,tt=4K-r j r
iffS^r**"
f
t
1. Very high swimming performance of dolphins at successive jumping. 1. Very high swimming performance of dolphins at successive jumping.
2. Standing swimming of a pacific white-sided dolphin. Only the caudal fin supports all the body weight.
3. Streamline-shaped body of a pacific bottle-nosed dolphin. The vertical flat tail meets with the horizontal flat caudal fin.
Thinking Fluid Dynamics with Dolphins
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Thinking Fluid Dynamics with
Minoru Nagai
Ohmsha P r e s s
Thinking Fluid Dynamics with Dolphins ©2002 Minoru Nagai All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, without the prior written permission of the publisher. ISBN 4-274-90492-X (Ohmsha) ISBN 1 58603 231 3 (IOS Press) Library of Congress Control Number 2001099011 Translated from the original Japanese edition: "Techno-life Series Iruka ni Manabu Ryutai Rikigaku" published by Ohmsha, Ltd. ©1999 Minoru Nagai Publisher Ohmsha, Ltd. 3-1 Kanda Nishiki-cho Chiyoda-ku, Tokyo 101-8460 Japan Distributor USA and Canada IOS Press, Inc. 5795-G Burke Centre Parkway Burke, VA22015 USA Fax: +1 703 323 3668
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LEGAL NOTICE The Publisher is not responsible for the use which might be made of the following information. Printed in Japan
Preface This small volume is the English edition of a Japanese book entitled 'Learning fluid dynamics from dolphins.' The title is derived from the fact that "Dolphins swim too fast to be explained scientifically." The first person to clearly describe this phenomenon was the English biologist Sir James Gray (J. Gray, 1936), and this mystery is known among physicists and specialists in marine engineering as 'Gray's paradox' or simply as the 'Mystery of dolphins.' In addition to dolphins, tuna, marlin and some other fish are also famous for swimming at extraordinary high speeds. Treating both dolphins and fish together in the same title is difficult because both animals have far different taxonomy dolphins belonging to oceanic mammals and tuna and marlin belonging to teleostei fish. This book uses dolphins as symbolic animals that perform high-speed swimming. Furthermore, 'dolphins' are chosen as the main character in this English edition because western readers feel an affinity for them. This book focuses on young readers, who are interested in technology and science and who hope to specialize in technological occupations. This book aims to introduce the developing history of fluid dynamics, and then outlines the research history and the present recognition of 'Gray's paradox.' Finally, the author's research, through about three decades, is reviewed. This paradox suggested in the early 20th century has carried over into the 21st century without finding a complete solution. It would be a great pleasure for the author if the readers find interest in fluid dynamics, a discipline that has developed by overcoming numerous paradoxes, or feel the mood at the forefront of the 'intelligence' of mankind. The original Japanese edition was published in the autumn of 1999, as a selected book of 'Techno-life' series by the Japanese Society of Mechanical Engineers. Fortunately, having the financial support from the Japan Society for the Promotion of Science, this English edition was realized within three years. Mr. Takayuki Kawamura, a post MSc researcher at University of the Ryukyus, was in charge of the language translation from Japanese to English, and Dr. George Yates who is a friend of the author finally inspected the written English. Dr. Yates was a coresearcher who was studying animal swimming under Professor T. Y. Wu when the author temporally stayed at the California Institute of Technology as a visiting researcher in 1984 - 1985. If this book catches the heart of many readers and is favored by them, the main contribution is owed to the joint-translators
vi
Preface
Mr. Kawamura and Dr. Yates, and the author deeply appreciates their efforts. Finally, the author dedicates this book to the late Takefumi Ikui, a Professor Emeritus at Kyusyu University, who was a master of the author, and to my wife, Megumi Nagai. Teacher Ikui warmly watched the author when the author stepped into the unknown academic area of 'bio-fluid dynamics,' and cheered up the author. Megumi has supported the author's irregular researcher life for more than 30 years since their marriage. None of this book would have appeared without their existences. February 2002
Minoru Nagai
Contents Preface
v
Chapter 1 Gray's Paradox Birth of Gray's Paradox Swimming Speed of Aquatic Animals Observations of Small Fresh-Water Fish - Discovery of the Swimming Number
1 2 5
Chapter 2 Early History of Fluid Dynamics - From Aristotle to Newton and D'Alembert Aristotle's Paradox 9 Newton's Fluid Dynamics and his Paradox 10 Euler's and Bernoulli's Equations and the Paradox of D'Alembert 14 Lifting Theory 18 Chapter 3 Modern Fluid Dynamics Navier-Stokes Equations Effects of Small Viscosity Reynolds' Law of Similarity Boundary Layer Theory Laminar Flow and Turbulent Flow Numerical Simulations and Physical Experiments Dynamics of Wings in Real Fluids
21 23 24 26 28 29 32
Chapter 4 Principles of Thrust Generation Momentum Theory Slender Body Theory Oscillating Wing Theory
38 40 43
Chapter 5 Research on High-Speed Swimming Performance Dimensional Analysis Possibility of Drag Reduction - Flexibility of the Surface Skin Toms Effect Effect of Riblets Studies in Japan
49 57 53 57 60
x
Contents
Chapter 6 High-Speed Swimming Method of Carp and Dolphins Small Circulating Water Tunnel and the Measurement of Drag on Fish Swimming Motion of Fresh-Water Fish Mechanical Fish - An Invention of Oscillating Wing Propulsion Mechanism Swimming Motion of Dolphins Maximum Swimming Speed of Dolphins Estimation for the Power of Fresh-Water Fish and Dolphins Episode - Rapid Increase of the Body Temperature of Fish Chapter 7 Robot Fish - Development of Ocean Engineering Hertel's Research Studies in Japan Mechanical Fish and Oscillating Wing Propulsion Ships of University of the Ryukyus Robot Fish of M.I.T.
63 69 71 76 80 84 87
97 93 97 99
Epilogue - The Silver Lining of Solving the Paradox New Challenges of University of the Ryukyus The Key Has Shifted to Unsteady and Three Dimensional Flow Fields
103 104
References
707
Author and Translators Profile
709
Index
777
Chapter 1 Gray's Paradox Aquatic animals such as tuna and dolphins have been observed to swim much faster than the maximum speed that is predicted from both the power generating capacity of their muscle and the estimated hydrodynamic drag. This discrepancy is known as 'Gray's paradox' or the 'Dolphins' mystery,' and the problem remains to be fully explained. The author has been wrestling with this problem for a quarter century since his early efforts at University of the Ryukyus, Okinawa, Japan. Examples of the work were initially 'Research on hydrodynamic drag of soft bodies,' 'Observation of swimming motions of several fresh-water fish in a water tunnel' and 'Observation of swimming motions of Dolphins.' Then the research was developed to include Trial production of a small scale automatic mechanical fish,' Theoretical/experimental research on oscillating wing propulsion mechanism' and 'Trial production of an oscillating wing propulsive boat and a large-scale mechanical fish.' Hence, present achievements on this issue are due to the surveys of many enthusiastic past students of the university. This chapter commences with the birth of Gray's paradox, and follows its history until the latest achievements. Although the backgrounds of paradoxes are often veiled, scientists and researchers retain their interest with a belief that there must be rational mechanisms or reasons that explains the paradoxes clearly. In fact, fluid dynamics was developed by overcoming various paradoxes. However, Gray's paradox, submitted in the early 20th century, seems to await complete resolution in the new century.
Birth of Gray's Paradox Sir James Gray (1891 - 1975), a famous biologist at Cambridge University in the U.K. introduced the observations of Thompson in the Indian Ocean. Thompson reported that a dolphin (Delphinus Delphus) 6 to 7 feet in length swam parallel to his ship, which was moving at 20 knots (approx. 10 m/s). Gray assumed the following physical specification for the dolphin: the body length was 6 ft (1.83 m), the muscle weight was 35 Ib (15.9 kg), the total weight was 200 1b (90.7 kg), and
2
Chapter 1 Gray's Paradox
the body surface area was 15 ft2 (1.39 m 2 ). Then the hydrodynamic drag was calculated as 42.5 1bf (189 N) and the necessary power (i.e. drag x velocity) was calculated to be 2.6 HP (1,940 watts). In this case, the dolphin's power per unit muscle weight amounted to 0.074 HP/lb (122 watts/kg), which was much grater than that of a well fit human or a dog whose unit power was reported to be about 0.01 HP/lb (16 watts/kg). It seemed unrealistic that the muscle of oceanic mammals could generate seven times as much power as the muscle of onshore mammals. If this consideration were true, the dolphin's muscle power, oxygen supplying system and the heat radiation method would be marvelous. This problem was submitted as a paradox. Furthermore, Professor Gray stated that if the flow in the boundary layer developed on the surface of a dolphin maintained laminar flow, the necessary power to swim could be decreased. Since the Reynolds number of the flow is about 107 the flow should already have transformed to a turbulent boundary layer, and the laminar flow hypothesis seems impossible (1936). Later, Professor I. Tani (1907 - 1990), a pioneer of fluid dynamics in Japan, assumed the ratio of muscle mass to the whole body mass of a mammal should be 40% and the average maximum power output per unit mass of muscle was 29 watts/kg in his explanation of Gray's paradox in 1964. Tani estimated that the output per unit body mass would be 6 to 12 watts/kg. The output of the above dolphin divided by the body mass is 21 watts/kg, which is still two to three times larger than Tani's suggested value. Gray's paradox has stimulated the interest of physicists, zoologists, and shipbuilding and fluid dynamics engineers all over the world, and has challenged them to explain the paradox from many perspectives. However, a clear solution has not been obtained yet. In 1985, Professor A. Azuma of the University of Tokyo, an authority in aeronautical engineering, introduced Dr. T. G. Lang's observations written in 1974, and he explained that the paradox had finally been solved. However, Dr. Lang's study actually reported that a pacific spotted dolphin (Stenella Attenuata) with a body length of 1.86 meters had a maximum swimming speed of 11.05 m/s, and Lang stated that the power of the muscle of a dolphin would be 2.5 times greater than that of a human, which was not unreasonable. Therefore, he had not proved the phenomenon biologically. Meanwhile, Lang's experiment recorded the maximum swimming speed of a dolphin with precision and under carefully controlled laboratory conditions by a scientist. This agreed with Thompson's report (1.8 meters body length at 10 m/s swimming speed). On the other hand, the British Guinness Book of Records shows the maximum swimming speed was 30 knots, i.e. 15.4 m/s, by a killer whale with a body length of 6 - 8 meters observed in the east Pacific Ocean on October 12, 1958.
Swimming Speed of Aquatic Animals Figure 1.1 shows reported swimming speeds of numerous fish, mammals and a submarine. For each swimmer the velocity is plotted versus the body length. The single-dot chain line and the double-dot chain line in the figure are the theoretical
Swimming Speed of Aquatic Animals nuclear submarine
body length / (m) Fig. 1.1 Speeds of Various Swimmers (Nagai, 1982)
maximum speed if the boundary layer flow on the body surface is laminar flow and turbulent flow respectively. Although most swimmers move at high Reynolds number, their swimming speed can far exceed the theoretical limits of both laminar and turbulent flow. The concept of Reynolds number and its relation with boundary layer flow will be discussed in Chapter 3. Figure 1.1 shows that high-speed swimmers can swim faster than the theoretical expectations, and the maximum speed is almost proportion to the body length, as shown with two solid lines of U ¥ /. There are two groups of dolphins plotted in this figure. The upper group is the data from Dr. Lang and the lower group is the one from the author. The data for a killer whale ( 6 - 8 meters body length) is from The Guinness Book of Records. The yellowfin tuna and the marlin, of only 1 to 2 meter in body length, are among the fastest reported speeds and indicate speeds of about 40 knots. The three frontispieces of this book are photographs of pacific white-sided dolphins and pacific bottle-nosed dolphins taken by the author, which illustrate their superb swimming ability and fitness. There is no doubt that their ability is due to their strong caudal fins. Readers can easily see the following physical characteristics: the sectional profile at the end of the body is vertically extended, and the caudal fin joins perpendicularly. A German aeronautical engineer, Professor Hertel, observed this characteristic with interest, and he recorded the profiles as silhouettes in Fig. 1.2 from his book. Figure 1.3 shows the side and the top views of a killer whale (Orcinus Orca} quoted from the book 'Whale & Dolphin, Seals' (1965) by M. Nishiwaki who is an emeritus Professor of the University of Tokyo and a respected marine scientists. Both the dolphins and the killer whales have slender bodies that give the impression of low hydrodynamic drag and both have
4
Chapter I Gray's Paradox
distinctive well-developed dorsal fins. A typical caudal fin has a relatively small vertical thickness and a relatively large horizontal width and joins the body where the body has a thin horizontal width and a relatively large vertical dimension. The caudal fin is a high aspect ratio wing with a crescent shaped outline and a horizontal span. Furthermore, it is well known that cross sections of caudal fins are symmetric wing profiles.
Fig. 1.2 Silhouette of a Dolphin (Hertel, 1966)
Fig. 1.3 Killer Whale (Orcinus Orca, M. Nishiwaki, 1965)
Observations of Small Fresh-Water Fish - Discovery of the Swimming Number
5
Observations of Small Fresh-Water Fish - Discovery of the Swimming Number Figure 1.1 gives only the swimming speed. It shows the distance each swimmer can propel forward in a second. The figure does not specify the swimming method or the propulsive efficiency. To investigate the essence of their fast swimming ability, the swimming mode - i.e. method of propulsion - of these animals must be researched precisely. Professor Gray, the proposer of the paradox, who might consider the same aspects of problem, constructed large- and small-scale rotational ring pools in his laboratory at Cambridge University. One of the experimental pools which was later called a 'Fish Wheel' is shown in Fig. 1.4. The ring pool rotates around the center, and observers can watch and take photographs of fish swimming against a water flow as if the fish is swimming stationary in a flowing stream. Readers can also find that there are fish swimming in the tank and that there are some other devices including a rotational indicator in the figure.
Fig. 1.4 Fish Wheel Used by Bainbridge et al. (Gray, 1957)
Using the fish wheel, Dr. Bainbridge, a successor of Gray, investigated correlations of the swimming speed with body length, the amplitude and the frequency of the caudal fin oscillation of swimming fish. He studied the motion of dace, trout and goldfish. He clarified that their instantaneous maximum swimming velocity was approximately 10 body lengths per second. That was about twice the continuous swimming speed that could be maintained for more than ten seconds. The data group shown at the lower-left hand side in Fig. 1.1 is taken from Bainbridge (1958). In addition, the magnitude of the swimming speed under highspeed swimming was found to be proportional to both the body length and the
6
Chapter I Gray's Paradox
caudal fin beat frequency. He also reported that the amplitude of the tail oscillation was almost constant at approximately 20% of the body length regardless of the species and the tail beat frequency. Professor Breder (Breder, C. M., Jr., 1926), an American zoologist and researcher, first classified the various swimming motions of fish into three styles or modes of propulsion as follows: (1) Anguilliform (eel style of propulsion) - moving the whole body which is long and narrow (2) Carangiform (mackerel style of propulsion) - moving the caudal fin and the rear half of the body (3) Ostraciiform (boxfish style of propulsion) - moving only the caudal fin and the body is rigid Figure 1.5 shows the pictorial classification of these three styles. Highspeed fish such as dace, goldfish, carp, mackerel and tuna belong to the carangiform mode. According to Bainbridge, high-speed fresh-water fish oscillate only the rear third of their body in the horizontal direction, and they are classified as carangiform style swimmers. Tuna are often regarded as the most highly evolved fish using the carangiform mode of propulsion, and their crescent shaped caudal fin, that joins perpendicular to the horizontally extended section of the body, resembles the shape of the caudal fin of dolphins. The dolphins' style of swimming closely resembles that of tuna despite the fact that the caudal fin oscillates in a different direction.
1 Anguilliform
(2)
Carangiform
(3)
Ostraciiform
Fig. 1.5 Classification of Swimming Style of Fish (C. M. Breder, Zoologica, 4-5, N.Y. Zoological Society, 1926)
As described later, the author also carefully observed the swimming speeds of fresh-water fish such as carp, goldfish and tilapia. As a result, an interesting fact appeared. The swimming speed of these fish is proportional to both the body length and the frequency of the caudal fin oscillation. The proportionality constant depended on the species. Although Bainbridge also pointed out this proportionality, he only specified it in the high-speed range. He did not find the linear proportionality in the low-speed and low-frequency range. The proportionality
Observations of Small Fresh-Water Fish - Discovery of the Swimming Number Table 1.1 Swimming Number Species
Swimming Number (Observer)
Dolphin Carp Dace Trout Goldfish Tilapia
0.82 (Nagai) 0.70 (Nagai) 0.63 (Bainbridge) 0.62 (Bainbridge) 0.61 (Bainbridge) 0.58 (Nagai)
Note: For Bainbridge's data, the tail beat frequencies of dace and trout are more than 5 Hz, for goldfish it is more than 3 Hz.
constants are shown in Table 1.1, which also shows data for dolphins obtained by the author. From the table, it is clear that, among the fresh-water fish, carp is a distinctively faster fish compared to dace and goldfish, and dolphins have superb swimming ability. The superior swimming ability of carp might seem reasonable for all Japanese, however, there is an interesting story about carp. When Professor M. J. Lighthill (1924 - 1998), a leader in modern fluid dynamics, met a Japanese researcher, Mr. Y. Watanabe, who visited him with films of his carp typed robot fish, Professor Lighthill remarked, "Carp are regarded as dull fish in England." The author concluded that if the constant number in Table 1.1 shows the extent of swimming ability of aquatic animals, the number should be recognized as a physically meaningful number. The author named it the 'Swimming Number,' and proposed it in a Japanese seasonal periodical 'Flow,' the former journal of the Japan Society of Fluid Mechanics, in 1979. The swimming number is defined in equation (1.1). Definition of the Swimming Number : Sw = — (1.1) fl Since the Swimming Number is obtained by dividing the swimming velocity U (m/s) by the tail beat frequencyf(1/s) and the body length / (m), the number has no dimension. The swimming number is also interpreted as the distance traveled per body length during one caudal fin oscillation. Frankly speaking, the author was not confident that this new dimensionless number would be accepted by researchers throughout the world. However, the hypothesis has been well known among researchers in Japan since the proposal was introduced in 'Handbook of Fluid Dynamics' (1987) and in a book 'Fluid Dynamics of Drag & Thrust' jointly authored with Professor Emeritus I. Tanaka of Osaka University (1996), both in Japanese. The experimental results shown in Fig. 1.1 indicate that the maximum swimming speed of high-speed aquatic animals is proportional to the body length. The data for marlin are on an extended empirical line made with data of the instantaneous speed of carp. Nevertheless, a complete understanding of the phenomenon is not as simple as it seems. When considered in terms of fluid dynamics or animal physiology, the wide range of validity of this proportionality is
8
Chapter 1 Gray's Paradox
still a mysterious phenomenon. In the following chapters, we will look back to the origins of fluid dynamics, and prepare for the challenge of elucidating Gray's Paradox.
Chapter 2 Early History of Fluid Dynamics - From Aristotle to Newton and D'Alembert Since the drag force opposes the forward motion of an object in water or air, it seems evident that a force acting forward on the object (a thrust force) is essential to continue the motion. It took thousands of years for human civilization to arrive at the correct relationship between the drag and the thrust forces. A careful tracing of history makes it clear that science developed to provide rational explanations of many such mysterious phenomena in the world. Therefore, science can be considered as a series of battles between humankind and paradoxes. Even today, fluid dynamics, as one branch of physics, has retained such a strong tendency.
Aristotle's Paradox Aristotle (BC 384 - 322), a famous Greek philosopher, tried to rationally explain all things in the world. He interpreted them as following: "All substances have their own laws (which he called as nature) and the phenomena such as fish swimming in the water and birds flying in the sky are subject to 'the Nature of fish1 and 'the Nature of birds.'" Thus he could not rationally explain the phenomenon in which an arrow shot from a bow continued going upward (Fig. 2.1). According to his philosophy of the world, phenomena such as a rock dropping downward, water finding its own level and fire rising upward are subject to their own nature. So, if a rock is moved upward against the law, there must be a certain violent force acting on the rock to maintain the motion. Unfortunately, he could not find the force acting on the arrow. Although he developed an erroneous explanation under the pressure of necessity, he interpreted it as 'The air is slit in front of the arrow and is closed behind it.' This explanation suggested the concept of a vacuum, but he remained committed to his famous belief that "Nature dislikes a vacuum," and refused to accept the existence of a vacuum. Nowadays, people studying the principles of dynamics are familiar with the law of inertia: 'Without an external force acting on an object, the object will continue its uniform linear motion.' It took about 2,000 years to reach this understanding that was explained by Galileo Galilei (1564 - 1642). Thus, in the latter half of the Renaissance, when the Heliocentric system replaced the Ptolemaic system, it was correctly recognized that 'An arrow
10
Chapter 2 Early History of Fluid Dynamics
Fig. 2.1 An Arrow Shot from a Bow (What force can keep the arrow moving upward?)
does not necessary need an external force to continue its forwarding motion.' For instance, when a bird or an airplane flies straight forward at constant speed, the net magnitude of the external force is zero, thus the drag and the thrust forces, which have been described at the beginning of this chapter, exactly balance each other. As Isaac Newton (1642 - 1727) formulated later, "If the drag force exceeds the thrust force, the object will decelerate, and the converse makes the object accelerates." It is needless to say that a static condition is one of uniform linear motions. It is also an interesting fact that the year of Galileo's death coincided with the birth of Newton.
Newton's Fluid Dynamics and his Paradox Newton, the famous discoverer of the universal law of gravitation, reviewed numerous discoveries in astronomy and physics that occurred during the 100 years before his birth, and summarized them in his great book 'Principia' (1687, the philosophy of mathematical principles). This book became the foundation of modern science based on experiments and examinations, and it replaced Aristotle's so-called religious understanding of the world that had endured for about 2,000 years. The law of inertia that Galileo had discovered was taken as the first law of Newton's three laws of motion. Newton's three laws, which perform the most important roles even in modern fluid dynamics, are shown as equations (2.1) - (2.3), in which the character f represents all external forces (in vector), m is the mass of an object and a is the acceleration (in vector). Newton's laws of motion: The first law; the law of inertia The second law; f =ma The third law; the law of an action and the reaction D'Alembert's principle: /+ (-ma) = 0
(2.1) (2.2) (2.3) (2.21)
Examining the three laws, it is clear that they express three aspects of one
Newton's Fluid Dynamics and his Paradox
11
theorem regarding motions and forces. The three laws always correlate with each other and never apply independently. For instance, if the force f is taken as zero in equation (2.2), the second law requires that the acceleration a is also zero, and this is an expression of the first law, i.e., the law of inertia. Furthermore, the third law, stating that an external force always accompanies a reaction force, requires that the external force be the same magnitude as the reaction force but in the opposite direction. This, in turn, helps us to understand the concept of inertia as a reaction force against an external force, and enables us to rewrite equation (2.2) as a force balancing equation (2.2). This equation is called D'Alembert's principle. Figure 2.2 shows four forces acting on an airplane flying at a constant speed. In this case, the thrust T and the drag D acting on the plane balance exactly, and the lift L and the weight under gravity W (= mg} also exactly cancel each other. Therefore, the net force is zero as described previously. Since the direction of the thrust is perpendicular to gravity, both forces are independent of one another, and it is possible for the thrust force to be smaller than the gravity force. This implies that only one ton of thrust force can move an airplane weighing 10 tons.
thrust;
gravity; W = mg
Fig. 2.2 An Airplane during Steady Cruising (Holding T= D and L = W. T is independent of W, hence T<W is possible.)
It is no exaggeration that the Newton's Principia included such wide and deep essence that it was able to replace Aristotle's world concept, and hence, it influenced all areas in modern science. The concept of 'Fluid Dynamics' also appeared in the Principia, and it was Newton who first identified equation (2.4), which is known as 'Newton's law of viscosity.'
