Comments on "CRITIQUE OF 'DISK BALLOONS'"
William J. Beaty
William J. Beaty
First let me say that, in my opinion, John Denker's critique of my "Disk Balloons" article has exposed no fatal flaws. True, there are flaws, but "flaws" exist upon a spectrum ranging from tiny quibbles which only a nitpicker would notice, all the way up to glaring errors which stab into the very heart of a concept and kill it dead. I see nothing in John's critique which "punctures" the disk-balloons. :) No fatal flaws have been discovered as yet. Mr. Denker writes that my "Disk Balloons" model is arbitrary. I'm sorry, I don't understand: in what sense "arbitrary?" They are based upon reality. This "balloon model" matches certain features of the real air movements around a real aircraft. I have read various descriptions of these air movements, so I took it upon myself to construct a simplified visual model in order to clarify certain elements of the physics of flight. I look at the air-flows around an aircraft in flight, I mentally slice them into thin segments, then I replace each segment with a pair of counterrotating disk-balloons where the center of each balloon is aligned with the wingtip (just as the thread of vorticity connects near the wingtip in a realworld aircraft.) If such a process must be denegrated as being arbitrary, when why aren't most other descriptive physical models arbitrary as well? I do try to take criticism seriously, but in this case I'll need more. I'll need an explanation on how to make my model *NOT* be "arbitrary." Please tell me how to alter the disk-balloons model so it is not arbitrary, and yet it still approximately describes the actual air motions surrounding a real-world aircraft. Perhaps the problem is that my model is DESCRIPTIVE. It is not a full-blown physical theory which predicts the airflows from first principles. The "disk balloons" are obviously a simplified mental model of the vortex-wake which is produced behind a real-world aircraft. I chose this model in order to accentuate certain features of the physics of flight, while covering up others. My goal was to create a highly simplified and "Newton-compatible" description of what happens when a wing applies a downwards force to the air passing near it. Why don't my "balloons" rotate chordwise? Because they are a model for the aircraft wake; for the trailing circulation patterns *behind* a wing, and the air trailing for hundreds of yards behind a wing does not rotate chordwise about the wing. Surely this is obvious. If we slice up the vortex-wake found behind an aircraft, we obtain something that resembles the magnetic field around a DC transmission line consisting of two conductors with a space between them: \ | / \ | / ______ | | | / ___ \ | | | / ___ \ / / \ \ | | | / / \ \ | | o | | | | | | | o | | \ \___/ / | | | \ \___/ / \_______/ | | | \_______/ / | \ / | \ The cores of the two vortices are centered on the wingtips of the aircraft. The pattern also vaguely resembles a pair of counter-rotating disks of air, therefore I chose these pairs of disk-shaped bouyant gasbags as an appropriate model. Are the radii of the disks a random choice? No, the radii are fixed by the shape of the vortex-wake behind the aircraft, which in turn are fixed by the geometry of the aircraft. The center of each disk is aligned with a wingtip of the aircraft, and the diameter of the huge disks is adjusted so that they nearly touch each other in the center of the span where the fuselage would normally lie. Hardly an "arbitrary" shape. I thought this was obvious to anyone, but apparently I was wrong. Perhaps I need something better than ASCII art to show how disk-balloons work. Why balloons? I used balloons because I wanted to create a mass-bearing "conceptual object" to which I could apply Newton's laws without much grief, (and hopefully without introducing any bizarre artifacts into the concepts.) A balloon entrains mass and carries momentum. So does a rotating vortex-pair. Throw either a balloon or a vortex-pair downwards, and you will feel a reaction force which pushes you in the upwards direction. OK, if we must call the balloons a "visual model" of real-world air flows, rather than an "explanation" of the lifting force, that's fine, because that's what they are. If the model has faults, I'll listen to the details of any complaints. Perhaps the reasons for various features of the model seem obscure. If so, please ask for clarification and I'll try to explain. *WHY* does the air behind a wing organize itself into a pair counter- rotating cylindrical structures? Similarly, WHY does a ring-vortex appear whenever a parcel of moving viscous fluid is injected into a non-moving volume of fluid? Drip a droplet of dye into a glass of water, and WHY