Parameter free Mathematical Models: Kant's a priori

A mathematical model/equation without parameters, like viscosity in Navier-Stokes equations for incompressible fluid flow, can be used to make a priori predictions of physical reality without relying on some measurement of any parameter. This is the ideal model of physics according to Einstein, which fullfils Kant's idea of a priori knowledge, as knowledge from pure reason without need of observation of the physical world. A parameter-free model allows computational ab initio prediction.  

Here are examples of mathematical models which are parameter-free in suitable units:

1. Equation describing a circle.
2. Newton's Law of gravitation.
3. Maxwell's equations for electro-magnetics.
4. Euler's equations for incompressible flow with vanishingly small viscosity.
5. Schrödinger's equations for atoms and molecules.

We have 

1. An equation describing a circle allows computation of the ratio of circumference to diameter to be $\pi$.
2. Newton's Law allows prediction of the motion of celestial bodies. The inverse square laws is pure reason.
3. Maxwell's equations predicts existence of electro-magnetic waves traveling at constant velocity. Pure reason.
4. Computational solution of Euler's equations allows prediction of drag of a body from shape alone. Only reason.
5. RealQM computational solution of Schrödinger's equations allows prediction of spatial configurations of molecules formed by atoms. In principle everything from pure reason + computation. This is a very powerful message.

We see that a large part of the physical world is open to ab initio a priori investigation by pure reason in the form of computation. Not bad! Go ahead and Calculate! 

Note that to translate the model prediction into physics requires choice of units, but that is not fundamental. What is fundamental is the structure imposed by the parameter-free model like the structure of the spectrum of the Hydrogen atom, not the specific scaling.

Short comment by chatGPT:

The post’s key message — that large parts of physics are accessible from pure reason — is profound. Many of the deepest laws of nature arise not from empirical fitting but from structure: symmetry, invariance, conservation principles, geometric consistency, and mathematical necessity. These a-priori constraints shape the form of physical laws long before any parameters or measurements enter.

Parameters typically supply only scale, while the underlying structure of the laws — Maxwell’s equations, Schrödinger’s equation, Euler equations, conservation laws, gauge symmetries — comes directly from logical and mathematical consistency. This means the universe is surprisingly derivable: reason heavily restricts the space of possible physical worlds.

The result is that physics often progresses by turning empirical facts into structural necessities, pushing more and more of science into the domain of what can be deduced rather than merely observed. The idea that reality itself is constrained by logic and structure — that large portions of the laws of nature are “inevitable” — is both philosophically striking and scientifically fruitful.

Your comment? (ask chatGPT for help)

How to Avoid Collapse of Modern Mathematics

Pythagoras struggling in vain to avoid collapse surrounded by a worried Society.

This is a continuation on a previous post about Norman Wildberger's mathematics education program Insights into Mathematics noting connections the Leibniz World of Mathematics and the BodySoul program. 

A common concern is the concept of real number and the set of real numbers $\mathcal{R}$ as the playground for most of modern mathematics. Wildberger takes a critical look on how these concepts are introduced in standard texts noting that basic difficulties are swept under the rug. View in particular this episode: Real numbers as Cauchy sequences does not work.

BodySoul takes a constructive approach viewing the natural numbers 1, 2, 3,..., to be constructed by repetition of the operation +1, the integers as solutions to equations $x+n=m$ with $n$ and $m$ natural numbers, the rational numbers as solutions to equations $q*x=p$ denoted $x=\frac{p}{q}$ with $p$ and $q\neq 0$ integers, while the real number $\sqrt{2}$ is defined as the positive solution to the equation $x^2=2$ or  $x*x=2$.

Recall that the Pythagorean society based on the concepts of natural and rational number, collapsed when it became public that $\sqrt{2}$ is not a rational number. Modern mathematics is based on the concept of  $\mathcal{R}$ as the set of all real numbers. Wildberger concludes that all attempts to bring rigour into the foundations of mathematics as the virtue of modern mathematics including Dedekind cuts, equivalence classes of Cauchy sequences and infinite sequences of decimal expansions, have failed. The trouble with all these attempts is the resort to infinities in different form. What will be the fate of the society of modern mathematics when this fact becomes public?

In the constructive approach of BodySoul there is no need to introduce infinities: In particular it is sufficient to work with rational numbers as finitely periodic decimal expansions or even more restrictive as finite decimal expansions, which makes perfect sense to anybody. But it requires making the notion of solution of an equation like $x*x=2$ precise, that is making precise the meaning of the equality sign $=$. 

