Nowacki H. Leonhard Euler and the theory of ships (794398), страница 7
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Regarding trim and heel the progress made in shiphydrostatics prior to 1750 had furnished important prerequisites (cf. Section 3.1). Onfurther details in the development of sailing theory during this period, see also LudwigRank [55].Although Euler’s results on rowing, paddle wheel and screw propulsion still sufferedfrom their reliance on impact theory for force predictions and hence failed to be ofquantitative value, they still made a lasting contribution to propulsion theory which liesprimarily in the analysis of the acting physical principles and mechanisms, which arefundamentally based on the momentum balance. Euler’s scientific courage must beadmired to study the mechanical principles of propulsion systems which in his day werenot yet ready to be technically realized.In 1753 on the occasion of another prize contest by the Parisian Académie Royale desSciences Euler had submitted a treatise [22] that dealt with the propulsion of shipswithout windpower, i.e., without sails („De promotione navium sine vi venti“, E.413).This award winning treatise written at a time when sail propulsion was still by far thedominant propulsion method for all major ships gave a basic overview of alternativepropulsion methods, whose power was to be provided essentially by the humans onboard, be it by known means like rowing, be it by mechanisms to be newly developedsimilar to the paddle wheel or the screw propeller, yes, even by jet propulsion aspropagated earlier by Daniel Bernoulli.
Though Euler did not develop such newpropulsion systems to technical maturity as patents, he still qualitatively describedcorrectly their physical principles of operation, also for propulsion systems which couldonly be realized in the 19th c. by means of steam power. Thus he cannot be regarded asthe inventor of such later solutions, but he anticipated by more than half a centurybefore their realization the physical explanation of the performance of paddle wheel,screw propeller and jet propulsion.Human Performance LimitsEuler began his considerations with the question of how much mechanical power ahuman according to his physiological capacity can provide continuously.
He estimatedthat a man under favorable load conditions may be able to move a load of about 15 kp ata velocity of about 0.65 m/sec for an extended period of time. This in today’s unitscorresponds to providing a power of about 100 Watts continuously. This power capacityagrees surprisingly well with current data on the continuous power performance ofhumans, if we disregard sportive peak performances in shorter time intervals. Euleradapted the power absorption of his ship propulsion systems to this human performancepotential.Oar PropulsionThe principle of ship propulsion by rowing was schematically regarded by Euler as if asubmerged planar plate (FF, Fig.
7) was attached to a linkage system gliding on a rollerC and was arranged forward of the bow (or abaft the stern) so that it could behorizontally shifted in a direction opposite to the motion of the ship. Thereby watermass is accelerated backward and a reaction force is generated on the plate which drivesthe ship forward as a thrust.
After this work cycle the plate must be raised from thewater, transferred back through the air and lowered into the water again (as the oar inrowing).- 18 -Fig. 7: Ship propulsion by backward shifting of a plate in water (Euler [22])Let the ship velocity be C, the horizontal velocity of the plate relative to the ship be V,hence the horizontal velocity of the plate relative to water is V – C. Then if V is greaterthan C, a positive thrust is acting on the plate which drives the ship and in steadymotion exactly overcomes the resistance. Euler thus equated thrust and resistance anddetermined the achievable velocity C of the ship from the power input by the crew at anassumed velocity V of the plate or “oar blade”. Unfortunately Euler, as was stillcommon practice at this time, used Newton’s impact theory for estimating the influenceof hull form and velocity upon the resistance coefficient, which was unrealistic andresulted in quantitatively misleading conclusions, which were later also rejected byEuler himself.
Qualitatively it was correct that an increase in the area of the plate and areduction of the hull resistance would improve the propulsion of the ship and increasethe achievable ship speed.Euler expected an improvement in the efficiency of this propulsion method if the bladewas fitted with rotatable slats like Venetian blinds (Fig. 8) which during the returnstroke of the blade would be turned into a horizontal position and hence would have alow resistance. Thus this kind of unidirectionally permeable blade could be arranged onboth sides of the hull and connected to a system of levers OA-NN so that the bladescould be continuously moved back and forth.Propulsion System with Cranked ShaftThe previous idea was further simplified if the blades on both board sides were drivenby a horizontal shaft DD (Fig. 9) whose center part CC is cranked so that the propellingforces can act on the crank.
The blades by their slats are again unidirectionallypermeable and are intended to move continuously back and forth in their submergedcondition.Fig. 8: Venetian blind blade withrotatable slats and lever system forattachment and position control ofthe blades [22].Fig. 9: Drive of Venetian blindblades by means of rotatable, crankedshaft [22].- 19 -The paddle wheel principleFig. 10: Ship propulsion by two “paddle wheels” with plates as blades [22]In order to avoid this pendulum like motion, which was not very practical as Eulerprobably also realized, in order to make the permeable blades dispensable and in orderto operate in a continuous motion, it was almost cogently necessary to arrive at theprinciple of the paddle wheel (Fig.
