December 10, 1999 Stalls and Dr. B.

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by	Roger Long

About the Author ... Roger Long is a 120-hour private pilot who lives in Cape Elizabeth, Maine and flies a Cessna 172 out of PWM. He pays for his flying by designing and consulting on small (under 200 feet) commercial boats.

Dr. Bernoulli and Master Euler

No wonder Master Euler put somebody else's name on this business. Few scientific ideas have ever been as widely misapplied. Poor Dr. B. must be rolling over in his grave every time a school textbook pops off a printing press. What did he do to deserve this?

The principle now blamed on the good doctor enables us to make a simple prediction about the behavior of fluid flow. Where the flow speeds up due to encountering objects, the pressure will be lower. When the flow slows down, pressure will rise. The cause of the worn upholstery in the doctor's coffin is widespread confusion between "prediction" and "cause." It is as if Euler was describing the life cycle of chickens. They start as an egg, they hatch, they grow, they become Sunday dinner. That description has been converted into conventional wisdom that eggs cause chickens.

Simply making air go faster will not lower its pressure! What Euler said is that a change in either speed or pressure will be accompanied by a change in the other ... that's "accompanied," not "caused." A balance will be maintained between pressure and flow. This fact is based on the inviolate law of conservation of energy, of which both velocity and pressure are forms.

Bernoulli's prediction does not apply in cases where energy is added to or subtracted from the system. If you speed up the air by adding energy from a source other than the flow, the pressure will not go down. Most textbook and popular press errors arise out of confusion over this fact. The reverse is also true. The speed of the air in a thin layer near the wing is slowed to nearly zero by the friction of the surface. Friction means the energy of the airflow is being converted to heat. Since the energy is being drained by a mechanism outside of the Bernoulli predictions, it is not necessary for pressure to increase and, it doesn't.

A wing moving through the air sets up a pattern of air motion, called "Circulation," in which there is faster flow above the wing and slower flow below it. These changes in speed must be accompanied by changes in pressure. The primary mechanism of lift is the reaction between the wings moving air downwards and being pushed upwards in return. You can't move air without having pressure differences somewhere. All of the changes in flow speed, pressure, and direction are linked manifestations of the same process. None of these changes cause another any more than eggs cause chickens.

Separation

A flat plate will create lift but a cambered airfoil does it more efficiently. Air has weight and inertia that set limits on how fast the direction of a flow can change. Look at the flow over this flat plate wing:

Note first that the circulation process creates an upwash ahead of the wing. The air goes over the leading edge and is then unable to turn the corner fast enough to follow the surface. Since a vacuum in this situation is impossible, two things happen. First, the overall pressure of the atmosphere pushes the flow down as far as it can. Second, the air between the main flow and the wing breaks off into slower moving eddies that fill the space. The first result causes a vertical displacement of air from A to B. This is part of the downwards displacement of air that is lift. The second result is that the air in the turbulent region, moving slower, has a higher pressure than if it had remained part of the faster moving main stream. This results in a lower pressure differential between the top and bottom of the wing, i.e., less lift. Look at point (C). If the flow had been able to follow the line of the top surface of the wing, the air that went through (B) would pass through (C), which is lower. This would be greater downwards displacement and more lift. The region of turbulent, slower moving air would also be thinner. The pressure differential above and below the wing, and the vertical displacements created by the flow patterns, are inextricably linked. These are different aspects of the same process and can not be thought of separately.

Now let's add camber to our wing.

The camber basically fills up much of the space in which the eddies formed over the flat wing. This keeps the air in the region close to the surface moving faster, and thus the pressure must be lower. An additional factor now comes into play. We have a gentleman named Coanda to thank for its first recognized description. Once a fluid flow is following a curved surface, it will tend to remain stuck to the surface and follow it. The approximate explanation is that if it didn't, a space or vacuum would have to form between the flow and the surface. It remains stuck like a suction cup to a window. The real answer is much more complicated, of course, but we don't need to get into it any deeper. What we do need to notice is that the flow, following the surface, goes through point (D). This is even lower than (C) in the previous figure. The camber, by picking up the flow and holding it to the surface, creates even greater downward displacement of air and greater lift.

Another aspect of the whole process, applicable to both the flat plate and the cambered foil, is the curved nature of the flow path. You can't swing something on a string without it trying to break the string. The air, forced around the curve, is trying to move away from the wing, lowering the pressure on the top. This is not a separate phenomenon. It is part of everything discussed above but the relationship is harder to grasp intuitively. If you want to know more, click here.

