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By David Ison
I always joke about different scenarios whenever my students scoff at the idea of learning more than just the basics about lift. One of my favorites is to describe that they are about to have heart surgery, and their surgeon comes in to have a chat about the procedure. When asked about a detail of the heart, the doctor responds, Im not really sure how it worksI mean, the heart beats due to electrical signalsbut the rest is kind of like magic.
How would they feel going under the knife after that conversation? The same is true for passengers boarding an airplane. What confidence should they have in a pilot who doesnt truly grasp lift? And trust me, Ive heard less-detailed descriptions of how an airplane flies than that dreadful medical example.
Sadly, even experienced flight instructors struggle with the concept of lift. Yet a great, but often overlooked, tool that can be used to explain it to all levels of pilots resides within the lift equation. While the use of something that seems so complex on the surface may seem counterintuitive, the equation can actually be a simplerand more accurateway to convey instruction on aerodynamics than more commonly used methods.
Representing Lift
A highly instructional equation allows for the mathematical computation of the amount of lift being generated by an airfoil under specific conditions. Heres the technical construction of the equation:
L = CL x x V2 x S
The variables are defined as: L is lift, CL is the coefficient of lift, represents density, V is velocity, and S is the area of the wing.
From this point, many options exist for the level of detail that instructors can choose to pursue. For those who want to keep things simple, no numbers are needed, and units of measure can be ignored. In other words, you only need to use the concept of relationships of one side of an equation to the other to explain aerodynamic concepts. If you prefer to go the truly mathematical routeplugging in numbers to solve real-world problemsyou will need to know what units each variable must come in to get accurate answers. An example of this will come shortly.
For the time being, well stick to basics. When you introduce most students to an equation, their eyes are likely to glaze over quickly. Some encouragement will no doubt be necessary to assure them that they will be Jedi masters of lift by the time theyve listened to your spiel (candy and coffee are good motivators, too).
Once you have redirected their attention to the equation, rather than on daydreams of medieval torture, give an explanation of each variable in the equation. Start with the easier stuff. S is pretty simple to grasp; its simply the size (the area) of the wing. V is the speed the wing is encountering. This part of the equation is a distant cousin of the Bernoulli equation and has to do with the term dynamic pressure.
Next, mention the harder stuff; is the Greek letter rho, which represents air density. Of course density changes with altitude and temperature. Ask your students if they knew that altitude and temperature affected lift. If they did, great; if not, that is a bonus lesson (be sure to assure your students you won t charge them extra for that). For all of the following examples, assume that density is equivalent (i.e., all of the illustrations occur at the same density altitude) and therefore it can be neglected.
CL is an aerodynamicists way to water down something that is pretty complex into a simple numerical value. I like to think of it as an indicator of the efficiency of an airfoil. There are a variety of factors that determine CL, including viscosity, compressibility, wing shape, and wing inclination (aka angle of attack, or AOA). For this discussion, viscosity and compressibility will be disregarded (viscosity has to do with boundary layer issues and is mostly a concern when experimenting with airfoil designs; compressibility is only a concern at high speeds). Thus, CL is related to shape and AOA. In general, the more cambered a wing is, the higher the CL. Also, when AOA is increased, CL also increases. Now that the basics of the equation are on the table, use it to explain a range of aerodynamic concepts.
Solving Scenarios
The most straightforward notions are best brought forward first. Since the CL increases with more camber, and the effective camber of a wing increases when flaps are extended, its no surprise that lift increases with the addition of trailing-edge devices. Simply, as one of the multipliers on the right-hand side of the equation grows, so the final mathematical product, lift, must also grow larger. The reverse is true, as well. If flaps are retracted, then CL gets smaller, and, in turn, lift does the same. Since leading-edge devices, such as slats, have a similar effect on camber, the same premises are true for these lift augmenters.
In a similar fashion, if AOA is increased, CL will grow larger. Again, because something on the right-hand side of the equation is enlarged, then the left sideliftwill wax. When a pilot reduces the AOA, CL will decrease, resulting in a lower total value (i.e., lift will wane).
The next variable in the equation is rho, or density. When density decreasesas it does with increases in altitude and temperaturethe total result on the right side of the formula will shrink. Thus, the left-hand side, lift, will also become smaller.
If students struggle with the reasoning behind why decreased density does in fact reduce lift, rely on Sir Isaac Newtons concept of equal and opposite reaction (after all, Daniel Bernoulli did). There are fewer air molecules in terms of a volume of air at 30,000 feet than at sea level (remind them thats why they need oxygen up there). Fewer air molecules means less equal and opposite reaction as they quickly make their way off the trailing edge of the wing. If you must get even simpler, you can digress by saying air up high isnt as thick, thus it cant provide as much support.
Your students eyebrows will likely rise at this point. They may turn to you and say, But if you create less lift at altitude, why dont we have to fly faster? You told me to use the same approach speed in Denver, a mile high, as I would at sea level.
Its true that less lift is generated at altitude. But the velocity (V) in the equation is true airspeed (TAS). Recall for the same indicated airspeed (IAS), TAS is higher at higher altitudes. Thus the airplane must fly at a higher TAS (V in the equation) to produce the same IAS in Denver, and it ends up creating lift similar to what one produces at sea level (mathematically it is very close, but not exact).
Velocity is the subsequent variable, and it should be no surprise that increases in speed have a dramatic affect on available lift. Keeping all other variables the same, if speed is increased, the resulting effect is, as noted in the formula, the square of the velocity. Does this sound familiar? It should; doubling airspeed results in a quadrupling of the effect of the speed increase in terms of the equationa concept harped upon on many an FAA written exam. Just as important, however, is the dramatic effect that a speed reduction has on lift. The eyes of students will open wide when you explain the difference between lift produced at cruise speeds and that generated near stall. This concept should make it clear why we have and use flaps for normal landings (and for some planes, for takeoff, too).
