FINITE ELEMENT CENTER - KTHjhoffman/archive/papers/preprint-2006-09.pdfResolution of Scienti c Paradoxes by Computation Johan Ho man and Claes Johnson Finite Element Center Preprint

FINITE ELEMENT CENTER

PREPRINT 2006–09

Resolution of Scientific Paradoxes by

Computation

Johan Hoffman and Claes Johnson

FINITE ELEMENT CENTER

PREPRINT 2006–09

Resolution of Scientific Paradoxes by

Computation


Finite Element Center

http://www.femcenter.org/

Resolution of Scientific Paradoxes by Computation


Finite Element Center PreprintNO 2006–09ISSN 1653–574X

This preprint and other preprints can be found athttp://www.femcenter.org/preprints/

RESOLUTION OF SCIENTIFIC PARADOXES BY COMPUTATION

JOHAN HOFFMAN AND CLAES JOHNSON

Abstract. We propose new resolutions of the classical paradoxes of d’Alembert, Som-merfeld, Loschmidt and Gibbs, by computational solution of the inviscid Euler equa-tions for compressible and incompressible flow. The resolutions are all based on the factsupported by both mathematical analysis and computation that inviscid flow always isturbulent with kinetic energy irrecoverably being transformed into heat.

How wonderful that we have met with a paradox. Now we have some hope ofmaking progress. (Nils Bohr)

...the whole procedure was an act of despair because a theoretical interpretationhad to be found at any price, no matter how high that might be... (Max Planckon the statistical mechanics basis of his black-body radiation law)

Introduction

In this note we propose new resolutions of the following scientific paradoxes, which havehaunted scientists over centuries:

(a) d’Alembert’s paradox : Zero drag of inviscid flow.(b) Sommerfeld’s paradox : Stability of inviscid Couette flow.(c) Loschmidt’s paradox : Irreversibility in reversible Hamiltonian systems.(d) Gibbs’ paradox : Inextensivity of Boltzmann’s entropy.

The above citation by the famous physicist Nils Bohr illustrates the role of a paradox inthe development of a physical theory, as a striking formulation of a contradiction withinthe theory, or between a prediction of the theory and a factual observation. Scientificprogress can be made by showing that the contradiction is only apparent, thus improvingthe understanding of the meaning of a theory remaining correct, or that the contradictionis real, thus showing that the theory is not correct and thereby opening for a new bettertheory to be developed. A theory predicting that an apple let free to fall, will stay whereit is, will be contradicted by Newton’s observation that the apple falls to the ground, andthus will have to be abandoned.

The first known paradoxes were formulated by Zeno of Elea (490-430 BC) on the (still)puzzling nature of motion: Zeno asked e.g. how it can be that a flying arrow, which at each

Date: September 28, 2006.Key words and phrases. d’Alembert’s, Gibbs’, Loschmidt’s, Sommerfeld’s, paradox.Johan Hoffman, School of Computer Science and Communication, KTH, SE–100 44 Stockholm, Sweden,

email : [email protected] Johnson, School of Computer Science and Communication, KTH, SE–100 44 Stockholm, Sweden,

email : [email protected]

2 JOHAN HOFFMAN AND CLAES JOHNSON

instant looks just the same as a motionless arrow, can be moving? Modern physics harborsmany unresolved paradoxes including the wave-particle duality of quantum mechanics andtime dilation of special relativity.

Since an unresolved paradox is a deadly threat to a theory, it has to be resolved byshowing that the contradiction is only apparent and not real, “at any price, no matter howhigh that might be...” from the above citation of the famous physicist Max Planck facingin 1900 the paradox of the ultra-violet catastropy in the classical theory of black-bodyradiation.

In each of the cases (a)-(c) analytical mathematics predicts effects which are not observedin reality, while (d) is a contradiction within a mathematical theory. If a paradox cannotbe resolved by showing that the contradiction is only apparent and not real, it can becovered up by presenting new information. When asked where the tortoises can be seen,which according to a certain (ancient) theory support the Earth, the cover up would be toclaim that they are invisible (but still there).

In d’Alembert’s paradox formulated in 1752 [3], mathematics predicts that a body maymove through an incompressible fluid with very small (zero) viscosity, like air and water,with very small (zero) resistance or drag. But everbody (except a mathematics specialist)knows that this is impossible; the drag increases roughly quadratically with the velocityand becomes very substantial for higher velocities.

In Sommerfeld’s paradox from 1908 [20], mathematics predicts that the simplest of allflows, Couette flow with a stationary linear velocity profile, is stable and thus should exist.But nobody (except a fluid mechanics specialist) has ever observed this flow in a fluid withsmall viscosity.

In Loschmidt’s paradox formulated in 1876 [15], mathematics of Hamiltonian systems(with zero viscosity) predicts that time reversal and a perpetum mobile is possible. Buteverybody (except a physics specialist) knows that time is always moving forward and thata perpetum mobile is impossible, even though ultimately the World is based on Hamiltonian(quantum) mechanics.

In Gibbs’ paradox formulated in 1875 [5], statistical mechanics predicts that removing amembrane separating two volumes of equal gases at rest in the same state (same density,temperature and pressure), will result in a significant increase of entropy. But everybody(except a statistical mechanics specialist) understands that removing (or inserting) a mem-brane in such a situation changes nothing.

The cover up of d’Alembert’s paradox by Prandtl [18] is to blame the assumption ofzero viscosity for the erronous prediction: In reality there is always some possibly veryvery small viscosity (of some nature), which changes everything (in some mysterious way).We will below argue that such an explanation is not scientific and we shall instead presenta new resolution based on computational turbulence in the inviscid Euler equations, whichis scientific.

The cover up of Sommerfeld’s paradox by Schlichting [19] is to say that the linear velocityprofile of Couette flow is too simple (no inflection point) for the mathematical theory toapply, which is not scientific. We will study Couette flow analytically and computationally

RESOLUTION OF SCIENTIFIC PARADOXES BY COMPUTATION 3

and find that it is not stable, just as observed, and we will point to the mathematicalmistake made by Sommerfeld.

The cover up of Loschmidt’s paradox by Boltzmann [2] is to introduce statistical me-chanics based on microscopic games of roulette. We will argue that such an explanation iscumbersome scientifically, and we shall instead present a new resolution based on compu-tational turbulence in the compressible Euler equations.

