Airfoil

The dream of soaring in the sky like a bird has captivated the human mind for ages. Although many failed, some eventually succeeded in achieving that goal. These days we take air transportation for granted, but the physics of flight can still be puzzling.

In this article we’ll investigate what makes airplanes fly by looking at the forces generated by the flow of air around the aircraft’s wings. More specifically, we’ll focus on the cross section of those wings to reveal the shape of an airfoil – you can see it presented in yellow below:

We’ll find out how the shape and the orientation of the airfoil helps airplanes remain airborne. We’ll also learn about the behavior and properties of air and other flowing matter. In the demonstration below, you can see a fluid flowing around a gray cube. Using the slider to change just one property of this substance, we can end up with vastly different effects on the liveliness of that flow:

Over the course of this blog post we’ll build some intuitions for why these different effects happen to airfoils and other objects placed in flowing air. We’ll start this journey by looking at some of the methods we can use to visualize the motion of the air.

Visualizing Flow

If you’ve ever been outside in a grassy area on a windy fall day, you may have witnessed something similar to the little scene seen below. The slider lets you control the speed of time to observe in detail how the falling leaves and the bending blades of grass are visibly affected by the wind sweeping through this area:

We intuitively understand that it’s the flowing air that pushes the vegetation around, but note that we only observe the effects that the wind has on other objects – we can’t see the motion of the air itself. I could show you a similarly windy scene without the grass and leaves, and I could try to convince you that there is something going on there, but that completely empty demonstration wouldn’t be very gratifying.

Since the air’s transparency prevents us from tracking its movement directly, we have to come up with some other ways that can help us see its motion. Thankfully, the little outdoor scene already provides us with some ideas.

Notice that as the wind hits a blade of grass, that blade naturally bends in the direction of the blowing gust, and the faster that gust, the stronger the bending. A single blade indicates the direction and speed of the flow of air in that area.

In the next demonstration we’re looking at the same grassy field from above. When seen from this perspective, all the blades form short lines that are locally aligned with the wind. The more leaned over a blade of grass is, the longer the line it forms. We can mimic this behavior with a collection of small arrows placed all over the area, as seen on the right side:

Each arrow represents the direction and the speed of the flow of air at that location – the longer the arrow, the faster the flow. In these windy conditions the flow varies from place to place and it also changes over time, which we can clearly see in the motion of the arrows.

Note that we have some flexibility in how the speed of wind corresponds to the length of an arrow. I adjusted the lengths of the arrows to prevent them from visually overlapping, but I also made sure to maintain their relative lengths – if one arrow is twice as long as the other, then the flow at that location is also twice as fast.

For visual clarity I’m also not packing the arrows as densely as the blades of grass are placed, but it’s important to note that every point in the flow has its own velocity which contributes to the complete velocity field present in this area. If we wanted to, we could draw a velocity arrow at any of the seemingly empty spots on the right side.

The arrows are convenient, but the grassy scene also has another aid for visualizing flows. Many light objects like leaves, flower petals, dust, or smoke are very easily influenced by the motion of the surrounding air. They quickly change their velocity to match the flow of the wind. We can replicate the behavior of these light objects with little markers that are pushed around by that flow. You can see them on the right side:

These little markers also show us the motion of the air. Each marker represents an object so small and light that it instantly picks up the speed of the surrounding airflow. We’d have a hard time seeing these miniscule specks at their actual sizes, so I’m drawing the markers as visible dots.

In fact, the motion of each marker is equivalent to the motion of the parcel of air right around it. If you slow down time, you’ll be able to see how each marker just moves in the direction of the arrows underneath it. I also made each marker leave a little ghost trail behind it – this lets us track the path the air, as represented by the marker, took on the way to its current position.

Let’s pause for a second to emphasize what the grass-like arrows and leaf-like markers represent – they both show the velocity of the flow of air, but in slightly different ways. An arrow is attached to its fixed point in space, so it represents the current direction and speed of the flow at that location. The whole collection of arrows lets us easily see what the entire flow is doing at the moment.

On the other hand, the little markers are actively following the flow, letting us see how the air is actually moving through space, with the ghosty trails giving us some historical overview of where this parcel of air has come from.

The two methods we’ve seen so far are very versatile, but sometimes we don’t care about the local direction of the flow, only its speed – in the middle of this grassy field one might get cold from a fast blowing wind regardless of the direction from which that wind is coming. This brings us the third way of visualizing flow:

In this method we show the speed of the airflow using colors of varying brightness – the faster the wind, the brighter the color. You can see the whole spectrum of colors in the scale below the plot.

This method shows the speed of the flow at all locations giving us a more fine-grained insight into the motion of air at the cost of the directional information. To help with that I’ll sometimes overlay the regular arrows on top to let us know where the flow is going as well.

You may have noticed that all these methods present a flat, two dimensional view of the flow. It’s based on the assumption that the wind in our little scene doesn’t change with elevation, and that it also doesn’t blow towards or away from the ground.

In reality, the air velocity could vary in all three dimensions, and that air could also flow upwards or downwards. Thankfully, the air flows we’ll consider in this article will be two dimensional and the simple flat drawings will suffice.

Before we finish this section, let me bring up visualization of a simple airflow, but this time I’ll give you some control over its direction, which you can change using the second slider. The first one once more controls the speed of time:

Don’t be misled by the frozen arrows, the wind is actually blowing there. Remember that the arrows represent the local velocity of the flow of air, so while the velocity doesn’t change, the position of each packet of does. You can see those changes by tracking the markers moving around with the flow. This demonstration represents a steady flow, which means that its properties don’t change over time.

So far we’ve been exploring the notion of airflow’s velocity on a more intuitive level, with a general understanding that’s it’s “the air” moving around in some direction and at some speed. I illustrated that concept using simple arrows↑, markers •, and varying colors, but we’re now ready to investigate the details hiding behind those straightforward graphical representations.

To do that, we have to look at individual particles of air. Although I briefly discussed the particle nature of air before, this time around we’re going to take a closer look at the motion of these molecules, and what it means for airflow as a whole.

Velocity

Let’s take a look at the air particles in a small, marked out volume of space seen in the demonstration below – you can drag the cube around to change the viewing angle. The slider controls the speed of time:

You’re witnessing the motion of over twelve thousand air particles. It may seem like a lot, but this cube is extremely tiny, its sides are only 80 nanometers long. To put this in perspective using more familiar sizes, if that cube’s side measured just 1 inch1 centimeter, it would contain around 410 quintillion, or 4.1×102025 quintillion, or 2.5×1019 particles.

The particles are zipping around in random directions, constantly entering and leaving this region. However, despite all this motion what you’re seeing here is a simulation of still air.

To understand how all this movement ends up creating still conditions, we first have to look at the velocity of each particle – I’ll visualize it with a small arrow in the direction of motion. To make things a easier to see, I’ll also highlight a few of the particles while fading out the rest of them:

The length of an arrow is proportional to the speed of a particle, so when you freeze the time you should be able to see how some particles are slower and some are faster. This speed variation follows a certain distribution that’s related to temperature – the warmer the air, the faster the motion of its particles.

At room temperature the average speed of a particle in air is an astonishing 1030 mph1650 km/h, which is many times higher than even the most severe hurricanes. Given the size of the cube, this means that even at the fastest speed of simulation everything happens 11 billions time slower than in real life.

If you paid close attention, you may have also noticed that sometimes the particles randomly change direction and speed of their motion – this happens when molecules collide. Each particle experiences roughly ten billion collisions per second. We’ll get back to these interactions later on, but for now let’s try to figure out how all this turmoil creates still air.

