Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 88
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If thecurvature is too high, then viscous drag is not sufficient to 'pump off the excess of deadwater, and a discontinuity surface is formed. Lanchester thus made separation depend onthe competition between external pressure gradient and internal viscous stress, as Prandtlhad done differently in 1904. 59Lanchester also investigated skin friction along a plate advancing with the velocity Uthrough a still fluid.
Following Rankine and Froude, he balanced the frictional force onthe plate with the momentum increase in the boundary layer. In Prandtl's symbols, thisgives8ujJ, [)y= d.x J p(u - U) dy.d+oo2(7.10)0Lanchester then assumed that the flow caused by the velocity a U only differed from theflow induced by the velocity U through the rescaling uau, y -+ {3y of the velocity-+58Ibid.
p. 1606.59Lanchester [1907] pp. 27-30. Cf. Ackroyd [1992], (1996].290WORLDS OF FLOWprof!le. Since the previous balance between inertial and frictional forces must be preservedin the new flow, the relation a{32 = 1 must hold. The friction is therefore multiplied bya312, which means that the resistance is proportional to U312 . Dimensional homogeneityfurther requires the resistance to have the formR=Cb.jf.LplU3,(7. 1 1)in conformance with Prandtl's result (7.9). 60Besides this remarkably simple derivation of the form of the laminar resistance law foran edgewise moving plate, Lanchester explained that the U312 dependence correspondedto a form of resistance intermediate between purely viscous and purely inertial.
In purelyviscous cases, such as Stokes's pendulum, the resistance is entirely due to the energydissipated by the viscous stresses and is therefore proportional to the velocity. In purelyinertial cases, such as eddy production at a blunt stem, the resistance corresponds to thekinetic energy of a continually-generated wake of eddies and is therefore proportional tothe velocity squared. For the edgewise moving plate, both effects are combined becauseboth heat and wake are generated. Lanchester further noted that his derivation of the U3 12law required the motion around the plate to be laminar. As he knew from Rankine andFroude, the flow is in fact turbulent along a ship hulL In this case, Lanchester's intuitionled to a velocity exponent intermediate between 3/2 and 2, in conformance with Beaufoy'sand Froude's measurements.
6 1Lanchester's derivation ofthe U312 law intrigued Lord Rayleigh. In 191 1 , this championof dimensional reasoning commented that the only changes in space and velocity scale thatled to geometrically-similar motions were those for which the Reynolds number Ul/v ofthe plate was left invariant. Lanchester's special rescaling assumption only made senseif the plate was so long that its length did not significantly affect the structure of theboundary layer. Rayleigh went on to derive this structure on the basis of an analogy with aproblem that Stokes had long ago solved in his pendulum memoir, namely, the flowinduced by an infmite plate suddenly set into constant motion in its own plane.
62As in the similar problem treated by Boussinesq, the only nonzero component of theinduced fluid motion satisfies the equation8u82uOt = V [}y2 '(7.12)which has the same form as the equation for the propagation of heat. For the givenboundary condition (if t < 0 then u = 0 everywhere, and if t2:: 0 then u = U for y = 0 andu = 0 for y = oo), Stokes obtained the solution by Fourier analysis as60Lanchester [1907] pp.
50-2. As Lanchester implictly kept the ratio b I I constant, he wrote s3/4 instead of bv'i(with S = bl).61 Lanchester [1907] pp. 70-5. Joukowski's brief discussion of 1aminar boundary layers ([1916] pp. 1 20-1) waslargely erroneous, for he assumed S ex 1/ U in order that the experimental law R ex U2 should result from the wallstress -r "' p. U(S.62Rayleigh [19 1 1 ] pp. 39-40.DRAG AND LIFTu=�J291+ooe-TJ'd'lJ.(7. 13)y/2../ViThe corresponding resistance per unit area islP-au = pU /v .ay y=oV -;;i(7.14)By convolution, Stokes then determined the resistance induced by any given motion U(t) ofthe plane. His purpose was to refine the determination of a fluid's viscosity through Coulomb's old measurements of the viscous damping of a disc oscillating in its own plane, bytaking into account the fact that the fluid is at rest at the beginning ofthe first oscillation.