The cover up of Gibbs’ paradox by Gibbs’ himself [5], is to change the counting ofmicrostates underlying Boltzmann’s definition of entropy in statistical mechanics, to giveno change of entropy for equal gases, but retaining a substantial entropy change as soonas the two gases differ slightly. We will below argue that such a discontinuous changeis not scientifically convincing, and present a different resolution based on computationalturbulence avoiding tricky statistics.

Our new resolutions all come out of the new methodology of computational turbulenceexposed in detail in the upcoming books [10] and [14], using the fact that flow with smallor zero viscosity always is turbulent with substantial kinetic energy being transformedinto heat energy. Our resolutions are controversial by questioning fundamental beliefs ofestablished scientific disciplines and thus generate strong opposition by referee’s in leadingscienticfic journals [12, 11]. We hope the reader is open to scrutinizing the presentedevidence without prejudice.

1. d’Alembert’s Paradox

Thus, I do not see, I admit, how one can satisfactorily explain by theory theresistance of fluids. On the contrary, it seems to me that the theory, in all rigor,gives in many cases zero resistance; a singular paradox which I leave to futureGeometers for elucidation. (d’Alembert in 1752)

...do steady flow ever occur in Nature, or have we been pursuing fantasy allalong? If steady flows do occur, which ones occur? Are they stable, or will asmall perturbation of the flow cause it to drift to another steady solution, oreven an unsteady one? The answer to none of these questions is known. (MarvinShinbrot in Lectures on Fluid Mechanics, 1970)

1.1. Separation of Theory and Practice. d’Alembert’s paradox unfortunately sepa-rated fluid dynamics from the start into a theoretical mathematical part explaining phe-nomena, which could not be observed in practice, and a practical part observing phe-nomena, which could not be explained theoretically. Today, computational simulation ofturbulent flow based on mathematical theory opens to a new fruitful synthesis of theoryand practice.

To illustrate the new possibilities we now present a resolution of d’Alembert’s paradox,which is fundamentally different from the accepted resolution from 1904 by Ludwig Prandt[18], called the father of modern fluid mechanics, building on boundary layer effects fromvery small viscosity. We base our resolution on computational solution of the Euler equa-tions describing inviscid incompressible fluid flow, which shows that the potential exact


solution with zero drag considered by d’Alembert is unstable, and instead a turbulent ap-proximate solution develops with substantial non-zero drag. This solves the paradox inthe original setting of d’Alembert and Euler, assuming the fluid to be inviscid, by showingthat the zero drag potential solution cannot be realized physically, because it is unstable,and thus cannot be observed. We solve the Euler equations with a slip boundary conditionat solid boundaries prescribing the normal velocity to be zero but letting the tangentialvelocity be free, which means that no boundary layers are created. Nevertheless, a turbu-lent (approximate) solution to the Euler equations with non-zero drag develops, startingfrom the potential solution. Our resolution is thus completely different from Prandtl’s andwe claim that our resolution is more to the point for the very small viscosities met in awide range of turbulent flows in aero- and hydro-dynamics. Our resolution is better froma scientific point of view, because we do no suggest that a very small cause (very smallviscosity) can have a large effect (change the drag), as Prandtl does, which is close tosaying that anything can happen from virtually nothing, and which can be very hard toeither prove or disprove. We do not claim that boundary layer effects never can influencethe global flow e.g. by affecting the location of separation points, but we do claim based oncomputational evidence that these effects often are small for sufficiently small viscosities,which also fits with the experimental observation that the skin-friction in the boundarylayer decreases to small values as the viscosity gets small.

To solve the Euler equations numerically we use an adaptive finite element method withautomatic control of the error in the drag ([10, 6, 8, 9, 13, 4]), which we refer to as GeneralGalerkin (G2) and applied to the Euler equations as Euler/G2, or EG2 for short. Wefind that the computed drag is stable under mesh refinement reflecting an a posteriorierror estimate presented below. In [7] we also use a skin-friction boundary condition tomodel the effect of turbulent boundary layers, with zero skin-friction corresponding to aslip boundary condition. Letting the skin friction tend to zero, we obtain good agreementwith experimental drag coefficients for varying viscosity (Reynolds number) including theso-called drag crisis occuring for very small viscosities ([21]).

We now recall the Euler equations for ideal incompressible fluid flow and the stationaryirrotational potential solution with zero drag representing d’Alembert’s paradox, as wellas Prandtl’s resolution. We then present EG2 results suggesting a new resolution.

1.2. The incompressible Euler equations. We consider the motion of an inviscid in-compressible unit density fluid occupying a fixed volume Ω in R

3 with boundary Γ. Wewant to find the fluid velocity u(x, t) and pressure p(x, t) for points x = (x1, x2, x3) ∈ Ωand time t > 0, assuming that the fluid flow through the boundary Γ and the initial ve-locity u(x, 0) are given. We assume that Γ is divided into a part Γ0 corresponding to asolid (inpenetrable) boundary, and a remaining part corresponding to inflow and outflow.The mathematical model for the motion of the fluid takes the form of the Euler equationsformulated by Euler in 1755 [?] expressing conservation of momentum (Newton’s law) andconservation of mass (incompressibility), combined with a boundary condition (g) and an


initial condition (u0) for the velocity: Find u = (u, p) such that

(1.1)

u + (u · ∇)u + ∇p = 0 in Ω × I,∇ · u = 0 in Ω × I,u · n = g on Γ × I,

u(·, 0) = u0 in Ω,

where I = (0, T ] with T a final time. Here n is the outward unit normal to Γ and thegiven boundary flow g satisfies

∫

Γg ds = 0 and g = 0 on Γ0. Requiring u · n = 0 on Γ0

corresponds to a slip boundary condition (bc for short) with the normal velocity vanishing,while the tangential velocity is free.

1.3. G2 for the Euler Equations. EG2 takes the form (assuming for simplicity Γ = Γ0

thus with no in/out-flow): Find U = (U, P ) ∈ Vh such that

(1.2) (R(U), v)Q + (hR(U), R(v))Q = 0, for all v ∈ Vh,

where Vh is a space of piecewise polynomials satisfying the slip bc on a mesh in space/time

with mesh size h, (·, ·)Q is the =L2(Q)4-scalar product with Q = Ω× I, R(U) = (U + (U ·∇)U+∇P−f,∇·U) and R(v) = (v+(U ·∇)v+∇q,∇·v). It is here not necessary to go intodetails, which in particular contain jump terms arising from discontinuous approximationin time, or a modification of the test functions in case U is kept continuous in time.