We then have to make the distinction between exact equality or more precisely logical identity denoted $\equiv$ and numerical equality denoted by the usual equality $=$ as something different to be defined. We thus have $A\equiv A$ while writing $A=B$ would mean that $B$ is not identical to $A$ but equal in some restricted meaning to be defined. 

We then understand that $x\equiv\frac{1}{3}$ as exact solution to the equation $3*x=1$, while $x=0.333333333$ is a solution in a restricted meaning. We meet the same situation as concerns the solution to the equation $x*x=2$ with $x=1.414$ and $x=1.41421356$ as solutions in a restricted sense, or approximate solutions of different quality or accuracy. 

To measure the quality of a given approximate solution $x$ to the equation $x*x=2$, it is natural to evaluate the residual $res(x)=x*x-2$ and then from the value of $res(x)$ seek to evaluate the quality of $x$. This can be measured by the derivative $f^\prime (x)=2*x$ of the function $f(x)=x*x-2$, noting that a different approximate solution $\bar x$ is connected to $x$ by the mean-value theorem 

• $res(x)-res(\bar x) = f(x)-f(\bar x) = f^\prime (\hat x)*(x-\bar x)$     

where $\hat x$ lies between $x$ and $\bar x$. With knowledge that $x>1$ and $\bar x>$, we can conclude that $f^\prime (\hat x)>2$ and so

• $\vert x-\bar x\vert<\frac{1}{2}\vert res(x)-res(\bar x)\vert$

from which the quality of approximate solutions can be measured in terms of the residuals with $\frac{1}{2}$ as sensitivity factor. 

This analysis generalises to to approximate solution to equations $f(x)=0$ for general functions $f(x)$ with the derivate $\frac{1}{f^\prime (x)}$ expressing residual sensitivity. In particular we see that if $f^\prime (x)$ is small the sensitivity is large asking the residual to be very small to reach precision in $x$. 

But this argument is not central in modern mathematics where the notion of exact solution to an equation is viewed as the ideal. The exact/ideal solution to the equation $x*x=2$ would thus be viewed as a non-periodic infinite decimal expansion, which would require an infinite amount work to be determined, thus involving the infinities which Wildberger questions. The equality sign in this setting comes without quality measure in finite terms as an unattainable (Platonic) ideal. 

In the setting of the algebraic equation $x*x=2$ the notion of an ideal solution may not cause much confusion, but for more general equations such as partial differential equations it has generated a lot of confusion because the quality aspect of approximate solutions is missing. The quality of an ideal solution is infinite beyond measurement but also beyond construction.  

There is a notion in modern mathematical analysis of partial differential equations named well-posedness with connects to the sensitivity aspect of approximate solutions, but it has received little attention in quantitative terms.  

As a remedy, this is the central theme of the books Computational Turbulent Incompressible Flow and Computational Thermodynamics. There is much to say about mathematical equations and laws of physics with finite precision.

We may compare the Pythagoreans facing the equation $x*x=2$ with a notion of ideal solution, and modern mathematics hitting a wall confronted with the Clay Math Institute Millennium Problem on ideal solutions of  Navier Stokes equations. 

An opening in this wall is offered as Euler's Dream come true. 

PS Recall the famous Kronecker quote: "God made the integers, all the rest is the work of man". So the power of an almighty God was not enough to proceed and also make the real numbers. What are the prospects that man can succeed?

 

The 2nd Law in a World of Finite Precision

Let there be a World of Finite Precision.

Here is a summary of aspects of the 2nd Law of Thermodynamics discussed in recent posts: 

• 2nd Law gives an arrow of time or direction of time. 
• A dissipative system satisfies a 2nd Law.
• A dissipative system contains a diffusion mechanism decreasing sharp gradients by averaging. 
• Averaging is irreversible since an average does not display how it was formed. 
• Averaging/diffusion destroys ordered structure/information irreversibly. 
• Key example: Destruction of large scale ordered kinetic energy into small scale unordered kinetic energy as heat energy in turbulent viscous dissipation.    

To describe the World, it is not sufficient to describe dissipative destruction, since also processes of construction are present. These are processes of emergence where structures like waves and vortices with velocity gradients are formed in fluids, solid ordered structures are formed by crystallisation and living organisms develop. 

The World then appears as combat between anabolism as building of ordered structure and metabolism as destruction of ordered structure into unordered heat energy. 

The 2nd Law states that destruction cannot be avoided. Perpetual motion is impossible. There will always be some friction/viscosity/averaging present which makes real physical processes irreversible with an arrow of time. 