Figure 2.3 explains the meaning of equation (2.4). Taking the model of two parallel flat plates stretching horizontally with a viscous fluid between them and with the upper plate moving in the x-direction, the stress (T : a force per unit area), resulting from the motion of the plate against the viscous fluid force, is in direct proportion to the velocity gradient du/dy perpendicular to the plate. The
12
Chapter 2 Early History of Fluid Dynamics
t
Fig. 2.3 Definition of a Viscous Force
proportionality constant jU is called the viscosity coefficient, and is a physical property of the fluid (with units of Pa.s). Nowadays, a fluid whose viscosity m is constant regardless the external stress is called a 'Newtonian Fluid,1 and all other fluids are called 'Non-Newtonian Fluids.' Body fluids and most secretions of living creatures tend to be Non-Newtonian fluids. There is an influential hypothesis that such Non-Newtonian characteristic of fluids secreted on the body surface can dramatically decrease the swimming drag of fish and dolphins. This will be described in detail in Chapter 5. Furthermore, Newton appears to have correctly recognized that the drag acting on a moving object in a fluid has two elements: one element is proportional to the moving speed of the object, and another is proportional to the square of the speed. Nowadays, the former is called the viscous drag and the latter the pressure drag (or profile drag). Since the pressure drag is induced by the change of the momentum of the fluid, Newton correctly suggested that the pressure drag is in proportion to the square of velocity, the density of the fluid and the surface area (the square of a characteristic body dimension) of the object. For example, he showed that the drag D acting on a sphere of diameter d in a uniform flow is given by equation (2.5). Da1/2-pU2S, S = -p d2 (2.5) 2 4 Although equation (2.5) is considered correct even today, Newton could not proceed with further development because he could not find the proportionality coefficient. Newton considered the motion of each particle organizing the fluid. When fluid particles collide with an object, the slant angle of the surface of the object is crucial for computing the drag. He calculated drags in several cases, however, he could not logically obtain solutions that fit with the observations. Figure 2.4 shows a typical problem of an inclined flat-plate in a flow, which is occasionally quoted as an example of Newton's calculation. The surface area of the plate is taken as 5, the density of the fluid as p, the uniform velocity as U and the angle of attack to the flow as a. The mass of the fluid colliding with the flat plate in a unit time is pUSsina. Assuming that the fluid particles flow smoothly along the
13
Newton's Fluid Dynamics and his Paradox
Fig. 2.4 Fluid Force on an Inclined Flat-Plate
plate after the collision, the component of velocity perpendicular to the plate before the collision is Usina and becomes zero after the collision. The component of velocity parallel to the plate Ucosa does not change. Thus, the change of momentum of the fluid in a unit time is the product of pUSsina and Usina. From the equation of motion (2.2), the force acting perpendicular to the plate is given by equation (2.6), which is a special case of equation (2.5). F = pU2Ssinn2a
(2.6)
According to equation (2.6), the force Fis proportional to the square of the sine of the angle of attack a. Therefore, if the angle a becomes vanishingly small, the force F becomes negligible. Furthermore, multiplying F by cosa gives the lift acting on the plate. Surprisingly, measured results of the lift force far exceed those predicted by Newton's theory. This variance between observation and theory is known as 'Newton's Paradox.' Figure 2.5 shows the dimensionless normal forces on a two-dimensional 1.0 -
Fig. 2.5 Fluid Force Normal to an Inclined Flat-Plate (Theodore von Karman, Aerodynamics, Cornell University Press, 1954) Legend 1. Newton's theory 2. Dead-Water theory of Kirchhoff and Rayleigh 3. Modern Lifting Theory
14
Chapter 2 Early History of Fluid Dynamics
slanted flat plate. Parameter c is the width of the plate. Curve 1 is the result of Newton's theoretical equation (2.6). In the case of small angles of attack, the force acting on the plate becomes almost zero. Curves 2 and 3 are the dead-water theory of Kirchhoff-Rayleigh and the current lifting theory respectively. The last two theories are able to explain the large force generated on the plate at small angles of attack. Measurement results coincide very well with curve 3 when the angle a is small.
Euler's and Bernoulli's Equations and the Paradox of D'Alembert Newton's laws of motion have such good accuracy that they are still applied today, however, Newton's considerations were basically limited to the dynamics of point masses. Therefore, the flows of air and water were analyzed in terms of a collection of numerous mass points. The actual air and water are not simple collections of mass points, but they have an aspect of a continuum that cannot be separated. Thus, as Aristotle's belief that "Nature dislikes a vacuum," both the existence of a vacuum and an opposite concept of the superimposing of two fluid particles (intersecting of stream lines) had to be eliminated to obtain correct solutions. Standing on such philosophy, physicists after Newton (most of them were also mathematicians) ignored the molecularity of real fluids (such as air and water), and they researched an imaginary fluid model that was a continuum without viscosity and compressibility. Such imaginary fluids are called 'Ideal Fluids.' Equations that reflect these characteristics and the conservation of mass (equation of continuity) of an ideal fluid are written as equations (2.7) - (2.9). In the equation of continuity, A is the cross-sectional area of an imaginary flow path (called a stream tube). Figure 2.6 shows the definitions of a stream line and a stream tube. It is clear that if the cross-sectional area becomes smaller (A2
v,). Ideal fluid model: No-viscosity m = 0 Incompressibility p = const. Equation of continuity pvA = const.
(2.7) (2.8) (2.9)
The momentum equation and the law of energy conservation of an ideal fluid were formulated by Euler (Leonhard Euler: 1707 - 1783) and his close friend Bernoulli (Daniel Bernoulli: 1700 - 1782), and are shown as equations (2.10) and (2.11) respectively, where, s is the coordinate along a stream line, z is the height and g is the acceleration of gravity. Euler's equation (2.10) is an application of Newton's second law, equation (2.2). The acceleration term is on the left hand side (LHS) and is put equal to the force per unit mass on the right hand side (RHS). In this equation, the LHS comprises the terms of unsteady acceleration and convective acceleration, and the RHS comprises the gravity force and the pressure gradient force along a stream line. Unsteady acceleration means that a fluid element is
15
Euler's and Bernoulli's Equations and the Paradox of D'Alembert
Fig. 2.6 Stream Line and Stream Tube
accelerated by the unsteadiness of the flow, i.e. time to time. On the other hand, convective acceleration means that the fluid element is accelerated along the flow direction, i.e. place to place. Euler's equation of motion: dt
ds
ds
p ds
(2.10)
(Acceleration = Gravity force + Pressure gradient force) Bernoulli's equation: —H
h gz = const.
(2.11)
(Pressure energy + Kinetic Energy + Potential energy = Constant) In the case of steady flow, Bernoulli's equation is obtained by integrating Euler's equation along s. It conlains an importanl proposition, namely, that the total amount of energy per unit mass of fluid is conserved. Thus, it represents 'the law of conservation of energy.' Equation (2.11) is one of the most important equations in modern fluid dynamics. For example, the water velocity and the volumelric flow rale from a lap on a water lank can be exactly calculated with this equation. The Pilot tube invented by Pilot (Henri de Pilot: 1695 - 1771) in 1732, which is still used to measure the speed of modern aircraft, is based on this equation. One of the first persons to apply the above equations of an ideal fluid for the flow around an object of arbitrary profile and to try to rationally explain the aerodynamic drag was D'Alembert (Jean le Rond D'Alembert: 1717 - 1783) who is also the first editor of the 'Encyclopedia.' In spile of his precise calculations, the results he obtained surprised and disappointed him. His written article 'dune nouvelle theorie de la resistance des fluides' concludes with the sentences as following passage: "/ do not see then, I admit, how one can explain the resistance of fluids by the theory in a satisfactory manner. It seems to me on the contrary that this theory, dealt with and studied with profound attention, gives, at least in most cases, resistance absolutely zero; a singular paradox which I leave to geometricians to explain."
16
Chapter 2 Early History of Fluid Dynamics
D-0! (a) Stream line diagram
(b) Surface pressure distribution (solid line; ideal fluid, broken and single dot chain lines; real fluids)
Fig. 2.7 Flow of an Ideal Fluid around a Cylinder and the Pressure Distribution
This was the first declaration of the famous 'Paradox of D'Alembert.' Euler's equation of motion was written above in one-dimensional form for motion along a stream line. However, the equation can also be written for two-dimensional and three-dimensional flows, and many flows under numerous boundary conditions have been calculated since the appearance of the equation. Figure 2.7 shows an example of a two-dimensional flow of an ideal fluid flow around a circular cylinder. Figure (a) shows the stream lines, and figure (b) shows the distribution of the static pressure on the cylinder surface. Considering the stream line that runs on the center line of the cylinder (from a to e, called the stagnation stream line), the flow approaches the cylinder with reducing velocity. Then, the flow collides with the cylinder at the stagnation point b, and the velocity becomes zero while the pressure takes on a maximum value. Then, the flow splits into two parts and is accelerated to pass the point c (or c') where the maximum velocity is 2V and the pressure is minimum. Next, the flow velocity is reduced as the stream lines meet at the rear stagnation point d. Here the velocity is zero and the pressure is maximum once more. Finally, the flow velocity gradually increases to the previous velocity of V far downstream. This is the mathematical concept of an ideal flow that is obtained exactly. The pressure distribution is drawn with a solid curved line shown in Fig. 2.7 (b). The front and the rear stagnation points have positive pressure whose force acts inward on the surface of the cylinder. In contrast, a large negative pressure is generated at the shoulder point c (or c') where the pressure is minimum, and the pressure force pulls the surface outward. In fact, the behavior of the negative pressure is frequently experienced in our daily life. Now, if the surface pressure on the cylinder is integrated over the entire surface of the cylinder, the net force (in vector) acting on the cylinder can be obtained. As it is clearly seen from the symmetry of the flow and the pressure distribution, the resulting net force is zero. This result conflicts with our daily experiences. Even D'Alembert recognized that the explanation of this paradox
Euler's and Bernoulli's Equations and the Paradox ofD'Alembert
17
comes from the ideal fluid modeling, especially the hypothesis of no-viscosity. Although the viscosity of air or water is small, the conclusion of zero-drag force of an ideal fluid seemed too dramatic to explain. In figure (b), the pressure distributions shown as a dash line and a single-dot chain line are obtained from experiments using real fluids. As it is clearly shown in the figure, the experimental result of the pressure distribution near the front stagnation point almost coincides with the ideal fluid theory. In contrast, the situation at the rear stagnation point differs greatly from the ideal fluid, and the measured pressure takes on negative values there. As a result, the cylinder is pushed on its front and drawn backward on its downstream side. Therefore, the cylinder suffers a large drag force. Figure 2.8 shows the flow of an ideal fluid around an inclined flat-plate. This is also an integrated solution of the two-dimensional Euler's equation of motion. Comparing this ideal flow field to Newton's flow shown in Fig. 2.4, the former seems to have a more realistic stream line pattern. However, the stagnation stream line in this case runs through a® b® c (or c')® d® e. Thus the flow is perfectly closed and the stream line pattern on the upper and the lower streams have beautiful point-symmetry. Therefore, the net force acting on the plate calculated by integrating the pressure distribution on the plate in this case is exactly zero. However, the moment acting to make the plate rotate in the clockwise direction still exists.
Fig. 2.8 Flow of an Ideal Fluid around an Inclined Flat-Plate
We conclude that an object put in an ideal fluid has a total fluid force equal to zero regardless of the shape of the object. Keeping the concept of an ideal fluid model, there is another theory, the 'Dead-water region theory,1 proposed by Kirchhoff (G. Kirchhoff: 1824 - 1887) and Rayleigh (Lord Rayleigh: 1842 - 1919), which was an attempt to avoid the paradox of D'Alembert. Figure 2.9 shows an example. Comparing Fig. 2.9 and Fig. 2.8 it is clear that the stagnation stream line running through a ® b ® c (or c') does not close again behind the plate but separates and splits the flow from both edges of the plate. There is a dead-water region between the free stream lines in which the flow velocity relative to the plate is zero. Provided the pressure in the dead-water region equals the static pressure p^, the pressure at infinity, a large force acting on the plate can be explained by means of the pressure difference between the dead-water region and the front face of the
18
Chapter 2 Early History of Fluid Dynamics
Fig. 2.9 Flow around an Inclined Flat-Plate with Dead-Water Region Theory
plate on which the stagnation stream line collides. Curve 2, in the previous Fig. 2.5, is the computed result of the dimensionless normal force based on the dead-water theory. Although this theory can avoid the paradox of D'Alembert and seems to be more realistic than Newton's model, the result does not agree with experimental results. Moreover, the width of the deadwater region needs an assumption and its length is assumed to be infinite. Thus, this theory includes some controversial elements and does not fully explain D'Alembert's Paradox.
Lifting Theory Using ideal fluid theory in 1878, Rayleigh explained that the fluid force acts perpendicular to the flow direction (lift). The principle of lift is illustrated in Fig. 2.10. This flow diagram is composed of a uniform flow around a cylinder shown in Fig. 2.7 (a) and a flow with circulation F
(=fv.ds)
where R is the
radius of the cylinder and V is the speed of the undisturbed flow. r (=
-ds)=2p
RV(2.12)
Circulation is a physical quantity that measures the intensity of the vorticity
p
—i Vx
pbo
Fig. 2.10 Superposition of a Clockwise Circulation on a Uniform Ideal Fluid Flow around a Cylinder
Lifting Theory
19
in the flow field. The circulation is calculated by a circular integration of the tangential velocity component vds along an arbitrary closed curve C. In this case, there is a vortex with circulation 2pRV inside the cylinder. The flow approaching the cylinder is first pulled upward, and then pushed downward by the induced velocity of this vortex. Since the momentum of the incident flow is turned downward, the cylinder will receive a large upward force, or lift, as a reaction. Rayleigh showed that the direction of the lift is perpendicular to both the velocity vector V and the circulation vector F (taken positive in the clockwise direction along the vortex center), and the vector lift L is expressed in equation (2.13) by a vector cross product. The generation of lift can also be explained by Bernoulli's equation, and by considering the stagnation stream line from a to e in Fig. 2.10. The velocity is zero at the stagnation points, b and d, where the pressure is maximum. The locations of the stagnation points are moved downward on the cylinder compared to the zero circulation condition. The flow speed at c is accelerated from 2V to 3V , and the flow velocity at c is reduced from 2V to V . The intensity of the negative pressure at point c is larger than that at c' and the imbalance of the pressure results in a lifting force acting upward on the cylinder. The trajectory of a curve ball or a screwball in baseball and a cut ball in tennis follows large curved lines, and the 'Magnus Effect' is due to the lift. However, it must be noticed that even in this case, the drag force acting opposite to the flow direction is zero since the pattern of the stream lines is perfectly symmetric. Thus, as far as the drag is concerned, the paradox of D'Alembert has not been resolved. L = pV.xT
(2.13)
Figure 2.11 shows the flow field around a wing with a finite angle of attack. This pattern of stream lines is in accordance with modern wing theory (lifting theory), and the pattern is almost identical to the experimental results for small angles of attack. Comparing the stagnation stream line from a to e in this figure to the pattern drawn in Fig. 2.8, it is clearly seen that the rear stagnation point d has moved from the upper surface of the wing to the trailing edge of the wing. Kutta (Wilhelm Kutta: 1867 - 1944) and Joukowski (Nicolai Egorovich Joukowski: 1847 - 1921) superimposed a circulation on the uniform flow around a wing, and they determined the intensity of the circulation so that the location of the rear stagnation point exactly coincides with the trailing edge. This idea is called the Kutta condition or the Kutta-Joukowski condition. Thus, they clarified that a wing
Fig. 2.11 Flow around a Wing with an Angle of Attack
20
Chapter 2 Early History of Fluid Dynamics
moving forward is equivalent to a vortex, and why a large lift is generated. They gave the final solution to this difficult problem of calculating the lift force. Their solution also allows a flying wing to be considered as a bound vortex. Even in this case, the drag acting on the wing is theoretically zero. It has been clarified that the drag on an object cannot be explained by means of an ideal fluid model. Although Euler's equation of motion can be integrated and can be used to solve numerous flow fields beautifully by introducing circulation, it is totally in vain for dealing with drag. As a consequence, theoretical fluid dynamics was of little use for engineers working on the design of ships, pumps and fans. It was generally thought that the explanation of D'Alembert's paradox was the presence of the viscosity of real fluids. The equations of motion that considered the internal friction of fluids (viscosity) were known in the middle of the 19th century, however, because of their complexity, their solution and understanding was not accomplished until the 20* century.
Chapter 3 Modern Fluid Dynamics The 20th century is often thought of as the 'age of aeronautics' because of the dramatic progress made in the field since the first powered flight in 1903. This is correct in many ways. However, scientists and engineers, including the author, would prefer to think of this century as the 'age of fluid dynamics.' This is because the appearance of boundary layer theory in 1904 - the year after the success of powered flight - led to a solution of D'Alembert's paradox. Is it destiny that some new scientific discoveries occur at the dawn of a century? Modern fluid dynamics has grown from the boundary layer theory to become an essential basis of modern aeronautical engineering. This chapter attempts to clarify the developments and achievements of modern fluid dynamics.
Navier-Stokes Equations The first and most fundamental equations of motion involving fluid viscosity (meaning the viscosity of a Newtonian fluid as given in equation (2.4) in Chapter 2) were formulated independently by Navier (C. L. M. H. Navier: 1785 - 1836) and Stokes (G. G. Stokes: 1819 - 1903), and are thus known as the Navier-Stokes Equations. The equations, given as equaions (3.1) and (3.2), describe the motion of an incompressible fluid, which are represented in vector format in equation (3.1) and in x, y, z components in equation (3.2). The Navier-Stokes equations (p = const.): dv/dt + (v.V)v = K- — Vp + (3.1) dt P P (Acceleration = External force + Pressure gradient force + Viscous force)
22
Chapter 3 Modern Fluid Dynamics
x component; du + du + du dt dx dy
dz
p dx
p\dx2
dy2
y component; dv + M dv +v dv +vv dv=yv dt dx dy dz component; dw dt
dw dx
dw dy
dw _ ~ dz
1 dp f l ( d 2 v d -5^ + — -5^ + 3 P dy p\0x~ dy2 1 dp p dz
^l(d2w p\dx2
d2w dy2
(3.2) 2
dz
d2w dz2
Equation of continuity: ry
V-v =nQ or du -r- +, dv ^—- +dw -— =n0 dx dy d z
(3.3)
Comparing equation (3.1) to Euler's equation of motion (equation (2.10) in Chapter 2), it can be seen that the acceleration terms on the LHS of each are identical. The external force term on the RHS, represented by gravity in equation (2.10) and more generally by X, Y and Z in equation (3.2), and the pressure gradient terms, are also comparable. The only difference is the addition of a viscous force as the third term on the RHS. Despite its apparent simplicity, the Navier-Stokes equations are one of the most difficult non-linear partial differential equations known to mathematicians. Even using modern super-computers, it is impossible to obtain general solutions of these equations. Even the existence of a singular solution is doubted. Since the viscous force (third term on the RHS) is in proportion to the coefficient of viscosity, it appears that this term is negligible for fluids that have very low viscosity, such as air and water. And we might approximate such fluids as no-viscous fluids. However, the fact that the difference between both solutions of real and ideal fluids becomes significantly large is the critical basis of D'Alembert's paradox. Figures 3.1 and 3.2 are photographs of the flow field made by a cylinder moving from right to left in a water tank. Figure 3.1 was produced by Prandtl (Ludwig Prandtl: 1875 - 1953) and his disciple Tietjens. It can be seen that the pattern of stream lines behind the cylinder is completely different from that of an ideal fluid and of the dead-water region model. The experiments show large vortices downstream of the cylinder. The stream lines that should theoretically run along the wall-surface of the cylinder actually separate from the trailing surface of the cylinder, and form large vortices downstream. These vortices are generated alternatively from the top and bottom of the cylinder, forming a staggered pattern. Figure 3.2 shows various vortex streets generated downstream of the cylinders under six different flow conditions.
Effects of Small Viscosity
23
Fig. 3.1 An Actual Fluid Flow around a Cylinder (Visualized photograph) (L. Prandtl, J Roy. Aero. Soc., 31-730,1927)
Fig. 3.2 Karman Vortex Streets Downstream of a Cylinder (Visualized photograph) (F. Homann, Forsch. Ing-Wes., 7-1,1936)
Karman (Theodore von Karman: 1881 - 1963), another of Prandtl's disciples, theoretically proved the stability of the vortex street - this effect is now known as a 'Karman vortex street.' The flow fields shown in Figs. 3.1 and 3.2 should serve as reference states for any mathematical solutions of equation (3.1). However, such solutions have not been found yet.
Effects of Small Viscosity Why does neglecting the viscosity of air and water produce such large differences between a real flow and an ideal fluid flow? The answer is hidden once again in the molecularity of a real fluid. Water and air are collections of innumerable molecules that move freely relative to each other. A 'solid' body is also a collection of individual molecules or atoms, with the distance between the molecules being far larger than the size of each molecule. Since the velocity of molecular motion under normal temperature and pressure reaches 400 m/s, the molecules of the fluid can
24
Chapter 3 Modern Fluid Dynamics
easily move in direct contact with the surface or move away from the body surface. What then are the boundary (velocity) conditions that a real fluid should satisfy on average at the surface of a solid body? In the ideal fluid model, the only boundary condition is that the velocity component perpendicular to the wall be zero. Hence the velocity component parallel to the wall need not be zero and the fluid may slip along the body surface. In fact, for all ideal fluid flows described in the previous chapter, it was assumed that a stagnation stream line was located on the body surface. More strictly speaking, the solid body surface itself was a stagnation stream line. If the molecularity of a real fluid is considered, then the molecules of the fluid lose their average momentum by means of their 'free access' to the solid surface. Thus both the perpendicular and tangential velocity components should be zero. For those who have studied the dynamics of a real fluid flow, this commonly accepted concept is known as the 'no slip' boundary condition. Fluid friction is therefore very different from the concept of solid friction that appears in text books on dynamics: Two solid bodies slip against each other, and the friction force is in proportion to the normal force.1 The reader needs to be aware that this boundary condition holds irrespective of the magnitude of the viscosity. It holds even if the viscosity is very small. The general expression of the description above is shown in equation (3.4), where vb is the velocity vector of the body, which equals the fluid velocity v. This condition must always be satisfied, even, for example, if a dolphin is swimming very fast and is rapidly wiggling its body. No slip condition: v = vh on the body surface
(3.4)
Reynolds1 Law of Similarity Since the LHS of equation (3.1) is associated with acceleration, it can be thought of as inertia force per unit mass. In fluid dynamics the ratio of this term to the viscous force (the third term on the RHS) is known as the 'Reynolds Number.' The Reynolds number is defined in equation (3.5), where the density of the fluid is p, the coefficient of viscosity m a 'characteristic' body length L and the velocity is V. It is a dimensionless number, and varies widely from 10 ' to 108 depending on the ratio between the inertia force and the viscous force in the flow field. For example, the Reynolds number is quite small in the aquatic world of an amoeba or a paramecium. and quite large for flows around a whale or a ship. ( Inertia force Reynolds number: Re = —-V Viscous force
pV2 1 L \ pVL = -- - = -V/L ) M
(3.5)
If each term of the Navier-Stokes equation is non-dimensionalized and if gravity is neglected then equation (3.6) is obtained. Where, all physical quantities including the time t in the equation are non-dimensionalized with representative quantities. In accordance with theory, the solution of the Navier-Stokes equation is
25
Reynolds' Law of Similarity
only determined by the dimensionless coefficient 1/Re (the second term on the RHS). In other words, 'flow fields whose boundaries are geometrically similar have similar flow fields under the same Reynolds number.' This theorem is known as 'Reynolds' law of similarity.' Non-dimensional Navier-Stokes equation: ^ + (V • V) v =- Vp + — Av m Re
(3.6)
Figure 3.3 demonstrates the great value of Reynolds' law of similarity. The figure shows the coefficient of drag force acting on a sphere for a wide range of Reynolds numbers. The drag coefficient CD is defined in equation (3.7). All the experimental results lie on a single curved line, thus showing that the flow fields are identical under the same Reynolds number, regardless of the sphere diameter or flow velocity. Once again, the general solution of the Navier-Stokes equation for arbitrary Reynolds numbers does not exist. Stokes and Oseen (C. W. Oseen) produced theoretical solutions for very low Reynolds numbers less than 1 (line (1) in 1851 and line (2) in 1910 in Fig. 3.3). In addition, there is a phenomenon where the drag coefficient at a Reynolds number near 3xl05 suddenly drops from about 0.4 to 0.1 - this is described later. Drag Fluid dynamic pressure x Sphere projected area )
—
-
• L ebster
-----
lien ©
}Meselsbtrger
- « 1 926
\
..A.
Sl
- ' N**" ^
V
, > =*,
jufo,
MM i. >
\ \ ,<
— -»>
(3.7)
1/
iNSarat if: chiller -Schmiedel
.,
vis —
D
4e
v?
46
V
? 45
V
V6S
j
?
4 ff
)
t
?
4 6^
70* '
V6a
;i
Re =Vd/v
Fig. 3.3 Drag Coefficient for a Sphere; (D Stokes1 theory, (2) Oseen's theory (H. Schlichting, Boundary Layer Theory, 7th ed., 17, McGraw Hill, 1979)
26
Chapter 3 Modern Fluid Dynamics
Boundary Layer Theory According to equation (3.6), the greater the Reynolds number the lower the influence of the viscous force (on the RHS in equation (3.6)) compared to other terms. However, as already described, even if the Reynolds number is very large, the flow velocity at the surface of the object must satisfy the 'no slip1 condition due to the molecularity of the fluid. Prandtl noted this point, and proposed his boundary layer theory in 1904. He divided the flow around an object into two parts. He considered the influence of viscosity only in a very thin layer near the surface of the object, and considered the main stream away from the body as a no- viscous fluid. This is the Prandtl's boundary layer theory. Using boundary layer theory, the Navier-Stokes only applies in a very thin layer of fluid near the body surface. Thus, for the flow around a two-dimensional object, only the equation of motion for the x-direction (tangential to the surface of the wall) remains from the three equations of (3.2). The other two equations can be ignored because of conditions (3.8). Furthermore, in the equation of motion for the x-direction, the viscous force term can be simplified significantly. Finally, equation (3.9) is obtained - this is the boundary layer equation. Assumptions of the boundary layer equation: near the wall, w = 0, <1 (3.8) d hence, du ,
du , du
1 dp . H d2u
dt
dx
p dx
— + M— - + v— = -- ~^ + — -TT
dy
p dy
.. 3 n.9) <-
Equation of continuity: ^ +^ =0
ox
ay
(3.10)
The flow outside the boundary layer can be analytically derived using Ruler's equation and the equation of continuity. Thus, the flow field of a real fluid around a body can be separated into two problems, (1) solving the boundary layer equation with the conditions of no slip (u = v = 0) at the wall surface, and (2) solving the ideal fluid flow outside the outer edge of the boundary layer. The edge of the boundary layer becomes a boundary condition for the main stream, along with the velocity in the jc-direction at the boundary layer edge that corresponds to slip. In 1908 another of Prandtl's disciples, Blasius (H. Blasius), first calculated equation (3.9) for a boundary layer that developed on a flat-plate positioned parallel to a uniform flow. Blasius' solution has since been verified experimentally with good accuracy. Using boundary layer theory, difficult flow fields of actual fluids could have been treated mathematically. The theory has brought about remarkable progress in
Boundary Layer Theory
27
fluid dynamics during the 20th century. In particular, the theory was essential for the progress of aeronautical engineering that began with the Wright brothers' first powered flight (W. Wright: 1867 - 1912, O. Wright: 1871 - 1948). D'Alembert's paradox also revealed its hidden essence: the existence of the no-slip condition, the development of the boundary layer and the separation of the flow from the wall surface. In a flow with an adverse pressure gradient (dpldx > 0), such as the flow behind a cylinder where the pressure increases along the flow direction, fluid particles inside the boundary layer have significantly slower speed than for an ideal fluid near the surface. The fluid cannot move forward and is pushed away from the surface and into the main stream. Thus, large separation flows may exist wherever there is a large positive pressure gradient. In addition, boundary layer separation never occurs under a normal pressure gradient where the pressure decreases along the flow direction (dpldx < 0). The fact that the flow pattern in front of a cylinder in Fig. 3.1 and the pressure distribution of the relevant area in Fig. 2.7 (b) are identical to the solution of an ideal fluid flow can be seen as the proof of this. One important result of boundary layer theory is that to reduce the drag of an object in a fluid, the outline of the object should have a streamlined or a spindle shape. This means that the rear part (the shape behind the cross-section of maximum thickness) of a wing should be drawn smoothly to reduce the adverse pressure gradient dpldx. One of the main topics in aeronautical engineering in the early part of 20th century was the development of wing shapes that had large lift and small drag. Many countries made efforts to develop low drag wing profiles individually and in secret throughout the two World Wars. Figure 3.4 shows cross sections of a laminar wing profile and a circular cylinder (steel wire), both of which have the same drag force. As shown, for a wire of diameter d, a wing whose cord length is 167 times d and whose thickness is 35 times d has the same small drag as the wire itself. This fact illustrates how the many guide wires used in early biplanes created very large drag, and also makes it easy to understand why such biplanes were subsequently replaced with modern monoplanes.
