Page 51 - 48Fundamentals of Compressible Fluid Mechanics

1.3. HISTORICAL BACKGROUND 13

Euler equations) were considered unsolvable during the mid 18xx because of the
high complexity. This problem led to two consequences. Theoreticians tried to
simplify the equations and arrive at approximate solutions representing specific
cases. Examples of such work are Hermann von Helmholtz’s concept of vortex
filaments (1858), Lanchester’s concept of circulatory flow (1894), and the Kutta-
Joukowski circulation theory of lift (1906). Practitioners like the Wright brothers
relied upon experimentation to figure out what theory could not yet tell them.
Ludwig Prandtl in 1904 explained the two most important causes of drag
by introducing the boundary layer theory. Prandtl’s boundary layer theory allowed
various simplifications of the Navier-Stokes equations. Prandtl worked on calculat-
ing the effect of induced drag on lift. He introduced the lifting line theory, which was
published in 1918-1919 and enabled accurate calculations of induced drag and its
42
effect on lift .
During World War I, Prandtl created his thin–airfoil theory that enabled
the calculation of lift for thin, cambered airfoils. He later contributed to the Prandtl-
Glauert rule for subsonic airflow that describes the compressibility effects of air at
high speeds. Prandtl’s student, Von Karman reduced the equations for supersonic
flow into a single equation.
After First World War aviation became important and in 1920s a push of
research focused on what was called the compressibility problem. Airplanes could
not yet fly fast, but the propellers (which are also airfoils) did exceed the speed of
sound, especially at the propeller tips, thus exhibiting inefficiency. Frank Caldwell
and Elisha Fales demonstrated in 1918 that at a critical speed (later renamed the
critical Mach number) airfoils suffered dramatic increases in drag and decreases
in lift. Later, Briggs and Dryden showed that the problem was related to the shock
wave. Meanwhile in Germany, one of Prandtl’s assistants, J. Ackeret, simplified
the shock equations so that they became easy to use. After World War Two, the
research had continued and some technical solutions were found. Some of the
solutions lead to tedious calculations which lead to the creation of Computational
Fluid Dynamics (CFD). Today these methods of perturbations and asymptotic are
43
44
hardly used in wing calculations . That is the “dinosaur ” reason that even today
some instructors are teaching mostly the perturbations and asymptotic methods in
Gas Dynamics classes.
More information on external flow can be found in , John D. Anderson’s
Book “History of Aerodynamics and Its Impact on Flying Machines,” Cambridge
University Press, 1997
42 The English call this theory the Lanchester-Prandtl theory. This is because the English Astronomer
Frederick Lanchester published the foundation for Prandtl’s theory in his 1907 book Aerodynamics.
However, Prandtl claimed that he was not aware of Lanchester’s model when he had begun his work
in 1911. This claim seems reasonable in the light that Prandtl was not ware of earlier works when he
named erroneously the conditions for the shock wave. See for the full story in the shock section.
43 This undersigned is aware of only one case that these methods were really used to calculations of
wing.
44 It is like teaching using slide ruler in today school. By the way, slide rule is sold for about 7.5$ on
the net. Yet, there is no reason to teach it in a regular school.