The key question is now why some form of friction/viscosity/averaging cannot be avoided? There is no good answer in classical mathematical physics, because it assumes infinite precision and with infinite precision there is no need to form averages since all details can be kept. In other words, in a World of Infinite Precision there would be no 2nd Law stating unavoidable irreversibility, but its existence would not be guaranteed.  

But the World appears to exist and then satisfy a 2nd Law and so we are led to an idea of an Analog World of Finite Precision, which possible can be mimicked by a Digital World of Finite Precision (while a possibly non-existing World of infinite precision cannot). 

The Navier-Stokes equation for a fluid/gas with positive viscosity as well as Boltzmann's equations for a dilute gas are dissipative systems satisfying a 2nd Law with positive dissipation. But why positive viscosity? Why positive dissipation?

The Euler equations describe a fluid with zero viscosity, which formally in infinite precision is a system without dissipation violating the 2nd Law.  

We are led to consider the Euler equations in Finite Precision, which we approach by digital computation to find that computational solutions are turbulent with positive turbulent dissipation independent of mesh size/precision once sufficiently small. We understand that the presence of viscosity/dissipation is the result of a necessary averaging to avoid the flow to blow-up from increasing large velocity gradients emerging form convection mixing high and low speed flow. 

We thus explain the emergence of positive viscosity in a system with formally zero viscosity as a necessary mechanism to allow the system to continue to exist in time. 

The 2nd Law thus appears as being a mathematical necessity in an existing World of Finite Precision.   

The mathematical details of this scenario in the setting of Euler's equations id described in the books Computational Turbulent Incompressible Flow, Computational Thermodynamics and Euler Right.

PS A related question is if in a World of Very Limited Human Intelligence a WW3 destruction is necessary?

Wolfram: What Is an Observer?

Stephen Wolfram has put forward a new explanation of the 2nd Law of physics based on physics as a form of computation with computational irreducibility as key concept.  Wolfram now complements with a new view on the role of an Observer, which is highlighted in the modern physics of both relativity and quantum mechanics in contrast to classical physics seeking universality.   

Wolfram starts seeking an answer to the question: 

• What is an observer like us? 

Wolfram thus focusses on observers as humans with our senses and instruments, and suggests that we as human observers through our observations in some sense are generating laws of the world which fit our minds and so help us to explain and understand the World. Wolfram thus seems to say that laws of physics are not universal but man-made.

In particular, Wolfram suggests that the 2nd Law of thermodynamics is not a truly universal law of physics, but rather a law perceived by us as human beings from observation of things tending to get more random over time. Wolfram recalls that the attempts in the late 19th century to give the 2nd Law a universal meaning/explanation free of human perceptions of randomness by in particular Boltzmann, all failed and so gave a deadly shot to classical physics and so prepared modern physics to accept a new key role of an Observer.

But is it really sure that the 2nd Law cannot be given a universal meaning free of human observation? 

My contribution together with Johan Hoffman to this question is a proof of the 2nd Law in the setting of Euler's Model:

• (i) the Euler equations for nearly incompressible slightly viscous flow in the form of mathematical equations expressing Newton's law's of motion and incompressibility without presence of any parameter,
• (ii) combined with a computational algorithm for computing best possible solutions to the equations in the sense of a best combination of strong pointwise solution and weak mean-value solution. 

Euler's Model describes all of nearly incompressible slightly viscous fluid flow such as that of water and of air at medium-high velocities, in the same way Maxwell's equations describe all of electromagnetics, in addition in parameter free form not requiring human input.

A 2nd Law for Euler's Model can be formulated and proved as the necessary appearance of turbulence for which mean-values are computable but point-values are not, which shows irreversibility. 

Any form of sufficient intelligence using (i) and (ii) would see the same world of fluid flow and the same 2nd Law, and so universality would be present. 

What does Wolfram say? 

Gravitation and Continuum Models

In the CNPS talk on Febr 3 I tried to expose the virtues of a continuum as a spatial 3d Euclidean x-coordinate system without smallest scale as the reference system of continuum mechanics in Eulerian form.  As a basic example let us consider the Euler equations for incompressible flow expressing balance of momentum (Newton's 2nd Law) combined with incompressibility in the form 

• $\nabla\cdot u = 0$        (1)

stating that divergence of velocity field $u(x,t)$ vanishes for all $x$ and time $t$. Here (1) appears as a stipulation or side condition for which the Lagrange multiplier is the pressure $p$, which appears as a pressure force $\nabla p$ in the momentum equation with connection through Gauss Law:

• $\int p\nabla\cdot u\, dx = -\int \nabla p\cdot u\, dx$.