The first term in (1.2) is a Galerkin term asking the residual R(U) to vanish in a weaksense, and the second term is a weighted least squares stabilization with weight equal tothe mesh size h restricting the pointwise residual. Choosing v = U , we obtain the basicenergy estimate:

(1.3)1

2‖U(T )‖2

Ω+ ‖

√hR(U)‖2

Q =1

2‖U(0)‖2

Ω,

where ‖ · ‖ω denotes an appropriate L2(ω)-norm. For a turbulent solution the stabilization

term ‖√

hR(U)‖2

Q is not small, indicating that the residual R(U) is large pointwise (∼h−1/2), and that the total kinetic energy is strictly decreasing with corresponding increaseof internal energy. This shows that that time reversal of a turbulent solution is impossible.

We show in [10] that the difference in a macroscopic mean-value output M such as

drag/lift of two different EG2 solutions U and W , can be estimated in terms of theirrespective residuals as

|M(U) − M(W )| ≤ S(‖hR(U)‖Q + ‖hR(W )‖Q),

where h represents the larger of the two mesh sizes, and S represents a stability factor ofmoderate size obtained by solving a dual linearized problem. We notice the presence ofthe crucial factor h multiplying the residuals, which makes it possible for the difference inoutput to be small, although the residuals are not small pointwise. Of course, this requiresthe stability factor S to be of moderate size, which in general is true for mean-value (butnot pointwise) outputs, as a consequence of crucial cancellation effects [10].

We notice again that it is the combination of Galerkin and weighted least squares thatproduces a reasonable compromise between Scylla and Carybdis: Only least squares will


not work because the residual cannot be small in the L2(Q)-norm, and from only theknowledge that the residual is large nothing can be concluded. Further, only Galerkin willnot work either because the residual control is too weak to produce any sensible output.

1.4. Potential Flow around a Circular Cylinder. Following d’Alembert, we considerstationary (time-independent) laminar potential flow around an (infinitely) long cylinderof diameter 1 oriented along the x3-axis and immersed in an inviscid incompressible fluidfilling R

3 with velocity (1, 0, 0) at infinity. The potential velocity is given as u = ∇φ, whereφ satisfies Laplace’s equation ∆φ = 0 outside the cylinder and appropriate conditions atinfinity, that is,

(1.4) φ(x1, x2, x3) = (r +1

r) cos(θ),

where (x1, x2) = (r cos(θ), r sin(θ)) is expressed in polar coordinates (r, θ). Using standardCalculus one verifies that u is irrotational, that is ∇ × u = 0, since ∇ × ∇φ = 0, andthat (u, p) solves the Euler equations, where the pressure p is determined by Bernoulli’slaw 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 cylinder, which are the curves followed by fluidparticles, and the pressure. We notice that the potential flow (in each section) has oneseparation point at the back of the cylinder, where the flow separates from the cylinderboundary. We also notice that both the velocity and pressure are symmetric in the flowdirection (x1-direction), which means that the drag of the cylinder is zero; the build upof pressure in front of the cylinder is balanced by the same strong pressure behind, andthus the drag is zero. The cylinder thus seems to be ”pushed through the fluid” by thestrong pressure behind, which of course is counter-intuitive and in fact is never observedin practice, where the pressure behind always is much lower than up front, with resultingnon-zero drag. According to d’Alembert’s potential solution there would be no wind loadon a high-rise and no force on a bridge pillar from a strong current, which is in contradictionwith all practical experience.

One can extend this result to flow around a body of arbitrary shape, since there is alwaysa corresponding potential laminar solution. We have thus met a scientific paradox, withobservations of substantial drag contradicting theoretical predictions of zero drag, whichhas to be resolved “at any price”, to save fluid mechanics as a mathematical science fromcollapse. Evidently, something must be wrong with the potential solution of the Eulerequations expressing Newton’s second law and mass conservation, but what can it be?

1.5. Prandtl’s 1904 Resolution. Prandtl claimed in 1904 that an arbitrarily small vis-cosity can substantially change the global flow including the drag by creating transversalvorticity in a thin boundary layer at a solid boundary, where the fluid velocity quicklychanges from zero at the boundary to the free stream value outside the layer and therebyis “tripping” the flow. Prandtl thus claimed that it is necessary to consider the Navier-Stokes equations with possibly very small but yet non-zero viscosity with no-slip bc, whereboth the normal and tangential components of the flow velocity are precribed to zero.Prandtl discarded the potential solution on the ground that it satisfies only a slip bc and


Figure 1. Potential solution of the Euler equations for flow past a circularcylinder; colormap of the pressure (left) and streamlines together with acolormap of the magnitude of the velocity (right) .

not a non-slip bc. Prandtl further claimed that because of the retardation of the flow inthe boundary layer, due to an adverse pressure gradient combined with the no-slip bc,the flow would separate away from the boundary somewhere at the rear of the cylinder.Prandtl thus claimed that because of symmetry there must be two separation points (ineach section) at the rear of the cylinder, one above and one below the x1-axis (althoughhe could see only one in his experiments). Prandtl thus discarded the zero drag potentialsolution because it did not satisfy a no-slip bc and claimed that the solution observed inexperiments was a Navier-Stokes solution with two separation points at the rear of thecylinder connecting to a turbulent wake. Several generations of fluid dynamicists haveallowed themselves to be convinced by Prandtl’s argument (see the standard view [17]).But is it correct?

