D'Alembert's paradox

[1] D'Alembert proved that – for incompressible and inviscid potential flow – the drag force is zero on a body moving with constant velocity relative to (and simultaneously through) the fluid.

[3] D’Alembert, working on a 1749 Prize Problem of the Berlin Academy on flow drag, concluded: 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 [i.e. mathematicians - the two terms were used interchangeably at that time] to elucidate.

Fluid mechanics was thus discredited by engineers from the start, which 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 – in the words of the Chemistry Nobel Laureate Sir Cyril Hinshelwood.

In conjunction with scientific experiments, there were huge advances in the theory of viscous fluid friction during the 19th century.

With respect to the paradox, this culminated in the discovery and description of thin boundary layers by Ludwig Prandtl in 1904.

Even at very high Reynolds numbers, the thin boundary layers remain as a result of viscous forces.

[5][6][7][8][9][10] A formal mathematical proof is lacking, and difficult to provide, as in so many other fluid-flow problems involving the Navier–Stokes equations (which are used to describe viscous flow).

First steps towards solving the paradox were made by Saint-Venant, who modelled viscous fluid friction.

Saint-Venant states in 1847:[11] But one finds another result if, instead of an ideal fluid – object of the calculations of the geometers 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 or 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.Soon after, in 1851, Stokes calculated the drag on a sphere in Stokes flow, known as Stokes' law.

[12] Stokes flow is the low Reynolds-number limit of the Navier–Stokes equations describing the motion of a viscous liquid.

In the second half of the 19th century, focus shifted again towards using inviscid flow theory for the description of fluid drag—assuming that viscosity becomes less important at high Reynolds numbers.

The model proposed by Kirchhoff[16] and Rayleigh[17] was based on the free-streamline theory of Helmholtz[18] and consists of a steady wake behind the body.

[22] It was readily known that such steady flows are not stable, since the vortex sheets develop so-called Kelvin–Helmholtz instabilities.

Rayleigh asks "... whether the calculations of resistance are materially affected by this circumstance as the pressures experienced must be nearly independent of what happens at some distance in the rear of the obstacle, where the instability would first begin to manifest itself.

[21][23] Moreover, the observed pressure differences between front and back of the plate, and resulting drag forces, are much larger than predicted: for a flat plate perpendicular to the flow the predicted drag coefficient is CD=0.88, while in experiments CD=2.0 is found.

This is mainly due to suction at the wake side of the plate, induced by the unsteady flow in the real wake (as opposed to the theory which assumes a constant flow velocity equal to the plate's velocity).

[20][21][25] The German physicist Ludwig Prandtl suggested in 1904 that the effects of a thin viscous boundary layer possibly could be the source of substantial drag.

This leads to the generation of vorticity and viscous dissipation of kinetic energy in the boundary layer.

The energy dissipation, which is lacking in the inviscid theories, results for bluff bodies in separation of the flow.

The full problem of viscous flow, described by the non-linear Navier–Stokes equations, is in general not mathematically solvable.

Boundary-layer theory is amenable to the method of matched asymptotic expansions for deriving approximate solutions.

Form drag is due to the effect of the boundary layer and thin wake on the pressure distribution around the airfoil.

[32] The importance and usefulness of the achievements, made on the subject of the d'Alembert paradox, are reviewed by Keith Stewartson thirty years later.

His long 1981 survey article starts with:[9] For many paradoxes in physics, their resolution often lies in transcending the available theory.

[33] In the case of d'Alembert's paradox, the essential mechanism for its resolution was provided by Prandtl through the discovery and modelling of thin viscous boundary layers – which are non-vanishing at high Reynolds numbers.

[26] The three main assumptions in the derivation of d'Alembert's paradox is that the steady flow is incompressible, inviscid and irrotational.

Substituting this all in the equation for momentum conservation yields Thus, the quantity between brackets must be constant (any t-dependence can be eliminated by redefining φ).

Assuming that the fluid is at rest at infinity and that the pressure is defined to be zero there, this constant is zero, and thus which is the Bernoulli equation for unsteady potential flow.

Then the velocity field of the fluid has to follow the body, so it is of the form u(x, t) = u(x − v t, 0), where x is the spatial coordinate vector, and thus:

The right-hand side is an integral over an infinite volume, so this needs some justification, which can be provided by appealing to potential theory to show that the velocity u must fall off as r−3 – corresponding to a dipole potential field in case of a three-dimensional body of finite extent – where r is the distance to the centre of the body.