Having just seen the small velocity arrows of individual particles, let’s calculate the average velocity of a group of three particles, using the process shown below. We first take the velocity arrows from each particle and place them head to toe, one after another. Then we connect the start of the first arrow with the end of the last arrow to create the sum of all velocities. Finally, we divide, or scale down, the length of this sum by the number of particles to get the average velocity:

In the next demonstration we’re repeating this whole procedure by tallying up all the particles inside the red box. You can change the size of that region with the second slider. The large arrow in the middle shows the average velocity of particles in the box. To make that central arrow visible, I’m making it much larger than the tiny arrows tied to particles:

The counter in the bottom part of the demonstration tracks the current number of particles in the red cube. That value fluctuates as the molecules enter and leave that region. While aggregating over a small number of particles creates a very noisy readout, it doesn’t take that many particles to get a much steadier measure.

Recall that the scale of the large central arrow is much larger than the scale of individual tiny arrows attached to each particle. Despite that increase in size, the arrow practically disappears when we average out a larger number of particles and we can clearly see that the average velocity of particles is more or less zero even in this extremely small volume.

In still conditions, all these motions in different directions average out to nothing. As some particles enter the area from a random direction, the others also leave it in a random way. The bulk of air doesn’t really go anywhere and the particles just meander in a random fashion.

An imperfect, but convenient analogy is to imagine a swarm of bees flying in the air. While all the individual insects are actively roaming around at different speeds, the group as a whole may steadily stay in one place.

All these experiments form the key to understanding what happens when wind sweeps through an area. In the demonstration below, we’re once again watching a small volume of space, but this time you can control the speed of the blowing wind:

Notice the mphkm/h speedometer in the bottom of the demonstration. This is not a mistake – even with hurricane-level wind speeds it’s very hard to see any difference in the motion of the particles. Perhaps you’ve managed to see the tiniest shifts in the small particle arrows as you drag the second slider around with time paused, but it’s difficult to even perceive from which direction the wind is blowing.

However, when we use the procedure of averaging the velocity of all the particles, we can reveal the motion of their group in the box of a given size, at a specific speed of the flow:

Because the motion of each individual particle is so disordered, we have to look at many of them at once to discern any universal characteristics. And when we do just that, from all the chaos emerges order.

It’s important to note that with this approach we’re tracking the velocity of the flow within the same region of space outlined by the red box – the molecules keep entering and leaving this area as the flow moves and the arrow in the middle shows the average velocity of the air’s particles in that area.

This is exactly what the grass-like arrows we’ve played with in the previous section represent – each one shows the average velocity of air particles in that local region of space. The big arrow we just saw in the middle of the swarm in the averaging red box is equivalent to each of the arrows seen below:

Naturally, the averaging box needs to be large enough to avoid the jitteriness related to aggregation of too few particles, but at any scale that we could care about the noisy readout completely disappears.

The average motion of particles is very different than the motion of each individual molecule. Even in very fast flows, many of the molecules move in the opposite direction than what the arrow indicates, but if we tally up all the particle motion, the air as a whole does make forward progress in the direction of velocity.

Up to this point, we’ve mostly looked at the flow of air by looking at wind and the way it moves through space, but what we consider a motion of air is relative. Let’s see how, by merely changing the point of view, we can create a motion of air in otherwise windless conditions.

Relative Velocity

Let’s zoom away from the world of microscopic particles to look at the motion of larger bodies. In the demonstration below, you can see two different views of the same car driving in the left direction. In the top part, the camera stays firmly on the ground, but in the bottom part, the camera tracks the motion of the vehicle. If needed, you can restart the scene with the button in the bottom left corner or tweak the speed of time with the slider:

These two views show the exact same scene – we’re just changing what the camera is focusing on. As seen in the top part, from the perspective of the static camera, it’s only the car that has some velocity in the left direction.

On the other hand, from the perspective of the camera focused on the vehicle, the car doesn’t move, but everything else does. The poles and road markings all move to the right with a speed equal to that of the car. This shouldn’t come as a surprise from daily experience in any form of transportation – when you’re sitting in a moving vehicle, static things in the surrounding environment seem to move towards and past you.

The very same rules apply to any region of air – I’ve outlined some of them with dashed boxes up in the sky. For the observer on the ground that air is still, but from the car’s perspective, that air is moving.

With that in mind, let’s see the same scene, but this time I’ll add the familiar small arrows showing the air’s velocity as “seen” by the camera:

From the point of view of the car, as seen in the bottom view, the air is moving to the right, as if there was some wind blowing right at the vehicle. You’ve probably felt this many times by sticking your hand out the window – it feels no different than if you were standing still on the ground with the wind hitting your fingers.

In fact, there is absolutely no difference between “regular” wind and wind experienced by the car or your hand sticking out the window – both are simply a motion of air relative to some object. This means that we can use our arrows to represent any motion of air, as long as we note what that motion is relative to.

You may have also noticed that the moving car affects the motion of air in its vicinity. Let me bring up the previous demonstration one more time:

In the top view, we can see how the front of the vehicle pushes the air forward, and how the air “bends” and speeds up around the shape of the car to roughly follow its shape, only to end up circling right behind the machine.

The same effects are seen in the bottom view – they’re just experienced differently. For example, the air right in front of the car slows down, while the air on top moves even faster than the rest of the undisturbed, distant air.

We’ll soon explore why the air behaves this way when flowing around an object, but for now let’s raise above the ground to see the motion of an airplane flying in the sky. We’ll use the familiar setup of a camera kept steady relative the ground, as seen in the top part, and a camera that follows the airplane, seen in the bottom part:

Before we continue, notice that it’s getting a little hard to pay close attention to what happens to the moving objects in the ground-fixed camera view – the bodies quickly leave the field of view of the demonstrations. For the rest of this article I’ll stick to the camera style seen in the bottom part of the demonstration – this will let us directly track the interaction between the object and the air that flows around that object.

From the point of view of the airplane, it also experiences a flow of incoming air as seen by the air “boxes” approaching the plane, which is very similar to the car example. What’s completely different from the car example is the fact that the airplane somehow stays suspended in the air, despite gravity pulling it down towards the ground. This means that there must be some other force acting on it to prevent the plane from falling from the sky.

Let’s compare these two vehicles by looking at the basic forces affecting their motion, starting with the diagram of forces acting on the car:

The down-pulling gravity force is counteracted by the reaction forces from the ground – they act through the car’s tires to prevent the car from sinking. The air drag and other forms of resistance push the car back, but the car’s tires powered by the engine keep propelling the car forward.

In my previous article I presented a more elaborate description of the interplay between forces and objects, but to briefly recap here, if forces acting on an object are balanced, then that object will maintain its current velocity.

All forces on the car are balanced and the vehicle moves forward with constant speed, and it doesn’t move at all in the up or down direction – the object’s velocity is indeed constant.

Let’s draw a similar diagram of forces for the flying plane:

We still have the air drag that pushes the vehicle back, and the plane’s propeller powered by the engine keeps pushing it forward. As a result the plane moves forward with constant speed.

We also have the down-pulling gravity. This time, however, that gravity is not countered by the reaction forces from the ground, but instead it’s balanced by lift, a force that pushes the plane up. When gravity and lift are equalized, the plane doesn’t move up or down either.

Airplanes create most of their lift with wings, which are carefully designed to generate that force. While length, area, and the overall geometry of the wings are very important, in this article we’ll focus on the shape of the cross-section of a wing which I highlighted below in yellow:

This is an airfoil, the protagonist of this article. This airfoil has a smooth, rounded front and a sharp trailing edge. Let’s take a closer look at the flow of air around this airfoil using the grass-like arrows that show the velocity of air at that location:

These arrows paint an interesting picture, but in the demonstration below I’ve also added the little leaf-like markers that track the motion of air parcels in the flow. I steadily release a whole line of them from the left side, but you can also clicktap anywhere in the flow to drop a marker at that location. You can do this in any demonstration that has a little hand symbol in the bottom right corner:

The markers show that the flow splits ahead of the airfoil, then it gently changes direction to glide above and below the shape. Moreover, the markers right in front of the airfoil gradually slow down and lag behind their neighbors. The air somehow senses the presence of the body.

