How wonderful that we have met with a paradox. Now we have some hope of making progress. (Niels Bohr)
Zero Drag of Potential Flow
d’Alembert’s Paradox  formulated by the mathematician d’Alembert in 1752  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):
Symmetric pressure with zero drag. Streamlines close behind the body.
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.
Potential flow predicts that motion through water and air is without resistance, which is against all experience. This 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
The official resolution of the paradox propagated in fluid mechanics text books and on wikipedia, puts the blame on the assumption of inviscid flow. Following a suggestion by Ludwig Prandtl (called the father of modern fluid mechanics) from 1904  going back to an idea of Saint-Venant from 1846 , it is argued that the presence of the slightest viscosity will generate substantial drag from effects of a thin viscous boundary layer. However, despite strong efforts no real evidence that the official resolution is correct has been presented. It remains as a speculative cover-up of a yet undetected flaw in theory .
Recently a new resolution has been presented   which shows that the reason the zero-drag potential solution is not observed in practice, is that it is unstable, and under small perturbations develops into a time-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 . The time-dependent turbulent solution has the following form (see movie):
Low pressure in wake behind. Unsymmetric flow pattern.
We see that the pressure is low in a wake behind the cylinder, which creates substantial drag. We show below that the wake consists of low-pressure counter-rotating tubes of 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).
We present first the official resolution and then the new resolution including some applications.
Mathematics of Fluid Mechanics
Everybody knows by experience that the drag or resistance to motion through a slightly viscous (small viscosity) flow such as air or water, becomes very substantial as the velocity of motion increases. It is also widely known by experience that the drag increases roughly quadratically with the speed, so that doubling the speed increases the drag by a factor four.
Further, it is known that water is nearly incompressible, as is the flow of air with subsonic velocities below say 200 miles an hour. To fly an upward force or lift has to be generated by the air flow around a wing, and the enigma of gliding flight is how to create a ratio lift/drag of size 10-20.
Fluid mechanics as a science started with the development of Calculus by Leibniz and Newton in the late 17the century, and the basic equations for fluid flow were formulated by the son and father Bernouilli and the great mathematician Euler in the 1740s. The prospects of using Calculus to describe the world looked very good (and still do), in particular for the fluid mechanics of slightly viscous and almost incompressible flow such as air and water, which compactly can be described by a system of differential equations namned the Euler equations , 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
The first real test of Calculus in fluid mechanics was formulated in 1749 as a prize problem of the Berlin Academy of Science asking for computation of the drag of a body moving through an inviscid and incompressible fluid. This was (and still is) of great importance in the design of boats, and today also of cars and airplanes, replacing time-consuming and expensive experimental testing in wind tunnels and test tanks. The problem was quickly solved by the mathematician d’Alembert, who computed special solutions of the Euler equations named potential solutions with the additional properties of being (a3) steady (independent of time) and (a4) irrotational (zero vorticity or rotation). D’Alembert was much surprised to find that the drag (and also lift) of a potential solution is zero! The prediction of Calculus was thus that it []:
- 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.
Euler had come to the same conclusion of zero drag of potential flow in his work on gunnery  from 1745 based on the observation that in potential flow the high pressure forming in front of the body is balanced by an equally high pressure in the back, in the case of a boat moving through water expressed as
- ….the boat would be slowed down at the prow as much as it would be pushed at the poop...
Mathematical fluid mechanics thus predicted that it should be possible to cross the ocean without both sail or engine. Equally surprising was that the lift of potential flow around a wing was also zero, and thus the gliding flight of birds could be proved mathematically to be impossible.
The predictions of zero drag and lift were of course laughable and d’Alembert was not awarded any prize. The first test of the applicability of Calculus in fluid mechanics was not successful and thus from start mathematical fluid mechanics was discredited by engineers. In the words of the chemistry Nobel Laureate Sir Cyril Hinshelwood this resulted in an unfortunate split between the field of hydraulics, observing phenomena which could not be explained, and theoretical fluid mechanics explaining phenomena which could not be observed. A split like that of course is catstrophical from scientific point of view, but we shall see that it remains into our days.
Something is wrong, the question is what?
