Can I draw simple conclusions from thin airfoil theory, like how much lift and drag relate to airfoil thickness, camber, and angle of attck
PK Bot’s answer:
Thin-airfoil theory is very very good at helping with some of the things you asked about, and useless for the others.
Thin-airfoil theory assumes the airfoil is infinitely thin, so it cannot inform the effects of thickness. You need vortex-panel methods for that.
Thin-airfoil theory is inviscid theory, so it cannot predict viscous drag or separation. It is helpful for slender airfoils at modest angles of attack. It is useful to predict lift. It is not useful to predict drag. You need boundary-layer analysis to predict drag.
Thin-airfoil theory is great at linking lift directly to angle of attack and camber. In fact, it tells you that one component of lift is linearly proportional to the angle of attack, and a second component of lift is linearly proportional to the camber, expressed as the mean concavity of the airfoil. These two factors are independent and sum to create the total lift generated by the airfoil.
Thin airfoil theory replaces the effect of an airfoil with a distribution of vorticity along the chord line. You also do a coordinate transform and a Fourier-series expansion of the local airfoil slope. When you do that, you get a pretty amazing simplification.
Fourier components
The zeroth-order component, A_0, is just the angle of attack of the airfoil.
The first-order component, A_1, is a measure of the mean concavity of the airfoil centerline.
The higher-order components, A_2 through infinity, measure the nonuniformity of the concavity of the airfoil centerline.
The lift is given by the equation:
C_{\mathrm{L}} = 2\pi \left(A_0 + \frac{1}{2} A_1 \right)
Higher-order components play a role with the pitch moment coefficient and aerodynamic center, but they do not affect the lift.
So thin-airfoil theory tells us that (to the extent the model is correct i.e., for slender airfoils with attached flow) the lift of an airfoil comes from angle of attack as well as mean concavity. Details of concavity distribution matter only if we are calculating moments or other factors that are a function of the location of lift rather than just the sum of the lift.
To formulate this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2). Below, I have organized the sections based on their relevance. If they are marked as relevant, I encourage you to explore them:
Textbook sections:
- Chapter 9 Engineering Models: Thin-airfoil theory (HIGHLY RELEVANT)
- Chapter 9 Fundamentals: Derivation of Thin Airfoil Theory (HIGHLY RELEVANT)
- Chapter 9.1: How Airfoils Work Part 2 (HIGHLY RELEVANT)
- Chapter 9 Fundamentals: How does a plane fly (HIGHLY RELEVANT)
- Chapter 9.3: Why We Use Irrotational Flow Analysis Part 1 (HIGHLY RELEVANT)
Disclaimer:
See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.
Thin-airfoil theory also IMO provides an immediate and clear explanation for why airfoils, in practice, have blunt leading edges and sharp trailing edges.
Thin-airfoil theory leads to an infinite vorticity at the leading edge – unless you line up the airfoil at its “magic angle”, it predicts that the fluid accelerates infinitely fast at the leading edge, which also implies that the fluid pressure goes to negative infinity at the leading edge. At that is all not physically possible. We ignore that problem when we do thin-airfoil theory, because it turns out that even though that detail is totally wrong, the predicted lift value is quite good. We take the win where we can get it.
However, this tells you that in practice, if you are going to have a lifting surface that actually works, you can’t have a sharp leading edge, because a real airfoil with a sharp leading edge would just lead to separation. This problem does not occur at the trailing edge – a sharp trailing edge dictates the circulation of the airfoil and does not (at reasonable angle of attack and camber) lead to separation.