Mathematics of flight

Summary: Aerodynamics is necessary to understanding the flight of objects through three-dimensional space and the forces acting upon them.

Human flight involves moving in a three-dimensional environment within the atmosphere in a stable, controlled way. Aerodynamics is the study of forces and the resulting motion of objects through air. It comes from Greek aerios, meaning “air,” and dynamis, meaning “force.” Mathematics is fundamental to understanding flight and in the design of different flying devices and machines, including kites, balloons, helicopters, and airplanes. From Orville and Wilbur Wright’s initial experiments with gliders at the beginning of the twentieth century, to the breaking of the sound barrier in the middle of the century, to the development of suborbital craft at the start of the twenty-first century, airplanes have been constructed in many different forms.

However, the ability to fly for all fixed-wing aircraft ultimately depends on a differential movement of air above and below the wings to generate positive lift. Control depends on three parameters, known as “pitch,” “yaw,” and “roll,” that are angles of rotation in three dimensions or axes about the plane’s center of mass. Mathematicians and others continue to study flight in order to more fully understand the mathematical and scientific principles that keep heavier-than-air craft in the air and to produce designs that are faster, safer, and more efficient. They also explore related issues in air travel, such as optimal strategies for loading passengers onto planes and the scheduling of aircraft flight crews.

Mathematical History

Stories from many cultures around the world suggest that humans have been interested in flight for thousands of years. There is evidence that the Chinese used kites well before the first century c.e.Leonardo da Vinci recorded his studies of flight in the fifteenth century with more than 100 drawings, including his theoretical ornithopter. Air is a fluid, and so much of the mathematics of flight science derives from fluid force studies, such as those performed by mathematician Daniel Bernoulli in the seventeenth century. Bernoulli’s principle is one foundation of flight mechanics.

Mathematical models for flight rely on the Navier-Stokes equations, named for mathematicians Claude-Louis Navier and George Stokes, which are fundamental partial differential equations describing fluid flow. They have many extensions. The Darcy–Weisbach equation, derived by dimensional analysis and named for engineer Henry Darcy and mathematician Julius Weisbach, is important to understanding the dissipation of energy because of friction, such as drag. Working in the early twentieth century, mathematician Otto Blumenthal studied the theory of complex functions, which he also applied to problems such as stress in airplane wings. Mathematician Selig Brodetsky studied equations of airplane motion, including three-dimensional phugoids, which are extensions of common, undesirable oscillatory motions where a plane pitches up and climbs, then pitches down and descends, with changes in airspeed. Peter Lax studied a class of nonlinear equations that can develop singularities, which have applications in aerodynamics that are related to phenomena like the shock waves that result from breaking the sound barrier.

Principles of Flight

Balloons are an example of lighter-than-air craft that use buoyancy to ascend and descend within the atmosphere, and hot air balloons are known to have been explored and used in the eighteenth century. There is also evidence that miniature hot air balloons were used in China for several centuries.

Heavier-than-air craft use the principle of lift to overcome gravity. There have been various mathematical and physical theories posed regarding how lift in airplane wings is accomplished. Aerodynamicists have analyzed how the motion of the air over an airplane wing creates circulation and differential pressure above and below the wing, which creates lift. Lifting forces on the airfoil are perpendicular to the motion of the lifting surface through the air and, in level flight, they counteract gravity. An observable example is the “sing” or hum that occurs in telephone wires in a steady wind, which is a repeating pattern of swirling vortices. This effect is because of the oscillations induced by a phenomenon called “vortex shedding,” which causes the wires to oscillate perpendicular to the wind flow.

Studies and models suggest that an airfoil produces circulation in a similar manner. Airfoils can be optimally designed to take advantage of this effect by allowing a smooth flow to develop over the surface of the airfoil, called “laminar flow.” The Reynolds number, named for mathematician Osborne Reynolds, quantifies laminar flow. Without laminar flow over an airfoil, turbulence is produced and vortex shedding occurs. Others suggest that aircraft lift is a Newtonian reaction force, named for Isaac Newton, coupled with the Coandă effect, named for engineer Henri Coandă, which is the tendency of a fluid to be attracted to a surface, like an airplane wing. The wing pushes the air down, so the air pushes the wing up.

