Foreword to the book The Secret of Flight:
- I have not the smallest molecule of faith in aerial navigation other than ballooning or of expectation of good results from any of the trial we hear of. (Lord Kelvin refusing to join the Aeronautical Society of London)
The book presents a solution to the enigma of flight in a New Theory of Flight as a spin-off of the resolution of d’Alembert’s paradox of zero resistance or drag in inviscid incompressible flow motion from 1752, which was published by the authors in 2008. D’Alembert’s paradox paralyzed theoretical fluid mechanics from start and the birth modern fluid mechanics is signified by the resolution presented in 1904 by Ludwig Prandtl, named the father of modern fluid mechanics.
The official doctrine or accepted truth of the fluid mechanics community is that Prandtl resolved d’Alembert’s paradox by suggesting that drag originates from a thin boundary layer of slightly viscous flow resulting from imposing a no-slip boundray condition. Our new resolution shows that Prandtl’s resolution does not describe correct physics, by showing that drag instead results from separation instability of inviscid flow satisfying a slip boundary condition without boundary layer.
- As an introduction to our New Theory of Flight it is instructive to read the History of Hydrodynamics from the Bernoullis to Prandtl by O. Darrigol, with a shift of perspective from the official doctrine that Prandtl resolved d’Alembert’s paradox to our new resolution, that is, with shift from a false to a true answer which fundamentally changes light and shadows of the picture.
Darrigol: What distinguishes the history of hydrodynamics from that of other physical theoies is not so much the tremendous effect of challenges from phenomenal world, but rather it is the slowness with which these challeneges were met. Nearly two centuries elapsed between the first formulation of the fundamental equations of the theory (Euler equations for inviscid incompressible flow) and the deductions of laws of resistance in the most important case of large Reynolds numbers.
Comment: This refers to the official resolution of d’Alembert’s paradox attributed to Prandtl.
Darrigol: The reason for this extraordinary delay are easily identified a posteriori. They are the … nonlinear character of the fundamental equations…Moreover, instability often deprives the few known exact solutions of any physical relevance….almost every theoretical description of a natural or manmade flow involves instabilities.
Comment: This connects to the New Theory with instability as fundamental property of solutions to the Euler equations.
Darrigol: These difficulties have barred progress along purely mathematical lines. They have also made physical intuition a poor guide, and a source of numerous paradoxes…..Hydrodynamicists therefore sought inspiration in concrete phenomena. Challenged to understand and act in the real world, they developed a few innovative strategies. One was to modify the fundamental equations, introducing for instance Navier’s viscous term. Another was to give up the continuity of the solutions of Euler’s equations, and to study the evolution of the resulting singularities. Helmholtz pursued this approach without leaving the realm of the perfect fluid.
Comment: Both approaches miss the main aspect pf instability.
Darrigol: None of these strategies sufficed to fully master the real flow for which they were intended. Prandtl’s ultimate success depended on combining them within the asymptotic framework of high Reynolds numbers /quasi-inviscid flow) and large aspect ratios (quasi-2d-flow). The role of small viscosity, Prandtl reasoned, is to produce boundary layers of high shear, and vortex sheets to which Helmholtz’s theory of vortex motion may be applied in a second step. Vortex sheets are always unsatable, and boundary layers ofteh are so. Thg instabilities lead to turbulence… When separation occurs, the hydrodynamicist is left with Columbus’s egg, unless strong resistance is desired, in which case he can appeal to model measurements combined with similitude arguments.
Comment: This is the official mantra presented as a Columbus egg, that is, a solution which is not a true solution but only trick (to make the unstable egg stable).
Darrigol: The evolution from paper theory to an engineering tool thus depended on transgression of the limits between academic hydrodynamics and applied hydrodynamics. The utilitarian spirit of Victorian science, the Polythechnique ideal of a theory-based engineering, a touch of Helmholtz’s eclectic genius, and the Göttingen pursuit of applied mathematic, all contributed to the fruitful blurring of borders between physics and engineering. The “sagacious geometers” who answered d’Alembert’s ancient call for a solution to his resistance paradox all visited the real worlds of flow.
Comment: This is the official picture as a mish-mash of theory (mathematics) and practice (real world engineering flow) with a touch of genius and utilitarian spirit.
Darrigol: On Wing Theory: In the 1890s, interest in flying contraptions grew tremendously, partly as a consequence of Otto Lilienthal’s invention of the man-carrying glider in 1889. The prospects of building a motor-powered, piloted airplane seemd high in some engineering quarters. The materialized in 1903 when Wilbur and Orville Wright flew th first machine of that kind. Theory played almots no role in this spectacular success….The contemporary flight frenzy prompted theoretical comments and reflections from flat rejection to elaborate support. … In summary, Rayleigh, Lamb and Kelvin knew too much fluid mechanics to imagine that circulation around wings was the main cause of lift. The two men who independently hit upon this idea lacked training in theoretical physics. One of the them was an engineer (Lanchester), and the other was a young mathematician (Kutta).
Comment: Lanchester circulation theory was not credible to people with training in theoretical physics.
Darrigol: By the end of the 1st World War, Prandtl and his collaborators could legitimately claim a mathematical, quantitative solution to the wing problem. The leftover was to justify the various approximations that Prandtl had introduced at various steps of the reasoning. Although Prandtl’s justifications for these assumptions lacked rigor, experiments performed during the war in the Göttingen wind tunnel vindicated them. Post-war British and American experiments further confirmed Prandtl’s theory….After some hesitation on the British side, by the mid 1920s, it became routine…Under the stimulus of the rising field of aeronautics and with the strong support of Göttingen institutions, Prandtl’s group put an end to the engineer’s legitimate distrust of the theoretical predictions of fluid mechanics.
Comment: Kutta circulation theory is a non-physical mathematical theory, as well as Prandtl’s extended version of circulation theory boosted with a boundary layer theory. Prandtl’s institutions boomed under the German preparations for the 2nd World War in the 1920-30s and lack of theory was compensated by powerful wind tunnels and steel.