This note captures my ongoing effort to understand how aeroplanes fly. What is actually going on is quite complicated and I am far from alone in finding it difficult to reduce to simple principles. For the present, this investigation is no better than any of the others. I leave out many complications, such as compressibility, viscosity, temperature and so on, which are not fundamental to the mechanisms at work. Nevertheless, the absence of understanding appears to be endemic.

There are three aspects to any description of lift. At the heart of the problem are the numbers measured in the wind tunnel, the phenomena to be explained. In the usual scientific manner, we use mathematics to model and predict those numbers. But then we demand an intuitive explanation of what those equations represent – of how and why it all works. This last is the subject of this essay.

There are many such intuitive explanations out there, and most of not all of them are either flawed or incomplete or both. There is a body of basic physics which underlies lift and represents a level of irreducible complexity to any explanation. It is all too common for one aspect or another to be oversimplified, forgotten, misrepresented or misunderstood in these explanations. Of particular merit is Doug McLean's in his *Understanding Aerodynamics: Arguing from the Real Physics* (Wiley 2013). But even he gets tangled up in the complexities and ends up with his handwaving moments, and some commentators believe that nobody fully understands as yet. I seem to be joining their camp.

The basic condition for flight is a smooth flow, in which any small drop of fluid or parcel of air follows a smooth and continuous path we call a stream tube. An arbitrarily tiny particle follows a streamline. Any disturbance creates a small pressure pulse which, like any other pressure pulse, dissipates at the speed of sound. Under conditions of subsonic flow, such pressures never gets a chance to build up and even a gas is essentially incompressible.

The flow has by definition two characteristics at every point; speed and direction. Together these are called its velocity, and we map them out as a velocity field throughout the flow. These are changed by any disturbance. A continuous disturbance, such as flowing past a wing, creates a pressure field around the wing. In particular, we can define a given disturbance by the changes in speed, direction and pressure that it presents. These changes; acceleration, turning and pressure gradients, interact with each other in the flow and are the fundamental physical mechanism by which lift is created. Understand how and why these changes happen, and you will understand lift.

The conservation of energy is fundamental to the resolution of many of the puzzles that confuse people over why things happen the way they do. It underlies the gas laws and other principles of fluid dynamics, and as such it dictates the basic conditions of smooth, steady flow that provide the backdrop to these principles. You will also find it popping up in the discussions of all the other mechanisms which follow; Newton, Bernoulli and circulation. If your explanation is not consistent with energy conservation, you are lost.

One key gas law worth highlighting is that, under constant flow, pressure × volume is constant. Or, to put it another way, they have an inverse relationship; double the pressure and the volume will halve. Since we are dealing with incompressible flow, any drop of fluid or parcel of air must maintain a constant volume throughout.

A fluid flow past a stationary object and an object travelling through a calm fluid are, from this point of view, the same phenomenon. The difference is only relative to the observer (called a reference frame). However both involve local sideways and other movements of the fluid as it meets the body, so it is usually easier to discuss the whole thing from the point of view of a fixed body with fluid flowing around it, say a pilot's view out of the cockpit, rather than that of a bystander watching a moving aeroplane fly by.

Fluid dynamics, like any other mechanical system, is dominated by Newton's laws of motion. For the present purpose, I will state these as:

**N.1:**A body moves steadily in a straight line unless acted on by an external force. This is a direct consequence of the law of conservation of energy.**N.2:**When an external force acts on a body, it accelerates in the direction of that force. This is the famous*F*=*ma***N.3:**A force exerted on one body is accompanied by an equal and opposite force on another body. Commonly expressed as, "Every action has an equal and opposite reaction."

One can think of a small drop or parcel of fluid being carried along in the flow, its passage marked by a streamline past the wing. By N.2, the energy of such a parcel can only change if a net force is exerted across its boundary.

Ultimately, lift is an upward force on a wing and by N.3 it is obtained by exerting the opposite force on the air. By N.2 this accelerates the air downward.

