onsdag 17 oktober 2012

The Enigma of the Aerofoil 3

                                                         Sir Leonard Bairstow (1880-1963)

The Enigma of the Aerofoil by David Bloor describes the dispute in England after the 1st World War between Leonard Bairstow representing Cambridge mathematics and Hermann Glauert influenced by German aerodynamics engineering led by Ludwig Prandtl, concerning the fundamental problem of aerodynamics of flight with the following highlite:
  • The International Air Congress for the year 1923 was held in London. It provided a further occasion for assessing the advances that had been made in aeronautics during the war years and for addressing unresolved problems. It was a highly visible platform on which the supporters and opponents of the circulatory theory could express their opinions and, in some cases, air their grievances. 
  • In the morning session of Wednesday, June 27, there were three speakers: Leonard Bairstow, Hermann Glauert, and Archibald Low. The first to speak was Bairstow, whose talk was titled “The Fundamentals of Fluid Motion in Relation to Aeronautics.” 
  • Bairstow was explicit: his aim was nothing less than the mathematical deduction of all the main facts about a wing from Stokes’ equations and the known boundary conditions. 
  • The work of Stanton and Pannell had shown that eddying motion did not compromise the no-slip condition and had established kinematic viscosity as the only important variable. “These experiments appear to me,” said Bairstow, “to remove all doubt as to the correctness of the equations of motion of a viscous fluid as propounded by Stokes and the essential boundary conditions which give a definite solution to the differential equations. 
  • The range of these equations covers all those problems in which viscosity and compressibility are taken into account, and from them should follow all the consequences which we know as lift, drag etc. by mathematical argument and without recourse to experiment. Such a theory is fundamental”.
  • The boundary conditions were empirical matters, but thereafter everything should follow deductively: lift, drag, changes in center of pressure, the onset of turbulent flow and stalling characteristics, along with a host of other results. 
In short, Bairstow stated that theoretical aerodynamics could be reduced to solving the Navier-Stokes equations for slightly viscous flow using mathematics. Bairstow thus reduced aerodynamics to mathematics, in the same way as celestial mechanics was by Newton reduced to solving the equations of motion for a set of n bodies subject to gravitational attraction, referred to as the n-body problem. 

Newton showed that the equations of motion could be solved analytically for a sun with one planet (n=2) and by followers using perturbation techniques for a planetary system like ours (n=10), which made Newton immensely famous. 

But Bairstow (and nobody else at his time) could solve the Navier-Stokes equations in any case of interest of aerodynamics,  which made his grand plan useless. Bairstow thus had nothing to match Glauert's The Elements of the Aerofoil and Air Screw Theory published in 1926 based on circulation theory for inviscid flow with a fix-up in the form of the Kutta condition at the trailing edge, which gave some results and set the text book standard into our time. 

Today it is possible to solve the Navier-Stokes equations using computers and Bairstow's grand plan can finally be realized.  This shifts the weight back to fundamentals without fix-up boosted by computation, and circulation theory with fix-up has no longer any role to play, as explained in detail on The Secret of Flight.

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