And Hypersonic Flight

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From Leonardo to BusemannBefore Leonardo (1452-1519), no one had really studiedfluid motion as a subject in itself and with a general methodology,for reasons indicated above. We wish again toemphasize the crucial role that Cusa directly played in shapingthe method Leonardo used, in particular the idea oflooking upon the behavior of water, air, and light as a singlephenomenon: fluid motion. This was possible only becauseCusa imposed upon the humanists the injunction toseek that which is invariant in the universe as a whole. Thisseemingly abstract idea allowed Leonardo to use water toform hypotheses concerning air and light, with obviouslyastonishing results replicated only much later by Prandtl.Newton could not grasp the same interrelation of thecommon properties of fluids. However, the same principleintroduced to Leonardo the concept of reversibility in fluidmotion: Either an object moves in water or water movesagainst the object—the effect will be the same. Again, thisis an abstract point that allowed him major experimentalopportunities. He constructed artificial birds against whichhe moved air to experiment with flow patterns.Why Leonardo took on this study is apparent from hisunderlying motivations: the combination of his concern forirrigation projects and his sense of the necessity for man toconquer space. This optimism is the driving force of hisgenius. This is visible for most people through his artisticwork. For the moment, the essential point in Leonardo'sfounding of true fluid dynamics is his unambiguous indicationof the importance of the formation of singular discontinuousphenomena. These can take the form of vortices,hydraulic jumps, breakers, vortex-filaments, and soforth, out of apparently continuous wave motions. We willemphasize this tendency, this methodological aspect, andnot the details of Leonardo's work. The relevance of theformation of discontinuity in a fluid is not purely a philosophicalissue. It implies the creation of the right or wrongtechnology.Also essential in Leonardo's method is the scarcely hiddenimplication that when a principle of formation of discontinuityis understood for one fluid, it must be true withsome modification for other fluids too. As he emphasized,with the added characteristic of compressibility, up to acertain point the experiments in water are valid for air. Inthat sense Leonardo's method is still important. What heindicated as relevant phenomena in water has been verifiedin air with the shock wave theory and now in plasma physics.He himself projected and experimented with results heobtained in water to understand the behavior of blood circulationand tree growth with obviously astonishing "anticipation"of later scientific discoveries. Any erroneous resultswere caused mostly by his lack of experimental facilitiesrather than a deficit in method."After Leonardo, this outlook in hydrodynamics and opticswas crried through, although only partially, by SimonStevin (1548-1620), Christiaan Huygens (1629-1695), andBlaise Pascal (1623-1662). Mathematically, the importanceof discontinuous processes was emphasized by GottfriedLeibniz (1646-1716) and Leonhard Euler (1707-1783). M Withthe addition of Daniel Bernoulli (1700-1782), this is the trueschool of hydrodynamics, which was opposed to Newton,d'Alembert, and Laplace. Historically, the work of Leonardois known only through the distribution of manuscriptcopies by Diirer and Desargues, which came into the handsof Pascal, Leibniz and others in their networks. Beginningin 1798, Leonardo's manuscripts were republished inFrance—at least what was left of them—by the Italian physicistGiovanniBattistaVenturi (1746-1822), under the directionof Monge and Carnot. Venturi himself is famous forhis reexamination of Leonardo's discoveries: the Venturitube, the importance of hydraulic jumps, and the velocitydistribution in vortices.After Venturi there was a renewed interest in real fluidmotion especially on the part of Ernst and Wilhelm Weber(1795-1878 and 1804-1891).' 5 The Weber brothers along withGauss opened the way for the physical-mathematical innovationof Bernhard Riemann (1826-1866) and his school,including Felix Klein, Prandtl, and Busemann in Germany;Betti, Beltrami, Crocco, and Ferri in Italy. This school hadto fight with the heirs of d'Alembert, Kelvin and friends.While the Leonardo school would emphasize the importanceof the formation of discontinuities and of rotationalmotion, the Newton-d'Alembert-Kelvin school would denythe formation of such discontinuities on the assumption ofthe conservation laws. Out of this outlook comes the attemptto look at turbulence as only a chaotic, static localphenomenon. Even if the "paradoxes" of this school haveoften been made ridiculous by the actual creation of technologiesit claims are