Maximal Speed of Underwater Locomotion
Background. An increasing interest in unmanned underwater vehicles continues to draw attention to swimming in aquatic animals. Their high speed continues to surprise researchers. In particular, the high dolphin speed caused a series of attempts to explain its paradox, which continues to this day. Some researchers believe that even rigid bodies, shaped like water animals, provide an attached flow pattern, as opposed to the widespread view of the inevitable separation. The possible explanation may be in the perfect body form, which provides an attached flow pattern (without boundary layer separation). Elongated unseparated shapes can not only reduce the pressure drag but also delay the laminar-to-turbulent transition in the boundary layer, significantly reducing the friction drag. Thus, the highest possible swimming speeds are expected in aquatic animals.Objective. We will try to prove that the low drag and the high speed of aquatic animals can only be ensured by their unseparated shape (as a rigid body), neglecting flexibility and compliance.
Methods. We will use: a) shape calculations of special bodies of revolution with negative pressure gradients near the tail similar to fish trunks with the use of the developed before approach; b) the known drag estimations of such shapes for laminar and turbulent cases; c) the swimming power balance and the theory of ideal propeller; d) statistical analysis of available data about the length, the speed and the aspect ratio of aquatic animals.
Results. The swimming speed of most aquatic animals is proportional to the length of the body in power 7/9. The exception is whale locomotion that occurs in turbulent mode at supercritical Reynolds numbers.Conclusions. The perfect body shapes of most aquatic animals provide an attached laminar flow pattern. Estimated maximum speeds for laminar and turbulent cases show that the special shaped unseparated hulls can greatly increase the speed of underwater vehicles and SWATH ships. Further increase in speed can be achieved by using supercavitation and greater than animal capacity-efficiency.
Gray J. Studies in animal locomotion VI. The propulsive powers of the dolphin. J Exp Biol. 1936;13:192-9.
Greiner L, editor. Underwater missile propulsion. Arlington: Compass Publications; 1967.
Fish FE, Rohr J. Review of dolphin hydrodynamics and swimming performance. San Diego: SPAWARS, 1999. Technical report 1801.
Fish FE. The myth and reality of Gray's paradox: implication of dolphin drag reduction for technology. Bioinspiration Biomimetics. 2006;1:1. DOI: 10.1088/1748-3182/1/2/r01
Fish FE, Legac P, Williams TM, Wei T. Measurement of hydrodynamic force generation by swimming dolphins using bubble DPIV. J Exp Biol. 2014;217:252-60. DOI: 10.1242/jeb.087924
Bale R, Hao M, Bhalla AP, Patel N, Patankar NA. Gray's paradox: A fluid mechanical perspective. Sci Rep. 2014;4:5904. DOI: 10.1038/srep05904
Aleyev YuG. Nekton. W Junk, The Hague; 1977.
Rohr J, Latz MI, Fallon S, Nauen JC, Hendricks E. Experimental approaches towards interpreting dolphin stimulated bioluminescence. J Exp Biol. 1998;201:1447-60.
Nesteruk I. Rigid bodies without boundary-layer separation. Int J Fluid Mech Res. 2014;41(3):260-81. DOI: 10.1615/interjfluidmechres.v41.i3.50
Nesteruk I, Passoni G, Redaelli A. Shape of aquatic animals and their swimming efficiency. J Marine Biol. 2014;2014:470715. DOI: 10.1155/2014/470715
Nesteruk I. Efficiency of steady motion and its improvement with the use of unseparated and supercavitating flow patterns. Naukovi Visti NTUU KPI. 2016;6:51-67. DOI: 10.20535/1810-0546.2016.6.81605
Nesteruk I, Brühl M, Möller T. Testing a special shaped body of revolution similar to dolphins trunk. Naukovi Visti NTUU KPI. 2018;2:44-53. DOI: 10.20535/1810-0546.2018.2.129140
Landau LD, Lifshits EM. Fluid mechanics. 2nd ed. Butterworth-Heinemann; 1987. Volume 6. Course of theoretical physics.
Nesteruk I. Peculiarities of turbulization and separation of boundary-layer on slender axisymmetric subsonic bodies. Naukovi Visti NTUU KPI. 2002;3:70-6.
Nesteruk I. Body forms of minimal drag. Dopovidi AN USSR Ser A. 1989;4:57-60.
Lutz T, Wagner S. Drag reduction and shape optimization of airship bodies. J Aircraft. 1998;35(3):345-51. DOI: 10.2514/2.2313
Goldschmied FR. Integrated hull design, boundary layer control and propulsion of submerged bodies: Wind tunnel verification. In: AIAA (82-1204). Proceedings of the AIAA/SAE/ASME 18th Joint Propulsion Conference. 1982. p. 3-18. DOI: 10.2514/6.1982-1204
Nesteruk I. Experimental investigation of axisymmetric bodies with negative pressure gradients. Aeronaut J. 2000;104:439-43.