The variable S represents wing area. The basic aerodynamic concept that a larger wing produces more lift can be easily demonstrated, yet it doesnt take a rocket scientist to know that we cant realistically produce monster-sized wings due to structural and practical limitations. It might make more sense to explore the effects of wing area from a more practical standpoint.
Some general-aviation airplanes have a means of changing the wing area by using Fowler flaps. Recall that this type of flap increases camber and wing area when extended; its possible to manipulate S in some cases. As a Fowler flap extends, S will become larger, thus resulting in a larger value for lift.
Adding Complexity
Once the basics of the formula have been explained, it can be used to investigate more complex concepts. One particular notion that students have difficulty grasping is the reasoning behind changes in stall speed in a variety of flight conditions and weight and balance scenarios.
Since its of interest to demonstrate the variability of stall speed, some ground rules should be established for using the lift equation in this context. For all examples, assume that the AOA is constant and is just short of the critical AOAthe airplane is just on the verge of stall. Aircraft flap configuration is also fixed; thus, the combination of the former and latter means that CL remains constant. Wing area also must be assumed to remain invariable. Therefore the only thing that can change on the right side of the equation (i.e., the only thing that can be assumed to influence lift) is velocity.
Using the aforementioned rules, lets consider the difference between power-off and power-on stall speeds. In a Piper Arrow, the power-off stall speed with gear and flaps up is 65 knots, but the power-on stall speed is 54 knots. Why the difference?
According to the Jeppesens Principles of Flight (a great aerodynamic text that is part of the JAA air transport pilot prep series), at high power settings, some of the thrust counteracts weight (since power is canted upward). This reduces the lift requirement of the wing, since a component of power is supporting the aircraft. Consequently, the value of L in the lift equation is smaller.
Since we already agreed that the only variable on the left side of the equation that could be changed is V, the velocity required under high power circumstances to stay just above stall is lower. Voil! Thats why power-on stall speed is lower. (There is another factor that influences power-on stall speed, the accelerated airflow from the propeller, but that is a whole other discussion.)
Another topic to tackle is the change in stall speed that occurs with changes in the weight of the aircraft. In a Beechcraft Duchess, the stall speed at maximum gross weight is 70 knots (flaps up), and at a lighter weight, its around 65 knots. In this case, the premise of why the lift requirement is reduced is a little more clear: If the airplane weighs less, then it doesnt need as much counteraction to the upward force. If AOA is kept right at the edge of stall and the wing area remains the same, a reduction in the lift requirement allows for a lower velocity. Again, this is why stall speed is decreased.
Its well-known that stall speed increases as bank angle increases. For example, the stall speed in level flight for a Cessna 152 is 36 knots, but at a 60-degree bank, it rises to 51 knots. Another well-known fact is that at 60 degrees of bank, the load factor is twice that found in level flight. Simply, the plane weighs twice as much; hence the lift requirement imposed on the wing is also doubled. Keeping the coefficient of lift and area the same, V would have to increase to compensate. Point out to students that stall speed does not double; the effects of velocity are squared and therefore more compensatory for increased lift requirements.
While books note that center of gravity (CG) location affects stall speed, there are more than a few students who struggle with the concept. But the difference does exist. According to the Cessna 152 information manual, the stall speed when the aircraft is loaded at the forward CG limit is 40 knots and is 36 knots when loaded at the aft limit. So what gives?
It all has to do with tail-down force. When there is a lot of weight in the front of the airplane, more elevator deflection must be used to hold the nose up. This results in lift being generated toward the ground on the underside of the horizontal stabilizer. Think of it as negative lift in terms of the total lift demand imposed on the wing. In short, some of that tail-down force counts against the lift generated by the wing (i.e., canceling part of it out). Thus, the total lift demands on the wing increase as tail-down force increases. To compensate using the formula ground rules, V would need to increase to make up the difference in L. So its no surprise that the forward CG stall speed is greater than the aft CG stall speed.
By the Numbers
For the more curious student, or just to hit things home even faster, do some actual calculations with real numbers. Lets say were flying a Cessna 172. It has a wing area of 175 square feet. At sea level, the density of the atmosphere in slugs per cubic foot is 0.002378 (see HYPERLINK "http://www.USAToday.com/weather/wstdatmo.htm" \t "_top" www.USAToday.com/weather/wstdatmo.htm) At the critical AOA, the coefficient of lift is about 1.6. If the aircraft is flying at its maximum gross weight of 2,300 pounds, what is stall speed?
In this case, the unknown is V. Plugging in the numbers to the formula L = CL x x V2 x S, we have a resultant formula of 2,300 = 1.6 x 0.002378 x V2 x 175. This results in 2,300 = V2 x 0.33292, then 6908.5 = V2. With the square root taken, V = 83.11.
Remember this is in feet per second. Converting this to knots requires dividing it by 1.68, which results in 49.24 knots (TAS). Compare that to the 172 manual, which states that VS1 is 48 knots (IAS). Math works pretty good, eh?
While playing with numbers, its interesting to look at the relationship between CL, airspeed, and lift. As the AOA is reduced, as it is when speed increases, CL
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This relationship can be examined in a different way. As airspeed decreases, AOA must be increased to improve CL so as to compensate and maintain lift. This explains the basic premise of flight from low-speed operations to cruise.
Although seemingly perplexing at first glance, the lift equation is actually a simple tool for helping to augment student understanding of aerodynamics. By taking some time to explain the basics about the formula, you can use it as a powerful means of exploring some of the finer points of how lift is generated and why common changes in aircraft configuration, weight, and balance can influence this process. Its a welcome addition to your arsenal of teaching tools, and it doesnt rely on magic to make it tick.
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