impossible, its outlook remains dominantin the universities.Now let's get into the substance of fluid mechanics.ELEMENTARY FLUID DYNAMICS FROM THELEONARDIAN STANDPOINTThe Geometric Method: Nature Seen with the 'Artist's Eye'Natural science was born by man's trying to answer problemsposed either by the evolution of human society or byfundamental but simple and visible natural phenomena, forwhich he sought the causal relations. Progress in sciencehas occurred not by giving definite and final answers, butby achieving successively better approximations to the solutionsof the two categories of problems seen above.Therefore, it is not a banality to say that one side of scientificprogress comes by "walking with one's eyes open." In otherwords, there are still a lot of everyday natural phenomenathat have not yet received real explanations.The current trend in science tends to reduce physics topure logical-mathematical analysis. Although mathematicsis a crucial aspect of the comprehension of invisible causalityby the use of our conceptual powers, we must neverforget whence it came and what we are looking for—thatis, reality. The real world is not a simple symbol. It exists,and our own existence and evolution depend on our masteryof its laws. The point is that the objects we see are notthemselves reality, but the causal processes that createdthem. Such objects exist as singularities, but we can graspthem only as space-time transformation. That is why geometryis the way to comprehend reality. Leonardo andthen Kepler initiated what today is called topology: lookingat physics from the standpoint of characteristic singulari-18 January-February 1986 FUSION
ties, whether shapes or numbers. Such geometry does hotneed the notion of mass or force initially. We look only atwhich kind of transformations are possible and what theboundary conditions are that define the transfinite limit forthe kind of transformation we are observing.Why are there only certain specific forms that naturechooses in shaping its own transformations? Why the commonpatterns in the sky and in the water? Why the symmetryin snowflakes and crystals? Why do only certain proportionsappear when growth is the primary parameter? Theseare the kinds of questions Leonardo posed to himself, andthey are the kind of questions anyone can pose and thenpursue the answers. The answers will reflect the level ofscientific knowledge we have mastered, but the point ofdeparture is that of being interested and motivated to know,and thus to pose the right questions.There is no numerology or Masonic mystery in the formsand proportions of nature itself, or for us to employ inacting upon nature. There is no symbology in spiral formsor in the Fibonacci series. Such knowledge is not the secretof Isis or anybody else. It is only the attempt to banalize andgive quick answers to deep scientific questions that accountsfor the mystic currents of Sufists and others turningnatural forms of growth and transformation into symbolism.Even so, if we simply look around, we will not see whatLeonardo and Kepler captured in water flow, tree growth,and snowflakes, because they were looking for the causesbehind these phenomena, the manifold "invisible" to thesimple eye. That is the meaning of Leonardo's notion of the"artistic eye." A real artist—one who recognizes beauty asnatural law—knows what to look for.As Leonardo described his paintings, an artist does notpaint lines or points. If he is a humanist, he paints souls—moral beautiful souls—whatever the external form. This isthe infinite in the finite that we see in the best painting ofthe Italian Renaissance or in Rembrandt. This is the notionof aesthetic transfinite developed later by Schiller.' 6Projective geometry was invented as the only way suchinfiniteness in the finite could be visualized. Beauty, then,is not simple external proportion; that is, the normal law oftopological necessity obeyed by any natural transformation.Beauty instead is defined by the specific individuality,the moral individual character that is also synonymous withlife. This concept of individuality—singularity—withinconstant proportions sparks our interest and fascinationwith man and nature. Natural forms may look the same, butthey are never precisely the same. The stronger this individuality,the nearer we are to being real human beings. In thatsense beauty is not an arbitrary notion defined purely subjectively,but is natural law, and the real artist is he alonewho grasps beauty.The geometric, artistic eye, then, although recognizingthe self-similarity of natural forms, is not satisfied with theSufist fundamentalist notion that everything is anything innature in one big blah. The artistic eye looks for the universalprocesses that characterize individuation. We wouldlook for the soul of natural, visible forms. This method—intheological terms the recognition