Fig. 3.4 A Cylinder and a Laminar Aerofoil with Equivalent Drag (H. Schlichting, Boundary Layer Theory, 7th ed., McGraw Hill, 1979)
28
Chapter 3 Modern Fluid Dynamics
The same reasoning accounts for the fact that high-speed swimming animals, that are the main theme of this book, have such streamlined shapes or spindle shapes. They arrived at their current shapes through biological evolution over tens of thousands of years or even longer. The phenomenon that 'fish swim smoothly1 means that only a small propelling force is needed to accelerate them, and once moving the swimming speed is not reduced easily due to the body inertia and the very low drag.
Laminar Flow and Turbulent Flow By understanding the concept of a boundary layer, the behavior of real fluids can be understood. However, there remains one un-avoidable problem: Laminar flow and turbulent flow occur not at the level of molecular motion, but in macro-dynamic fluid motion. For instance, even in the case of water flow in a pipe that appears to be uniform, there are two kinds of flow patterns: Laminar flow and turbulent flow. Reynolds (Osborne Reynolds: 1842 - 1912) first clarified this phenomenon experimentally in 1883 in England. Figure 3.5 shows his experimental apparatus at Cambridge University. Using this quite simple equipment Reynolds made water flow at several different speeds through a transparent glass tube fixed in a tank. The flow was visualized by injecting colored water through an inlet tube. Reynolds discovered that when the average velocity va was small, the colored water flowed in a straight line with no mixing. When the flow velocity exceeded a certain critical velocity, the colored line disappeared as fluid particles mixed with each other. Thus Reynolds first clarified that a transition from laminar flow to turbulent flow occurred at a Reynolds number exceeding a certain critical number (about 2,000). Sommerfeld (A. Sommerfeld: 1868 - 1952), however, first introduced the term Reynolds number, Reynolds did not use the term himself.
Fig. 3.5 Reynolds' Experimental Apparatus
Figure 3.6 shows the difference between laminar and turbulent flow in Reynolds' experiment. In the case of laminar flow, fluid particles do not mix with each other and thus form a single line parallel to the pipe. In contrast, for turbulent flow the particles mix with each other and form a complex pattern. In the figure, v =m/p is the 'Kinematic viscosity1 or 'Coefficient of kinematic viscosity,1 which is obtained by dividing the viscosity fi by the density of the fluid p. For example, the kinematic viscosity of water, at 20° Celsius and at one atmosphere of pressure, is
Numerical Simulations and Physical Experiments
29
colored liquid
colored liquid
V
r
r (a) Re - -<
2,000
(b) Re -
- > 2,000
Fig. 3.6 Laminar and Turbulent Flow in Reynolds' Experiment
approximately 0.01 cm2/s. If the diameter of the water pipe is 1 cm and the average velocity of the flow is 20 cm/s, the Reynolds number is 2,000. For a larger pipe diameter or a faster flow speed, the flow will become turbulent. Most flows that we experience in everyday life have a Reynolds number of 104 or greater, and thus most flows are turbulent. The Navier-Stokes equations for a real fluid (shown in equation (3.1)) are valid for both laminar and turbulent flows, however, the detailed flow fields for laminar and turbulent flows are very different. Since the stream lines in laminar flow are parallel each other, a coordinate axis is taken parallel to the flow direction, and the convective acceleration, the second term on the LHS, becomes exactly zero or leaves only the term u (duldx) like the one dimensional Euler equation. As a consequence, equation (3.1) can then be easily integrated. Indeed, for a laminar flow including the flow in the boundary layer, most flow patterns can been solved nowadays. In the case of a turbulent flow, even if the time-averaged velocity is steady (dvldt = 0), the instantaneous velocity of each fluid particle (a lump of molecules) is unsteady (dvldt ¹0). Thus, the three components of the velocities u, v and w are functions of the space variables x,y,z and the time t. This means that all terms in the momentum equation (3.1) or (3.2) are equally important, and none of them can be ignored. This is the essential complexity of turbulent motion. Although modern fluid dynamics uses statistical strategies and the concepts of vortex flow modeling or Vortex viscosity' to analyze the turbulent motion, the reality is that no approach has proceeded beyond the area of an experimental or a semi-experimental theory.
Numerical Simulations and Physical Experiments The recent development of super-computers has enabled precise and realistic flow simulations by means of direct solutions of the Navier-Stokes equation without any turbulent models. Figure 3.7 shows one such trial example. The flow field around a two-dimensional cylinder was divided into 32,761 (= 181x181) O-type grid meshes. The uniform flow passing the cylinder was discretized and then calculated directly from time to time. The no-slip condition on the cylinder was also used as the boundary condition, and the meshes near the cylinder (in the boundary layer area) were made as small as possible. Three different Reynolds numbers 102, 104 and 3 x 105 were computed. The results seem surprisingly identical to the actual flow. They show an instantaneous view of the boundary layer flow separating behind the cylinder and
Chapter 3 Modern Fluid Dynamics
30
flowing backward to generate large vortices. This seems to agree with the photographs of an actual flow, as seen in Fig 3.1. Thus a super-computer can simulate precisely non-uniform (fluctuating) flows that change with time, and compare favorably with the physical experiments in water tanks or wind tunnels. Simulations made by super-computers are called 'Numerical experiments' nowadays, and are treated separately but with the same regard as physical experiments. The time-averaged drag forces on a cylinder, obtained by numerical simulations, are compared with the results of physical experiments in Fig. 3.8. The solid line in the figure is the results of physical experiments. Numerical results were obtained at seven different Reynolds numbers. In the case of a relatively small Reynolds numbers, the various numerical solutions are nearly identical and agree well with the physical experiments. This is not the case for larger Reynolds numbers. The numerical results show different drags, and depend on the calculation conditions and the size of the time step, AT, used in the calculation. The numerical
Fig. 3.7 Numerical Calculation of the Flow around a Cylinder (H. Matsumiya et al., 1993)
2.0
AT" 1.0 . +0.1 X 0.05 V 1.67 X10-3 2 ©3.5X10- 3 • 1.67 X10-3 A 0.83 X1Q0.0
10'
10'
10"
10*
Fig. 3.8 Numerical Calculations of the Drag Coefficient for a Circular Cylinder (Solid curved line shows the results of physical experiments after H. Matsumiya et al., 1993)
Numerical Simulations and Physical Experiments
31
solutions generally have a higher drag than the physical experiments for Reynolds numbers greater than 102. Thus, even for calculations made with a super-computer, the results must be examined and compared with physical experiments that have the same boundary conditions. It should be noted that physical experiments are not always correctly compared to the numerical simulations. For instance, there is no assurance that the flow, in the physical experiment for the flow around a cylinder, is perfectly twodimensional, as is assumed in the numerical simulation. Thus, the essential difficulty of real fluid motions can be understood, and the importance of experiments in fluid dynamics is also emphasized. As previously mentioned, the sudden decrease of drag seen in Fig. 3.8 for Reynolds numbers near 3x 105 is quite an interesting phenomenon. It is no exaggeration to say that modern fluid dynamics has solved the entire mechanism of this phenomenon. As the Reynolds number increases, the boundary layer flow near the wall reaches a critical velocity, and the flow in the boundary layer undergoes a transition from laminar to turbulent flow. The key to this phenomenon is that, when the boundary layer becomes turbulent, the fluid particles obtain kinetic energy from the main flow in addition to the one-dimensional kinetic energy along the wall. As a result, the robustness against the adverse pressure gradient that was previously described becomes stronger, and the separation of the boundary layer is delayed compared to a laminar boundary layer. Thus the breadth of the separated flow area becomes narrow compared to the case of laminar flow separation, and consequently the value of the drag coefficient decreases to about one third of the laminar flow case. This sudden decrease in drag is also observed in Fig. 3.3 for the flow passing a sphere at Reynolds number of around 3x 105. Prandtl, who proposed the boundary layer theory, first proved the phenomenon physically. He set a tripping wire in front of a sphere, and artificially created a turbulent boundary layer in the lower Reynolds number region. This effect is so well known that nowadays most golf players know the reason for the dimples on a golf ball - the drag of a ball with dimples is much smaller than without the dimples, and thus the driven distance is three to five times longer. It is interesting to note how much fluid dynamics has contributed to sporting technology, and there are many books on the subject. An additional important fact must be considered when discussing turbulent flow in pipes and turbulent boundary layers, namely, that there still exists a laminar flow region very near to the wall (the layer under the turbulent boundary layer). Since the boundary condition on a solid wall must be the no-slip condition, the flow next to a smooth wall should be laminar, and a turbulent boundary layer exists above this laminar layer. This sub-layer is known as a Viscous' or 'laminar' sublayer. Considering the smoothness or roughness of a pipe wall, if the height of roughness exceeds the thickness of the viscous sub-layer, then the wall surface is judged to be 'rough' - otherwise it is termed 'fluid dynamically smooth.' The viscous sub-layer is considered again in the section on Toms effect' in Chapter 6.
32
Chapter 3 Modern Fluid Dynamics
Dynamics of Wings in Real Fluids This final section of the chapter describes the motion of a wing in a real fluid, and attempts to explain how a wing moving forward in air or water produces lift. Figure 3.9 shows sequential (by time) animations of the flow around a wing that suddenly starts to move from rest. Figure (a) shows the instantaneous flow field as the wing just starts to move. This flow field is nearly identical to Fig. 2.8 (a flow pattern around an inclined flat-plate), which was already described in the previous chapter for an ideal fluid. Initially, circulation does not exist in the fluid or around the wing, and the fluid flows as if it is an ideal fluid that generates neither lift nor drag. Point A in the figure is the front stagnation point and point S is the rear stagnation point. The fluid particle initially existing at the trailing edge of point B flows upstream to the point S on the wing. Figure (b) shows the next instantaneous flow field in which fluid particles try to pass around point B to the upper surface of the wing. However, the flow fails to go around the trailing edge and separates at point B. Point B becomes a 'singular 1 point - in ideal fluid dynamics, the acceleration required to negotiate the 180° turn is infinity. However, this is not possible in the case of a real fluid. The author explains the reason in his lecture, by saying "a flow cannot suddenly change its direction." Figure (c) shows the flow field at a later time. Despite the separation of the flow from the lower surface, no dead-water region is generated downstream. In this case, the rear stagnation point S moves downstream to the trailing edge. From this time onwards the upper and the lower flows meet at the trailing edge. The fluid portion between points B and 5 forms a counter clockwise vortex that moves
Fig. 3.9 Flow around a Wing Starting from Rest
Dynamics of Wings in Real Fluids
33
downstream. Since the flow velocity above the wing is faster than the flow velocity below the wing, this phenomenon can be regarded as the generation of a 'circulation' around the wing. The intensity of the circulation is obtained from equation (2.12). As shown in Fig. (d), the vortex that is generated downstream has exactly the same magnitude as the circulation around the wing but with negative value. Therefore, the integration along the entire closed curve C is zero, which is the same as when the wing was at rest. The vortex attached to a wing is known as a 'bound vortex,1 and the vortex downstream is known as a 'starting vortex.' The phenomenon described above can be confirmed by both flow visualization and computer simulation. The important thing is that the rear stagnation point moves to the trailing edge due to the molecularity or viscosity of a real fluid, and circulation is generated as a consequence. As has been described in the previous chapter, in ideal fluid dynamics the intensity of the circulation is mathematically determined so that the position of the rear stagnation point coincides with the trailing edge (Kutta-Joukowski condition), and the lift is explained on this basis. Thus, circulation occurs only in case of a real fluid flow. The above explanation is for a two-dimensional fluid flow. What about the case of a wing that has finite width? Let us consider an airplane taking off the ground, as in Fig. 3.10. For a finite wing, vortex lines are drawn from both ends of the wing and are left in the atmosphere as free vortices. These free vortices from the wing tips are connected with the starting vortex that is also left behind. The bound vortex, wing tip vortices and the starting vortex constitute a large vortex loop. The intensity of the free vortices exactly equals that of the bound vortex. To put it a different way, an airplane consumes fuel in generating energy for the free vortices that extend continuously during flight. Since the intensity of a vortex is in proportion to the lift of an airplane, i.e. the weight, it is understood that there are extremely strong free vortices and a downward flow behind a large airplane. This means there is dangerous turbulence for any small airplane flying nearby. bound vortex
free vortices
airport starting vortex
Fig. 3.10 Relationship among a Bound Vortex, a Starting Vortex and Free Vortices
34
Chapter 3 Modern Fluid Dynamics
Figure 3.11 shows an example of the characteristics of a wing. The horizontal axis is the angle of attack a while the vertical axis is the lift coefficient CL, the drag coefficient CD and the lift-drag ratio CLICD. The relationships between lift and drag and their coefficients are defined in equations (3.11) to (3.13). L = CL-^-pV2A
(3.11) (3.12) (3.13)
D .12
-40 8 16 angle of attack OL (°)
24
Fig. 3.11 Characteristic Curves of a Wing (I. H. Abbott and A. E. Doenhoff, Theory of Wing Section, Dover, 1958)
From the figure, the lift coefficient at an angle of attack of zero degrees is positive because the wing has camber. For small a, CL increases in proportion to a, which agrees with the circulation theory described previously. On the other hand, it is well known that if the angle of attack increases beyond a certain angle, there is a sudden decrease in lift and a rapid increase in drag. This is known as the 'stall1 phenomenon, and in the figure it happens for a exceeding 18 degrees. The circulation around a wing in a stall tends to zero, and the existence of a dead-water region on the upper wing surface causes a large pressure resistance. According to Fig. 3.11 the lift-drag ratio LID at an angle of attack of 1 degree (relatively small) has a maximum value of 32. If the attack angle of a wing is kept at this angle during steady cruising, the airplane can realize the most economical flight, i.e., the plane can carry the maximum load with the minimum propelling force (minimum fuel consumption). Therefore, in addition to the maximum lift coefficient, the maximum lift-drag ratio is a particularly important
Dynamics of Wings in Real Fluids
35
property for a wing. The shape of a wing defines the characteristic curves for a wing: the thickness and camber relative to the cord length and the thickness distribution along the cord. These characteristics are measured mainly by wind tunnel experiments. It must also be mentioned that the Reynolds number affects the results. Thus the same wing profile will have very different characteristics when used for a small propeller airplane, and for a high-speed jumbo jet airplane. In other examples, when a wing is used for a blade cascade of a water wheel or of a wind turbine, an engineer has to develop an appropriate wing profile that is optimized to the conditions of use and the Reynolds number. This chapter has described the physics of real fluids, the boundary layer theory, laminar and turbulent flows, numerical and physical experiments and the movement of a wing in a real fluid. It is a pleasure for the author not only to introduce this basic knowledge of fluid dynamics to beginners, but also to allow senior readers who have studied fluid dynamics to rediscover the difficulties, and thus the interesting complexity and deepness of this field of the science.
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Chapter 4 Principles of Thrust Generation As already described in the introduction of Chapter 2, the Greek philosopher Aristotle considered fish swimming in water and birds flying in the sky and concluded that their motion was governed by the nature of fish and the nature of birds. He could not explain why an arrow - which is not a living animal - continues to fly against gravity. This mystery was finally solved by Galileo and Newton. Since the only external forces acting on the arrow are gravity and air resistance (drag), the flying speed of the arrow is gradually reduced. In this chapter, we will consider how fish swim against the drag and how birds continue to fly against gravity. Another question arises: how is the propelling force acting on a fish or a bird obtained? The correct explanation for this problem was accomplished after the development of Newtonian dynamics, especially after the establishment of fluid dynamics. Using Newtonian dynamics (Newton's three laws expressing three aspects of one theorem for motions and forces), an explanation of thrust generation can be obtained from the second and the third laws. When a fish or a bird applies a force to the surrounding fluid (water and air) by moving its body, the magnitude of the force equals the product of the affected fluid mass multiplied by its acceleration as seen in equation (2.2). On the other hand, according to the law of action and reaction, exactly the same magnitude of force acts on the fish or bird but it is oppositely directed. This is the principle of thrust generation. A fish pushes the surrounding fluid backward by wiggling its body and caudal fin, and thus obtains thrust. A bird pushes air backwards and downwards by flapping its wings, and thus obtains both thrust and lift simultaneously as a reaction to the air movement. It is easier to understand the concept of action and reaction by imaging a reader to jump up by giving a kick on the ground. The thrust and lift forces on an airplane have been described in Fig. 2.2, and the forces on a bird flying steadily are similar to those for an airplane. However, describing the acceleration of the flow around an easily deformable body is very different from that around a solid body, and the further developments in fluid dynamics for flexible shapes (especially real (viscous) fluid dynamics) are needed. The full and final solution of this problem has not yet been obtained just as real fluid dynamics has not been completed. In this chapter, momentum theory, slender body theory and
38
Chapter 4 Principles of Thrust Generation oscillating wing theory are explained within the limits of current understanding.
Momentum Theory A force acting on a fluid can be described by the change of momentum of the fluid. Thus even if the thrust that an object generates or the drag acting on it is not directly measured, the magnitude of the force can be determined by considering an arbitrarily sized control volume surrounding the object and investigating the net change of fluid momentum that flows in and out of the volume. This is the momentum theory. Figure 4.1 shows the momentum theory for a screw propeller. The propeller exists as an actuator disk at the center of the figure. The relative velocity of the fluid that flows into the propeller is denoted as V, and the velocity of the fluid flowing out is V+v. Provided the screw propeller moves in a static fluid, V is the forward speed, and v is the speed of the jet of fluid directed backward by the propeller. The area of the disk is S, the pressure in front of and behind the disk are p1, and p2 respectively, the pressure far from the disk is p_, and the fluid velocity passing through the disk is V+v'. The thrust T that the disk generates is given by equation (4.1) or (4.2). Equation (4.1) is due to the pressure difference between the front and the back of the disk, and equation (4.2) is the increase of fluid momentum through the control volume. Ignoring the energy loss due to viscosity and the rotational movement, Bernoulli's equation for the stream lines in the flow upstream of the disk is given by equation (4.3) and for the flow downstream of the disk by equation (4.4).
T=(p2-Pl)S
(4.1) (4.2) (4.3) (4.4)
(c.v)
V+v'
V+v
Fig. 4.1 Momentum Theory for a Screw Propeller
Momentum Theory
39
From the above equations it can be induced that v' is just one half of v (in equation (4.5)). The a shown in equation (4.6) is an important variable in the momentum theory, and is known as the axial interference factor. Finally, the thrust coefficient, power and thrust efficiency of a screw propeller are defined in equations (4.7), (4.8) and (4.9). v' =
(4.5)
v
V
(4.6)
2V
Thrust coefficient: CT
T =1/1
= 4a (1 + a)
(4.7)
Power: P = -pS(V + v'){(V + v)2 -V2} = ~pSV3-4a(l + a)2 TV
1
P
l +a
Thrust efficiency: 77 = - = -
(4.8) =-
2
,
(4.9)
Figure 4.2 shows characteristic curves of thrust efficiency and thrust coefficient versus the axial interference factor. The figure shows that the thrust efficiency is 100% when a is 0. Indeed, the propeller does not produce any force when a = 0 , and since the drag is also zero, the efficiency is 100%. To keep a high thrust efficiency, the accelerated flow speed v should be small compared to the advancing speed V. High-speed swimming fish seem to satisfy this condition. For a propeller on a ship, the thrust must overcome the drag on the ship body, and a cannot have a very small value. As a compromise, if a takes a value of 0.5 (thus making v equal to V), then CT is 3 and 77 is 2/3. It can be found that a modern highspeed boat such as a jet-foil tends to have a large a to obtain a large thrust. However, this is not a wise strategy in terms of fluid dynamic efficiency. To correlate the required thrust by momentum theory with the needed mechanical shaft torque of a propeller, it is necessary to use wing theory. Thus each blade of the propeller must first be divided into a number of blade elements along its longitudinal direction. Next, the partial thrusts and the partial torque of the blade element in each annular flow volume (whose rotational center is always the same) must be calculated by means of the circulation generated in each blade element. Finally, the net mechanical torque and power are determined by integrating over all the blade elements. This is known as blade element theory. Whereas momentum theory is a solution considering the momentum change in a macro view of a control volume, the blade element theory is a solution considering the flow around each
40
Chapter 4 Principles of Thrust Generation 1.0 rv
i
i
0.2
0.4
i
i
0.6
0.8
71 o
0.5
a Fig. 4.2 Thrust Efficiency and Thrust Coefficient of a Screw Propeller (Momentum theory)
blade element in a micro view. Therefore, the momentum theory can be viewed as a compulsory theory to determine the boundary condition in a combined application of blade element and momentum theory.
Slender Body Theory Although momentum theory precisely holds even for fish and birds, defining the actuator disk is extremely complicated because the mechanism of thrust generation depends on the body motion of the individual animal. As introduced in Chapter 1 , there are two swimming styles among fish that are considered to have high thrust efficiency. The first is the Anguilliform (eel swimming) mode, which involves oscillating (wiggle) the whole body from the head to the tail. The second is the Carangiform (mackerel swimming) mode, which limits the oscillation to the rear half or third of the body, and also utilizes a welldeveloped caudal fin. Both styles impart backward momentum to the surrounding fluid and the resulting reaction propels the fish forwards, which was discussed in the introduction of this chapter. The fluid dynamic efficiency is defined by equation (4.10), which uses the average thrust T generated by the fish, the swimming speed V and the average power P (the power that the fish puts into the surrounding fluid). This definition is completely identical to equation (4.9). Lighthill applied slender body theory (assuming that the ratio of the body width to the body length and the ratio of the flow velocity perpendicular to the direction of the motion to the swimming speed are both small) to an Anguilliform swimming motion to analyze the propelling mechanism as follows. Figure 4.3 shows a coordinate system with the origin fixed at the head of the fish. The fish is assumed to have symmetry about the x-z plane and to propel itself forwards with a horizontal oscillatory movement. If the displacement in the ydirection of the body centerline along the x-axis is denoted as h(x,I), then the lateral
41
Slender Body Theory
flow velocity of the surrounding fluid in y-direction is approximated by equation (4.11). This cross flow velocity v consists of an unsteady component plus a convective component. If the virtual mass of the cross section with area S per unit length is ma(x), then the sectional momentum in the j-direction is mavdx and the necessary force for the movement of this section is equal to the net change of the momentum, and equation (4.12) holds. / *\ = — v(x,t) dt
F =-
dh — dx -^~ dx
dt
(4.11) , , ,1 m a v) v \ dx J
(4.12)
displacement h(x,t)
cross sectional area S(x)
Fig. 4.3 Coordinate System for Swimming Motion Analysis (Slender body theory)
Multiplying the force times the lateral velocity of the body, dh/dt, and integrating the product with respect to x from 0 to / gives the total power P(t) that the fish provides_to the surrounding fluid, which is shown as equation (4.13). The average power P is obtained by integrating this equation over time, resulting in equation (4.14). Since this power consists of the effective power to propel the fish forward and the kinetic energy that is released and wasted downstream, the average thrust and the thrust efficiency of the fish can be calculated using equations (4.16) and (4.17). The above is a summary of Lighthill's theory. dh f'dh ( d_ d ^,, d\ ., -^-=7 -^-•\-z- + V—\(mav)dx dt Jo dt \dt .dt dx)
^
A
-^- + V—\\ — mav \dx-l —~mavdx ° \dt dx)\dt J ^o dt = 4-1 f'^m.vdx-lf'm.Sdx dt\Jo dt 2Jo tl Vv-^ " dt
1+vr^Lm.vT J idt Jo
(4.13) (4.14)
42
Chapter 4 Principles of Thrust Generation
(4.15)
17 = 1-
&L dt
(4.17)
Next, Lighthill considered the swimming motion as a progressive wave moving along the fish body of the form given by equation (4.18). Heref(x) is the distribution of lateral amplitude along the body axis x, and c in the function g is the propagating wave velocity. In this case, the average thrust and the thrust efficiency are remarkably simplified and are given by equations (4.19) and (4.20). (4.18) Figure 4.4 shows the thrust and efficiency in this case as functions of Vic. From the figure, thrust generation is a maximum at a swimming speed of zero with an efficiency (effective work ratio) of only 0.5. In addition, the efficiency is 100% when V/c is equal to 1 with a thrust of zero. According to momentum theory in the previous section, this corresponds to the situation where a is zero in Fig. 4.2 and thus the effective work ratio is also zero. In mechanical systems, such as pumps or waterwheels, this condition is known as 'runaway.' If as a compromise, V/c is taken as 0.8, then the efficiency becomes 90%. It is clear that this theory also assumes that the fluid is ideal and ignores various losses. Thus it should be noted that equations (4.19) and (4.20) represent the upper limits of thrust and efficiency.
