dAlembert Paradox

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Finally: Resolution of dAlemberts Paradox

Johan Hoffman and Claes Johnson

January 20, 2006

Abstract

We propose a new resolution to dAlemberts Paradox from 1752 com-paring the mathematical prediction of zero drag (resistance to motion)through an ideal (zero viscosity) incompressible fluid, with massive ob-servations of non-zero drag in fluids with very small viscosity, such as airand water. Our resolution is fundamentally different from the acceptedresolution suggested by Prandtl in 1904 based on boundary layer effectsof vanishing viscosity. We base our resolution on computational solution

of the Euler equations describing ideal incompressible flow by noting thatthe zero drag potential solution considered by dAlembert, is unstable andinstead a turbulent (approximate) solution develops with non-zero drag,even without boundary layer effects. We claim that our resolution is bet-ter than Prandtls in the case of very small viscosity.

1 Introduction

How wonderful that we have met with a paradox. Now we have somehope of making progress. (Nils Bohr)

We present a new resolution of dAlemberts Paradox from 1752 ([1]), whichstates that a body can move through an ideal (zero viscosity) incompressible

fluid without any resistance (drag) to the motion. This statement is paradox-ical because all experience shows that motion through fluids with very smallviscosity, such as air and water, meets a drag which is far from zero. Our res-olution is fundamentally different from the accepted resolution from 1904 byLudwig Prandt ([2]), called the father of modern fluid mechanics, which buildson boundary layer effects from very small viscosity.

We base our resolution on computational solution of the Euler equations de-scribing ideal incompressible fluid flow, which shows that the potential solutionwith zero drag considered by dAlembert is unstable and instead a turbulentsolution develops with non-zero drag. This solves the paradox in the origi-nal setting of dAlembert and Euler, assuming the fluid to be ideal with zeroviscosity, by showing that the zero drag potential solution cannot be realizedphysically, because it is unstable, and thus cannot be observed. We solve theEuler equations with a slip boundary condition at solid boundaries prescribing

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the normal velocity to be zero but letting the tangential velocity be free, whichmeans that no boundary layers are created. Nevertheless, a turbulent solutionto the Euler equations with non-zero drag develops, starting from the poten-tial solution. Our resolution is thus completely different from Prandtls and weclaim that our resolution is more to the point for the very small viscosities metin a wide range of turbulent flows in aero- and hydro-dynamics.

We also claim that our resolution is more satisfactory from a scientific pointof view than Prandtls, because we do no suggest that a very small cause (verysmall viscosity) can have a large effect (change the drag), as Prandtl does,which is close to saying that anything can happen from virtually nothing, andwhich can be very hard to either prove or disprove. We show instead thatthe potential solution is unstable and that a turbulent solution develops, evenwithout influence from boundary layer effects of very small viscosity. We donot claim that boundary layer effects never influence the global flow, e.g. byseparation, but we do claim that these effects may be small for very smallviscosities, which fits with the observation that the so-called skin-friction tendsto zero as the viscosity tends to zero.

To solve the Euler equations numerically we use an adaptive finite elementmethod with automatic control of the error in the drag ([8, 4, 5, 7, 9]), and we

find that the computed drag is stable under mesh refinement. In [6] we also usea skin-friction boundary condition to model the effect of turbulent boundarylayers, with zero skin-friction corresponding to a slip boundary condition. Let-ting the skin friction tend to zero, we obtain good agreement with experimentaldrag coefficients for varying viscosity (Reynolds number) including the so-calleddrag crisis occuring for very small viscosities ([10]).

An outline of this note is as follows: We first recall the Euler equations forideal incompressible fluid flow, and the stationary irrotational potentional solu-tion with zero drag considered by dAlembert. We then present computationalresults and point to basic features. Finally, we compare our new resolution ofdAlemberts Paradox with that of Prandtl, and leave to the reader to judgewhich resolution may be closer to the truth.

2 The Euler equations

We consider the motion of an ideal incompressible fluid occupying a fixed volume in R3 with boundary . We want to find the fluid velocity u(x, t) and pressure

p(x, t) for all points x = (x1, x2, x3) and time t > 0, assuming that the fluidflow through the boundary and the initial velocity u(x, 0) are given. Weassume that is divided into a part 0 corresponding to a solid (inpenetrable)boundary, and a remaining part corresponding to inflow and outflow. Themathematical model for the motion of the fluid takes the form of the Eulerequations formulated by Leonard Euler in 1755 ([11]) expressing conservationof momentum (Newtons second law) and conservation of mass, combined with

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a boundary condition (g) and an initial condition (u0) for the velocity:

u + (u )u + p = 0, in I, u = 0, in I,u n = g, on I,

u(, 0) = u0, in .

