Predictive aerodynamics - Euler was right
- Real Flight Simulation in the Digital Math framework
Johan Jansson (email@example.com)
We show that computing turbulent solutions to Euler’s equations with a
slip boundary condition offers a Theory of Everything ToE for slightly viscous
incompressible fluid flow as a parameter-free model, covering a vast area of
applications in vehicle aero/hydrodynamics including airplanes, ships and cars. This
work resolves the Grand Challenges of fluid dynamics described in NASA Vision
The foundation of the methodology is an extremely efficient Direct FEM
Simulation (DFS) method. We describe a breakthrough in efficiency, allowing
extrenely small numerical dissipation by choosing very small stabilization
coefficients, while allowing very large time step size.
This work is developed as part of the Digital Math framework  - as
the foundation of modern science based on constructive digital mathematical
We show that Euler CFD by the scientific method in Digital Math predicts
drag, lift and pressure distribution in close correspondence with observations for
real problems with complex geometry and so can serve to deliver complete realistic
aero/hydro-data for simulators without input from model experiments in wind
tunnel and towing tank or full-scale experiments, as a new revolutionary capability.
1 Euler CFD overview
The methodology is a Direct FEM Simulation (DFS) [1, 3] of the first principle Euler
equations with a slip boundary condition - here denoted Euler CFD. The methodology is
realized according to the scientific method in the Digital Math framework. We call this
realization Real Flight Simulator (RFS).
These first principle equations are discretized by the Direct FEM approach, meaning
Galerkin-Least-Squares (GLS) stabilization.
The Galerkin part of the method is formulated as below in FEniCS notation:
and in corresponding strong form in Latex notation:
An overview of the main ingredients of Euler CFD with the Digital Math RFS realization
are given below, a more detailed description is in section 6:
Free slip boundary condition with 3D rotational slip separation
No thin boundary layers to resolve.
We show that the flow can separate with 3D rotational slip separation, at high velocity. See
the 3D cylinder benchmark below for an illustration.
Our detailed validation of the reference benchmarks in the field: HiLiftPW2-4, NACA0012
wing, etc. all show that with only the pressure drag our results are within 5% of the
experiment. This means skin friction drag is a small/negligible effect, which either can be
omitted, or added as a minor adjustment.
In  we give an overview of both experimental and Euler CFD evidence, of low
dependence of drag from Reynolds number in untripped configurations, consistent with free
slip, from e.g. Abbott.
Automatic turbulence modeling by residual stabilisation
Through weighted least squares residual stabilisation generating turbulent dissipation
, as a
solution to the open problem of turbulence modeling [?]. In particular, the weighted strong
residual measures the turbulent dissipation as a mesh independent quantity meeting
Kolmogorov’s K41 conjecture of finite turbulent dissipation .
Adaptive adjoint-based a posteriori error control
Guaranteeing mesh-independence of drag and lift, and accuracy to a few percent in the
Guaranteeing the scientific method, allowing inspection, falsification, modification.
There is today a reproducibility crisis in science which we resolve with the Digital Math
framework, and specifically here for Euler CFD with RFS.
Automated Digital Math
We leverage our Open Source FEniCS framework, which automated the solution of partial
differential equations by FEM, taking the mathematical notation as input. This allows an
automation of Digital Math, described in a bird’s eye view below:
1.1 Extremely efficient stabilized Direct FEM
The predictive adaptive stabilized Direct FEM Simulation method takes the form:
F = F_Galerkin + F_Stab
where F˙Stab is a residual-based Least Squares Galerkin stabilization of the form
proportional to the mesh size h, controlled by the duality-based adaptive error control.
F_Stab provides numerical dissipation for the unresolved subscales in a Direct predictive
and adaptive setting.
In previous work on DFS we have chosen
In this work we show a key breakthrough, choosing
for the velocity
component U and
for the pressure component P. This corresponds to an efficiency increase of many
orders of magnitude, which is key for the resolution of the Grand Challenges in fluid
2.1 High Lift Prediction Workshop 4
2.2 NACA0012 wing at aoa=0
We can now present breakthrough results validating RFS for the NACA0012 aoa=0
case, demonstrating that RFS predicts the regime of cruising aircraft and ships
3 Euler CFD as a solution to NASA Vision 2030
We see that Euler CFD with the Real Flight Simulator (RFS) realization already today in
2021 satisfies the goals of the NASA Vision 2030 challenges:
1. Emphasis on physics-based, predictive modeling
Euler/RFS is predictive by being first-principles, parameter-free and mesh/discretization
independent by adjoint-based adaptive error control.
