The Trauma of Paradoxes of Modern Physics

• How wonderful that we have met with a paradox. Now we have some hope of making progress. (Niels Bohr)

Science appears to be filled with paradoxes, which it itself is a paradox, because true science should be free of paradox. For a true scientist a scientific paradox is thus something unbearable, which requires immediate action, because one paradox is enough to kill a whole theory.

There are logical paradoxes as contradiction between words and there are physical paradoxes as contradictions between theory and observation.

One logical paradox is enough to kill a mathematical theory. Thus Russell's paradox killed set theory as the foundation of mathematics in the early 20th century.

Zeno's paradox (still unresolved) of the arrow which is moving although it is not moving at every instant, triggered the development of Calculus, but with a delay of 2000 years!

One physical paradox is enough to kill a physical theory as a mathematical theory about phenomena of physics, all according to the famous physicist Feynman:

• It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong.

If theory does not at all fit with reality, then something is fundamentally wrong with the theory, not the other way around.

A paradox may be thus devastating to existing theory, while leading to new better theory by focussing on weak points.

Yet, the list of physical paradoxes has remained through the development of modern physics and in fact have multiplied since modern physics is loaded with many more paradoxes than classical rational physics, as if modern physics is irrational. Thus the pillars of modern physics in the form of relativity theory and quantum mechanics are both filled with paradoxes, which have remained unresolved for 100 years. This has formed the deep trauma of modern physics with no escape from ever more paradoxes.

Niels Bohr was a master of handling the many paradoxes of quantum mechanics lifting sophistry to a new level with his "complementarity principle" addressing the wave-particle contradiction with murky statements like:

• The opposite of a correct statement is a false statement. But the opposite of a profound truth may well be another profound truth.

While classical physicists had to come to grips with paradoxes, in one way or the other, modern physicists appear to welcome paradoxes as a sign of deep magical physics as opposed to shallow understandable classical physics.

The first defence line for a classical physicist is to simply deny the existence of a paradox formulated by some renegades. The next is to accept that there is indeed a paradox and then come up with an ad hoc explanation for the contradiction between theory and reality, showing that the contradiction is in fact only apparent, but not really real. If the ad hoc explanation is refuted, a new ad hoc explanation is presented and so on.

For a modern physicist, a paradox thus poses no real problem, but of course it is some kind of nuisance and so occasionally may get some attention. Like the Twin Paradox of special relativity discussed in earlier posts, unresolved since 100 years.

The prime paradox of fluid mechanics is d'Alembert's paradox comparing the prediction of zero resistance to motion through a fluid from potential flow solutions to Euler's equations of slightly viscous flow like air and water, with the observation of heavy resistance increasing quadratically with velocity.

The paradox was formulated by d'Alembert in 1755 but nobody was able to come up with a resolution until the young German fluid mechanician Ludwig Prandtl in 1904 came up with the ad hoc solution to discriminate the zero drag potential solution of Euler's equations because potential flow does not satisfy a no-slip boundary condition coming with a thin boundary layer. With the potential solution thus eliminated form the discussion, the paradox simply disappeared. But the act of discrimination of solutions of the Euler equations of course was not so glorious. Discrimination of prefect exact solutions on formal grounds carries the same weakness as discrimination of good citizens on purely formal grounds.

In 2008 we gave a different resolution of d'Alembert's paradox than Prandtl's, based on the fact that potential solutions of the Euler equations are unstable and thus turn into turbulent solutions with substantial pressure drag. This was not discrimination on formal grounds, but on real grounds; an unstable solution does not persist over time. This opened to a revolution in computational fluid dynamics freed from a perceived necessity to computationally resolve unresolvable thin boundary layers.

You find on this web site, if you are interested and make a search, resolutions of the following paradoxes:

• D'Alembert's paradox and other paradoxes of fluid mechanics.
• The Reversibility paradox of classical and quantum mechanics (Loschmidt's paradox) 
• Paradoxes of special relativity including the Twin paradox.
• Paradoxes of wave-particle and collapse of the wave function of quantum mechanics.

