# Mathematically model airflow over an aerofoil

• ChrisHarvey
In summary: With these simple shapes, you can approximate the airflow without having to worry about the more complex equations.

#### ChrisHarvey

Hi!

I'm looking for someone who could explain how I can mathematically model airflow over an aerofoil using the "correct theory of lift" - Turning airflow.

I have tried several approaches but all fail because of one step... all textbooks I have read on the subject say that the airflow velocity parallel to the horizontal is constant. Therefore the air must accelerate over the aerofoil because velocity must change. The forces required for this to happen act on the aerofoil through the viscous forces of the air. I have however not even got this far in my model because...

Taking any equation for one surface of an aerofoil, at least two parts of it have a 0 gradient. When I differentiate this equation to get velocity, at these points I therefore have infinite velocity, which is obviously not the case.

Please can anybody guide me in the right direction as I am obviously doing something terribly wrong!?
Thanks, Chris

Last edited by a moderator:
Come on now, you know bloody well this isn't atomic physics.

I'm moving this to the Aeronautical Engineering Forum, where it will probably get more attention.

Originally posted by ChrisHarvey

I'm looking for someone who could explain how I can mathematically model airflow over an aerofoil using the "correct theory of lift" - Turning airflow.

Not exactly sure what you mean. Try looking up "Potential Flow Theory" or "Stream Function Theory", both of which are complementary (really just perpendicular to each other) theories in low speed inviscid aerodynamics.

Originally posted by ChrisHarvey

I have tried several approaches but all fail because of one step... all textbooks I have read on the subject say that the airflow velocity parallel to the horizontal is constant.

For a start, the velocity parallel to a surface need not be constant. Don't forget viscosity . The textbooks you were reading were probably (no offense) lower level aerodynamic ones or physics texts which don't deal with issues like separation or shear layers...

Originally posted by ChrisHarvey

Therefore the air must accelerate over the aerofoil because velocity must change. The forces required for this to happen act on the aerofoil through the viscous forces of the air. I have however not even got this far in my model because...

Taking any equation for one surface of an aerofoil, at least two parts of it have a 0 gradient. When I differentiate this equation to get velocity, at these points I therefore have infinite velocity, which is obviously not the case.

Please can anybody guide me in the right direction as I am obviously doing something terribly wrong!?
Thanks, Chris

See above. The maths can get quite detailed for all but the simplest geometries. At at introductory level, you are probably best off modelling your 'aerofoil' as a flat plate with a small inclination, or a semi-circle.

## 1. How is the airflow over an aerofoil mathematically modeled?

The airflow over an aerofoil is mathematically modeled using a combination of fluid dynamics equations, such as the Navier-Stokes equations, and numerical methods, such as computational fluid dynamics (CFD). These equations and methods allow for the prediction of air velocity, pressure, and turbulence patterns around the aerofoil.

## 2. What factors affect the mathematical model of airflow over an aerofoil?

The mathematical model of airflow over an aerofoil is affected by several factors, including the shape and size of the aerofoil, the angle of attack (the angle at which the aerofoil meets the oncoming air), the air density, and the air viscosity. Other factors such as surface roughness, temperature, and altitude may also have an impact.

## 3. How accurate is the mathematical model of airflow over an aerofoil?

The accuracy of the mathematical model of airflow over an aerofoil depends on several factors, such as the complexity of the aerofoil shape, the precision of the input data, and the quality of the numerical methods used. Generally, the more detailed and accurate the model and data, the more accurate the predictions will be.

## 4. Can the mathematical model of airflow over an aerofoil be used in real-life applications?

Yes, the mathematical model of airflow over an aerofoil is commonly used in various real-life applications, such as designing aircraft wings and optimizing wind turbine blades. It is also used in the development of aerodynamic vehicles, such as cars and bicycles, and in the analysis of air flow in buildings and structures.

## 5. Are there any limitations to the mathematical model of airflow over an aerofoil?

While the mathematical model of airflow over an aerofoil is a powerful tool, it is not without limitations. The accuracy of the model can be affected by factors such as viscous effects, boundary layer separation, and complex flow phenomena. Additionally, the model may not account for certain real-life factors, such as manufacturing imperfections or external disturbances.