onsdag 16 oktober 2019

Paradoxes of Fluid Mechanics

The book Hydrodynamics A study in logic, fact and similitude (1950) by Garrett Birkhoff gives a long list of paradoxes of fluid mechanics including the following concerning incompressible flow:
  1. D'Alembert's paradox (zero drag of (potential) inviscid flow). 
  2. Reversibility paradox (reversion of flow direction does not reverse flow).
  3. Fatness paradox of Kutta-Joukowsky theory (lift decreases with thickness of wing).
  4. Magnus effect (lift of backspin tennis ball opposite to that of table tennis ball).
  5. Eiffel paradox ("drag crisis" as sudden drop of drag). 
  6. Dubaut paradox (smaller drag of stationary pole in streaming fluid than pole moving through stationary fluid).  
Birkhoff describes the role of paradoxes:
  • These paradoxes have been the subject of many witticisms. Thus, it has recently been said  that in the nineteenth century, " fluid dynamicists were divided into hydraulic engineers who observed what could not be ex­plained, and mathematicians who explained things that could not be observed." (It is my impression that many survivors of both species are still with us.)—Again, Sydney Goldstein has observed that one can read all of Lamb without realizing that water is wet!  
  • I think we should welcome the discovery of hydrodynamical paradoxes—recognizing frankly the inadequacy of existing mathematics (and logic) to analyze the complex wonders of Nature. Experience shows that man's imagination is far more limited than Nature's resources: as Pascal wrote, "l'imagination se lassera plutot de concevoir que la nature de fournir.
Solving paradoxes thus may open roads to progress. The mother of all paradoxes of incompressible flow is d'Alembert's paradox. Prandtl was crowned Father of Modern Fluid Mechanics because he saved the face of fluid mechanics confronted by the paradox, by coming to rescue in a short 1904 article selling the idea that drag somehow is an effect of a boundary layer caused by an imagined necessity of a no-slip boundary condition forcing a fluid to "stick" to a solid wall. However, Prandtl's boundary layer theory came with a serious side effect as the impossibility to computational resolve thin boundary which has paralysed Computational Fluid Dynamics CFD throughout the 20th century.
The paralysis was lifted only in 2008 with a new resolution of d'Alembert's paradox (check video) showing that inviscid flow modeled by the Euler equations can be described as potential flow modified by 3d rotational slip separation into turbulent flow, which can be resolved computationally by Direct Finite Element Simulation DFS. This is described in Computational Turbulent Incompressible Flow with subsequent elaborations, resolving all the paradoxes listed by Birkhoff, and more as shown in the previous post with references including revealing The Secret of Flight (check video).

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