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)!



Restart: Take Off

takeoffThis 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:

  1. DFS = CFD based on first principle physics 1995 –.
  2. Book: Computational Turbulent Incompressible Flow 2007.
  3. Resolution of d’Alembert’s Paradox 2008.
  4. New Theory of Flight 2008 –.
  5. High Lift Workshop: DFS for full aircraft 2017.
  6. DFS as MOOC 2018.
  7. DFS as open source CFD launched by Icarus Digital Math 2018.
  8. Presentation of New Theory of Flight to Boeing 2019.
  9. 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.

Model of Flow Separation


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!

Role of Shear Layer: No-Slip vs Slip

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.

Continue here.

Bypass Transition from No-Slip Laminar Boundary Layer to Slip Boundary Condition


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.

Continue here.


Update of New Theory of Flight

Here is a short update of the New Theory of Flight as concerns the slip/small friction boundary condition which is instrumental, with reference to the last sequence of posts:

  1. The boundary layer of a wing initialised as laminar at stagnation point at leading edge, effectively turns into (acts like) slip with very small skin friction.
  2. This is because transition to a turbulent boundray layer on the leading edge is blocked by wall and damped by acceleration.
  3. The flow once turned into slip on leading edge stays with slip, because transition to turbulent boundary layer is not triggered by slip (no shear).
  4. The net is that the flow around a wing effectively acts as having slip, because transition to a turbulent boundray layer is not triggered by artificial device on leading edge.
  5. The large skin friction from flat plate experiments with artificial tripping should not be used for a wing. If used they give much too big skin friction drag.
  6. The new flight theory builds on slip. With no-slip (laminar or turbulent) the flow separates on crest destroying the functionality of the wing.
  7. We now can see slip as a limit form of a laminar boundray layer with very small skin friction (without the negative aspect of no-slip of 6.), not as a limit form of a turbulent boundary layer with large skin friction, because of “by-pass” as discussed in previous post.
  8. The correct way to add skin friction to DFS is by the friction coefficient of laminar flow, which is an order of magnitude smaller than that of a turbulent bounder layer (used in RANS et cet).
  9. Comparison between experiments for a wing with and without tripping (and other experiments) show skin friction coefficient of size 0.002-3, much bigger than laminar skin friction as shown in this plot:


On the dream of a “laminar wing”

Without tripping the flow around a common wing under pre-stall conditions thus effectively satisfies a slip boundary condition with the very small friction of a laminar boundary layer, and then without the destructive crest separation from vanishing normal pressure in a laminar boundary layer.

This means that already a common wing realises the dream of very small skin friction drag associated with a “laminar wing” as a wing with a laminar boundary layer.  This explains why the search for further skin friction reduction by e g blowing or suction has not been successful.  To reduce something which is already very small can be very difficult.

By-Pass from Laminar No-Slip Boundary Layer to Slip without Layer


Artificial vibrating ribbon generating artificial Tollmien-Schlichting waves.

When theory does not fit experiment, one approach is to change the experiment. This is an established technique in fluid mechanics since the discovery of d’Alembert’s paradox in 1755 separating from start fluid mechanics into theory explaining what cannot be observed in reality, and real observation which cannot be explained theoretically.

There are thus basic experiments in fluid mechanics which are manipulated in the form of artificial forcing containing:

  1. Artificial generation of Tollmien-Schlichting waves by a heavily vibrating ribbon in experiments on transition from laminar to turbulent flow in a shear layer.
  2. Artificial tripping of the flow over a wing by a fixed rib or wire to generate a turbulent boundary layer with substantial skin friction to fit Prandtl’s boundary layer theory.

Computational Turbulent Incompressible Flow presents a different non-artificial real scenario for transition to turbulence in a shear later such as a laminar boundray layer. The scenario is that weak streamwise vorticity always present from small perturbations, acting over long time by non-modal linear growth restructures the flow in a laminar shear layer into high and low speed streamwise streaks (increasing transversal velocity gradients) which when big enough triggers transition to turbulence. This effect is damped in streamwise accelerating flow, but not so in constant or decelerating flow.