2. Management of errors and uncertainties resulting from all possible sources
Euler is first-principles, and does not have explicit modeling parameters. RFS relies on
adjoint-based adaptive error control to guarantee mesh/discretization-independence, and
additionally automatically generates the low-level source code from mathematical notation
(just a few lines), thus eliminating the possibility of human bugs.
3. A much higher degree of automation in all steps of the analysis process
RFS relies on automated mesh generation based on adjoint-based adaptive error
control, and additionally automatically generates the low-level source code from
mathematical notation (just a few lines) including the adjoint formulation and the adjoint
4. Ability to effectively utilize massively parallel, heterogeneous, and fault-tolerant HPC
We demontrate that Euler/RFS has extremely cheap and fast performance ( 200 core hours),
which allows an extreme effectiveness by being able to run a large number of simulations on in
principle any parallel computer (also e.g. any virtual machine in a cloud setting,
even in a web browser), at a cost affordable to any engineer, researcher or even
5. Flexibility to tackle capability- and capacity-computing tasks in both industrial and
The same answer as above. We demontrate that Euler/RFS has extremely cheap and fast
performance ( 200 core hours), which allows an extreme effectiveness by being able to run a
large number of simulations on in principle any parallel computer (also e.g. any virtual
machine in a cloud setting, even in a web browser), at a cost affordable to any engineer,
researcher or even student.
6. Seamless integration with multidisciplinary analyses that will be the norm in
Euler/RFS is realized in the Digital Math framework in FEniCS, taking the mathematical
notation (just a few lines) as input and automatically generating the low-level source code. We
have demonstrated general fluid-strucure interaction (FSI) generalizations in a very simple and
automated way, and other multidisciplinary generalizations are possible or have been
done in a similar way. Digital Math means an Open Source setting, where it’s easy
and natural to merge and integrate different formulations for e.g. different physical
4 Euler’s Dream
Read Euler, read Euler, he is the master of us all. (Laplace)
In 1755 the German mathematician Euler formulated a mathematical model describing the
flow of air (subsonic) and water with the following prophetic declaration of Euler’s Dream
My two equations contain all of the theory of fluid mechanics. It is not the principles
of mechanics we lack to pursue this analysis but only Analysis (computation), which
is not sufficiently developed for this purpose.
These are Euler’s equations for (unit density) slightly viscous
incompressible fluid flow formulated in terms of fluid velocity
and fluid pressure
as an expression of force balance (Newton’s 2nd Law) and incompressibility complemented by
a slip boundary condition. They read :
is a spatial domain occupied by the fluid with boundary
acting like a solid wall impenetrable to the fluid as expressed by
only forces acting on the fluid (without gravitation) are the internal pressure gradient
as a volume force in
combined with a surface pressure
force from the wall acting
in the normal direction on .
Formally there are no internal viscous shear forces (zero viscosity) and no force tangential to
the boundary (zero skin friction).
In bluff body flow
is the domain filled by fluid flowing past a volume occupied by a solid body at rest in a
coordinate system with the flow velocity being constant at large distance from the body as a
far-field condition. The basic problem in bluff body flow is to determine the pressure
distribution from the fluid on the body with drag and lift as net forces opposite and
perpendicular to the main flow direction in normalized form appearing as coefficients of drag
This is the basic problem of vehicle aero/hydrodynamics including airplanes, ships and cars.
We shall see that computing turbulent solutions of Euler’s equations allows accurate prediction
of drag and lift for a body of arbirtary shape, as a realisation of Euler’s Dream by
As is clear from (2) the Euler equations are parameter-free since viscosity and skin friction
parameters are set to zero. This means that the Euler equations/Euler’s Dream represent
Einstein’s ideal mathematical model as a Theory of Everything ToE for a certain
range of physics (slightly viscous incompressible) flow, that is a mathematical theory
capable of making predictions about reality (drag and lift) without any input of
parameters such as viscosity and skin friction. We give below massive evidence that
computation of turbulent solutions to Euler’s equations is a ToE for fluid mechanics,
and as such very remarkable and useful. But it took 250 years to make computing
powerful enough to make Euler’s Dream come true, and the start for Euler in 1755 was
Eulers French adversary mathematician d’Alembert namely quickly crushed Euler’s grand
plan by showing that Euler’s equations admitted certain solutions (potential solutions)
showing zero drag and lift of a body moving through air or water, in direct contradiction to
observation [22, 32, 33, 29]. This was coined d’Alembert’s Paradox (in fact realised by Euler
before 1755 ), which from start as expressed by Chemistry Nobel Laureate Hinshelwood,
separated practical fluid mechanics (hydraulics) describing phenomena (drag, lift), which
cannot be explained, from theoretical fluid mechanics explaining phenomena (zero drag, lift),
which cannot be observed.
