D'Alembert's paradox

D'Alembert's paradox is a contradiction reached by French mathematician Jean le Rond d'Alembert in 1752 [1] using inviscid theory in the form of potential solutions of the incompressible Euler equations, to prove that the drag of a body of any shape moving through an inviscid fluid is zero. This result was in direct contradiction to an abundance of evidence of substantial drag in fluids of very small viscosity (high Reynolds number) such as air and water. Thus, from the start, mathematical fluid mechanics was discredited by engineers, which resulted in an unfortunate split between the field of hydraulics, observing phenomena which could not be explained, and theoretical fluid mechanics explaining phenomena which could not be observed (in the words of the Chemistry Nobel Laureate Sir Cyril Hinshelwood).

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Solutions to the paradox

The paradox is considered within the fluid mechanics community to have been resolved by the German physicist Ludwig Prandtl in 1904 who in the short report Motion of fluids with very little viscosity [2], suggested that the effects of a thin viscous boundary layer possibly could be the source of substantial drag. A reading of the report shows that Prandtl does not claim to have solved the paradox and that evidence to this effect is missing. In fact, it seems difficult to find original research claiming to resolve the paradox. What can be found is second hand information suggesting that a no slip boundary condition causes a retardation (tripping) of the flow near the boundary, which possibly may lead to generation of transversal vorticity and separation of the flow with a large attached wake. Evidence that this actually occurs in fluids with very small viscosity is missing in Prandtl's report and elsewhere. The nature of the resolution attributed to Prandtl in the form of a vanishingly small cause (vanishingly small viscosity) having a large effect (substantial drag), makes the resolution to the paradox difficult, or even impossible, to either verify or disprove by theory, computation or experiment. This is illustrated by Stewartson in the long 1981 survey article [3]: "...great efforts have been made during the last hundred or so years to explain how a vanishingly small frictional force can have a significant effect on the flow properties.". However, Stewartson does not present any clear resolution of the paradox and in the summary he states that: "Much remains to be done; in particular, the development of a rational theory for three-dimensional, and possibly also unsteady two-dimensional flows may be in its infancy...".

Recently the following alternative resolution of d'Alembert's paradox was presented [4]: The reason the zero drag potential solution of the Euler equations is not observed in experiments, is that this solution is (exponentially) unstable at separation, and develops into a turbulent Euler solution (with a slip boundary condition and thus no boundary layer prior to separation) with the drag arising from low-pressure tubes of streamwise vorticity generated at separation. This is a completely different resolution from that by Prandtl and it is supported by both theory, computation and experiment.

The mathematician Garret Birkhoff in the opening chapter of his book Hydrodynamics from 1950 [6], addresses a number of paradoxes of fluid mechanics (including d'Alembert's paradox) and expresses a clear doubt in their official resolutions: "...I think that to attribute them all to the neglect of viscosity is an unwarranted oversimplification The root lies deeper, in lack of precisely that deductive rigor whose importance is so commonly minimized by physicists and engineers...". In particular, on d'Alembert's paradox, along the lines of [4], he critizises the lack of stability analysis of potential solutions: "the concept of a "steady flow" is inconclusive; there is no rigourus justification for the elimination of time as an independent variable. Thus though Dirichlet flows (potential solutions) and other steady flows are mathematically possible, there is no reason to suppose that any steady flow is stable...".

In his 1951 review [7] of Birkhoff's book, the mathematician James J. Stoker sharply critizises the first chapter of the book: "The reviewer found it difficult to understand for what class of readers the first chapter was written. For readers that are acquainted with hydrodynamics the majority of the cases cited as paradoxes belong either to the category of mistakes long since rectified, or in the category of discrepancies between theory and experiments the reasons for which are also are well understood. On the other hand, the unitiated would be very likely to get the wrong ideas about some of the important and useful achievements in hydrodynamics from reading this chapter.". Assuming that "the majority of the cases" includes d'Alembert's paradox, this standpoint of a long since well understood resolution appears to be contradicted by the review article [3] by Stewartson 30 years later.

The official standpoint of the fluid mechanics community seems to be that the paradox in principle can been solved along the lines suggested by Prandtl, even if concrete evidence is still to be provided, and the new resolution is (very) controversial.

Boundary condition: slip or no-slip?

Experiments show that the skin friction from a turbulent boundary layer decreases towards zero as Re^-0.2 as the Reynolds number Re increases [5]. This indicates that for large Reynolds numbers (small viscosity) a slip boundary condition (or more generally a friction boundary condition with small friction), is a better model than a no-slip boundary condition. Computational simulation [4] of drag crisis supports this approach, which opens entirely new possibilities for simulation of high Reynolds number flow without resolving very thin boundary layers. This is contrary to Prandtl's claim that even for very high Reynolds numbers thin boundary layers need to be resolved (which is impossible) to get a correct drag. This indicates that a correct resolution of d´Alembert´s paradox can have important consequences also in applications.

References

[1] Jean Le Rond dAlembert, Essai d'une nouvelle the'orie de la re'sistance des fluides, 1752.

[2] Ludwig Prandtl, Motion of fluids with very little viscosity, NACA Technical Memorandum 452, 1904.

[3] Keith Stewartson, D'Alembert's Paradox, Siam Review, Vol 23(3), pp. 308-343, 1981.

[4] Johan Hoffman and Claes Johnson, Computational Turbulent Incompressible Flow, Springer, 2007.

[5] Herrman Schlichting, Boundary layer theory, McGraw Hill, 1979.

[6] Garret Birkhoff, Hydrodynamics: a study in logic, fact, and similtude, Princeton University Press, 1950.

[7] James J. Stoker, Review: Garrett Birkhoff, Hydrodynamics, a study in logic, fact, and similitude, Bull. Amer. Math. Soc. Vol. 57(6), 1951, pp. 497-499.