Lift and Thrust

In general, a pilot taking off from the ground initially accelerates directly into oncoming wind whenever possible, since there is agreement based on observation and mathematics that relative forward motion of the plane’s wings with respect to the air is required for flight. Usually, the plane itself is in motion, though a strong wind over a stationary wing can also generate some lift. To maintain a steady, level flight path after takeoff, without any added acceleration, two mathematical relations must be maintained: thrust = drag and lift = weight. Early aircraft engines were powered by gasoline, similar to automobile combustion engines. A fundamental problem of weight, which inhibited lift, was solved by using aluminum as a construction material. Although oxygen is needed to burn gasoline, it is not carried by the aircraft but extracted from the atmosphere so that it does not add to the mass of the aircraft. Jet engines compress and discharge a fast-moving jet of air to generate thrust, using the same principles of fluid dynamics that govern other aspects of aircraft flight, according to Newton’s third law of motion. In contrast, a rocket must carry propellants, both fuel and oxidizer, and can thus fly outside of the atmosphere. The added force helps compensate for the extra weight.

Flight Speed

The types of speeds of flight are typically classified as slow subsonic flight, fast subsonic flight, trans-sonic flight, and supersonic flight. The Bell X-1 rocket-propelled airplane is credited as the first piloted aircraft in the world to break the sound barrier, under control of test pilot Charles Yeager. Other planes have been thought to have broken the sound barrier during steep dives, which many do not consider flight. The joint United Kingdom and France plane known as the Concorde, which flew from the 1970s until its retirement in 2003, was the only commercial supersonic aircraft. Commercial jets of the early twenty-first century typically achieve speeds in the range of 80% to 85% of the speed of sound, the slower end of trans-sonic flight.

The design speeds tend to avoid compressibility effects in air, which occur above roughly 80% of Mach 1. The Mach number is a ratio of the speed of the aircraft to the speed of sound at the aircraft’s altitude. Supersonic flight requires much more energy to sustain, and generally only military aircraft conduct sustained supersonic flight within the atmosphere. The Prandtl-Glauert equation, named for scientists Ludwig Prandtl and Hermann Glauert, is used to help correct computations of fluid flow at high speeds a function of compressibility, while the Prandtl-Glauert singularity is observed as a visible cloud of vapor that results from air pressure changes around a trans-sonic airplane. The pressures can be modeled as an N-wave, named because a plot of pressure versus time resembles the letter N.

A mode of atmospheric flight explored with experimental aircraft at the beginning of the twenty-first century is hypersonic flight, which starts at speeds approximately 5–10 times the speed of sound. Special engines must be developed to make this speed possible. Previously, the Lockheed Aircraft SR-71 held the speed record at greater than Mach 3. It was powered by a special fuel and was air breathing. In 1974, the SR-71 set a speed record flying across the Atlantic from Beale Air Force Base in Louisiana to London in less than two hours. This flight occurred many decades after aviator Beryl Markham’s speculations about flying the Atlantic in an hour. Hypersonic aircraft flying at speeds greater than Mach 5 likely will be powered by different forms of air breathing propulsion systems, such as turbine-free engines known as “scramjets,” which at very high speeds use ram air compression to ignite a fuel in the engine. In principle, such designs have the capability of going at very high speeds at high altitude and form a transition to spaceflight.

Bibliography

Anderson, David, and Scott Eberhardt. Understanding Flight. 2nd ed. New York: McGraw-Hill, 2009.

Tennekes, Henk. The Simple Science of Flight: From Insects to Jumbo Jets. Revised and expanded ed. Cambridge, MA: MIT Press, 2009.