The result is that a laminar shear layer over a flat plate (without acceleration) turns turbulent if the Reynolds number is big enough and the plate long enough

On the other hand, in the accelerating flow on the upper part of the rounded leading edge of a wing, the transition does not take place. Instead the laminar no-slip boundary layer present at the stagnation on the leading edge stays laminar (as well as on the lower pressure side of the wing) and if the Reynold’s number is big enough effectively acts and can be modeled as a slip boundary condition without boundary layer.

The change from laminar no-slip boundary layer to effectively slip without boundary layer, thus without transition to a turbulent boundary layer, can be connected to a Reynolds number of size 1000000 with thus a laminar boundary layer of thickness 0.001 with free stream velocity and size normalized to 1.

Slip would then result when the thickness of the boundary layer is about 0.1% of the gross dimension. For a wing with chord 1 m this would be 1 mm.

We thus add theoretical evidence that the slip condition used in DFS as well as the New Theory of Flight has a sound rationale.

In particular DFS shows that total drag is more than 90% form/pressure drag and skin friction drag less than 10%, while standard theory and computation says that skin friction dominates form/pressure drag.

Connecting to 2. above, the direct passage from laminar no-slip boundary to slip without boundary layer, thus in real cases “bypasses” the generation of a turbulent boundary from artificial forcing.

Likewise, without the artificial vibrating rib transition to turbulence is not by Tollmien-Schlichting waves, but instead through the scenario presented after 2

In short, reality does not do what standard theory says reality should do. Reality “bypasses” standard theory, but standard theory is nevertheless claimed to be correct because it fits experiments with artificial forcing! This is state of the art. Something to think about.


Dear Reader, Student, Pilot, Air Passenger, Engineer, Scientist, Interested Layman…:

Welcome to follow the resolution of a scientific riddle which has baffled human imagination since the dawn of science into present time, formulated by Kenneth Chang in New York Times on Dec 9 2003 in Staying Aloft: What Does Keep Them Up There? as follows:

  • To those who fear flying, it is probably disconcerting that physicists and aeronautical engineers still passionately debate the fundamental issue underlying this endeavor:
  •  What keeps planes in the air?

1. Old Theory with Old Answers

You will find that the answers to the question what keeps planes in the air, which you can find in classical books and articles in science and popular science, fall into one of the following categories:

  • non-trivial but incorrect (for educated)
  • trivial and incorrect (for uneducated)
  • correct but trivial (for uneducated).

To see how this Old Theory is presented to uneducated as explaination of observations, it is instructive to watch:

Take your time, watch with amusement knowing that you soon will understand much more than The Department of Defense did in 1957 and does even today.

In short you will be able to explain the secret of flight by understanding the flow around the airfoil as

  1. potential non-turbulent flow before the trailing edge,
  2. non-potential turbulent flow with 3d rotational slip separation at the trailing edge without pressure rise/drop,
with 2. the crucial new element of the New Theory.

2. New Theory with New Answers

You will be led to a real answer to the question with the following characteristics:

  • non-trivial and correct (for both educated and uneducated)
  • reveals the real physical mechanism of flight.

The New Theory has been submitted to AIAA, The World’s Forum for Aeronautics Leadership, as the article New Theory of Flight which should generate lively debate to be reported on this site: It contains new mathematics and physics, and is highly controversial by challenging the very foundations of the ruling paradigm defined by Ludwig Prandtl as the Father of Modern Fluid Mechanics 100 years ago.

You will find that the New Theory is based on the following new discoveries, which bring fundamental changes to the science of high Reynolds number flow:

  1. It is possible to computationally solve the Navier-Stokes equations in the case of high Reynolds number of aerodynamics, using a slip boundary condition as a model of a turbulent boundary layer with small skin friction.
  2. It is possible to theoretically understand high Reynolds number bluff body flow as potential flow modified by rotational slip separation.