It may be hard to see, but the top and bottom sections of this airfoil aren’t symmetric. This asymmetric design is very important, but right now it will needlessly complicate our discussion on how the flow around this shape arises.

To simplify things a little, let’s use a less complicated shape of a symmetric airfoil – you can see it in the demonstration below. I overlay the previous asymmetric shape with a dashed outline to show the difference between the two:

The motion of air around this airfoil is very similar – the flow changes its direction and speed when it passes around an object. Until now we’ve simply been observing that the flow changes to adapt to the shape of the body, but it’s finally time to understand why it happens. To explain that behavior we need to go back to the world of air particles to discuss the concept of pressure.

Pressure

As we’ve discussed, even in the seemingly steady conditions the particles of air are zipping around at high speeds colliding with each other at an incredible rate. The surface of any object placed in the air will also experience these bounces.

In the demonstration below, you can see air particles bombarding a small box. Every time a collision happens I briefly mark it with a dark spot on the surface of that cube:

To understand the implications of these collisions, let’s first take a look at objects with more ordinary sizes. In the demonstration below, tennis balls are hitting a large cardboard box from the left and right side. By dragging the slider you can change the intensity of both streams of balls:

When a tennis ball hits the box, the collision imparts some force on it, causing the box to move. However, in this simulation the collisions from all the balls on each side balance each other out, so the box doesn’t make any consistent progress in either direction.

In real air, the situation is similar, but at vastly different scales. The mass of each particle constituting air is absolutely miniscule, so the impact of an individual collision on any object of meaningful size is completely imperceptible.

Moreover, each air particle hitting an object has a different speed, and it strikes the surface of that object at a different angle – some hit the object straight on, but some barely graze it. Due to the enormous number of these collisions happening at every instant of time, all these variations average out, and even a small section of surface of any body experiences uniform bombardment.

In aggregate, we say that the air exerts pressure on any object present in that air. The magnitude of this pressure depends on the intensity of these collisions across an area.

Let’s see how this pressure manifests on our tiny cube. In the demonstration below, you can use the second slider to control the number of air molecules present in this volume:

The black arrows you see on the sides of the cube symbolize the magnitude of pressure on these walls. As we uniformly increase the number of particles in this volume, the intensity of collisions, and thus the pressure, also increases. Because the collisions happen at more or less the same rate on every side of the box, the net balance of forces is also maintained and the cube doesn’t move, regardless of how big or small the overall pressure is.

This is exactly what happens in the Earth’s atmosphere – everything is constantly squeezed by relatively high pressure caused by the barrage of countless air particles. That pressure is typically balanced either by an object’s material, which resists compression like a spring, or by the air itself that fills the insides of the object. When that inner air is removed, the seemingly innocuous atmospheric pressure reveals its might.

The underlying particle nature also shows us that pressure is never negative. Without any particle collisions, we reach the lowest possible pressure of zero. Beyond that, any impacts on the surface of an object create some amount of positive pressure.

In the demonstrations we’ve seen so far, the balanced number of collisions on each wall was very important for keeping the objects steady. Unsurprisingly, more interesting things happen when this harmony isn’t maintained. Let’s first investigate this scenario using the tennis balls. In the demonstration below, the slider controls if it’s the left side or the right side that’s shooting more balls:

As you can see, if one of the sides has a higher number of collisions, the forces acting on the box are no longer balanced and the box starts to move.

The very same situation happens in air, which you can witness in the simulation below. Notice that the volume in which the tiny cube exists has more particles on one side than the other. Observe what happens to cube once you let the time run using the slider:

The higher number of particle collisions on one side of the cube creates higher pressure forces on that wall. The uneven forces end up pushing the block to the side. In this demonstration, the pressure re-balances after a while and the cube stops moving.

Intuitively, the air exerts an imbalanced net force on the cube only when different parts of that object experience different pressure – it’s the spatial variation in pressure that creates an acting net force. When the difference in pressure between any two points increases, the net force acting on the object also grows.

It’s easy to see that a larger number of collisions on the left side of an object would start to exert a net force pushing that object to the right, but, perhaps surprisingly, the same rules apply to any chunk of air itself.

In the demonstration below, I once again made one half of the test volume contain more particles than the other half. As you unpause the demonstration, observe the average velocity of molecules in the marked out section of air:

The particles on the more occupied side can easily travel to the less crowded side, because there are fewer particles there to collide with and bounce back from. Additionally, each particle in the less populated section is more likely to hit a particle in the more populated section, which will typically cause that particle from the desolate side to bounce back where it came from.

The particles end up, on average, traveling from the area of high pressure to the area of lower pressure. Even though we don’t have any clean borders between different sections, we can still see the bulk of particles getting accelerated towards the less dense section.

Once again, the initial pressure differences in the test volume dissipate after a while. On their own, these freely suspended pressure variations quickly disappear, but we will soon see how, with the aid of airflow, these areas of different pressure can be sustained indefinitely.

In the examples we’ve been playing with, the notion of increased pressure came from an increased number of collisions, which in turn came from an increased number of particles in the area. This shows that, all other things being equal, pressure is tied to the local density of the air, which was very easy to perceive in an increased concentration of molecules.

However, the pressure can also grow due to increased average speed of the particles, which in turn comes with increased temperature. As particles get faster, each collision gets more impactful and it pushes on an object or other particles a bit harder, causing the overall pressure to also increase. In the demonstration below, we can simulate this with tennis balls hitting the cardboard box at the same rate, but with different speeds, which you can control with the slider:

As we make the balls on one side of the box faster, their impacts also become stronger and the package starts moving to the right, even though the number of collisions per second is equal on both sides.

The important point from these discussions is that air pressure exerts force on everything inside it, be it a solid object or any parcel of air. It’s a little unintuitive that the air itself both exerts the pressure and it also “feels” the pressure, but it’s all just a consequence of very rapid motions of particles and the collisions between them happening at an enormous rate.

Recall that even in small volumes of air there are billions of billions of particles, and each particle experiences roughly ten billion collisions per second. What we’ve simulated at a micro scale and in slow motion as countable, individual interactions, very quickly smooths out into a uniform and uninterrupted notion of force-exerting pressure.

This fact lets us abandon the molecules and their collisions yet again. It’s not a big loss, since counting the number and intensity of collisions was never convenient in the first place, but we can now investigate some other ways of visualizing pressure in a region of air.

Visualizing Pressure

As we’ve seen in the particle simulations, pressure can vary from place to place. One of the most convenient ways to express this variation is to use colors of different intensities. Let’s see how that simple approach could work here. In the demonstration below, the dashed circles represent regions of high and low pressure – you can drag them around to change their position:

This map of pressure is colored with varying shades of red as indicated by the scale below – the redder the color, the higher the pressure. The small triangle in the middle of the scale indicates the location of the base, static pressure present in the atmosphere.

In this simulation we have complete control over where the different locations of lower and higher pressure are. To make things more interesting, each draggable pressure circle has a different strength and range. You can infer this variation from color changes around these points.

Let’s put an airfoil in this area to see how it’s affected by the pressure of the surrounding air. The arrows seen below symbolize the force that pressure exerts on the surface of the airfoil at that location. They’re the exact same arrows that we’ve seen acting on the walls of the tiny yellow cube, here we just see them at a larger scale:

As you move around the locations of lower and higher pressure, the forces acting on the surface of the airfoil also change, matching what we’ve seen with little cubes bombarded by air particles. The static pressure always exerts some base load, but in the areas of higher pressure the surface forces are higher, and in the areas of lower pressure the surface forces are lower than these base forces.

Note that you can also move the pressure circles into the airfoil, but it only serves as a convenience to let you customize the shape of the air pressure field around that body – we don’t particularly care about the pressure inside the solid itself.

When we tally up all the pressure forces acting on each piece of the airfoil’s surface, we end up with the net force acting on that object. In the demonstration below, I’m showing it with the big arrow at the center of the airfoil:

By changing the distribution of pressure around the airfoil, we can affect the total force that this object feels.