The paradox showed to resist all attempts of resolution by numbers of most able
mathematicians, but zero lift is in particular incompatible with flight, and so had to be
resolved at the dawn to modernity when powered human flight was shown to be possible by
the Wright brothers in 1903. The young ambitious fluid mechanician Ludwig Prandtl took on
the challenge by presenting a resolution in a 10 minute presentation at a 1904 mathematics
conference at Heidelberg of a sketchy 8-page note On flow motion with very small viscosity
[59, 26, 8]. Prandtl discriminated potential flow with zero skin friction claiming that a real
fluid always meets a solid wall through a boundary layer with zero tangential relative velocity
named no-slip, then apparently as a result of sufficient skin friction supposed to cause ”flow
separation by adverse pressure” and drag. Prandtl thus ”resolved” d’Alembert’s
Paradox by declaring that the Euler equations with slip had to be expanded to the
Navier-Stokes equations including boundary layer effects from positive viscosity
with in particular a no-slip boundary condition as somehow an effect of viscosity,
although admittedly ”very small”. But no-slip was an ad hoc assumption which Prandtl
could not justify, since the exact nature of the microscopic or effective macroscopic
contact between fluid and wall was unknown to him and so has remained into our
Prandt started out boldly declaring I have now set myself the task to investigate
systematically the laws of motion of a fluid whose viscosity is very small  (same as
Euler) with the plan of showing a big effect (substantial drag and lift) from a very
small cause (very small viscosity), however as scientific problem something very
delicate by asking for very detailed analysis. Large scale instability is a different
Navier-Stokes equations as presented by Navier in 1823 (clarified by Saint-Venant 1930 
and Stokes  in 1842) were combined with a slip-friction boundary condition (shown below)
and so Prandtl’s no-slip was by no means a necessity, only a convenient assumption (to get rid
of potential flow) as expressed by Prandtl :
By far the most important question in the problem area is the behaviour of fluids at
the walls of solid bodies. One does sufficient justice to the physical processes in the
boundary layer between the fluid and the solid body if one assumes that the fluid
adheres to the wall and that the velocity there is zero or correspondingly equal to
the velocity of the body.
Anyway, the fluid mechanics community was with the help of Prandtl relieved from a
seemingly unresolvable most disturbing paradox as expressed by Hinshelwood, and
so Prandtl in the 1920s was named Father of Modern Fluid Mechanics [61, 67]
based on the Navier-Stokes equations with no-slip and not Euler’s equations with
But there was one main caveat: The Navier-Stokes equations with no-slip have solutions
with boundary layers so thin that computational resolution is impossible with any forseeable
computational power . Prandtl’s resolution thus came with the cost of making
Computational Fluid Dynamics CFD into an impossibility asking for resolution of atomistic
scales in a macroscopic setting, or complicated modeling.
In 2010, Hoffman and Johnson published  a different resolution of d’Alembert’s
Paradox showing that the reason zero-drag/lift of potential flow cannot be observed, is that
potential flow as gradient of a potential satisfying Laplace’s equation, is large scale unstable at
separation as exact solution to Euler’s equations, and so is replaced by solution with a
turbulent wake after separation with drag/lift. In this article we now present Digital Math
reproducible proof of this resolution, by computing turbulent solutions to Eulers
equations with slip from a principle of best possible approximate solution with drag
and lift in close agreement with observations, supported by mathematical analysis
[38, 39]. Here large scale instability is not a small cause in the same sense as very
small viscosity, and so is open to mathematical understanding. The new resolution
was anticipated by Euler expecting separation being different from attachement
As a spin off a New Theory of Flight [37, 36] was developed revealing the true Secret of
Flight  in physical terms, very different from the unphysical lifting line theory
advocated by Prandtl as a follow up of the unhysical Kutta-Zhukowski circulation theory
In 2017 Johan Jansson et. al. finally resolved the NASA Vision 2030 grand challenge by
completing the methodology, and with Digital Math demonstrated prediction of stall
in the Third High Lift Prediction Workshop . Jansson showing that the critical
addition of noise guaranteed the triggering of the instabilities in the 3D slip separation
Stokes  suggested the possibility that a given flow motion does not imply its necessity
 as expression of instability: There may even be no steady mode of motion possible, in
which case the fluid would continue perpetually eddying. This was shown to be a reality in the
new resolution 2010 of d’Alembert’s Paradox. Stokes idea was expressed by the