63Rayleigh saw in eqn (7 . 1 3) the velocity profJ.!e of a boundary layer that has developed inthe time t, and he guessed that a similar profile and a similar resistance per unit arearoughly applied to Lanchester's boundary layer if the time t in Stokes's problem wasidentified with the time x I U taken by the fluid to travel the distance x from the cutwater atvelocity U. The resulting resistance for a blade of length l and width b is(:;) 1 12 U312 J x-112 dx = 2bVP-PlU3,IR = bp(7. 1 5)0in conformance with Lanchester's result (7.1 1). Rayleigh had no illusions about thepractical usefulness of this result:64The fundamental condition as to the smallness of v would seem to be realized innumerous practical cases; but any one who has looked over the side of a steamer willknow that the motion is not usually of the kind supposed in the theory. It wouldappear that the theoretical motion is subject to instabilities which prevent the motionfrom maintaining its simply stratified character.
The resistance is then doubtlessmore nearly as the square of the velocity and independent of the value of v.7.3.4Slow receptionNeither Lanchester nor Rayleigh were aware of Prandtl's paper of 1904. Yet it did not gocompletely unnoticed. The towering Giittingen mathematician Felix Klein told Prandtlthat his Heidelberg communication was 'the most beautiful' he had heard in the wholecongress. Since the 1890s, Klein had been very active in promoting applied mathematics atGiittingen, securing private funds and recruiting competent personnel for this purpose.Since 1900, he had had an eye on Prandtl as a potential contributor to this effort.
In June1904, Prandtl accepted a call to a chair of technical physics at Giittingen. The followingyear, he assumed the directorship of the Technical Physics Section of the new Institute forApplied Mathematics and Mechanics. Thus, he enjoyed excellent conditions for develop-is63Stokes [1 850b] pp.
130-2, 102-3.64Rayleigh [191 I] p. 40 (I have corrected a transposition of b and [). The more exact coefficient ofB!asius [1 908]1.33 (Prandtl's estimate was 1.1).292WORLDS OF FLOWing his research, having brilliant graduate students to help him develop theoretical ideas,and first-class experimental facilities to test the results. 65Historians of boundary-layer theory all agree that this theory remained mostly a smallGottingen affair until the 1920s. Most frequently, they hold the concision of Prandtl'spaper of 1904 and the boldness of its contents responsible for this sluggish reception.
Thisis part of the Prandtl myth. In reality, Prandtl's short paper did not have much to dazzlecontemporary experts on hydrodynamics. Its two main novelties, namely, the computation of the laminar boundary-layer profile and a plausible separation mechanism, werelargely irrelevant to resistance prediction in concrete cases. As Rayleigh emphasized in191 1 , laminar boundary layers are rarely encountered in nature. A critical reader couldalso doubt that Prandtl's separation mechanism sufficed to determine the separationpoint in the final separated flow, the ensuing turbulent motion, and the resistance. As isnow well known, the tentative separation condition &ujay = 0 usually implies a failureof approximate integration procedures near this point. Even if this difficulty was solved,a more fundamental one would remain, namely that the potential gradient along theboundary layer is not a priori known, for it depends on the separated flow.
Lastly,the instability of the separation surface leads to essentially unpredictable motions in thewake. 66In summary, Prandtl's early insights into boundary-layer theory did not bring him muchcloser to a practical solution of low-viscosity resistance problems. The difficulties of thedetermination of separated flow remain unsolved to this day. Most of the thirteen paperson boundary layers published before 1930 were mathematical studies of the laminar caseunder Prandtl's supervision.
In the first of these, published in 1908, Heinrich Blasiusskillfully integrated the boundary-layer equation through power series, for a flat plateand for a synunetric cylinder. In the latter case, he managed to approximately determinethe separation point in permanent, suddenly started, and uniformly-accelerated flows. 67The limited value of such calculations soon became evident when testing experimentsperformed by Karl Hiemenz in Prandtl's laboratory led to unexpectedly violent but quiteregular oscillations in the wake of the cylinder.
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