The reddish plots we’ve been looking at are correct, but a little inconvenient. Recall that final net force on the object depends only on the differences of pressure – when we uniformly increased the number of collisions on the walls of the tiny cube, it steadily remained in place.

This means that the static background pressure doesn’t matter for the cumulative forces acting on an object. It’s only the differences relative to that static pressure that affect the overall balance. This lets us overhaul our visual representation of pressure – we can use no color where the pressure has the static value, use blue color when the pressure is lower than the static pressure, and use red color when the pressure is higher than the static pressure:

This is the exact same distribution of pressure that we’ve just seen. All the pressure demos in this section are connected, and here we simply changed the reference point against which we present the pressure variation.

If we then throw in the airfoil back into the mix we can now also adjust the arrows representing the forces that the pressure exerts on the surface of that object:

The areas of higher pressure still seem to push on the surface of the airfoil, but the areas of lower pressure now seem to pull it. However, I need to emphasize once more that pressure always pushes on the object, and we can only talk about a pulling force when we discard that uniform, pushing contribution coming from the static pressure. In those “pulling” areas the pressure is still pushing, it just pushes less intensely.

I will also use the convenient terms of positive and negative pressure, but remember that this refers to their difference from the static pressure. The phrase “pressure lower than static pressure” is a mouthful, so the expression “negative pressure” is very handy, even when it hides the fact that pressure is always positive.

While the color variations used here show the true nature of the smoothly varying pressure changes, they make it a little hard to see how quickly those changes happen. To fix this, I’ll also draw the contour lines that join the locations of the same pressure – they’re very similar to lines of the same altitude you may have seen on maps:

Every point on one of those contour lines has the same value of pressure, and each subsequent line is drawn at the same increment of pressure – you can see this in the scale placed below the plot. This means that the closer the lines are together, the more quickly the pressure changes in that area.

The mathematical concept that describes the direction and rapidness of these changes is known as a gradient. Informally, gradient describes how some property changes from one point to another, and, thankfully, this notion tracks closely with how this word is used in graphic design to describe smooth color changes. Wherever you see a color gradient , this also implies that there is a pressure gradient – the pressure changes from place to place.

This spatial variation is particularly important for the motion of air. Recall that the air pressure differences don’t just exert forces on solid objects, but also on the air itself – any small parcel of air is subject to the same whims of pressure forces.

Those spatial variations in pressure end up pushing the air around, changing its velocity. Let’s see this in action using the little leaf-like markers that are moved around by pressure differences. In the demonstration below, I’m steadily releasing the markers from the left side – notice how their trajectory changes when you modify the pressure field:

You may still find it a little difficult to grasp how pressure differences affect the motion of a parcel of air. Luckily, we can draw parallels between the contour lines of pressure seen on these pressure maps and the contour lines of elevation seen on traditional maps. This lets us build a little pressure-landscape analogy.

In the demonstration below, the very same distribution of pressure is expressed as a mountainy landscape. Positive pressure lifts the ground above the base level and negative pressure depresses it below the base level. A parcel of air moves like a marble that loses speed when climbing uphill and accelerates when rolling downhill. You can drag the demo around to change the viewing angle:

Notice that when the pressure changes more rapidly and the contour lines are closer, the steepness of the corresponding hill or valley also increases, and so do the forces acting on a parcel of air. If the pressure is increasing by a large amount, it may even make the marker go back. This landscape analogy also shows that the static pressure doesn’t matter for the motion of air parcels, as any changes in static pressure would just lift all the areas by the same amount without changing their steepness.

When watching these air parcels move around, you may have noticed that things were a little bit off. For example, it’s possible for air parcels coming from different directions to arrive at the same location, and then continue to travel in different directions. You can see an example of that on the left side of the demonstration below, with the slider letting you scrub back and forth in time:

Recall that the markers always follow the local velocity of air, so the motion seen in the left part implies that the air at the location of the meetup of the two markers has two different velocities at the same time, which is not realistic.

It’s worth pointing out that the situation seen on right side, where one marker merely intersects the historical path of the other, can be realistic, as long as we’re dealing with an unsteady flow, where the velocity of the air at the crossing location has changed since the first marker was there. For steady conditions in which no changes occur over time, the scenario seen on the right is also not physically correct.

We’ll look at some unsteady flows later in the article, but for now we’re interested in steady conditions so the crossing paths of our markers indicate implausible velocities. Even more dubious result happen when we simulate the motion of these markers with an airfoil present in the flow:

For most distributions of pressure, the air markers will flow right through the body. This is clearly wrong! The demonstrations we’ve seen so far correctly represent what would happen to individual air parcels and bodies placed in these pressure fields, but those pressure fields themselves were completely made up and didn’t correspond to any physical reality. Our mistake was that we completely ignored any interactions between the pressure of the air and the motion of that air.

The flow of air, the pressure of air, and the shape of the objects placed in that air are all tied together – for a given incoming flow speed and the shape of the object, we can’t just arbitrarily arrange the pressure field like we did in our artificial demonstrations. Instead, that pressure field will arise on its own.

Let’s see a real distribution of pressure around this airfoil and witness how it affects the motion of air parcels around it:

The behavior of air parcels now matches our intuitive expectations – the markers don’t go through the body, and in these steady conditions they also don’t cross paths.

We’re now one step closer to understanding how the flow of air takes its shape to move around an airfoil – it’s the pressure differences that cause the flow to change its direction and speed.

The pressure field we’ve just seen clearly works – regions of lower and higher pressure guide the air around the airfoil. However, it’s still unclear how these areas emerged in the first place. Let’s try to follow nature’s path to see how this pressure distribution is created and sustained in a flow.

Airfoil Flow

Before we start building the correct pressure field from scratch, let’s first establish two guiding principles that the flow around any object has to follow.

Firstly, the air can’t penetrate solid walls. A valid pressure field should either completely stop the flow at the surface of the object, or redirect that flow to make it travel in the direction perpendicular to the walls. This means that the markers that we track can never get inside the object.

Secondly, we also have the restrictions on the relative motion of the markers. For now we’ll only be interested in steady conditions, which means that the markers can’t cross their paths – we expect the ghostly historical trails to never intersect.

Let’s first focus on the pressure field in front of the airfoil. In the demonstration below, I created an artificial pressure field in that frontal region, you can control it using the slider:

It should quickly become clear that to prevent the approaching air from getting into the object, the pressure in the frontal region has to be positive, so that it pushes the incoming air away.

If that positive pressure in front is too low the air can still erroneously flow through the object. If that pressure is too high, the air parcels arriving at the airfoil will turn back and incorrectly cross paths with the incoming air. When the pressure is just right, the air parcels don’t go through the wall, and, at least in front of the object, they also don’t cross their paths.

The faster the incoming flow, the higher the pushing force required to slow down and redirect the incoming air. In the demonstration below, you can also control the speed of that incoming air using the second slider:

While for slow flows, only a small amount of positive pressure is enough to stop the incoming air, for fast flows, the pressure in front of the airfoil has to become much higher.

The pressure needed to stop air at a given velocity is known as stagnation pressure and it’s proportional to the square of that velocity – twice as high speed requires four times larger pressure. Naturally, when there is no flow, no pressure is required as the air no longer tries to flow through the object.

In the previous two demonstrations, we manually adjusted the pressure to get the correct result, but in nature this process happens on its own – it’s the flow itself that creates this region of increased pressure in front of the object.

As the incoming parcels of air arrive at the surface of the airfoil, they can’t continue going forward, but air parcels from further up ahead continuously want to keep flowing into this region. This compresses the air close to the object, which causes the pressure in front to increase, which then helps to slow down the incoming flow.