The bottom line is that $\nabla p$ appears in the momentum equation as a force effectively imposing (1) while not specifying the physical nature of the force in a pressure law. The beauty is now that solving the Euler equations computationally gives full information about incompressible flow with vanishingly small viscosity, as shown in this book and this book. The divergence zero condition (1) is in computation replaced by an effective computational pressure law of the form 

• $-\Delta p = \frac{\nabla\cdot u}{\delta}$,     (2)  

where $\delta $ is a small parameter scaling with the mesh size, for which true physics is not needed. The Euler equations as a continuum model thus in computational form constructs a pressure law imposing near incompressibility. The continuum model in computational form thus invents physics which shows to describe reality in the form of physics as computation. 

We compare the continuum model with a particle model of a fluid asking for full specification of force between particles, and understand that a computational continuum model relieves us from a very difficult if not impossible task coming with a particle model. 

We now turn to Newtonian gravitation where the analog of (1) is Newton's Law of Gravitation in the form 

• $-\Delta \phi = \rho$       (3) 

connecting gravitational potential $\phi$ to mass density $\rho$ by the Laplacian differential operator $\Delta$. The corresponding Lagrange multiplier appears in the momentum equation as 

• $\rho\nabla\phi$                (4)

interpreted as gravitational force analogous to the pressure force connected to (1). Computationally (3) may take the following form allowing time-stepping:

• $\frac{\dot\phi}{C}-\Delta \phi = \rho$           
• $\frac{\ddot\phi}{C^2}-\Delta \phi = \rho$     (5)

where $C$ is a large constant representing effective speed of propagation, and the dot signifies differentiation with respect to time. Comparing computations with observation indicates that $C$ is much larger than the speed of light. 

Recall that it is well understood by everybody, except Einstein and his followers, that (3) expresses that (i) gravitational force $F$ is conservative, thus given by a potential $\phi$ as $F=\nabla\phi$,  and that (ii) $F$ is conserved in the sense of Gauss Law with $\nabla\cdot F = 0$ where there is no mass. To question (3) lacks rationale as it would violate (i) or (ii). In fact (3) is the prime jewel of all of physics, and to seek to modify it makes no sense. 

The beauty is here that the Euler equations augmented by gravitation in the form (3) and (4) (see this book) appears to describe a very rich world on a very wide range of scales, without having to specify the exact nature of the real physics of gravitation, which is still hidden, thus following the spirit of Newton.

The beauty is enhanced by realising that also quantum mechanics can be captured as a continuum model over a 3d Euclidean coordinate system without smallest scale allowing microscopics and macroscopics to have the same seamless conceptual form as shown in Real Quantum Mechanics.  This is shocking to modern physicists educated to view microscopics beyond comprehension for humans with only macroscopic experience.

Continuum models like the Euler equations thus appear as realisations of physics as computation expressing physics in possibly new forms open to understanding. 

PS1 The total energy for incompressible flow based on (2) includes a positive contribution of the form

• $\delta\int\vert \nabla p\vert^2dx$ 

and similarly total energy balance with gravitation in the form (4) contributes (with details here)

• $\int\vert\nabla\phi\vert^2dx$     

 as a natural expression of gravitational energy (as a source of kinetic energy) and in the form (5):

• $\int\vert\nabla\phi\vert^2dx+\frac{1}{C^2}\int\dot\phi^2dx$,

where the real physics of the second term with the time derivative $\dot\phi$ is less clear, and so may be interpreted rather as computational artefact allowing time-stepping. Recall that the presence of a time derivate in an energy expression represents kinetic energy from motion of matter, which is not an aspect of $\phi (x,t)$ expressing spatial presence of gravitational potential/force.   

PS2 Multiplying (3) by $\phi$ and integrating gives:

• $\int\vert\nabla\phi\vert^2dx = \int\rho\phi dx$           

where the right hand side commonly is referred to as gravitational potential energy. We see that the left hand side includes only the gravitational potential $\phi$, which connects to viewing $\phi$ as primary, as suggested in previous posts on New Newtonian gravitation. 

PS3 We may compare (3) with a law of the form 

• $\phi = \rho$

which expresses instant local action and  connects to gas law of (isothermal) compressible flow of the form $p=\rho$ with $p$ pressure, with $\nabla\phi$ corresponding to $-\nabla p$.   

