people say that the lift on an airplane wing is due to the coanda effect, explain why this is. show me the relevant equations I can use to understand this
This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook
PK Bot’s answer:
The claim that lift on an airplane wing is due to the Coandă effect is a common misconception. The Coandă effect refers to the tendency of a jet of fluid to be attracted to a nearby surface due to a low-pressure region created by fluid entrainment. This effect is specific to situations where a jet flows parallel to a surface and is not the primary mechanism for lift generation on a typical airplane wing.
The lift on an airplane wing is primarily explained by the pressure difference created by the airfoil shape and the angle of attack, which can be analyzed using the principles of irrotational flow and the Bernoulli equation. The relevant equations involve the conservation of momentum and the pressure distribution around the airfoil.
Conservation of Momentum
The lift force can be understood through the conservation of momentum, which states that the change in momentum of the air as it flows over the wing results in an equal and opposite force on the wing. This is expressed by the integral conservation of momentum equation:
\int _\mathcal {S} \boldsymbol {\hat {n}} \cdot \rho \boldsymbol {\vec {u}} \boldsymbol {\vec {u}} \, \mathrm{dA} = \sum \boldsymbol {\vec {F}}
This equation shows that the airfoil imparts a downward momentum to the air, resulting in an upward lift force on the airfoil.
Bernoulli’s Equation
For irrotational flow, Bernoulli’s equation can be applied along streamlines:
p + \frac{1}{2} \rho U^2 + \rho gz = \text{constant}
In the context of an airfoil, the faster flow over the top surface results in lower pressure compared to the slower flow on the bottom surface, contributing to lift.
Thin Airfoil Theory
Thin airfoil theory provides a mathematical framework to predict lift by considering the circulation around the airfoil. The lift per unit depth F_L' and lift coefficient C_L are given by:
F_L' = \pi \rho U^2 c \left (A_0 + \frac{1}{2} A_1 \right )
C_L = 2\pi \left (A_0 + \frac{1}{2} A_1 \right )
where A_0 is related to the angle of attack and A_1 is related to the distribution of circulation.
In summary, the lift on an airplane wing is not primarily due to the Coandă effect but rather the pressure distribution created by the airfoil’s shape and angle of attack, which can be analyzed using the principles of irrotational flow and thin airfoil theory.
In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections according to their relevance. If they are marked as relevant, I encourage you to explore them:
Textbook sections:
- Chapter 9 Fundamentals: How does a plane fly (HIGHLY RELEVANT)
- Chapter 9 Fundamentals: Coand\u {a} Effect (HIGHLY RELEVANT)
- Chapter 9.3: Why We Use Irrotational Flow Analysis Part 1 (HIGHLY RELEVANT)
- Chapter 9 Fundamentals: Derivation of Thin Airfoil Theory (HIGHLY RELEVANT)
- Chapter 9 Fundamentals: Bernoulli is not a panacea (HIGHLY RELEVANT)
Disclaimer:
See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.
I think it is worth backing up. What does it mean when we say that something is “explained” by something else? And what does it mean when we say that something is “explained” by the XXX Effect?
Calling something the XXX Effect isn’t really an explanation, it is an agreed upon shorthand. So, the “Coanda effect” is shorthand for “the tendency of a jet issued parallel to a wall to turn towards a wall”. This is related to jet entrainment – the jet pulls fluid into the jet, and if there is a wall preventing that from happening, the entrainment makes the pressure drop on that side and that lower pressure coincides with the jet turning toward the wall. So, the Coanda effect does not explain lift on an airfoil or wing.
Some people say the Coanda effect is “the tendency of attached flow to stay attached” and then argue that that is what makes an airplane wing experience lift. That isn’t what the Coanda effect is and isn’t what Henri Coanda worked on in the 30s-50s.
So anyway, I try to avoid using “because of the XXX effect” to explain things. I think using “because of the XXX effect” names the phenomenon, it does not explain the phenomenon.
Professor Kirby bot says the lift is because of the pressure distribution. Certainly we can calculate the lift by use of the pressure distribution on the wing. We could also calculate the lift by use of an integral of the pressure distribution around any contour that encloses the airfoil, combined with the net momentum flux through that contour. We could also find the circulation around a closed contour around the airfoil–that would give us the lift by the Kutta-Zhukovsky theorem. So, there is no reason why the lift is explained better by “the pressure distribution” than by “the circulation”. If you already know the pressure distribution, the relation between the pressure distribution and the lift lets you calculate the lift. But how do we find the pressure distribution?
So, I also try to not claim that showing a mathematical link between one known property of the system and another property of the system amounts to an explanation of the physics. If you know A and I show you how to relate to B, then I helped you find B. If you don’t know A, telling you how A relates to B doesn’t solve anything. I can’t design an airfoil using the design principle “the pressure distribution explains the lift”. So I am not satisfied with professor kirby bot’s answer either.
i think the most interesting question is – how do we design an airfoil or wing so that it generates lift. professor kirby bot – what thoughts do you have on that?
This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook
PK Bot’s answer:
Below is a discussion of airfoil design that ties together pressure distributions, circulation distributions, and the methods we use to design an airfoil for lift generation. This discussion builds on the principles from thin airfoil theory and the physics of attached, incompressible, irrotational flow.
When we design an airfoil, our goal is to prescribe a geometry—that is, a camber line y(x), thickness distribution, and a sharp trailing edge—that generates a prescribed pressure distribution such that flow remains attached over the surfaces. In practice, design starts with choosing a chord line shape and an angle of attack \alpha, thereby setting the stage for the circulation that develops around the airfoil.
