Propulsion Strategy Analysis of High-Speed Swordfish

Trans. Japan Soc. Aero. Space Sci. Vol. 52, No. 175, pp. 11–20, 2009

Propulsion Strategy Analysis of High-Speed Swordﬁsh

By Hsing-Juin LEE, Yow-Jeng JONG, Li-Min CHANG and Wen-Lin WU

Department of Mechanical Engineering, National Chung-Hsing University, Taichung, Taiwan, R.O.C.

(Received February 12th, 2008)

Fish have appeared since Precambrian more than 500 million years ago. Yet, there are still much untamed areas for fish propulsion research. The swordfish has evolved a light thin/high crescent tail fin for pushing a large amount of water backward with a small velocity difference. Together with a streamlined forward-enlarged thin/high body and forward- biased dorsal fin enclosing sizable muscles as the power source, the swordfish can thus achieve unimaginably high propulsion efficiency and an awesome maximum speed of 130 km/h as the speed champion at sea. This paper presents the innovative concepts of ‘‘kidnapped airfoils’’ and ‘‘circulating horsepower’’ using a vivid neat-digit model to illustrate the swordfish’s superior swimming strategy. The body and tail work like two nimble deformable airfoils tightly linked to use their lift forces in a mutually beneficial manner. Moreover, they use sensitive rostrum/lateral-line sensors to detect upcoming/ambient water pressure and attain the best attack angle to capture the body lift power aided by the forward- biased dorsal fin to compensate for most of the water resistance power. This strategy can thus enhance the propulsion efficiency greatly to easily exceed an astonishing 500%. Meanwhile, this amazing synergy of force/beauty also solves the perplexity of dolphin’s Gray paradox lasting for more than 70 years and gives revelations for panoramic fascinating future studies.

Key Words: High-Speed Swordﬁsh, Propulsion Strategy, Super-Eﬃciency/Kidnapped Airfoils, Circulating Horsepower/ Fin Embryology/Evolution, Solving Gray Paradox/Biomimics

1. Introduction detect upcoming velocity/pressure, sustain good agility, and be a lethal weapon during hunting and defence.9) The Primitive fish-like creatures first appeared about 500 Scombroidei are also characterized by a forward-enlarged million years ago in the Precambrian. Now, there are more thin/high body and a forward-biased first dorsal fin as well than 57 orders, 482 families and 24,618 species of known as a light thin/high crescent tail fin. fish. This number is more than half the known 48,000 spe- The broadbill swordfish rostrum is generally about four cies of vertebrates. Fishes can survive widely in frigid times longer than the length of the mandible, or more 0 C polar seas, scorching tropical deserts, 5,200-m high than half the body length.10) The back color is dark gray mountain springs, and other harsh environments. They have gradually becoming gray to yellowish downward. The tail evolved diverse specializations and swimming speeds to fin has a keel on each side of the base. This species also become the fittest to survive.1,2) Typical representative lacks jaw teeth, pelvic fins and scales.10) Age and growth maximum swimming speeds of some fishes are shown of broadbill swordfish is studied from counts of growth in Fig. 1.3) bands on cross-sections of the second ray of the first anal Figure 2 shows the 12 species in the Scombroidei subor- fin. A swordfish can reach 90 cm at 1 year old, increasing der.4,5) The broadbill swordfish (Xiphias gladius) is the only 10–20 cm annually for a few years. When the anal fin reach- species in the Xiphiidaeis (Fig. 3).6) The Pacific sailfish es 70–80 cm, it separates into two parts. The largest known (Istiophorus platypterus) shown in Fig. 4 has a partly/fully broadbill swordfish was 450-cm long and weighed 540 kg, retractable dorsal fin and is one of the best non-amputated but other species in the Scombroidei can reach more than living models for demonstrating superior swimming strat- 900 kg.11) Embryology and larval studies show that at about egy.5) Further, the seemingly not-so-streamlined mahi-mahi 23 cm in length, the swordfish dorsal fin extends (Coryphaena hippurus) in Fig. 5 is another species in the almost the entire length of the body (Fig. 7).12) With further Coryphaenidae family of the same Perciformes order with growth, this fin develops a single large lobe, followed by a an amazing 93 km/h maximum speed.7) For comparison short part that still reaches the tail peduncle. At about the fastest marine mammal is the bottlenose dolphin 52 cm in length, the second dorsal fin is well developed, (Fig. 6).8) Fish in the Scombroidei are generally character- and at about 150 cm, only the first large lobe dorsal fin ized by a prolonged upper jaw extending beyond the lower remains. Although it is relatively difficult to find hard fossil jaw, evolving into a long flat rostrum like a two-bladed evidence of dorsal fin evolution, alternatively some clues sword in swordfish, or a rounded spear in striped marlin of dorsal fin forward-biased evolution can be found from (Tetrapturus audax).5) This multi-function bill full of embryology/larva footprint (Figs. 712)) in order for well- pressure sensors can cut through water with great ease, preparing to capture circulating horsepower together with forward-enlarged thin/high body, as explained later. Ó 2009 The Japan Society for Aeronautical and Space Sciences 12 Trans. Japan Soc. Aero. Space Sci. Vol. 52, No. 175