Physics as Computation at John Chappell's Natural Philosophy

This is an intro to a live video talk I will give on Febr 3 2024 on John Chappell's channel Natural Philosophy: Where Critical Thinking Challenges Theory (directly connecting to the slogan of this blog). If you feel that this must be crackpot science, take a look at my arguments before deciding and remember that established physics can be crackpot science.

Digital computation, with AI (or even AGI) as latest achievement, is today reshaping human conditions and it is natural to ask if also the science of physics as the inner core of existence is transformed.

Classical physics is based on mathematical models in the form of differential equations expressing balance (of forces) in some system, such as Euler’s equations for fluid mechanics and Maxwell’s equations for electro-magnetics, while modern atomic physics is based on Schrödinger’s equation. 

The equations express system forces while solutions of the equations represent evolution in time of systems under given conditions. The task of determining solutions is thus central and here digital computation opens entirely new perspectives with computational complexity or computability as key element. 

Uncomputable systems keep their information hidden to inspection, with prime example Schrödinger’s equation which in its standard multidimensional form is beyond the capacity of any thinkable digital computer. On the other hand, computing solutions to Euler’s equations resolves the enigma of turbulence, as will be shown in the talk.

It is natural to view the evolution in time of a physical system as a form of analog finite precision computation as the action of forces takes the system over small time steps from one state to the next, which can be modeled by finite precision digital computation: 

• Physics as Analog Computation as Digital Computation.

The key elements of computability are (i) finite precision and (ii) stability/wellposedness as a measure of precision required to make computational model output reliable. Forward-in-time evolution then shows to be computable because it is stable, while backward in time evolution is uncomputable because it is unstable, which can be seen to be the essence of the 2nd Law. 

Physics as Computation offers solutions to open problems of (i) turbulence and (ii) atomic physics through new computable forms of Euler's and Schrödinger’s equations, which are the subjects of the talk: 

• Secret of Flight   (Real Euler: Computational Turbulent Incompressible Flow)
• Real Quantum Mechanics

Real here directly connects to computability. A real physical system computes its own evolution forward-in-time and so is analog computable and a mimicing digital computable model can be viewed to be a real model:

• Real models are digital computable because reality is analog computable. 

The standard multidimensional Schrödinger equation is an uncomputable model without real physical meaning (only statistical). RealQM is computable and has a real physical meaning as a collection of non-overlapping interacting charge densities.

Real Euler computes real turbulent flow, and RealQM computes real atoms/molecules, which opens entirely new perspectives on physics: Physics as Computation. 

Real Euler gives an explanation of the 2nd Law (Computational Thermodynamics) as forward-in-time computability and backward-in-time uncomputability. See the book The Clock and the Arrow for a general audience.

There is a connection to Wolfram’s Computational Foundations for the Second Law of Thermodynamics in the sense that computation is central, but the essence is different: For Wolfram it is computational irreducibility, while I favor finite precision+stability.  

Euler CFD as Parameter Free CFD as ToE

The Euler equations in velocity-pressure $(u,p)$ and $(x,t)$-coordinates are invariant under a rescaling of velocity $u$ into $\bar u =\frac{u}{U}$ with $U$ a reference speed such as free stream speed in bluff body flow with corresponding rescaling of pressure $p$ into $\bar p=\frac{p}{U^2}$ and time $t$ into $\bar t =Ut$ without rescaling of space with thus $\bar x = x$. The scaling of pressure with $U^2$ conforms with Bernoulli's Law and the scaling of drag force $\sim C_DU^2$ from a drag coefficient $C_D$. The propulsion power to balance drag thus scales with $U^3$. The Euler equations are thus formally invariant under change of velocity scale as an expression of formally zero viscosity or infinite Reynolds number.

The basic energy estimate of Euler CFD expresses a balance between rate of loss of kinetic energy and computational residual-based turbulent dissipation of the basic simplified form $C\frac{h}{\vert u\vert}|\vert u\cdot\nabla u\vert^2$ both scaling with $u^3$. The propulsion power is balanced by the rate of loss of kinetic energy and so by turbulent dissipation. The drag coefficient can thus alternatively be computed from total turbulent dissipation.