But this mechanism alone is insufficient to explain the amount of lift we actually obtain. Sir George Cayley built a whirling-arm test rig, attached flat plates angled to the airflow and measured their lift. He and others found that the actual lift is around four times more than a simple model of bullet-like molecules or parcels hitting the plate and bouncing off downwards could account for. Or, to put it another way, there is four times as much air being deflected as a simple application of Newton's laws predicts. Evidently there is some kind of multiplier effect going on. In fact, as we shall see, there are two such effects and all three of them work tightly together to create the total lift.

Bernoulli's principle is key to how wings create lift, but it is notoriously hard to gain an intuitive understanding. Daniel Bernoulli punctured a tube, ran water through it and wathced the flow rates of the leaks through the holes. He realised that a faster leak was caused by higher internal pressure. He then squashed the tube in various places to restrict the main flow and noticed that *an increase in the speed of the flow is accompanied by a fall in pressure (and vice versa)*. This is the principle which is named after him.

It was first published by Jacob Bernoulli, who dervied it from Newton’s Second Law, on the basis that the acceleration of the fluid was related to the pressure difference along its path as it speeds up.

It is easy enough to picture how a fall in pressure might speed up the flow. Fluid flowing from a high-pressure zone to a low-pressure zone will be sucked towards the low-pressure one and will accelerate in that direction. If a tyre gets a puncture, air accelerates to flow out and not in. Similarly, fluid flowing from low to high pressure will be sucked backwards and slowed down.

But why should it also work the other way? Why should speeding up the flow *cause* a fall in pressure? It is easy to confirm that it does. The atomiser is a simple gadget used by artists and perfumiers. A short tube dips into your bottle of liquid. A second tube blows air across over that end. As the air flows, it draws the fluid up and sprays it out of the end. That drawing-up can only be explained by low pressure created by the moving air above. But *why* does that happen?

A venturi tube has a wide mouth and tapers back, constricting the flow. The air has constant volume so has to speed up to keep the flow rate constant. The tube has to be carefully shaped or the flow instead slows down and chokes; the central mystery of Bernoulli's observations is why this does not always happen.

And over a wing, why should the flow speed up anyway? Why should the flow over a wing behave as if it were in a Venturi tube, and how can the air maintain constant volume if its pressure is changing significantly?

The first part of this, how a single-sided channel can narrow the flow, can be understood as a slight narrowing of the stream tubes. By definition, flow does not cross the boundary of a stream tube, so it can be thought of as a virtual Venturi tube. The narrowing happens because the pressures associated with it fade away as you go further from the wing. The air a moderate distance above flows straight across, so the stream tubes have to narrow a little to fit between the far flowfield and the wing.

That and the incompressible flow explain the increase in flow rate. But what then causes the drop in pressure that Bernoulli observed and every atomiser makes use of? Doesn't incompressible flow mean that there is by definition no change in the pressure?

To answer the inconsistency over volume, we need to takle a step backwards here and realise that there are different kinds of pressure, and people often fail to make it clear which kind they are talking about. Bernoulli's "pressure" is very different from the gas laws' pressure.

Although Bernoulli derived his ideas from Newton's laws, the great Leonhard Euler later reformulated it around the various energies in the flow. This can be easier to explain and to apply. So mostly nowadays we just apply Euler's maths and argue interminably about Bernoulli's key insight.

Euler's model is based on the conservation of energy. Pressure stores potential energy. When you pump up a bicycle tyre, it takes a lot of energy. The higher the pressure builds, the harder you have to work to get more in. All this energy gets stored in the tyre and if it springs a leak the energy and air are released with a loud hiss. A big car tyre can burst with a loud bang.

Another source of energy in a fluid flow is its kinetic energy of motion, derived from the mass of the air and its velocity.

Add these two energies together and you get the total energy of the flow. In a smooth flow, the only transfer of energy is of that being carried along in the flow. Each drop of fluid in the flow carries its total energy along with it, and the energy of the drop does not change; its total energy remains constant.

Thus, if the energy of either motion or pressure changes, the other must counterbalance that by changing the other way. As a fluid flows faster, its kinetic energy increases and so its pressure energy falls. Conversely, if the local pressure falls, the fluid responds by increasing its energy of motion.

Other things being equal, as they are in steady flow, this yields Bernoulli's principle, that an increase in speed and a pressure drop are inseparable.

