Nowacki H. Leonhard Euler and the theory of ships (794398), страница 5
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5: Flow characteristics in parallel onset flow of a flat plate according to Newton andDaniel Bernoulli:A. Newtons impact theory in an elastic, thin medium (reflected particles), inflownormal to the plate.B. The same theory, oblique inflow.C. Daniel Bernoulli’s theory of deflected streamlines, normal inflow.D. The same for oblique inflow.- 13 -This was the state of knowledge that Euler, who corresponded frequently with bothBernoullis, found by around 1742, before he began to base his own ideas on it and toarrive at an original new view of fluid mechanics.
At the time when his Scientia Navaliswas written, viz., between 1737 and 1741, Euler was still an adherent of Newton’simpact theory of the resistance. Therefore in this principal work of his [1] and also inthe later adapted French translation, the Théorie Complète [21], we find only deductionsbased on impact theory. Euler expressed his later contrasting views in other places.Nonetheless it is worthwhile to briefly reexamine Euler’s thoughts in Scientia Navalis,also with respect to resistance, at least because of his systematic approach.In Scientia Navalis, vol.
I, Chapter V, Euler first considered the resistance of planarfigures or planar cross sections of cylindrical bodies (Example: A rectangular rudderwith constant profiled cross sections). He determined the resulting resistance in normalor oblique inflow in three different ways: By Newton’s impact theory in an elastic or aninelastic medium or thirdly by the energy principle.
All three results have the samestructure (in modern notation):R = CD ρV2 S ,as stated by Newton above. Euler did not commit himself to any of the CD values, butargued that it was only the structure of the expression which mattered, while thecoefficient had to be brought into agreement with experiments anyway. Besides thetheoretical CD value represented an upper bound because the theory neglected that thefluid particles were not strictly repelled, but could reach the rear side of the body.
Thiswas an elegant way out of the dilemma, but not a quantitative solution.Euler then proceeded to determine the resistance coefficient for several simple shapes ofcross sections (triangle, circular and elliptical segment etc.). It is of interest that healready searched for an optimal profile form with minimal resistance with certain profiledimensions (length, thickness) as given constraints. Newton had already posed theproblem of an axisymmetric body shape of least resistance and had formulated avariational problem for this special case. Euler went one step further and specified herethe general problem statement for a variational problem with equality constraints whosegeneral theory he published later in 1744 [39], hence after completing the manuscriptfor the Scientia Navalis.
He solved it here for special cases. Further examples by Eulerwere related to the effects of oblique inflow on foils (under an angle of incidence),hence on their resistance, lift and moment. His systematic approach is admirable,though unfortunately the quantitative results based on impact theory are useless.In a corresponding way in Scientia Navalis, vol. I, Chapter VI, Euler approached theresistance problem for spatial solid shapes. He started out with the basic expression forthe resistance and moment of a solid in parallel inflow according to impact theory, butthen restricted himself to the special case of inflow parallel to the axis of symmetricalshapes.
His procedure in generating families (or systematic series) of shapes isremarkably modern. He created mathematical hull form representations that can besystematically varied so that a whole range of typical shiplike form parameters iscovered. The ship form or hull surface was represented in parameter form by r(u,v) ={x(u,v), y(u,v), z(u,v)} and was applied to the special case where separation ofvariables is feasible, e.g., x(u,v) = f(u) g(v), where u and v are parametric, curvilinearcoordinates in the surface.
Then it is easy to generate families or types of shapes inwhich all stations or all buttocks or all waterlines are affine curves. (A similar approachwas frequently taken much later in ship theory for systematic investigations of hull formvariation, e.g., by Weinblum [40] in wave resistance studies).
Euler then examined theresistance of various types and treated some of them by optimization based onvariational calculus. Further he discussed special shapes in which the forebody, the onlypart relevant to resistance in impact theory, was conical, pyramidal or cono-cuneiform(cono-cuneus by John Wallis [41]) or was an axisymmetric shape (with circular sectionshapes).
It is clearly evident that by these variations in ship form he was aiming at asystematic overview, almost a compendium, on the dependence of resistance on shape- 14 -design, a very practical purpose. His plan was again brilliantly logical, we may call it“systematic engineering”. His results regrettably shared the fate of impact theory whichsoon became obsolete. Euler recognized the weak points early, but during his lifetimethere did not yet exist a viable alternative for resistance prediction. A fundamental newstart was required.The first steps in this new direction were still taken by Euler himself, inspired by a fewdiscoveries by his contemporaries.
An important impetus came from the ballisticexperiments performed by the Englishman Benjamin Robins, whose results weredescribed in his book “New Principles of Gunnery” [42] in 1742. Robins had shot withspherical projectiles against a target disk suspended as a pendulum and had found outthat Newton’s impact theory was completely untenable for the resistance of hisprojectiles. Euler [43] had the opportunity in 1745 to translate Robins’ book intoGerman and provided its text with many elaborate comments and footnotes. He fullyagreed with Robins in his criticisms, fundamentally rejected the idea of a “thinmedium”, because such a fluid was not found in nature, and he developed his own firstideas, inspired by the Bernoullis, on how the flow about a body could be described morerealistically by “stream tubes” that enclose the body shape and remain attached to itsafterbody. (On this episode, cf.
also Truesdell [3], Szabó [11] and Calero [14]).3.3Field Theory of FluidsIn the following years Euler turned above all to general fundamentals of fluidmechanics, which were not yet applicable to ship theory, but were necessary to create anew basis on which later also the theory of ships could be safely founded. Euler was nodoubt strongly influenced by the results obtained by Daniel Bernoulli (stream tubeconcept, energy conservation theorem, deflection of the flow before impact), by JohannBernoulli (Second law of dynamics applied to a volume element, internal pressure,Bernoulli equation for a fluid filament or in a stream tube) and by Robins (refutation ofimpact theory).
D’Alembert [44] somewhat earlier had already formulated a first “fieldtheory” for the flow about bodies in a fluid. Euler likewise wanted to establish the lawsof continuum mechanics in fluids, i.e., the laws for the state variables pressure andvelocity in any desired point of the fluid flow, e.g. in the flow about a body.D’Alembert’s field theory had consistently avoided the use of concepts like pressure asa state variable and force according to Newton’s dynamics. Euler based his approach onNewton’s laws and chose pressure and velocity as state variables.
He thus created thefoundation of the modern field theory of fluids, as we know it and use it today.Euler pursued the goal of replacing Newton’s impact theory by a new theory whichdealt with the physical state variables in the whole domain of the fluid treated as adeformable medium. His axiomatic premises were:-Newton’s laws as principles of dynamics.Constitutive equations, i.e., information about the material properties of the fluid,initially regarded as an ideal fluid, and on the boundaries of the fluid domain.Conservation laws, especially on the conservation of mass (hence the continuityequation).From this information alone pressure and velocity can be determined. These variables ofstate are multivariate functions of the location in the field.
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