does it form a ring-vortex? These are very interesting questions, but they're a distraction. The goal here is to achieve a gut-level "feel" for the mechanism behind lift, not to explore the depths of every detailed facet of viscous fluid dynamics. If we concentrate on details, it distracts us into ignoring the overall picture. (If for some strange reason we wish to *intentionally* ignore the overall picture, then the tactic of concentrating on the details is one method to fulfill this goal, but it is not honest, and it is not science.) Only after we understand the overall picture of the production of lifting-force should we take a look at the interesting details. I make this assertion: "When the wing throws air downwards all along its trailing edge, a long series of rotating 'disk balloons' are created, but no downward-moving blocks of air are created." If we wish, we can drip a droplet of blue food-coloring into a cup of water and note that the downward-moving dye organizes itself into a ring of moving fluid. This simple phenomena shows how vorticity can arise. I do not understand the details behind this event (perhaps somebody can explain how viscosity and turbulence creates vortex-rings?) For some obscure reason, viscous interactions usually cause these particular patterns of flow to appear behind aircraft as well as surrounding their wings. Knowing this, we can note the general shape of the downward-moving regions of air, figure out the mass that happens to be entrained within them, and end up with a simple way to apply Newton's laws to the lift-generating process: disk-shaped balloons. About "Qualitative Inconsistencies". Mr. Denker claims that a "real" vortex is a flow pattern where the (tangential) velocity is greater near the center of the vortex in a 1/r profile of velocity. NOT TRUE. While John's is a clear description of a mathematically-simple textbook vortex, (where the "vorticity" is entirely concentrated as an infinitely thin thread at the center of the the flow pattern,) this type of vortex is not "real," if by "real" we mean that it's the only pattern allowed in any real-world situation. In reality, many different sorts of vorticity distributions are perfectly possible, each with a different profile of tangential velocity (and each with a resulting different profile of distributed vorticity which is not necessarily concentrated at the center of the rotating mass of fluid.) If the vorticity is concentrated as an infinitely thin thread running down the center of the rotating fluid, then the tangential velocities are distributed as Mr. Denker describes. In that case, for a lone vortex ring or thread, each parcel of fluid will have a tangential velocity which is inversely related to its distance from the vortex-thread. However, whenever a quantity of rotating fluid instead happens to resemble a rotating solid cylinder, then obviously the vorticity is not concentrated in a central "vortex thread." Instead it must be distributed throughout the volume of the rotating air. Why is this a problem? It's a problem because aircraft vorticies don't have thin vortex-threads down their centers! It's well known that the vorticity behind an aircraft IS NOT concentrated totally at the wingtips. (I hope that Mr. Denker knows this. After all, he's the airplane expert here, not me.) If we use our hand to stir a bucket filled with water, we *don't* tend to produce one of these textbook-ideal vortex-thread patterns (although it's not impossible to do so if that is our goal.) OK, let us see who is right: Where is the vorticity concentrated in the wake of a *real* aircraft? Is it exclusively concentrated in two thin threads extending from the wingtips as John seems to imagine? Or does vorticity appear elsewhere as well? What do textbooks say? I'm not too experienced in such things, but I do recall several descriptions in books I read long ago. They mentioned an "elliptical" distribution of tangential (downwards) velocity, not a 1/r distribution, and they described a "sheet of vorticity" which pours off the trailing edge of a wing. If this is accurate, then the air behind a wing does not move like Mr. Denker imagines. It does not have a 1/r tangential velocity profile relative to the wingtip. Instead, the vorticity behind a wing exists as a sheet-like structure, and it's NOT totally contained in two narrow threadlike structures which extend from the wingtips. Therefore the motions of the air-parcels within the rotating cylinders of air behind a real wing do *not* have huge downwards velocity near the wingtip and much lower downwards velocity in the center of the span, Their their downwards velocity profile does not at all resemble the 1/r rule for ideal vorticies. So, these 'disk balloons' are