We are now in a position to understand the different kinds of pressure. Explanations of Bernoulli distinguish static, dynamic and total pressures. They usually forget to explain that none of these is the ambient air pressure. So for the moment I will define four kinds of pressure:

**Ambient pressure:**The pressure in still air. It is what meterologists measure.**Static pressure:**Local pressure in a moving fluid flow. This is the pressure described by Bernoulli's principle.**Dynamic pressure:**Pressure reduction corresponding to the lowering of potential energy in a moving flow.**Total pressure:**Equals the static + dynamic pressures, and is constant throughout the flow.

From these we can see some relationships between them that seldom get explained properly:

The pressure referred to in Bernoulli's principle, as stated above, is the local static pressure and not the ambient pressure described by the gas laws. In his equations, the total pressure remains constant.

When you are standing still, the dynamic pressure is zero. Therefore the ambient, static and total pressures are all equal. As you start to move and the dynamic pressure builds, the static pressure falls but the total pressure remains constant. But so too does the ambient presure, in fact these two pressures, total and ambient, are one and the same. The total pressure in Bernoulli's theory is just the pressure described by the gas laws. This explains how the volume of a fluid packet can stay constant under a decrease in static pressure.

The total pressure may be measured simply by pointing a tube, closed at the back, straight into the oncoming flow. If a similar tube is closed at both ends but has holes in its side, it will measure the static pressure. By measuring the difference between these pressures, i.e. at the back ends of the two tubes, the dynamic pressure is obtained and the airspeed may be calculated. This is the principle of the Pitot tube.

This is perhaps the least intuitive aspect of the theory, but it is as essential as the others. Circulation around a wing is a two dimensional flow back over the wing, down behind, forwards underneath and up again at the front, back where it started. It is essential to understand that this is not the actual flow seen, but is a mathematical *component* of the overall flow.

The various components of a flow are all the effects on the overall pattern that each individual mechanism has. The most immediately obvious is the free-stream flow. Another obvious one is the parting of the air in front to allow passage of the wing, and its closing again behind. A third is the net downward deflection of the flow, against which the lift reacts. Less obvious (and ignored here) is a net forward motion given to the air in reaction to the drag which holds the wing back.

The circulation component is really a composite of several others. Much is made, in other explanations, of the increased speed of the airflow over the wing causing a drop in static pressure. But how does that all get started?

When a plane is about to take off, it rotates its nose upwards so that the wing angles upwards. The underside of the wing must push air away to make room, and the necessary force is manifest as an increase in pressure. At the same time the air above finds less resistance to pressing down on the wing as it flows towards the back, and this manifests as a reduced pressure.

These pressure fields tend to dissipate at the speed of sound (which itself is just repeated pulses of pressure), so they extend in front of the wing. This creates a pressure gradient drawing the approaching air upwards over the wing. Meanwhile at the back, the air coming off the wing now has a pronounced downwards movement, and the same pressure gradient does no more than push some of the downflow further back in the wake of the wing).

The slowing of the air under the wing can be understood as a *forward component* of the flow, even though overall it is backwards. Thus, we now have four components all feeding into each other: forward under the wing, up at the front, backward over the top and down behind. They form a closed loop, the circulation component of the overall flow.

You might object to this principle, on the grounds that if you just created a circulatory component artificially, that would also magic up lift. In fact that is exactly what happens, although the magic comes at a price; you have to have a power source. Up to a point, it works just fine.

The circulation theory of lift describes how the amount of circulation corresponds directly to the amount of lift created. But it still cannot predict how strong the circulation will be in the first place; that still has to be derived by other means.

Among those who accept both Bernoulli and circulation, this is perhaps the most contentious area of all. The problem with all that has gone so far is that, as a full explanation of lift, it still falls short. Bernoulli explains *what* happens, Euler explains *how* and circulation summarises the consequences, but none explain *why* high speed creates a low pressure.

Many hold that we simply do not know how it all fits together, that something is still missing. Others disagree. If you program all these phenomena into a simulator, it will correctly predict the amount of lift. The programmer certainly told the computer how to fit it all together correctly! So really, there can be no mystery, no missing piece of theory, it is all there in the maths. It is just that we have not quite pictured to ourselves how the various equations all feed back on each other from one mechanism to another, which loopbacks in the computer code are the crucial ones. It's a bit like trying to solve a dozen complicated simultaneous equations; we can solve each step individually and, given a sound method, we can solve them all eventually. But it's too much to hold all the steps in your head while you are working on each individual one.