1.6. The Potential Solution is Unstable. To seek an answer we first analyze the stabil-ity of the potential solution through the linearized Euler equations based on two solutionsu = (u, p) and w = (w, r) of the Euler equations (1.1) with different initial data u0 andw0. Subtracting the two versions of the Euler equations, we see that v = (v, q) = u − wsatisfies:

(1.5)

v + (u · ∇)v + (v · ∇)w + ∇q = 0 in Ω × I,∇ · v = 0 in Ω × I,v · n = 0 on Γ × I,

v(·, 0) = u0 − w0 in Ω,

which is a linear convection-reaction problem for v with u a given convection velocity,and ∇w a given matrix of reaction coefficients. Now, the stability of a given solution uis determind by the growth properties of the solution v of the linearized Euler equations(1.5) with w = u viewing v as a perturbation. The stability of a solution u thus directly


relates to the reaction term (v · ∇)u with ∇u as reaction coefficient, while the convectionterm (u · ∇)v intuitively does not seem to influence the growth of v, since it just “shiftsv around”. We thus expect the eigenvalues of ∇u to connect to the growth properties ofv, with eigenvalues with positive real part corresponding to eigenmodes with exponentialdecay and with negative real part to eigenmodes of exponential growth. Because of theincompressibility, the trace of ∇u will be zero, and thus the sum of the eigenvalues will alsobe zero, and thus ∇u normally will have eigenvalues with real parts of both signs. Thus,normally we expect to see some exponential growth, unless all the eigenvalues are purelyimaginary or zero. We thus expect perturbations of Euler solutions to grow exponentially,and thus any Euler solution would be expected to be unstable! In particular, a stationarysolution given by an analytical formula would be expected to be unstable.

As a basic example we consider the solution u = (u, p) of the Euler equations withu(x, t) = (2x1,−2x2, 0) and p = −2(x2

1+x2

2) in the halfplane x1 > 0, which is a model of

the flow at a separation point at x = 0. The potential solution (1.4) has this form at therear separation point. In this case ∇u is a diagonal matrix with diagonal (2,−2, 0), thuswith one positive (stable) and one negative (unstable) eigenvalue. This presents analyticalevidence that potential flow around a cylinder is unstable and thus cannot be observed.

1.7. Resolution by EG2 Simulation of Cylinder Flow. We now turn to EG2 simula-tions initiating the flow from rest by sudddenly imposing an inflow condition. We then seethe zero-drag irrotational potential solution quickly developing during the first time steps,but then the potential solution gradually changes into a turbulent solution with large dragand vorticity, see Fig. 2. We observe that the computed Euler solution has the followingkey features: (i) no boundary layer prior to separation, (ii) one separation point in eachsection of the cylinder which oscillates up and down and (iii) strong vorticity in the stream-wise direction. The computed drag is ≈ 1.0, which is consistent under mesh refinement,and which fits with the observation ([21]) that the drag increases from ≈ 0.5 to about 1.0beyond the drag crisis occuring for ν ≈ 10−6. We see in Fig. 3 that the streamwise (x1)vorticity dominates the tranversal (x3) vorticity, and that the pressure is low inside tubesof vorticity in the x1-direction behind the cylinder, which creates drag. Details on thesimulation, including movies for velocity, pressure and vorticity are available at [10, 1].

1.8. Comparing the New and the Old Resolution. Prandtl discarded the potentialsolution because it does not satisfy the no-slip bc, while we say based on both analyticaland computational evidence, that it will have to be discarded because it is unstable andthus cannot be observed.

Prandtl claims that that the no-slip bc generates substantial transversal vorticity, whilewe say based on both analytical and computational evidence, that strong streamwise vor-ticity is generated at the separation point. Prandtl’s transversal vorticity is of strength1/δ in a layer of width δ, giving a total vorticity of unit size with corresponding smallturbulent dissipation reflecting a small skin friction. Prandtl’s resolution thus is unable toaccount for a substantial drag, while a strong streamwise vorticity is.

Prandtl claims that there are two separation points in the rear, while we say based onboth analytical and computational evidence, that there is only one.


Figure 2. Computational solution of the Euler equations for flow past acircular cylinder; colormap of the pressure (left) and streamlines togetherwith a colormap of the magnitude of the velocity (right) .

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


Prandtl claims that it is the boundary layer prior to separation which causes the drag,while we say based on both analytical and computational evidence, that for small viscos-ity the effect on the drag of the boundary layer is small. Altogether, our resolution isfundamentally different form Prandtl’s including in particular the aspects (i)-(iii).

2. Sommerfeld’s Paradox

For a time after this negative result (stability of Couette flow for all Reynold’snumbers), it was thought that the method of small oscillations (classical theory)was unsuitable for the theoretical solution of the problem of transition. It tran-spired later that this was not justified, because Couette flow is a very restrictedand special example (Schlichting in Boundary Layer Theory 1955 [19]).

The sudden transition from smooth, laminar flow to turbulence as the fluid ve-locity is gradually increased remains one of the least adequately explained phe-nomena in all of classical physics (James Case, SIAM News, 2002).

2.1. Couette Flow and Sommerfeld’s Modal Analysis. Couette flow between twoparallel plates (with normals in the x2-direction) is the simplest of all laminar flows witha linear velocity profile u(x) = (x2/δ, 0, 0) and p = 0 with δ > 0 a constant, which isa stationary solution of the Euler (and also the Navier–Stokes) equations. It representsparallell shear flow in the x1-direction, which may occur inside a flow or in a boundarylayer along a boundary at x2 = 0. The streamlines are parallell to the x1-axis and theu1 velocity increases linearly with x2, and the coefficient δ represents the thickness of theshear layer and 1/δ the strength of the shear.

Figure 4. Couette flow: parallell shear flow.

To determine the stability of Couette flow, Sommerfeld considered the linearized equa-tions (1.5) with reaction coefficient matrix ∇u = (0, 1/δ, 0; 0, 0, 0; 0, 0, 0) with rows sepa-rated by a semi-colon, noting that the eigenvalues of ∇u are all equal to zero. Sommerfeldconcluded that Couette flow should be stable, since exponentially unstable eigenmodes arelacking. However, Sommerfeld then forgot that the presence of the off-diagonal coefficient1/δ allows for linear growth in t with slope 1/δ. This is referred to as non-modal growth asopposed to modal growth of eigenfunctions, occuring because the matrix ∇u is non-normal(in particular non-symmetric) with degenerate eigenmodes corresponding to the multiple


zero eigenvalue. More precisely, we expect to see that v1(t) ∼ tv02/δ corresponding to large

growth in v1(t) of an initial perturbation v0

2over a time interval of unit length if δ is small,

and over a long time interval if δ is of unit size. In particular, in a boundary layer of smallthickness δ, perturbations grow quickly. The net result is that we have obtained analyticalevidence that the laminar Couette flow may turn turbulent (for small or zero viscosity), aconclusion we support by computational evidence below.