10). Here several plates are attached to the spokes ofthe driving shaft on both sides of the hull which are driven by the shaft AA, which inturn is kept in steady rotation by means of a whim gear (E, D) by the crew rotating thearms M about the vertical axis OO. The blades are not profiled like paddles, but they doprovide a steady thrust while they are immersed in the water. The “paddle wheel” as asteadily rotating engine according to Euler is a logical further advance of the idea of the“oar blade”, which works only intermittently in a horizontal translation.
Thus Eulersucceeded immediately in generalizing the balance of input power vs. usefully deliveredthrust power from the oar blade to the paddle wheel. “Paddle wheel propulsion” is thusregarded as a generalized, continuously operating form of “oar blade propulsion” withimproved efficiency.The “screw propeller principle”Encouraged by the proven idea of the windmill Euler in his next step arrived at apropulsion system whose configuration resembled a modern screw propeller (Fig. 11).A system rotatable about the longitudinal axis AB is arranged in front of the bow orabaft the stern of a ship to whose spokes planar blades FF are attached with some angleof inclination relative to the longitudinal direction.
Thereby in their rotation theyexperience a longitudinal force (thrust) and a circumferential force, similar to a modernscrew propeller, though without the helical curvature of the blade surface and withoutmodern profiled blade sections.Fig. 11: Propulsion by propeller withplanar blades [22]Fig. 12: Velocities and forcesacting on the blade [22]- 20 -In his analysis of propeller operation, Euler took into consideration the mean effectsacting on the blade, lumped into its area centroid G. He combined the components ofinflow in the direction of advance (αG) and in the circumferential direction (GL) intothe resultant GN (Fig.
12), acting with an angle of incidence γ to the blade. In thiscontext, too, Euler remained prepossessed by Newton’s impact theory of resistance andtherefore in his analysis of the force acting on the blade section (through G) accountedonly for the normal force GH, perpendicular to the blade, which has components in thecircumferential and advance directions. Thus he neglected the tangential forces of theblade section and all effects of foil theory acting on the blade, which are known today.Thereby his analysis remained crudely approximative. Qualitatively his theory didcorrectly explain the chain of phenomena by which a planar blade propeller or later ascrew propeller in its rotation, in order to overcome blade resistance in thecircumferential direction, absorbs propulsive power and at the same time generates athrust in the direction of advance.Jet propulsionThe efflux from a containment vessel or the flux through a vessel or pipe causes areaction force acting on the boundaries of the vessel as is known from the garden hose.The flow vessel or pipe can be arranged in a ship in such a way that the resultingreaction force provides a thrust for ship propulsion.Euler picked up this idea following suggestions by Daniel Bernoulli(“Hydrodynamica”, 1738), analyzed the acting forces and conceived a jet propulsionsystem for a ship (Fig.
13). The flow through the system of pipes is induced either bythe pressure head of a tank arranged in a high position or by a reciprocating piston pump(EE) in the pipe. This pump in principle can be driven by human operator power.Euler first calculated the resulting reaction force on the boundaries of the vessel for asystem with arbitrary cross section distribution and in some arbitrary spatial position.He demonstrated that this force depended only on the flow rate and on the area andorientation of the inlet and outlet cross sections, but not on the cross section variation ofthe vessel between the end sections.Fig.
13: Jet propulsion system according to Euler [22] with piston pump (EE)He then addressed the most favorable case for ship propulsion (Fig. 13) where the fluidenters the system through a horizontal pipe of cross section EE from abaft and after adeflection of 180 degrees leaves the system rearward via a horizontal pipe nozzle withthe orifice FF. The cross section of the orifice FF can be made very small relative to theintake cross section at EE, thus very high jet velocities can be achieved.- 21 -Mechanically the process was subdivided by Euler into two cycles (Fig. 13), the intakecycle and the ejection cycle: During the intake cycle fluid is sucked into the pipe systemfrom below (at the suction funnel at B), the valve at n is open, the valve at m closed.Because of the deflection of the fluid by 90 degrees at A a certain horizontal forcealready arises which acts as thrust.
During the ejection cycle the piston is advancedforward, the valve at n closes, the one at m opens, the jet can now exit at FF. This cyclegenerates a very high thrust due to the great jet velocity and the complete deflection by180 degrees. In order to alleviate the thrust fluctuations between cycles, Euleradvocated two parallel jet systems in counter rhythm.In his example powering calculations Euler then recognized the limitation that jetpropulsion can profitably operate probably only at great propulsive power whichexceeded the powering potential from human energy, i.e., from the energy sourcesavailable in his day.AftereffectsAlthough first experiments with and patents for paddle wheels, screw propellers and jetpropulsion had existed for some time before Euler’s publications, the realization ofthose ideas still failed in Euler’s time, essentially due to the lack of a power source ofsufficient capacity aboard ships.
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