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.

- 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!

]]>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.

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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.

]]>- 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.
- This is because transition to a turbulent boundray layer on the leading edge is blocked by wall and damped by acceleration.
- 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).
- 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.
- 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.
- The new flight theory builds on slip. With no-slip (laminar or turbulent) the flow separates on crest destroying the functionality of the wing.
- 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.
- 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).
- 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:

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.

]]>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:

- Artificial generation of Tollmien-Schlichting waves by a heavily vibrating ribbon in experiments on transition from laminar to turbulent flow in a shear layer.
- 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.

]]>The total drag of a body moving through air (airplane) or water (ship) consists of

- form/pressure drag + skin friction drag.

It is generally believed from experiments dragging a plate through water, that for an airplane and ship skin friction may be 50-70% of total drag. Experiments are performed with (i) untripped/free transition and (ii) tripped/forced transition to a turbulent boundary layer, with (ii) showing a bit bigger drag than (i).

Tripping is done e g by mounting a rib along the upper part of the leading edge of a wing. Its effect of creating a thick turbulent boundary layer is illustrated in the above image.

Computations with DFS Direct Finite Element Simulation with zero skin friction (slip boundary condition on wall) shows drag in close accordance with drag experiments with free transition.

The DFS results thus show total drag as pure form/pressure drag with zero skin friction, in accordance with free transition experiments. This gives evidence that drag with free transition has very little contribution from skin friction, and further that the measured (small) difference between tripped and untripped drag can be used to assess the skin friction, which is forced by tripping and is thus absent without tripping.

Now, a real airplane is not equipped with tripping devices on wings or fuselage since that would increase drag for no use, and DFS with slip shows close correspondence to experiments with free transition.

Altogether, there is strong evidence that skin friction drag for an airplane or ship is an order of magnitude smaller than that commonly used based on experiments from tripping. The results indicate that what is believed to be a thick turbulent boundary layer forced by tripping with substantial skin friction, in fact is absent i reality without tripping and thus that the interaction between fluid and solid acts as slip/small friction (without boundary layer to resolve computationally).

Obviously, if skin friction in reality is less than 10% of total drag, instead of an unreal tripped imagination of 50-70%, the design of an airplane or ship will work from different premises.

DFS with slip makes CFD computable, whereas std CFD with no-slip tripped boundary layers is uncomputable.

Why is then tripping used in experiments if in reality not? This is to make experiments fit with the boundary layer theory of Prandtl as the Father of Modern Fluid Mechanics tracing drag to the presence of a thick turbulent boundary layer. But to fit unreal experiments to theory is opposite to the idea of of real science where theory is fitted to real experiments.

The effect of fitting experiment to theory is that standard CFD is calibrated to a skin friction of 50-70% of total drag, which with the evidence from DFS means that standard CFD underestimates form/pressure drag, and thus gives wrong input to design.

The plot below shows drag coefficients for NACA0012 by Ladson with free and tripped transition. Note the small dependence on Reynolds number for free transition and that difference between tripped and untripped drag is about 0.001 as about 10% of total tripped drag as an estimation of skin friction drag.

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