Notice how the circulation is driven by the pressure fields around the wing, and also interacts with the other effects discussed. On the other hand, the pressure fields are driven by the overall flow pattern established by all these effects. Broadly, we can say that flow velocities create pressure fields, which in turn create changes (accelerations) in flow velocity, which in turn modify the pressure fields. The whole thing pulls itself up by its mathematical bootstraps, once you first get some speed up and then tilt your wing upwards to start deflecting air down. But this feedback is non-linear; eventually it reaches an equilibrium where the flow stabilises. To borrow a metaphor from Spinal Tap, the volume levels off at around 4 – well actually, these days aerofoils have improved, it's probably nearer 11.

I would suggest the following summary. The basic mechanism is the Newtonian reaction understood by Cayley and his contemportaries. The pressure fields arising from this also create a circulatory flow component. Bernoulli's principle then comes into play, increasing the circulation and accelerating more air downwards more effectively, thus significantly enhancing lift.

The equilibrium state this reaches is characterised by a flow characteristic known as the Kutta condition, but that is both too complicated and not necessary to understand at this level, so here is a good place to stop.

Unfortunately, the fundamental reason why high speed lowers static presure, allowing the humble atomiser to work in the first place, the Venturi tube to avoid flow choking, or a wing to rotate for takeoff without stalling, does indeed appear to remain a mystery. There are after all many ways for a flow to change while conserving energy. Flow choking is just one such, and does in fact happen unless the conditions are right. Or the air temperature could fall as it speeds up, much as it does when it expands under reduced total pressure, or its volume could change anyway. So why do the laws of physics choose instead to lower the static pressure in a smooth airflow? We know they do, we know the atomiser works, but we still appear to have no idea why.

Let's take a closer look at a typical bulb atomiser, commonly used for example as a medial or cosmetics spray.

Operation is driven by mechanically squeezing the bulb, which creates an increase, say M, in the total bulb pressure T_{b}. Given the ambient air pressure A, it is clear that T_{b} = A + M. This creates a pressure gradient along the connected tube, causing air to flow along it. M reduces along the tube due to friction, so that at the exit, located at its junction with the reservoir tube, M = 0 and T_{j} = A.

We now find that the liquid in the reservoir rises up the reservoir tube. The pressure at the top of the fluid column C_{top} has fallen below A by an amount represented by the weight (strictly, weight per unit area) of the column, C_{w}, such that C_{w} + C_{top} = A.

Bernoulli explains this by saying that C_{top} equals the static pressure at the junction S_{j}, and this has fallen below T_{j} by an amount we define as the dynamic pressure at that point, D_{j} = C_{w}. Since T_{j} = S_{j} + D_{j}, we confirm that A = C_{w} + C_{top}.

We are still faced with the question, how can puffing under pressure create an even greater pressure difference? Why does an excess pressure in the bulb create a deficient pressure in the reservoir column, such that T_{b} > A > C_{top}? More specifically, why does only S_{j} affect C_{top} and not D_{j} as well? Why do we not see changes in temperature, volume and/or airspeed instead? These phenomena all occur in various related circumstances, why not here also?

I think the answer must lie, at least in part, in thermodynamics and the statistical mechanics which underlies it. The atomiser appears at first sight to reverse the laws of thermodynamics, where gases are supposed to flow to make pressures even out, and thus maximise entropy. Here we see the height of the column representing a decrease in entropy of the fluid; how has this enabled entropy to be increased elsewhere, above that of the airflow alone?

The innate compressibility of air still makes the situation complicated. Water experiences the same effects but we intuitively appretiate its incompressibility, so it might be best to seek an explanation there, and then transfer it to air. In water, propellers can experience a phenomenon known as cavitation, in which small bubbles of near-vacuum form in the regions of lowest pressure. It happens when the static pressure falls to zero (well, almost; it is in fact the vapour pressure of water, which varies but is typically less than 1 lb/ft^{2}, and thus insignificant compared to the total water pressure). Maybe that has something useful to tell us. I need to think about it.

Updated 15 July 2023