There is a limit to how quickly a flow can change direction without leaving the surface and creating a void of slower moving turbulent flow. If we increase the angle of attack on our cambered wing, the same events discussed for the flat foil will take place as the flow separates. As long as the flow can remain attached, each increment of increase in angle of attack will increase lift by a similar amount. Once the flow breaks loose from the surface, increases in angle of attack will result in smaller and smaller gains in lift. Eventually, the gains will become reductions.

Boundary Layer

Even in the absence of flow separation due to excessive angle between the wing surface and the flow, there will be a thin region of turbulent flow over much of the wing. The air in this fraction of an inch thick Boundary Layer is slowed (and warmed up) by friction. The slowing increases as you measure closer to the surface until the air touching the wing is nearly motionless. The airflow at the leading edge is able to move in a smooth and organized way for a short distance but the difference in speeds between layers eventually causes the lowest ones to trip over into rotating eddies. The smoothly moving flow then rides over the top of the turbulence and the effect on lift is much as described above for flow separation. The result of separation due to excessive angle of attack is essentially to increase the thickness of the boundary layer over the after portion of the wing.

This is why laminar flow has been such a Holy Grail of aircraft designers. Keeping the boundary layer thin and avoiding turbulence increases the efficiency of the wing. Another approach is the installation of vortex generating vanes near the leading edges of the wings. These look like small vertical wings and convert some of the energy of the flow into tip vortices with a direction of rotation roughly 90 degrees to the eddies naturally created in the boundary layer. This has two effects. First, it locally increases the flow speed without adding any energy to the system. Dr. B's rule tells us that this must be accompanied by a decrease in pressure. The increase in flow speed is largely sideways but it still has the desired effect. Second, the vortexes created by the vanes tend to suck in and organize the meandering eddies of the boundary layer. This shrinks the turbulent layer, allowing the main flow to follow a steeper path and create more of the downward air movement that is lift.

Ice on the leading edges of the wings also creates eddies, but they are not the tightly organized and located vortexes just described. Their rotation is aligned roughly with the airflow so, on the bottom where the rotation is towards the leading edge, the air is slowed relative to the wing surface and pressure must rise accordingly. This disturbance thickens the boundary layer sooner. The result is a wing that must operate at a higher angle of attack to generate the same lift. The increased drag of this higher angle of attack is one of the most significant factors requiring higher power settings to maintain airspeed in an iced-up airframe.

Coefficient of Lift

In order to know the amount of lift generated by a particular wing, you have to know the following things: air density, speed, and angle of attack. We will pretend for the rest of this discussion that density is always the same. We're all at a sea level airport on a standard day and we'll forget about it. The basic physics of the universe dictate that speed be squared. A 40 mph wind has four times the force of a 20 mph wind. All else being equal, a wing will generate four times as much lift at 200 knots as at 100 knots.

Take two wings of different shapes but the same area and they will probably generate different amounts of lift. We introduce another number to the equation to account for the difference. It is roughly a measure of the wing's efficiency; thus "CoEFFICIENT of lift," hereafter called C_L

For an idealized wing in a wind tunnel:

We're forgetting about density, remember?

In the real world, C_L will be different at each part of the wing. Pressure differences cause air to spill around the tips into vortices and flow effects of the fuselage and engines effect the efficiency of the wing nearby. For now, we will keep things simple by considering C_L to be the average over the entire wing.

This graph shows Coefficient of Lift, C_L, at various angles of attack.

To determine lift at any point, you need to know the C_L from the graph, airspeed (squared), wing area, and air density. Area will be the same for our hypothetical airplane and we are pretending that density is always the same. That leaves airspeed. You probably know that airspeed is closely related to angle of attack and that, except near stall, the airspeed gauge is pretty much an Angle of Attackometer.

Here is our plane in cruise at 80 knots: (Yes, I'd like it to be a Malibu but AVweb isn't paying me enough for these articles.)

Cruise

Here is our plane gliding at 80 knots:

Glide

Here is our plane climbing at 80 knots:

Climb

If you look closely, you will see that these are the same drawings copied to different pieces of paper and simply rotated. The difference is the power input from the engine.

This brings us to some concepts that are rather difficult to visualize. It is tempting to think that you climb by increasing lift and descend by reducing it. As with most things, the real story is much more complicated. Lift is always measured perpendicular to the relative wind or flight path; except for momentary disturbances, it is always equal to the weight of the airplane. Since lift is measured perpendicular to the flight path, you actually have to say "the component of the aircraft's weight that is perpendicular to the flight path," in order to be strictly accurate. When climbing, the pull of the engine offsets some of the aircraft's weight. When gliding, drag does the same thing. Minor corrections such as these are more significant to being completely accurate than to understanding the more basic principle. If the flight path is straight, then lift essentially equals weight.

