# Abstract

A new resolution of d’Alembert’ s paradox from 1752 is presented.

The new resolution is based on computational solution of the incompressible inviscid Euler equations with slip boundary condition showing that zero-drag potential flow is unstable and develops into a turbulent flow with substantial drag. The new resolution is entirely different from the official resolution supported by the fluid dynamics community based on Prandtl’s boundary layer theory, and is supported by mathematical analysis, computation and experiment.

**How wonderful that we have met with a paradox. Now we have some hope of making progress. (Niels Bohr)**

**d’Alembert’s Paradox**[1][2] formulated by the mathematician d’Alembert in 1752 [3] compares observation of substantial drag (resistance to motion) in nearly incompressible and inviscid (small viscosity) fluids such as water and air at subsonic speeds, with the theoretical prediction of

**zero drag**of

**potential flow**, which is

**inviscid**(zero viscosity),

**incompressible,**

**irrotational**(zero rotation or vorticity) and

**steady**(time-independent) flow. The pressure and velocity with streamlines in a section of potential flow around a three-dimensional circular cylinder takes the form (with the flow from left to right):

We see full symmetry of in particular the pressure resulting in zero drag. We see the streamlines close behind the body creating high pressure (red) which pushes the cylinder through the fluid without resistance.

**paradox indicates a flaw in the basic theory of fluid mechanics, which must be corrected to maintain scientific credibility**. Evidently something is wrong with the potential solution, and the enigma is what?

## Official and New Resolution

**zero-drag potential solution**is not observed in practice, is that it is

**unstable**, and under small perturbations develops into a

**t**

**ime-dependent**

**turbulent solution with substantial drag.**The new resolution comes out from recent progress in computational turbulence offered by finite element methods with a posteriori error control [9]. The time-dependent turbulent solution has the following form (see movie):

**strong streamwise vorticity**generated by a

**mechanism of instability at rear separation.**We show that the drag can be viewed as a cost associated with separation (well known from divorce).

Mathematics of Fluid Mechanics

**Euler equations [11][9]**, assuming that the fluid is (a1)

**inviscid**(zero viscosity) as an approximation of slight viscosity, and (a2)

**incompressible**. The basic idea was (and still is) to solve the Euler equations and thereby predict fluid flow, which is how weather predictions are made today.

## The Paradox: Zero Drag of Inviscid Flow

*It seems to me that the theory (potential flow), developed in all possible rigor, gives, at least in several cases, a strictly vanishing resistance, a singular paradox which I leave to future Geometers to elucidate.*

- ….
*the boat would be slowed down at the prow as much as it would be pushed at the poop..*.

*, and theoretical fluid mechanics*

**observing phenomena which could not be explained***. A split like that of course is catstrophical from scientific point of view, but we shall see that it remains into our days.*

**explaining phenomena which could not be observed**### Something is wrong, the question is what?

### Suggestion by Saint-Venant

*But one finds another result (non-zero drag) if, instead of an inviscid fluid - object of the calculations of the geometers Euler of the last century – one uses a real fluid, composed of a finite number of molecules and exerting in its state of motion unequal pressure forces having components tangential to the surface elements through which they act; components to which we refer as the friction of the fluid, a name which has been given to them since Descartes and Newton until Venturi.*

**Navier-Stokes equations including forces from friction or viscosity.**In particular the

**slip boundary condition of the Euler equations**letting fluid particles slide along the boundary of the moving body without friction, should be replaced by a

**no-slip boundary condition forcing fluid particles on the boundary to slow down to zero velocity.**A zero-drag potential solution of the Euler equations with slip boundary condition, does not satsify the Navier-Stokes equations with no-slip, and thus can be discarded on this ground, which was argued to be an indirect resolution of the paradox.

**turbulent**without analytical representation. In particular analytical computation of lift and drag was (and is) impossible.

### Resolution by Prandtl

__boundary layer__[17] [18] possibly could be the source of substantial drag, which effectively was Saint-Venant’s idea. And this is still the official resolution of the paradox propagated in all text books of fluid mechanics: What is wrong is the assumption (a1) of inviscid flow; even the slightest viscosity will change the solution completely and give both drag and lift.

**Is the physics described by Kutta-Zhukovsky and Prandtl presented in text books, the correct real physics?**What precisely is the evidence that the official resolution of the paradox by Prandtl is correct?

### Criticism of Official Resolution

*…**great efforts have been made during the last hundred or so years to explain how a vanishingly small frictional force can have a significant effect on the flow properties.*

*Much remains to be done; in particular, the development of a rational theory for three-dimensional, and possibly also unsteady two-dimensional flows may be in its infancy…*

*Hydrodynamics*[19] from 1950 several paradoxes of fluid mechanics including d’Alembert’s paradox, and expresses a clear doubt [20] in their official resolutions:

*I think that to attribute them all to the neglect of viscosity is an unwarranted oversimplification The root lies deeper, in lack of precisely that deductive rigor whose importance is so commonly minimized by physicists and engineers….*

*The concept of a ”steady flow” is inconclusive; there is no rigorous justification for the elimination of time as an independent variable. Thus though Dirichlet flows (potential solutions) and other steady flows are mathematically possible, there is no reason to suppose that any steady flow is stable…*

**is unstable and non-physical,**as shown by the new resolution, but it has taken more than 50 years to come to this conclusion.

