Streamlining and Drag

    [Contrary to intuition, the shape of the front is less critical. Perhaps our intuition owes too much to the bows of ships, which need a sharp edge to slice through surface waves. Deep-sea nuclear submarines have blunt spherical fronts, just as airships.do.]

    The proportionality of the drag of a streamlined wing to v² was found from observations, but it can also be supported by the rough ("hand-waving") argument below. Follow it slowly!.

    A wing of an airplace facing an airstream loses energy mostly by shoving aside air whose space it proceeds to occupy. If the pattern of flow lines around the wing at low and high speeds is the same (reasonably true), when v doubles, the velocity of air shoved aside is also doubled, and from this process alone, the (kinetic) energy imparted to it, proportional to mv², should increase 4 times.

   How about m? With the doubled velocity, the wing advances twice the distance as before each second, so the mass of air m shoved out of the way also doubles. The overall rate at which energy is imparted by the wing to the surrounding air, therefore, increases 8 times.

    That rate should match the mechanical work done each second by the force D (i.e. the power required): it, too, should increase 8-fold. Since the distance covered per second is v, the work done each second is Dv. If v is doubled and Dv increases 8-fold, then D must grow 4 times --a growth proportional to v².

Lift

The Wright brothers used a wind tunnel to test models of different wing profiles, identifying cross sections suited for various types of flying. They also found by observations that the lift force produced by a wing was roughly proportional to the density d of the air and to the square of the velocity v of the airflow over it:

    One can increase the lift by increasing the angle of attack (as one does with a kite), but at the price of a much greater drag. Furthermore, if the angle is too steep, the orderly flow above the wind is disrupted and the wing "stalls, " suddenly losing much of its lift. Many airplane crashes have been traced to sudden stalls.

How high, how fast?

    How high and how fast should the airliner fly? Passengers want to reach their destination quickly, so designers aim at a high "cruising speed." However, passengers also value safe landings, and hence the landing speed should be slow.

    Speed is the main reason why airliners fly at an altitude above 30,000 feet or close to 10 kilometers. Air density decreases to about 1/2 with each added 5 km of altitude, so at 10 km, d is about 1/4 its sea-level value and an airplane can double its speed to generate the same lift, with the same drag D (which, as shown, also grows like dv²). The main reason airliners have pressurized cabins is to allow them to fly higher, in order to fly faster.

    How fast? The practical limit seems to be around 600 mph (960 km/hour). Any closer to the speed of sound (1200 km/h = 746 mph, varies with temperature) and the air flow above the wing creates shock fronts which increase drag and reduce lift. Even to get that far, swept-back wings are needed.

Landing Safely

    The practical solution is to increase the angle of attack during landings, and to temporarily increase the size of the wing. If you ever sat near the window of a landing airliner, you may have noted extra wing surfaces sliding out during the final approach, to provide more lift--and if you watched such an airliner from the ground, you would see that these surfaces extend downwards from the wing, at a steeper angle. All that of course makes drag much larger, but a landing airplane must anyway get rid of its extra speed.

    All these allow the airliner to land at about 150 mph. Landing is actually a precision maneuver, in which the airliner (ideally) runs out of airspeed just before its wheels touch the ground. Glide slope radars and other navigation aids make it possible, thousands of times each day.

Non-stop around the world--How high? How fast?

    Initially the hope was to have a pressurized cabin and to fly at 25,000 ft., but weight limitations precluded that, so that a slower, lower flight was undertaken, lasting 9 days. The take-off weight of "Voyager" was 9700 pounds, and to lift such a heavy airplane, it used two engines--one pushing and one pulling--to reach the required airspeed of 138 mph.

    Halfway through the flight, as fuel was used up, less lift was needed. Therefore one engine was shut off, airspeed was allowed to drop to 79 mph, and to avoid reducing it even more, the flight altitude was raised to 11-12,000 feet. As a result the second half of the flight was much slower than the first, and much harder on the sleep-deprived pilots. They had, however, no other choice--flying faster would have required a less efficient wing angle, and would have wasted too much fuel.

Note:     The story of the non-stop flight around the world is told in "Voyager" by Jeana Yaeger, Dick Rutan and Phil Patton, viii + 337 pp, Alfred A Knopf, New York, 1987. Note: in 2005 Steve Fossett flew solo around the world, nonstop, on the jet-propelled Virgin Atlantic Global Flyer. The flight took less than 3 days because his cabin was pressurized and the airplane could reach around 340 mph (plus tailwind) at altitudes near 40,000 ft.