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Boundary-Layer Theory
An Italian translation of Ludwig Prandtl’s seminal paper Über Flüssigkeitsbewegung bei sehr kleiner Reibung is offered to engineering students in stratolimite212. The original text and photos are taken from Vier Abhandlungen zur Hydrodynamik und Aerodynamik by L. Prandtl and A. Betz, Göttingen, 1927, while figures belong to an (inaccurate) English translation by NASA, published in 1928 as report TM-452.
Rarely a technical approach can fill a theoretical gap for a century! While the approach was clearly a success, the gap reflected the fact that Euler equation was unable to “attach” an inviscid fluid to a boundary. This no-slip condition was ignored since it would force the applied mathematicians of the time to abandon the glorious irrotational motion. Lamb’s Hydrodynamics is the proud chronicle of this irrotational “drunk”. Preferences of applied mathematicians apart, the gap was quite severe. Despite Euler insights, Newton’s second law did not apply to fluids with boundaries.
In 1904 Ludwig Prandtl was an obscure professor (background in solid mechanics), but duly aware of this theoretical issue. He submitted the above paper to a congress of mathematicians, probably to get their attention. At the same time (he might be felt himself unable to solve it) he turned to the Navier-Stokes equation, i.e. the Euler equation plus a viscous term containing velocity second derivatives that could provide the required no-slip condition at the boundaries. Much later, in a lecture delivered to students at Göttingen University Prandtl made a straightforward example that might give the cue to his approach. He showed that the solution of a second degree differential equation (damped oscillations of a point-mass) fails to satisfy an arbitrary initial condition when the coefficient (the mass of the point) of the highest derivative (second degree) is made to vanish.
The limit of Prandtl’s approach rests on its technicality, on its lack of theoretical breath. The bad result of this successful approach is that the scientific community is still unable to clarify the origin of the lift which keeps a flying object in the air. More specifically, we learned thank to Prandtl’s approach how to let an object flying, but we do not know yet why it is so. Both Euler and Navier-Stokes equations determine the pressure once the velocity field is known. Unfortunately, the viscous term in the Navier-Stokes equation, raising the degree of the Euler’s one, gives the false hint that this order can be also reversed: apparently, Prandtl applied the same “kind” of equation to determine both the pressure (by Euler equation) and the velocity field (by Navier-Stokes equation).
To fill the gap, a further independent equation for the (non-irrotational) velocity field is required, beyond the continuity equation which must be certainly satisfied, but sets very little constraints to the possible motions.