Sommerfeld thus made a completely elementary error in the stability analysis based onthe linearized problem, by forgetting that for a non-normal matrix, a simple eigenvalueanalysis does not tell the whole truth (which may be comforting to all of us who strug-gle with mathematics in one form or the other). However, Sommerfeld’s mistake passedthrough several generations of fluid dynamicists, and was questioned first in the 1960s andthen only by a few. Most textbooks in fluid mechanics still today present Sommerfeld’serronous conclusions [21], and the debate in the fluid dynamics community between a mi-nority of proponents of non-modal analysis and a majority following Sommerfeld is stillgoing on, see [10] for more details.

0 10 20 30 40 50 600

0.02

0.04

0.06

0.08

0.1

0.12

0 10 20 30 40 50 600

5

10

15

20

25

0 10 20 30 40 50 600

5

10

15

20

25

0 10 20 30 40 50 600

5

10

15

20

25

Figure 5. Couette flow (random initial perturbation): Perturbations ‖ϕi‖as functions of time (upper left), derivatives ‖∂ui/∂x1‖ (upper right),‖∂ui/∂x2‖ (lower left), and ‖∂ui/∂x3‖ (lower right) as functions of time.

2.2. EG2 for Couette Flow. We now turn to EG2 simulations of flow between twoparallel plates at unit distance, initating the flow as Couette flow with the streamwisevelocity varying linearly in the direction normal to the plates from -1 to 1. EG2 shows thephenomenon of transition from the laminar Couette flow to turbulent flow after a certainlength of time depending on the form and size of the perturbation. The non-modal stability


Figure 6. Streamwise velocity isosurfaces for |u1| = 0.2 in Couette flow(random initial perturbation) for t = 0, 1, 4, 5, 7, 10


Figure 7. Streamwise velocity isosurfaces for |u2| = 0.015 in Couette flow(random initial perturbation) for t = 0, 1, 4, 5, 7, 10


0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

3

0 5 10 15 20 25 30 35 400

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Figure 8. Couette flow (κν = 0.5): Perturbations ‖ϕi‖ as functions of time(upper left), derivatives ‖∂ui/∂x1‖ (upper right), ‖∂ui/∂x2‖ (lower left), and‖∂ui/∂x3‖ (lower right) as functions of time.

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

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3.5

4

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Figure 9. Couette flow (κν = 0.1): Perturbations ‖ϕi‖ as functions of time(upper left), derivatives ‖∂ui/∂x1‖ (upper right), ‖∂ui/∂x2‖ (lower left), and‖∂ui/∂x3‖ (lower right) as functions of time.


Figure 10. Streamwise velocity isosurfaces for |u1| = 0.2 in Couette flow(κν = 0.5) for t = 0, 5, 10, 15, 20, 30


Figure 11. Transversal velocity isosurfaces for |u2| = 0.2 in Couette flow(κν = 0.5) for t = 0, 5, 10, 15, 20, 30


Figure 12. Transversal velocity isosurfaces for |u3| = 0.2 in Couette flow(κν = 0.5) for t = 0, 5, 10, 15, 20, 30


analysis indicates that transition to turbulence may take place when the perturbationv1(t) ∼ tv0

2(with δ = 1) reaches a certain threshold value, and this is also what EG2

shows: If the initial perturbation v0

2is not too small, then transition to turbulence will

occur within the available time. The perturbations with maximal growth are streamwiseconstant transversal velocities forming “rolls” in the streamwise direction with the effectof mixing high and low speed streamwise velocities to a streamwise velocity profile withstronger gradients, which eventually triggers the transition to turbulence. The initial non-modal mixing phase of the transition process just described, is the most critical one andalso the one open to analysis because the base flow is so simple with a linear velocity. Itis thus possible to understand the basic initial mechanism of the complex phenomenonof transition. In the computations we use periodic boundary conditions in the spanwisedirection, and also in the streamwise direction, which allows simulation over the long timeinterval required in the present case with δ of unit size

In Fig. 5-12 we present computational results for Couette flow with a (i) random ini-tial perturbation of maximal size 1, centered in (0.5, 0.5, 0.5), and (ii) transversal “roll”perturbations. We see most of the random perturbations being quickly damped leavingslowly decaying transversal rolls, which if strong enough may cause perturbation growth inthe streamwise direction leading to transition to turbulence. The simulations thus exhibitboth the perturbations with maximal growth as well as the full complex phenomenon oftransition.

2.3. Resolution of Sommerfeld’s Paradox. Sommerfeld claimed based on a modaleigenvalue analysis forgetting the non-normality of the linearized problem, that laminarCouette flow is stable, which is at variance with observations of transition to turbulencein Couette flow with very small viscosity. We have seen that a correct non-modal analysisshows that certain small transversal perturbations may over long time cause substantialperturbation in the streamwise velocity by mixing high and low speed flow, which we inEG2 simulations see triggering transition to turbulence. We thus resolve Sommerfeld’sparadox by correcting the mathematical stability analysis to conform with observations.

3. Loschmidt’s Paradox

There are great physicists who have not understood it. (Einstein about Boltz-mann’s statistical mechanics)

There is apparently a contradiction between the law of increasing entropy andthe principles of Newtonian mechanics, since the latter do not recognize any dif-ference between past and future times. This is the so-called reversibility paradox(Umkehreinwand) which was advanced as an objection to Boltzmann’s theory byLoschmidt 1876-77. (Translators foreword to Lectures on Gas Theory by Boltz-mann).

3.1. The Arrow of Time and the 2nd Law of Thermodynamics. In classical Hamil-tonian mechanics there is no preferred direction of time or no arrow of time; the future aswell as the past of a Hamiltonian system is uniquely determined by the state at a giventime instant. Hamiltonian mechanics allows the existence of a perpetuum mobile, e.g. in


the form of a solar system with planets for ever orbiting a sun, forward or backward intime, or electrons for ever orbiting atomic kernels.

Hamiltonian mechanics served as a basis of science until the later half of the 19th century,when statistical mechanics was developed by Boltzmann [2] and the beginning of the 20thcentury when the first steps towards quantum mechanics was taken by Planck [16], as a wayto handle the irreversible nature of various observed physical phenomena with the arrowof time always pointing forward, including the impossibility of constructing a perpetuummobile. Irreversibility was formulated by Clausius in 1850 as a summary of experimentalobservations in the form of the 2nd Law of Thermodynamics stating that the entropy of aclosed system cannot decrease, noting that certain closed systems have a strictly increasingentropy and thus are time irreversible. Yet the origin and nature of the 2nd Law remaineda mystery, despite the tendency of Nature to always move forward in time, never backward.

