### Ludwig Prandtl

**Ludwig Prandtl (1875-1953)**

Born in Freising, Germany, Ludwig Prandtl started his life with much engineering exposure since his mother died early in his life; as a result he was always with his father, who was an engineering professor. Encouraged by his father and his surroundings, he was very critical and curious about nature, striving hard to learn from his observations in the real world. He attended the University of Munich in 1894, graduating in 1900 with a Ph.D. From then on, his life was nothing short of great accomplishment.

At this time, fluid mechanics was understood only to the point of describing mathematical "potential flows" - inviscid models of flow fields that did little to describe the full reality of what was going on. Potential flow also "proved" D'Alembert's Paradox, which stated that an inviscid flowfield generated no net force on the object in the field, particularly drag. This is obviously incorrect, as everyone knew that fluids generated friction on a body that is immersed in said fluid, but only the Navier-Stokes equations could really solve this kind of problem. Unfortunately, there were very few closed-form solutions to the Navier-Stokes equations at this time, and those that were discovered were only for extreme special cases.

Prandtl's 1904 paper "Fluid Flow in Very Little Friction" described a very important theory that no one had thought of to that point - how to incorporate friction (viscous effects) into a fluid flow regime while still allowing potential flowfield solutions to work with aerodynamic theory. This theory, known as Boundary Layer Theory, says that only in a very thin region around the immersed object is there the presence of a boundary layer which possesses viscous effects - everywhere outside this field, the flowfield can be assumed inviscid with reasonable accuracy. Boundary Layer Theory opened the way to explaining many physical phenomena including surface drag, pressure drag, flow separation, and stall.

After this, most of Prandtl's work was done at the University of Gottingen. In 1908, he worked with his student Theodor Meyer to describe the concept of expansion waves, which are now named in their honor. This allowed for additional analytical work in the field of supersonic flowfields. In the 1910's, he worked to create mathematical models to describe airfoil (infinite wing) and finite wing theory that incorporated viscous effects (through means of the Kutta condition) to effectively provide analytical solutions. Known as Thin Airfoil Theory and Lifting Line Theory, these two powerful mathematical conclusions allowed for analytical solutions of lift, angles of attack, aerodynamic moments, induced drag, and other very important aerodynamic values that were necessary in the design of airfoils and wings. These theories were published in 1918-1919, and are still extremely useful to this day.

Prandtl's work in aerodynamics helped push Germany at the forefront of aerospace engineering, and his and his students contributions have been extremely important for the development of aerodynamics. For this reason, Prandtl has been deemed the father of modern aerodynamics. A nondimensional coefficient which describes the ratio of viscosity to thermal diffusivity has been named the Prandtl number in his honor.

**Resources**

http://www.fluidmech.net/msc/prandtl.htm

http://www.centennialofflight.gov/essay/Theories_of_Flight/Prandtl/TH10.htm

http://www.allstar.fiu.edu/aero/prandtl.htm

Modern Compressible Flow (book) by John D. Anderson

Fundamentals of Aerodynamics (book) by John D. Anderson

**Pictures of Prandtl**

http://rclsgi.eng.ohio-state.edu/courses/Korpela/images/htransfer/prandtl.gif

http://www.mlahanas.de/Physics/Bios/images/LudwigPrandtl40s.jpg

**Pictures of Prandtl's Theories**

Boundary Layer: http://www.centennialofflight.gov/essay/Theories_of_Flight/Prandtl/TH10G1.jpg

Thin-Airfoil Theory:

http://www.aeromech.usyd.edu.au/aero/liftline/llt1.gif

Prandtl-Meyer Expansion Waves:

http://www.aeromech.usyd.edu.au/aero/gasdyn/pm1.gif

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