Given how deeply air transportation is woven into modern life, it’s surprising that the precise workings of aerodynamic lift remain a topic of debate among the experts. To sort all this out, I met on a video call last month with Paul Bevilaqua, retired from Lockheed Martin Skunk Works, and Haithem Taha of the University of California, Irvine. I learned about several myths and at least one collapsing theory. Here is our discussion, lightly edited and compressed.
Ben Iannotta: I know I’m not the only one who’s looked out an airplane window and marveled at the power of aerodynamic lift. A 787-8 weighs about 230,000 kilograms. That’s the equivalent of 38 African bull elephants. And yet, most of us have no fear of getting on an aircraft of that size, and, in fact, maybe those are the among the safest out there. So let’s learn more about lift from two experts. First, Paul Bevilaqua is a former chief engineer of Lockheed Martin Skunk Works in California. Paul, what projects were underway during your tenure?
Paul Bevilaqua: I played a leading role in developing the Joint Strike Fighter. The Marines asked me if I could come up with something to replace both the subsonic Harrier and the supersonic F-18, and so I invented the propulsion system that gives the supersonic F-35B vertical capabilities. I also realized you could take that propulsion system out and develop Navy and Air Force variants of the same airplane.
Iannotta: Next, Haithem Taha is an associate professor of mechanical and aerospace engineering at University of California, Irvine. Haithem, you and a student published a paper, “A Variational Theory of Lift.” I’ve heard that maybe the textbooks now need to be rewritten.
Haithem Taha: Well, I received a few congratulating comments from some colleagues all over the world, and at least some of them plan to add 10 minutes in their class on the Kutta condition about this theory.
Iannotta: You’re from Egypt. How did you come over here to UC Irvine?
Taha: I did my Ph.D. at Virginia Tech, and it was in engineering mechanics on flapping flight, simultaneously with a master’s degree in mathematics. Then I searched in the job market and became lucky to come to UC Irvine.
Iannotta: Let’s get into it. I thought aircraft generate lift because the air pressure over the wing is less than under it. Is that much correct?
Bevilaqua: It’s partially correct. There are several aspects of lift, and different people have different views. Some people say an airfoil is shaped to develop a pressure difference. That’s the action, and the reaction is a downwash behind the wing that satisfies Newton’s law of action and reaction. Other people say the airfoil is shaped to push the air down behind the wing. That’s the action, and the reaction is a lift on the wing. But both those ideas are wrong.
What we’re dealing with is something like the Hindu parable about the blind men and the elephant. An elephant comes to town, and the blind men have no experience with an elephant. So one of them grabs the trunk and says, “Aha! An elephant’s like a snake.” Another one grabs a leg and says, “Aha! An elephant’s like a tree.” Another one grabs the tail and says, “Aha! It’s like a rope” and the last one says, “No, no” — he grabs the tusk — “it’s like a spear.” And that’s what we’re dealing with when it comes to lift. In one version of the parable, they end up fighting each other, beating each other with their canes. In another version, they sit down and listen to each other and put together a complete picture by collaborating. And I hope that’s what we’re going to do.
There’s a beach in St. Martin in the Caribbean, where I was vacationing several years ago, where you can stand at the end of the international airport runway. The airplanes are coming in for a landing a hundred feet above you, but there was no huge downwash. I’m thinking, “Well, wait a minute. How does this really work?”
Iannotta: Haithem, does your variational theory of lift address this downwash question?
Taha: The theory confirms what Paul was saying. The theory is more to complement the picture that Kutta has formulated rather than to challenge the Kutta theory itself. For the lack of better words, Kutta’s theory becomes a special case of the new theory. There is no contradiction, really.
Iannotta: You’re talking about Martin Kutta, the German mathematician from the early 1900s.
Taha: 1902, yes, his famous paper.
Iannotta: To what degree is the Kutta condition a complete explanation, and to what degree is the Kutta condition incomplete?
Bevilaqua: I think it is complete. In one view, it’s the boundary layer viscosity that adds up and prevents the flow from going around the trailing edge. Another view is that it requires an infinite velocity to go around a sharp trailing edge, and because the flow is inviscid [meaning there is no viscosity], viscosity doesn’t have anything to do with it.
Taha: On the Kutta condition, I’d like to address the completeness versus incompleteness. Yes, it is indeed the dominant theory of lift that is being taught in every single aeronautical engineering school throughout the world and discussed in every single textbook in aerodynamics.
Iannotta: Briefly explain what the Kutta condition is.