3.2. Statistical Mechanics. Boltzmann developed statistical mechanics to counter Loschmidt’sParadox [15] comparing the macroscopic irreversibility of Boltzmann’s equations expressedin the H-theorem reflecting the 2nd Law, with their supposed microscopic basis as a re-versible Hamiltonian system of particles colliding elastically. Boltzmann based the deriva-tion of the collision term in his equation for the particle density interpreted as a probabil-ity distribution, and thus the H-theorem, on statistical independence of particle velocitiesbefore collision, but not after, which removed the time symmetry and opened up for irre-versibility. Prandtl was led into quantum mechanics stimulated by statistical mechanicsas a way the handle the observed irreversible nature of black-body radiation, in which anon-reflecting body absorbs light of all frequencies but only emits lower frequencies, whichcould not be explained by reversible Hamiltonian mechanics.

Statistical mechanics poses difficulties of falsification required by Popper in a scientifictheory, since the validity of Boltzmann’s basic microscopic assumption of statistical inde-pendence in a gas with each mole consisting of 6 · 1023 molecules, seems to be beyond thepossibility of any kind of conceivable experiment or mathematics; only indirect evidencein the form of macroscopic observations seem to be possible, which is far from enough. Infact, it is known that Boltzmann’s assumption can only be (nearly) true in the very specialcase of a very dilute gas with rare collisions, and the derivation of Boltzmann equationsfor more general situations seems to pose unsurmountable problems.

3.3. Resolution by Finite Precision Hamiltonian Mechanics. We now proceed topresent a new resolution of Loschmidt’s paradox on the occurence of irreversibility inreversible Hamiltonian mechanics, using a certain form of finite precision computation,instead of statistics. Our resolution is not trivial in the sense that the finite precisioncomputation simply adds some friction leading to irreversibility, but deeply connects tofundamental features of turbulence of macroscopic mean-value determinism combined withmacroscopic pointwise indeterminism, based on microscopic determinism and resultingfrom a combination of finite precision computation and certain aspects of stability. Wethus find that pointwise values (but not mean-values in space-time) of turbulent solutions,based on deterministic finite precision computation, are indeterminate, and therefore maybe viewed as representing a macroscopic game of turbulence roulette reflecting the pointwise


complexity of turbulent solutions. Statistical mechanics is based on microscopic games ofroulette, each of which seemingly requiring a microscopic resolution, which would seemto lead to a never-ending chain of microscopics of microscopics. In contrast, we assumedeterministic microscopics and discover macroscopic pointwise indeterminism and mean-value determinism, without any form of statistics. In particular, we replace statisticalensemble mean values with mean-values in space-time and do not need any properties ofergodicity connecting these quantities, as in statistics.

As Hamiltonian model we consider the Euler equations for an inviscid compressibleperfect gas expressing the 1st Law of Themodynamics as conservation of mass, momentumand energy. We solve the Euler equations with finite precision computation in the form ofG2 and refer to the computational model as EG2. We find that EG2 solutions are turbulent,identifying turbulence as an irrecoverable transformation of kinetic energy into heat energyexpressing a certain form of the 2nd Law. We show both analytically and computationallythat EG2, based on the 1st Law and finite precision computation, automatically satisfiesa certain form of the 2nd Law expressing an irrecoverable transformation of kinetic energyinto heat energy. In short, we obtain the 2nd Law as a consequence of the 1st Law andG2 finite precision computation, without any reference to statistics, thus eliminating themystery of the 2nd Law and resolving Loschmidt’s paradox.

The Euler equations lack exact pointwise solutions, because any exact solution is unstableand thus develops into an approximate solution with non-smooth features of turbulenceand shocks. EG2 is based on satisfaction of the Euler equations in a certain weak mean-value sense to a precision proportional to h1/2, combined with a satisfaction in a strongpointwise sense proportional to h−1/2. EG2 represents a midway between the Scylla ofweak solutions and the Carybdis of strong solutions, both being non-functional alone. Theturbulent Euler/G2 solutions thus do not (even approximately) satisfy the Euler equationsin a pointwise sense, but nevertheless may produce correct mean-values.

We thus resolve Loschmidt’s paradox by showing that the exact solutions of the Eulerequations, which would have been reversible had they only existed, do not exist, andby showing that the approximate solutions, which do exist, are irreversible because theyirrecoverably transform kinetic energy into heat energy. We thus show that in Hamiltoniansystems, which are sufficiently complex to allow turbulence, there is an arrow of time, andwe do this without any reference to the concept of entropy. We thus do not have to givethis concept a physical meaning, which led Boltzmann to invent statistical mechanics.We believe this amounts to a substantial simplification of thermodynamics as scientificdiscipline, which has shown to be very difficult to both learn, teach and apply.

3.4. The Compressible Euler Equations. We consider an inviscid compressible perfectgas enclosed in a fixed (open) domain Ω in three-dimensional space R

3 with boundary Γover a time interval [0, t ] with intial time zero and final time t. We thus assume thatthere are no viscous forces (inviscid flow) and we also assume that there is no heat flowfrom conduction (zero heat conductivity). We seek the density ρ, momentum m = ρu withu = (u1, u2, u3) the velocity, and the total energy e as functions of (x, t) ∈ Ω∪Γ×[0, t ], wherex = (x1, x2, x3) denotes the coordinates in R

3 and ui is the velocity in the xi-direction.


The Euler equations for u ≡ (ρ, u, e) read with Q = Ω × I and I = (0, t ]:

(3.1)

ρ + ∇ · (ρu) = 0 in Q,mi + ∇ · (miu) + p,i = fi in Q, i = 1, 2, 3,

e + ∇ · (eu + pu) = 0 in Q,u · n = 0 on Γ × I,

u(·, 0) = u0 in Ω,

where p = p(x, t) is the pressure of the fluid, p,i = ∂p∂xi

is the partial derivative withrespect to xi, the dot indicates differentiation with respect to time, n denotes the outwardunit normal to Γ and f = (f1, f2, f3) is a given volume force (like gravity) acting on thefluid, and u0 = u0(x) represent initial conditions. Further, the total energy e = k + θ,where k = ρ|u|2/2 is the kinetic energy and θ = ρT is the internal energy with T thetemperature scaled so that cv = 1, where cv is the heat capacity under constant volume,and p = (γ−1)ρT where γ = cp/cv > 1 is the gas constant with cp the heat capacity underconstant pressure.