mathematician Garrett Birkhoff in the 1st edition 1950 of his Hydrodynamics , but
was removed in the 2nd edition because of harsh critcism from the fluid dynamics
community and so only managed to resurface 55 years after Prandtl’s death in the new
Darrigol  (also ) gives a detailed exposition of early work on hydrodynamic stability
by Stokes, Helmholtz, Lord Kelvin and Reynolds, as well as attempts to resolve
d’Alembert’s paradox by Rayleigh, Poncelet and Saint-Venant followed by that by Father
Prandtl, which became the resolution serving the modern fluid mechanics of the 20th
century, although: In summary, Prandtl’s early insights into boundary-layer theory
did not bring him much closer to a practical solution of low-viscosity resistance
problems. The difficulties of the determination of separated flow remain unsolved to this
We now present massive evidence collected in this article in Digital Math reproducible
form showing that computing turbulent solutions of Euler’s equations with slip, which we will
refer to as Euler CFD, opens basically all of slightly viscous nearly incompressible flow
to predictive simulation without parameter input and need to resolve thin no-slip
boundary layers, thus with readily available computing power, all along Euler’s visionary
Euler thus was right in predicting that his dream would come true once the computing
power (Analysis) was strong enough to compute turbulent solutions of the Euler
equations with slip, and Prandtl was wrong claiming drag and lift to be effects of
unresolvable thin no-slip boundary layers making CFD into an impossibility. It took 250
years, but now it is here and sets a new standard in engineering by computational
mathematics opening a window to the Clay Millennium Problem on Navier-Stokes
equations  presented as follows:Mathematicians and physicists believe that an
explanation for and the prediction of both the breeze and the turbulence can be found
through an understanding of solutions to the Navier-Stokes equations. Although these
equations were written down in the 19th Century, our understanding of them remains
minimal. The challenge is to make substantial progress toward a mathematical theory
which will unlock the secrets hidden in the Navier-Stokes equations. It seems that the
secrets were hidden already in the Euler equations and can now be revealed by Euler
5 Digital Math: Scientific Automated Flow Simulation
The scientific process has not kept up with digital technology, and there is today a
reproducibility crisis. Lorena Barba representing NASEM describes the situation
“The widespread use of computation and large volumes of data have transformed most
disciplines of science and enabled new and important discoveries. But this revolution is not yet
reflected in the ways that scientific findings are published and shared with the relevant
communities. Extending the scholarly record to data, software, and computational
environments and workflows is a must to ensure the robustness of science in this digital
We present the Digital Math framework as the foundation for modern science
based on constructive digital mathematical computation, and as a solution to the
reproducbility crisis. The computed result (coefficient vector, FEM function, plot,
etc.) is a mathematical theorem, and the mathematical Open Source code, here in
the FEniCS framework, and computation is the mathematical proof. We can also
derive additional constructive proofs from the FEniCS and FEM formulation, such as
Digital Math represents digitalization of science, mathematics, society and industry in
the form of automated and easily understandable computation of mathematical
models. It is here realized in the Open Source FEniCS framework with world-leading
performance and recognized at the highest level of science and industry together with
an effective pedagogical concept with combined abstract theory and mathematical
interactive programming in a ”one-click” cloud-HPC web-interface, accessible to
anyone: Here is the full adaptive Euler CFD methodology for a standard 3D cube
benchmark case  for you to inspect, run, modify, and reproduce, just as all examples
Computational solution of turbulent solutions of Euler’s equations as Euler CFD is
automated using FEniCS  for automation of the finite element discretisation used to
express the principle of best possible approximate solution. This brings a new tool of
Automated Flow Simulation with only geometry input, which in particular allows for the first
time computation of complete aero-data (forces) for any given airplane/car/ship for design,
Digital Twins, interactive simulators, etc. Euler CFD can include (small) positive boundary
friction allowing also flow before drag crisis to be computed, but it introduces friction as a
coefficient to be fittted to experiments. For high Reynolds number beyond drag crisis - the
regime relevant to aerodynamics - this is not needed, and Euler CFD is completely
6 Euler CFD: Turbulent Solutions of Euler’s Equations
When I meet God, I am going to ask him two questions: why relativity?
And why turbulence? I really believe he will have an answer for the first.