This is a complex system of vectors with different reference points that can shift in three dimensions. You do not rise or descend by changing lift but by altering your flight path so that the sum of all these vectors will work out to lift equaling weight. The aircraft does not care as much as ATC does whether you are climbing or descending, it just wants to maintain the lift/weight equality. If lift becomes less than weight, the aircraft will sink. As we will see below, this will increase the angle of attack. C_L will go up and lift will increase. This process will stop when lift again equals weight and the plane will then proceed on a straight flight path (assuming, of course, that no one is moving the controls).

The initial action that alters your flight path is a brief change in lift. You change one or both of the two basic factors you control, angle of attack or power. This creates an imbalance in lift and weight. The inequality is only momentary however and the plane quickly seeks a new flight path where lift equals weight. If you change only one of the lift factors, the new flight path will have a different angle to the ground. If you change both, the new flight path can have the same angle but at a different airspeed.

It is beyond the scope of an article of this length to prove this, or even begin to fully describe the relationships. If you would like to know more, please read John Denker's excellent web book, "See How It Flies."

Stalls (Finally!)

It is fairly intuitive that drag increases with Angle of Attack along with C_L. After all, it is easier to make something go through the air edgewise than to drag it at an angle. This is closely related to the correspondence between speed and AOA.

The exact shape of the C_L curve will depend on the cross sectional shape of the airfoil. Almost all shapes used as airfoils share the common characteristic that the first part of the C_L curve is nearly straight. This is the region in which the flow stays mostly attached to the after portion of the wing. At these modest angles, doubling the AOA will double the lift, if you could maintain the same airspeed. But, you can't. Pull back gently on your Angle-of-Attack-O-juster (yoke or stick, personal preference) and C_L will increase while airspeed decreases. The amount of lift will remain the same. The throttle setting will determine whether the flight path that forms one side of the attack angle is a climb, a decent, or level. A slight power reduction will need to be made so that the flight path that maintains lift equal to weight will remain parallel to the earth's surface and you don't get a call from ATC reminding you about your altitude.

Pull back farther and some sort of alarm will go off, depending on where your plane was built. Pull a little farther and there will be some control buffet. In simple planes, this my be your first warning. Pull a little farther and, YIKES! Whew, it sure felt like you just lost all your lift. But, just moments ago you had the same lift as you did at cruise and C_L was at maximum. When the nose dropped, you were about at point Y (for "Yikes!") on the curve above. Plenty of C_L there.

Now get your plane moving along really briskly, right up near the red line. Yank hard on your Angle-of-Attack-O-juster, all the way back. One of two things are going to happen. Either the tail will break off, in which case you have no further interest in this article and are excused. Still with us? Good. The plane is not going to be able to change flight path that quickly and AOA will go right up and past point Y. Airspeed and C_L will both be high and you may actually be developing more lift than the plane ever did before. You might even over-stress the wings and face some huge repair bills on landing (if the wings don't come off). Still, the plane will have stalled and felt like it just lost all its lift. Clearly, a stall is more complicated than the wings suddenly giving up and stopping their downwards movement of air.

This animation shows what happens to Angle of Attack when airspeed suddenly decreases and the aircraft sinks. Watch the streamer on the frame ahead of the wing as a tailwind gust reduces the airspeed.

Headwind gusts will have the opposite effect.

This change in angle of attack with rising or sinking of the plane from the flight path is the key to understanding basic flight dynamics. An excess or deficit of lift over weight will result in the plane rising or sinking accordingly. This will change the AOA so that lift is adjusted in the opposite direction (at least in unstalled flight). If the plane is sinking, lift will increase. If the plane is rising, lift will decrease. The aircraft, with controls held steady, will settle quickly on a flight path where lift equals weight.

Here is our C_L curve again:

We are flying along at point A on the left side when a tail gust hits the plane. Airspeed drops so lift decreases and the plane sinks. The sinking increases the angle of attack and, with it, the lift, restoring the equilibrium. There's a bobble and your passenger yelps, but it's no big deal.

Now put the plane at the top of the C_L curve and bring on the same tail gust. The sink-induced increase in Angle of Attack now REDUCES the C_L at the same time that it is already being reduced by the change in airspeed. Further sinking causes even greater reduction in C_L. With these changes in AOA come increases in drag, slowing the plane, increasing sink, decreasing lift, slowing the plane. The whole system grabs its tail and starts going around, faster and faster, until either the pilot or the ground intervenes. For many planes, simply letting go of the controls will be sufficient intervention. Holding the nose up in a panic is truly a "death grip."

