Thinking Fluid Dynamics With Dolphins (Stand Alone)

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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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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 (A2v,). 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 (=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