Nowacki H. Leonhard Euler and the theory of ships (794398), страница 4
Текст из файла (страница 4)
By integration of these effects over the entire ship length Euler obtainedthe restoring moment for the whole ship. Therefrom resulted after a few intermediatesteps the expression for the restoring moment (for a symmetrical ship), which is verywell known today:-9-Fig. 3: Derivation for a cross section (according to Euler [1]): Displacement of thevolume centroid O to the immersed side,G = center of gravityMREST = Δ (GB + IT/V) = Δ (GB + BM) = Δ GM,whereΔ = γ V = displacementV = volume under the water surfaceGB = distance from the center of gravity to the center of buoyancy, positivefor B above GIT = area moment of inertiaBM = IT/VGM = GB +BM = “transverse metacentric height”Thus the floating position is stable at small angles of inclination provided that therestoring moment Δ GM is positive.
This is called positive initial stability of the ship.This result is entirely equivalent to Bouguer’s in [2], who for a stable ship postulates apositive metacentric height GM. The name “metacenter” stems from Bouguer and wasnever used by Euler, who was not familiar with this terminology. But in fact both derivethe magnitude of GM from the ship form by the same expression in order to judge theinitial stability. Euler uses a physical quantity, Bouguer a geometric one in formulatingthe stability criterion.Further details on the derivation of the stability criterion by Eulerund Bouguer, also in comparison, are given by Nowacki and Ferreiro [21].Extensions:The knowledge of the metacentric height GM for the transverse stability and by analogyof the longitudinal metacentric height GML for the longitudinal stability of the ship nowalso permitted deriving the equilibrium floating condition of the ship provided that theinternal weight distribution in the ship and the external loads, e.g.
wind loads in thesails, were known by magnitude and direction. Thereby one could predict the angles ofheel and trim for any desired internal weight distribution and external load case. Thusboth Bouguer and Euler closed a knowledge gap still existing in their 1727 prize contesttreatises.Further Euler demonstrated in his Scientia Navalis, Vol. I, Ch. IV, how the stability of aship can be improved, e.g., by lowering the center of gravity, by raising the center ofbuoyancy or by broadening the design waterplane (raising of the metacenter).
Theeffects of weight displacements aboard the vessel or changes of the cargo distributionduring loading and unloading as well as the effects of ballast placement by quantity andposition were analyzed as practical questions. These results have not lost any practicalrelevance and accuracy until today.- 10 -3.2Ship ResistanceIn order to better appreciate Euler’s contributions to the theory of ship resistance, a fewremarks on the earlier developments on this subject will be useful. The interest inpredicting the resistance of a hull form and in improving the design of the hull shape inorder to reduce the resistance and to increase the achievable ship speed in practicalapplications is probably as ancient as seafaring. Theoretical methods for resistanceprediction, however, based on scientific grounds were not developed before the stage ofthe “Scientific Revolution” in the 17th and 18th c.In experimental work Christiaan Huygens (1629-1695) and Edmé Mariotte (16201684) must be named as important precursors.
Huygens [32] already in 1669 in a smalltowing tank performed resistance tests with simply shaped ship models, which weretowed by a falling weight apparatus (Fig. 4), in order to determine the dependence ofthe resistance R on the speed V. He found a quadratic resistance law: R ~ V2. Mariotte[33] investigated other simple shapes, which he exposed to a current, e.g., in a currentin a river or in air, in order to measure their resistance. He found the relationship R ~ρV2, hence also a proportionality to the density of the fluid medium ρ. Both sets ofexperimental results were published only posthumously (1698 for Huygens, 1686 forMariotte).Newton who first published his Principia in 1687 certainly was not familiar with theseresults at that time. But in his own way he quite independently arrived at correspondingconclusions. Thus at the beginning of the 18th c.
from various sources there wasagreement that the resistance law for objects in parallel inflow had the followingstructure:R ~ ρV2 SorR = CD ρV2 S ,where S = reference area, e.g., the projection of the maximum cross section (midshipsection)ρ = density of the fluidCD = resistance coefficientFig. 4: Towing test apparatus with falling weight according to Huygens [32]- 11 -If this assumption was accepted, the most important open question was the dependenceof the resistance coefficient on the body shape, the direction of inflow and otherpotential influences. Resistance research, at least in the 18th c., concentrated on this keyquestion, strongly motivated by the goal of finding favorable shapes in fluid flow.Newton devoted the second volume of his Principia completely to fluid mechanicswhose theory he newly conceived from fundamentals.