(4 20)
'
Lighthill also considered the moment of force about key points along the body axis. He concluded that the most effective motion should: © limit the motion of the body axis to the rear half of the fish, and take the fish's mass there to be small. (D take Vic close to 1 so that v is not large compared to dhldt. This prevents large momentum changes in the rear half of the body where the virtual mass is large. (D most importantly, manage the wave form of the motion so that one positive phase and one negative phase always exist in the rear half of the body.
43
Oscillating Wing Theory 1.0
0.5
0.5 V/c
1.0
Fig. 4.4 Thrust and Efficiency vs. V/c (f'(/) = 0)
This minimizes the angular momentum reaction along the body axis. An example that fails to fill the third condition is the swimming motion of mosquito larvae. The more effective swimming style that primarily uses the tail or the caudal fin is the Carangiform (mackerel swimming) motion. The profile of the caudal fins of fish belonging to this style (horse mackerel, mackerel, tuna, dolphins and whales - although these last two are not fish) is largely protrusive in z-direction, having a crescent shaped wing with a high-aspect ratio. Thus the assumptions of a slender body cannot hold. For the swimming motions of such caudal fins, oscillating wing theory discussed below should provide more realistic analysis.
Oscillating Wing Theory Figure 4.5 shows the principle of thrust generation by a two-dimensional oscillating wing. The wing moves in a static fluid from right to left with speed U. In this case the wing performs a heaving motion, and a simultaneous pitching motion that has a phase difference of about p/2 relative to the heaving. As shown in the figure, if the heaving speed is v, then the flow speed relative to the wing is w which is found by vector addition of v and U. The direction of the relative velocity w coincides with the tangent of the locus of the wing movement shown as a dashed curve in the figure. Provided that the wing position relative to the vector w has a non-zero angle of attack, a circulation is generated around the wing and a lift force L perpendicular to w is generated. The vector component of the lift L in the forward direction gives the thrust T on the body of the fish, and the oscillating wing generates a thrust with two peaks per period. Since the relative flow around the wing changes periodically, it is easy to imagine that the flow downstream of the wing is complex, especially for unsteady changes of circulation. As described in the previous chapter, a negative starting vortex is generated downstream due to the Kutta-Joukowski condition as the wing obtains a positive circulation (a bound vortex). When the positive circulation
44
Chapter 4 Principles of Thrust Generation
U
L
Fig. 4.5 Principle of Oscillating Wing Propulsion
reduces its intensity a bound vortex is released and left behind downstream. Next, a positive starting vortex is generated downstream when the wing obtains a negative circulation, and a negative vortex is released downstream when the negative circulation reduces its intensity. This motion repeats to generate two pairs of vortices in each period. The generated vortex row develops into a 'Reverse Karman vortex street' with an induced velocity opposite the swimming direction. This unsteady state naturally affects the characteristics of the wing, and there is no guarantee that the generated lift on the wing has the same magnitude as the lift in a steady flow. For a two-dimensional oscillating wing, if the fluctuating speed V is small compared to the main speed U, there is an analytical method. However, in the case of large amplitude oscillations that may be accompanied by flow separation there is no general solution to date. Thus, computer analysis seems to be suitable for such flow fields, and these kinds of research have been performed. Figure 4.6 shows a two hinged oscillating wing mechanism used by the author. The first hinge and the second hinge are connected with an arm (an oscillating plate), and the arm oscillates with a rotational motion with constant amplitude centered about the first hinge. The wing is pin-jointed at the second hinge of the arm to make a rotation-free motion following the arm's motion with a relative angle of ¹. As described in the next chapter, this mechanism allows an elastic body such as a coil spring to be inserted inside the second hinge. Since the spring acts to reduce the relative angle b to zero, the wing can always keep an angle of attack relative to the flow velocity U.
Fig. 4.6 Model of Thrust Generation for Two Hinge Oscillating Wing Mechanism
45
Oscillating Wing Theory
Since the arm makes a sine-wave motion centered around the first hinge, the arm angle aa can be defined as equation (4.21). Where, A is the amplitude and tt>0 is the angular frequency (2p times the frequency f ).
aa = A sin ( w 0 T )
(4.21)
Although the relative displacement angle b of the wing to the arm is determined by the fluid force and the spring reaction, solving this problem to establish momentum equations is quite difficult because the fluid force cannot easily be determined (despite this, the actual fish performs it easily!). The author took a strategy of initially assuming a proper motion with an angle b ( t ) , and then determining the respective forces. This may not be an elegant method to deal with large amplitude motions, however, direct computer simulations adopt the same strategy as far as information is available. Assuming that the advancing speed U» is constant for a calculation means that the average thrust generated by an oscillating wing is exactly balanced by the average drag forces on the fish. This is a reasonable physical assumption. Equation (4.22) is a newly defined swimming number for this calculation, having the arm length R as a representative length. If aa(i), b (t), U, arm length R, half of the cord length c and wing's characteristics CL and CD are given, then the relative velocity, acceleration and respective forces on the wing can be calculated as functions of time t.
SwR =
f'R
(4.22)
\Rw0
Figure 4.7 shows the calculation results of thrust efficiencies. The results were extracted from the data of the most efficient motions b(t)oPT among an infinitive number of b(t) corresponding to given SwR. Both a flat-plate wing and a symmetric shape wing are shown in the figure. As a reference, other calculation results are shown in the figure in which the virtual mass is neglected and only the 100
80
6
°
40 20
4
6
SwR Fig. 4.7 Thrust Efficiency
10
46
Chapter 4 Principles of Thrust Generation
lift/drag components are considered. From the figure it can be seen that both a flatplate and a symmetric wing have maximum efficiency at a non-dimensional velocity SwR of less than 1 , and the efficiency linearly decreases in accordance with an increase of the velocity. A symmetric wing has an extremely high efficiency - above 80% for values of SwR of less than 10. In addition, the contribution to the efficiency by the virtual mass force is large for relatively small SwR. In contrast, the contribution becomes almost negligible for large SwK. From a number of calculations, the author notes that the lift and drag coefficients for larger SwR almost entirely govern the thrust force and thrust efficiency, and recently estimated the contributions analytically. It was found that for large SwK the average thrust coefficient can be expressed as equation (4.23), with the maximum value given by equation (4.24), and the efficiency by equation (4.25). It was thus clarified that the elements of thrust performance at high swimming numbers are the lift coefficient CL and the drag coefficient Cn of the wing. Generated thrust for the case of large SwK (analytical solution): An averaged thrust coefficient
4pA-
V
1 C: The maximum valueCTAmax~ —-— —
n~ Cn
Efficiency
T] « 1 - ,/*'*
(SwK)ra
CD
(4.23)
(4.24)
(4.25)
Where (Sw R ) ra is the swimming number at runaway (S W R )
=4A^-
(4.26)
CD
Figure 4.8 shows the estimated results described above. According to the figure, the averaged thrust coefficient of the oscillating wing has a maximum value at half the runaway speed, and the efficiency at this point is only about 50%. To obtain a reasonable efficiency 77 of 80%, SwR is taken to be about a fifth of the runaway speed. Applying this study to the previous Fig. 4.7, SwR is approximately 10. Furthermore, the study implies that if an arm of length R (corresponding to the length of the fish body undergoing lateral oscillation) is assumed to be a third of the body length, then even if fish swim with a swimming number of about 3.0, the fluid dynamical thrust efficiency could be kept at about 80%. It can therefore be confirmed that the cross-sectional profile of a dolphin's caudal fin has evolved into a beautiful symmetric shape in order to have a high liftto-drag ratio, and the Carangiform (mackerel swimming mode) type of oscillating wing mechanism is based on the lift force generated by the caudal fin. Figure 4.8 corresponds with Fig. 4.4 that described the slender body theory. Comparing both
47
Oscillating Wing Theory
- 100
500
-
80
400
[-
-60
300 200
h
0
20
0 0
*
100
0
10
20
30
40
SwR Fig. 4.8 Estimation of Average Thrust Coefficient and Thrust Efficiency (for large SwR)
theories, the slender body theory only considers the virtual mass force while the oscillating wing theory takes into account the lift and drag force characteristics, the contribution of which becomes definitive for large SwR. Comparing both figures at the runaway speed, both thrust factors become zero. However, the theoretical efficiency of the slender body theory at this point is 100%, while that of the oscillating wing theory is zero. Conversely, the theoretical efficiency of the slender body theory at zero velocity is 50%, while that of the oscillating wing theory is 100%. These results produce a new and interesting agenda for us. This concludes our description of the basic and current situation of fluid dynamics and the generating principles of the thrust requirements for fish and dolphins. The readers should now have the physical and mathematical background required to understand Gray's paradox.
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Chapter 5 Research on High-Speed Swimming Performance The reports that dolphins swim so fast that a theoretical explanation is impossible and that the U.S. Navy and the top fluid dynamics researchers were systematically investigating this mystery, made the author, who began working at University of the Ryukyus in his homeland in 1972, feel as if a mountain climber had heard of the existence of an unexplored mountain near his homeland. The Islands of the Ryukyus, also called Ryukyus Arc, form the fringe of the East China Sea. In 1972, University of the Ryukyus was the only university in Okinawa (located in the most southwest part of Japan) who had faculties of science and technology and it was promoted to the status of a national university in that year when Okinawa reverted to Japan from the U.S. As a researcher there, the author could not pass up the opportunity to choose this topic as a research theme at the university. The author's specialized category in fluid dynamics during his doctorial degree at Kyusyu University was high-speed aerodynamics, investigating super-sonic fluid flow and shock wave generation. The motion of a body in water was not a specialty, and neither was zoological biology. Thus, at the beginning, the author mainly managed his own degree thesis while collecting bibliographic references relevant to Gray's paradox. However, he found in a short time that there were only a few researchers in engineering fields who were wrestling with this problem, and his interest changed to a conviction that he could make his own contribution. One of the methods used to approach a new engineering problem is dimensional analysis. This chapter initially considers dimensional analysis, and then reviews remarkable research regarding the possibility of drag reduction.
Dimensional Analysis For fish to swim quickly, the fluid drag acting on the surface of the fish body must be small, or the power generated by the muscle must be large. This is because, as described earlier using an example of a cruising of tuna, the net force on a body that performs a constant linear motion has to be exactly zero, and thus the drag acting on
50
Chapter 5 Research on High-Speed Swimming Performance
a fish body and the thrust generated by a fish must balance each other. The muscles of fish must generate more power than the value of the 'thrust1 multiplied by the 'velocity' - thus the 'drag' multiplied by the Velocity.' The relationship between the maximum speed of a fish and its body length is investigated by using dimensional analysis in this section. Firstly, it is seen that fast swimming fish, from small fresh-water fish (such as trout and carp) to large aquatic mammals (such as dolphins and whales) have similar body profiles. The common characteristics among these swimmers are that the body shape is streamlined or spindle shaped to reduce drag, and the thrust is generated by means of 'Carangiform' (mackerel propelling) motion or 'oscillating wing motion.' Provided that the body shapes of these animals are similar, the body surface area S and the body mass m are in proportion to the square and the cube of the body length / respectively. The power generating capability of a fish is in proportion to the volume of muscles, and if the ratio of muscle volume to weight of these animals is assumed constant, then the generated power P is proportional to the mass. In addition, the fluid drag is assumed to consist only of surface friction drag. Then, if the relationship between the friction drag coefficient cf and the Reynolds number is distinctly expressed for the two cases of laminar and turbulent boundary layer flow on a fish body surface, the relationships between the velocity and the body length of fast swimming fish can be conclusively expressed as equations (5.8) and (5.9). It is generally true that larger fish swim faster, however, according to these equations, it is not a linear relationship, and the velocity increases in proportion to the body length to the power from 0.4 (for turbulent flow) to 0.6 (for laminar flow). Dimensional Analysis (Relationship between swimming speed and body length of fast swimming fish):
Sa/2 ma/3 Pam
(5.1) (5.2) (5.3)
D = cr~pU2S
(5.4)
where, for a laminar flow, c, <* Re --2 =(I Ul — \~* V V )
(5.5)
_\_
and for a turbulent flow, c,
\ v)
while,P=7- U = D. U thus, the following two results are obtained for a laminar flow, U a / 3/5 for a turbulent flow, U a 1 ^
a—-I
(5.6) (5.7) (5.8) (5.9)
Possibility of Drag Reduction - Flexibility of the Surface Skin
51
Figure 1.1 in Chapter 1 plotted the relationship between body length and velocity for various swimmers on a double logarithmic graph. The single-dashed line and the double-dashed line show the two cases described above. In addition, five lines that decrease towards the right of the graph are each lines of constant Reynolds numbers. It is seen from the figure that most data are in the range of high Reynolds number of between 105 and 109. As described in Chapter 3 the boundary layer on a flat-plate transitions to a turbulent boundary layer if the Reynolds number exceeds 3xl05. From the figure, for fish whose body length is larger than 30 cm and whose swimming speed is greater than 1 m/s (carp for example), it is reasonable to assume that the flow in the boundary layer on the fish's body surface is turbulent. According to the figure, a whale with a body length of 30 meters swimming at 10 m/s can be explained by extrapolating the data of carp, trout and dace. However, this extrapolation does not apply to dolphins of around 2 meters in length that swim as fast as whales. The speeds of dolphins seem to have a relationship of U a / with the continuous (cruising) swimming ability of carp. Reported maximum speeds of bonito, tuna and marlin have another relationship U a / with the instantaneous (burst) swimming ability of small fresh-water fish. As described in Chapter 1, the dimensionless swimming number (discovered and named by the author) takes an empirical value of about 0.6 for dace and goldfish, 0.7 for carp and 0.8 for dolphins. However, the observed results that the maximum speed is proportional to the body length regardless of species are still not explained by the analysis. This is another expression of Gray's paradox. To understand why the swimming speed is larger than the estimation, three possible explanations are suggested as follows: (1) The drag coefficient on the fish body surface is less than the value calculated by equation (5.5) or (5.6). (2) The power per unit weight of an aquatic animal is not constant as assumed by equation (5.3). Thus the power increases at a rate greater than the cube of the body length. (3) The fluid dynamic efficiency of Carangiform (mackerel propelling) motion or oscillating wing propelling motion may be higher for larger animals. For a number of researchers who wrestled with this problem, the possibility of reduced drag was the first to gain focus and to be investigated. Let us now discuss this possibility in detail.
Possibility of Drag Reduction - Flexibility of the Surface Skin In 1960 in Los angels, U.S.A., Dr. Kramer (Max O. Kramer) manufactured a special skin using silicon rubber and silicon oil to imitate the skin of dolphins. He covered a rocket shaped model with the special skin and performed towing tests in seawater. Subsequently, he obtained the surprising result that the test model exhibited a reduced drag of up to 60% compared to a model with a solid surface. Figure 5.1 shows the structure of the special elastic surface skin. Silicon rubber comprised the
52
Chapter 5 Research on High-Speed Swimming Performance
skin's three layers, of which the inner and the outer layers were joint-less hoses. A circular rubber film with cylindrical protrusions was sandwiched between the inner and outer layers. The rigidity of the film was varied by controlling the viscosity of silicon oil that filled the space between the protrusions. Figure 5.2 shows the results of the frictional drag coefficients for various Reynolds numbers. The two straight lines in the figure are the theoretical values for a laminar boundary layer and a turbulent boundary layer. The curved line A is the case of an experimental solid model. From line A, it can be seen that the experimental Reynolds numbers are in the transition region, where a turbulent boundary layer exists downstream of a laminar boundary layer. The special elastic skin was set to different levels of softness in order of B, C and D. An obvious drag reduction was found in case C, which was regarded as preventing the boundary layer transition to turbulent. Kramer's experiment caught worldwide attention, and a number of similar experiments were executed. Regrettably, however, such distinct
mm. Fig. 5.1 Structure of Special Elastic Surface Skin (Kramer's rubber film)
4
6 8 10
20
40
60 80100
Fig. 5.2 Results of Kramer's Experiment (A is a rigid body, B, C, D are soft surface skins in order of increasing softness)
Toms Effect
53
results have not been repeated. In Japan, Professor Taneda (S. Taneda, Kyusyu University) et al. attempted towing experiments in a tank with a two-dimensional flat-plate covered with a Sofrun (chemical sponge) sheet. Although the experiments were conducted with a Reynolds number of 105 - 107, the results showed that the drag coefficient of a flatplate with a soft skin was always larger than that of a solid surface. Professor T. Tagori (1927 - 1990, the University of Tokyo) et al. conducted experiments on various cloths used for swimming suits and reported that for a cloth containing elastic fibers, the drag of a flat plate covered with extended cloth in high tension was always less than the drag of a plate covered with loose cloth. The author also attempted an experiment on a vinyl flag set in a flow-circulating tank that will be described later. However, the phenomenon that the softness or fluttering of the flag reduced the flow drag has not been confirmed. Kramer's experiment also caught the attention of researchers in theoretical fluid dynamics. The stability problem of small fluctuations with a flexible boundary in accordance with stability theory for turbulence was researched, and also gave negative results. A soft surface skin and a flag passively change their profiles when a fluid flows over them. If this is defined as 'Passive control,' then to actively change the surface profile to control the flow is known as 'Active control.' An attempt was made to use active control of a boundary layer flow in order to produce a drag reduction. Taneda, previously noted, added propagating waves to a plate surface and precisely observed the boundary layer flow. As a consequence, he reported that if the propagating speed of the waves was slower than the main flow speed, the flow separated downstream of the wave amplitude peaks. In contrast, when the wave speed was faster than the main flow speed, the flow did not separate and periodically released reverse Karman vortices downstream of the plate. For an actively controlled skin, as demonstrated by Taneda's experiments, it is possible that a surface skin, that performs a propagating wave motion at a speed exceeding the main flow speed, accelerates the main flow, and thus obtains a thrust due to the reaction. This result is similar to that for the thrust generating mechanism of slender body theory. However, measuring only the drag force on a skin that is generating thrust is impossible so far. Similarly, a jet exhausted from the gills of a fish seems to accelerate the main flow, however, the details of the effect are not clear. For the next possibility of drag reduction on a fish body, the effects of the secretion of mucus are raised. How does the slimy substance on a fish relate to drag?
Toms Effect It is well known that solutions of certain kinds of long-chain molecules have quite low flow drag - this is known as the Toms effect. In 1948 in the Netherlands, Toms (A. B. Toms) performed a pipe flow experiment with solutions of polymethlmethacrylate-monochlorobenzene (0.625 - 2.5 g//), and clarified that these solutions have a reduced pressure loss compared to solutions of simple mono-
54
Chapter 5 Research on High-Speed Swimming Performance
chlorobenzene. The Toms effect excited researchers, and many experiments followed the initial discovery. In the field of ship fluid dynamics, Dr. Hoyt (J. W. Hoyt) and Dr. Fabula (A. G. Fabula) performed a disc rotating experiment using a number of polymer solutions to investigate the reduction of frictional torque to determine what kind of polymers exhibited the Toms effect. According to their results, the common characteristics of effective polymers such as Polyethylene oxide are that the molecular weight should be of more than 106, the molecular structure should be a linear chain connection, and the polymer should be water-soluble. In 1969, Professor Tagori wrote an explanation of 'Frictional drag reduction by a polymer solution1 in an article in the journal of the Society of Naval Architects of Japan, in which he precisely described the achievements of his empirical and theoretical research over the past 20 years. Since Tagori also confirmed the Toms effect for mucus on the skin of loach, and directed field tests with a motorboat, he was a pioneer in this field of study in Japan. In his commentary chapter, he tells about an episode where he obtained and was cultivating single-celled algae from Hoyt that exhibited the Toms effect, and he had lost the culture during a school conflict at the University of Tokyo. In 1973, two Americans, Drs. S. C. Ling and T. Y. Ling, sampled body surface mucus from several kinds of fish living in the Chesapeake Bay in the State of Maryland. They found that the mucus from a small-flat fish (Trinectes Macutatus: a kind of wrasse) was made of glycol-protein and exhibited similar physical properties to a polyethylene-oxide solution (Polyox 301) made by Union Carbide Co. They used both solutions to conduct experiments to confirm the Toms effect. The mucus of fish was obtained by putting many fish one by one in a shallow bucket to extract the mucus naturally. Since the amount of mucus was limited, and since many experiments used large volumes of mucus, polymer solutions were used as well. The experiments were conducted by releasing fish mucus or polymer into a pipe flow of water through 40 small holes (diameter at 0.06 cm) drilled in the pipe wall. The inner diameter of the pipe was 1.27 cm (1/2") and the length was 2 m. They measured the pressure drop along the length of the pipe. Figure 5.3 shows their results. The experiment used 5 different Reynolds numbers from 5,500 to 74,000, which generally belong to turbulent flow regimes. The curved line at the top of the figure is the case where only fresh-water is released, and the pipe-friction coefficient / corresponds quite well to the Prandtl equation (5.10). The two curved lines at the bottom of the figure are the results of two cases where mucus or polymer solutions were released from the wall. The upper curve is for a 50 ppm polymer solution and fish mucus at 120 ppm (6%). The lower line is for a polymer solution greater than 500 ppm and fish mucus at more than 1,500 ppm (75%). As a result of these measurements, it was confirmed that the former solution reduced drag by 40% while the later solution reduced drag by 60%. In the former case, with a comparative low concentration, the drag is reduced dramatically with an increase of polymer concentration. In the latter case, with high polymer concentration, an increase in concentration does not further reduce drag but
55
Toms Effect 10-1
turbulent flow
10"
103
104 Re
10s
Fig. 5.3 Experiment Results of the Pipe Friction Drag Coefficient for Fish Secretions and Polymer Solutions
converges at a 60% reduction. In addition, this experiment used an extremely accurate laser-Doppler flow speed indicator that had the ability to measure the velocity distribution in the viscous sub-layer existing under the turbulent boundary layer. According to the results, the drag reduction by a low concentration polymer solution could be achieved without increasing the thickness of the viscous sub-layer. Prandtl Equation
p= = 2.
(5.10)
Figure 5.4 shows the velocity distribution adjacent to the wall on a singlelogarithm graph. The vertical axis u+ and the horizontal axis v+ are the nondimensional velocity and non-dimensional distance from the wall respectively, and they are defined by equations (5.11) to (5.13). In these equations, vw is the kinematic viscosity of water, and r0 is the wall shear stress. In all three cases, (fresh-water, low concentration polymer or low concentration fish mucus, and high concentration polymer or high concentration fish mucus), the velocity distribution adjacent to the wall falls on the same curve regardless of the different Reynolds numbers. Thus, the velocity distribution is self-similar and has a typical turbulent boundary layer structure. The point at which u+ equals to y+ adjacent to the wall is the viscous sub-layer, and the region far from the wall where u+ is proportional to lny + is typical of a turbulent boundary layer. u =
(5.11)
y -
(5.12)
56
Chapter 5 Research on High-Speed Swimming Performance
/ ^ -"1 p ^7 / ^-
N
1*=
pm.Pblyox301 -I >l500ppm. ^»H "(>75%) mucus
30
~
1000
Fig. 5.4 Dimensionless Velocity Distributions in a Boundary Layer
where, u* = to.
(friction velocity)
(5.13)
P
Ling and Ling discussed the thickness of the viscous sub-layers, and they discovered an 'anomalous' layer in which the velocity gradient du+ldy* was greater than 1. This occurred at a y+ of around 10 for a low concentration solution and around 20 for a high concentration solution. They presumed that this anomalous layer was generated by the presence of polymer or fish mucus, and it became a major cause of drag reduction in accordance with an increase of the viscous sublayer thickness. Furthermore, since the viscous sub-layer was quite thin and became thinner with increasing flow velocity, the rate of diffusion of mucus was almost constant regardless of the flow velocity, and the volume of mucus required would be quite small. If the suggestion of Ling and Ling is correct and if live bonito and tuna reduce their swimming drag by 60% because of their mucus, then the power required to achieve the same velocity with the mucus would be two-fifths of that required without the mucus, and thus the gap of Gray's paradox would become much smaller. However, further detailed research, following these or successive reports of this kind of engineering applications, have not been found to date. On the other hand, for dolphins, the existence of such mucus or similar effects has not been reported.