supposedly faulty because the velocity of air within them is not inversely related to the distance from the center of rotation? Apparantly air doesn't behave that way in a real-world aircraft either. Therefore any description which only uses vortex-threads is faulty as well. Mr. Denker is totally wrong when he states that this aspect of the disk-balloons "VIOLATES THE LAWS OF MOTION." The laws of motion do not require that vortex-motions have threads of vorticity running down their centers. With the disk-balloons, do we find a discontinuity at the edge of the balloon where the balloon meets the outside air? Yes, but not in the way Mr. Denker imagines. Look at the region found near the center of the wingspan. The two disk balloons are almost in contact, and they shield each other from the effects of the outside atmosphere. On the other hand, at the two regions on the far edges of the disk-balloons, far to the left and right of the wing, the rotating disk-balloons must meet the unmoving atmosphere with no "shielding effect" whatsoever. However, the vertical relative velocity of the surface of the disk balloon is zero there, since the pair of disk-balloons is moving downwards as a system, while at the same time they rotate so as to move their outer edges upwards. When the rotating motion is added to the net downwards motion, the result is a near-zero velocity between the air and the balloon surfaces in the region where the balloons are in direct "unsheilded" contact with the outside atmosphere. This effect is obvious to anyone who can see these balloons rotating in their minds. I did not realize that I had to specifically explain it. Are most people unable to visualize the "rotating disk balloons." Maybe my article is biased towards visual thinkers. Maybe I need to create an animated image of the disk-balloons moving downwards while the "little man" runs forwards along them. [NOTE: I ADDED AN ANIMATION MONTHS LATER.] The balloon-pair behaves in much the same way that a real-world wake- vortex pair does. Because it moves downwards as a whole, the rotation of each vortex eliminates any gross discontinuity at the far edges of the rotating mass of air. Actually, this conversation gives me quite a bit of insight into one particular phenomenon in fluid dynamics which I had lacked before. It has been clear to me that the wake vorticies do not obey a 1/R velocity rule. But if this is so, and if the tangential velocity of each parcel of air which is far from the vortex-core is actually very large, then shouldn't there be a discontinuity where the rotating air meets the environment? Don't we REQUIRE a 1/R velocity distribution in order to avoid a huge shear at the "surface" of these cylindrical vortices? Not necessarily. Think of it this way. When an automobile tire meets the road, the tire does not "skid." Instead the velocity of the tire matches the velocity of the road. It's clear that the wake-vortices of an aircraft must do something very similar: they "roll against" each other as they descend, and their outer regions "roll against" the unmoving outside atmosphere and push downwards through it. The entrained air is separated from the outside air by a circular-shaped "separatrix", and the air at this "surface" can match the velocity of outside air which it "touches." This well-organized motion must lead to very low friction, and as a result, the wake-vortex pair probably travels for quite a large distance downwards before it is halted by viscosity. Aha! THAT'S why ring-vorticies in general can travel with such low friction. They are like a row of wheels threaded upon a metal ring. A row of solid wheels. Rotating solid cylinders. Like "disk balloons." One thing I've noticed in many explanations of flight: they almost universally concentrate on inviscid flow. However, vortex patterns cannot be created in an inviscid fluid. If viscosity is zero, then parcels of fluid cannot drag against each other, and therefore we cannot create smoke-rings, nor tornados, nor can we build wings which leak vorticity from their trailing edges at the same time that they create the necessary new vorticity associated with chordwise circulation. The entire topic of trailing wake-vorticies requires a VISCOUS model of air. If the usual explanations of flight exclusively focus on inviscid aerodynamics and simplified computer simulations, then they are missing something important. They are ignoring an entire section of aerodynamics where Newton connects with Bernoulli. The other points raised in Mr. Denker's article are totally accurate. Real-world physical balloons cannot superpose upon each other, yet patterns of air-motion can and do superpose, as when aircraft fly with wingtips nearly touching. As with any other physical