is not on the grid, interpolation can be used to estimate the solution at that point. However, it may not always be practical to include every specific point of interest, as the computational load may become too high. Strategic placement of grid points is necessary to balance accuracy and computational efficiency.

Q: What are some challenges in translating the equations onto the grid?

One of the challenges in translating the equations onto the grid is ensuring that the grid is well-suited to the specific problem and accurately represents the flow behavior. Skewed grids, where the angles of the grid lines are too extreme, can lead to computational issues and inaccuracies. It requires careful planning and consideration to design a grid that effectively captures the flow characteristics and produces accurate solutions.

Q: How can computational mathematics be used in practical applications like sail design?

Computational mathematics can be used in practical applications like sail design by simulating the behavior of fluid flow around the sail. By solving the equations on a grid and approximating the solution, designers can analyze different designs and configurations without the need for physical prototypes. This allows for faster and more efficient testing of new sail designs to improve performance and optimize their shape.

Q: How does the accuracy of the solutions depend on the grid and interpolation?

The accuracy of the solutions depends on the grid density, with denser grids providing more accurate solutions. Interpolation is used to estimate the solution at points not on the grid, and the accuracy of interpolation also affects the overall accuracy of the solutions. A balance must be struck between grid density and computational efficiency to achieve accurate solutions while managing computational resources.

Takeaways

Computational mathematics plays a crucial role in understanding and analyzing fluid flow problems. By translating complex equations onto a grid and approximating the solution, researchers and engineers can simulate and study various flow systems. The use of grids allows for efficient computation and enables interpolation to estimate the solution at points not on the grid. This approach has applications in sail design, aircraft wing optimization, and many other fields where fluid flow is a critical factor.

Summary & Key Takeaways

• Professor Margot Gerritsen directs Stanford's Institute of Computational Mathematical Engineering, which has 140 graduate students, 70 masters and 70 PhDs, no undergraduates, and teaches 20 courses accounting for 4,000 student units across the university. She earned her own PhD in energy resources at Stanford and returned as faculty in 2001.

• The central message is that a single set of equations governs all fluid flow processes. Whether modeling climate, weather, coastal oceans, wind turbines, yacht sails, airplane wings, or groundwater and oil reservoirs, the mathematics is fundamentally the same, which is why she has worked across a dozen seemingly different projects.

• The equations describe how quantities of interest, called U, change over space and time due to forces. Key quantities include velocity, pressure, momentum, and energy, while the main driving forces are pressure gradients, gravity, and boundary friction, illustrated by air stopping to zero velocity at a wing surface.