Turbulent solutions to Euler’s equations as Euler CFD are computed as best possible
approximate solutions in the sense of having residuals which are small in a weak
sense and not too large in a strong and sense, in a situation when all solutions with
small strong residuals (laminar solutions and potential solutions in particular) are
unstable (as solutions to Euler’s equations) and thus do not persist over time. We
here face a new situation where only turbulent flow is computable and laminar not,
as an expression of the fluctuating nature of turbulence as a consequence of local
exponential instability, as seen in a waving flag showing the only motion which can
More precisely, the best possible aspect is realised by a finite element method augmenting
small weak residuals from Galerkin orthogonality with weighted least squares control
of strong residuals, the latter introducing a viscous effect as a form of turbulent
viscosity set by computation alone without need to model or measure turbulent
viscosity beyond human comprehension. It connects to Leibniz idea of the Real
World as a Best Possible World; the flag is doing its best possible as well as Euler
CFD showing mesh-independence of drag and lift with readily available computing
Euler CFD takes the following space-discrete variational form: Find
are finite element spaces of continuous piecewise linear velocity-pressure functions
and appropriate far-field
conditions, local finite
element size, and
are constants chosen from a principle of best possible with respect to output measured by the
solution to an associated linearised dual problem. Time stepping is performed using continuous
piecewise linear trial functions and piecewise constant test functions in time (Crank-Nicolson
type). Euler CFD shows insensitivity within quite wide margins to the precise choice of
well as mesh size once small enough to resolve geometry and separation.
Euler CFD performs automatic turbulence modeling through weighted
least squares residual stabilisation of the momentum equation combined
with pressure stabilisation with corresponding turbulent dissipation
as a representative part of the full momentum residual
when not small. Turbulence is then identified by substantial turbulent dissipation
over time (showing mesh size independence) as an expression of impossbility
of computing a solution over time with small momentum residual in a strong
regions of turbulence.
Residual stabilisation of the momentum equation is necessary because weak satisfaction in the form
for all smooth
, does not itself give control
of the kinetic energy ,
the reason being that it is not feasible to choose
Euler CFD includes adaptive duality based a posteriori error control guaranteeing drag and
lift up to a few percent. A key element is here the oscillating character of turbulent flow
leading to cancellation in the linearised dual problem making the dual solution computable and
properly bounded .
Euler CFD includes automatic turbulence modeling through weighted strong residual
control as a dissipative effect with a complex flow dependence beyond viscous shear
stress. It appears as a solution to the open problem of turbulence modeling [?]. In
particular, the weighted strong residual measures the turbulent dissipation as a mesh
independent quantity meeting Kolmogorov’s K41 conjecture of finite turbulent dissipation
7 Predictive Euler CFD - Resolution of the paradox
We show that predictive Euler CFD resolves the paradox. The potential solution with zero
drag is unstable - shown by stability analysis and computational evidence with adaptive
error control. We illustrate the resolution by the basic cylinder model problem,
and also by the most advanced benchmark in the world representing vehicles and
aerodynamic devices - the High Lift Prediction Workshop, where we show that Euler CFD
predicts the experiment to 5% with mesh independence, and predicts they key stall
In Figure 14 we show our resolution of the paradox with Digital Math Euler
CFD: the potential solution is unstable and develops streamwise vortices on the
downstream side of the cylinder, generating “3D slip separation” - separation at high flow
In Figures 7 and 8 we show prediction of stall in the Third High Lift Prediction
Workshop , both CD and CL within 5% of the experiment and mesh-independent (the
experimental drag had a systemic error of appx. 10%). This represents the resolution
of the NASA Vision 2030 grand challenge by completing the methodology with
the critical addition of noise guaranteeing the triggering of the instabilities in the
3D slip separation mechanism., and with Digital Math guaranteeing the scientific
7.1 Car benchmark - DrivAer
7.2 Wing benchmark - NACA0012
7.3 Cylinder benchmark - 3D slip separation
We have showed that computing turbulent solutions to Euler’s equations with a slip boundary
condition offers a Theory of Everything ToE for slightly viscous incompressible fluid flow as a
parameter-free model, we are now able to predict a vast area of applications in vehicle
aero/hydrodynamics including airplanes, ships and cars. This work resolves the Grand
Challenges of fluid dynamics described in NASA Vision 2030.
Key specific results are breakthrough results validating RFS for the High Lift Prediction
Workshops and the NACA0012 aoa=0 case, demonstrating that RFS predicts the regime of
cruising aircraft and ships (hydrodynamics). Additionally we show Digital Math simulations of
similar applications such as the full car DrivAer automotive standard benchmark,
9 Digital Math: Reproducible results and open source software
The software for reproducing the results in the paper is available as part of the distribution for
the Digital Math framework.
Contact Johan Jansson (firstname.lastname@example.org) for instrutions how to run the software in a Google
Cloud virtual machine in an easy web interface (free credits).
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