Mathematics is rational, so either (I) there is something wrong with the assumptions, or (II) there is something wrong with the solution of the equations expressing the assumptions. It must be either (I) or (II) or both. If we go for (I), then at least one of the assumptions of potential flow: (a1) inviscid, (a2) incompressible, (a3) steady, (a4) irrotatonal, must be wrong.
Suggestion by Saint-Venant
Calculus flourished as the industrial revolution picked up in beginning of the 19th century, but nobody could solve d’Alembert’s paradox. The next attempt was made by the French mathematician Saint-Venant in 1846  who targeted (a1):
- 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.
In other words, Saint-Venant suggested that instead of the Euler equations one should consider the 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.
However, a direct proof showing that the a solution of Navier-Stokes equations with small viscosity has substantial drag, was out of reach, because solutions of slightly viscous flow are turbulent without analytical representation. In particular analytical computation of lift and drag was (and is) impossible.
Resolution by Prandtl
In 1903 the brothers Orwille and Wilbur Wright demonstrated that motored heavier-than-air flight was possible, in contradiction to potential flow with zero lift, and the mathematicans Kutta and Zhukovsky then quickly modified the potential solution by adding a large scale rotation of air around a wing, and in this way obtained lift , but still no drag. Fluid mechanics under increasing pressure was then rescued by the German physicist Ludwig Prandtl, called the father of modern fluid mechanics, who in 1904 in the short report Motion of fluids with very little viscosity , suggested that the effects of a thin viscous boundary layer   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.
We are thus supposed to believe, when we lean back before take-off in a jumbojet, that the physics of flying is well understood on the basis of of Kutta-Zhukovsky modified potential flow for lift and/or Prandtl’s boundary layer theory for drag. But is it? 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
A reading of Prandtls report shows that Prandtl does not claim to have resolved the paradox, and there is in fact no original research claiming a resolution. What can be found is second hand information suggesting that a no slip boundary condition causes a retardation (tripping) of the flow near the boundary, which possibly may lead to generation of transversal vorticity and separation of the flow with an attached low-pressure wake. Evidence that this actually occurs in fluids with very small viscosity is missing in Prandtl’s report and elsewhere.
The nature of the resolution attributed to Prandtl in the form of a vanishingly small cause (vanishingly small viscosity) having a large effect (substantial drag), makes the resolution of the paradox difficult, or even impossible, to either verify or disprove by theory, computation or experiment. This is illustrated by Stewartson in the long 1981 survey article :
- …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.
Despite the great efforts, Stewartson cannot give much hope and in the summary he states that:
- 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…
The mathematician Garrett Birkhoff addresses in the opening chapter of his book Hydrodynamics  from 1950 several paradoxes of fluid mechanics including d’Alembert’s paradox, and expresses a clear doubt  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….
In particular, on d’Alembert’s paradox, he critizises the lack of stability analysis of potential solutions:
- 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…
Birkhoff evidently suggest that a resolution possibly can be sought under (II) instead of (I): Is potential flow stable and physical? We shall see below that Birkhoff was on the right track: Potential flow 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
In his 1951 review  of Birkhoff’s book, the mathematician James J. Stoker sharply critizises the first chapter of the book:
- 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.
Stoker’s critique served its purpose and eliminated Birkhoff from fluid mechanics, but Stoker’s claim that the official resolution is well understood is contradicted by the review article by Stewartson 30 years later . Nevertheless, the official  standpoint of the fluid mechanics community today is that the paradox can be solved along the lines suggested by Prandtl, even if concrete evidence is still to be provided more than 100 years after Prandtl and 250 years after d’Alembert.
An example of how Stoker’s attitude to criticism of the official resolution is today exercised, is given in the referee´s report of the article  first submitted to Journal of Fluid Mechanics, and then accepted by Journal of Mathematical Fluid Mechanics.
Official Resolution on Wikipedia
- 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.
New Resolution: Turbulent Euler Flow
A different resolution based on computational solution of the Euler equations with slip is presented in a sequence of publications . The new resolution focusses on (II) and shows that 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.
It is understandable that the early enthusiasts of Calculus were blinded by the mathematical beauty of potential solutions, and therefore were not ready to accept that they represented 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.
But it is not enough to show that a potential solution is unstable, because it remains to explain 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
The new resolution is intimately connected to the appearance of 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.