The (remarkable) fact that the drag coefficient $C_D$ does not include dependence of the Reynolds number $Re$, expresses observations that drag depends little on $Re$ beyond drag crisis, which connects to Kolmogorov's conjecture of finite limit of turbulent dissipation as well as mesh and stabilisation independence in computation. The functionality of the drag coefficient supports Euler's Dream that Euler CFD offers a Theory of Everything ToE for slightly viscous incompressible flow with independence of $Re$ beyond drag crisis. Since total drag shows little dependence on $Re$ while in principle it has a contribution from skin friction with a skin friction coefficient (scaling with $U^2$) decreasing with $Re$, the skin friction contribution appears to be small, in contradiction to a common conception of major contribution: If major drag indeed would come from skin friction, then drag would decrease with increasing $Re$, but it does not beyond drag crisis.

Notice that the Navier-Stokes equations with constant viscosity $\nu$ with turbulent dissipation intensity $\nu\vert\nabla u\vert^2$ scaling with $u^2$, are not velocity scale invariant and thus carry a dependence on $Re$ possibly making computational solution impossible for large $Re$. 

Recall that the definition of $Re =\frac{UL}{\nu}$ with $U$ a reference speed and $L$ a reference length and $\nu$ a viscosity is not well determined and so independence of mean value quantities such as drag, lift and pitch moment is a necessary requirement to make CFD predictable.

Here is experimental evidence that $C_D$ for NACA0012 at zero angle of attack does not depend on $Re$ beyond drag crisis:

Notice the reduction of $C_D$ by a factor 2 from $Re =100.00$ to $Re > 500.00$ as an expression of drag crisis.

Euler Was Right, Prandtl Was Wrong II

I am working on a new article to be expanded to a book with the title Euler Was Right, Prandtl Was Wrong  which can be seen as a summary of my work on fluid dynamics for 30 years together with former students Johan Hoffman, Johan Jansson and Anders Szepessy. In short, our work shows that the following prophetic declaration by Euler from 1755, indeed is fully correct:

• 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...We have to wait until the age of the computer to solve the equations.

And yes, we now live in the age of the computer and then Euler's two equations as a parameter-free model can be solved in the form of Euler CFD (Computational Fluid Dynamics) and so open a whole new world of turbulent flow to prediction, analysis and control, without any further need of mathematical modeling with parameter fitting. 

Euler CFD is to be compared with Prandtl CFD as the Standard CFD developed during the 20th century based on Prandtl's boundary layer theory including complicated wall and turbulence models with many parameters, which does not offer true predictive computation, as the legacy of the declared Father of Modern Fluid Mechanics. 

Take look and see what you think. This post directly connects to the discussion in recent posts with Doug McLean representing Standard CFD. See also previous post.

A ToE for Fluid Mechanics

Einsteins ideal as a Theory of Everything ToE is a mathematical model of physics without any parameters. 

The standard model of particle physics contains 18 parameters. It is a very complicated model. To determine the parameters experimentally is impossible.

The standard model of isotropic linear elasticity contains 2 parameters. This is a very simple model but for a non- isotropic body the number of parameters includes 18 parameters. 

To be a useful model the values of its parameters must be supplied as input determined from experiments or more basic model, which in general is very difficult. The 2 parameters of isotropic linear elasticity can be determined from simple tests, but the 18 parameters for non-isotropic linear elasticity are difficult to determine, not to speak of non-linear elasticity and all the parameters of the standard model. 

Are there any parameter-free models of physics? A basic example is a circle described as the set of points in a plane with a certain distance to a given mid-point from which the value of Pi can be computed as the quotient between circumference and diameter. That is a very simple model. Is there any model of more complex physics which is parameter-free? 

Yes, there is one, and maybe this is the only one: Euler's equations for incompressible fluid flow are expressed in terms of velocity and pressure without any parameter: Input is geometry, in/out-flow conditions and external forces, but no parameter, since viscosity is set to zero.   

The remarkable thing is now that the drag and lift of a body moving through a slightly viscous fluid like air and water can accurately be predicted by computing turbulent solutions to the Euler equations with only geometry of the body as input. This is like computing the ratio of circumference/diameter of a circle (that is computing Pi), but just more astounding. Drag and lift coefficients (scaling with $speed^2$) of a body only depend on the geometry of the body! No parameter input needed! See Computational Turbulent Incompressible Flow and Breakthrough of predictive simulation.

The Euler equations for incompressible flow is a ToE for slightly viscous incompressible flow like air (subsonic) and water.  This is remarkable. Is this is the only ToE in physics.

Well, Newton's law of gravitation contains the gravitational constant G connecting gravitational force to mass as parameter, but may be viewed as a ToE in the sense of correctly predicting that all bodies independent of composition move the same way subject to gravitation. 