Newton took an experimental andtheoretical approach, but in his theory had to confine himself to simple cases. In hismodels of thought he introduced many distinct cases and for each set of assumptionsargued very cautiously and with incisive simplifications. Although he clearlyrecognized that in fluid mechanics the influences of inertia forces, gravity forces andviscous effects all play a certain role, in studying resistance he much favored thesituation with pure inertia effects and for this case developed a corpuscular resistancetheory with the following further assumptions:Let the onflow be composed of mass particles (corpuscles) that move on parallelpaths with uniform velocity V toward the body (or obstacle).-Let the fluid medium be so „thin“, i.e., of such low density, that the particlesmaintain a small, but finite distance from each other without colliding with orinfluencing each other.-The fluid be either elastic so that the particles bouncing on the obstacle are repelledas in an elastic impact without loss of kinetic energy (Fig.
5A); or the fluid be inelasticso that the particle motion upon impact is completely stopped. In the event of obliqueinflow the particle paths are deflected and mirrored relative to the body normal at thepoint of contact (Fig. 5B).-According to the laws of impact in the elastic or inelastic case the resistance of theobject can be determined by the momentum balance of the fluid mass stream.
Theresistance in the elastic case is twice as high as in an inelastic fluid. Newton found theresistance coefficients for several simple body shapes on the basis of these laws andassumptions.-Newton himself was very cautious when justifying this experiment of thought for abody in a “thin”, corpuscular medium subject only to inertia forces. He never claimedthe existence of thin media in nature.
He explicitly pronounced that water was not a thinmedium. Rather he considered this case as a hypothetical scenario and perhaps as alimiting case that could never be reached. In a different place he directly mentionsviscous and gravity effects. But unfortunately his disciples and adherents were not socautious.
They quickly and uncritically proceeded to apply Newton’s theory to thinmedia, which they called “impact theory”, to real fluids and e.g. to bodies in water andair. The results were entirely disappointing, but due to Newton’s authority suchmisleading concepts were widespread for a considerable time. The main deficits ofimpact theory became clearly evident in such applications which were well outside therange of validity which Newton had claimed:The corpuscular theory does not permit any particles to reach the rear side of thebody in its inflow.
Rather they are all reflected from the front side. Thus only the frontside will incur any resistance. The orbits of all particles near the body are unrealistic.-The neglect of viscous and gravity effects gravely impairs the prediction ofresistance. If inertia effects are assumed to be the only existing forces, then each bodyshape will have its own, speed independent CD value. In a real fluid, however, as weknow today, the resistance coefficients depend on several categories of forces andhence on several parameters of similitude.-- 12 -These critical insights were not pronounced in the beginning of the 18th c.
Rather for atleast half a century impact theory remained the only available, even though increasinglydistrusted method for predicting the ship resistance.The beginning of a critical reanalysis and reformulation of fluid mechanics was made in1727 by Daniel Bernoulli [35], when in experiments he measured the force exerted by ajet impinging on a flat plate and detected fundamental contradictions between hisresults and Newton’s impact theory. He then developed a new theory for pipe flows,based on the energy conservation principle, which for this case yielded a newrelationship between pressure and velocity in a „stream tube”, the predecessor ofBernoulli’s equation.
In its further development this led to a new paradigm for fluidmechanics for the parallel onset flow of a flat plate [36], where in contrast to impacttheory the streamlines (and thus the particle orbits) are no longer reflected upon impact,but are deflected laterally before impact (Figs. 5C and 5D). This concept was alsoelaborately presented in Daniel Bernoulli’s famous book “Hydrodynamica” [37].His father Johann Bernoulli in his no less famous book “Hydraulica” [38], whichappeared in 1742, carried these ideas a little further in a more general vein and basedhis derivations on Newton’s Lex Secunda (in place of the energy principle), which heapplied to a free volume element in the flow. He also introduced the concept of“internal fluid pressure” in a moving fluid, whereby the Bernoulli equation for a“stream tube” obtained the form well known today.Fig.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