Effect
ofRiblets
57
Effect ofRiblets When one seeks numerous possibilities to solve the problem that the friction drag of fish is much smaller than imagined, he or she will face a mysterious phenomenon the shark. As is widely known, sharks are so aggressive that they sometimes attack humans, and people have a fearful image of them. However, here we will consider the shark as an academic object. There is a native expression for 'sharkskin' (that means sandy texture) in Japanese, and the shark's sandy skin has been well known for a long time. Figure 5.5 shows a photograph of sharkskin published in Dinkelacker's (A. Dinkelacker) report. As can be seen in the photo, there are many small protrusions lying side by side and the direction of the protrusions seem to align with the flow direction. For this shark, the height of the protrusions is about 0.1 mm, and the space between them is the same as the height. These minute protrusions cover the whole shark's body. A model of the whole shark body is shown in Fig. 5.6. The arrows in the figure indicate the direction of the fine longitudinal protrusion (from front to rear; the arrows do not show the number of
Fig. 5.5 Photograph of Sharkskin (shield scales) (A. Dinkelacker, et al., Proc. IUTAM Symp., Bangalore, India, 1987)
Fig. 5.6 Conceptual Picture of Protrusions on a Sharkskin (A. Dinkelacker, et al., Proc. IUTAM Symp., Bangalore, India, 1987)
58
Chapter 5 Research on High-Speed Swimming Performance b
Fig. 5.7 Modeled Figure of Surface Riblets
protrusions; there are many more of them). Figure 5.7 is a cross sectional view of the protrusions, taken perpendicular to the flow direction. These features of protrusions are sometimes called 'grooves' to emphasis the indented parts on the surface, and they are also known as 'longitudinal ridges' or 'riblets.' Why does a shark have such a skin? The conventional idea is that since the protrusions can be felt with the hand (even though they are minute) their existence creates a roughness that increases drag. Therefore, the protrusions would not exist to reduce the drag but for another reason. For instance, considering the aggressive nature of sharks, the surface skin may have become tough like a shield for protection while fighting with enemies. Actually these protrusions are named shield scales in Japanese and placoid scales in English. Since the latter topic belongs to zoo-ecology, let us leave it and go ahead. Regarding the former question, the answer is an unexpected conclusion - the protrusions reduce the turbulent friction drag. Dinkelacker et al. manufactured a surface imitating sharkskin inside a circular pipe, and performed drag experiments. Such pipe drag experiments are the most basic method for measuring flow drag as described in the research of Ling and Ling. The friction drag on the inner surface of a circular pipe equals the pressure difference between the inlet and outlet times the cross sectional area of the pipe in the case of fully developed flow. Measuring the pressure difference precisely, the effect on frictional surface drag with and without protrusions can be determined. The results of Dinkelacker et al. are shown in Fig. 5.8. For Reynolds numbers between 104 and about 4 x 104, the flow with protrusions has slightly lower drag than that without (in this case the difference is reported as about 3%). According to the figure, the surface with protrusions shows larger drag at high Reynolds number, and shows little difference from the smooth pipe at low Reynolds numbers. The height of the protrusions and the space between them is also a factor, and thus the situation is quite complex. Considering the original conception that the protrusions create roughness and therefore the drag should increase and not decrease, this is considered as a very mysterious phenomenon. The relationship between the height of the protrusions, their spacing and the velocity range that gives drag reduction, is somewhat sensitive. The Reynolds numbers based on the friction velocity u and the protrusion spacing b (refer to Fig. 5.7) and the Reynolds number based on the friction velocity and the protrusion height h should both have a value of several tens. In that case, the swimming speed of a shark would be between 5 - 1 0 m/s, and this is the velocity corresponding to quick movements when a shark feels in danger or is feeding. After all. there is
Effect ofRiblets
59
Fig. 5.8 Flow Drag for a Pipe with its Inner Surface Coated with Microscopic Protrusions 1: measured for a smooth surface circular pipe 2: theory for a smooth surface circular pipe O, X: measured for a coated surface (A. Dinkelacker, et al., Proc. IUTAM Symp., Bangalore, India, 1987)
nothing like the mystery of nature. Research on riblets was begun in 1970 by a team lead by Walsh (M. J. Walsh) at NASA Langley Research Center. Walsh and Weinstein (L. M. Weinstein) presented their first thesis in the AIAA Journal in 1978. In their report, the fluid drag and heat transfer of a longitudinally ribbed surface (in the flow direction) were investigated. The aim of the experiment was clearly described to control the flow by limiting expansion of turbulent bursts in the lateral direction within small areas near the wall. Later, results were quoted in many theses, and the influence of the profile, spacing and height of the minute protruding ridges lying parallel to the flow direction was precisely investigated. Subsequently, the characteristic properties have been almost fully clarified, and it has been confirmed that, if the flow properties meets certain conditions, the drag can be reduced by a maximum of about 8%. The reason for this drag reduction is considered to be the same reason that initially motivated the studies by Walsh and Weinstein. At the beginning of a turbulent burst, the protrusions regulate the width and growth of three-dimensional
60
Chapter 5 Research on High-Speed Swimming Performance
hairpin vortex filaments (also known as banana vortex). Thus the turbulent structure is changed in a way that leads to a drag reduction. The Americas Cup yacht race is a practical example of the application of riblets. This race has a 150 year history, with the first event in 1851. An American club had won the race every year, and the Americas Cup had not moved outside the U.S. until 1983. In 1983 an Australian boat beat the undefeated American team by using a yacht named 'Australia-II,' which was constructed with high-tech designs, and finally brought the cup to the southern hemisphere. This achievement became big news. This yacht was equipped with a fin attached to the keel, which had been tested but never used until the 1983 race. It was said that the fin greatly improved performance under the maximum waterline regulations. This incident inspired interests in European style boats. An honorable American yacht club re-designed their ship profiles using numerical fluid dynamics in addition to more traditional design tools. The ship subsequently built was named 'Stars and Stripes' and was fully covered with a riblet film. The film was made of fine grooves, and was developed by NASA in conjunction with 3M Co. in the U.S. The film used on 'Stars and Stripes' was an adhesive tape-sheet (12 inches in width, 75 inches in length and 0.18 mm in thickness) and was designed for a speed of 4 - 8 knots. It was attached so that the longitudinal grooves followed the flow direction. The mechanism of the frictional drag reduction has been described before, and the fluid should flow parallel with the longitudinal grooves. The effect of riblets became famous because they made it possible for an American club to bring the cup back to the U.S. Since the incident, the use of riblets on the surface of yachts has been prohibited in the Americas Cup races. The reader may also remember the sharkskin swimming suits used by the Australian team at the Sydney Olympics in 2000. Of course, they aimed the riblet effect to make the swimming motion more effective.
Studies in Japan In 1988, I. Tani reassessed the possibility of drag reduction using microscopic surface roughness distributed on the surface (described above as riblets), by carefully reanalyzing the data from the experiments of Nikuradse (J. Nikuradse) for sand roughness on the inside surface of a circular pipe that was reported 50 years previously. The roughness height was no more than y+ = 5 in the viscous sub-layer of a turbulent boundary layer, which was regarded as smooth enough in conventional fluid dynamics. Since the success of the 'Stars and Stripes' and Tani's reappraisal, reports that confirmed the riblet effect have been recently produced in Japan. Professor H. Ohsaka et al. (Yamaguchi University) and Professor Y. Kohama et al. (Tohoku University) have independently confirmed that, by using a netpatterned roughness, the turbulent friction drag coefficient could be reduced by 2 5% at a roughness Reynolds number of about 2.0. Furthermore, Kohama suggested that the small spaces formed between riblets catch slow speed fluid-blocks to create an effect whereby the blocks are not lifted by longitudinal horseshoe vortices. This
Studies in Japan
61
subsequently weakens the bursts (ejections) that are the main cause of turbulent energy generation. In addition, Professor T. Nakahara et al. (Tokyo Institute of Technology) reported that after coating the inner surface of a pipe with fine fur the pipe drag was dramatically reduced. The effect of riblets as well as the Toms effect seems to be controlled by still veiled fluid dynamics phenomena of quite delicate structure. The difference between ideal fluid dynamics and real fluid dynamics, the difference between slip and no slip on a wall, has been clearly explained by considering the molecularity of the fluid. The reduction of the frictional drag of a turbulent boundary layer over rough surfaces (a mysterious phenomenon at first sight) will also be understood by considering the molecularity of the fluid. However, it should be remembered that molecularity in this case refers not to the single molecule level (e.g. H2O) but rather to the 'fluid particle1 level (a collection of an infinite number of molecules). This means that the behavior of 'fluid particles' in fluid dynamics should be considered (refer to Chapter 3 for a definition of the fluid particle). It is suggested that the molecular weight for polymers that show Tom's effect is approximately 1,000,000 and this is far greater than the molecular weight of 18 for a single H2O molecule. The friction drag on a ship is affected by small changes in surface conditions, such as the paint quality on the bottom of the hull or the presence of algae, and also by the properties of seawater. This has been a difficult problem for many years in the shipbuilding association and related industries. If there is a dramatic explanation of the flow (as Ling and Ling suggested for polymers) in an anomalous layer adjacent to the viscous sub-layer and if it is consistent with the dimensions of polymers having a molecular weight of 106, modern science will also unveil its mechanism in the near future.
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Chapter 6 High-Speed Swimming Method of Carp and Dolphins As Aristotle reasoned, the swimming of fish in water is governed by 'the Nature of fish.' There must be a maxim of "Learn swimming from fish." The first effort that the author undertook to solve Gray's paradox was to prepare a special water tank as Gray's investigations inspired. The author's tank was a full-scaled circulating water tunnel constructed with engineering principles. This chapter introduces several research efforts conducted by the author over the last 25 years.
Small Circulating Water Tunnel and the Measurement of Drag on Fish Figure 6.1 shows a circulating water tunnel for observation of fresh-water fish. The design and construction were done by undergraduate students, K. Shinzato and K. Tokashiki (graduated in 1974, Mechanical Engineering Department, Faculty of Science & Engineering at University of the Ryukyus). Shinzato devoted an additional year as a postgraduate researcher to complete the tunnel. The test section has the dimensions of 210 mm x 210 mm and a length of 1,000 mm, and is made of transparent acrylic that enables the flow inside to be observed. The flow velocity is variable from 0 to 2.3 m/s. This tunnel can also measure the fluid drag acting on a fish body. Figure 6.2 shows a specially made towing-drag detector. The theme of Shinzato's dissertation was 'Research into the drag on a soft plate put parallel to the flow.' He put a vinyl flag in the test section of the tunnel, and investigated the drag and flapping movements of the flag. As described before, the flapping flag in a flow has a larger drag than the flag with no flapping (rigid plate) in every case, which was against our expectation. Successive juniors to Shinzato have measured the drag on spindle shaped fish models and real fish that had a soft surface skin. Figure 6.3 shows the scene while measuring the drag on a carp. Prior to this experiment, the authors tried to measure the drag on seawater fish (sea bream, yellowtail, saury, etc.) purchased in the fish market and anesthetized fresh-water fish. However, reliable data was not obtained. Controlling the posture of dead or anesthetized fish was so difficult
64
Chapter 6 High-Speed Swimming Method of Carp and Dolphins
—1600-j-U: I 1™
2100
MOOO-M 000^1096 -L-1500~ LL° ..._ _ 8556
Fig. 6.1 Small Circulating Water Tunnel (University of the Ryukyus)
r
Oscillograph
Fig. 6.2 Towing Type Drag Force Sensor
because the body immediately started a pitching or rolling motion with an increase of the flow speed, and only unstable data resulted. In contrast, since living fish control their posture themselves, stable data was obtained. It must be noted that the adopted drag data was the data that was sampled when fish were in a static state. To calculate the drag coefficient of the tested fish, body profiles such as body length, sectional and surface areas must be accurately measured. Students started on photographing and sketching the fish body.
Small Circulating Water Tunnel and the Measurement of Drag on Fish
65
Fig. 6.3 Carp Undergoing Drag Measurement (The towing force while the specimen is motionless in the flow was regarded as the drag)
Figure 6.4 shows an example sketch of a carp body. Based on this data, the cross-sectional area A (= 7tabl4) and the perimeter / were calculated (/ must be calculated by a computer) from the cross-sectional profiles (sliced in distances of 1 2 cm). Each section was regarded as an ellipse with major diameter a and minor diameter b. Finally, the 'hydrodynamic equivalent diameter' de (the fluid dynamically equivalent circle diameter) was determined using equation (6.1). „ . , ,. . , Equivalentt diameter de
4 x A(cross-sectional area) nab ^ —- = / (wetted perimeter) /
Fig. 6.4 Sketch of a Carp (/ = 28.6 cm)
(6.1)
66
Chapter 6 High-Speed Swimming Method of Carp and Dolphins
Figures 6.5 and 6.6 are the profile diagrams of carp and tilapia. The dashed lines in the figures are the actual body profiles (outlines of the major diameters a), and the solid lines are those of the computed hydrodynamic equivalent diameters. The solid lines are, in other words, profiles of axial symmetric (spindle shaped) objects that are fluid dynamically equivalent to the tested fish.
Fig. 6.5 Body Profile of a Carp (Solid line is the equivalent spindle shape) l=23.4cm
1
Fig. 6.6 Body Profile of a Tilapia (Solid line is the equivalent spindle shape)
The surface area S of the fish was estimated by equation (6.2) - the surface area of the spindle shaped object plus the area of the dorsal, 5,, anal, S2, and caudal, S 3 ,fins. Table 6.1 shows measurement profiles for 22 examples of the tested fish. The data is sorted in order of increasing body length for each species. The weights are based on actual measurements. It was found that the average ratio of the maximum equivalent diameter demax to the body length / was about 0.21 regardless of the species. This value will be discussed later and almost coincides with the ratio of the amplitude of the oscillating caudal fin to the body length. In addition, if the ratio of the cube of the body length to the fish weight is calculated as an index of the slenderness of the fish, then the average value is 60 for tilapia and 55 for carp as seen in equations (6.4) and (6.5). We note that there is a slight difference in the slenderness index between these two species.
dedx den
+ S, +S,
= 0.21
Tilapia: — -60 cm 3/ gf W
(6.2) (6.3) (6.4)
Table 6.1 Profiles of the Tested Fish
Species Tilapia Tilapia Tilapia Tilapia Tilapia Tilapia Carp Colored carp Colored carp Carp Carp Carp Carp Carp Carp Carp Carp Carp Colored carp Carp Carp Carp
Body length /(cm)
Max. equ. dia. de max (cm)
Surface area S (cm2)
21.3
4.27
23.2
5.94
23.4
4.56
23.8
4.77
Max. area Sab (cm2)
24.4
4.78
27.7
6.16
12.2
2.52
230.13 322.73 261.42 352.03 283.62 401.07 87.69
4.04
188.57
13.47
4.02
186.17
14.19
16.38 30.61 18.96 21.04
20.98 33.37 5.47
17.0 17.5
223.1 351.2
34.5 114.2
l3/w
S//2
demaxll
(cmVgf)
0.507 0.600 0.477 0.621 0.476 0.523 0.589
0.200 0.256 0.195 0.200 0.196 0.222 0.207
60.43
0.616
0.231
0.527
0.214
122.2
18.5 18.8
Weight W(gf)
123.0
60.52 52.63 40.24 46.93 51.48
19.5
134.3
55.21
19.8
134.4
57.76 59.48
4.10
211.16
14.27
132.5
27.7
6.25
27.9
6.02
423.15 490.15
32.30 30.10
28.6
5.31
396.68
36.2
7.00
19.9
0.533
0.206
458.1
0.551
388.9 480.2
0.630
0.226 0.216
0.485
0.186
25.7 27.5
316.0
28.0
40.0
23.81 43.51
842.2 1,109.0
0.193
65.81 46.40 55.84 45.71
56.33 57.71
68
Chapter 6 High-Speed Swimming Method of Carp and Dolphins
Carp:l/w= w
55 cm 3/ gf
(6.5)
Figure 6.7 shows plots of the friction drag coefficient on fish bodies measured in the circulating water tunnel. The horizontal axis is the Reynolds number whose representative length is the body length. The drag coefficients on the vertical axis are calculated by equation (6.6), where S is the surface area of the fish determined by equation (6.2). The Reynolds numbers for tilapia was limited in a narrow range compared with carp because at lower Reynolds number flows (low speed), tilapia tended to make caudal fin and the body oscillations and thus provided few stable data. At high Reynolds number flows (high speed), posture control became difficult and the body started rolling and measurements also became impossible. D
(6.6)
These oscillations result because the major axis of tilapia's cross-sectional profile is comparatively larger than that of carp. Considering the fluctuation of the data for tilapia, the measurement results plotted for tilapia at each flow speed are the average values with their dispersion indicated. According to the figure, for Reynolds numbers of less than 2x 105, the drag coefficient for both carp and tilapia decreases with an increase of the Reynolds numbers. For Reynolds numbers of more than 2x 105, both approach a constant value of about 0.02. The measurement values of CfD for carp tend to be smaller than the average for tilapia. Although this tendency seems to be considerable at higher Reynolds numbers, it is not decisive U1U
l
Q05
i
J • Carp $ Tilapia i A Mechanical fish ~j
* • r
-
•i V 1,1 f 1 A
<§
^^sCfD=0.\ 10(/te/)" ""
<* A
n IT ft ^ft S fe^r
• •
4
h
0.005
10
Fig. 6.7 Frictional Drag Coefficients on Fish Bodies (Obtained by circulating water tunnel measurements)
!
Swimming Motion of Fresh-Water Fish
69
due to the limited data. For a mechanical fish having a metal surface, the drag coefficient is clearly less than that of carp and tilapia. This is especially remarkable at higher Reynolds numbers where the friction drag coefficient drops to around 0.008 at a Reynolds number of about 106. The straight line lying below the line for the mechanical fish in the figure is the friction drag coefficients of a flat-plate that was described in Chapter 3, and is a semi-empirical equation for a turbulent boundary layer. According to the equation (C fD = 0.074(Ret)-1/5 ), the friction drag coefficient is 0.0064 at a Reynolds number of 2xl05. The measurement values for carp and tilapia are three times of this value. Readers might wonder about the justification of a direct comparison between the drag of a plate and that of a spindle shaped object. We, anyway, could not confirm that fluid drag on a living fish was less than that on a rigid body in this experiment. Nevertheless, the drag measurements for sharks and dolphins whose Reynolds numbers can reach no less than 107 have not been performed, and this problem still remains as a future agenda.
Swimming Motion of Fresh-Water Fish Next, we attempted direct and precise observations of the swimming motion of fresh-water fish by using the circulating water tunnel. The experiments were performed for all carp and tilapia shown in Table 6.1. A fish was put in the test section of the tank and allowed to swim freely. The current speeds were increased by steps and the swimming motions were recorded by a movie camera. The test fish were observed to swim easily upstream against the flow as expected. A flowrectifying mesh upstream and a metal net downstream of the test section restricted the fish to swim in the test section. The swimming motion was recorded on 16 mm color film at 48 frames per second from the same angle as shown in Fig. 6.3. As shown in Fig. 6.3, a mirror inclined at a 45 degrees angle was set beneath the test section to simultaneously record the side and bottom view of the swimming fish. Figure 6.8 shows the time-split swimming movements of a carp. The figures were extracted from every fourth frames in the film and the oscillation period of the fish body was 0.6 seconds. The relative flow speed (i.e. swimming speed) was measured at 45.90 cm/s, and the wave speed of the longitudinally propagating wave along the body was measured at 60.66 cm/s. If the ratio of the flow speed to the wave speed (0.76) is put into equation (4.20) for the efficiency in slender body theory in Chapter 4, the efficiency becomes 0.88. Figure 6.9 is a superimposed picture that consists of the centerlines of the carp from the same recorded swimming sequence. From this observation, the following four points are clarified: (1) The oscillating amplitude of the caudal fin is about 20% of the body length, which coincides with Bainbridge's observations for dace and trout. (2) The maximum hydraulically equivalent diameter of the body and the oscillating amplitude of the caudal fin have approximately the same magnitude.
70
Chapter 6 High-Speed Swimming Method of Carp and Dolphins
Fig. 6.8 One Period of the Swimming Motion of a Carp (U = 45.90 cm/s, f= 1.7 Hz)
18 20
{/=45.90cm/s =7.00cm b/l=0.199 Fig. 6.9 Superimposed Images of a Carp During Swimming (Time between each image is 1/48 sec)
(3) The movement of the body axis mainly occurs along the rear part of the body and the gradient of the oscillating amplitude at the tail is almost zero, which coincides with that of the efficient slender body motion that Lighthill suggested. (4) According to the observations, the mouthpart of the swimming fish also oscillates slightly from left to right. Figure 6.10 shows the swimming speed of carp, in which the horizontal axis is the oscillation frequency of the caudal fin, and the vertical axis is the specific speed (the ratio of the swimming speed to the body length). Although the body lengths of the fish varied from 12 to 40 cm, it was found that the experimental results could be arranged by the specific speed regardless of the body length.
Mechanical Fish - An Invention of Oscillating Wing Propulsion Mechanism
1
2
3
4
5
6
71
7
Fig. 6.10 Swimming Speed of Carp (Body length 1=12-40 cm)
Although the measurement data were scattered, the average value could be approximated with a linear regression almost passing through the origin. Bainbridge reported that there are many cases in which the linear regression equation had a positive cut-off on the x-axis (i.e. the caudal fin is idling at zero speed). Another obvious tendency is that the scatter of the data points from the averaged linear line is greater at higher oscillating frequencies. This is due to the unsteadiness of the swimming motion at higher frequencies. When fish are accelerating against the stream, the specific speed is lower than the average specific speed; when fish are decelerating by the stream, the measured speed is higher than the average. The above is the story of how the 'Swimming number1 Sw was found. In addition to carp, experiments for tilapia were also conducted. As a result, we concluded that the swimming numbers of these two fish (carp and tilapia) were 0.695 and 0.576 respectively. On the other hand, Bainbridge reported swimming numbers for dace, trout and goldfish under faster swimming condition as 0.63,0.62 and 0.61 respectively. Thus, the three fish tested by Bainbridge have a superior swimming ability when compared to tilapia but a poorer swimming ability when compared to carp. Since the swimming number of carp is 10% greater than that of the other fish tested, carp can be considered to have superior swimming ability.
Mechanical Fish - An Invention of Oscillating Wing Propulsion Mechanism The next step of our research on small fresh-water fish was to produce a mechanical fish that swam similar to actual fish. The idea of a mechanical fish arose naturally from mechanical engineers who joined the research. Figure 6.11 shows a drawing of a small-scale mechanical fish that was first made at University of the Ryukyus in
72
Chapter 6 High-Speed Swimming Method of Carp and Dolphins
Fig. 6.11 Small-Scale Automatic Mechanical Fish (University of the Ryukyus, 1978)
1978. The body shell consisted of the head, the body, the tail and the caudal fin, each of which was made from an aluminum block or plate. The head and the body shell have spaces for batteries, a motor and a Scotch-yoke mechanism, which were joined with screw bolts to make a waterproof seal. There were two hinges: one connecting the body and the tail, and another connecting the tail and the caudal fin. Both hinges were free to rotate around the joints. A silicon rubber film (part (?) in the figure) with the thickness of 1 mm was attached between the body and the tail to seal against water. In addition, a coil spring (part ©) was inserted between the tail and the caudal fin to reduce the relative rotational angle between them. The elastic support of the caudal fin with springs is the characteristic of this mechanism, and was the first invention in the author's research life. The details of the thrust generating mechanism were described in the section on 'oscillating wing theory' in Chapter 4. The body shape and the caudal fin of this mechanical fish are based on an actual tuna. The model has an overall length of 416 mm (body length of 390 mm) and a maximum diameter of 78 mm, which is almost the same size as a carp. The oscillation frequency of the tail and the caudal fin (i.e. the rotational speed of the DC motor) is remote-controlled by a radio on shore. A purpose of producing this mechanical fish was that, if a 'fish with a known propelling force' was available, then the difficulty of making measurement on a 'self-propelling body' that accompanies a complex correlation between the drag force and the propelling force might be avoided. The completion of this mechanical fish was mainly accomplished by graduate students T. Ikemiya, K. Ohta, and H. Nakachi (all are now working for Okinawa Electric Power Co.). The construction was especially benefited by Ohta's craftsmanship, as well as technology provided from the workshop of the Department. Figure 6.12 shows a series of pictures taken during one period of the swimming motion in the small circulating water tunnel obtained by an 8 mm-film camera. The period of the body oscillation is about 0.2 seconds (frequency of 5
Mechanical Fish - An Invention of Oscillating Wing Propulsion Mechanism
73
Fig. 6.12 Swimming Motion of a Small-Scale Mechanical Fish (U » 1.2 m/s, VII « 3.1 cm, /= 5 Hz)
Hz), and the advancing speed is about 120 cm/s. The single dashed line in the figure indicates the advancing speed. Comparing this figure with Fig. 6.8, it is seen that the mechanical fish advances by nearly identical movements as the carp. However, looking carefully at the motions, it is found that the lateral displacement from the centerline at the head and the body is larger than that of the carp. The absence of the dorsal fins in the mechanical fish seems to cause the increased lateral motion. The crescent shaped fins attached to both sides of the tail stabilize the straightforward movement of the mechanical fish. Figure 6.13 ((a) - (d)) shows the relationship between the specific speed and the oscillating frequency of the caudal fin. These experiments were done under free-swimming conditions in a competition size swimming pool. The oscillating frequency was adjusted by radio control. Figure (a) is the result using a coil spring whose spring constant k is 0.78 gf/mm, which is the softest among the springs used. According to the figure, the fastest specific speed of 1.25 is recorded at an oscillating frequency of 2.8 Hz, and the specific speed gradually decreases with an increase of the oscillating frequency to become 0.78 at the maximum oscillating frequency of 6.34 Hz. Thus, if the oscillating frequency increases, the advancing speed does not increase, which is contrary to the case of carp and dolphins. As
74
Chapter 6 High-Speed Swimming Method of Carp and Dolphins
Fig. 6.13 Specific Speed versus the Oscillating Frequency of the Caudal Fin of a Small-Scale Mechanical Fish (k is the spring constant)
clearly seen from comparing with the other cases, this behavior is because the spring constant is too small (i.e. the fluid force acting on the caudal fin exceeded the reaction of the spring) to retain the relative angle of attack within an effective area. As a reference, the case without a coil spring was also attempted, and then the mechanical fish moved slightly backward. Figure (b) is the case for the spring constant of 7.67 gf/mm with the frequency between 5.3 - 6.3 Hz, in which 13 experiments were conducted. In this case, it is seen that the advancing speed increases in proportion to the frequency. Since there is a considerable scatter of the data, the average gradient (average swimming numbers) by the least-square method and their divergence were calculated and shown in the figure. In this case, the average swimming number was determined as 0.538 ± 0.039. Figure (c), the case of a harder spring with a constant of 13.5 gf/mm, shows a wider frequency range from 3.7 to 5.75 Hz and more stable data with an average swimming number of 0.596 ±0.040. In the case of the strongest spring constant of 22.4 gf/mm shown in Fig. (d), it is obvious that the obtained frequencies are plotted in a very narrow region, and the scatter of data is relatively large. The average swimming number in this case was recorded at 0.668 ±0.087, which was the largest among all tests performed. The reason why data exceeding 5 Hz could not be obtained is that the electric motor had insufficient power to overcome the increased fluid reaction forces when the average wing angle of attack to the relative water flow became large. The small mechanical fish that has a body length of 39 cm achieved a maximum swimming speed of 130 cm/s at an oscillation frequency of 5 Hz. It was
Mechanical Fish - An Invention of Oscillating Wing Propulsion Mechanism
75
found that this swimming ability cannot reach the ability of dolphins (Sw = 0.818) and carp (Sw = 0.695) but can exceed trout (Sw = 0.62) and dace (Sw = 0.63). On the other hand, it was also found that the efficiency (the net power to tow the mechanical fish divided by the power of the electric motor) did not exceed 23%, which is not a surprisingly high efficiency. In other words, even if the power transmission efficiency is estimated as about 0.5, the fluid efficiency for the caudal fin is no more than 50%, which is not a very high efficiency. Although we expected that these experiments and calculations for swimming efficiency would provide hints to unveil Gray's paradox, the expectation was in vain. One of the reasons is considered to be that the generated specific power relative to the weight of mechanical fish was still too small when compared with the maximum power estimated from the weight of an actual fresh-water fish, and thus the flow speed used in the experiments did not reach the flow speed at which Gray's paradox would occur.
