— Aviation proverb.
The main purpose of this chapter is to clarify the concepts of lift, drag, thrust, and weight. Pilot books call these the four forces.
It is not necessary for pilots to have a super-precise understanding of the four forces. The concept of energy (discussed in chapter 1) is considerably more important. In the cockpit (especially in critical situations like final approach) I think about the energy budget a lot, and think about forces hardly at all. Still, there are a few situations that can be usefully discussed in terms of forces, so we might as well learn the terminology.
The relative wind acting on the airplane produces a certain amount of force which is called (unsurprisingly) the total aerodynamic force. This force can be resolved into components, called lift and drag, as shown in figure 4.1.
Here are the official, conventional definitions of the so-called four forces:
In later chapters we will sometimes find it convenient to choose a frame comoving with the airplane, in which case our notions of “gravity” and and “downward” are heavily modified. That’s because these notions are totally dependent on the choice of reference frame. For details on this, see section 19.6.
It is ironic that according to convention, the total aerodynamic force is not listed among the four forces.
In general, the lift is a vector in two dimensions, i.e. a vector in the plane perpendicular to the relative wind; this plane is shown in yellow in figure 4.1. In general, you get to specify the magnitude and direction of this vector. However, in normal coordinated flight, to a good approximation, the lift is perpendicular to the wingspan (as well as perpendicular to the relative wind). Therefore, when people speak of the lift in one-dimensional terms, they are probably talking about the component of lift in the direction perpendicular to the wingspan (unless context demands otherwise).
Figure 4.2 shows the orientation of the four forces when the airplane is in “slow flight”, i.e. descending with a nose-high attitude, with the engine producing some power. Similarly, figure 4.3 shows the four forces when airplane in a high-speed descent. The angle of attack is much lower, which is consistent with the higher airspeed. Finally, figure 4.4 shows the four forces when the airplane is in a climb. I have chosen the angle of attack, the lift, and the drag to have the same magnitude as in figure 4.3.
Note that the four forces are defined with respect to three different coordinate systems: lift and drag are defined relative to the wind, gravity is sometimes defined relative to the earth, and thrust is defined relative to the orientation of the engine. This makes things complicated. For example, in figure 4.2 you can see that thrust, lift and drag all have vertical components that combine to oppose the weight. Meanwhile the thrust and lift both have forward horizontal components.
Let’s temporarily imagine you are flying straight and level, maintaining constant speed and constant attitude, through still air. We further imagine that the axis of the engine happens to be aligned with the straight-ahead direction, for this chosen attitude. Then all three coordinate systems coincide, in which case thrust is opposite to drag, and lift is opposite to weight.
In reality, it isn’t safe to assume that lift always matches weight, or thrust exactly matches drag. Consider a bomb falling straight down (figure 4.5) – it has no lift and no thrust; when it reaches terminal velocity its weight is supported purely by drag. Another interesting case is a moon lander hovering on its rocket plume (figure 4.6) — it has no lift and no drag; its weight is supported by its thrust.
Figure 4.5: Bomb (Weight = Drag) | Figure 4.6: Moon Lander (Weight = Thrust) |
You may think lift, thrust, weight, and drag are defined in a crazy way, but the definitions aren’t going to change anytime soon. They have too much history behind them, and they actually have advantages when analyzing complex situations.
The good news is that these subtleties usually don’t bother you. First of all, the angles in figure 4.2 are greatly exaggerated. In ordinary transportation (as opposed to aerobatics), even in climbs and descents, the pitch angle is always rather small, so thrust is always nearly horizontal. Also, the relative wind differs from horizontal by only a few degrees, so drag is always nearly horizontal, and lift is nearly vertical except in turns.
If we don’t like the technical definitions of lift, drag, thrust and weight, we are free to use other terms.
This description is nice because it shows things from the pilot’s point of view, which is consistent with the general spirit of this book. Note that in figure 4.7 the parallel component of weight is pulling the aircraft forward along the path of flight. Indeed, in this situation, this component of weight is a larger contribution than engine thrust.
Before going on, let me mention a couple of petty paradoxes. (1) In a steady, low-speed, high-power, nose-up climb, lift is less than weight. That’s possible because thrust is supporting part of the weight. It sounds crazy to say that lift is less than weight during climb, but it is technically true. (2) In a steady, low-power, high-speed, nose-down descent, lift is once again less than weight. That’s possible because drag is supporting part of the weight.
These paradoxes are pure technicalities, consequences of the peculiar definitions of the four forces. They have no impact on pilot technique.
There is some additional discussion of the balance of forces in section 19.1.
The most important non-aerobatic situation where you have to worry about the forces on the airplane is during a turn. In a steeply-banked turn, the lift vector is inclined quite a bit to the left or right of vertical. In order to support the weight of the airplane and pull the airplane around the turn, the lift must be significantly greater than the weight. This leads us to the notion of load factor, which is discussed in section 6.2.3.
