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)!
This web site started in 2012 (total 165.000 views) now makes a restart representing the recent breakthrough of the New Theory of Flight revealing for the first time the Secret of Flight. The time line is the following:
- DFS = CFD based on first principle physics 1995 –.
- Book: Computational Turbulent Incompressible Flow 2007.
- Resolution of d’Alembert’s Paradox 2008.
- New Theory of Flight 2008 –.
- High Lift Workshop: DFS for full aircraft 2017.
- DFS as MOOC 2018.
- DFS as open source CFD launched by Icarus Digital Math 2018.
- Presentation of New Theory of Flight to Boeing 2019.
- New Flight Simulator based on DFS initiated 2019.
After a long incubation period the new theory and new computation of DFS thus enters into the real world of flight from design to pilot training, offering entirely new possibilities. This website will track the further development and offer updates of background material on both theory, computation and application.
Its is a travesty that all through the modern era of aviation a physically correct theory of flight has been lacking. NASA reports on its home page 3 incorrect theories, but no theory claimed to be correct. It is now time for the New Theory to replace the text book theory and form a new practice.
The holy grail of CFD as computational fluid mechanics is:
- Turbulence modeling.
- Flow separation.
DFS as Direct Finite Element Simulation offers answers to these problems:
- Turbulence captured as best possible computational solution to the Euler equations.
- Flow separation described as 3d rotational or parallel slip separation.
To see details of this picture ponder the above picture of the dynamics of a tornado with air sucked horizontally along the land surface towards a low pressure center being redirected into a raising swirling air thus separating from the surface. Then read this post and think!
Read this post on the Kutta condition as the core of the Kutta-Zhukovsky (unphysical) theory of lift of a wing.
The book Computational Turbulent Incompressible Flow (Chap 36) describes in theory and computation the transition to turbulence in parallel shear flow such as Couette flow between two parallel plates and in a laminar boundary layer. The basic mechanism is the action of streamwise vorticity, generated from perturbations in incoming flow, which slowly redistributes the shear flow transversally into high and low speed streamwise flow streaks with increasing transversal velocity gradients, which trigger turbulence when big enough.
The New Theory of Flight is supported by Direct Finite Element Simulation DFS as best possible computational satisfaction of Euler’s equations expressing first principle physics in the form (i) incompressibility, (ii) momentum balance and (iii) slip boundary condition on solid walls.
Observations and experiments (connecting to the so-called drag crisis) indicate that at a Reynolds number Re of about 1 million the boundary condition at a solid wall changes from no-slip at the wall accompanied with a thin laminar boundary layer, to effectively a slip condition as a thin film without layer.
This is the generic text book picture of the flow around a wing according to Prandtl as Father of Modern Fluid Mechanics. To this picture in new light go to this post.