This mechanism is self-balancing – if the pressure is too low to push away the incoming air parcels, the air parcels will compact the existing air more, causing an increase in pressure. If the pressure is too high, it will easily push the incoming air away, which relieves the frontal area, causing the pressure to decrease. Any fluctuations quickly settle to an equilibrium that balances the pressure in the entire frontal region.

Let’s look at the distribution of the positive frontal pressure once more:

Notice that the positive pressure isn’t limited to just the close vicinity of the airfoil, but it spreads out much further ahead to gradually reach the value of the static pressure, far away from the airfoil itself.

All in all, we have a large area of increasing pressure that starts far away from the body and ends at its surface. Those pressure differences create a pressure “hill” that not only gradually slows the incoming air down, but it also redirects that air to flow around the object.

It seems that with our frontal pressure field we’ve easily completed our goal of preventing the air from flowing through the walls of the body. However, our second guideline of non-crossing marker paths is still not fulfilled – this condition is broken above and below the airfoil.

Let’s first try to rectify this manually. In the demonstration below, you can control the pressure in these two regions using the slider:

While positive values of pressure in those zones make the problem worse, negative values get us much closer to the expected behavior – in the top and bottom areas the markers no longer veer off into different directions. However, that pressure can’t bo too low, otherwise it will pull the markers back into the body.

In real flow, these regions of lower pressure arise on their own, but the explanation for this phenomenon is a little less straightforward than what I’ve described for the area of positive pressure in the frontal region. We can get some, albeit a bit hand-wavy, understanding by observing what happens to the air markers when those negative regions are missing.

In that scenario, the incoming air parcels no longer reach those areas above and below the airfoil, causing some local depletion of air that has since left those zones. This decreases the pressure in those regions, and that lower pressure attracts the surrounding air to flow into those less occupied spaces.

If that lower pressure is too negative, more air will come in and the pressure will rise. If the pressure is not negative enough, those region will get depleted again. Once again, it’s the flow itself that creates the balancing system – without the flow no pressure differences would arise.

As we’ll see later on, in more extreme scenarios that negative pressure can alter the flow more dramatically, and the regions of “missing” air get filled through other means, but for now let’s close things up by tweaking the pressure in the rear part of the airfoil:

Some amount of positive pressure in the rear prevents the air parcels from smashing into each other after leaving the airfoil. Intuitively, this pressure arises naturally from the flow, because as the air slides off from the ends of the top and bottom sides, it all arrives into the same region, creating some compression.

If that compressive pressure in the rear is too low, more air will manage to get in, which will further increase the pressure. If that pressure is too high, it will push the incoming air away, which depletes the area and the pressure decreases. The system balances itself yet again.

The quite informal description of these balances that I’ve presented can be formalized mathematically using the Navier–Stokes equations. These equations describe the motion of liquids and gasses, collectively known as fluids, subject to various forces like gravity, or, most importantly for us, pressure.

Navier–Stokes equations are notoriously difficult to solve analytically, but a lot of insight about the behavior of fluids can be gained with computer simulations with various degrees of complexity.

In this article, I’m also employing simulations to investigate the flow of air around objects. However, the computer models used here are quite simplified and they don’t reflect the full richness of physics involved in the motion of air. These slow-motion demonstrations are intended to present the broad strokes of the delicate interaction between the air and the airfoil, but I would advise against relying on them when building an airworthy airplane.

With all of these caveats in place, let’s get back to the pressure distribution around a symmetric airfoil. We’re done recreating the nature-made pressure field, but there is one small aspect that we haven’t yet accounted for.

For our experiments, I kept the pressure steady in time so that we could focus on its general outlines. In practice, a pressure field imposed by a fast flow around any object will experience some degree of instability, which you can see in the demonstration below. You can once more drop the markers at any location to track the flow in the area:

As the pressure builds up on one side, it redirects the flow, which changes the pressure again. The pressure ends up oscillating back and forth like a swing. The pressure distribution and the flow direction are once again at the mercy of their mutual balance, one affecting the other. We’ll soon see some other examples of these unstable behaviors.

As we’ve just seen, the variation in pressure doesn’t just happen in the close vicinity of the airfoil, but it stretches quite far away from the body itself. This means that the velocity of the flow is also affected quite far away from the shape.

However, when it comes to the forces exerted on the airfoil, it’s only the pressure right at the surface of the airfoil that matters. Let’s bring back the two tools we’ve used before: surface arrows that show how the air pushes or “pulls” on the airfoil, and the net force arrow that tallies up the net results of these forces:

As the pressure field fluctuates, the resulting net force also moves around. Let’s decompose this force into two different components, one perpendicular to the flow, and one parallel to it:

The force acting in the direction perpendicular to the flow is known as lift, and the one acting in the direction of the flow is known as pressure drag, or form drag. As the name implies, this component of drag is created by the distribution of pressure around the shape.

For this airfoil, the pressure drag is very tiny. While airfoils are specifically designed to minimize the overall drag, most of that force hindering their motion comes from another source – we’ll discuss it soon enough.

Notice that as this flow fluctuates, the lift force jumps around, but averaged over time the upward and downward swings of that force end up balancing each other. This airfoil in this configuration doesn’t generate any continuous lift.

This shouldn’t come as a surprise since this situation is completely symmetric, so the pressure forces on the upper and lower sides of the airfoil are, on average, completely balanced. However, there is an easy way to disturb that symmetry. In the demonstration below, we’re once again meeting the plain, symmetric airfoil, but this time we can gently tilt it using the slider:

The slider controls the so-called angle of attack, which is spanned between some reference line on the body, like the one joining the front and back, and the direction of the incoming flow. I’m showing this angle right in the middle of the airfoil.

As we change the angle of attack, the shape that the airflow “sees” is no longer symmetrical relative to the incoming direction of that flow. The velocity and pressure fields adapt in their mutual push and pull to form a new, asymmetric distribution. Notice that the stagnation point of high pressure has moved around, and the little markers that indicate the motion of air now travel on very different paths below and above and below the airfoil.

If we then put the pressure arrows back in, we can tally them all up to get the resulting lift and pressure drag. When compared to the previous simulation, I’m scaling down all the arrows to make them fit in the bounds of the demonstration:

When this symmetric airfoil is tilted up, the asymmetric pressure distribution generates a lift force that pushes the object up. Conversely, for a downward tilted airfoil, the pressure forces push the airfoil down.

Naturally, we’re typically interested in upward-pointing forces, and when the lift generated by the wings is equal to the weight of the plane, the plane will stay in the air without raising or falling to the ground – we’re finally flying.

Let’s plot the dependence between the lifting force and the angle of attack of an airfoil – you can see it in the right side of the demonstration below. Note that this plot presents time-averaged and settled values, so you may have to wait a little for the flow to normalize and the lift to start oscillating around the expected value:

Clearly, as the angle of attack increases, so does the generated lift. The same thing happens on the other end of the spectrum, where a more negative angle of attack creates more negative lift. Note that for this symmetric airfoil the positive and negative sides of the diagram are just mirror images of each other, so let’s focus only on positive angles of attack.

One could naively hope that we could keep increasing the angle of attack to generate more and more lift. Let’s see what happens in practice:

Initially, the lift force indeed keeps increasing with the angle of attack, but at some point it plateaus. Once that critical angle of attack is surpassed, the lift force starts to fall after the flow fully develops.

What we’re witnessing here is known as a stall. The onset of a stall imposes limits on how much lift the wings of an airplane can generate from merely increasing the angle of attack.

Notice that when the stall happens, the pressure distribution on the upper part of the airfoil becomes very erratic – it’s not only the surface pressure arrows that are changing rapidly, but the whole pressure field in that area is very disturbed.

Let’s bring in the velocity arrows and markers to get a better feel on what’s going on in that region:

At high angles of attack, the flow above the upper part of the airfoil becomes very complicated. If you clicktap in that region to drop a few markers, you’ll notice that the air is trapped in various swirling eddies that are eventually shed to fly away with rest of the flow.