Taha: If we have a regular wing shape, there are several possibilities for how the air flows over. It could rotate around the trailing edge from the lower surface to the upper surface, or vice versa. The flows could come smoothly together off the trailing edge, and there is no flow from the lower surface to the upper surface or vice versa. The Kutta condition simply states that this is the case in reality: There is no flow from the lower surface to the upper surface, or vice versa. But luckily, our undergraduate students do not ask the following question: What happens if we don’t have a sharp edge? The dominant theory that we teach all over the world — it immediately collapses.
There is no theoretical model nor a physical explanation now that can give you how much lift is generated if we don’t have a sharp trailing edge, or if we have multiple sharp edges or if an airfoil with a single sharp edge is doing an unsteady maneuver, like in the “Top Gun” movie by Tom Cruise. We already know in this case that during the transient period before reaching steady state, the flow goes from around the edge from the lower surface to the upper surface — so the Kutta condition will not apply. So, it’s quite bothering that the dominant theory that we teach all over is quite fragile.
Iannotta: Are there aircraft flying today that have non-sharp trailing edges or multiple sharp trailing edges?
Bevilaqua: There are supersonic airfoils that have sharp leading and trailing edges, and so does the airfoil on the F-117. In fact, what we say in aerodynamics is that you can make a barn door fly if you have enough thrust in your engine. The purpose of shaping the airfoil is to get lift efficiently.
Iannotta: Haithem, can you take apart the Hertz principle of least curvature discussed in your theory for us?
Taha: It’s actually very simple and intuitive: If we have a free particle, it moves along a straight line. But if you add a constraint on the particle, it will deviate from a straight line. So Hertz’s principle asserts that the particle will deviate from the straight line only by the amount to satisfy the constraint. Nature will not overdo it because the deviation from a straight line is curvature, hence the name “least curvature.” So if I have a particle moving anywhere, it will always try to minimize its curvature. If I have a collection of particles together moving in a system, they minimize the curvature of the entire system. You place a wing in their way, so now you have forced them to flow around. The only option that will minimize the total curvature of the system is that they come smoothly together from the trailing edge, matching the Kutta condition. So this is how we find the circulation on any smooth shape without sharp edges. We simply minimize the total curvature with respect to circulation.
Iannotta: You mentioned the idea of air from the bottom circulating around the trailing edge.
Taha: I’m saying nature will prevent that. Engineers always like to ask why. “Why is there lift?” Because there is pressure difference. “Why is there pressure difference?” Because there is velocity difference. “Why is there velocity difference?” Because there is circulation. “Why is there circulation?” Well, the answer has always been “Because of the Kutta condition.” But the Kutta condition did not come from first principles. We stop asking why only when we encounter first principles. So now we are saying, “No, there is circulation because that is the minimum curvature solution, and the minimum curvature is a first principle.”
Bevilaqua: I would like to add that the variational principle is really also another way of stating Newton’s law of inertia. The body moving in a straight line will keep moving in a straight line. So you’ve got these flows coming along the top and bottom of the surface, moving along the surface. They want to keep doing that, and the flow will make an adjustment to preserve that motion. It’s the same principle, really.
Iannotta: Am I correct that it’s a myth that the air crossing beneath the wing and over the wing must come together at the same time?
Taha: It’s a myth. Nature is not that kind to keep people who are together, together forever.
Bevilaqua: Well, I don’t know where it came from. It’s a simple explanation that satisfies most people. They go away, and they stop bothering you.
Iannotta: In reality, the air is going faster over the top of the wing, and it gets past the trailing edge sooner than the air underneath.
Taha: Yeah.
Bevilaqua: The circulation actually is an exponential decay as you move away from the airfoil. So when you get far upstream and far downstream, the velocities do come back together. It’s just locally where the airfoil is that you have the difference. But far downstream they do come together.
Iannotta: I’m picturing a rock in a river.
Bevilaqua: Right. After a while, the flow forgets there was a rock. You get back to uniform flow.
Iannotta: Let’s say there’s an airfoil, and above it we know the pressure is less than below it. So there’s nothing above the wing to restrain it. The air beneath the wing naturally pushes it upward, and that’s lift. Have I offended reality?
Bevilaqua: The whole thing is the circulation around the wing. The air is going faster above and slower below it, so the pressure is lower above and higher below.
Iannotta: And that’s Bernoulli’s principle.
Bevilaqua: Yeah. And if you look upstream and downstream, the air in front is going up, and the air in back is going down. And so, if you look at a momentum change, there’s Newton’s second law of motion: The mass of air is accelerated downward, but there is no net downwash. You’re changing the up velocity to a down velocity. So, the whole thing is happening together all at once — the pressure force and momentum change. The elephant is the circulation.