THE ARIZONA REPUBLIC
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Fig. 6.14 Interview Article with the Author at the ASME Meeting (Arizona Republic, Nov. 19 1982)
In a regular winter meeting of the American Society of Mechanical Engineers (ASME) held in November 1982 in Phoenix, Arizona, the author presented a series of research works including the small mechanical fish. The Arizona Republic, a local news company, interviewed the author and published an article describing his research. This article made the attendance in the society focus on the author's research. Professor Triantafyllou (M. S. Triantafyllou) at Massachusetts Institute of Technology, described later, was one of them. Figure 6.14 is a copy of the article in the Arizona Republic. The heading and leading line of the article are humorous and also very inferable: Experts study why some fish swim faster than fact; "Nature's laws say some fish cannot swim as fast as they do. The fish haven't heard this, so they swim very fast indeed." Since the interviewed article written by Mr. Hodge was the only article relevant to the ASME meeting on that day among all newspapers, it seemed like an advertisement for the ASME meeting.
76
Chapter 6 High-Speed Swimming Method of Carp and Dolphins
Swimming Motion of Dolphins The next step, after gaining some fruitful achievements and confidence through observations for small fresh-water fish and producing a small mechanical fish, was the observation of living dolphins and producing a full-scale robot fish similar to a dolphin. Let us start by describing the Okinawa Memorial Park Aquarium in which observations of dolphins were conducted. In 1975, the International Oceanic Exhibition was held in Okinawa. The exhibition was located at the shore area in Motobu-town at the northern part of the mainland of Okinawa, and the site occupied an area of about 77 hectares. After termination of the exhibition, the entire site was rearranged and became the National Okinawa Memorial Park, in which an aquarium, a botanical garden (the Tropical Dream Center) and the Aqua-police (this was closed permanently and towed away in 2000) were the main facilitates. Both theaters, the 'Oki-chan's theater' and the 'Dolphin Studio,' are affiliated with the aquarium, and these facilities provided worthy opportunities to investigate the swimming motions of dolphins. Figures 6.15 and 6.16 are photographs of the 'Oki-chan's theater1 and the 'Dolphin Studio.' The former is an ellipse shaped seawater pool, with a major diameter of 25 m, a minor diameter of 15 m and a depth of 4 m. The latter is a
Fig. 6.15 Scene of Oki-chan's Theater and the Tank Dimensions
Fig. 6.16 Scene of Dolphin Studio and the Tank Dimensions
77
Swimming Motion of Dolphins
circular shaped seawater pool, whose diameter is 14 m with a depth of 3 m. Both facilities are connected to each other with a pipe (a bypass for dolphins). Fresh seawater is exchanged at a rate of 300 tons every hour, and is provided to both theaters and the main aquarium. As seen from Fig. 6.16, the sidewall of the Dolphin Studio facing the audience is a flat-plate wall with a width of 8 m so that the audience can observe the swimming movements of dolphins through a transparent acrylic window (width 7 m and height 1.8 m) fitted on the wall. Table 6.2 shows the body properties for the pacific white-sided dolphins, pacific bottle-nosed dolphins, bottle-nosed dolphins, and false killer whales observed. The designations, for example, such as Oki(l) and Oki(2) indicate the body properties as of June 1979 and November 1995 respectively. For Oki, it can be read that she grew and increased her body length by 24 cm and her weight by 59 kgf over the 16 years. By the way, the name of 'Oki-chan's theater' derived its nickname from the then popular baby dolphin. Table 6.2 Body Profiles of Dolphins Species
Nickname *
Sex
Body length /(cm)
Pacific white-sided dolphin (Lagenorthyncus obliquidents)
Moto (1) Toppo (2) Pal (2) Oki(l) Kuro(l) Poy(l) Poy (2) Oki (2) Muk(l) Dan (1) Dan (2) Kuro (2) Muk (2)
F M
200 202
F F M M M F M M M M M
Tok (2) Shiro(l) Pea (2) Gon (2)
Pacific bottle-nosed dolphin (Tursiops truncates aduncas)
Bottle-nosed dolphin (Tursiops truncates gilli) False killer whale (Pseudorca crassidens)
P/W (cmVgf)
206 209 225 230 235 233 236 237 258 258 259
Weight W(kgf) 89 96 102 114 130 138 168 173 120 148 202 171 175
F M
262 277
218 264
M F
347 361
483 483
82.50 80.51 86.51 97.40
89.89 85.86 85.75 80.08 87.62 88.17 77.25 73.12 109.54 89.95 85.02 100.43 99.28
* Numbers inside the brackets indicate the date measured ((1) June 1979, (2) November 1995).
From the table, the average value of 13/W (this is an index for the body profile and slenderness) for pacific bottle-nosed dolphins is 88.7. If the result is compared with the previously described tilapia (60) and carp (55), then these dolphins are more 'slender,' and this conclusion is against the author's expectation. Although the number of data is small, the indexes for bottle-nosed dolphins (81.5), false killer whales (92.0) and pacific white-sided dolphins (88.0) are more or less close to that of the pacific bottle-nosed dolphins. These results imply that these kinds of large sea animals (dolphins and whales) are sufficiently slender and they have almost similar body shapes.
78
Chapter 6 High-Speed Swimming Method of Carp and Dolphins
Fig. 6.17 Swimming Motion of a Dolphin During One Period (U = 2.4 m/s, /= 1.41 Hz) Figure 6.17 is a series of pictures of a dolphin (Poy(l)) swimming. Every fifth picture was extracted from a 16 mm color film that was recorded at 48 frames per second. The movie was taken in the Dolphin Studio. Although the actual swimming movement was performed upside down (backstroke), the pictures are shown as a normal view. For this example, the swimming speed is 2.4 m/s and the period is 0.708 second. Since the body length is 2.3 m, the swimming number is calculated as 0.74. Although measurement of the propagating wave speed along the body axis direction was attempted as it was for carp, it could not be reliably deduced due to the asymmetric movement of the body axis. Figure 6.18 is a superimposed picture that consists of instantaneous pictures (every third frame of the movie, thus 1/16 of a second between images shown) for the movements of the body axis during one period. From this picture, the following characteristics that are far different from fresh-water fish are observed:
Swimming Motion of Dolphins
79
flow
U=2.4m/s
f
= 1.41Hz
b/l = 0.332
Fig. 6.18 Superimposed Images of a Dolphin (Time between each image is 1/16 sec)
(1) The oscillation of the body center axis and the caudal fin is not horizontally symmetric. (2) The oscillating amplitude gradient of the rear part of the body is quite steep compared with fish. (3) The oscillating amplitude reaches 33% of the body length, which is quite large compared with fish. (4) The oscillating pattern of the body center axis is a first-mode oscillation, and a node occurs near the maximum sectional area. The amplitude of the head is comparatively large. In terms of evolution, dolphins are mammals that may have lived on shore at one time in their evolving process, and their bones are far different from those of fish. Such biological background may have caused the differences in swimming method as described above. It is clear that the swimming method of dolphins does not agree with the effective movement that Lighthill suggested in his slender body theory. The author understands the swimming motion of dolphins as a typical form of oscillating wing propulsion - the swimming method does not generate a thrust force along the whole body as is adopted among slender fish but generates thrust only by heaving and pitching motion of the caudal fin. The author also considers that the part of the body behind the dorsal fin acts as an arm that provides the heaving motion to the caudal fin. As seen from the frontispiece-3, this arm is flattened to reduce the drag against its vertical movement. The caudal fin acts as an oscillating wing and is attached perpendicularly to the end of the 'arm-plate.' By the way, it should be noted that the dolphin's pectoral fins evolved from the mammalian hand and thus they have bones inside. In contrast, the dorsal and the caudal fins do not have such bones. In the process of evolution, dolphins and whales have retained high-speed swimming, and such demands made muscles develop independently to obtain large dorsal fins and wing shaped caudal fins without bones. In addition, the cross-sectional shapes of both fins are fluid dynamically high-performing symmetric wing profiles. Figure 6.19 shows the relationship between the oscillating frequency of the caudal fin and the specific speed, which was derived from the filmed records in the Dolphin Studio. Although the number of dolphins and the measurements were limited compared with those of carp shown in Fig. 6.10, this figure also supports the
80
Chapter 6 High-Speed Swimming Method of Carp and Dolphins
f (Hz)
Fig. 6.19 Swimming Speed of Dolphins (Measured in : 1979. •: 1995)
study that the swimming speed of dolphins is in proportion to the body length and the oscillating frequency of the caudal fin. In the figure the white marked plots (unfilled symbols) are the results measured in 1979, and the black marked plots (filled symbols) are the results obtained in 1995. There are some groups of points from the 1979 data that have the same specific speed but different frequencies. This means that a specified dolphin varied the oscillating period of the caudal fin and maintained the same swimming speed. In other words, dolphins might keep a fixed swimming speed while changing the oscillation frequency over a range of values. From the figure, the swimming number, which shows the extent of the swimming ability of pacific bottle-nosed dolphins, was determined as 0.818. This result is comprehensively greater than goldfish and tilapia (both about 0.6) and carp (about 0.7), which verifies a higher swimming ability for dolphins as expected.
Maximum Swimming Speed of Dolphins Following the observations in the Dolphin Studio, movies were taken and analyzed for numerous jumping behaviors in the Oki-chan's theater. Figure 6.20 shows superimposed pictures when a dolphin jumped up and over a bar. The body outlines were extracted from every tenth frame (208 millisecond interval between images) of a film taken at 48 flames per second. The observed dolphin (Poy(l)) had a body length of 2.3 m and a weight of 138 kgf, and the maximum height of the center of gravity of the body was measured at 2.73 m. The maximum instantaneous speed during the jumping occurs when the dolphin escapes from the water surface, and the three calculation methods to determine this speed are as follows: (1) by the maximum reachable height h (v0 = 2gh ) (2) by the duration of the jumping T (vn = gT/2)
Maximum Swimming Speed of Dolphins
81
Fig. 6.20 Superimposed Pictures of a Bar Jump by a Dolphin (/ = 2.30 m, W = 138 kgf)
Fig. 6.21 Vertical Jump of a Dolphin (/ = 2.77 m, W = 264 kgf)
(3) by the duration needed for the whole body to escape the water surface (the body is used as a scale) We attempted all three methods, and confirmed that the deviations among them were about 10% at maximum. Since the third method seemed to be the surest, this method was adapted finally. This method took the smallest value among the three measurement methods, and the maximum speed was judged as 6.50 m/s. Thus, the specific speed was 2.83. Figure 6.21 is another picture for a high-jumping dolphin (Shiro(l)) that has a body length of 2.77 m and a weight of 264 kgf. In this picture, although the instance of escaping from the water was not caught, the initial speed calculated by
82
Chapter 6 High-Speed Swimming Method of Carp and Dolphins
the maximum reachable height (4.52 m) of the center of gravity of the body was 9.41 m/s, which gives a specific speed of 3.40. As introduced in Chapter 1, Lang recorded a maximum speed of 11.05 m/s for a pacific spotted dolphin whose body length was 1.86 m. Allegedly, dolphins in the open sea are said to swim at no less than 15 m/s. The author requested the aquarium staff for special performances to record the maximum possible speed close to the hearsay evidence. However, an escape velocity of 9.41 m/s recorded for a high jump by Shiro was the maximum record, which means, contrary to our expectation, data exceeding 10 m/s could not be obtained at that time. There were two periods of earnest measurements conducted at the Okinawa Memorial Park Aquarium in 1979 and 1995. The second measurement period was conducted to obtain additional data needed for an earlier book of 'Fluid Dynamics of Drag and Thrust' that was produced by the Ship and Ocean Foundation. The authors' interest since the beginning of the publishing plan was in the maximum speed of dolphins. They visited the aquarium again after 16 years, and collected much new information. Professor Tanaka, Assistant Professor I. Teruya (University of the Ryukyus), Y. Kina, T. Nakai, Y. Harada and Z. Yosimine (all graduates/ postgraduates of University of the Ryukyus), Mr. Y. Fukui and Ms. Y. Kikuchi (a head of Ship & Ocean Foundation and a secretary) and the author constituted 9 members of the visiting staff to the aquarium on November 23 and 24 in 1995. Mr. S. Uchida (the director of the aquarium), Mr. T. Nagasaki (the vice-chief of the breeding section) and other staff, who had established numerous records such as the Guinness record for the longest breeding of a whale shark, showed comprehensive attitudes for the series of investigations, and responded to data collection need in the same good spirit as they did 16 years ago. Some surprises for the research team were that 'Oki' had grown up and increased her body length and weight, and that most dolphins continued as performers for 20 years. According to Mr. Uchida, the most difficult thing is to retain the performance level over the long years. As far as observations, the aquarium's breeding skills are at the top level when compared to others all over the world. Table 6.3 shows data for the swimming ability of dolphins obtained during the second measurement period. The measurements were conducted in the Dolphin Studio, and the performers were Kuro and Muk (pacific bottle-nosed dolphins). Videotapes were used to record and analyze the swimming ability this time. The method used to determine the escape speed was unified to adopt the third method in the previous description (a dolphin's body length was put as a scale, and the speed was calculated from the duration to pass a certain point). The instantaneous speed and the frequency at jumping were determined when the dolphin escaped from the water surface. From this table, the average swimming number of dolphins was 0.79 for backstrokes, and 0.88 for high jumps. The average swimming number for the total of 8 performances is 0.81, and this value agrees well with the average swimming number of 0.818 shown in Fig. 6.19. Through these experiments, the superb swimming ability of dolphins and the reliability of the measurement methods
Maximum Swimming Speed of Dolphins
83
Table 6.3 Swimming Ability of Dolphins (at Dolphin Studio, 1995) Dolphin name Kuro(2) , „ ~o / = 2.58 m Muk (2) , „ ^r, / = 2.59 m
Performance Backstroke
, f , Backstroke
D
Average of Backstrokes Kuro (2) High jump Muk (2) High jump Average of High jumps
Speed U (m/s)
VII
/(Hz)
Sw
4.46 3.56 3.20 5.17 4.50 4.33 4.20 7.74 7.35 7.55
1.73 1.37 1.24 2.00 1.74 1.67 1.63 3.00 2.83 2.92
2.01 2.07 1.88 2.26 2.10 2.07 2.07 3.17 3.44 3.31
0.86 0.67 0.66 0.88 0.83 0.81 0.79 0.95 0.82 0.88
were confirmed. In addition there was a new observation that the swimming numbers for high jumps (under higher oscillating frequency) are considerably high. The swimming numbers at high frequencies in the 1979 measurements were very low and led to reduce the average value. This is due to the difference of the measuring conditions and the methods. During the second occasion (1995), we measured the speed and frequency at jumping from the underwater side. Assuming that the 1995 measurements are more reliable than the 1979 measurements, dolphins retain a swimming number of about 0.82 up to the high frequency of 3 Hz. Naturally, the maximum speed of animals including tuna and marlin is determined by the maximum frequency and the swimming number. The record for a killer whale's (body length of 7 m) maximum speed is 15.4 m/s in the British Guinness Book of Records, and this leads to an oscillating frequency of the killer whale's tail of only 2.7 Hz, with an assumed swimming number of 0.82. If the swimming number is smaller, then the oscillating frequency must be larger. Table 6.4 Maximum Speed of Dolphins (at Oki-chan's theater, 1995) Nickname
Body length Km)
Performance
Speed U(m/s)
Moto (2) Toppo (2) Tok (2) (1)* Gon (2)
2.00 2.02 2.62 2.77 3.61
Vertical jump Screw jump Bar jump Vertical jump Vertical jump
8.57 8.99 7.74 9.41 7.47
Estimated frequency (Hz) (Sw = 0.82)
4.29 4.45 2.95 3.40 2.07
5.23 5.43 3.60 4.14 2.44
* Data for Shiro was obtained in 1979.
Table 6.4 shows the maximum speeds of dolphins recorded by a video camera in the Oki-chan's theater. The judgment method to determine the swimming speed was the same as for the previous table (the escape speed from the water surface). Each speed shown in the figure is the maximum speed among plural measurements. In the table, the data for Shiro was collected in 1979, as described with Fig. 6.21 (Shiro passed away regrettably in 1983).
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Chapter 6 High-Speed Swimming Method of Carp and Dolphins
The maximum speed measured in the 1995 tests was 8.99 m/s, and no speeds exceeding 10 m/s were obtained. However, this record made by Toppo (a pacific white-sided dolphin) was the biggest fruit obtained this time, and the specific speed of this small dolphin (with a body length of about 2 m) reached 4.45 body lengths per second. Assuming that the swimming number is 0.82, which may be a high value, the estimated frequency is calculated as 5.43 Hz. Indeed, there was information that this dolphin swam fast, and this was the first opportunity for the author to collect data for this species. Since the two pacific white-sided dolphins were brought into the facility after the previous investigation in 1979 and since they joined the other players after training in 1984, the measurements for them were of keen interests in this research. The readers might be immediately aware that if the maximum oscillating frequency of 5 Hz is achievable for other larger dolphins, their maximum swimming speed would easily exceed 10 m/s. Using an oscillating frequency of 5 Hz for Shiro (bottle-nosed dolphin, body length of 2.77 m) and Gon (false killer whale, 3.61 m), their maximum swimming speeds are estimated as 11.4 m/s and 14.8 m/s respectively. Another point the author noticed at the second investigation was the difference in the size of the Dolphin Studio and the Oki-chan's theater. Not only the depth but also the area of the pool considerably affects the maximum escape speed. The influence is not so clear when comparing Table 6.3 with Table 6.4. Nevertheless, it was found that in the Oki-chan's theater, when dolphins were ordered to perform higher jumps, they took longer approach distances and times. When Toppo was ordered to perform a screw-jump, he took so long of an approach distance and time that the audience could not imagine where he would came out of the water; and he suddenly flew into the air and drew a big arch using the whole theater space (refer to the frontispiece-1). If the observation regarding the approaching distance strikes the essence of the issue, both the estimated maximum frequency for the pacific white-sided dolphin and the estimated maximum speeds for Shiro and Gon described above can have their basis.
Estimation for the Power of Fresh-Water Fish and Dolphins Let us estimate the power of fresh-water fish and dolphins in this last section of this chapter. The drag and power of a carp with a body length of 28 cm and a weight of 390 gf is estimated in Table 6.5. Assuming that the oscillating frequency of the caudal fin is 5 Hz and the swimming number is 0.7 at maximum speed, then the swimming speed and the Reynolds number are 0.98 m/s and 2.74x 105respectively. According to Fig. 6.7, the friction drag coefficient and the fluid drag acting on the carp are 0.0179 and 43 gf respectively, and the power that the carp consumes to propel itself and the power per unit weight are estimated as 0.413 W and 1.06 W/kgf respectively. If the calculated results are approximated, a carp can achieve a swimming speed of 1 m/s by expending energy at a rate of about 0.5 W. Even considering the
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Table 6.5 Typical Body Profile and Estimated Power of Carp Weight: W= 390 gf Body length : / = 28 cm Maximum cross sectional area of the body : Sab = 30.0 cm2 At maximum cross sectional area, major dia. (a), minor dia. (b) : a - 7.5 cm, b - 5.1 cm At maximum cross sectional area, equivalent dia.: de = 6.0 cm S = 490 cm2 Body surface area (including fins) : 3/ 3 / W=56cm /gf S/WM = 9.18 /=5Hz Oscillating frequency £7 = 0.98 m/s Speed (U = Swxfxl) Re = 2.74xl05 Reynolds number D = 43 kgf Drag C>=0.0179 Friction drag coefficient (C/= 0.219fo,-'/5) Thrust power (P = DxU) P = 0.413 W P/W= 1.06 W/kgf Generated power per unit weight
efficiencies for energy transmission to the muscles and for the swimming motions, the energy consumption per unit weight of a carp is estimated to be around 2.0 W/kgf, and the described swimming speed is achievable. Since the power generated by a human or a dog per unit weight is about 10 W/kgf, there is enough margin in the power consumption for a carp to swim fast. Therefore, Gray's paradox for carp does not exist. Table 6.6 Typical Body Profile and Estimated Power of Dolphins Body length : / = 2.3 m
Weight W =138 kgf / 3 /W=88.2cm 3 /gf Estimated body surface area : 5 = 2.45 m2 (calculated from S/W23 = 9.18) Swimming number : Sw = 0.82 Oscillating frequency of the caudal fin : / 4 Hz 5 Hz 7.54 m/s Speed : U 9.43 m/s 1.73x 107 Reynolds number : Ret 2.17xl0 7 3 5 7.81 x 10Friction drag coefficient : C/ (= 0.219 Ret " ) 7.44 x lO-3 Drag : D 52.9 kgf 82.7 kgf Thrust power : P 3.91 kW 7.64 kW 28.3 W/kgf 55.4 W/kgf Generated power per unit weight : P/W 2.89 m/s 5.65 m/s
Next, let us consider a dolphin with a body length of 2.3 m and a weight of 138 kgf with a typical body profile. Table 6.6 shows these properties including other elements. As described earlier, if 13/W in this table and that in Table 6.5 are compared, then dolphins are more slender than carp. To estimate the body surface area, we adopted the method that the surface area was not in proportion to I2 but to W2'3. Thus, the body surface area S in Table 6.6 was determined by calculating from the value of S/W2'3 for carp. As a consequence, this method prevents
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overestimation for the generated power per unit weight of dolphins. As seen in the previous section, the swimming number for dolphins is estimated to be about 0.82, and the maximum speeds are scattered between 7 - 1 0 m/s. Here, let us estimate the power of a dolphin under two different oscillating frequencies of the caudal fin, namely, 4.0 and 5.0 Hz. According to Table 6.6, the Reynolds number of the dolphins at maximum speed is about 2 x 107, and the friction drag coefficient is about 0.007 or 0.008 and the total fluid drag is between 50 - 80 kgf. For the caudal fin frequency of 4.0 and 5.0 Hz, the power consumed for swimming is 3.91 kW and 7.64 kW respectively. The speed ratio between the two frequencies is 1.25 (7.54 m/s at 4 Hz, and 9.43 m/s at 5 Hz), while the ratio of power demand is enlarged to 1.95. The reason for the difference between the speed ratio and the power ratio is that the power is in proportion to the cube of the speed. The power per unit weight is calculated as 28.3 W/kgf (4 Hz) and 55.4 W/kgf (5 Hz), which are 27 and 52 times as large as that of carp (Table 6.5). Lang, et al. managed observations for three dolphins (body length of 1.86 2.09 m, weight of 52.7 - 91 kgf), which were slightly smaller than the dolphin discussed here. They estimated the power consumptions to be between 2.5 - 4.5 kW. Assuming the average weight is 90 kgf, the power generated per unit weight would be 28 - 50 W/kgf, which coincides with the author's estimation. The traditional rule of thumb in the physiological sciences is that the power generated per unit weight by a human ranges between 8 and 10 W/kgf. For example, if a sport player, whose weight is 60 kgf, pedals a bicycle connected to an electric generator, then he can generate a maximum power of about 500 W. Tani suggested that humans are able to instantaneously generate 1 HP (736 W) using their whole body. Using the same 60 kgf human and Tani's power estimate, the power per unit weight is 12.3 W/kgf. The author also attempted his own examination by having a student (23 years old. with a height of 177 cm and a weight of 74 kgf) dash up the external stairs of a building at University of the Ryukyus. As a result, the student generated a maximum power of 589 W. Thus, the estimated maximum power is no more than 750 W for humans or other mammals such as dolphins. It is interesting to note that a power of 736 W is known as a horse power! It is almost certain that the dolphin illustrated in Table 6.6 could swim at a speed of 7.5 m/s, and retaining this speed for 1 minute seems likely. In this case, the generated power per unit weight is 28.3 W/kgf, which is three times as large as that for humans, and Gray's paradox still exists here. Figure 6.22 was prepared to rearrange the problem. The variables taken on the vertical and the horizontal axes are the same as Fig. 1.1. Data points are shown for only 4 species; typical carp (from Table 6.5), dolphins (from Table 6.6), blue whales and martin (both from Fig. 1.1). The solid line running to the upper right side is an estimated line of the maximum swimming speed under the assumption that the power per unit weight is 10 W/kgf. Since all body shapes are assumed similar to
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Estimation for the Power of Fresh-Water Fish and Dolphins
30
10
•8 8,
0.5
1
5
10
50
body length / (m)
Fig. 6.22 Relationship between Swimming Speed and Body Length (Dashed and solid lines are estimation of maximum speeds on condition that the unit power of swimmers is 10 W/kgf)
dolphins (or carp), and since the boundary layer on the body surface is considered as a turbulent boundary layer, the relationship between the maximum speed and the body length is approximated as U = 3.55/3/7. The slope of this line is the same as the corresponding line in Fig. 1.1. Comparing the estimated line and the observation results in the figure, the speeds of carp and blue whales are lower than the estimation, which implies that their muscle power is sufficient to overcome the drag. For dolphins and the marlin, the observed speeds are two or more times faster than the estimation. Replacing the friction drag coefficient Cf= 0.219/te"5 (Table 6.6) with that for a rigid body model (the mechanical fish in Fig. 6.7) of Cf = 0.110/?e 1/5, the estimated speed is approximated as U = 4.54/3/7 which is shown as a dashed line in the figure. By assuming a rigid body, the expected speed increases by 30%, however, the observed values for dolphins and the marlin cannot yet be explained. Therefore, as a conclusion of this chapter, the author again declares that Gray's paradox is not yet solved. Towards further research - although it sounds very difficult - there remains two agendas: proving that the drag on these aquatic animals under high-speed swimming condition is very low, and/or showing that these animals can generate extremely large power per unit weight. Episode - Rapid Increase of the Body Temperature of Fish Tani asked himself whether fish could generate a power of 60 - 70 W/kgf. He then described that "some people wonder that if a fish generates several times as much power as a human, then the heat generation and the oxygen demand must become
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Chapter 6 High-Speed Swimming Method of Carp and Dolphins
extreme, and the heart and the blood vessels must be overloaded." From his description, he seems to deny the possibility for his own question. Concerning this topic, the author obtained surprising information that bonito are heated as if they are burning after high-speed swimming. An account of the circumstances of these observations is given here. Every year Mr. H. Teruya (the chief of the fish breeding section of the Aquarium in the National Okinawa Memorial Park) goes aboard a bonito-fishing boat owned by a nearby organization to collect bonito and tuna for exhibition use in the aquarium. The following sentences are descriptions about the fishing scene, quoted from part of a column named 'Kara-jishi (an imaginary lion which was inherited from China)' in the Okinawa Times (It is a slightly long passage to make the readers feel the reality of the fishing events): "The sea surface started agitating with water sprays. Bonito were just there. The first bonito was caught at the bow. All other fishing rods also started moving up and down, and bonito were thrown on the deck - or rather the expression of falling down from the sky might be right. Of course, all aquarium staff were wearing helmets for perfect protection. The surface of the bonito is felt by hands getting sore as if it is burnt. The caudal fin of the bonito that struggles violently on the deck would decay after a few days. For bonito that are fast swimmers, getting hurt on the caudal fin would be fatal damage. The staff have to adjust themselves to the rolling of the boat and to straddle on the deck in order to catch the bonito with special vinyl baskets so that the fish do not get hurt by falling on the deck, and put them into a prepared water tank." The expression of "The surface of bonito ••• as if it is burnt" excited the author and caused hot discussions. The author spent three months awaiting the details of the scene from Mr. Teruya. In July 1996, the author succeeded in directly asking him about this incident, and he replied: "I might have led you to misunderstand. The temperature was not so hot that I could not touch the skin." the comment disappointed the author. Teruya instead introduced the following important bibliographic information: "Generally the body temperature of fish that are so-called cold blood animals is within ±0.5t of the surrounding water temperature. However, lean meat species of fish are reported to have a higher body temperature than the water temperature." "Tuna often have a body temperature of around 30oC when they are caught, and it takes several hours (no less than 5 hours) to cool them down" (there are two kinds of reported seawater temperatures of l0oC and 16°C in a figure of the book that was introduced by Teruya). "Bullet mackerel have also been observed to have a body temperature increased by about 10oC during struggling for some tens of seconds." This rapid increase of the body temperature is considered to result from the sudden glycolytic reaction within the red muscle. According to Teruya, "Recent research into pacific mackerel and bullet mackerel has shown that the red muscle in the tail of fish that moves aggressively during
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swimming is more developed than that in the body, and the tail contains more myoglobin than the body." Although the deduction from these testimonies contradicts Tani's expectation, the author has confirmed the importance of the caudal fin to perform as a propelling device and convinced that the fish body is an extremely high power engine, through Teruya's sensitive observations and description. In addition to the above description, the author also intends to estimate the power of dolphins by analyzing the performance of Tail-walking' that is performed when the dolphin 'stands' on its caudal fin (photo-2 in frontispiece). However, this has not been accomplished yet due to time limitation. These thoughts and plannings are given as an attractive agenda not only for the author but also for readers to solve in the future.