model, this balloon-based model has areas of proper application, as well as particular limits beyond which it becomes foolish to employ them. However, if "disk balloons" are to be disparaged because they are not perfect, or because they are only a model, then we should also disparage all other models for the same reason, since a "model" is not reality. The same sort of attack could easily be launched against any models which use inviscid flow or 2D flow: these models are unrealistic because the real world is three-dimensional and viscous. They are unrealistic because Circulation must be added artificially by way of the Kutta condition. Also, they do not explain induced drag, and they suggest that upwash equals downwash even in a 3D aircraft. As a result, if 2D inviscid models are applied beyond their limits, they can lead us to profound misconceptions regarding how aircraft really work. They can even lead to blatant violations of Newton's laws. Yet these models are incredibly useful when they are applied correctly. Models are tools. A screwdriver is a very useful tool, but we should not attack it because it makes a lousy wrench. And if it shatters when we try to use it as a chisel, we should blame ourselves and not the tool. The "disk balloons" model totally ignores one important facet of flight: the mechanism which forms the vortex-pair. I provide a pair of human feet stepping onto a platform and spinning the balloons up to speed. In any realworld aircraft the balloons (the vortex pattern) envelop the entire wing, and the "disk balloons" connect smoothly to the pattern of chordwise circulation associated with the wing. My goal was not to explain how that part works. My goal was to focus everyone's attention on the DECENDING MASS that is launched by the wing, and to show why there is no need for a an aircraft to reach downwards for miles and create a direct force-pair between itself and the earth. This whole article is a response to a common incorrect statement: "airplanes fly because they push upon the Earth." No, an airplane does not have to push upon the earth. This is because it can give the nearby air a net acceleration and leave it descending afterwards. The decending air eventually applies its force to the earth, but the airplane does not participate in that exchange of forces. As a model aimed at K-6 textbooks, 'disk balloons' are a vast improvement compared to the usual faulty models presented in those books. It is impossible to explain flight to little children if we limit ourselves to circulation-based models. Even high-school physics teachers have trouble making sense of explanations based upon circulation, so we should think twice before insisting that children learn these models. At the same time, if "disk balloons" were used to replace full-blown aerodynamics theory, explanations would be ridiculously limited and distorted. They are a child's version of vortex theory. The right tool for the right job? What misconceptions do 'disk balloons' breed? Mr. Denker has found a few. Undoubtedly there are more. Hopefully any new misconceptions are as minor as the current ones being spread by textbooks, and hopefully there are none which cause a fundamental misunderstanding of how aircraft work. They are not without misconceptions, but we must compare them to the damage being done by the current crop of K-6 textbook explanations, and even the damage being done by 2D inviscid-flow models used in more advanced textbooks. 2D wings and 3D wings do not use the same mechanism to generate lift. The disk balloons are not "real" in that they don't match reality perfectly. Is a mathematically sophisticated, inviscid-fluid-based description of flight "real"? Of course not. No single model is. If we wish to understand the world, then we must employ a variety of separate models which "overlap" in order to cover all possibilities. We also need to be very aware of their individual flaws and their ranges of applicability. One issue might have been missed here. Perhaps the 'disk balloons' are a disgusting and unsophisticated oversimplification which are appropriate only for little children? Correct. Everyone please read the title of my website. "K-6" means *children* (or specifically, science teachers who work at those grade levels.) Some of the material does occasionally extend up to the K-12 grades (e.g. trying to use disk-balloons to calculate lift, instead of only using them as a visual intuitive picture of the lift-generating process.) If we desire a simplified, basic explanation of flight which gives us a feel of how Newton's laws apply to aircraft, then "disk balloons" work fairly well. If we want to explain why migrating geese should fly in "V" formations for efficiency, or what happens when two