The Mathematical Secret of Flight 4

The secret of flight is hidden in the above picture showing the flow separation at the trailing edge of a wing, as explained in detail in the article The Mathematical Secret of Flight and the upcoming book The Secret of Flight.

Mathematical analysis shows that the swirling flow separation shown in the picture results

from an instability of opposing flows meeting behind the trailing edge, and the swirling motion

allows the flow to separate with little retardation requiring high pressure. Instead low pressure

develops inside the swirling flow which does not like high pressure destroy the high lift/suction established on the crest of the wing, while causing only small drag because of the small diameter of the trailing edge.

This is the miracle of flight revealed by mathematical analysis, whether you like it or not in the words of Richard Feynman.

The swirling motion is similar to that used by noble men when backwards leaving the king after an audience; an elegant form of separation without high pressure destruction of what was gained during the meeting.

I have asked Antony Jameson about a comment but not received any response so far.

Compare with 50 ways to leave your lover: The most elegant and thus best way is with a swirling motion avoiding build up of high pressure.

The Mathematical Secret of Flight 3

After my talk The Mathematical Secret of Flight at Svenska Mekanikdagar 2011 an remark was made by Laszlo Fuchs recalling early attempts to compute the lift and drag of a wing by solving the Euler equations by Antony Jamseon (left) and Art Rizzi at KTH in the 1980s.

In essence what Jameson and Rizzi did was to solve the Navier-Stokes equations with a slip boundary condition at very high Reynolds number, essentially the same as we are doing, and observing the appearance of unsteady fluctuating solutions.

However, these solutions were regarded with suspicion by the fluid dynamics community as some kind of ghost solutions with unclear physical significance. In particular the slip boundary condition without any boundary layer was against the dictate of Prandtl that no-slip is the only physically correct boundary condition and that boundary layers have to be resolved because the truth is to be found there.

The result was that the (incompressible) Euler solvers of Jameson and Rizzi became marginalized, just as Birkhoff had been in the 1950s when asking if there were any steady solutions at all.

What we have shown is that Birkhoff, Jameson and Rizzi were on the right track and that the effective suppression of their ideas has delayed the advancement of computational fluid mechanics by several decades. The solutions computed by Jameson and Rizzi were not ghost solutions but true turbulent solutions carrying important information of real physics.

The suppression of correct science is often more harmful than the promotion of wrong science.

Mathematical Secret of Flight 2

Computed turbulent flow velocity around a NACA0012 wing at 15 degrees angle of attack in beginning stall with separation on top of the wing.

An updated version of my talk on June 15 at Svenska Mekanikdagar 2011, is now available for preview as

• The Mathematical Secret of Flight

describing joint work with Johan Hoffman and Johan Jansson.

This work shows that computation of mean values such as lift and drag of an airplane, car or boat can be accurately computed without resolving thin boundary layers. Based on the computations a new mathematical theory for flight is presented which is fundamentally different from that by Kutta-Zhukovsky-Prandtl filling text books. The new flight theory was first published in Normat 57:4 (2009) as The Mathematical Secret of Flight.

The spell of Prandtl as the father of modern fluid mechanics of attributing both lift and drag as effects of thin boundary layers requiring unreachable quadrillions of mesh points for computational resolution, can thus be broken. This opens a wealth of applications of computational fluid dynamics suddenly reachable with millions of mesh points.

The new theory of flight is presented in Mathematical Simulation Technology, a book which was officially banned by KTH in November 2010, as described in posts on KTH-gate. The reader can act as referee and decide if the ban is motivated from a scientific point of view. The last time a math book was banned was in 1632.

Mathematical Secret of Flight 1

Computed Lift and Drag of a 3d NACA0012 wing for different angles of attack by Unicorn (blue) compared with different experiments.

My talk on June 15 at Svenska Mekanikdagar 2011, is now available for preview as

• The Mathematical Secret of Flight

describing joint work with Johan Hoffman and Johan Jansson.

Based on accurate solution of the incompressible Navier-Stokes equations we identify the true mechanism for the generation of large lift L at small drag D of a wing with lift to drag quotient L/D of size 10 - 50, which is not described in the literature.

We combine the Navier-Stokes equations with a slip boundary condition on the wing motivated by the experimental fact that the skin friction is small for a slightly viscous fluid such as air or water, and we exhibit the role the slip condition in two crucial aspects:

• prevention of separation at the crest of the wing generating large lift
• 3d slip-separation at the trailing edge not destroying large lift and causing small drag.

Text books claim following Prandtl, named the father of modern fluid mechanics, that both lift and drag result from a boundary layer arising from a no-slip condition.

We obtain lift and drag in full accordance with experiments by solving the Navier-Stokes equations with a slip condition, which does not generate any boundary layer, and we thus present strong evidence that lift and drag do not originate from any boundary layer.

In short, we show that solutions to the Navier-Stokes equations with slip are computable and

correctly capture the physics of (subsonic) flight. See also

• The Mathematical Secret of Flight (Normat 57, 2009)
• Why It Is Possible to Fly (Google Knol) .

Compare with state-of-the-art as expressed by Moin and Kim in Tackling Turbulence with Supercomputers:

• To solve the Navier-Stokes equations for, say, the flow over an airplane requires a finely spaced computational grid to resolve the smallest eddies.
• Consider a transport airplane with a 50-meter-long fuselage and wings with a chord length (the distance from the leading to the trailing edge) of about five meters. If the craft is cruising at 250 meters per second at an altitude of 10,000 meters, about 10 quadrillion (10^16) grid points are required to simulate the turbulence near the surface with reasonable detail.

Kim and Moin express the necessity dictated by Prandtl to resolve thin boundary layers to correctly compute lift and drag of a wing or an entire airplane, which would require 50 years of Moore's law to increase the computing power with a factor 10^10 to reach the dictated 10^16 points.

We show that this is possible already today using 10^6 points by using slip without boundary layers to resolve.

Svenska Mekanikdagar June 13 - 15

The abstract of my talk: Why It Is Possible to Fly and Sail on June 15 at Svenska Mekanikdagar 2011, is now available for inspection, here with references.

For an introduction with a link to my presentation, see The Mathematical Secret of Flight.

The question is if Prandtl is still to be viewed as the father of modern fluid mechanics, with one answer given in Dr Faustus of Modern Physics.

A previous talk on the subject can be downloaded from my home page under News: The Mathematical Secret of Flight, BAIL 2010.