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Chapter 7 Robot Fish - Development of Ocean Engineering The fact that some fish and dolphins swim very fast has attracted attention of many engineers and researchers as already described. On the other hand, there has not been a long history of people successfully adopting the propulsion methods of fish into boat designs. Except for ancient sailing ships the propelling methods of boats in Europe used oars or paddles powered by humans, and similar tools in the Orient are called 'kai' and 'ro.' The author thinks that only 'ro' which utilizes the fluid dynamical lift force is the most similar propelling device to the caudal fin of fish. After the appearance of steam engines, the majority of propulsion devices changed to paddle wheels and then to screw propellers. Application of the propulsion methods of fish does not seem to have been considered until the establishment of the 20th century's modern fluid dynamics, because the principles of these propelling methods were too difficult to understand. As described in the beginning of Chapter 2, Aristotle believed that fish swimming was explained by 'the nature of fish.' However, design and manufacturing prototypes that have propelling devices with a similar mechanism to the caudal fin of fish have been attempted in many places, only during the most recent decades.
Hertel's Research Dr. Hertel (H. Hertel), who left a bright footprint in the aeronautic history of Germany and Europe from the 1930's to the 1960's, argued for the necessity of collaboration between biology and engineering, and wrote a famous book, 'Structure, Form and Movement.' This book shows a number of beautiful photographs, and explains how the structures of plants and animals have important functions by introducing the surprising aerodynamics of insects or bats, and the high-speed swimming ability of fish and dolphins. Since it was first published, the book has fascinated researchers for more than 30 years. The book also introduced the research on fish swimming by Gray, Bainbridge, and Wu (T. Y. Wu: California Institute of Technology). It is not too much to say that the topics discussed by Hertel touch all the various issues related to drag and thrust considered in this book.
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Hertel correctly evaluated the role of the caudal fin for fish swimming at highspeed. Figure 7.1 shows an oscillating plate propelling boat, 'TUB-TUB-1.' Hertel proudly introduced the origin of the name of this boat: 'Technik Und Biologic an der Technischen Universitaet Berlin-1.' In this figure, part-1 is a thin oscillating plate made of steel. This plate does not oscillate like a rigid wing. Instead, a propagating wave is transmitted downstream when a stroke-shaft (part 2) and twist-shaft (part 3) provide heaving and pitching motions to the head part of the oscillating plate. Thrust force is obtained in reaction to the water being pushed downstream.
Fig. 7.1 Oscillating Plate Propelling Boat TUB-TUB-1' (H. Hertel, Structure. Form and Movement, Reinhold Publishing Co., N.Y., 1966)
Since TUB-TUB-1 performed quite well, a series of prototypes were produced until TUB-TUB-3. The oscillating plate propagated a progressive wave motion similar to that observed in eels or sea snakes, and the boat achieved a speed of 65 cm/s. Professor Wu estimated the thrust efficiency to be 50 - 60%. The TUBTUBs were probably the first boats to apply fish-like motions in their propulsion methods. Hertel also wrote the following quite inferable passage after the description of the TUB-TUBs in his book: "It is still unclear how far we can compare the threedimensional flow around these animals with the flat potential flow. Future thought might be given to the question of the extent to which an isolated evaluation of the 'propuls|on efficiency' of fishes only is appropriate, when actually the propulsion and drag mechanisms are intertwined, as will be brought out in the section titled 'solution of Gray's Paradox.'" This indicates that Gray's paradox may exist because fish motions were analyzed by separately considering the drag on the body and the thrust efficiency. Professor Wu also produced a fish-like model and set it in a circulating water tank to measure the drag and to observe the flow field around the model - probably the motive was on the same basis. Unfortunately, quantitative data was not presented in public. When the author stayed at the California Institute of Technology in 1984 as a visiting researcher, Professor Wu and his laboratory's interest had shifted to other themes such as research on 'solitons.' that are. solitary waves.
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Studies in Japan Subsequent to the explanation by Dr. Tani (Ichiro Tani: late Emeritus Professor of the University of Tokyo) presented earlier, a number of studies on the high-speed swimming ability of fish were performed. In the field of fluid physics, researchers such as Professors S. Taneda (Kyusyu University), Y. Narasako (Kagoshima University), T. Tagori, A. Azuma and T. Kambe (all at the University of Tokyo) have performed theoretical or experimental investigations, and given superior explanations. Representative results have already been introduced in the previous chapters. On the other hand, since the propelling methods like fish are very interesting theme, there are some engineering studies that originated from the pioneering work by Dr. Isshiki (Syouji Isshiki, an Emeritus Professor at Tokyo Institute of Technology). Several examples of mechanical fish or fish-fin type ship propellers are now presented. Figure 7.2 shows a mechanical fish made by Mr. Watanabe (Y. Watanabe; from the former Mechanical Engineering Laboratory in the Ministry of Trade and Industry*1 now AIST). Although Watanabe was basically a specialist in (D Head : balanced by lead inside polystyrene form (2) External Skin: polyethylene (3) Motor ® Power Transducer: changing the rotational motion into reciprocal motion ® Battery : 8 AA size batteries (6)Tail: Phosphorus bronze with a thickness of 0.2mm (7) Caudal Fin: loosely tensioned Polyethylene membrane
Fig. 7.2 Mechanical Fish Made by AIST at MTI (Japan, Y. Watanabe) (Mainichi newspaper, April 7,1976)
control engineering, he made the model because of his interest in the swimming motion of carp. The body, which is 42 cm in total length and 950 gf weight, is made of copper plates and plastics, and the caudal fin is driven by a DC motor. According to the record, the mechanical fish swam just like carp in a pool in front of the laboratory. After filming the model in operation, Watanabe visited Europe with the experimental movie records, and met with Professor Lighthill at Cambridge University as mentioned in Chapter 1. The movement to solve problems related to fish motions from engineering perspectives had commenced in the former AIST and began attracting much attention. Professor Tsuchiya (K. Tsuchiya: Waseda University) et al. produced a two*1
This organization changed from a governmental organization to a public research organization, and newly started as the National Institute of Advanced Industrial Science and Technology (AIST) in 2001.
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Chapter 7 Robot Fish - Development of Ocean Engineering
dimensional model imitating a boxfish, and analyzed the motion on an experimental basis. Their model consisted of three parts: the head, the body and the caudal fin. These components moved relative to each other. In their experiment the following three kinds of motions were given to the body, and measurements such as the forces were executed. (1) A lateral oscillation when the head and the caudal fin have a coincidental phase. (2) A lateral oscillation when the body oscillation has a phase advanced by p/2 relative to that of the caudal fin. (3) A lateral oscillation only by the caudal fin. As a result, case (D had the largest values for the generated speed, the thrust and the power, while case (3) had the best efficiency among the three cases. Isshiki considered applying a caudal fin-type mechanism for a ship-propeller since the early stage. Figure 7.3 shows an initial polliwog-shaped mechanical fish made in his laboratory. Although the body length was very short, it was confirmed that the 'polliwog' swam well using rubber band power and a caudal fin made of plastic plates. Figure 7.4 shows another propelling machine that replaced a
Fig. 7.3 Polliwog-Shaped Mechanical Fish (Propelled by rubber band power and plastic fins) (S. Isshiki, Kinzoku, 44-11, 1974) support arm upper fin (rubber plate)
I Nmetalribj connecting rod crank
lower fin Fig. 7.4 Structure Diagram of Oscillating Caudal Fins (S. Isshiki, Kinzoku, 46-12, 1976)
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motorboat screw propeller. The caudal fins are made to oscillate by the supportarms connected via cranks that are jointed with the crankshaft (screw shaft), and the elastic force of the caudal fins, which were made of rubber plates, generates the thrust force. However, since the rotational speed of the crankshaft was very high, and the oscillatory amplitude was limited to quite small values, less thrust force was generated than expected. These two propelling mechanisms have the same characteristic as Watanabe's mechanical fish, and the thrust force relies on the elastic force of the caudal fin. Isshiki commented that if the thrust generation was expected only by the flexibility of wing materials such as rubber plates, the choice of material, especially for the fatigue destructive strength, would be decisively important. Next, Isshiki and his disciple, Associate Professor Morikawa (H. Morikawa: Shinsyu University), succeeded in developing a full-scale oscillating wing propelling ship in which a two-dimensional rigid horizontal wing was mechanically provided heaving and pitching motions to propel the ship. Figure 7.5 shows a photograph of the experimental cruising of this ship. This invention was also introduced by the press as the appearance of a new, safe, and highly efficient ship named 'Fin-ship.' As a result of the trial tests, they reported that the maximum speed reached was about 2 m/s and the average thrust efficiency was quite high at 65±10%. The mirror effect of the horizontal oscillating wing caused by the bottom plates of the ship is regarded as the reason why the average thrust efficiency is higher than that (40%) of conventional screw propellers. Isshiki stressed the significance for the development of oscillating wing propellers as follows: (1) It has high thrust efficiency and has energy saving effects at slower speeds. (2) The propulsion mechanism has minimal possibility of accidents that roll up fishing nets or ropes and has little danger of injuring aquatic animals and swimmers that is occasionally seen for conventional small boats. He also occasionally stresses "the necessity of reexamination for engineering in terms of the increase of safety, environmental conservation and biological insight" and the significance of "making too much evolved mechanical technology recur to the origin." His philosophy in many points seems to coincide with that previously described by Hertel, who established a 'Department of Engineering and Biology1 at Berlin University. When the author met with Dr. Isshiki, who was working as the president of the Japanese Society of Mechanical Engineers, in an academic meeting in Tokyo in 1982 or 1983, the author still remembers well that Isshiki told him with great pleasure as if the incident was his private happiness: "Mr. Nagai, our fin-ship has recorded 60% in its efficiency!" He also introduced an interesting episode: "When Perry visited Uraga*2, he was surprised that an indigenous oriental oar Perry was the first formal U.S. representative (admiral), who opened the door of Japanese modernization. Uraga was a satellite town of the then capital city of Edo (former Tokyo).
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Chapter 7 Robot Fish - Development of Ocean Engineering
Fig. 7.5 Cruising Scene of Oscillating Wing Propulsion Ship (S. Isshiki and H. Morikawa, SNAJ Journal, vol.642. Dec. 1982)
(known as 'Hatcho-ro,' meaning eight 'ro's) of a native boat easily performed highspeed." There are a number of scientific attempts to investigate or to apply the fish type propelling method at many universities in Japan since the research at Waseda University and Tokyo Institute of Technology. Dr. Tsutahara (M. Tsutahara at Kobe University) et al. invented an oscillating wing based on the Weis-Fogh mechanism, and tried to apply it to a small ship model. Dr. Shimizu (Y. Shimizu at Mie University) attempted to produce a small mechanical fish, as a biologically copied machine, to experimentally clarify the characteristics of an oscillating wing propelling mechanism. Dr. Nakashima (M. Nakashima at Tokyo Institute of Technology) is enthusiastically continuing theoretical and experimental research on 'Optimized motions for body bending-typed water propelling mechanism.1 Professor Kato (N. Kato at Tokai University) focused his work on the control methods of swimming motions of fish and experimentally analyzed the motions of the pectoral fins. He also accumulated basic research with the target of completing a well controllable automatic mechanical fish. In March, 1995, the author et al. visited Professor Kato at the Department of Oceanography at Tokai University, Shimizu-city, Shizuoka to collect information for his research. At that time, we also had a precious opportunity to look at numerous oceanic animal robots called 'mechanimal's that were on open displays in the departmental oceanographic science museum. If the author's memory is correct, these 'mechanimal's were developed for demonstration at the Okinawa International Oceanic Exhibition in 1975. In March, 1997, Dr. Morikawa, Dr. Kato and the author organized an 'Aqua Bio-Mechanism Studying Group' in Japan. The study group regularly meets twice a year, and includes a wide variety of members whose specialties span across many disciplines from engineering to biology. For example, one participant is Associate Professor Kamimura (S. Kamimura at the University of Tokyo) who investigated the 'bending/inner slipping motions of the sperm flagella of sea urchin.'
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Mechanical Fish and Oscillating Wing Propulsion Ships of University of the Ryukyus As already described, the author has luckily obtained a patent for a new oscillatory propelling mechanism, the 'Fish-Fin-Type Propeller,' through his research into fish motions. Figure 7.6 shows a large-scale mechanical fish made subsequent to the small-scale mechanical fish. In contrast to the small-scale fish that adopted a radiocontrolled system, the large-scale fish employed an on-line controlling system fixed on the ship. The new system was expected to be able to collect more quantitative data than the former system. As is obvious from a comparison with Fig. 6.11, and since the large-scale mechanical fish was also referred to the body profiles of a tuna, it has a similar shape to the small-scale mechanical fish. However, since the size of the large-scale model was almost the same as a dolphin (with a body length of 2.3 m and a maximum body diameter of 44 cm), all components required major changes in their design. The body and the hatch were made of FRP, and the tail was workprocessed from an aluminum block. The flexible skin joining the body and the tail employed a rubber sheet (with a thickness of 3 mm). The main driving device consists of a starter-motor (7 kVA, axial output of about 5 kW) and batteries (24 V, 120 Ah), both of which were for automobiles. The rotational speed of the motor was reduced by 1/10 to drive a Scotch-yoke mechanism.
Fig. 7.6 Large-Scale Mechanical Fish (University of the Ryukyus, 1980)
This large-scale mechanical fish was fixed via a strut to a mother boat (for 2 passengers), and it was manipulated (by controlling the electric current) on board. Numerous data such as the electric voltage, the electric current, the thrust force and the boat speed were measured. As expected, the mechanical fish succeeded to move and propel the mother boat. However, the maximum speed was no more than 1.5 m/s and the swimming number was no more than 0.40, which left many technical problems. The reason for the low maximum speed was a low gear ratio: the motor could only rotate at a lower rotational speed than its design speed, and thus the oscillating frequency was so low that the 'fish' could not perform well. This
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mechanical fish provided numerous technical exercises and studies to the research team at University of the Ryukyus. The next activity that we wrestled with was to design and produce an oscillating wing propulsion ship. The aim was to test the oscillating wing propulsion mechanism in a quantitative way and to verify the practical feasibility. The fact that Isshiki et al. had succeeded in making a horizontal wing and vertically oscillating typed fin-ship naturally provided insight. The intention of this project was to propose and develop a new ship-propeller as an alternative to the screw propeller by collaborating with Isshiki's team. The accumulation of quantitative data for the prototype ship would enable the team to make comparative studies with oscillating wing theory that was examined in parallel. Figure 7.7 shows the overall drawing of the first ship using the oscillating wing propulsion. A gasoline engine from a motorcycle (193 cc displacement, 13 HP output) was mounted on a hand made small boat. The axial rotational speed of the engine was reduced by a chain, and transmitted via an eccentric plate cam combined with its receiving device. A rotational oscillatory motion (via a new Scotch yoke that was joined with the cam mechanism) was provided to an oscillatory arm located at the bottom of the boat. As seen in the figure, the engine and the main driving mechanism of the ship associate a built-in unit in the external cylinder of the main shaft so that the whole unit can rotate by manipulating on board. This means the engine assembly can be rotated relative to the boat and can also function as a rudder. There was a centering keel attached to the front part on the bottom of the boat to prevent undesired yawing motion. The achievements of this experimental ship were presented at the ISOPE 2635
Fig. 7.7 Oscillating Wing Propulsion Ship - 1 (University of the Ryukyus, 1988)
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99
international conference held in Seoul, 1989. The maximum speed was 1.12 m/s. Thus, the effectiveness of the oscillating wing mechanism by means of producing full-scale ships was verified, and numerous interesting characteristics of the mechanisms were recorded quantitatively. It was clarified from both the theory and the experiments that designers should not optimistically use flat plates for oscillating wings but should use symmetric wings that have as high a lift-drag ratio as possible. The 3rd prototype (produced in 1996) that employed lighter weight components, an open-deck canoe and used a symmetric wing finally achieved a maximum speed of 2.0 m/s (4 knots, as fast as Olympic swimmers) at 2.0 Hz. This prototype had good controllability and thus the development of oscillatory wing propulsion ships is considered more or less in the practical stage.
Robot Fish of M.l.T. In January 1995, there was an article with the headline 'Robotic Fish Gotta Swim, Too!,' which was the closing article in a magazine of the American Society of Mechanical Engineers. The article said that a team led by Professor Triantafyllou (M. S. Triantafyllou at Massachusetts Institute of Technology) had developed a Robotic Tuna named 'Charley' measuring 52 inches (1.3 m) in length. According to the article, Triantafyllou had the idea of producing Charley three years before the production when he was investigating vortices behind a circular cylinder (the Karman vortices) and their inducing vibrations. By the way, the author received a private letter from him in August of 1992. In the letter he invited the author to collaborate with him, and said that he was making a mechanical fish with a body length of 5 feet (60 cm). In a later article, he correctly suggested, "the vortices formed behind the fish are similar to Karman vortices. Nevertheless, different from the case of the cylinders, the vortices behind the fish do not become the cause of drag but generate thrust force due to the induced velocity downstream." Figure 7.8 shows the structure of Charley. Obviously, the structure is different from the authors' mechanical fish but it does resemble a real tuna. Charley has an aluminum backbone that consists of 8 pairs of links and hinges, which are driven and controlled by 6 brushless servo-motors (the main motor generates 2 HP). Thus, the backbone accompanies a multiple jointed structure. The relative motions of the components including the caudal fin are independently controlled by the six motors and belt drives. The ribs are made of polystyrene and are distributed at 1inch intervals, and they are connected with the backbone by flexible beams. The body surface is covered with a special elastic fiber (Lycra®). The reason for using Lycra® is to reduce the drag. Charley, at present, seems to be hung from an electric train by a strut, and is towed or swims in an experimental pool to perform tests. Although the detailed data has not yet been presented, according to the previous article, Charley has achieved a swimming speed of about 2 m/s. The target is said to be about 20 m/s (40 knots). It is interesting that Charley achieved a speed almost the same as those of the mechanical fish and the oscillating wing ship tested in the authors' laboratory in spite of the differences in their methods.
100
Chapter 7 Robot Fish - Development of Ocean Engineering
Starting from Charley, their first mechanical fish, the M.I.T. team is planning to produce other kinds of mechanical fish and to produce a seabed-exploring robot with a length of 4.5 m. The production of Charley is a relevant circumstance that the author and Professor Tanaka accidentally became aware of while editing their former writing of 'Fluid Dynamics of Drag & Trust,' and made contact with Professor Triantafyllou. This event finally enabled a color copy of Charley to appear as an opening photograph of the book. As seen above, research and analysis of high-speed swimming ability has accumulated various endeavors for drag and thrust. The research has reached across
Fig. 7.8 Robotic Tuna 'Charley1 of M.I.T. (M.S. Triantafyllou and G. S. Triantafyllou. Scientific American, pp.272-273, March 1995)
Fig. 7.9 Large Mechanical Fish and Graduates (Andou. Ashimine and Usami, in order from the left to right)
Robot Fish of M.I.T.
101
the stages of engineering application of oscillating wing propelling mechanisms or of constructing practical fish shape robots and experimental ships. Professor Tanaka has given a concrete view for the future perspectives about oscillating wing propulsion ships. Although Gray's paradox has not yet been solved, it can be clear that some attempts such as Charley of M.I.T. excite world engineers and researchers, and accelerate their efforts. As an example, the appearance of an 'automatic mechanical fish for visual pleasure use' made by Mitsubishi Heavy Industry Co. (presented in the spring of 1999 just before publishing this original Japanese book) can be raised. Commencing from Hertel's TUB-TUB-1, Isshiki groped for harmonizing human technology with nature. The author is convinced that in the early days of this century, fish-typed robots that can swim in water and high-speed oscillating wing propulsion ships will appear. He also believes that continued research will pave the way for a new period of oceanic culture, symbolized by novel oceanic transports and numerous other activities.
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Epilogue - The Silver Lining of Solving the Paradox The seven foregoing chapters have explained the birth and history of Gray's paradox, observations for swimming motions of carp and dolphins, the possibility of drag reduction, oscillating wing theory and their applications respectively, in terms of fluid dynamics. The paradox has not been fully solved yet. However, even according to observations by the author, it is a sure fact that dolphins of 2 m in length can swim at 10 m/s by oscillating their tails at 5 Hz. It is also an undeniable fact that a 7 m long killer whale can swim at 15 m/s, and a 1.5 m marlin can swim even faster. Therefore, Gray's paradox still exists, and neither the fluid drag acting on these swimmers nor their generating power can be explained by modern fluid dynamics and zoo-physiology.
