Nowacki H. Leonhard Euler and the theory of ships (794398), страница 6
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Euler’s approach, theapplication of Newton’s Lex Secunda to a fluid volume element, yielded the equationsof motion of the element, the famous Euler equations in the form of a set of partialdifferential equations. The continuity equation is of a comparable form.The Euler equations and the continuity equation constitute the foundation of the fieldtheory of fluid dynamics. Euler developed this theory in general form in a few classicalpapers [45], [46], [47], [48], [49] between 1752 and 1756.
His theory holds for planarand spatial flows in incompressible and compressible fluids.- 15 -Thereby the equations of state of the flow problem are established. Therefrom a form ofthe Bernoulli equation can also be derived that holds for any arbitrary streamline in theentire fluid domain. If the boundary conditions are added, a complete problemformulation results in a form that is called boundary value problem today. In Euler’stime the mathematical theory for solving partial differential equations and boundaryvalue problems was developed concurrently with the treatment of this flow problem offield theory. Euler himself created important foundations although initially solutionmethods for arbitrary given body shapes were still missing.
Euler showed that the fieldunder certain conditions may possess a velocity potential for which solutions can morereadily be built up. This was the starting point for the later development of singularitymethods to solve this type of boundary value problem of potential theory.If in some specific case the solution for the state variables is known, a streamline fieldmap can be developed and therefrom, as Euler suggested, especially for the streamlineson the body surface velocity and pressure distributions can be derived. Pressureintegration then yields the resulting force on the body. In an ideal fluid, as D’Alembert’sparadox anticipated and Euler was able to confirm by pressure integration, the resultwhich is obvious to us today is achieved, viz., that the resistance of a deeply submergedbody in a steady flow vanishes.
Thus field theory proved the following facts to startwith:-The assumptions of Newton’s impact theory for the resistance are not tenable.-A flow that remains attached to the entire body surface and whose afterbodytherefore contributes to the resistance is feasible.-In an ideal, loss-free fluid the resistance vanishes.By these results fluid mechanics had overcome a difficult stalemate and was able todevelop further without impairing contradictions. For application in ship theory the newinsights gained were apt to produce a rich harvest later in the 19th and 20th c.
Today thepractically successful analytical and numerical methods for calculating the flow aboutan arbitrary ship form in an ideal fluid all are based on Euler’s equations of fluidmechanics on the one hand, and on Euler’s field theory on the other hand.The other major obstacle, which prevented the development of a realistic theory ofresistance and which unfortunately could not be removed in the 18th c. any more, wasthe lack of understanding the causes of wavemaking and viscous resistance. For theresistance cannot be realistically estimated without taking into account the energy lossescaused by the effects of gravity and hence wavemaking on the one hand, and thosecaused by the viscosity of the fluid on the other hand. In the case of a real fluid theresistance law must account for several parameters of similitude (later called Froudenumber, Reynolds number etc.) and therefore must provide more than one freelyallocable resistance coefficient.
In the 18th c. the influence of both resistancecomponents was underestimated. The effects of wavemaking were first more clearlyrecognized in model experiments (Juan [50], Chapman [51]) and there followed first,immature theoretical hypotheses [50]. The friction on the body surface was judged aslow [cf.
also Euler [52]) and was regarded as negligible in the interior of the fluid. Itwas only by the thorough experiments performed by Beaufoy [53] beginning in 1793 onthe frictional drag on plates that the relatively considerable significance of frictionalresistance, also on ships, was realized. It still took a long time before William Froude[54] presented a method that took into account several parameters of similitude in shipresistance and thereby permitted faithful prediction of full scale ship resistance frommodel tests.
The theories of wave resistance and viscous resistance could not reachmaturity before that insight.- 16 -3.4Ship propulsionIn the 18th c. ship propulsion by wind energy in sailing and by human energy in rowingwere the predominant methods and energy sources. Euler investigated the mechanicalprinciples for both methods of propulsion. Besides he analyzed certain innovative, notyet practically applicable propulsion systems like the paddle wheel, the screw propellerand jet propulsion.Already in his prize contest treatise [18] of 1727 Euler -not unlike Bouguer- devoted hisattention to sail propulsion. In the spirit of Newton and other precursors he analyzed theforces acting on a sail and the resistance of the hull by means of impact theory.
Later inScientia Navalis and even still in his Théorie Complète -like many of hiscontemporaries- he remained committed to this approach. This unfortunately led torather misleading results regarding the forces acting on the hull and the sails. Fig. 6shows the component decomposition in the forebody for resistance and thrust.According to impact theory the hull resistance acts on the forebody normal to itssurface, and hence obliquely upward. In the example of the figure Euler assumed aspherical segment bow shape so that all impact force contributions acted through thecenter W of the spherical shell.
The resultant WR acted in a direction normal to thespherical segment in its area centroid. Now it was the dominant opinion in the 18th c.that masts and their sails in design should be arranged in such a way that the resultantsail force would have a horizontal forward thrust component, intended to act alsothrough the point W, called “point vélique”, and then in steady motion should be inequilibrium with the horizontal component of the resistance.
Thereby it was intended toprevent a trimming moment formed by the couple of resistance and thrust. Howeverthese considerations on the point vélique were grossly misleading and also superfluous.For firstly the resistance resultant in actual fact by no means acts obliquely upward,secondly it is possible to compensate the inevitable trimming moment by the nose of thesailing vessel by means of ballast or other design measures.Fig. 6: Hull resistance and sail forces with point of intersection in the point vélique W( Euler [1])In addition it caused difficulties that the sail forces could not be accurately estimatedaccording to impact theory.
It was assumed that the sail area of several masts could belumped in the centroid of all sail areas and that the resultant sail force would act throughthis point, as impact theory without accounting for any sail interactions would suggest.Aerodynamic effects as the cause of lift and drag of the sails were still unknown.Therefore the predictions of sail forces were unrealistic in magnitude and direction. Thefurther assumption of impact theory that the sail force resultant in oblique inflow wouldvary with the square of the angle of incidence (sin2 α law) was false, which was not- 17 -recognized before some experiments in the second half of the 18th c.
In conclusion inEuler’s time the propulsive forces acting on hull and sail and hence the resulting shipspeed could not be realistically predicted.It must be acknowledged as a positive element of understanding that the forcecomponents acting on ship and sail and their interactions were correctly identified, andthe equilibrium position of the sailing vessel by drift angle, trim and heel angle werequalitatively properly understood.
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