aircraft approach each other wingtip-to-wingtip, then disk balloons are worse than useless because they make crazy predictions. However, the inviscid-fluid computer models which can correctly treat the problem of multiple-aircraft interactions will make crazy predictions when used to calculate fuel usage at various low airspeeds. In my opinion, they're also useless when it comes to explaining airplanes in a clear, easily-understood manner which even a child can understand. The right tool for the right job. As with newspapers and science museums, my target audience is the general public. Therefore I do the same thing as the New York Times: I try to aim everything at the 6th-grade level. Yes, there are many other models which could be used to explain how aircraft fly. But at the 6th grade level, most of them are hideously distorted (e.g. that stuff about parcels of air rushing to meet each other at the trailing edge of an airfoil.) At higher grade levels, sophisticated models employ abstract concepts and mathematics, and they tend to obscure the simple, basic ideas behind flight. Such models are totally unsuitable for explaining flight to non-physicists. BOTTOM LINE AGAIN Let's review John Denker's table of predictions made by "Disk Balloons" compared against the predictions regarding the behavior real vortices: DISK-BALLOON REAL VORTEX The wing imparts energy and downward momentum yes yes to the air. The wing affects a swath of air comparable yes yes in width to the wingspan. The wing imparts some rotational motion yes yes to the air. The wing affects a swath of air up to a yes yes height that depends on the wingspan. The theory requires an arbitrary assumption determined emergent about the size of the region of rotating air. empirically phenomenon The theory correctly and naturally describes NO yes interactions with the ground and with other wings. The theory is consistent with the known laws yes yes of motion. It appears that the disk-balloons theory stands up in every case but one: it cannot be easily used to explain the ground-effect phenomena, nor the changes in lifting force which occur when airplanes fly in formation. And ,as I said earlier, I give no explanation for why the wing can make those "balloons" start moving. In my opinion, this is pretty impressive for a "theory" which was developed in order to explain flight to children! If we are required to make things visible, basic, and to eliminate from our explanations as many abstract concepts and mathematics as possible, then "disk balloons" is one result. It's a "toy explanation," and it might cause any important and sophisticated aerodynamicist to laugh scornfully. Exactly. That's what I *intended* it to be from the start. Too bad about any scornful laughter though. Perhaps the important aerodynamicists should develop a small bit of humility and think of themselves not as experts who are unable to make embarassing errors, but as students who are always learning and who are expected to make huge errors as part of the trial-and-error process. I still find it amazing that the "disk balloon" description contains features of which some aerodynamicists are apparantly unaware: it points out that the gravity-fighting flow of momentum must end up in the descending wake-vortex behind the aircraft. It shows how the lifting force upon a wing is the result of a 3rd-law "reaction motor" effect. It illustrates how a pair of non-1/R vorticies can interface with the unmoving atmosphere without any high-shear regions. It allows an ambitious highschool kid to derive the equation for induced aerodynamic drag. And it clearly illustrates the fact that a two-dimensional description cannot hope to explain the flight of a real aircraft: the patterns of flow before and behind a 3D airplane are dissimilar, and also the lifting mechanism originates in the creation (and the downward acceleration) of a vortex-wake in three dimensions. None of this fits into a 2D simulation, and it cannot exist at all in an inviscid simulation unless we artificially place it there by hand. As a result of our 2-D thinking, we often do something which is unfortunately equivalent to this: ignoring the exhaust from a rocket motor, but then attempting to explain rockets by applying Bernoulli's equation to the tangential velocity of gases across their solid surfaces. Yet rockets cannot be explained if we close our eyes to the existence of their exhaust stream!!! The downwards-moving wake vortices behind an aircraft are the "exhaust" from the reaction motors, where the wings are the reaction motors. In a 2D simulation, all of these simple 3D concepts are impossible and unphysical. The right tools for the right job, and sometimes the incorrect mental tools can make the job impossible.