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d'alembert

D'Alembert's paradox

In fluid dynamics, d'Alembert's paradox is a contradiction reached by French mathematician Jean le Rond d'Alembert in 1752, who proves that for — incompressible and inviscidpotential flow the drag force is zero on a body moving with constant velocity through the fluid. Zero drag of inviscid flow is in direct contradiction to measurements finding substantial drag for bodies moving through fluids, such as air and water, especially at high velocities (corresponding with high Reynolds numbers).

D'Alembert concluded, working on a 1749 Prize Problem of the Berlin Academy on flow drag: "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". And a physical paradox indicates flaws in the theory.

Fluid mechanics was thus from start discredited by engineers, 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.

The occurrence of the paradox is due to the neglect of the effects of viscosity, according to scientific consensus. In interplay with scientific experiments, there were large advances on 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. The viscous effects in the thin boundary layers remain also at very high Reynolds numbers — they result in friction drag for streamlined objects, and for bluff bodies the additional result is flow separation and a low-pressure wake behind the object, leading to form drag.

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.

Viscous friction: Saint-Venant, Navier and Stokes

First steps towards solving the paradox were made by Saint-Venant, who modelled viscous fluid friction. Saint-Venant states in 1847:

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

However, when the flow problem is put into a non-dimensional form, the viscous Navier–Stokes equations converge for increasing Reynolds numbers towards the inviscid Euler equations, suggesting that the flow should converge towards the inviscid solutions of potential flow theory — having the zero drag of the d'Alembert paradox. Of this, there is no evidence found in experimental measurements of drag and flow visualisations. This again raised questions concerning the applicability of fluid mechanics in the second half of the 19th century.

Thin boundary layers: Prandtl

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. Prandtl put forward the idea that a no-slip boundary condition causes — also at high velocities and high Reynolds numbers — a strong variation of the flow speeds over a thin layer near the wall of the body. 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 low pressure in the wake region results in a large form drag — larger than the friction drag due to the viscous shear stress at the wall.

Evidence that Prandtl´s scenario occurs for bluff bodies in flows of high Reynolds numbers, can for instance be seen in impulsively started flows around a cylinder. Initially the flow resembles potential flow, after which the flow separates near the rear stagnation point. Thereafter, the separation points move upstream, resulting in a low-pressure region of separated flow.

Open questions

To verify, as Prandtl suggested, that a vanishingly small cause (vanishingly small viscosity for increasing Reynolds number) has a large effect — substantial drag — may be very difficult.

The mathematician Garrett Birkhoff in the opening chapter of his book Hydrodynamics from 1950 , addresses a number of 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 considers another possible route to its resolution: instability of the potential flow solutions for the Euler equations. Birkhoff states:
"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...".

In his 1951 review of Birkhoff's book, the mathematician James J. Stoker sharply critisizes 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 uninitiated would be very likely to get the wrong ideas about some of the important and useful achievements in hydrodynamics from reading this chapter."

The importance and usefulness of the achievements, made on the subject of the d'Alembert paradox, are reviewed by Stewartson thirty years later. His long 1981 survey article starts with:

"Since classical inviscid theory leads to the patently absurd conclusion that the resistance experienced by a rigid body moving through a fluid with uniform velocity is zero, great efforts have been made during the last hundred or so years to propose alternate theories and to explain how a vanishingly small frictional force in the fluid can nevertheless have a significant effect on the flow properties. The methods used are a combination of experimental observation, computation often on a very large scale, and analysis of the structure of the asymptotic form of the solution as the friction tends to zero. This three-pronged attack has achieved considerable success, especially during the last ten years, so that now the paradox may be regarded as largely resolved."
For many paradoxes in physics, their resolution often lies in transcending the available theory. 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.

Zero drag in potential flow

Potential flow

The three main assumptions in the derivation of d'Alembert's paradox is that the flow is incompressible, inviscid and irrotational. An inviscid fluid is described by the Euler equations, which for an incompressible flow read

begin{align}
& boldsymbol{nabla} cdot boldsymbol{u} = 0 && text{(conservation of mass)} & frac{partial}{partial t} boldsymbol{u} + left(boldsymbol{u} cdot boldsymbol{nabla}right) boldsymbol{u} = - frac{1}{rho} boldsymbol{nabla} p && text{(conservation of momentum)} end{align} where u denotes the flow velocity of the fluid, p the pressure, ρ the density, and is the gradient operator. The assumption that the flow is irrotational means that the velocity satisfies × u = 0.

