Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 97
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They do not lead to the overthrow of the theory, yet they entailtransformations of such a magnitude that the word 'application' sounds inadequate.The phenomena are not passively subjected to a rigidly established theory, but insteadreact upon the content and structure of the theory. They challenge the theory and may thusinduce important adaptive transformations.What distinguishes the history of hydrodynamics from that of other physical theories isnot so much the tremendous effect of challenges from phenomenal worlds, but rather it isthe slowness with which these challenges were successfully met.
Nearly two centurieselapsed between the first formulation of the fundamental equations of the theory andthe deductions of laws of fluid resistance in the most important case of large Reynoldsnumbers. In contrast, the theories of mechanics, electrodynamics, and thermodynamicswere almost immediately useful in making predictions in the intended domains of application. Hydrodynamics is probably the only theory whose promises to comprehend arange of phenomena took so long to be fulfilled.The reasons for this extraordinary delay are easily identified a posteriori.
They are theinfinite number of degrees of freedom and the nonlinear character of the fundamentalequations, both of which present formidable obstacles to obtaining solutions in concretecases. Moreover, instability often deprives the few known exact solutions of any physicalrelevance. Although unstable solutions also occur in ordinary mechanics, they do notinterfere with the most common applications. In contrast, almost every theoretical description of a natural or man-made flow involves instabilities.These difficulties have barred progress along purely mathematical lines. They have alsomade physical intuition a poor guide, and a source of numerous paradoxes.
Hydrodynarnicists therefore sought inspiration in concrete phenomena. Challenged to understandand act in real worlds, they developed a few innovative strategies. One was to modify thefundamental equations, introducing for instance Navier's viscous term, or still other termsof higher order (as a few French engineers tried to do).
Another was to give up thecontinuity of the solutions of Euler's equation, and to study the evolution of the resultingsingularities. Helmholtz pursued this approach without leaving the realm of the perfectliquid. The instability of laminar solutions was also evoked, and the resulting turbulencesubjected to a statistical analysis or absorbed in the parameters of semi-empirical, effectivetheories oflarge-scale flow.
Rules of similarity were used, either to predict the properties offull-scale flows from model measurements or to limit the form of resistance and retardation laws. When none of that worked, the Columbus-egg method was still available,where the hydrodynarnicist could try to determine the concrete conditions under which thefew flows he could predict would actually occur. This 'streamlining' strategy proved quitefruitful, because the computable flows happen to be those for which fluid resistance isa minimum.None of these strategies sufficed to fully master the real flows for which they wereintended. Prandtl's ultimate success depended on combining them within the asymptoticframework of high Reynolds numbers (quasi-inviscid fluid) and large aspect ratios (quasitwo-dimensional flow).
The role of a small viscosity, Prandtl reasoned, is to produceboundary layers of high shear, and vortex sheets to which Helmholtz's theory of vortexmotion may be applied in a second step. Vortex sheets are always unstable, and boundaryDRAG AND LIFT325layers often are so. These instabilities lead to turbulence. Similitude and statistical considerations allow a quantitative determination of the average effects of turbulence in casesof non-separated flow. When separation occurs, the hydrodynamicist is left with Columbus's egg, unless strong resistance is desired, in which case he can appeal to modelmeasurements combined with similitude arguments.Engagement with and challenges from the real worlds of flow were essential to thedevelopment of the above-mentioned strategies.
The challenged theorists strove to findnew solutions and to develop new methods of approximation. Experience indicated somegeneral properties of the motion, such as the existence of boundary layers, the randomcharacter of turbulence, the sudden character of the Reynolds transition, or the formationof trailing vortices. Experimentation on ship models induced reflection on the conditionsunder which similitude applied. The focus on specific systems, such as Stokes's 'boxes ofwater' or Helmholtz's organ pipes, permitted instructive comparisons between explicitsolutions of the fundamental equations and real flows. Altogether, there were many waysin which practical concerns oriented theorists in the conceptual maze of fluid dynamics.The evolution from a paper theory to an engineering tool thus depe�ded on transgressions of the limits between academic hydrodynamics and applied hydrodynamics.
