Aerodynamics, the study of air movement, has fascinated humans since ancient times. Early civilizations gazed at the skies, eager to decode the secrets of flight, setting the foundation for a field that would transform the world. Aerodynamics reached its maturity in the early 20th century, spurred by the advent and subsequent rapid evolution of powered airplanes. One groundbreaking theory developed during this period was Prandtl's Lifting-Line Theory. Developed by the German scientist Ludwig Prandtl, this theory provided profound insights into the mechanics of flight, revolutionizing how airplanes were designed and understood. In this blog post, we will use Prandtl’s Lifting-Line Theory, as visualized through SIMNET’s Virtual Wind Tunnel, to develop an intuitive understanding of how airplanes soar through the skies.

## Multiple Perspectives

Aerodynamics is interesting because it can be interpreted and modeled through different theories. Perhaps the most popular explanation for flight relies on Bernoulli’s principle. An airplane's wing is shaped such that it causes air to move faster over it than underneath. Bernoulli’s principle states that as air speeds up, its pressure drops. Therefore, the air pressure above the wing is lower than below it, creating an upward force that lifts the aircraft. We can visually explore this using SIMNET’s Virtual Wind Tunnel, observing that airspeed indeed increases above the wing.

Another common perspective is based on Newton’s Third Law. The shape of a wing is such that it deflects incoming air downward. By Newton’s Third Law, if the wing is pushing the air downwards, the air must be equally pushing the wing upwards, generating lift. Indeed, the Virtual Wind Tunnel shows that the air is deflected downwards as it flows around the wing.

## Circulation

However, Ludwig Prandtl recognized that there is yet another way to understand how wings generate lift: through the concept of circulation. Martin Kutta and Nikolai Zhukovsky, scientist contemporary to Prandtl, had independently reached a significant discovery in aerodynamics: lift forces always result in the fluid ‘swirling’ around in a vortex. Furthermore, they established that the intensity of the vorticity, called circulation, can be measured and is directly proportional to the magnitude of the lift. This relationship is now known as the Kutta-Zhukovsky theorem.

Here L’ is the lift per span, ρ∞ is the air density, V∞ is the flow speed, and 𝚪 is the circulation intensity.

It was Prandtl who realized how this theorem can be used to mathematically model wing aerodynamics. Recall that air tends to flow faster over the wing than under it. Assuming effects like air viscosity and compressibility are negligible, Prandtl realized that you can model this flow as a superposition (or the sum of) of two flows: a uniform speed flow, and a circulating flow around the wing, as shown in the image below.

In other words, wings generate lift by producing a circulation flow around them, which results in lift. Prandtl used this insight to create a method to estimate the lift generated by a wing.

## From 2D to 3D Aerodynamics

It’s fairly straightforward to measure the aerodynamic characteristics of a 2-dimensional section of a wing, called an airfoil. You can place an airfoil in a wind tunnel, and measure the lift and drag generated, as a function of the wind flow speed, and the airfoil’s angle to the flow (the angle of attack).

Calculating the lift produced by a wing of finite span using this data is a bit more challenging. A naive approach would be to divide the wing into many sections and add the lift generated by each section using the airfoil data, and the wind speed and direction the wing faces. However, this would result in a significant over-prediction of the lift and aerodynamic efficiency. The reason is that wings of finite spans deflect the incoming wind in more complex ways. To obtain a better prediction you must account for the fact that the local flow speed and angle of attack experienced at any specific location along the wing span is not the same as the flow speed and direction far away from the aircraft, because the presence of the wing itself deflects the incoming flow into different directions.

This is where Prandtl's method comes in. We can model the 3D flow around the vehicle as a superposition of a uniform flow, and multiple vortices passing through each section of the wing. We can then use the Kutta-Zhukovsky theorem to tune the intensity of each of these vortices, such that the lift strength at each airfoil matches the expected airfoil lift given its local wind speed and angle. In other words, we estimate the total lift of the wing by adding up the lift at each 2D section of the wing using the airfoil predictions, but the local wind speed and angle are calculated accounting for the vortices, which strength was tuned to match the lift generated by the airfoils themselves using the Kutta-Zhukovsky theorem. Clever!

Mathematically, you can model a vortex using what is called a bound vortex. It is a line around which the air swirls with a given intensity. Prandtl placed multiple vortices along the span of the wing on a ‘horseshoe’ configuration as shown below.

## Flow Around a Wing

The lifting line method does have some limitations, such as being limited to subsonic speeds and high aspect-ratio wings. Other aerodynamic methods such as Computational Fluid Dynamics (CFD) can produce much higher fidelity results at the expense of increased compute times. Therefore the Lifting Line method’s strength lies in that it can be solved very quickly, and even in real-time when using modern computers, providing a fast estimate of the aircraft aerodynamics, as opposed to CFD methods that can take on the order of hours to provide a solution.

The results below are produced using SIMNET cloud-based uncrewed aircraft (UAV or drone) simulation software, which has a modern real-time implementation of the Lifting Line theory. As you can see in the image below, wings tend to produce two large vortices at the wingtips. These vortices deflect the incoming air downwards, causing the angle of the wind relative to the wing to vary along the span. The yellow lines represent the lift force generated by the wing. This force is reduced as you move towards the wingtips, due to this reduced angle of attack. Modeling this effect is crucial to properly capture the aerodynamics of the aircraft, which is what the Lifting Line method does.

In conclusion, aerodynamics has significantly evolved from ancient curiosity to Prandtl's Lifting-Line Theory and beyond, shaping our understanding of flight. This blog post's exploration of Prandtl's theory, particularly through the lens of SIMNET’s Virtual Wind Tunnel, offers an intuitive grasp of how airplanes achieve their majestic flight. Aerodynamics remains a testament to human ingenuity and our relentless pursuit to conquer the skies.