(1)

Here n is the outward unit normal to and the given boundary flow g satisfies

g ds = 0 and g = 0 o n 0. Requiring u n = 0 o n 0 corresponds toa slip boundary condition (bc) with the normal velocity vanishing, while thetangential velocity is free. This is to be compared to the no-slip bc u = 0in NavierStokes equations (with non-zero viscosity > 0 including the termu in the momentum equation), where also the tangential velocity is requiredto vanish reflecting that the fluid sticks to a solid boundary. Prandtls resolutionof the Paradox is connected to the no-slip bc, whereas we advocate that the slipbc is more relevant, in the case of very small viscosity.

3 Potential Flow around a Circular Cylinder

Following dAlembert we consider stationary (time-independent) potential flowaround an (infinitely) long cylinder of diameter 1 oriented along the x3-axisand immersed in an ideal incompressible fluid filling R3 with velocity (1, 0, 0) atinfinity. This models, for example, the flow of air around a tall cylindrical high-rise subject to a strong wind, or the flow of water around a pillar of a bridge ina strong current. The potential velocity is given as u = , where satisfiesLaplaces equation = 0 outside the cylinder and appropriate conditions atinfinity, that is,

(x1, x2, x3) = (r +1

r)cos(), (2)

where (x1, x2) = (r cos(), r sin()) is expressed in polar coordinates (r, ). Us-ing standard Calculus one verifies that u is irrotational, that is u = 0, since

= 0, and that (u, p) solves the Euler equations, where the pressure p isdetermined by Bernoullis Law stating that 1

2|u|2 +p is constant for stationary

irrotational flow. In Fig. 1 we plot the streamlines of u in a section of the cylin-der, which are the curves followed by fluid particles, and the pressure. We noticethat the potential flow (in each section) has one separation point at the backof the cylinder, where the flow separates from the cylinder boundary. We alsonotice that the both velocity and pressure are symmetric in the flow direction(x1-direction), which means that the drag of the cylinder is zero; the build up ofpressure in front of the cylinder is balanced by the same strong pressure behind,and thus the drag is zero. The cylinder thus seems to be pushed through thefluid by the strong pressure behind, which of course is counter-intuitive andin fact is never observed in practice, where the pressure behind always is muchlower than up front, with resulting non-zero drag. According to dAlemberts

potential solution there would be no wind load on a high-rise and no force on a

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bridge pillar from a strong current, which is in contradiction with all practicalexperience.

One can extend this result to flow around a body of arbitrary shape, sincethere is always a corresponding potential solution. We have thus met a scientificParadox, which we have to resolve to save fluid mechanics as a mathematicalscience from collapse. But what do to? Evidently, something must be wrong

with the potential solution, since it gives zero drag. But what? It cannot beNewtons second law or mass conservation.

Figure 1: Potential solution of the Euler equations for flow past a circular cylin-der; colormap of the pressure (left) and streamlines together with a colormapof the magnitude of the velocity (right) .

Prandtl in 1904 claimed that the Paradox is due to the assumption of zeroviscosity. Prandtl stated that even if the viscosity is very small, it is not equalto zero, which means that one has to consider the Navier-Stokes equations withno-slip bc (instead of Eulers equations with slip bc), for which in a thin boundarylayer close a solid boundary the fluid velocity will change quickly from zero atthe boundary to the free stream value outside the layer. Prandtl remarked thatthe potential solution does not satisfy the no-slip bc and thus should be dis-carded. The no-slip bc would generate strong vorticity (fluid rotation) transver-sal to the flow direction by tripping the flow. Prandtl further claimed thatbecause of the retardation of the flow in the boundary layer, due to an ad-verse pressure gradient combined with the no-slip boundary condition, the flowwould separate away from the boundary somewhere at the back of the cylinder.Prandtl thus claims that there must be two separation points (in each section)at the back of the cylinder, one above and one below the x1-axis (although hecan see only one in experiments for very small viscosity/high Reynolds number).

Prandtl thus gets around the Paradox by claiming that even an arbitrarily

small non-zero viscosity will substantially change the drag through boundarylayer effects. Several generations of fluid dynamicists have allowed themselves

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to be convinced by this argument (see the standard view [3]). But is it correct,for very small viscosities (large Reynolds numbers)?