Theutilitarian spirit of Victorian science, the Polytechnique ideal of a theory-based engineering, a touch of Helmholtz's eclectic genius, and the Giittingen pursuit of applied mathematics all contributed to the fruitful blurring of borders between physics and engineering.The 'sagacious geometers' who answered d'Alembert's ancient call for a solution to hisresistance paradox all visited the real worlds of flow.APPENDIX AMODERN DISCUSSION OF D'ALEMBERT'S PARADOXA solid body is set into motion within an infinite, homogeneous, perfect liquid and keptmoving at the constant velocity U.
According to a theorem by Lagrange, the resulting flowadmits a potential cp (as long as the fluid motion remains thoroughly continuous). Owingto the incompressibility of the fluid, this potential must satisfy Laplace's equation !lcp = 0.Consequently, at every instant it is completely determined (up to a constant) by theboundary conditions that the velocity v should vanish at infmity and that the normalcomponent ofv - U should vanish on the surface of the body. The velocity field thereforefollows the body in its motion, which means that the flow pattern is steady from the pointof view of an observer bound to the body.The first non-constant term in the multipolar expansion of the potential at a largedistance from the body is dipolar, since a single pole would imply a divergent flux from thebody, in contradiction with the incompressibility of the fluid.
Hence the fluid velocityvaries asymptotically as the inverse cube of the distance from the body.The most direct way to determine the force impressed on the body by the fluid is tocompute the pressure integral(A. I)over the surface u of the body. According to a theorem by Green, this is also equal to theintegralR=JfluidJV'Pdr - P dS,(A.2)};where the second integral is taken over a spherical surface 2: surrounding the body, and thefirst over the volume of the fluid contained between the surfaces u and 2: (see Fig. A. l).The surface integral tends to zero as r-4 when the radius r of the sphere approaches zero,because Bemoulli's law applies to the pressure P.
Euler's equation gives8vV'P = -p t - p(v Y')v = p[(U - v) V']v,B··(A.3)or, using the incompressibility condition V' v = 0,·(A.4)Ostrogradski's theorem then gives... ----- .. .. ...... ....,,.. ... ...DRAG AND LIFT'L(''II\''" ""............ ......Fig. A. l .\CJr_ _ _ _ _ _ _ ..."',..327'')\'II'/'"Integration surfaces for a discussion of d'Alembert's paradox.J \1P dr=J-p v[(U - v)·JdS] + p v[(U - v) dS] .:z·(A.5)-The integral over the surface of the body vanishes since the normal component of U vvanishes.
The integral over the sphere l tends to zero as r- 1 when its radius r tends toinfinity. Consequently, the resistance vanishes.The former reasoning amounts to applying the momentum principle to the fluid contained between the surfaces <r and l in a reference system bound to the bo.dy: as the flow issteady in this system, the sum of the pressures applied to the fluid on these two surfacesmust be equal to the flux of the momentum tensor p(v; - U;)(vj - Uj) across them. Takinginto account the incompressibility of the fluid, this flux is identical to the right-hand side ofeqn (A.5).Although it is tempting to apply the momentum principle to the whole, infmite volumeof the fluid in the reference system for which the fluid is at rest at infinity, this is notpossible because the total momentum of the fluid diverges logarithmically.
In contrast,Borda's application of the energy principle (conservation of live force) turns out to beperfectly legitimate, because the total energy of the fluid is finite and well defined.According to Borda's simple reasoning of 1766, the work of the resistance during themotion of the body must be equal to the variation of the energy of the fluid motion, whichis nil since the flow pattern is invariant.
This reasoning only proves the nullity of the dragcomponent of the resistance. Recourse to the momentum principle is necessary to provethe nullity of the lift component. 1The derivations given above of d'Alembert's paradox crucially depend on the infiniteextent of the fluid. If there is a wall or a free surface in the vicinity of the body, then thesurface l can no longer be rejected to infinity, and the resistance generally takes a finitevalue. The only exception is the case of a body moving in a direction parallel to acylindrical wall. In the vicinity of a free surface, the body experiences a resistance even ifit moves in a direction parallel to the surface, owing to the constant production of surfacewaves.Another way to escape the nullity ofthe resistance within a perfect liquid is to reduce thedimensionality of the space. In two dimensions, the irrotational character of the motion no1Borda [1766].
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