New Challenges of University of the Ryukyus The last figure in Chapter 7 of this book is a photograph of a large-scale mechanical fish that was recently built and is being examined by performance tests. This robot fish is the second machine among large 'fishes' produced by the author and his coworkers, which employed not only the experience of the first prototype but also much experience derived from oscillating wing propulsion ships. The principle of a two-point hinged and oscillating wing mechanism has not changed, nevertheless, the structure of the secondary hinge, the assembling method of the spring materials and the forming method of the wing have undergone many improvements. Furthermore, the addition of two pairs (a total number of four) of small servomotors to the main drive motor enabled adjustment of the rotational angle of both the left and right pectoral fins, the second dorsal fin and the anal fin to control the posture of the body. Since this robot fish is equipped with lighting and a CCD camera, the operator can safely control the robot by watching a monitoring screen. The oscillating frequency of the caudal fin is 3.0 Hz. This robot fish was roughly completed by March 1998, and a postgraduate student J. Makiya repeatedly conducted performance tests through the period of writing this book. Although research has commenced with the objective to solve Gray's paradox, it is naturally expected that this mechanical fish will become one of the pioneers of ocean probing robots and will contribute to the development and progress of marine engineering.
104
Epilogue - The Silver Lining of Solving the Paradox
The Key Has Shifted to Unsteady and Three-Dimensional Flow Fields In the summer of 1998, the author joined the fluid dynamics meeting of ASME held in Washington, D.C., U.S.A. with Makiya. The meeting involved the latest progress in fluid dynamics, research in turbulence, direct-simulations of three-dimensional flow fields and the rapid improvement of visualization and measurement technology, which intellectually excited the author. Makiya and the author presented their research in a session for unsteady flow. There were many interesting research studies on unsteady and the three-dimensional flow fields at this meeting. Among the presentations, Professor Platzer (M. F. Platzer) et al. from the U.S. Naval Postgraduate School, Monterrey, California presented an interesting speech about 'Small flapping wings in the wake of a stationary main airfoil.' Their experimental equipment consisted of assemblies of two pairs of static airfoils and oscillating sub-wings in tandem. The small sub-wings were 1/10 the size of the main wings and oscillated independently in the wake of the main wings. The experiments were performed in a circulating water tunnel. According to Platzer, the flow downstream of the main wings was improved by simply adding a heaving (their so-called 'plunging') motion to the small wing. Their main wings (airfoils) resemble the body of a dolphin or a whale that generates a wide wake for the author. However, the wake structure was almost erased when the small sub-wings oscillated at more than 5 Hz. Platzer showed the velocity distribution along the mainstream measured by a laser-Doppler velocity meter. The results clearly proved that the velocity distribution adjacent the trailing edge of the main wings changed from a pattern of a conventional wake to that of a jet. According to him, the elimination of the wake was also confirmed by visualization of the flow. Improvement of the flow around the main wings directly suggested the flow drag of the system was reduced. The author's question after their presentation was natural: "Professor Platzer, did you measure the drag on the main wings?" He replied while smiling: "No, not yet." He seemed to have been investigating the influence of a finite width wing that imposes a two-dimensional oscillating movement upon a three-dimensional flow field. When the author heard about this research, he remembered the statement formerly made by Professor Hertel - "Future thought might be given to the question of the extent to which an isolated evaluation of the 'propulsion efficiency' of fishes only is appropriate, when actually the propulsion and drag mechanisms are intertwined, as will be brought out in the section titled 'solution of Gray's Paradox.'" The author completely agrees with this statement. It is no exaggeration to say that Hertel's and Platzer's hints can make us see the silver lining in the dark cloud to solve Gray's paradox. The interchange between Professor M. F. Platzer and the author is continuing. In August 2000, the author co-organized the '1st International Seminar on Aqua BioMechanisms' in Hawaii officially supported by the Japanese and the U.S. governments. The author requested Professor Platzer to join the meeting as well as
Epilogue - The Silver Lining of Solving the Paradox
105
Professor T. Y. Wu (California Institute of Technology), Professor P. W. Webb (University of Michigan) and others. The three Professors attended and gave keynote lectures. The representatives of Japan were the members of the 'Aqua BioMechanism Studying Group' that was introduced in Chapter 7. The Honolulu branch of Tokai University kindly provided lecture rooms and accommodation facilities. The seminar also welcomed European researchers, which certified that research in aqua bio-mechanism is becoming popular worldwide. Additionally, it is a most gratifying occurrence that Japanese Dr. M. Nakashima (Tokyo Institute of Technology) will study under Professor Platzer from the summer of 2001 for a year after an interchange held at this seminar. Thanks to the development of robot fish by M.I.T. and the author and his coworkers, and the enthusiastic research into turbulent drag and three-dimensional flow fields, the author firmly believes that the day on which a declaration that "Gray's paradox has been solved" will come sooner or later. No one knows whether such a declaration will be made by a researcher at University of the Ryukyus or at other Japanese institutes, or, after all, by American or European researchers. Towards solving Gray's paradox, the author has written everything to the best of his current understanding. It would be a great pleasure for the author, if this book excites the intellectual interests of readers, and if the readers comprehend the intellectual depth and interesting facets of fluid dynamics, and if they are stimulated to participate in the new marine engineering age that will definitely come to pass in this 21st century.
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References Chapter 1 J. Gray: Journal of Experimental Biology, 13, pp.192-199,1936. C. M. Breder: Zoologica, N. Y., 4, p.159,1926. I. Tani: KAGAKU, 34-9, pp.471-476,1964-9 (in Japanese). A. Azuma: Journal ofJSAA, 33-382, pp.617-625,1985-11 (in Japanese). M. Nagai: NAGARE, Journal of Japanese Society of Fluid Dynamics, 10-4, pp.47-55, 1979 (in Japanese). Japan Society of Fluid Dynamics: Hand Book of Fluid Dynamics, p.761, Maruzen, 1987 (in Japanese). I. Tanaka and M. Nagai: Fluid Dynamics of Drag and Thrust, Ship and Ocean Foundation, 1997 (in Japanese). Chapter 2 J. D. Bemal: Science in History, 3rd ed., C. A. Watts & Co., Ltd., London, 1965. T. von Karman: Aerodynamics, Cornell University Press, 1954. Chapter 3 H. Schlichting: Boundary Layer Theory, 8th English ed., Springer-Verlag, Berlin, 1999. G. K. Batchelor: An Introduction to Fluid Dynamics, Cambridge University Press, 1974. Y. Tomita: Hydraulics, Jikkyou-Shuppan, 1989 (in Japanese). H. Matumiya, K. Kishi, N. Taniguchi and T. Kobayashi: Transaction ofJSME(B), 59566, p.2937,1993-10 (in Japanese). Chapter 4 T. Kambe: Journal ofJSAA, 25-277, p.53,1977-2 (in Japanese). M. Nagai, I. Teruya, K. Uechi and N. Miyazato: Transaction of JSME(B), 62-593, p.200,1996-1 (in Japanese). M. Nagai, I. Teruya and T. Nakai: Transaction ofJSME(B), 62-604, p.113, 1996-12 (in Japanese). Chapter 5 M. O. Kramer: ASNE Journal, 72, p.25,1960-2. A. B. Toms: Proc. 1st Int. Rheol. Congr. Vol. 2, North Holland Pub. Co., Amsterdam, 1949. I. Tani: Proc. Japan Academy, 64-B, pp.21-24,1988.
108
References
Chapter 6 M. Nagai: Fish-Fin-Type Propeller, Patent No. 1275556 (1985-7), Japan. Tokyo Institute of Fisheries: TUNA, p. 155, Seizando-shoten, 1992 (in Japanese). Chapter 7 H. Hertel: Structure, Form and Movement, Reinhold Publishing Co., N.Y., 1966. T. Y. Wu, C. J. Brokaw and C. Brennen: Swimming and Flying in Nature, Plenum Press, N.Y., 1974. M. Nagai, I. Teruya and T. Nakai: Transaction ofJSME(B), 62-597, p.177, 1996-5 (in Japanese).
Author and Translators Profile Author Minoru NAGAI, Ph.D. Born in 1944, Naha, Okinawa, Japan. In 1972 he finished his Ph.D. study in Faculty of Engineering at Kyushu University and became Lecturer at University of the Ryukyus. He became Associate Professor at the university in 1973, and was a Visiting Researcher at the California Institute of Technology and the University College of London during 1984 - 1985. The author was promoted in 1987 to Professor at University of the Ryukyus where he is currently teaching. Since the beginning of his research work at University of the Ryukyus, Professor Nagai engaged himself in studies on fish swimming, oscillating wing propulsion mechanisms, wind energy, environmental issues and other topics in fluid dynamics. Now, his life work is marine engineering and natural energies. He also serves as a director of the Japan Wind Energy Association and as a director of the Okinawa Branch of the Japan Institute of Invention and Innovation. Translator Takayuki KAWAMURA, MSc Born in 1966, Tokyo, Japan. He graduated from the Department of Aerospace Engineering, Nihon University in 1988, and finished the MSc course in Renewable Energy and the Environment at the University of Reading, UK in 1999. He was a Researcher in Faculty of Engineering at University of the Ryukyus from 1999 to 2000. Now, he represents his own engineering workshop "Tiida crafts" in Tokyo, Japan. Translation reviser George Thomas YATES, Ph.D. Born in 1949, Youngstown, Ohio, USA. He received his Ph.D. in Engineering Science at the California Institute of Technology in 1977, and was a Research Fellow/Senior Scientist at the California Institute of Technology through 1992. Now, he works at Y's Engineering and is an Adjunct Professor of Mathematics at Kent State University.
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Index
•10,24 acceleration 14 acceleration of gravity 53 Active control actual flow 30 actuator disk 38 adverse pressure gradient • • • •27,31 aeronautical engineering • • • •21,27 21 aeronautics 59 AIAA Journal 37 air resistance 15 aircraft 33 airplane 61 algae American Society of Mechanical Engineers (ASME) 75,99,104 Americas Cup 60 amoeba 24 anal fin 103 anesthetized fish 63 angle of attack 12,19,34,44,74 Anguilliform 6,40 animal physiology 7 anomalous layer 56,61 aqua bio-mechanism 105 Aqua Bio-Mechanism Studying Group 96,105 Aqua-police 76 aquatic animals 2,7,87 Aristotle 9 Aristotle's paradox 9 Arizona Republic 75 arm-plate 79 ASME 104 aspect ratio 4 astronomy 10
Australia-II axial interference factor axial symmetric Azuma, A.
•••60 •••39 •••66 •2,93
B 78 backstroke Bainbridge, R. •5,69,71,91 60 banana vortex 19 baseball 97 batteries 14 Bernoulli, Daniel Bernoulli's equation ••• 15,38 biological evolution • • 28 95 biological insight 27 biplane blade element and momentum theory 40 39 blade element theory 26 Blasius' solution Blasius, H. 26 body length ••7,50 •64,77 body profile 24 body surface body temperature bonito •3,88 bonito-fishing 77 bottle-nosed dolphin bound vortex 20,33 boundary condition 29 boundary layer 2 26 boundary layer equation • • boundary layer flow ••3,29,53 27 boundary layer separation boundary layer theory •21,26,35 boxfish 94 6 Breder, C. M.
112
Index
breeding skill •• Bullet mackerel
•82
California Institute of Technology 92,105 camber 34,35 Cambridge University 1,5,28 Carangiform 6,40,50 carp 6,7,66,69,70,84,93 caudal fin 3,40,43,46,70,71,89 CCD camera 103 centering keel 98 change of momentum 13 characteristic body length 24 characteristic curve 35,39 characteristics of a wing 34 Charley 99 Chesapeake Bay 54 circular cylinder 16 circular pipe 58,60 circulating water tunnel 63,104 circulation 19,32,33,43 circulation theory 34 coefficient of drag force 25 coefficient of kinematic viscosity •• 28 coil spring 72,73 cold blood animal 88 compressibility 14 computer simulation 33 continuum 14 control volume 38 convective acceleration 15,29 crankshaft 95 crescent shaped 6 crescent shaped wing 43 critical velocity 31 cross-sectional area 14,65 curve ball 19 cutball 19 cylinder 29,30
D dace-
•3.5,6,7,71
D'Alembert, Jean le Rond 15 D'Alembert's paradox •20,21,22 D'Alembert's principle • • • 10,11 DC motor ••••72,93 dead-water region 18,34 17 dead-water region theory dead-water theory of KirchhoffRay leigh 14 density of the fluid 12 design speed 97 dimensional analysis 49 dimensionless number 24 dimples 31 Dinkelacker, A. 57 direct solutions of the Navier-Stokes equation 29 disc rotating experiment 54 divergence 74 dolphin 1,7,43,76,85 Dolphin Studio 76 dorsal fin 4.103 downward flow 33 drag 2,10,104 drag coefficient 25,31,34 drag force 9,17 drag mechanism 92 drag reduction 49,53.60
E 98 eccentric plate cam 69 efficiency 72 8 mm-film •53,99 elastic fiber 97 electric current 97 electric voltage 65 ellipse 15 Encyclopedia 95 environmental conservation 26 equation of continuity ••••14 Euler, Leonhard 14,15,16,22.26 Euler's equation • • • 79 evolution 31 experiment 10 external force
113
Index
Gray, James Gray's paradox 54 Fabula, A. G. •77,84 false killer whale fatigue destructive strength 95 33 finite wing 95 Fin-ship 5 FishWheel •53,63 flag 63 flapping movement 46 flat-plate flat-plate wing 45 32 flow field flow simulation 29 flow visualization 33 •63,84 fluid drag 39 fluid dynamic efficiency fluid dynamical lift 91 fluid dynamically smooth • 31 fluid dynamics 9,11,21,31,105 fluid force 45 fluid momentum 38 fluid particle 12,28,31,61 free vortex 33 frequency 6 friction drag coefficient • • '50,68,84,87 friction velocity 56,58 frictional drag coefficient 52 front stagnation point 32 FRP 97 fuel 33 fuel consumption 34 Fukui, Y. 82 fur 61
Galilei, Galileo •••• gasoline engine • • • gill glycol-protein glycolytic reaction goldfish gravity gravity force
•98 •88 •3,i
•37 •15
1,91
2,47,51,56,75,86,87,104 groove 58 Guinness Book of Records 2 Guinness record 82
H •61 H2O molecule •60 hairpin vortex Harada,Y. •82 •87 heat generation heat radiation 59 heat transfer 92 heaving 43 heaving motion 9 Heliocentric system • • •3,91,95,104 Hertel,H. 82,83 high jump 43 high-aspect ratio high-speed swimming ability • • • 91,100 high-speed swimming animal 28 high-tech design 60 horse mackerel 43 horse power 86 horseshoe vortex 60 Hoyt,J. W. 54 hull 61 hydrodynamic equivalent diameter 65
ideal fluid model •14,24 Ikemiya, T. 72 inclined flat-plate 17 induced velocity •44,99 24 inertia force International Oceanic Exhibition •76 invention •72 ISOPE •98 Isshiki, Syouji •93
114
Index
Japanese Society of Mechanical Engineers (JSME) 95 jet 38,53,104 Joukowski, Nicolai Egorovich 19
K Kagoshima University 93 kai 91 Kambe, T. 93 Kamimura, S. 96 Kara-jishi 88 Karman vortex 99 Karman vortex street • • 23 Karman, Theodore von 23 Kato, N. 96 keel 60 Kikuchi, Y. 82 killer whale •2,3,83 Kina, Y. 82 kinematic viscosity • • • •28,55 kinetic energy 31 17 Kirchhoff,G. Kobe University 96 Kohama, Y. 60 Kramer, Max O. 51 Kramer's experiment • • • •52,53 Kutta condition 19 Kutta, Wilhelm 19 Kutta-Joukowski condition • • • 19,33,43 Kyusyu University 49,53
laminar 50 laminar boundary layer •••31,52 laminarflow 3,28 laminar flow separation 31 laminar sub-layer 31 laminar wing profile 27 •2,82,86 Lang, T. G. laser-Doppler flow speed indicator ••55 104 laser-Doppler velocity meter
law of action and reaction 37 law of conservation of energy 15 law of energy conservation 14 law of inertia 9,10 lean meat species 88 least-square method 74 lift 18,19,32,33 lift coefficient 34 lift force 20,43,46 lift-drag ratio 34 lifting force 19 lifting theory 14,18 Lighthill, M. J. 7,40,70,79,93 Lighthill's theory 41 linear chain connection 54 linear regression 71 Ling,S. C. 54 Ling,T. Y. 54 living fish 64 loach 54 long-chain molecule 53 low drag wing profile 27 Lycra 99
M mackerel 6,43 macro-dynamic fluid motion 28 Magnus Effect 19 main flow 31 main stream 26 main wing 104 Makiya,J. 103,104 mammal 79 marine engineering 103,105 marlin 7 mass 50 mass of the fluid 12 mass point 14 Massachusetts Institute of Technology 75 maximum frequency 83 maximum power 86 maximum speed 81,83 maximum swimming speed 86
115
Index
measurement technology 104 Mechanical Engineering Laboratory 93 71,93,96,97 mechanical fish 96 mechanimal's •• 96 Mie University 69 mirror 95 mirror effect • • Mitsubishi Heavy Industry Co. •••101 modern fluid dynamics 29 molecular motion 28 molecular structure 54 molecular weight 54 molecularity 14,23,61 molecule 23 moment 17 momentum 38 momentum equation 14 momentum of the fluid 12 momentum theory 38 mono-chlorobenzene 53 monoplane 27 Morikawa,H. 95,96 mosquito larvae 43 Motobu-town 76 motorboat 95 motorcycle 98 movie camera 69 mucus 53,54 muscle 50 muscle power 87 myoglobin 89
N Nagasaki, T. Nakachi, H. Nakahara, T. Nakai,T. Nakashima, M. Narasako, Y. NASA nature Nature dislikes a vacuum nature of bird
82 72 61 82 •96,105 93 ••59,60 9 9 9
nature of fish nature's law Naval Postgraduate School Navier, C. L. M. H. Navier-Stokes equation Newton, Isaac Newtonian Fluid Newton's law of viscosity Newton's Paradox Newton's three laws of motion Nikuradse, J. Nishiwaki, M. no slip no slip condition non-dimensionalized Non-Newtonian fluid non-viscous fluid normal pressure gradient numerical experiment numerical fluid dynamics numerical simulation
9 75 •••104 21 •21,24 10 12 11 13 10 60 ••••3,4 •24,26 24 24 12 26 27 30 60 •30,31
o ocean probing robot .................. 103 oceanic mammal ........................ 2 Ohsaka,H. .............................. 60 Ohta,K. ................................. 72 Oki .................................... 77,82 Oki-chan's theater ..................... 76 Okinawa ................................. 76 Okinawa International Oceanic Exhibition ........................... 96 Okinawa Memorial Park Aquarium .......................................... 76 Okinawa Times 88 3 Orcinus Orca Osaka University 7 oscillating frequency 73,80,97 oscillating plate 92 oscillating wing 43,50 oscillating wing propelling ship ...... 95 oscillating wing propulsion ............ 79 oscillating wing propulsion mechanism .......................................... 98
116
Index
oscillating wing propulsion ship 98 oscillating wing theory 43,47,72,98 oscillation 6 oscillation frequency 70,72 oscillatory arm 98 Oseen, C. W. 25 Ostraciiform 6 oxygen demand 87 oxygen supplying 2
pacific bottle-nosed dolphin pacific mackerel pacific spotted dolphin pacific white-sided dolphin paddle wheel paint paradox of D'Alembert paramecium Passive control pectoral fin perimeter Perry, Admiral Phoenix photographing physical experiment physics physics of real fluid physiological science pipe drag pipe drag experiment pipe flow pipe-friction coefficient pitching pitching motion Pilot, Henri de Pilot tube Platzer, M. F. polli wog Polyelhylene oxide polyethylene-oxide solution polymer concentralion
3,77,80,82 88 2,82 3,77,84 91 61 16,17,19 24 53 79,103 65 95 75 64 30 10 35 86 61 58 53,54 54 64 43,92 15 15 104 94 54 54 54
polymer solution polymethlmethacrylatemonochlorobenzene power •2,: power generating power per unit weight • • • powered flight Prandtl equation Prandtl, Ludwig Prandtl's boundary layer theory pressure difference pressure distribution pressure drag pressure drop pressure gradient force Principia process of evolution profile drag propagating wave propelling mechanism propulsion efficiency protrusion Ptolemaic system
•53
50 84 •21,27 54 22
26 38,58 16,17 12 54 15 10 79 12 53,69 40 92 57 9
R 73 radio control Rayleigh, Lord ••••17,18 14,23,33 real fluid 32 rear stagnation point red muscle 88 relative velocity 38 72 remote-controlled • • 9 Renaissance Reverse Karman vortex street 44 Reynolds' experiment 28 Reynolds' Experimental Apparatus 28 Reynolds' law of similarity
25
Reynolds number • -2,24,25,26,30,35,50,52,58,68,84 Reynolds, Osbome 28 riblet 58,60 riblet film 60
Index
ro robotfish Robotic Tuna robustness rolling motion rotational indicator rotational ring pool rough roughness roughness Reynolds number rubber band rubber plate rudder runaway
91 •••103 99 31 64 5 5 31 31 60 94 95 98 •42,46
safety sand roughness saury Schlichting, H. school conflict Scotch-yoke mechanism • • • screw propeller screwball screw-jump sea bream sea urchin seabed-exploring robot seawater secretion self-propelling body self-similar semi-experimental theory • • separation flow separation of the boundary layer separation of the flow servo-motor sharkskin sharkskin swimming suits shear stress Shimizu, Y. Shinsyu University Shinzato, K. Ship and Ocean Foundation
95 60 63 27 54 •72,97 •38,91 19 84 63 96 •-100 61 •12,53 72 55 29 27 -"31 27 99 57 60 55 96 95 63 82
117
61 shipbuilding association • 52 silicon oil silicon rubber 51 104 silver lining single molecule 61 54 single-celled alga 32 singular point 69 16 mm color film sketching 64 slender body theory •40,47,79 77 slenderness 24 slip small fluctuation 53 small-flatfish 54 smoothness 31 Society of Naval Architects of Japan 54 Sofrun 53 soft surface skin 53 24 solid friction solitary wave 92 92 soliton 28 Sommerfeld, A. southern hemisphere • 60 specific speed 70,73 sperm flagella 96 25 sphere spindle shape 27,28,66 31 sporting technology • spring reaction 45 stability problem 53 stability theory 53 16 stagnation point stagnation stream line 16,18,24 stall 34 60 Stars and Stripes starter-motor 97 starting vortex 33,43 steam engine 91 Stokes, G. G. 21,25 stream line 14 stream tube 14 streamlined 27,28 stress 11
118 sub-wing •••104 super-computer ••••29 superimposed picture ••••69 surface area 12,64 surface friction drag ••••50 swimming method ••••79 swimming mode 5 swimming motion ••••69 swimming number •••7,46,51,71,74,97 swimming pool 73 swimming speed 2,70,84 swimming suit 53 Sydney Olympics 60 symmetric shape wing 45 symmetric wing 4,46,79
Tagori, T. ••• 53,54,93 72 tail Tail-walking 89 7,82,100 Tanaka, I. • • • 53,93 Taneda, S. •• Tani, Ichiro • •2,60,86,87,92,93 19 tennis 88 Teruya, H. • • 82 Teruya, I. • • 63 test section • • theoretical fluid dynamics • • • •20,53 thickness 35 Thompson 1 three-dimensional flow 16 three-dimensional flow field •••104 ••••60 3MCo. thrust 10,42,43 thrust coefficient 39 thrust efficiency 39,42,45,46,92,95 thrust force 9,46,92,95,97,99 thrust generation 40,43 Tietjens 22 tilapia 6,66,71 Tohoku University 60 Tokai University 96,105 Tokashiki, K. 63
Index
Tokyo Institute of Technology 61,93,96,105 Toms effect 53,61 Toppo 84 towing experiment 53 towing-drag detector 63 trailing edge 19,32,104 transition 31 transparent acrylic window 77 Triantafyllou, M. S. 75,99 tripping wire 31 Tropical Dream Center 76 trout 3,5,71 Tsuchiya, K. 93 Tsutahara, M. 96 TUB-TUB-1 92 tuna 6,43,72,88 turbulence 33,53 turbulent boundary layer ••• 2,31,52,60 turbulent boundary layer flow 50 turbulent boundary layer structure •••55 turbulent burst 59 turbulent flow 3,28,29 turbulent model 29 turbulent motion 29 two hinged oscillating wing 44 two-dimensional flow 16
u 82 Uchida, S. 18 undisturbed flow 54 Union Carbide Co. 10 universal law of gravitation ••••105 University of Michigan • • • University of the Ryukyus 63,82 University of Tokyo 2,53,93,96 unsteady acceleration 14
vacuum velocity distributionvinyl flag virtual mass virtual mass force •
9 •55,104 ••53,63 ••41,45 46
119
Index
viscosity viscosity coefficient viscous viscous drag viscous force viscous sub-layer viscous sub-layer thickness visualization visualized photographvolumetric flow rate • vortex vortex line vortex viscosity
12,14 12 31 12 22,24 •55,60,61 56 104 23 15 19,20 33 29
59 96 65 24,43 82 19,32 35 19,39 33 27 27 91,105
Y
w wake Walsh, M.J. Waseda University Washington, D. C. Watanabe, Y. wave speed Webb, P. W.
Weinstein, L. M. Weis-Fogh Mechanism wetted perimeter whale whale shark wing wing profile wing theory wing tip vortex Wright, O. Wright, W. Wu,T. Y.
••104 •••59 •••93 ••104 •7,93 •••69 ••105
yacht yacht race Yamaguchi University yellowtail Yosimine, Z.
60 60 60 63 82
z AQ
