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# Ludwig Prandtl, A great mechanician!

Ludwig Prandtl's contributions to fluid mechanics include his development of lifting line theory (to describe the lift and drag of wings of finite span), his work in turbulence, and his experimental and theoretical studies of gasdynamics. Prandtl was trained as a solid mechanist and continued to contribute to solid mechanics throughout most of his career. However, his discovery of the boundary layer is regarded as one of the most important breakthroughs in fluid mechanics of all time and has earned Prandtl the title of Father of Modern Fluid Mechanics.

Before Prandtl's description of the boundary layer in 1904, there was no lack of interest in the dynamics of fluids due to the practical problems of nautical engineering, ballistics, and hydraulics. Throughout the 18th and 19th century the top physicists and mathematicians of Europe examined flows from a mathematical point of view. Much of this work was to construct potential flows, i.e., incompressible, irrotational flows, over bodies. Examples recognizable to most undergraduates are flows over circular cylinders and other flows involving source-sink superpositions. Although the mathematics was elegant and the flows aesthetically pleasing, it was recognized that such flows failed to mimic "real" flows seen in nature. Furthermore, it was known since the time of d'Alembert that potential flows frequently resulted in zero drag; a prediction in clear contradiction with everyday experience!

What were these mathematicians to do? Thanks to Coulomb and Stokes, they were aware that a no-slip condition should be applied at solid bodies (we now realize that this condition holds at all fluid boundaries). However, standard external flow problems are ill-posed when the potential flow equations are combined with the no-slip condition. The correct approach would be to abandon the inviscid (small viscosity) approximation and solve the full Navier-Stokes equations. Stokes had done this himself for the problem of creeping flow around a sphere and derived a non-zero expression for the drag. However, the Stokes flow does not generate the large scale separation seen in most day-to-day flows and the predicted drag is always much less than what is measured for things like cannon balls and marbles in air and water. The reason for these discrepencies is the neglect of the fluid inertia in the creeping flow approximation. To include these terms is a daunting task, even today.

Thus, as the 19th century came to a close, a universal and practical application of fluid mechanics seemed far off. Prandtl's contribution was to realize that we can view the flow as being divided into two regions. The bulk of the flow can be regarded as a potential flow essentially the same as that studied by the mathematicians. Only in a small region near the body do viscous effects dominate. This thin layer is known as the boundary layer. Conceptually, Prandtl's boundary layer is the reason the potential flow theory is compatible with the exact physics. Furthermore, certain details of the structure of the boundary layer are the key to understanding both flow separation and the physical mechanism behind the Kutta condition. That is, a proper understanding of the boundary layer allows us to understand how a (vanishingly) small viscosity and a (vanishingly) small viscous region can modify the global flow features. Thus, with one insight Prandtl resolved d'Alembert's Paradox and provided fluid mechanists with the physics of both lift and form drag.

We should also note that the boundary layer is the region where the solid interacts mechanically and thermally with the surrounding flow. A practical spin-off of Prandtl's recognition of the boundary layer is the understanding of the mechanisms of skin friction and heat transfer.

Prandtl's place in history would have been secure with the discovery of the boundary layer alone. However, he also developed lifting line theory (along with Lanchester) which revealed the third source of subsonic drag, i.e., induced drag. This work, along with that of Max Munk on airfoil theory, gave engineers reasonably efficient tools to understand and improve the earliest aircraft in the pre-computer age. Without Prandtl's insights, the progress of manned flight would have been slowed considerably.

Amazingly, there are very few discussions of Prandtl on the web. He is regarded as an engineer and never seems to make it into lists of physicists.

## Indeed very few online

Indeed very few online discussion. If one wants to see Ludwig working in this water channel, one option is a site of DLR, mentioned once through in a cross-post:

http://alexl.wordpress.com/2007/03/21/dlr-site-with-original-photos-of-prandtl-and-sketches-from-his-work/

Panta Rhei - everything flows