Iannotta: We know from Bernoulli’s principle that pressure goes down when a fluid flows faster; that’s why the pressure over the wing is less. It’s not like there’s some gap there, correct?
Bevilaqua: Right. The wing is sucked up from the top and pushed up from the bottom.
Iannotta: I see. Haithem, you talk about viscosity in your paper.
Taha: I belong to the camp that believes viscosity is not essential to generating lift. There is a governing equation that is Euler’s [Swiss mathematician Leonhard Euler]. If you solve the Euler equation over an airfoil, it’s not unique. It has infinitely many solutions. It has a solution with zero lift. This doesn’t happen when you solve the Navier-Stokes equation with the friction term, which immediately gives many people the impression that viscosity is essential.
Bevilaqua: You said that the Euler equations do not develop lift, but Pradeep Raj [a professor at Virginia Tech] showed that they do.
Taha: Yes, I believe that Dr. Raj solved the compressible Euler equation and computed the lift. Even without invoking the Kutta condition, he found Kutta’s lift. However, he solved it using computational fluid dynamics by a numerical algorithm. Well, any numerical algorithm has some dissipation in it, and he argued that this numerical dissipation plays the role of an artificial viscosity. So that’s one explanation. But in our theory, there is no numerical scheme. There is no artificial viscosity, and we recover the Kutta condition, anyway. This is one of the strengths of our theory. Thanks, Paul, for reminding me of this point.
Bevilaqua: I think that’s the brilliant part of your contribution. It’s inertia, not viscosity, that enforces the Kutta condition and creates lift.
Iannotta: If engineers understood all this perfectly, could they make a better aircraft?
Bevilaqua: They wouldn’t make such a bad one. New hires often come in with a wrong conception. You have to explain to them: “Well, that’s not where lift comes from.” I have had engineers suggest blowing a jet over the top of the wing, “Because the air goes faster — that should produce lift.” No, that’s not going to produce lift, because the thickness of the jet is tiny compared to the huge atmosphere. Therefore, the pressure in the jet is the same as the atmospheric pressure. In fact, Einstein had a wrong idea about pressure. He thought that if you put up a huge hump on top of the wing, the hump would squeeze the air together and cause it to lower the pressure. So it makes it less of a hassle to design airplanes if people understand where the lift comes from.
Iannotta: Could someone make a better paper airplane if I knew the theory?
Bevilaqua: That’s a good question. The Wright brothers put thin wings on each side of a bicycle wheel and then rode it through the streets of Dayton. The wing that was pulled back had more drag. They also built a wind tunnel, and they tested little models of thin wings in there. And to his death in 1948, Orville Wright believed that a thin wing was the right way to go. Well, it is the right answer for a paper airplane or a model airplane. It’s the wrong answer for a large airplane. A breakthrough came in the 1920s when Clark came up with the Clark Y thick airfoil, and the Wright brothers both said, “That’s wrong. We have data that shows a thin airfoil is better.” But yes, it is better in a small wind tunnel or on a bicycle wheel balance.
Iannotta: We’ve been talking about wings with a curve on the upper surface, and probably asymmetrical. So how does a paper airplane fly?
Bevilaqua: Because a thin wing is optimal for very small airplanes. You put it at a large enough angle of attack, and you can get lift out of it even though it’s not shaped or cambered or thick or anything.
Iannotta: To close things out, I wanted to give each of you a chance to kind of wrap things up.
Taha: Back to your question about having a good theory. The Wright brothers flew something in 1903, but for 59 seconds. They were knowledgeable because they did an immense amount of tests. Without a theory, you can build a prototype and you can fly it for 59 seconds. You can push here and there and fly it for a few minutes. But to reach the level of maturity that we have nowadays, with millions of flight hours per year, this needs a good theory. And to go to the next phase, it needs an even better theory. So it’s just a one step along the right direction.
Iannotta: Paul, how would you wrap things up?
Bevilaqua: It’s circulation that creates the pressure difference between the top and bottom of the wing, and the momentum change between upstream and downstream of the wing. In a sense, an airfoil lifts itself by its own bootstraps. It’s just different aspects of the same phenomenon. And let me add something we have only touched on, which is the effect of a finite span wing. There is a downwash behind the wing that comes from the vortices that trail off the wing tips. They are the continuation of the circulation around the wing. So people say “Ah! F = ma. The downwash must be the reaction to the lift on the wing.” But it’s not, because the vortices induce an upwash outside of the wing, and there’s no net downward flow of momentum. It is not the Newtonian reaction to lift.
Iannotta: So another myth to bust. Thank you both. I think we all have more to think about when we get on our next airplane flight.