Babenko VV, Carpenter PW. Dolphin hydrodynamics. In: Carpenter PW, Pedley TJ, editors. Flow past highly compliant boundaries and in collapsible tubes. Fluid mechanics and its applications. Dordrecht: Springer; 2003. DOI: 10.1007/978-94-017-0415-1_13
Loitsyanskiy LG. Mechanics of liquids and gases. 6th ed. New York, Wallingford: Begell House; 1995. 961 p.
Cole JD. Perturbation methods in applied mathematics. Waltham, London: Blaisdell Pub. Co.; 1968.
Nesteruk I. Reserves of the hydrodynamical drag reduction for axisymmetric bodies. Bulletin of Kiev University Ser Phys Math. 2002;1:112-8.
Hoerner SF. Fluid-dynamic drag. Midland Park, N.J.; 1965.
Hansen RJ, Hoyt JG. Laminar-to-turbulent transition on a body of revolution with an extended favorable pressure gradient forebody. J Fluids Eng. 1984;106:202-10. DOI: 10.1115/1.3243103
Bainbridge R. Speed and stamina in three fish. J Exp Biol. 1960;37:129-53.
Lighthill MJ. Note on the swimming of slender fish. J Fluid Mech. 1960;9:305-17. DOI: 10.1017/s0022112060001110
Webb PW. Hydrodynamics and energetics of fish propulsion. Bull Fish Res Board Can. 1975;190:1-159.
Spakovszky ZS. 220.127.116.11 Typical propeller performance. In: MIT turbines; 2002. 16.Unified: Thermodynamics and Propulsion.
Draper NR, Smith H. Applied regression analysis. 3rd ed. John Wiley; 1998.
Gazzola M, Argentina M, Mahadevan L. Scaling macroscopic aquatic locomotion. Nature Phys. 2014;10:758-61. DOI: 10.1038/nphys3078
Gabrielly Y, Von Karman T. What price speed. Mech Eng. 1950;72(10):775-9.
Wave-making resistance [Internet]. En.wikipedia.org. 2019 [cited 2019 Apr 24]. Available from: https://en.wikipedia.org/wiki/Wave-making_resistance
Knapp RT, Daily JW, Hammitt FG. Cavitation. New York: McGraw Hill; 1970.
Franc JP, Michel JM. Fundamentals of cavitation. Dordrecht: Kluwer; 2004.
Iosilevskii G, Weihs D. Speed limits on swimming of fishes and cetaceans. J R Soc Interf. 2008 Mar 6;5(20):329-38. DOI: 10.1098/rsif.2007.1073
Logvinovich GV. Hydrodynamics of flows with free boundaries. Kyiv: Naukova Dumka; 1969.
Nesteruk I., editor. Supercavitation. Advances and perspectives. Springer; 2012.
Nesteruk I. Drag drop on high-speed supercavitating vehicles and supersonic submarines. Appl Hydromech. 2015;17(4):52-7. Available from: http://hydromech.org.ua/content/pdf/ph/ph-17-4%2852-57%29.pdf
Garabedian PR. Calculation of axially symmetric cavities and jets. Pac J Math. 1956;6(4):611-84. DOI: 10.2140/pjm.1956.6.611
Nesteruk I. Can shapes with negative pressure gradients prevent cavitation. In: Proceedings of FEDSM’03, 4th ASME_JSME Joint Fluids Engineering Conference; 2003; Honolulu, USA. Paper number FEDSM2003-45323.
Takahashi S, Washio S, Uemura K, Okazaki A. Experimental study on cavitation starting at and flow characteristics close to the point of separation. In: 5th Symposium on cavitation; 2003. No. Cav03-OS-3-003.
Washio S. Recent developments in cavitation mechanisms. A guide for scientists and engineers. Woodhead Publishing; 2014. 256 p.
Lockheed SR-71 Blackbird [Internet]. En.wikipedia.org. 2019 [cited 2019 Apr 24]. Available from: https://en.wikipedia.org/wiki/Lockheed_SR-71_Blackbird.
Bainbridge R. The speed of swimming of fish as related to size and to the frequency and amplitude of the tail beat. J Exp Biol. 1958;35:109-33.
Bottlenose Dolphin | Speed of Animals [Internet]. Speedofanimals.com. 2019 [cited 2019 Apr 24]. Available from: http://www.speedofanimals.com/animals/bottlenose_dolphin
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