11

Prandtl’s Resolution of d’Alembert’s Mystery

By denying scientific principles, one may maintain any paradox.
(Galileo Galilei)

11.1 Quotation from a Standard Source

To get the proper perspective, we present the standard view on the resolution
of d’Alembert’s Mystery as the one suggested by Prandtl (Fig. 11.1) in the
short report Motion of fluids with very little viscosity read before the Third
International Congress of Mathematicians at Heidelberg in 1904, in the form
of some quotations from the standard source [92]:

Ludwig Prandtl’s discovery of the boundary layer is regarded as one of the

most important breakthroughs in fluid mechanics of all time and has earned
Prandtl the title of Father of Modern Fluid Mechanics.

Before Prandtl’s description of the boundary layer in 1904, there was no

lack of interest in the dynamics of fluids due to the practical problems of
nautical engineering, ballistics, and hydraulics. Throughout the 18th and 19th
century the top physicists and mathematicians of Europe examined flows from
a mathematical point of view. Much of this work was to construct potential
flows, i.e., incompressible, irrotational flows, over bodies. Examples recogniz-
able to most undergraduates are flows over circular cylinders and other flows
involving source-sink superpositions. Although the mathematics was elegant
and the flows aesthetically pleasing, it was recognized that such flows failed to
mimic “real” flows seen in Nature. Furthermore, it was known since the time
of d’Alembert that potential flows frequently resulted in zero drag; a prediction
in clear contradiction with everyday experience!

What were these mathematicians to do? Thanks to Coulomb and Stokes,

they were aware that a no-slip condition should be applied at solid bodies (we
now realize that this condition holds at all fluid boundaries). However, stan-
dard external flow problems are ill-posed when the potential flow equations are

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11 Prandtl’s Resolution of d’Alembert’s Mystery

combined with the no-slip condition. The correct approach would be to abandon
the inviscid (small viscosity) approximation and solve the full Navier–Stokes
equations. Stokes had done this himself for the problem of creeping flow around
a sphere and derived a non-zero expression for the drag. However, the Stokes
flow does not generate the large scale separation seen in most day-to-day flows
and the predicted drag is always much less than what is measured for things
like cannon balls and marbles in air and water. The reason for these discrepen-
cies is the neglect of the fluid inertia in the creeping flow approximation. To
include these terms is a daunting task, even today.

Thus, as the 19th century came to a close, a universal and practical appli-

cation of fluid mechanics seemed far off. Prandtl’s contribution was to realize
that we can view the flow as being divided into two regions. The bulk of the
flow can be regarded as a potential flow essentially the same as that studied
by the mathematicians. Only in a small region near the body do viscous ef-
fects dominate. This thin layer is known as the boundary layer. Conceptually,
Prandtl’s boundary layer is the reason the potential flow theory is compat-
ible with the exact physics. Furthermore, certain details of the structure of
the boundary layer are the key to understanding both flow separation and the
physical mechanism behind the Kutta condition. That is, a proper understand-
ing of the boundary layer allows us to understand how a (vanishingly) small
viscosity and a (vanishingly) small viscous region can modify the global flow
features. Thus, with one insight Prandtl resolved d’Alembert’s paradox and
provided fluid mechanists with the physics of both lift and form drag.

11.2 Quotation from Prandtl’s 1904 report

We follow up with some paragraphs from Prandtl’s 1904 report which ap-
peared in English translation as Technical Memorandum 452 of the National
Advisory Committee for Aeronautics in 1928 [93]:

It is known, however, that the solutions of the Euler equations generally

agree very poorly with experience. I will recall only the Dirichlet sphere which,
according to the theory, should move without friction.

I have now set myself the task to investigate systematically the laws of

motion of a fluid whose viscosity is assumed to be very small. The viscosity is
supposed to be so small that it can be disregarded wherever there are no great
velocity differences nor accumulative effects. This plan has proved to be very
fruitful, in that, on the one hand, it produces mathematical formulas. which
enable a solution of the problem and, on the other hand, the agreement with
observations promises to be very satisfactory.

The most important aspect of the problem is the behavior of the fluid on

the surface of the solid body, assuming that the fluid adheres to the surface and
that, therefore, the velocity is either zero or equal to the velocity of the body.
In the thin transition layer, the great velocity differences will then produce
noticeable effects in spite of the small viscosity.

11.3 Discussion of Prandtl’s Resolution

83

The most important practical results of these investigations is that, in cer-

tain cases, the flow separates from the surface at a point entirely determined
by external conditions. A fluid layer, which is set in rotation by the friction
on the wall, is thus forced into the free fluid.

On the one hand, we have the free fluid, which can be treated as non-

viscous, while, on the other hand, we have the transition layers on the solid
boundaries, impart their characteristic impress on the free flow by the emission
of turbulent layers.

No. 7–10 show the flow around a cylindrical obstacle. No. 7 shows the

beginning of the separation; Nos. 8-9, subsequent stages. No. 10 shows the
permanent condition. The wake of turbulent water behind the cylinder swings
back and forth, whence the momentary unsymmetrical appearance (referring
to the pictures in Fig. 11.2).

Fig. 11.1. Ludwig Prandtl (1875–1953).

11.3 Discussion of Prandtl’s Resolution

The main point of Prandtl’s resolution of the d’Alembert’s Mystery is that
boundary layers always exist at solid boundaries, even if the viscosity is very
small, and that by the strong velocity difference in the boundary layer, vor-
ticity is created in the layer and is then ejected into the fluid. The important
feature is the direction of the vortex generation according to Prandtl, which
is parallel to the surface and perpendicular to the streamwise direction cor-
responding to “tripping” the flow by friction in the boundary layer. We refer
to vorticity in this direction as transversal vorticity. The accepted resolution
of the mystery according to Prandtl is thus that transversal vorticity is gen-
erated in the boundary layer, even if the viscosity is very small, and this
vorticity generation changes the global patterns of the flow, allowing non-zero

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11 Prandtl’s Resolution of d’Alembert’s Mystery

Fig. 11.2. Pictures 1–12 from Prandtl’s Technical Memorandum 452.

11.3 Discussion of Prandtl’s Resolution

85

drag to develop. Prandtl thus claims that the potential solution does not
occur in practice, but instead a different (turbulent) solution develops from
the generation of transversal vorticity in the boundary layer.

We will below show that Prandtl’s view on the potential solution is correct,

but we will question his explanation by transversal vorticity generation by
showing the importance of instead generation of vorticity in the streamwise
direction reflecting the stability analysis in Chapter 9.

Another important aspect concerns the separation points: It is clear from

Fig 5 and 6 that Prandtl believes that there must be two separation points, al-
though his remark on No. 10 (“The wake of turbulent water behind the cylinder
swings back and forth, whence the momentary unsymmetrical appearance.”),
indicates that he can see only one swinging back and forth. We will below
show that with very small viscosity, there is in fact only one separation point
(in each section perpendicular to the cylinder axis), which fits with Prandtl’s
experiment, but is contrary to Prandtl’s analysis.

Altogether, we will thus present a resolution of the mystery, which is

fundamentally different from the accepted resolution by Prandtl.