Figure 1: Illustration of Air Molecules Flowing Around an Airfoil (Source)
On December 17th, 1903, the Wright brothers made their first sustained flight. 120 years passed since humanity’s first ever flight but do we really understand how planes fly?
The short answer is yes. However, it is extremely difficult to predict lift using mathematical equations. Furthermore, the concept of lift has been a topic of controversy for many decades, and to this day, explaining it in simple terms is very difficult.
One of the most common misconceptions is the equal transit theory, which focuses on the fact that the upper surface of an airfoil is longer than the bottom surface. It erroneously assumes that air molecules meet up at the end of an airfoil. Since air molecules traveling on the bottom and upper surface reach the end at the same time, the theory states that air molecules moving along the upper surface must move faster than the molecules moving along the bottom surface. Therefore, applying Bernouli’s equation, the equal transit theory claims that there is a pressure difference between the bottom and upper surface.
Although it is true that there is a pressure difference and a difference in the velocity of air molecules, the air molecules do not reach the end at the same time. This has been proven by countless experiments that have analyzed air molecules passing through an airfoil inside a wind tunnel. Moreover, the counterexample to this theory are flat wings. For a flat wing, the bottom and top surface have equal lengths. Therefore, according to the equal transit theory, the velocity of air molecules must be the same, thus, no difference in pressure. However, this is not the case as flat wings could also create lift.
The correct explanation of lift is best explained using both the Newtonian and Bernouli explanation. The application of Bernouli’s equation comes from the concept of circulation. The motion of an air molecule can be divided into uniform irrotational and circulatory flow. Circulation explains the flow of air molecules as they leave the sharp end of an airfoil. Without circulation, the air molecules must turn a sharp corner at the end, which is impossible for a fluid. Circulation has the effect of going against the flow on the bottom and accelerating the flow above the surface. This phenomenon is referred to as the Kutta condition, and it explains for the difference in velocity above and below the airfoil. This difference in velocity could be applied to the Bernouli’s equation, which would explain for the pressure difference between the bottom and upper surface, creating lift. Furthermore, airfoils create a downwash, where air molecules are deflected downwards. Applying Newton’s third law, we know that the air is applying a force on the airfoil, and in reaction, the airfoil is applying a force on the air. These two action and reaction forces cancel out; therefore, momentum is conserved. Since momentum is conserved, air molecules directed downwards must result in an upward momentum on the airfoil, which creates lift. Finally, it is important to note that there is no direct cause and effect relationship between pressure difference, velocity difference, and downwash. All of these factors contribute to creating lift.
However, there are many complications when predicting lift. One limitation is turbulent flow. It is very difficult to predict the motion of turbulent air molecules and how it interacts with surrounding air molecules. Therefore, we often assume laminar air flow. Other assumptions that could be made are that there is no viscosity or that the fluid is not compressible. There are so many factors affecting lift that it is almost impossible to predict lift without making certain assumptions. Another limitation is the Kutta condition. Airfoils that have round trailing edges are hard to predict using the Kutta condition. A recent paper published from Cambridge University, A Variational Theory of Lift by Haithem E. Taha and Cody Gonzalez, explores Hertz’ principle of least curvature to overcome this limitation. This paper showed a possibility that our understanding of lift could once again be shifted.
As air travel becomes a part of our daily lives, lift feels like a simple concept. However, lift should not be underestimated for its complexity. With advancement in technology and new ideas, our understanding of lift could change in the future. Even to this day, researchers and theorists are working to strengthen our understanding of lift.
Q&A
Hannah: You mentioned that there is no direct cause-and-effect relationship between pressure difference, velocity difference, and downwash. However, do these factors still affect each other in certain ways? For example, it sounded like you find the pressure difference by applying the velocity difference in the Bernouli’s equation, which suggests the result of the pressure difference depends on what the velocity difference is.
Yes, these factors affect each other, such as the pressure difference caused by a difference in velocity, explained by Bernouli’s equation. However, it is not like there is a one way cause and effect relationship. For example, downwash could affect the velocity differences and explain for pressure differences. This is not only limited to downwash. Differences in velocity could also explain pressure differences and downwash, and the same applies for pressure differences. All of these phenomena are happening at the same time, and all of the factors interact with each other, and as a result of these interactions, lift is generated.
Works Cited
Beginner's Guide to Aeronautics - NASA. https://www.grc.nasa.gov/WWW/k-12/airplane/.
Gonzalez, Cody, and Haithem E. Taha. “A Variational Theory of Lift.” Journal of Fluid Mechanics, vol. 941, 2022, p. A58., doi:10.1017/jfm.2022.348.
Liu, T. Evolutionary understanding of airfoil lift. Adv. Aerodyn. 3, 37 (2021). https://doi.org/10.1186/s42774-021-00089-4
“Theory of Flight.” Theory of Flight, https://web.mit.edu/16.00/www/aec/flight.html.
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