3. The Secret Revealed in Four Basic Steps

  1. The flow being incompressible with a slip boundary condition can only (as potential flow) separate at stagnation, which cannot occur before the trailing edge (before stall).
  2. The flow above the wing is thus redirected downwards, which requires low pressure or suction peaking on top of the leading edge, which generates 2/3 of the lift with 1/3 from high pressure under the wing.
  3. Main drag is created by high pressure (positive) on the leading edge by low speed flow in accordance with Bernoulli’s Principle.
  4. Lift and drag from the leading edge are preserved by a specific flow separation pattern at the trailing edge with alternating high and low pressure with zero mean.

Here 1. and 4. are the new elements of the New Theory, with 4. the most surprising and intriguing, which are combined with the classical elements  2. and 3. relating to Bernoulli’s Principle from 1738 bridging over 280 years.

4. Understanding

You will discover that the resolution of the riddle can be expressed in basic mathematical terms which opens to understanding why flight is possible. This is because in physics real understanding can only be reached by understanding a mathematical model of the phenomenon under study. You will thus be able to understand what birds have understood since the Archaeopteryx as the first bird took off into the air some 150 million years ago, but man has not until very recently been able to grasp.

5. Ingeniuous Invention

You will find that the resolution is surprising and ingenious but also so simple that you will be able to understand why flight is possible so well, that you will be able to explain to anybody asking the question, from professors to school mates, friends and family members.

You will thus understand the cleverness of birds and why also human beings have been able to lift from the ground up into the air with proper equipment.

6. Incorrect Mathematics

You will discover a thrilling story of mathematical mistakes made by great mathematicians and scientists,

  • Newton followed by d’Alembert 1652 (incorrectly) proved that flight is impossible,
  • Kutta and Zhukovsky (incorrectly) proved in 1904 that flight is possible after the Wright brothers successful powered heavier-than-air flight in 1903 by connecting lift to circulation.
  • Prandtl (incorrectly) connected drag to thin boundary layers in 1904.
  • Aviation developed during the 20th century based on the (incorrect) theory of Kutta-Zhukovsky-Prandtl.

The fathers of the (incorrect) modern theory of flight: KuttaZhukovsky and Prandtl.

In short

  • Kutta-Zhukovsky developed a (incorrect) theory for lift without drag,
  • Prandtl developed a (incorrect) theory for drag without lift,

while what was needed was a correct theory for lift and drag.

7. Correct Mathematics

You will find that the miracle of flight can be explained from the following properties of the slightly viscous incompressible fluid flow characteristic of aerodynamics:

  1. slightly viscous flow can be simulated with a slip (zero skin friction) boundary condition
  2. potential flow satisfies slip and can only separate at stagnation
  3. slightly viscous flow undergoes 3d rotational slip separation without the high pressure of potential flow.

1-3 are carefully explained in on this site, and the explanations are not complicated. Yet they combine into the marvelous invention of flight.

8. Presentation: Overview

To get an overview you may browse the following Powerpoint presentation:

9. The Dream of Human Powered Flight

The Gossamer Condor pedaled and piloted by Bryan Allen won the Kremer Prize in 1977 as the first successful human powered flight. This was the dream of Icarus and Leonardo da Vinci which thus ultimately came true only recently, and then after seemingly endless trial and error.

10. Short History of Flight

A short history of the theory and practice of flight is as follows:

  • The flight of birds was intensely debated already by the Neanderthalers.
  • Newton proved that powered human flight is theoretically impossible by using incorrect mathematics.
  • The Wright brothers showed in 1903 that powered human flight is possible in practice.
  • The mathematicians Kutta and Zhukovsky proved in 1904 that powered human flight is possible by using incorrect mathematics.
  • In 2008 we proved that powered human flight is possible by using correct mathematics. Theory and practice eventually met in a happy marriage.

11. Watch and Ask Questions

12. Getting Ready for Take-Off

13. Take Off: Browse the Menu

You are now ready to in more detail browse the Menu including the Survey of posts.