Yes, it is wonderful to discover a paradox and even more wonderful to resolve it!

Towards Resolution of Gray's Paradox

Gray's paradox concerns the contradiction between standard fluid mechanics predictions of the resistance to motion (drag) of a dolphin and the observed speed of a dolphin. Gray estimated that the required muscle power would be seven times bigger than that available. A real paradox!

The search to resolve the paradox has tried different routes: One is to claim that the muscle power of dolphin in fact is much bigger (seven times) than what can be envisioned. Another is to claim that the skin of a dolphin has a magical composition somehow decreasing drag by a factor of seven.

None of the attempts has been successful.

Let us see if the new approach to computational fluid dynamics presented in Computational Turbulent Incompressible Flow and The Secret of Flight, offers a resolution.

We term the new approach DFS Direct Finite Element Simulation (of turbulent flow) based on computing turbulent solutions to the Euler equations for incompressible flow with slip/small skin friction boundary conditions. We have found that DFS predicts the drag of a wing, full airplane and car in close correspondence with observation, with slip as zero skin friction.

Our conclusion is that skin friction gives a minor contribution to total drag as pressure drag plus skin friction in many applications of aero- and hydromechanics, including the locomotion of a dolphin.

This is against the common view of the fluid dynamics community that skin friction is 50-90% of total drag. DFS thus gives design fundamentally new conditions to work from.

DFS in particular seems to offer a resolution of Gray's paradox, by showing that the drag of a dolphin is severely overestimated by conventional techniques as being based on a formula for flat plate drag.

The resolution is a spin-off of the resolution of D'Alembert's paradox (check video) as the mother of the paradoxes of fluid mechanics, a resolution which is intimately connected to DFS.

That skin friction drag predicted from flat plate experiments gives an overestimate of the drag of a streamlined body, like a dolphin, is supported by the article TURBULENT SKIN-FRICTION DRAG ON A SLENDER BODY OF REVOLUTION AND GRAY’S PARADOX, by Nesteruk and Cartwright (13th European Turbulence Conference (ETC13), Journal of Physics: Conference Series 318 (2011) 022042):

• The presented analysis shows that turbulent frictional drag on a slender rotationally symmetric body is much smaller than the flat-plate concept gives and the flow can remain laminar at larger Reynolds numbers. Both facts are valid for an unseparated flow pattern and enable us to revise the turbulent drag estimation of a dolphin, presented by Gray 74 years ago, and to resolve his paradox, since experimental data testify that dolphins can achieve flow without separation. The small values of turbulent skin-friction drag on slender bodies of revolution have additional interest for further experimental investigations and for applications of shapes without boundary-layer separation to diminish the total drag and noise of air- and hydrodynamic hulls.

We will now compute the drag of a dolphin by DFS and report the results shortly. Reducing prediction of skin friction from 70% to 10% may correspond to Gray's factor seven...

PS  From Passive and Active Flow Control by Swimming Fishes and Mammals by F.E. Fish and G.V. Lauder:

• Dolphins have the muscular capacity to swim at high speeds for short durations while maintaining a fully attached turbulent boundary layer. The turbulent flow conditions would delay separation of the boundary layer (Figure 1; Rohr et al. 1998). When the boundary layer separates from the skin surface and interacts with outer flow, this results in a broader wake and increased drag, so delaying separation is beneficial to the dolphin. Separation is more likely to occur with a laminar boundary flow, producing a greater drag penalty compared to turbulent boundary conditions. Thus, the turbulent boundary layer remains attached longer because it has more energy than the laminar boundary layer. The increased drag of a turbulent boundary layer is small compared to the increase in drag due to separation, which is more prone to occur with a laminar boundary layer.

This conforms with the theory and practice presented in Computational Turbulent Incompressible Flow showing in particular that flow with a slip boundary condition stays attached with small drag, while flow with a no-slip laminar boundary layer separates early with large drag. The observed small drag of a dolphin thus can be explained by the theory behind DFS, but not by any commonly accepted theory seeking the origin of drag in thin boundary layers following the legacy of Prandtl.