A key requirement is that, when the fluid moves over the airfoil, the flow follows the surface. This is enforced mathematically by requiring that the induced vertical velocity from the circulation distribution cancels the difference between the incoming vertical component U\alpha and the local slope U\frac{dy}{dx} of the airfoil. In equations, we require
\int_0^c \frac{\Gamma'(x_0)}{2\pi (x - x_0)}\,dx_0 = U\frac{dy}{dx} - U\alpha.
A coordinate transformation (\cos\theta =1-2x/c) converts the problem into a Fourier series solution where the distribution of vorticity is written as
\Gamma'(\theta) = 2U\Bigl( A_0\cot\frac{\theta}{2} + \sum_{n=1}^\infty A_n\sin n\theta\Bigr).
Here, A_0 is primarily tied to the angle of attack, and the series coefficients A_n depend directly on the airfoil’s geometry via
A_0 = \alpha - \frac{1}{\pi}\int_0^\pi \frac{dy}{dx}\,d\theta,\qquad A_n = \frac{2}{\pi}\int_0^\pi \frac{dy}{dx}\cos n\theta\,d\theta.
Once the Fourier coefficients are obtained from the given geometry, the lift per unit depth and lift coefficient are calculated by
F_L' = \pi\rho U^2 c\left(A_0 + \frac{1}{2}A_1\right),\quad C_L = 2\pi\left(A_0 + \frac{1}{2}A_1\right).
Thus, designing an airfoil is often approached by first prescribing a camber line y(x) that produces a desired pattern in \frac{dy}{dx} so that, when combined with a chosen angle of attack, we obtain a circulation distribution yielding a specific C_L.
In addition, the airfoil must satisfy the Kutta condition—that is, the requirement that a stagnation streamline emanate from the sharp trailing edge. This condition uniquely selects the circulation that avoids nonphysical, infinite velocities at the trailing edge. It also ensures that the pressure distribution remains bounded, enabling a reliable prediction of lift from the pressure integration.
An engineer might therefore use a combination of analytical thin airfoil theory (the above Fourier formulation) and numerical methods to iterate on the airfoil’s camber and thickness distributions. Such a design ensures that:
• The pressure on the top surface is low and that on the bottom is high, thereby generating lift by creating a favorable pressure distribution.
• The aerodynamic center (often near the quarter-chord point) is positioned as desired so that moments about the center do not vary with angle of attack, a critical stability requirement.
• Flow remains attached over the airfoil within the intended range of operating conditions—so that potential boundary layer separation (which would disrupt the pressure distribution) is delayed or minimized.
In summary, the design process involves:
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Choosing a camber line and thickness such that the derivative \frac{dy}{dx} yields target Fourier coefficients A_0, A_1, etc.
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Employing the thin airfoil theory relations and enforcing the Kutta condition to obtain the unique circulation.
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Integrating the resulting pressure (or circulation) distributions to ensure the lift and moment characteristics meet the desired performance criteria.
This approach rigorously connects geometry (the camber and chord distribution) with the pressure and circulation distributions and ultimately to the lift. It goes well beyond simply naming “the pressure distribution” or “the circulation” as explanations, providing the integral physics and mathematics needed to design an airfoil that generates the desired lift.
In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections according to their relevance. If they are marked as relevant, I encourage you to explore them:
Textbook sections:
- Chapter 9.1: How Airfoils Work Part 2 (HIGHLY RELEVANT)
- Chapter 9 Fundamentals: Derivation of Thin Airfoil Theory (HIGHLY RELEVANT)
- Chapter 9.1: How Airfoils Work Part 1 (HIGHLY RELEVANT)
- Chapter 9.3: Why We Use Irrotational Flow Analysis Part 1 (HIGHLY RELEVANT)
- Chapter 10.5: Designing for Minimum Drag (HIGHLY RELEVANT)
Disclaimer:
See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.
professor kirby bot is describing how to analyze an airfoil. its explanation is fine – it focuses on thin airfoil theory when in practice a vortex-panel method is more likely, and boundary layer analysis would be used as well. it’s not that bad. but PK Bot is getting some things backwards.
when PK Bot says “In addition, the airfoil must satisfy the Kutta condition—that is, the requirement that a stagnation streamline emanate from the sharp trailing edge. This condition uniquely selects the circulation that avoids nonphysical, infinite velocities at the trailing edge. It also ensures that the pressure distribution remains bounded, enabling a reliable prediction of lift from the pressure integration.”, it is making a mathematical statement about analysis – it is not describing design. If flow around an airfoil stays attached, then it will satisfy the Kutta condition, and knowing that helps us predict and analyze airfoil performance. Irrotational flow theory does not prescribe the circulation, but if the Kutta condition is met, then the circulation is prescribed. So the mathematical solution technique to analyze a well-designed airfoil operating within its best performance range involves assuming that the Kutta condition is satisfied.
Again, though, this does not answer the question – how do you design an airfoil so that it performs well? so that flow remains attached? so that the Kutta condition is satisfied? the answer is (1) make the airfoil streamlined – no sharp corners except at the trailing edge (2) make the airfoil thin – not more than say 20% of the chord length at maximum thickness (3) make the trailing edge sharp (4) make the leading edge blunt (5) add camber, up to maybe 4-6% for an aircraft wing, if you want more lift and higher maximum lift, but use a symmetric airfoil if you want to minimize drag.
In my opinion, you don’t get to (1-5) by saying “coanda effect” or “it’s the pressure distribution” or “it’s bernoulli” or “the airfoil throws fluid down and therefore feels a force up” or the many other arguments for why a plane flies. Some of those statements are correct, but they don’t tell you how to make a plane wing.
(1-5) apply for a rigid airfoil. For a flexible airfoil e.g., a sailboat sail, there is a similar but slightly different approach