The bottom line is that thrust is usually nearly equal (and opposite) to drag, and lift is usually nearly equal (and opposite) to weight times load factor.
In a turn, it is sometimes useful to express the total lift as a sum of two components.
In a steeply-banked turn, the horizontal component of lift is quite large. In the pilot’s frame of reference, that means the airplane is subject to very significant centrifugal forces. This important and interesting topic will be discussed in section 6.2.
We have seen that the total force on the airplane can be divided into lift and drag. We now explore various ways of subdividing and classifying the drag.
When a force acts on a surface, it is often useful to distinguish processes that act perpendicular to the surface (pressure against the surface) versus forces that act parallel to the surface (friction along the surface).
Figure 4.8 illustrates the idea of pressure drag. If the tea table is moving from right to left, you can oppose its motion by putting your hand against the front vertical surface and pushing horizontally.
Figure 4.9 illustrates the idea of friction drag. Another way to oppose the motion of the tea-table is to put your hand in the middle of the horizontal surface and use friction to create a force along the surface. This might not work too well if your hand is wet and slippery.
Figure 4.10 shows a situation where air flowing along a surface will create lots of friction drag. There is a large area where fast-moving air is next to the non-moving surface. In contrast, there will be very little pressure drag because there is very little frontal area for anything to push against.
Friction drag is proportional to viscosity (roughly, the “stickiness” of the fluid). Fortunately, air has a rather low viscosity, so in most situations friction drag is small compared to pressure drag. In contrast, pressure drag depends on the mass density (not viscosity) of the air.
Friction drag and pressure drag both create a force in proportion to the area involved, and to the square of the airspeed. Part of the pressure drag that a wing produces depends on the amount of lift it is producing. This part of the drag is called induced drag. The rest of the drag — everything except induced drag — is called parasite drag. The various categories of drag are summarized in figure 4.11.
The part of the parasite drag that is not due to friction is called form drag. That is because it is extremely sensitive to the detailed form and shape of airplane, as we now discuss.
A non-streamlined object (such as the flat plate in figure 4.12) can have ten times more form drag than a streamlined object of comparable frontal area (such as the one shown in figure 4.13). The peak pressure in front of the two shapes will be the same, but (1) the streamlined shape causes the air to accelerate, so the region of highest pressure is smaller, and more importantly, (2) the streamlined shape cultivates high pressure behind the object that pushes it forward, canceling most of the pressure drag, as shown in figure 4.13. This is called pressure recovery.
Any object moving through the air will have a high-pressure region in front, but a properly streamlined object will have a high-pressure region in back as well, resulting in pressure recovery.
The flow pattern^{2} near a non-streamlined object is not symmetric fore-and-aft because the stream lines separate from the object as they go around the sharp corners of the plate. Separation is discussed at more length in chapter 18.
Streamlining is never perfect; there is always at least some net pressure drag. Induced drag also contributes to the pressure drag whenever lift is being produced (even for perfectly streamlined objects in the absence of separation).
Except for very small objects and/or very low speeds, pressure drag is larger than friction drag (even for well-streamlined objects). The pressure drag of a non-streamlined object is much larger still. This is why on high-performance aircraft, people go to so much trouble to ensure that even the smallest things (e.g. fuel-cap handles) are perfectly aligned with the airflow.
An important exception involves the air that has to flow through the engine compartment to cool the engine. A lot of air has to flow through narrow channels. The resulting friction drag — called cooling drag — amounts to 30% of the total drag of some airplanes.
Unlike pressure drag, friction drag cannot possibly be canceled, even partially. Friction drag causes energy to be thrown overboard. The energy gets carried away by the relative wind, and there is no practical way for the airplane to recover that energy.
The way to reduce induced drag (while maintaining the same amount of lift) is to have a longer wingspan and/or to fly faster. The way to minimize friction drag is to minimize the total wetted area (i.e. the total area that has high-speed air flowing along it). The way to reduce form drag is to minimize separation, by making everything streamlined.
The word “drag”, by itself, usually refers to a force (the force of drag). Similarly, the word “lift”, by itself, usually refers to a force. However, there are other ways of looking at things.
It is often convenient to write the drag force as a dimensionless number (the coefficient of drag) times a bunch of factors that characterize the situation:
drag force = ½ρV^{2} × coefficient of drag × area (4.1) |
where ρ (the Greek letter “rho”) is the density of the air, V is your true airspeed, and the relevant area is typically taken to be the wing area (excluding the surface area of the fuselage, et cetera).
Similarly, there is a coefficient of lift:
lift force = ½ρV^{2} × coefficient of lift × area (4.2) |
We used these equations back in section 2.13 to explain why the airspeed indicator is a good source of information about angle of attack.
One nice thing about these equations is that the coefficient of lift and the coefficient of drag depend on the angle of attack and not much else. If you could (by magic) hold the angle of attack constant, the coefficient of lift and the coefficient of drag would be remarkably independent of airspeed, density, temperature, or whatever.