3.5. EG2 for Compressible Flow. We now present EG2 for the compressible Eulerequations written in a system as follows: Find u(x, t) such that

(3.2)R(u) ≡ ∂u

∂t+

∑

3

i=1fi(u),i = 0 in Q,

u(x, t) · n(x, t) = 0 in Γ × I,u(·, 0) = u0 in Ω,

where

u = ρ

1u1

u2

u3

ε/ρ

, fi(u) = uiu + p

0δ1i

δ2i

δ3i

ui

,

where δii = 1 and δij = 0 if i 6= j. EG2 takes the general form: Find u ∈ Vh such that forall v ∈ Vh

((R(u), v)) + ((hR(u), Ru(v)))

+ ((ν∇ui,∇vi)) + (u(·, 0) − u0, v(·, 0)) = 0,(3.3)

where Vh is a finite element space of mesh size h in space-time of functions v with velocitycomponents v satisfying the boundary condition v · n = 0 on Γ, ((·, ·)) and (·, ·) representL2(Q) and L2(Ω) scalar products, Ru(v) is the linearization of R(u) obtained by freezing theconvective velocity at u noting that Ru(u) = R(u), and ν = h2|R(u)| is a shock-capturingartificial viscosity.

3.6. Joule’s 1845 Experiment. For illustration, we recall Joule’s experiment from 1845with a gas initially at rest with temperature T = 1 at a certain pressure in a certainvolume immersed into a container of water, see Fig. 13. At initial time a valve was openedand the gas was allowed to expand into the double volume while the temperature changein the water was carefully measured by Joule. To the great surprise of both Joule and


the scientific community, no change of the temperature of the water could be detected, incontradiction with the expectation that the gas would cool off under expansion. Moreover,the expansion was impossible to reverse; the gas had no inclination to contract back to theoriginal volume.

We now simulate Joule’s experiment using EG2 and we discover the following as displayedin Fig.15-15: The mean temperature in the left container drops below 1 as the gas expandsinto the right container with increasing velocity, and shocks/turbulence appearing whichheat the gas in the right container, increasing the mean temperature in the right container.The total energy is, of course, conserved by the 1st Law, and in the final state with thegas at rest in the two containers, corresponding to zero kinetic energy, the temperature isback to T = 1. We can also understand that the rapidity of the expansion process makes itdifficult to detect any temperature drop in the water in the initial phase. Altogether, usingEG2 we can first simulate and then understand Joule’s experiment, and we thus see noreason to be surprised. We shall see below as a consequence of the 2nd Law that reversalof the process with the gas contracting back to the original small volume, is impossiblebecause the only way the gas can be put into motion is by expansion, and thus contractionis impossible.

Figure 13. The Joule-Thomson experiment

We now compare with an analysis of the experiment using classical thermodynamicsbased on statistical mechanics. We then recall that classical thermodynamics only considerssystems in equilibrium, that is we start with the gas at rest in the initial volume and weend up with the gas at rest in the double volume. By energy conservation, we understandthat the temperature cannot change, so Joule’s observation is after all not surprising evenwith a classical perspective. But how can we explain the irreversibility? Well, the basicidea of classical thermodynamics is to say that there is something, the entropy, which hasincreased from initial to final state. Boltzmann would say that because the volume of thefinal state is larger, the final state is “less ordered” or “more probable”, and this wouldmake the reverse process with the gas contracting back to the initial small volume, if not


Figure 14. EG2 simulation of the Joule-Thomson experiment: snapshotsof density (upper 2 figures) and temperature (lower 2 figures) at 2 differenttime instants.


0 5 10 15 20 25 30 350.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

0 5 10 15 20 25 30 350.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 5 10 15 20 25 30 350.75

0.8

0.85

0.9

0.95

1

1.05

1.1

0 5 10 15 20 25 30 350.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 5 10 15 20 25 30 350

1

2

3

4

5

6

7

8x 10

−4

0 5 10 15 20 25 30 350

0.005

0.01

0.015

0.02

0.025

Figure 15. The Joule-Thomson experiment: mean density (upper), meantemperature (middle) and mean kinetic energy (lower), for the left (left) andright chamber (right) in the initial phase of the EG2 simulation in Fig.15.


completely impossible, so at least very improbable. Of course, from scientific point thisis not easy to make sense of, and we would still wonder why the gas would be willingto expand but not contract. To say that expansion is “more probable” does not explainanything, as far as we can see at least.

In contrast, taking the true dynamics of the process into account including in particularthe second phase with heat generation from shocks or turbulence, we can easily intuitivelyunderstand the observation of constant temperature and irreversibility in a deterministicfashion without using any concept of entropy ultimately based on statistics.

3.7. The 2nd Law. We show in [10] that EG2 (up to a term of order h1/2) satisfies thefollowing global form of the 2nd Law:

K − W = −∆,

E + W = ∆,(3.4)

where

K =

∫

Ω

k dx, E =

∫

Ω

e dx, ∆ ≥ 0

and

W =

∫

Ω

p∇ · v dx,

represents the total work performed by the pressure p on the velocity v, and where ∆ > 0for solutions with shocks/turbulence. We note that W > 0 under expansion with ∇·v > 0.We see from the 2nd Law (3.4) that there is a transfer of kinetic energy to heat energy ifW < 0, that is under compression, and a transfer from heat to kinetic energy if W > 0,that is under expansion. Most remarkably, there is always a transfer from kinetic to heatenergy since always ∆ ≥ 0 with ∆ > 0 for shocks/turbulence.

Returning to Joule’s experiment, we see by the 2nd Law that contraction back to theoriginal volume from the final rest state in the double volume, is impossible because theonly way the gas can be set into motion is by expansion.

3.8. Resolution of Loschmidt’s Paradox. We have seen that EG2 produces approxi-mate solutions to the Euler equations which are irreversible in the general case of shocksand turbulence. Thus EG2 produces approximate solutions to the Euler equations, whichare irreversible, to be compared with exact solutions, which would have been reversiblehad they existed, but they don’t. This resolves Loschmidt’s paradox by finite precisiondeterministic computation, without reference to any form of statistics.