To the left side of the C_L curve, the airfoil is inherently stable in the airstream. Wind gusts and clumsy moves of the Angle-of-Attack-O-juster will cause the plane to quickly find a new equilibrium. There is plenty of lift to be had on the right side of the curve but the system is about as stable as a pencil balanced on its eraser. One little disturbance and it falls over into a descending feedback spiral where lift can only find equilibrium with weight in large diversions to a flight path so steep that it is hard to associate it with a concept like "lift." When the plane starts to sink after holding the nose too high, the same thing happens. Stall is not the cessation of lift but loss of stability of the airfoil in the airstream.

The C_L graphs above are for airfoils that are fixed in place, unable to change either flight path or airspeed. This is what the curve looks like in a wind tunnel but not the way the wing will behave in free air. The ability of the aircraft to radically change flight path is an essential feature of stalls. I still remember building a wind tunnel for a science project and being mystified as to why I couldn't demonstrate stall. My measurement apparatus only showed a slight decrease in lift at high attack angles, but nothing that I could relate to the behavior of the model planes I flew and crashed. If you could play with AOA while your plane was forced to stay in a straight line, like rolling along a runway, you would find stalls equally gentle. This is also why your hand out the car window doesn't seem to stall as you might expect.

Wingtip Dynamics

Stomp on your rudder pedals, whip those controls around. While you do, look out to your wingtips. All that motion, superimposed on that of the aircraft as a whole, is doing some really interesting things out there. Your manipulation of the controls, especially the rudder pedals, can add considerable motion to the wingtips that is independent of the flight path. This motion can have a big effect on the angle of attack by making it different at each portion of the wing.

Below is an animation that may take some study. The streamer, which shows the relative airflow (and thus angle of attack) is mounted on a frame attached to a wingtip of an aircraft on final. Only the wingtip is visible. The center of the wing is following the indicated flight path at steady rate of descent while the pilot pushes the rudder pedals. For clarity, the animation pretends that the pilot also keeps the wings exactly level as the plane yaws and that there is no change in airspeed due to the slip. The wingtip moves first forward and then aft. The overall angles of attack are exaggerated for clarity. Do not confuse the wingtip moving forwards and backwards from the flight path for sinking or rising above it. The center of the wing is maintaining a straight and steady line of flight as the aircraft yaws.

Note how forward motion of the wingtip decreases angle of attack and retreating motion increases it. The changes are the result of the yaw motion of the wingtip being added or subtracted from the overall airspeed of the plane. The vital fact is that these changes are rate dependent. Ease slowly into a slip and there will be very little change in angle of attack. Make a panicked or clumsy jab at the rudder pedals and there will be large change. This difference can have important consequences for your future.

The next animation is a graph of the Coefficient of Lift for the colored portions of the wingtips. The tips are important not only because they experience the greatest change in airspeed as the plane is yawed but also because they exert a great deal of leverage in rolling. That's why they put the ailerons out there. As the plane yaws, note how the colored dots corresponding to C_L for each wingtip move along the graph. The advancing wingtip has its angle of attack reduced so it moves down the graph. The angle of attack for the retreating wingtip increases, as does its C_L.

Once the plane is held in the slip, the rotational speed stops and the airflow over the ends of the wings returns to that of the nominal airspeed. The C_L's do not go back to matching values however. The end of one wing is now acting somewhat as a leading edge and the other as a trailing edge. Greater lift is created at leading than trailing edges so there is a residual imbalance between the wingtips. This is why you can get into trouble with crossed controls even in a steady slip. As the plane is yawed back to a straight course, the C_L's for each wingtip move in the opposite direction.

Again, the effect shown above is rate dependent. Ease slowly into the slip and the C_L values will change very little because the fore and aft speed of the wingtips will be a small proportion of the airspeed. The length of the wings is also a factor. Gliders are very sensitive to rudder work because the combination of their slow speed and long wings will let the pilot create very large angle of attack changes at the tips.

The animation shows average C_L for the colored portions of the wingtips. We could construct a similar graph for any other portion of the wings. Everything would be the same except the C_L dots would not move as much. The relative fore and aft speed of inboard wing sections will be less since they are not as far the center of rotation.

