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.