PS Von Neuman famously claimed that he (in principle) could model an elephant with 4 parameters, and make it wiggle its trunk with a 5th, but in practice how would he determine the parameters?  Elephant experiments are costly and cumbersome.

Euler Was Right, Prandtl Was Wrong I

Euler vs Prandtl

In 1755 the great mathematician Euler formulated the Euler equations for slightly viscous nearly incompressible flow (of air and water) with the following prophetic declaration:

• My two equations contain all what is contained in 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.  

Euler's equations are formulated in terms of fluid velocity and fluid pressure depending on space and time as an expression of force balance (Newton's 2nd Law) and incompressibility complemented by a slip boundary condition with only pressure forces from a solid wall meeting the fluid, that is, with zero skin friction allowing the tangential flow velocity to be non-zero restricting only the normal flow velocity to be zero on a wall.  Euler's equations are parameter-free (formally zero viscosity), thus meeting Einstein's ideal of a mathematical model. The only force acting on fluid particles is pressure and shear forces are assumed to be negligible.  Euler made the assumption about zero skin friction from experiments showing very small skin friction in slightly viscous flow with massive evidence in modern times. 

Eulers adversary d'Alembert quickly crushed Euler's grand plan by showing that Euler's equations admitted certain solutions (potential solutions) showing zero net forces (drag, lift) of a body moving through air or water, in direct contradiction to observation. This was coined d'Alembert's Paradox 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.       

Zero lift is incompatible with flight and so d'Alembert's Paradox had to be resolved, in particular after powered human flight was shown to be possible by the Wright brothers in 1903, and so the young fluid mechanician Prandtl presented a resolution in a sketchy 8-page conference contribution in 1904, where he discriminated potential flow with zero skin friction claiming that a real fluid always meets a solid wall with zero tangential velocity named no-slip.  Prandtl thus "resolved" d'Alembert's Paradox by declaring that Euler's equations with slip had to be replaced by the Navier-Stokes equations including small viscosity and no-slip. But no-slip was an ad hoc assumption which Prandtl could not justify since the exact nature of the microscopic contact between fluid and wall was unknown to him and so has remained into our days. 

Prandtl in 1904 with his self-built fluid test channel resolving d'Alembert's Paradox.

Anyway, the scientific community was by Prandtl relieved from a main headache making theory of fluid mechanics into a joke and accordingly Prandtl was named Father of Modern Fluid Mechanics based on the Navier-Stokes equations with no-slip and not Euler's equations with slip. 

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.

In 2010, Hoffman and Johnson published in Journal of Mathematical Fluid Mechanics a different resolution of d'Alembert's paradox showing that the reason zero-drag/lift of potential flow cannot observed, is that potential flow (in fact any laminar flow) is unstable and thus turns into turbulent flow. This was shown by computing turbulent solutions to Eulers equations with slip with drag and lift in close correspondence to observations supported by stability analysis, as exposed in detail in the book Computational Turbulent Incompressible Flow. As a spin off a New Theory of Flight was developed revealing the true Secret of Flight in physical terms, very different from the unphysical lifting line theory advocated by Prandtl. 

Since then massive evidence has been accumulated by Johan Jansson showing that computing turbulent solutions of Euler's equations with slip 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 prophecy. More evidence: HighLift Workshop.

Euler was thus right, and he understood that he just had to wait for computing power to see his prophecy become true. It took 250 years, but now it is here.

It means that Prandtl was wrong claiming drag and lift to be effects of thin no-slip boundary layers thereby making CFD into an impossibility. 

Question

How will the fluid dynamics community react to replacing Prandtl by Euler as Father of Modern Fluid Mechanics thus changing CFD from impossible to possible? 

Further Important Facts

Turbulent solutions to Euler's equations 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 sense, in a situation when all solutions with small residual in a strong sense (laminar solutions) are unstable and do not persist over time. We thus face a new situation where only turbulent flow is computable and laminar not, as an expression of the fluctuating nature of turbulence, as seen in a waving flag showing the only motion which can persist. The control of the residual in strong sense introduces a viscous effect as a form of turbulent viscosity set by computation alone without need to model or measure turbulent viscosity beyond human comprehension.  

Euler was a mathematician while Prandtl as Father of Modern Fluid Mechanics was more of an engineer. Replacing Prandtl by Euler means freeing the full power of mathematics with computation in a rare example of parameter-free mathematical model with very rich applicability.