### Elimination of Criticism

*The reviewer found it difficult to understand for what class of readers the first chapter was written. For readers that are acquainted with hydrodynamics the majority of the cases cited as paradoxes belong either to the category of mistakes long since rectified, or in the category of discrepancies between theory and experiments the reasons for which are also well understood. On the other hand, the unintiated would be very likely to get the wrong ideas about some of the important and useful achievements in hydrodynamics from reading this chapter.*

*The general view in the**fluid mechanics**community is that, from a practical point of view, the paradox is solved along the lines suggested by Prandtl.**A formal mathematical proof is missing, and difficult to provide, as in so many other fluid-flow problems modelled through the**Navier–Stokes equations**.*

Since the nature of the paradox is mathematical, the fact that a “formal mathematical proof is missing”, means that it is admitted that **Prandtl’s resolution is not a resolution.** Claiming that nevertheless the paradox is resolved, “from a practical point of view”, is not science, only practical politics. The debate between the official and new resolution can be followed on the Discussion of the wikipedia article and on the homepage of the book Computational Turbulent Incompressible Flow.

**Birkhoff´s conjecture of instability of potential flow is correct. The root of the paradox is thus not the assumption of inviscid flow, but instead that the potential solution of the Euler equations, is unstable and therefore cannot be observed in experiments. The potential solution is a mathematical solution which is not stable to small perturbations and therefore does not have any physical significance.**

**fictions without physical reality**. But it is very surprising that the lack of stability analysis of potential solutions identified by Birkhoff has carried the official resolution through the millennium shift.

**how a slightly viscous flow can generate substantial drag**. Prandtls suggests that transversal vorticity from a thin viscous boundary layer is the origin of drag, but evidence is lacking.

**The new resolution offers a completely different explanation, which is supported by theory, computation and experiment, while the official resolution is not.**

## The Origin of Drag and Lift

**turbulence in slightly viscous flow.**Turbulence is viewed to be the main unsolved problem of classical mechanics, and it has remained unsolved because analytical solution of the Navier-Stokes equations has shown to be impossible. But computational solution is today possible with desk top power and this opens to both a resolution of d’Alembert’s paradox and to computing lift and drag of vehicles.

**Computation shows that the zero-drag potential solution over time develops into a turbulent solution of the Euler equations with substantial drag,**which is also referred to as

**blowup**. Computational solution of the Euler eqations shows that the

**drag arises from low-pressure tubes of streamwise vorticity generated at separation,**which gives

**rear suction and thus drag. The streamwise vorticity is generated on the rear body surface from opposing retarding flows, and is enhanced by vortex stretching in accelleration after separation**.

## Wellposedness of Turbulent Euler Solutions

**perturbations of data**have to be taken into account. If a vanishingly small perturbation can have a major effect on a solution, then the solution is

**illposed,**and in this case the solution cannot carry any meaningful information and thus cannot be meaningful from physical point of view. According to Hadamard, only a

**wellposed**solution, for which

**small perturbations have small effects**on certain solution outputs, can be meaningful.

**drag and lift of potential solutions are illposed**, while

**drag and lift of a turbulent solution of the Euler equations are wellposed.**Although wellposedness in the form of hydrodynamic stability is a key issue in the classical fluid dynamics literature, a stability analysis of potential solutions is lacking as well as aspects of wellposedness of turbulent solutions.

## Turbulent Euler Flow around a Car

## Turbulent Flow around a Cylinder

**in the accellerating flow behind the cylinder, as shown on the sketch below. Recall that the pressure must be low**

**inside a vortex tube to accelarate fluid particles radially. We illustrate this scenario schematically as**

(i) formation of surface vorticity (ii) generating streamwise vorticity

### New Resolution vs Prandtl’s Experiment

- no boundary layer prior to separation
- strong counter-rotating low-pressure vorticity in the streamwise direction in the wake.

### Turbulent Flow around a Sphere

### We see that as the friction tends to zero the separation gets delayed and a pattern of four tubes of (counterrotating) streamwise vorticity is formed in the wake with a drag coefficient of about 0.2 with small friction.

### The Secret of Flying and Sailing

The resolution of d’Alembert’s paradox opens to understanding the mechanism of glidig flight as shown in the Knol Why it is possible to fly. Sailing against the wind is based on the same mechanism for both sail and keel.

## The Incompressible Euler Equations

∂u/∂t + (u · ∇)u + ∇p = f in Ω × I,

∇ · u = 0 in Ω × I,

u · n = g on Γ × I,

u(·, 0) = u0 in Ω,

### Computational Solution EG2