The yaw in the animation above is taking place with the plane well down on the C_L curve. The graph shows C_L, which is only one of the factors that determines overall lift. The advancing wingtip which is having its C_L reduced is also experiencing an increase in airspeed. This offsets the reduction in C_L and lift remains close to what it was in steady flight. The same is true on the other side although C_L increases and airspeed decreases. There will be some leftover rolling effect, depending on the plane's design, but the rolling force will usually not be very significant. Other factors, such as dihedral, blanketing of flow to one wing by the nose, etc. will have a greater effect on the aircraft's roll response.

Now envision this taking place up near the top of the C_L curve. The top of the curve is flatter so changes in angle of attack result in smaller changes in C_L. The effect of airspeed on lift is unchanged, however. The plane will be slow up here near the top of the curve so the wingtip speed that can be generated by rudder action is apt to be a greater proportion of the overall airspeed. The result is a much greater tendency for the advancing wingtip to rise and the retreating wingtip to sink. This is why, in very slow flight, the rudder is so effective in keeping the wings level. The reason you have to be aggressive with the rudder is again, the fact that the changes in wingtip airspeed are dependent on the rate of the yaw.

We now move into the dangerous territory right at the top of the C_L curve or a little to the unstable stalled side. As you envision the separation of the C_L dots, it's easy to see that the retreating wing is going to go over the hump into the region where its C_L will decrease; not increase with AOA. Now both C_L and airspeed are going south on that side of the plane. The wing is going to sink. The advancing wing is now right up at the top of the curve where its increased AOA will have almost no effect on C_L. The effect of increased airspeed will be unchanged and the wing will lift. The plane takes a sudden break to one side.

We saw above that sinking of an airfoil increases angle of attack. The sinking of the retreating wing tip further increases angle of attack as well as drag. The drag slows the wingtip and pulls the plane deeper into the yaw, increasing the airspeed over the now high wing and lifting it yet higher. At this point, the pilot may make matters even worse, a sudden movement of the yoke to raise the dropping wing with the ailerons is the last mistake many pilots ever make. An aileron is really just a mechanism for changing angle of attack by altering the camber of the airfoil. Lowering the aileron in an attempt to raise the wing increases the angle of attack on a portion of the wing that is already deep into stall where further increases in AOA result in large C_L reductions. The aileron sticking down into the clear flow below a wing already operating at a very steep angle is just like a speed brake. On the other side of the plane, the wing end is operating in a regime where angle of attack changes due to the aileron cause relatively little change in C_L. The upward raised aileron is partially blanketed in the flow behind a wing close to stall so it creates little offsetting drag.

The overall result of all these forces can be as dramatic as if the low, inside wingtip had suddenly pulled taut a rope attached to the ground. An eyeblink later, the plane is pointed towards the ground and possibly inverted. We all tend to do things faster when nervous, distracted, or panicked. A rapid rudder input will create large changes in C_L between the wings. If you are already low on airspeed, and therefore pitched to a high angle of attack, the speed of the rudder input can be the difference between life and death.

Rapid recovery from an indiscretion of this kind is something that can only be practiced in the air. It must be instinctive and decisive at low AGL or if a spin is to be avoided. Get the nose DOWN. That will slide the C_L of both wingtips up over the hump in the graph to the left and stable side of the curve. Keep the yoke level. Jab on the high side rudder pedal. The increased airspeed on the low wing will decrease its AOA, which now increases C_L. Airspeed and C_L rising together will lift the wing. You will still lose a lot of altitude because you will have slowed below the speed where you can establish an unstalled AOA in a flight path that is not inclined steeply downward. If this happens at low altitude, even the most rapid recovery will only change the shape of the hole the airplane makes in the ground.

The rate dependency of these wingtip dynamics is a vitally important given the human factors in loss of control accidents. Practice and practice putting your aircraft carefully on the edge of turning stall and recovering and you will gain a great deal of confidence. Get into a situation where you are panicked, or even just nervous, and your sense of timing will change. Just as student pilots often flare too quickly when they are nervous, you may perform the same action slightly faster. The timing will feel the same and the attitude and sight picture may be identical but the outcome can be very different. If things are tense, and you have made the mistake of getting backed into a corner where you really have to crank your plane around, slow yourself down. That will probably just bring your actions back to normal speed. More people have probably been killed in the air by doing things too fast then by doing them too slowly.

If you read this piece, you may conclude that I have above-average knowledge of the dynamics of stalls in uncoordinated flight. I must emphasize, however, that this knowledge does not make me any more likely to do the right thing if I stall out of a clumsy turn. When the windshield fills with trees, this knowledge will be locked away in a part of my brain that is no longer functioning. The only way to program the instincts that will be driving at that point is to go out with a competent instructor and do it, do it, do it.