Euler’s Dream Come True: DFS!


Euler’s Dream: No parameter needed as input = Einstein’s very rare ideal mathematical model of some physics with here u velocity and p pressure of an incompressible fluid with vanishingly small viscosity.

Euler formulated around 1750 a mathematical model of the dynamics of slightly viscous incompressible fluid in terms of first principle physics expressing (i) incompressibility and (ii) Newton’s 2nd law in the domain occupied by the fluid, combined with (iii) a boundary condition where the fluid meets a solid wall expressing that the fluid does not penetrate into the wall and on the wall acts with zero friction as a so called slip boundary condition. Euler was very happy with his model and expressed as Euler’s Dream:

  • My two equations include not only all that has been discovered by methods very different and for the most part slightly convincing, but also all that one could desire further in this science. 
  • Everything that the Theory of Fluids contains is embodied in the two equations formulated, so that it is not the laws of Mechanics that we lack in order to pursue this research but only the Analysis, which has not yet been sufficiently developed for this purpose.

Euler thus understood that all of fluid mechanics was hidden in his two equation model, but also that unfortunately it was impossible to reveal the true secrets of Euler’s Dream by analytical mathematics. What a wonderful insight and possibility offered by mathematics, but alas beyond catch.

Even worse, Euler knew that there was a whole family of analytical solutions in the form of potential solutions, which did not correspond to observed physical flow. This was coined as d’Alembert’s paradox  (video), which turned Euler’s Dream into Euler’s Nightmare.

It took 250 years to get out of the Euler’s Nightmare and make Euler’s Dream come true in the form of DFS Direct Finite Element Simulation as best possible computational solution of Euler’s equations, which showed to be in close agreement with observations and so revealed the Secret of Flight through mathematical analysis of DFS solutions.

This website describes the new world of CFD opened by DFS as Euler’s Dream come true.

More precisely, it is the wide world of slightly viscous flow with high Reynolds number beyond the drag crisis covering in particular flight, to discover from the previous post Restart: Take Off.

It is a world full of new possibilities different from the world of Prandtl as the Father of Modern Fluid Mechanics asking for costly computational resolution of thin boundary layers with then CFD in the form of RANS-LES restricted to Reynolds numbers well below the drag crisis, which in particular does not include flight.

The Real Euler Flight Simulator based on interactive DFS to be presented in 2020 opens entirely new possibilities in aviation design, testing, certification and pilot training in a realisation of Euler’s dream as predictive computational simulation from first principle physics.


Direct Finite Element Simulation DFS: Computes lift and drag of bluff or streamlined body from shape alone ( no parameter)!



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