måndag 15 oktober 2012

Feyerabend on the New Theory of Flight

Paul Feyerabend expresses the central message of Against Method as follows:
  • My intention is not to replace one set of general rules by another such set: my intention is, rather, to convince the reader that all methodologies, even the most obvious ones, have their limits. The best way to show this is to demonstrate the limits and even the irrationality of some rules which she, or he, is likely to regard as basic. 
Let us speculate about what Feyerabend might have said about the New Theory of Flight explaining the generation of large lift at the price of small drag of the wing of an airplane or bird, that is the secret of flight. Feyerabend could have posed the following questions:
  1. When and how was the secret of flight revealed in the form of the New Theory of Flight?
  2. Why did it take so long time (100 years after the first sustained powered flight)?
We could have answered Feyerabend as follows:

The New Theory of Flight came out as a natural consequence of the resolution of D'Alembert's paradox by Hoffman and Johnson in 2008 describing slightly viscous bluff body flow as potential flow modified by 3d rotational separation. This description was rationalized from a stability analysis of potential flow after inspection of computed turbulent solutions of the Navier-Stokes equations. 

The "method" used was to solve the Navier-Stokes equations computationally, understanding that slightly viscous flow should be well described by the Navier-Stokes with a slip boundary condition, and then look for features of the turbulent solutions which could be grasped by analytical mathematics. This "method" worked in this case because the main feature of 3d rotational separation could be connected to the basic instability mechanism of potential flow at separation.

We could thus "understand" slightly viscous bluff body flow, such as the flow around a wing, in analytical mathematical terms as potential flow modified by 3d rotational separation resulting from instability of potential flow separation.

We then used the general principle of science that "understanding" of a physical phenomenon means capturing the main features of a mathematical model of the phenomenon in "understandable" analytical terms.

We thus used a "method"consisting of
  • solving the Navier-Stokes equations computationally, 
  • comparing the solution with the potential solution,
  • analyzing the stability of the potential solution.  
We could have asked Feyerabend if this can be described as "anything goes" in the spirit of Against Method, or if it represents an approach which is less ad hoc.

Why did it take so long time to reach this relatively simple insight? Because computational solution of turbulent solutions of the Navier-Stokes equations became possible only 100 years after the first powered sustained flight. 


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