We’re witnessing flow separation, where the main part of the flow detaches from the surface and doesn’t follow its shape anymore. The interactions in the complicated flow right above the airfoil affect the pressure field, which then decreases lift.

There is a lot going on there, but to understand how these effects arise we have to talk about a property that affects the flow of every fluid: viscosity.

Viscosity

You might have heard the term viscosity used to describe “thickness” of different liquids, with a classic example that contrasts the slowness of the flow of honey to the rapidness of the flow of water.

Viscosity is also a property of gasses like air, but before I describe this concept more formally, we’ll first build an intuitive understanding of what viscosity is and what it does to the flow of different fluids.

In the demonstration below, the fluid flows in from the left side, but note that the flow in the top half is faster than the flow in the bottom half, which is reflected by the different lengths of the arrows. Dragging the slider to the left decreases the viscosity of the fluid, and dragging the slider to the right increases viscosity:

While we can see some changes to the arrows as we move the slider around, you probably agree that, for this flow, the arrow-based visualization isn’t very rewarding. Let’s add the color-based visualization of speed distribution in this flow:

We can now see how viscosity blends the speed variation between different sections of the fluid. For highly viscous fluids, this mixing behavior spreads very easily and the initially distinct velocities of the two layers average out quite rapidly.

At lower viscosity these two layers with different speeds remain quite separated. If you make the viscosity low enough, you may even notice that, after a while, the flow develops some interesting wave-like phenomena – we’ll get back to these soon.

All this mixing behavior may remind you of a diffusion process, where some quantity, like temperature or concentration, evens out over time. Let’s see some basic diffusion in action. In the simulation below, I filled half of the bottle with with red-dyed water, while the other half is filled with blue-dyed water. The slider lets you control the speed of time:

As time passes, the sharp difference between the two layer blends more and more to eventually completely disappear. Clearly, there is some similarity between the diffusion of differently colored dyes and the averaging of velocity that we’ve seen in the earlier example.

In our flow demonstrations, viscosity seemed to have controlled the diffusion of velocity. To define it more precisely, viscosity controls the diffusion of momentum, which is a product of velocity and mass. The simplified fluids we’re looking at have more or less constant density, so each equally-sized parcel of those fluids has the same mass. Therefore, if it makes things easier for you, wherever you see the word momentum you can think of velocity, but in more complex scenarios these differences can matter.

Let me bring in the previous flow simulation one more time:

You’ve probably noticed that, as the flow moves to the right, the size of this blended region increases. When the regions of fluid with different momentums meet for the first time, they barely have any time to average out, and the blending is minimal. As time passes, these regions of fluid get to average out more, similarly to how two different layers of dyed water mix more over time.

However, as time is passing, these parcels also keep moving, and that stronger blending happens further to the right. The downstream regions had more time to mix and average out, so the visible thickness of the blended region on the right side is also larger.

With higher viscosity, the size of blended region grows much more quickly, which lets us be more precise about our working definition – viscosity controls the rate of the diffusion of momentum.

So far we’ve only observed flows with nicely separated horizontal layers, but viscosity averages momentum between any two regions of fluids. In the demonstration below, you can witness how viscosity affects a swirly motion of fluid in a vortex:

Notice that with high viscosity any differences in velocity are very quickly diluted out into nothing, but with low viscosity the revolving motion can survive for quite a while.

Viscosity has a damping or smoothing effect that makes it much harder to sustain any large variation in a velocity field. Let’s see how this affects the motion of objects in fluids of various viscosity. In the demonstration below, we’re tracking a velocity field close to a very thin plate put directly in the stream of an incoming fluid of adjustable viscosity:

With high viscosity, there is a large region of slow down around the plate that regains its speed fairly quickly behind the object. At lower viscosity that surrounding region is much smaller, but it extends much further behind the plate. For very low viscosity we’re once again seeing some more unusual behavior that we’ll get back to in a minute.

From the dark colors we can easily see that right by the surface of the plate the fluid doesn’t move at all – it sticks to that surface. This velocity difference between the halted flow at the wall and the moving outer flow gets smoothed out over time by viscosity, similar to how it blended in the flow between two different layers of fluid.

As before, with higher viscosity, the velocity averaging process becomes more rapid, and the blended region becomes more widespread. This averaging effect doesn’t just alter the velocity of fluid, but it also affects the plate. In some sense, the viscosity also wants to make the velocity of the surface of the plate to be more like the velocity of the surrounding flow.

The viscosity makes the flow want to pull the plate with it, which creates a shearing force that tries to slide the surface of this object away. The net effect is that that viscosity creates additional drag known as skin friction drag that wants to slow down any object moving in it.

All of these effects underline why highly viscous fluids are “thick”. Viscosity not only quickly averages any local differences in velocity, which prevents those fluids from flowing easily, but it also represses motion of objects in those fluids – you’ve likely experienced the difficulty of moving a spoon through a jar of honey.

The flow of any fluid exhibits tiny, random disturbances. In fluids with high viscosity, these variations are very quickly dispersed, so their motion is rarely erratic. Fluids with low viscosity aren’t as effective at damping motion, and these disturbances can grow to create oscillatory patterns. We’ve seen glimpses of them in the previous simulations, but here is another example:

At lower viscosity the flow becomes quite wave-y. Those instabilities happen at the border of regions of fluid with different velocities, like where the slow wake behind a plate is in contact with the fast external flow. In those regions, any tiny random intrusion of slower flow into the faster flow can get magnified and rolled over like a wave.

In our discussion of the motion of air around an airfoil, we’ve seen how the flow, the pressure field, and the shape of the body have effects on each other. These influences can be quite dynamic in nature, with distributions of velocity and pressure swinging back and forth in a never-ending fight for dominance.

In the demonstration below, we can see a more dramatic example of these battles, where, depending on the viscosity, the flow around a gray cube can take many different forms:

With very high viscosity, the flow is completely stable, but as viscosity decreases, it starts to regularly oscillate from side to side, shedding vortices in the process. At very low viscosity, the motion becomes even more erratic.

While I can’t easily simulate it here, with further decrease in viscosity, the flow can develop full featured turbulence in which highly irregular and chaotic mixing motions occur at different scales. Turbulent flow stands in contrast to laminar flow, in which neighboring areas of fluid move in an orderly way past each other without any varying fluctuations.

Although we’ve put most of our focus on viscosity, which is often denoted with the Greek letter μ, the general behavior of the flow also depends on its velocity u, density ρ, and the size L of the body or container involved in the flow. These parameters are tied together by the Reynolds number Re:

Re = ρ · u · L / μ

Flows with the same Reynolds numbers exhibit similar behavior, which means that if we make the obstacle size L twice as large and we halve the speed of the flow u, the Reynolds number won’t change and neither will the characteristics of the flow – it will exhibit the same smooth or oscillatory motion.

The Reynolds number also “predicts” the onset of turbulence. When we increase the speed of the flow u, or decrease the viscosity μ, the Reynolds number rises. When it reaches a high enough value, turbulence is likely to occur.

Let’s quantify the difference in viscosity between different fluids. The precise values aren’t that important to us, but to briefly be a bit more formal, viscosity is expressed in units of pascal-seconds, or Pa·s. To let us use more manageable numbers, the following table uses millipascal-seconds, or mPa·s:

honey~10000 mPa·s
olive oil~100 mPa·s
water1.0 mPa·s
air0.018 mPa·s

These values are measured at 68 °F20 °C, but many fluids like oil get much less viscous with increased temperature. As expected, honey is significantly more viscous than water. Compared to water, the viscosity of air is around 50 times less still, but even a very low viscosity has effects on flow and its interaction with solid walls.

To understand how viscosity arises in gasses like air, we have to once more get back to the world of particles. So far we’ve been watching them from a distance, with individual collisions barely perceptible in the moving swarm. This time we’re going take a closer look at these interactions.

In the demonstration below, you can experience a simplified simulation of two molecules colliding in space. Each molecule represents nitrogen or oxygen – these two elements constitute the vast majority of air, and, in normal conditions, each one consists of two atoms.