of the immanence of theCreator within the created—is the specific characteristic ofthe Augustinian current in Christian Western civilization.Visual manifolds. From this standpoint, Leonardo noticedand differentiated two different visual manifolds.First, by working with Luca Pacioli, he studied very closelythe topological characteristics of the Platonic solids andtheir proportions. We do not see these Platonic solids innature unless we closely study crystal forms, but they representan important geometric principle in terms of topology,limits of forms, angles, and number of existent symmetricclosed forms. Such solids, according to Leonardo'spoint of view, represent the "residuum" of fluids, that is,semistatic crystals. In other words, the type of self-similargrowth characterized by the Platonic solid is more typicalof an inorganic, static growth.Second, Leonardo saw that the rest of the universe seemedto choose overtly to limit shapes either to pure spheres orto spiral-type circular action. Leonardo did an extensivestudy of such morphological growth patterns. He studiedand measured flowers, trees, horns, and so on and clearlypointed out that the Fibonacci proportions seemed to bethe main characteristic of life as a growth process. Leonardoalso went a step further: He began to look at these patternsof transformation not only as a by-product of life, but alsoas a general law of fluid motion—in water, air, magnetismor "fire," as he would call plasma-type phenomena. Normally,if we look at water, we would not see what Leonardosaw and represented in his famous drawings (see frontispiece).The reason is that he was looking for patterns thathe knew should be there according to his hypothesis of thegeometry of transformation in the universe.As Leonardo wrote: "If you wish to see the movementthe air makes when it is penetrated by a movable thing, takean example in the water, that is, underneath its surface, forit may have mingling with it thin millet or other minute seedwhich floats at every stage of height of the water, and youwill see the revolutions of the water which ought to be in asquare glass vessel shaped like a box" (Leicester 29v).This is the first known technique of visualization of waterflow patterns, to our knowledge. The importance shouldbe obvious to anyone familiar with hydrodynamic work;only Prandtl was able again to master such an approach,which proves that this was not a simple technical issue, buta methodological issue.Leonardo did the same thing to visualize wave motion ona vibrating plate (Figure 2). He wrote: "<strong>And</strong> when the dustytable is struck on one side, notice the manner in which themotion of the dust begins toward the creation of the mentionedmountains. ... In the same way describe the flexibleridges of particles of dry matter, that is the creation ofthe waves of sand carried by the wind" (Institut de FranceMs. F61r).Here and in other locations, Leonardo is describing whatwas reproduced in the 19th century by the German physicistErnst Chladni (1756-1827): the visualization of wave frequencieson a vibrating plate by putting powder on top ofit. We will describe below the implication of this preciseexperiment, but for the moment, let us stress the importanceof the clear-cut method of visualization that Leonardoinvented. Usually, this is possible only if one has in mind ahypothesis that the causal relations that shape an action arenot immediately seen, but can be visualized. In this wayFUSION January-February 1986 19
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Page 19: We can look at the universe as a co
Page 23 and 24: Figure 3THE COMMON TOPOLOGY OF ACTI
Page 25 and 26: Figure 6UNDISTURBED INTERPENETRATIO
Page 27 and 28: FLUID IN MOTION OR MOTION IN FLUIDT
Page 29 and 30: Figure 12FORCED AND FREE VORTICES-A
Page 31 and 32: Figure 15VORTICES IN THE FLIGHT OF
Page 33 and 34: Figure 20UNDERWATER AND SUBSONICRES
Page 35 and 36: e reduced to Newtonian mechanics no
Page 37 and 38: e educated to death," as Professor
Page 39 and 40: GEOMETRICAL VARIATION IN SHOCK WAVE
Page 41 and 42: If hot for the Gottingen Tradition
Page 43 and 44: e very familiar with that. Equation
Page 45 and 46: Figure 1PRANDTL'S APPROACH TO FLUID
Page 47 and 48: Ludwig Prandtl (above) is probably
Page 49 and 50: (a)(b)Figure 2FLUID FLOW AROUND AN
Page 51 and 52: flight was studied in supersonic wi
Page 53 and 54: ticipation of scientists from those
Page 55 and 56: Hypersonic Planes:by Charles B. Ste
Page 57 and 58: supersonic velocities, the "ramjet"
Page 59 and 60: erate equally well on transcontinen
Page 61 and 62: transmit them to collector plates v
Page 63 and 64: dius. Add corona trim to the plates
Page 65 and 66: ins are offended whenever electrici
Page 67 and 68: The most cnaiienging,controversial

And Hypersonic Flight

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