4. Gibbs’ Paradox

Neither Herr Boltzmann nor Herr Planck has given a definition of W ... UsuallyW is put equal to the number of complexions. In order to calculate W , oneneeds a complete (molecular-mechanical) theory of the system under considera-tion. Therefore it is dubious whether the Boltzmann principle has any meaningwithout a complete molecular-mechanical theory or some other theory which de-scribes the elementary processes (and such a theory is missing). (Einstein)


4.1. Boltzmann’s Entropy. Using assumptions about particle statistics Boltzmann couldprove his H-thorem stating that a certain quantity S named entropy could never decreaseand when increasing strictly would signify irreversibility and give the arrow of time a di-rection forward: Reversal of a process with strictly increasing entropy would violate the2nd Law and thus be impossible. On Boltzmann’s tombstone the famous formula

(4.1) S = k log(W )

is engraved, where k is Boltzmann’s constant (k ≈ 10−23 joules/kelvin) and W denotes theprobability of a certain state representing the number of ”complexions” or “microstates”corresponding to a “macrostate”. Increasing entropy would then reflect that Nature wouldtend to move from less probable to more probable states or towards states with morecomplexions. However, the crucial questions why and how Nature would seek to alwaysincrease entropy, was left without any answer.

4.2. Gibb’s Paradox and Its Resolution. Boltzmann’s statistical mechanics poses se-rious difficulties from scientific point of view and paradoxes line up. One of them is Gibbs’paradox [5] illustrating the difficulty of defining entropy by counting micro-states: Considera volume V divided by a membrane into two equal volumes V/2 filled with two differenttypes of gas at rest with the same density and temperature. Remove the membrane andlet both gases expand to the double volume and come to rest in a mixed state. Gibbsrecalls that according to Boltzmann the entropy of each gas would then increase by thefactor log(V ) − log(V/2) = log(2), since for constant temperature S ∼ log(V ) with V thevolume of the gas. The entropy of the system would then also increase by a log(2) factor.

Gibbs then compares with the case with the gases being of the same type in which caseremoval of a membrane would change nothing and in particular the entropy should notchange, in contrast to the above case. In other words, the entropy should be extensive inthe sense that the entropy of a gas over a volume V should be equal to the sum of theentropies over two volumes V/2. To achieve this, Gibbs suggested to change the counting ofmicrostates in the case of equal gases, taking into account permutations of equal molecules,but was then faced with a paradoxical discontinuos behavior of the entropy: No changefor the same type of gas, and a log(2) change as soon as the types of gas differ only theslightest.

With our experience from the Joule-Thomson experiment, we can easily resolve thisparadox. It suffices to note that an initial pressure difference resulting from different stateequations for two types of gas in the two volumes, would drive a process towards equalpressure in the full volume, with the associated turbulent dissipation (entropy production)being small if the initial pressure difference is small, that is, if the type of gas is nearlythe same. In particular, for equal gases nothing would happen and there would be nosudden entropy jump under a slight change of gas, as in Gibbs’ resolution by recountingmicrostates.


We understand that the Gibbs’paradox results from Boltzmann’s idea to define entropyby counting microstates, and thus is (one of many) indications that statistical mechan-ics creates more problems than it solves, and from scientific point of view therefore isquestionable.

References

[1] New resolution to d’alembert’s paradox, http://www.csc.kth.se/∼jhoffman/research/dal.html.[2] L. Boltzmann, Lectures on Gas Theory, Dover, 1964.[3] J. d’Alembert, Essai d’une nouvelle theorie de la resistance des fluides, Paris,

http://gallica.bnf.fr/anthologie/notices/00927.htm, 1752.[4] FEniCS, Fenics project, http://www.fenics.org, (2003).[5] J. W. Gibbs, On the Equilibrium of Heterogeneous Substances (1875), The Scientific Papers by J W

Gibbs, Dover, 1961.[6] J. Hoffman, Computation of mean drag for bluff body problems using adaptive dns/les, SIAM J. Sci.

Comput., 27(1) (2005), pp. 184–207.[7] , Computation of turbulent flow past bluff bodies using adaptive general galerkin methods: drag

crisis and turbulent euler solutions, Comput. Mech., 38 (2006), pp. 390–402.[8] , Adaptive simulation of the turbulent flow past a sphere, J. Fluid Mech., (accepted).[9] , Efficient computation of mean drag for the subcritical flow past a circular cylinder using general

galerkin g2, Int. J. Numer. Meth. Fluids, (accepted).[10] J. Hoffman and C. Johnson, Computational Turbulent Incompressible Flow: Applied Mathematics

Body and Soul Vol 4, Springer-Verlag Publishing, 2006.[11] , Finally: Resolution of d’alembert’s paradox, Finite Element Center preprint

(www.femcenter.org), (2006).[12] , Is the world a clock with finite precision?, Finite Element Center preprint

(www.femcenter.org), (2006).[13] , A new approach to computational turbulence modeling, Comput. Methods Appl. Mech. Engrg.,

195 (2006), pp. 2865–2880.[14] , Computational Thermodynamics: Applied Mathematics Body and Soul Vol 5, Springer-Verlag

Publishing, 2007.[15] J. Loschmidt, Sitzungsber. kais. akad. wiss. wien, math., Naturwiss. Classe 73, (1876), pp. 128–142.[16] M. Planck, Acht Vorlesungen uber Theoretischen Physik, Leipzig, 1910.[17] Prandtl, Prandtl standard view, www.fluidmech.net/msc/prandtl.htm, (2006).[18] L. Prandtl, On motion of fluids with very little viscosity, Third International Congress of Mathe-

matics, Heidelberg, http://naca.larc.nasa.gov/digidoc/report/tm/52/NACA-TM-452.PDF, (1904).[19] H. Schlichting, Boundary Layer Theory, McGraw-Hill, 1955.[20] A. Sommerfeld, Ein beitrag zur hydrodynamischen erklarung der turbulenten flussigkeitsbewegungen,

Atti del 4. Congr. Internat.dei Mat. III, Roma, (1908), pp. 116–124.[21] D. J. Tritton, Physical Fluid Dynamics, 2nd ed. Oxford, Clarendon Press, 1988.

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