You can drag the orange particle around, and once you let go I’ll automatically aim it so that it hits the blue particle. The speed of the orange molecule is four times larger than the speed of the blue one:

Notice that after the collision, it’s the orange molecule that’s slow, and it’s the blue one that’s fast. In this demonstration the two particles have the same mass and they collide straight on, so they simply end up trading velocities.

More generally, particles of different masses that strike each other at different angles will exchange some amount of momentum. Recall that the heavier the particle, or the faster it moves, the higher its momentum.

Let’s see how this behavior ends up affecting the average velocities of larger quantities of molecules. In the paused demonstration below, air molecules are grouped into two different parts. The air in the blue region has higher velocity than the air in the red region, which you can see in the black arrows showing the average velocity in those regions. Notice what happens to these averages as you let time flow by dragging the slider:

At the very beginning, the average velocities in these two sections are visibly different, but they quickly even out when fast particles from the blue region flow into the slower red region, and the slower particles from the red region move into the faster blue region, balancing the initial velocity differences.

Moreover, some of the faster particles collide with slower particles in the red region and some of the slower particles collide with faster particles from above. The faster particles lose some of their higher momentum, while the slower particles gain some of the momentum. All of these effects “dilute” some of those average velocity differences between the two regions.

You may also remember that when we observed a flow of fluid around a flat plate, that fluid wasn’t moving at all right on the surface of that plate, because it was stuck to it. Let’s see how this behavior may arise on a microscopic scale.

In the demonstration below, we’re watching the familiar air particles right next to the surface of an object. To make tracking easier, I’m highlighting some of the particles in the vicinity of this surface:

When seen at a very large magnification, this surface, like almost all surfaces, isn’t perfectly smooth and has various peaks and valleys. The particles hitting these irregularities get bounced in more or less random directions. Some of the unlucky molecules can even get stuck for a while in these local crevices.

Close to the surface, the random collisions with peaks and valleys prevent the particles from making bulk progress in any direction. The average velocity of the air flow by the wall is more or less zero. Some molecular interactions between the particles and the surface can also prevent the fluid from moving.

This sticking behavior is known as the no‑slip condition and it holds true for most typical flows of fluids that we experience day to day. It’s only in extreme conditions of very rarified gasses in the upper parts of the atmosphere or flows in microscopic capillaries that can break this assumption.

Let’s leave the world of particles behind for the last time and see how these two effects play an important role of influencing the airflow close to the surface of any object.

Boundary Layer

Let’s take another look at a thin plate placed in the stream of incoming fluid:

From this broader perspective, it’s hard to see how the flow interacts with the surface of that plate, because the effects of viscosity are limited to the region close to that surface. Let’s focus our attention on the small area that I’ve outlined with a dashed line, right in the top part of the plate. Here it is zoomed up close:

We can once more see that, due to the no-slip condition, the velocity is zero at the wall, and then it grows to meet the velocity of the flow further away from the surface itself. What we’re seeing here is known as the boundary layer, which spans the region between the surface of the object and the “outer” flow, which is mostly unaffected by the presence of the object.

Because the velocity in the boundary layer smoothly approaches the speed of the outer flow, it doesn’t have a well-defined end point. One of the choices is to agree that the boundary layer ends where the speed reaches 99% of the speed of the surrounding flow far away from the solid surface. Let me visualize this boundary in the flow using a dashed line:

As we move with the flow along the distance of the plate, the viscosity keeps averaging out the velocity differences, making the boundary layer thicker – this is similar to what we’ve seen at larger scales with highly viscous flows around objects.

Let’s quantify the distribution of speed in the boundary layer a little more precisely. In the demonstration below, I put the velocity arrows back in. I then connected the ends of these arrows with a thin line to show a profile of velocity at that location along the surface:

Notice that, initially, the velocity close to the wall increases almost linearly, but then it smoothly tapers to reach the speed of the external flow. The velocity profile close to the surface has a certain steepness, which I’m showing with the white dotted line. This line determines the amount of skin friction drag at that spot – the closer to the surface, or more horizontal, the line is, the higher the skin drag.

As the differences in velocity become less severe, the force with which viscosity wants to drag the surface with the flow also decreases. In the conditions present in the demonstration, the skin friction drag decreases over distance.

At this point you hopefully have an intuitive grasp of how viscosity affects the flow close to the surface of the object. From our earlier discussion, you may also remember that pressure differences also affect how the flow behaves, with parcels of air slowing down when climbing the hill of increasing pressure and accelerating on the downhill of the decreasing pressure.

In the boundary layer flows we played with, the pressure distribution was more or less constant in the investigated region. Let’s see how the flow changes when we vary that pressure.

In the top part of the demonstration below we see the exact same view of velocity we’ve experimented with so far. In the bottom part of the demonstration below you can see the pressure distribution in the boundary layer, which you can change using the slider below.

If the pressure decreases in the direction of the flow in the boundary layer, we say that the pressure gradient is favorable. Favorable pressure gradient accelerates the air, and the boundary layer doesn’t grow as quickly, since the slowdown caused by viscosity is opposed by that acceleration.

When the pressure increases in the direction of the flow, we say that the pressure gradient is adverse. Adverse pressure gradient pushes against the direction of motion of the air. Far away from the surface, the air has enough momentum that the adverse pressure merely slows the flow down. However, close to the surface, the flow in the boundary layer was slow in the first place, so a pushing adverse pressure gradient may even reverse the direction of the flow.

When the flow in the boundary layer gets reversed, we say that the boundary layer separates. This region of reversed flow can form a sort of wedge that can lift the rest of the flow away from the surface.

Let’s take a step back from the subtleties of boundary layers to see how what we’ve learned corresponds to behavior of a flow around an airfoil. Let me once more bring up the demonstration that brought us here in the first place:

As we move across the surface of the airfoil, the high pressure at the stagnation point up front gradually decreases to reach minimum close to the “peak” of that curved surface. Across this transition the pressure gradient is favorable, and that distribution works in our favor – the boundary layer stays nicely attached to the surface.

However, as the air reaches the valley of the lowest pressure, it then has to start climbing back up to reach the slightly positive pressure in the rear of the airfoil. For small values of the angle of attack, the pressure pit from which the air has to climb out is not very deep and the adverse pressure gradient isn’t very strong, so the boundary layer remains attached.

As we increase the angle of attack of the airfoil, the pressure on top becomes lower and lower. For even higher angles, the adverse pressure gradient becomes so strong that it eventually reverses the flow in the boundary layer, creating separation. Let’s look at this region up close to see how the arrows of velocity in the separated region point in the other direction:

If you clicktap to add markers in the bottom right corner of the simulation you’ll notice that many of them move against the bulk of the flow – the boundary layer and the flow have separated.

We’ll get back to looking at airfoils soon enough, but we still have a few things to wrap up in the world of boundary layers.

The boundary layers we’ve looked at so far were laminar – the layers of fluid with different velocities flowed in an orderly way on top of each other. However, at higher flow speeds and over larger distances, or at high Reynolds numbers, the flow in the boundary layer transitions to a turbulent flow:

Be aware that what you’re seeing here is a very simplified simulation of a turbulent boundary layer. Turbulence is inherently three dimensional and it contains various evolving structures of different sizes that are extremely computationally expensive to evaluate in detail. Thankfully, you can find many videos of computer simulations and real flows showing turbulent boundary layers.

While the laminar boundary layers we’ve seen in the past exhibited very organized flows, the turbulent one is very chaotic, with large and small swirls causing the flow to mix very rapidly. The transition from laminar to turbulent boundary layer happens spontaneously, but for a given flow speed, the location of the transition depends on surface roughness, steadiness of the flow outside of the boundary layer, and presence of pressure gradients.