4 Computational Solution of Eulers Equations

To seek understanding we go back to the roots, that is to the Euler equationswith slip boundary conditions, but now we solve these equations computation-ally ([9]) instead of using analytical mathematics as dAlembert did. We consideragain the circular cylinder case now put into a channel of finite dimensions withgiven inflow velocity (1, 0, 0) and choose the initial velocity u0 equal to zero.We see the zero-drag irrotational potential solution quickly developing duringthe first time steps, but then the potential solution gradually changes into aturbulent solution with large drag and vorticity, see Fig. 2. We observe thatthe computed Euler solution has the following key features: (a) no boundarylayer prior to separation, (b) one separation point in each section of the cylinderwhich oscillates up and down and (c) strong vorticity in the streamwise direc-tion. The computed drag is 1.0, which is consistent under mesh refinement,and which fits with the observation ([10]) that the drag increases from 0.5

to about 1.0 beyond the drag crisis occuring for 106

. We see in Fig. 3that the streamwise (x1) vorticity dominates the tranversal (x3) vorticity, andthat the pressure is low inside tubes of vorticity in the x1-direction behind thecylinder, which creates drag.

Figure 2: Computational solution of the Euler equations for flow past a cir-cular cylinder; colormap of the pressure (left) and streamlines together with acolormap of the magnitude of the velocity (right) .

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Figure 3: Computational solution of the Euler equations for flow past a circularcylinder; colormap of the pressure and isosurfaces for low pressure (upper left),colormap of the magnitude of total vorticity and isosurfaces for high magnitudeof the individual components: x1-vorticity (upper right), x2-vorticity (lowerleft), x3-vorticity (lower right).

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5 A New Resolution of dAlemberts Paradox

We have shown by computation that the zero-drag potential solution of the Eulerequations is unstable, and develops into a turbulent solution with substantialdrag. This resolves the Paradox.

Our resolution is fundamentally different with respect to the aspects (a)-(c)

from Prandtls resolution based on boundary layer effects in the Navier-Stokesequations. Our resolution does not involve a very small cause with large effectas Prandtls does, and thus from a scientific point of view is more satisfactory.

References

[1] Jean-le-Rond dAlembert, Essai dune nouvelle theorie de la resistance desfluides, Paris, 1752, http://gallica.bnf.fr/anthologie/notices/00927.htm.

[2] Ludwig Prandtl, On motion of fluids with very little viscosity,Third International Congress of Mathematics, Heidelberg, 1904.http://naca.larc.nasa.gov/digidoc/report/tm/52/NACA-TM-452.PDF

[3] www.fluidmech.net/msc/prandtl.htm.

[4] Johan Hoffman, Computation of mean drag for bluff body problems usingAdaptive DNS/LES, SIAM J. Sci. Comput. 27(1), pp.184-207, 2005.

[5] Johan Hoffman, Adaptive simulation of the turbulent flow past a sphere,accepted for publication in J. Fluid Mech., 2006.

[6] Johan Hoffman, Computation of turbulent flow past bluff bodies using adap-tive General Galerkin methods: drag crisis and turbulent Euler solutions, inreview for Computational Mechanics, 2006.

[7] Johan Hoffman and Claes Johnson, A new approach to Computational Tur-bulence Modeling, Comput. Methods Appl. Mech. Engrg., in press.

[8] Johan Hoffman and Claes Johnson, Computational Turbulent Incompress-ible Flow: Applied Mathematics Body and Soul Vol 4, Springer-Verlag Pub-lishing, 2006.

[9] www.fenics.org

[10] Tritton, D. J. Physical Fluid Dynamics, 2nd ed. Oxford, Clarendon Press,1988.

[11] Leonard Euler, Principes generaux du mouvements des fluides, lAcademiede Berlin, 1755.

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Figure 4: Snapshot of the velocity illustrating the single separation point, thatis the tangential velocity at the cylinder surface changes sign over one finiteelement (there is one arrow in each node in the section).

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Figure 5: High vorticity isosurfaces just after start-up (t = 0.5), showing thatstreamwise vorticity is generated along the separation line, whereas the othervorticity components are small: x1-vorticity (upper), x2-vorticity (middle), x3-vorticity (lower).

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0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Figure 6: Time series of the drag coefficient cD for a G2 solution to the Eulerequations (for the mesh with 153 440 nodes).

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Figure 7: Section through one of the meshes used for the computations: 74 247mesh points.

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Figure 8: Section through one of the meshes used for the computations: 153 440mesh points.

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