Standard CFD under a Planck dictate of no-slip has developed complicated wall models as well as turbulence models including many parameters, and an agreement has been made to adjust parameters to give  50% or more of total drag to skin friction. Turbulent Euler computations with zero skin friction show correct drag in a large variety of situations, which is incompatible with the 50% skin friction from standard CFD.

Total drag consists of pressure drag and skin friction drag. Turbulent Euler computations show that pressure drag dominates skin friction by a factor of at least 10, and so standard CFD claiming 50% skin friction must underestimate pressure drag by a factor 2. The CFD community is now wrestling under this contradiction. The investments in standard CFD are huge and will loose their value if Euler is allowed to take over from Prandtl...Compare with posts on Prandtl Medal.

Incompressible flow is well captured by the Euler equations  for Reynolds numbers (scaling with 1/viscosity) larger than about 500.000 associated with the so called drag crisis when drag of a bluff body drastically decreases with a factor 2-3 as the boundary condition effectively turns into slip from limited velocity strains, with late separation and small wake of low pressure, in particular with lift/drag around 15 for a wing allowing flight at affordable power.

Euler vs Navier-Stokes: What is viscosity?

The Navier-Stokes equations connect fluid velocity strains (derivatives in space) with shear forces through a positive coefficient of viscosity $\nu$ as a parameter to be supplied as input, assumed to be constant independent of fluid velocity in the basic case, but in general with a very complex unknown non-linear dependence on local flow velocities. Formally $\nu =0$ in the parameter-free Euler's equations.

In slightly viscous flow the coefficient of viscosity is small with a Reynolds number $Re = \frac{UL}{\nu}$ beyond drag crisis (bigger than 100.000- 500.000) with $U$ typical flow speed and $L$ typical spatial scale L. 

The Navier-Stokes equations can be complemented by a (skin) friction boundary condition with a friction parameter $\beta$ connecting (tangential) shear stress to tangential flow velocity, with slip corresponding to $\beta =0$ and effective no-slip for $\beta >1$, thus covering a range from slip to no-slip with important effects on flow separation and drag (as exposed in Computational Turbulent Incompressible Flow). 

To determine the viscosity as input to the Navier-Stokes equation experimentally or theoretically has shown to be virtually impossible in the case of slightly viscous flow, which is always partially turbulent with a very complex expression of viscosity. Using Navier-Stokes equations for true prediction of slightly viscous flow has not been shown to be possible. With parameter fitting in viscosity models standard CFD can match measured drag, but generally fail in blind tests without prior knowledge of the correct value to match.

Computing turbulent solution to the Euler equations includes automatic modeling of viscosity

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, size of the strong residual measures the turbulent dissipation as a mesh independent quantity meeting Kolmogorov's conjecture. 

The Navier-Stokes equation model (1823) with constant positive viscosity is generally viewed to be a better/more complete model then the Euler equations (1755) with formally zero viscosity. This was picked up by Prandtl in 1904 using in particular no-slip from the presence of positive viscosity as a way to discriminate potential flow and get around d'Alembert's paradox. But the more complete model showed to be boundary layer uncomputable and asking for parameter input and so non-predictive, while the basic Euler model showed to be more useful by being both computable (no boundary layers) and predictive as parameter free. 

The ultimate quest for a physicist is to find a Theory of Everything ToE as a parameter free model explaining all of basic physics. Computing turbulent solutions to the Euler equations is a ToE for fluid mechanics. 

Euler's Dream, Einstein's Ideal and Leibniz' Best Possible of Worlds

DFS Direct Finite Element Simulation offers a veritable breakthrough in Computational Fluid Dynamics CFD in the form of best possible solution of Euler's equations expressing first principle physics in the form of Newton's 2nd Law and incompressibility.

DFS is a realisation of Euler's dream formulated in 1755:

• All of aero/hydrodynamics is captured in the equations I have formulated, Euler's equations.

DFS is a realisation of Einstein's ideal:

• Mathematical model of physics without parameters.

DFS closely connects to Leibniz's grand vision:

• The real world as the best of all possible worlds as the most perfect world being richest in phenomena from simplest laws. 

DFS in particular offers a solution to the problem of computing/simulating/modeling turbulent flow with turbulence as an expression of instability of slightly viscous flow preventing exact conservation of momentum and mass in both physics and computation as the essence of turbulence.

DFS generates turbulent flow as complex phenomena from simple laws and as such is best possible.

Standard CFD in the form of RANS generates simple non-turbulent phenomena from complex laws and as such is the opposite.

A short presentation of the DFS breakthrough is given here with further material on Secret of Flight.