The coefficient of lift is a ratio^{3} that basically measures how effectively the wing turns the available dynamic pressure into useful average suction over the wing. A typical airfoil can achieve a coefficient of lift around 1.5 without flaps; even with flaps it is hard to achieve a coefficient of lift bigger than 2.5 or so. For data on real airfoils, see figure 3.16 and/or reference 26.
Figure 4.14 shows how the various coefficients depend on angle of attack. The left side of the figure corresponds to the highest airspeeds (lowest angles of attack). Note that the coefficient-of-lift curve has been scaled down by a factor of ten to make it fit on the same graph as the other curves. Airplanes are really good at making lots of lift with little drag.
In the range corresponding to normal flight (say 10 degrees angle of attack or less) we can use the basic lift/drag model. The details of this model are explored in section 7.6.3, but in most piloting situations all you need to know are the following approximations, which are the conceptual basis of the model:
In flight, we are not free to make any amount of lift we want. The lift is nearly always equal to the weight times the load factor. This leads us to rearrange the lift equation as follows:
coefficient of lift = (weight × load factor) / (½ρV^{2} × area) (4.3) |
On the right-hand side of this equation, the only factors that are likely to change from moment to moment are airspeed and load factor. (Weight can change, too, but usually only slowly. Extending the flaps can change the area somewhat.) Because of the factor of airspeed squared, the airplane must fly at a very high coefficient of lift in order to support its weight at low airspeeds.
These ideas can be used to explain the shape of the power curve, as shown in figure 4.15 (which is a copy of figure 1.13).
For more on this, see section 5.3.
Figure 4.16 is related to figure 4.14. It plots the same four curves against airspeed (rather than angle of attack). Now the left side of the plot corresponds to the lowest airspeeds (highest angles of attack).
At higher angles of attack (approaching or exceeding the critical angle of attack) the basic-model approximations break down. The coefficient of parasite drag will rapidly become quite large, and the induced drag will probably be quite large also. There will be no simple proportionality relationships. The details aren’t of much interest to most pilots, for the following reason: Typically you recover from a stall as soon as you notice it, so you don’t spend much time in the stalled regime. If you do happen to be interested in stalled flight and spins, see chapter 18.
Figure 4.17 shows the corresponding forces. We see that whereas the coefficient of parasite drag was more or less constant, the force of parasite drag increases with airspeed. If somebody says “the drag is a ... function of airspeed” you have to ask whether “drag” refers to the drag coefficient, the drag force, or (as discussed below) the drag power.
We can also see in the figure that the lift force curve is perfectly constant, which is reassuring, since the figure was constructed using the principle that the lift force must equal the weight of the airplane; this is how I converted angle of attack to airspeed.
The lowest point in the total drag force curve corresponds to V_{l/d}, and gives the best lift-to-drag ratio. Using the standard lift/drag model and a little calculus, it can be shown that this occurs right at the point where the induced drag force curve crosses the parasite drag force curve.
Figure 4.18 shows the amount of dissipation due to drag, for the various types of drag. Dissipation is a form of power, i.e. energy per unit time.
Dissipation is related to force by the simple rule:
power = force · velocity (4.4) |
In this equation, we are multiplying two vectors using the dot product (·),^{4} which means that only the velocity component in the direction of the force counts.
In the case of drag, we have specifically:
dissipation = force of drag · airspeed (4.5) |
The lowest point in the curve for total drag power corresponds to V_{y}, and gives the best rate of climb. Using the standard lift/drag model and a little calculus, it can be shown that at this speed, the minimum occurs right at the point where the induced drag power is 3/4ths of the total, and the parasite drag power is 1/4th of the total. Actually, in the airplane represented in these figures, V_{y} is so close to the stall speed that the standard lift/drag model is starting to break down, and the 3:1 ratio is not exactly accurate.
In the case of lift, the lift force is (by its definition) perpendicular to the relative wind, so there is no such thing as dissipation due to lift. (Of course the physical process that produces lift also produces induced drag, but the part of the force properly called lift isn’t the part that contributes to the power budget.)
There are several useful conclusions we can draw from these curves. For starters, we see that the curve of total power required to overcome dissipation has a familiar shape; it is just an upside-down version of the power curve that appears in section 1.2.5 and elsewhere throughout this book.
We can also see why the distinction between induced drag and parasite drag is significant to pilots:
In the high-speed regime (which includes normal cruise), the power required increases rapidly with increasing airspeed. Eventually it grows almost like the cube of the airspeed. The reason is easy to see: parasite drag is the dominant contribution to the coefficient of drag in this regime, and is more-or-less independent of airspeed.^{5} We pick up two factors of V from equation 4.1 and one from equation 4.4. Knowing this cube law is useful for figuring out the shape of your airplane’s power curve (section 7.6.2), and for figuring out how big an engine you need as a function of speed (section 7.6.4) and altitude (section 7.6.5).