One and One and One Make Four

I'm still reviewing my aerodynamics, paying far more attention to the jet performance side than I ever did before. I'm going to explain some background today. I will skip some parts though, so don't panic if I don't explain everything. If you want to know everything, you'll have to buy the textbook yourself.

An airplane in flight has three forces acting on it: weight, acting down as a result of gravity; thrust, acting forward in the direction the engines are pointing, as a result of the engines working; and the aerodynamic force, acting kinda up and kinda back, as a result of air whacking against the airframe. People who write aerodynamics textbooks like to be more precise than "kinda up and and kind back" but pause for a moment to envision a whole lot of air whacking against an airplane, and realize that reality isn't about neat lines. Aerodynamicists are clever, though, so they've defined all the air-whacking forces that act parallel to the direction of the onrushing air before the airplane disturbed it (equivalent to the direction of travel of the airplane) as drag and all the air-whacking forces that act perpendicular to that onrushing air as lift. This is easily achieved using a wand of trigonometry. That way they can draw a diagram of the airplane being acted upon by four forces: weight, thrust, lift and drag, depicted as nice straight lines. If you type "four forces airplane" (feel free to use your nationally approved spelling) into Google and select the image search, you will find hundreds of diagrams like this one. One is legally required to begin any discussion of aerodynamics with such a diagram.

In level flight, thrust counteracts drag. (At cruise speeds, it's generally considered okay to believe that all the thrust acts forward, and ignore the fact that the thrust line may be angled a little above the flight path.) Therefore the thrust required to maintain level flight at any given airspeed is equal to the total drag at that airspeed. Extra thrust over and above that matching the drag is used to accelerate or climb.

Drag is affected by all kinds of things: the airspeed (v), the air density (ρ), and the wing area (S), the number of bugs smashed on the windshield, the size and shape of the airplane, and the way the wings are positioned with respect to the onrushing air. Total drag is the sum of three sorts of drag. One is parasite drag, from air that impinges on, slides past, and swirls around the airplane, impeding its forward motion. You might guess that the faster the airplane is going, the more the air impedes forward motion, and you would be right. Parasite drag is proportional to the square of the airspeed. Another sort of drag, induced drag, results from downwash from the wingtip vortices changing the direction of the aerodynamic force, so that less of the force is lift and more is drag. The slower the airplane is going, the more the wingtip vortices interfere with the airflow, so induced drag is inversely proportional to the square of the airspeed. Plot both those sorts of drag against airspeed, on the same graph, and the sum of the two curves is vaguly U-shaped. The low-speed end of the U is capped at the drag, mostly induced, corresponding to the lowest speed the airplane can achieve without falling out of the sky. That means that at very low speeds, drag is high, because of the contribution of induced drag. Drag reaches a minimum at an intermediate speed where induced drag is reduced, but parasite drag has not yet become large. The high-speed end of the graph swoops upwards, showing that the parasite drag becomes large at high speeds. Add in the the third form, wave drag, attributable to compressibility effects as flight approaches the speed of sound, and the top end of the graph bends noticeably upwards. Before reading this book I would have said there were two sorts of drag, because the airplanes I fly have never even reached half the speed of sound, leaving compressibility effects negligible. If other factors are held constant, drag increases proportionate to air density or wing area.

Aerodynamicists have devised an equation that summarizes this: Drag = 1/2 C_D ρ S v². The size and shape of the airplane, and the number of smashed bugs are all incorporated into the coefficient of drag, C_D. If you're paying attention, you might be complaining that that above equation does not take into account induced or wave drag. But you see it does, because the value of C_D for any particular airplane changes with angle of attack. At low airspeeds (high angles of attack), C_D is large, and at high airspeeds it is much less, but its rate of decrease slows as airspeed approaches the speed of sound, so that the result of the equation matches the actual drag experienced by the airplane. In other words, C_D is a fudge factor. It's like the Russian judge in figure skating, adjusting everyone's marks so that the final scores come out the way she wants them. A constant of proportionality that is not constant may seem supremely useless, but trust me for a few minutes, while I produce a similarly dubious equation describing lift.

Seeing as lift and drag are perpendicularly resolved components of the same aerodynamic force, it stands to reason that they would be described by the same equations. Thus Lift = 1/2 C_L ρ S v². The only difference is that C_L, the coefficient of lift, replaces C_D. Naturally the coefficient of lift varies with angle of attack, too: it is low at high airspeeds, corresponding to small angles of attack, increases with angle of attack, up to the critical angle, and then falls off again. Swept wing aircraft, like the typical passenger jet, have a much gentler C_L curve than straight wing aircraft: the value does not increase as rapidly with AoA, and it falls off much more gradually, with no definable critical angle.

Now for the payoff to all these equations and fudge factors. In level flight, lift is equal to weight, and as weight is constant, lift must be constant across the speed range of the graph described above. But we know what the weight of the airplane is, so, despite all the fudge factors, we can know how much lift is being generated.

Next take the equation of lift and divide it by the equation of drag. (Just like high school: write one over top the other and cross out everything that is the same on the top and the bottom.) You discover that L/D = C_L/C_D. Everything else cancels. The ratio of the whacky coefficients is the same as the ratio of the lift to drag itself. Why is this a payoff? Because lift is what is holding us up ("good") and drag is what is holding us back ("bad"), we want to maximize lift while minimizing drag. The ratio of lift to drag becomes a maximum at the speed where drag is a minimum. This is where the induced and parasite drag curves intersect, and is known as (L/D)_max (I say "ell dee max"). The speed, and more precisely the angle of attack that produces the speed corresponding to (L/D)_max is very important to understanding aircraft performance. And it's one of the places where the jet and the propeller case diverge.

So tomorrow I can get to the good stuff.

[Little edit, in response to a comment: in both places where I equate two things "in level flight," I mean level, unaccelerated flight. No unbalanced forces.]