Leonardo da Vinci's ornithopter design.

Leonardo da Vinci sketched various flow fields over objects in a flowing stream.

Benjamin Robins, the British mathematician, proved that air resistance was a critical factor in the flight of projectiles in 1746. His apparatus consisted of a whirling arm device in which weight (M) turned a drum and rotated the test object (P).

#### Early Developments in Aerodynamics

Aerodynamics, literally “air in motion,” is the branch of the larger field of fluid dynamics that deals with the motion of air and other gaseous fluids. It concerns the forces that these gaseous fluids, and particularly air, exert on bodies moving through it. Without the science of aerodynamics, modern flight would be impossible.

The word “aerodynamics” itself was not officially documented until 1837. However, the observation of fluids and their effect on objects can be traced back to the Greek philosopher Aristotle in 350 B.C. Aristotle conceived the notion that air has weight and observed that a body moving through a fluid encounters resistance.

Archimedes, another Greek philosopher, also has a place in the history of aerodynamics. A hundred years later, in 250 B.C., he presented his law of floating bodies that formed a basic principle of lighter-than-air vehicles. He stated that a fluid—either in a liquid or a gaseous form—is continuous, basically restating Aristotle's theory of a hundred years earlier. He comprehended that every point on the surface of a body immersed in a fluid was subject to some force due to the fluid. He stated that, in a fluid, “each part is always pressed by the whole weight of the column perpendicularly above it.” He observed that the pressure exerted on an object immersed in a fluid is directly proportional to its depth in the fluid. In other words, the deeper the object is in the fluid, the greater the pressure on it. Deep-sea divers, who have to accustom themselves to changes in pressure both on the way down into the sea and again on the way up to the surface, directly experience this phenomenon.

A direct proportional relationship means that if one part increases, the other will increase by the same factor. Physicists and mathematicians use the Greek letter alpha (µ) to denote such a relationship. Applied to pressure and depth, if the depth of an object is doubled, the pressure exerted on the object would double as well (Depth µ Pressure). The opposite would also be true. As altitude increases (negative depth), pressure decreases. Archimedes also demonstrated that, in order to set a stagnant fluid in motion, the pressure on the fluid must be increased or decreased. The resultant movement will take place in the direction of the decreasing pressure.

The next contribution to aerodynamics did not occur until the end of the 1400s. In 1490, the Italian painter, sculptor, and thinker Leonardo da Vinci began documenting his aerodynamic theories and ideas for flying machines in personal notebooks. An avid observer of birds and nature, he first believed that birds fly by flapping their wings, and thought that this motion would have to occur for manmade aircraft to rise. He later correctly concluded that the flapping of the wings created forward motion, and this forward motion allowed air to pass across the bird's wings to create lift. It was the movement of the wing relative to the air and the resulting reaction that produced the lift necessary to fly. As a result of his studies, he designed several ornithopters—machines that were intended to copy the action of a bird's wing with the muscle power being supplied by man. But these designs did not leave the drawing board. His other designs included those for the first helicopter and a parachute.

Leonardo noticed another phenomenon that would prove useful in the study of aerodynamics. He noticed that water in a river moved faster—at a greater velocity—where the river narrowed. In numerical terms, the area of a cross-section of a river multiplied by the velocity of the water flowing through that section equals the same number at any point in the river. This is known as the law of continuity (Area x Velocity = constant or AV = constant). The law of continuity demonstrates the conservation of mass, which is a fundamental principal in modern aerodynamics. He also observed the different ways in which a fluid flowed around an object—called a flow field.

Leonardo also stated that the aerodynamic results are the same if an object moves through the fluid at a given velocity or if the fluid flows past the object at rest at the same velocity. This became known as the “wind tunnel principal.” For example, the results are the same aerodynamically whether a runner moves at 10 miles per hour in calm air and if the wind is blowing at 10 miles per hour past a stationary person. He also determined that drag on an object is directly proportional to the area of the object. The greater the area of an object, the greater the drag. Further, Leonardo pointed out the benefits of streamlining as a way to reduce an object's drag.

However, Leonardo's notebooks were not discovered until centuries later, and his ideas remained unknown until the 19th century.

Scientists working in the 17th century contributed several theories relating to drag. The Italian mathematician and inventor Galileo Galilei built on Archimedes' work and discovered that the drag exerted on a body from a moving fluid is directly proportional to the density of the fluid. Density describes the mass of an object per unit volume. A very dense fluid produces more drag on objects passing through it than a less dense fluid. The density of air (a fluid) changes with its distance from the Earth's surface, becoming less dense the farther it is above the Earth's surface and, as such, exerting less pressure. Thus, an object passing through air high above the Earth's surface will encounter less drag than the same object passing through air close to the Earth's surface.