Hence, we have

left(boldsymbol{u} cdot boldsymbol{nabla}right) boldsymbol{u} = tfrac12 boldsymbol{nabla} left(boldsymbol{u} cdot boldsymbol{u}right) - boldsymbol{u} times boldsymbol{nabla} times boldsymbol{u} = tfrac12 boldsymbol{nabla} left(boldsymbol{u} cdot boldsymbol{u}right) qquad (1)
where the first equality is a vector calculus identity and the second equality uses that the flow is irrotational. Furthermore, for every irrotational flow, there exists a velocity potential φ such that u = φ. Substituting this all in the equation for momentum conservation yields
boldsymbol{nabla} left(frac{partialvarphi}{partial t} + tfrac12 boldsymbol{u} cdot boldsymbol{u} + frac prho right) = boldsymbol{0}.
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
frac{partialvarphi}{partial t} + tfrac12 boldsymbol{u} cdot boldsymbol{u} + frac prho = 0, qquad (2)
which is the Bernoulli equation for unsteady potential flow.

Zero drag

Now, suppose that a body moves with constant velocity v through the fluid, which is at rest infinitely far away. Then the velocity field of the fluid has to follow the body, so it is of the form u(x, t) = u(xv t, 0), and thus:

frac{partial boldsymbol{u}}{partial t} + left(boldsymbol{v} cdot boldsymbol{nabla} right) boldsymbol{u} = boldsymbol{0}.
Since u = φ, this can be integrated with respect to x:
frac{partialvarphi}{partial t} = -boldsymbol{v} cdot boldsymbol{nabla} varphi + R(t) = -boldsymbol{v} cdot boldsymbol{u} + R(t).

The force F that the fluid exerts on the body is given by the surface integral

boldsymbol{F} = - int_A p, boldsymbol{n}; mathrm{d} S
where A denotes the body surface and n the normal vector on the body surface. But it follows from (2) that
p = - rho Bigl(frac{partialvarphi}{partial t} + tfrac12 boldsymbol{u} cdot boldsymbol{u} Bigr) = rho Bigl(boldsymbol{v} cdot boldsymbol{u} - tfrac12 boldsymbol{u} cdot boldsymbol{u} - R(t) Bigr),
thus
boldsymbol{F} = - int_A p, boldsymbol{n}; mathrm{d} S = rho int_A left(tfrac12 boldsymbol{u} cdot boldsymbol{u} - boldsymbol{v} cdot boldsymbol{u}right) boldsymbol{n}; mathrm{d} S,
with the contribution of R(t) to the integral being equal to zero.

At this point, it becomes more convenient to work in the vector components. The kth component of this equation reads

F_k = rho int_A sum_i (tfrac12 u_i^2 - u_i v_i) n_k , mathrm{d} S. qquad (3)

Let V be the volume occupied by the fluid. The divergence theorem says that

frac12 int_A sum_i u_i^2 n_k , mathrm{d} S = - frac12 int_V frac{partial}{partial x_k} left(sum_i u_i^2 right) ,mathrm{d} V.
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. The integrand in the volume integral can be rewritten as follows:
frac12 frac{partial}{partial x_k} left(sum_i u_i^2 right) = sum_i u_i frac{partial u_k}{partial x_i} = sum_i frac{partial(u_iu_k)}{partial x_i}
where first equality (1) and then the incompressibility of the flow are used. Substituting this back into the volume integral and another application of the divergence theorem again. This yields
- frac12 int_V frac{partial}{partial x_k} left(sum_i u_i^2 right) ,mathrm{d} V = -int_V sum_i frac{partial(u_iu_k)}{partial x_i} ,mathrm{d} V = int_A u_k sum_i u_i n_i ,mathrm{d} S.
Substituting this in (3), we find that
F_k = rho int_A sum_i (u_k u_i n_i - v_i u_i n_k) , mathrm{d} S.
The fluid cannot penetrate the body and thus n · u = n · v on the body surface. Thus,
F_k = rho int_A sum_i (u_k v_i n_i - v_i u_i n_k) , mathrm{d} S.
Finally, the drag is the force in the direction in which the body moves, so
boldsymbol{v} cdot boldsymbol{F} = sum_i v_i F_i = 0.
Hence the drag vanishes. This is d'Alembert's paradox.

References

Historical

Further reading

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