At any given moment, the velocity profile in the turbulent boundary layer is very unsteady, but it can be averaged over time to get the mean distribution of speed. Let’s compare the time-averaged profiles of the laminar and turbulent boundary layers:

In the dynamic simulation of the turbulent boundary layer, we saw how the slower flow close to the surface rapidly mixed with the upper regions of the flow. This slows down those faster sections, and we need to go farther away from the surface for these sluggish intrusions to stop affecting the flow. For this reason, the turbulent boundary layer is thicker and grows faster than a laminar boundary layer.

On the other hand, the strong turbulent mixing causes the fast external flow to get close to the body, so the overall velocity profile by the surface increases much more quickly in the turbulent case as opposed to laminar case – I’m showing that with white dotted lines.

Recall that the more horizontal the velocity profile at the surface of the object, the bigger the skin friction drag – a turbulent boundary layer has higher skin friction drag than a laminar layer. Despite the cost of increased friction drag, a turbulent boundary layer is often beneficial.

Because of that higher velocity closer to the surface, a turbulent boundary layer is more resistant to adverse pressure gradients and it can stay attached to the surface of an object for longer distances.

For some objects like golf balls, which purposefully make their boundary layer turbulent by roughing up the surface with little dimples, the delayed separation also decreases the pressure drag caused by uneven pressure distribution. That reduction more than compensates for the increased skin friction drag, making the dimply golf balls fly farther than equivalent smooth balls.

For airfoils, a turbulent boundary layer delays separation of the flow, which can help prevent stall at higher angles of attack, but at normal cruising conditions the increased skin friction becomes an important drawback. For many aerodynamic shapes in typical conditions, the skin friction frag is the primary contributor to the total drag that these objects experience.

As we’ve seen, by increasing the angle of attack on an airfoil, the lift force grows up to a certain limit, at which the boundary layer separates over most of the upper surface. By staying under this limit, a symmetric airfoil can safely generate lift force.

However, when it comes to angle of attack and lift, the shape of an airfoil isn’t particularly unique in its lift-creation capabilities. Most simple elongated shapes generate lift when put in a flow at an angle of attack. In the demonstration below, you can tilt a flat plate and see the forces exerted by the pressure field around it:

You may be surprised to see that, at small angles of attack, this flat plate also generates lift. An airfoil-like shape is not a requirement for lift generation. After all, paper airplanes with their flat wings can fly just fine. Lift is just an outcome of the pressure distribution created and sustained by the flow.

Although it doesn’t take a sophisticated shape to generate lift at an angle of attack, a well-designed airfoil can often create more lift and with lower drag. In the last section of this article, we’ll explore how other variations to the shape of an airfoil can affect its characteristics.

Airfoil Shapes

Let’s go back to the simple symmetric airfoil we’ve been playing with thus far. This time, however, we’re able to control its thickness using the slider:

Notice that as we increase the thickness of the airfoil, the pressure on the top and bottom sections of the shape becomes more negative. For this symmetric airfoil at 0° angle of attack the thickness doesn’t change much other than increasing the pressure drag.

However, if we break the symmetry of the shape, we can use thickness-dependence to make one side of the airfoil have a higher negative pressure than the other. In the demonstration below, you can control the “thickness” of the upper surface of the airfoil using the slider:

Notice that an asymmetric shape creates an asymmetric pressure distribution, which ends up creating lift without any changes to angle of attack. With some slight tweaking of this shape we finally recreated the asymmetric shape we first saw on the airplane in the early sections of this article.

Naturally, when combined with an increasing angle of attack, this airfoil will generate even more lift until it eventually reaches stalling conditions:

While symmetric airfoils are sometimes used in acrobatic airplanes, which often find themselves flying upside down, most typical planes use an asymmetric airfoil shape.

The underlying mechanism of lift generation by changing the angle of attack or by shaping the object differently is ultimately the same – we’re changing the placement and orientation of the surface of the body relative to the incoming flow. The flow reacts by changing the velocity and pressure distribution, and the resulting pressure field creates the forces on that object.

This all means that we have a lot of flexibility in how an airfoil is shaped, as long as the resulting pressure distribution fulfills the design goals of achieving a certain amount of lift while minimizing drag.

For example, in some applications it’s important to minimize the skin friction drag caused by a turbulent boundary layer. Some laminar flow airfoils achieve this by shaping the airfoil to move the “pit” of negative pressure further to the back of the airfoil:

The favorable pressure gradient between the front and the lowest pressure point extends over a longer distance across the surface of this airfoil, which, at least in principle, helps to keep the boundary layer laminar to keep the skin friction low.

Notice that even this unusual airfoil had a rounded front and a sharp back. The roundness of the front helps the air smoothly flow around this area at different angles of attack, and the sharp back reduces the pressure drag by avoiding the separation of the flow.

The velocity of the flow around the airfoil is also a contributing factor to the design of the shape. Let’s look at the speed distribution in the flow around a simple asymmetric airfoil using the varying colors and markers:

The flow above the airfoil is faster than the incoming flow as indicated by brighter colors. The markers that start in the same line don’t end up sliding off the airfoil in the same formation – the ones on top are further ahead. This is particularly visible for larger values of the angles of attack.

This acceleration in the upper part becomes another point of consideration for airfoil design. While commercial airliners don’t fly faster than the speed of sound, the accelerated flow in the top part of an airfoil can break that barrier. This creates a shockwave that can sometimes be seen in flight. Modern airliners use supercritical airfoils that are designed to reduce these drag-causing shockwaves by carefully controlling the speed of the flow around the wing.

Planes designed to fly above the speed of sound use supersonic airfoils that are quite different from the shapes we’ve seen. These airfoils have a thin profile and their front edge is sharp and not rounded. Supersonic flows of air are more complicated than what we’ve explored in this article, as variations in density and temperature become an important component of the behavior of the flow.

Many of the airfoils used today are designed specifically for the plane they’ll be used in. Moreover, that cross-sectional shape may change across the length of the wing. Real airplanes are three dimensional and the overall shape of the wings also significantly affects the lift and drag of an airplane, but ultimately all the resulting forces are an outcome of interactions between the flow and the body.

Further Reading and Watching

John Anderson’s Fundamentals of Aerodynamics is a very well-written textbook on aerodynamics. Over the course of over a thousand pages, the author presents a classic exposition of the motion of fluids and their interactions with bodies put in those flows.

Understanding Aerodynamics by Doug McLean is a great textbook that takes a different approach of explaining aerodynamic phenomena using physical reasoning. For me, the crowning achievement of the publication is showing that many popular explanations of the origins of lift are either incorrect or they’re based on merely mathematically convenient theorems. The author’s video lecture gives an overview of some of these misconceptions.

In this article, I’m using computational fluid dynamics to simulate the flow of air around different objects. For an approachable introduction to these methods I enjoyed Tony Saad’s series of lectures on the topic. For an alternative, and slightly more rigorous approach, Lorena Barba created 12 steps to Navier-Stokes. That website is also accompanied by video lectures.

Finally, YouTuber braintruffle created a series of beautiful videos that start with the behavior of fluids on a quantum scale and build up increasingly abstract models that can be used in more practical applications. The videos are packed with interesting takes on fluid mechanics, and they’re worth watching for their visuals alone.

Final Words

If you were to sit on a flying airplane and look out the window to glance at its wings, you’d often have a hard time seeing anything going on. However, in that crisp clearness of air whose invisible flow sustains the varied pressure field, lies the hidden source of lift that overcomes the might of gravity to keep the plane safely above the ground.

Since the first human flight, we’ve now mastered the art of soaring in the skies by bending the flow of air to our will, using physical quantities like pressure and velocity to help shape our designs. These tangible concepts are ultimately just a manifestation of motions and collisions of billions of inanimate air particles that somehow conspire to assemble the forces we need.

I hope this deeper, technical exploration of airfoils hasn’t diminished your appreciation of the greatness of flight. Perhaps paradoxically, by seeing how all the pieces fit together, you’ll find the whole thing even more magical.