In 1673, the French scientist Edme Mariotte demonstrated that drag is proportional to the square of the velocity of an object (D µ V2). Dutch mathematician Christiaan Huygens had been testing this theory since 1669 and published his results with the same conclusion in 1690. The English scientist and mathematician Sir Isaac Newton presented a derivation of the drag equation of a body in 1687: Drag µ ρSV2 (where ρ is density and S is cross-sectional surface area of the body).

In 1738, the Dutch scientist Daniel Bernoulli published his findings on the relationship between pressure and velocity in flowing fluids. Other scientists used his research as a foundation for further research. The French scientist Jean le Rond d'Alembert, an associate of Bernoulli's, introduced a model for fluid flows and an equation for the principle of the conservation of mass. He further presented the idea that velocity and acceleration can vary between different points in fluid flow. (Remember that air is a fluid.)

Swiss mathematician Leonhard Euler, also an associate of Bernoulli, derived equations from Bernoulli's and d'Alembert's principles. The most famous of these became known as “Bernoulli's Principle.” It states that, in a flowing fluid, as velocity increases, pressure decreases. This became a key concept for understanding how lift is created. Euler also introduced equations for fluid flow, though at the time they could not be solved and applied.

Italian mathematician Joseph Lagrange and French mathematician Pierre-Simon Laplace studied Euler's findings and tried to solve his equations. In 1788, Lagrange introduced a new model for fluid flow as well as new equations for calculating velocity and pressure. In 1789, Laplace developed an equation that would help solve Euler's equations. It is still used in modern aerodynamics and physics. Laplace also successfully calculated the speed of sound.

In addition to these theoretical advancements, experiments in aerodynamics were also producing more practical results. In 1732, the French chemist Henri Pitot invented the Pitot tube, a device that enables the calculation of velocity at a point in a flowing fluid. This would help explain the behavior of fluid flow. The English engineer Benjamin Robins performed experiments in 1746 using a whirling arm device and a pendulum to measure drag at low and high speeds.

In 1759, the English engineer John Smeaton also used a whirling arm device to measure the drag exerted on a surface by moving air. He proposed the equation D = kSV2, where D is the drag, S is the surface area, V is the air velocity, and k is a constant, which Smeaton claimed was necessary in the equation. This constant became known as Smeaton's coefficient, and the value of this constant was debated for years. Those making the first attempts at flight, including the Wright brothers, used this coefficient. The French scientist Jean-Charles Borda published the results of his own whirling arm experiments in 1763. Borda verified and proposed modifications to current aerodynamic theories and was able to show the effect that the movement of one object had on another nearby object.

Sir George Cayley of England is generally recognized as the father of modern aerodynamics. He understood the basic forces acting on a wing and built a glider with a wing and a tail unit that flew successfully. He realized the importance of the wing angle of attack and that curved surfaces (camber) would produce more lift than flat ones. Stability in his designs came with the use of dihedral—an important concept still used today He first made public the notion that a fixed-wing aircraft was possible in 1804 in his major publication, “On Aerial Navigation,” which described the theoretical problems of flight.

The contributions of all of these thinkers, mathematicians, and scientists are part of the foundation of the science of aerodynamics. They paved the way for the aerodynamic developments that would occur during the nineteenth century, as well as for those who would eventually achieve heavier than air flight.

—Dan Johnston

References:

Anderson, Jr., John D. A History of Aerodynamics, Cambridge, England: Cambridge University Press, 1997.

Jakab, Peter L. Visions of a Flying Machine, Washington DC: Smithsonian Institution Press, 1990.

Smith, Hubert “Skip.” The Illustrated Guide to Aerodynamics. 2nd edition. Blue Ridge Summit, Penn.: Tab Books Inc., 1992.

Wegener, Peter P. What Makes Airplanes Fly? New York: Springer-Verlag, 1991.

On-line References:

Denker, John S., “See How It Flies,” http://www.av8n.com/how/htm/4forces.html

 Educational Organization Standard Designation (where applicable) Content of Standard National Council for Geographic Education Standard 1 How to use maps and other geographic representations to acquire and process information. National Council of Teachers of Mathematics N/A Students should understand meanings of operations and how they relate to one another. National Science Education Standards Content Standard B Students should develop an understanding of motions and forces.