The new resolution of d’Alembert’s paradox suggests a resolution of the Clay Mathematics Institute millennium problem on the Navier-Stokes equations which can be viewed as a revival of the 1749 Berlin Academy of Science prize problem 
The new resolution also gives a new explanation of the generation of lift of a wing: It is shown in  that the high/low pressure on the upper/lower part of the wing at the trailing edge, canceling the low/high pressure at the leading edge in the zero-drag potential flow, is modified by the low-pressure streamwise vorticity at separation into instead a non-symmetric pressure distribution effectively generating substantial lift. This explanation presented below, is completely different from the explanation in text books based on Kutta-Zhukovsky modified potential flow.
Wellposedness of Turbulent Euler Solutions
The mathematician Hadamard explained in 1902 that solving differential equations, such as the Euler equations, 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.
The new resolution can be summarized by saying that 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
As an important first application of the new resolution we show turbulent Euler flow around a car with substantial drag in accordance with wind tunnel experiments [computations by Murtazo Nazarov, and geometry courtesy by Volvo Cars]:
State-of-the-art computational fluid mechanics  claims that it is impossible to compute the drag of a car, because drag is considered to originate from thin viscous boundary layers which cannot be resolved computationally. In the new resolution, drag has a different origin, which can be computationally captured in turbulent Euler solutions without viscous boundary layers. The slip boundary condition (or more generally a friction boundary condition with small friction) is motivated by the experiments and computations show that the skin friction of a turbulent boundary layer tends to zero with viscosity and for a car contributes at most to a few percent to the total drag.
Turbulent Flow around a Cylinder
(i) formation of surface vorticity (ii) generating streamwise vorticity
where we see (i) formation of alternating surface vortices from retarding opposing flows, and (ii) tubes of counter-rotating low- pressure streamwise vorticity by vortex stretching in accellerating flow. We retrace these phenomena in EG2 computations:
Surface vorticity. Streamwise voriticity from surface vorticity.
[Computations by Niclas Jansson]
We can thus identify perturbations of strong growth consisting of low-pressure streamwise streaks/tubes attaching to the rear of the cylinder, which generate drag by suction.
We can view the drag in turbulent flow as the result of a “separation trauma”, where the ideal (unstable) separation of potential flow without surface vortices and drag, is replaced by a real (stable) separation with surface vortices and drag. The attachment in the front is different because the retardation does not come from opposing flows and the accelleration phase is much shorter. This explains why separation in the back is so different from the attachment in the front, in the real stable turbulent flow, but not in the fictitious unstable potential flow, and why the cost of attachment is small as compared to separation.
New Resolution vs Prandtl’s Experiment
We observe the following key features of the turbulent Euler solution, which are not features of Prandtl’s resolution:
- no boundary layer prior to separation
- strong counter-rotating low-pressure vorticity in the streamwise direction in the wake.
The turbulent Euler solution is in  shown to be very similar to the experiments recorded by Prandtl in 
Turbulent Flow around a Sphere
Turbulent flow around a sphere with decreasing friction as a no-slip boundary condtion (left) is relaxed to a slip boundary condition (right) takes the form
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
The Euler equations  expressing conservation of momentum and mass of an incompressible inviscid fluid enclosed in an open set Ω in three-dimensional space with boundary Γ, read: Find the velocity u and pressure p depending on (x, t) ∈ Ω ∪ Γ × I such that
∂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,
where n denotes the outward unit normal to Γ, f is a given volume force, g is a given inflow/outflow velocity, u0 is a given initial condition, and I = [0, T] a given time interval. We notice the slip boundary condition expressing inflow/outflow with zero friction with g = 0 at a solid boundary.
It is truely remarkable that a wide variety of slightly viscous flow of strong importance in applications can be modeled in the extremely compact form of the Euler equations, which is a parameter-free model, since viscosity and heat conductivity is set to zero. The data required to compute the drag and lift of a body is only the geometrical shape of the body.
Computational Solution EG2
To solve the Euler equations computationally we use an adaptive finite element method referred to as EG2 with a duality-based a posteriori error control of drag/lift presented in detail in , and in executable form available from the Unicorn-project under FEniCS , allowing direct verification of the computational results of this note.