Ecologically, swordfish can be found in surface waters warmer than 13C, but the optimum temperature ranges from 18 to 22C.5) Sometimes, individuals are found in frigid waters. Today, swordfish are fished in tropical and temperate seawaters worldwide. During spawning season, they migrate to warmer waters.5) Their normal depth distri- bution ranges from surface to about 550 m.5) They feed on mackerel, herring, flying fish, squid and other small fish as

Fig. 1. Comparison of representative maximum fish speeds.3) Fig. 4. Pacific sailfish (Istiophorus platypterus, ) with partly/ fully retractable full-fledged dorsal fin has typical maximum speed 110 km/h.5)

Fig. 5. Mahi-mahi (Coryphaena hippurus, ) with the similar features of forward-enlarged thin/high body and retractable forward-biased dorsal ﬁn and maximum length 2:1 m, maximum weight 40 kg and typical maximum speed 93 km/h.7)

Fig. 2. Twelve species in Scombroidei suborder in four genus and two Fig. 6. Bottlenose dolphin (Tursiops truncatus, ) with forward- families.4,5) biased ﬂippers evolved from natural forearms with typical maximum speed 40 km/h.8)

Fig. 3. Broadbill swordﬁsh (Xiphias gladius, ) with typical maximum speed 130 km/h.6) May 2009 H.-J. LEE et al.: Propulsion Strategy Analysis of High-Speed Swordﬁsh 13

Fig. 7. The growth process of swordfish dorsal fin.12) Fig. 9. NACA64(1)-412 airfoil: (a) L=D ratio for different attack angles (b) variation in lift and drag coefficients with attack angle.29)

than its muscles supposedly provide, creating the so-called Gray paradox. In aquatic propulsion, there are a few important exemplary sources of innovative ideas, analytical meth- odologies and insightful visions.9,16,18–20) The tail fin has (a) (b) been extensively studied as the primary propulsor and flow Fig. 8. The trident harpoons for swordfish catch.14,15) fields around the tail peduncle/fin are especially complex with vortex rings21,22) in Scombrid fish. Philosophically, the biggest ‘‘vortex’’ with most recoverable rotational their everlasting dinner menu of all day long sashimi. Yet, energy is usually the fish itself as explained later. The swordfish itself is one of the most important resource of elongate lunate tail fin with high aspect ratio is a well- sashimi for people with an annual world catch of about known important efficiency feature for pelagic high-speed 97,110 tonnes.13) The major nations catching swordfish are fishes pushing large volumes of water with small velocity Taiwan 20%, Japan 10%, Spain 13%, the USA 7%, Italy difference23–26) in contrast to the generally broad/rounded 6%, Brazil 5%, and others 34%, locally the seasonal distri- tail fin of slower swimmers. On the other hand, the functions bution of swordfish is near the Kuroshio Current off eastern of dorsal, anal and other fins have been much less Taiwan usually from April to August. Except for devastat- studied.27,28) In particular, the function of the forward-biased ing fishing by trawling, longlining, and so on, in Taiwan dorsal fin of the swordfish and its cousins is relatively unex- one still-existing traditional conservational way of catching plored, not mentioning its great aid to propulsion the swordfish is by trident harpoons as shown in Fig. 8 efficiency. While the fish body is commonly considered only ( ),14,15) this technique is said to be imported from as a drag generator, it is really another nimble deformable Okinawa in earlier times. airfoil like the tail fin but bigger, and thus open a bright Videler (1993) said ‘‘Howdo fish swim? The short answer window for kidnapped airfoils/circulating horsepower is that we still do not know.’’16) Even today, we still don’t phenomena to explain the high-speed fish swimming clearly understand sophisticated propulsion strategies of strategy and also the long-lasting Gray paradox of marine high-speed fish, such as swordfish swimming at an astonish- mammals in the following. ing speed of 130 km/h. This condition motivates us to initiate this research with our especially advantageous 2. Innovative Concepts of Kidnapped Airfoils and background in aeronautical jet propulsion to address these Circulating Horsepower interesting problems. The 2004 Olympic record for the men’s 50 m freestyle 2.1. Airfoil lift and drag was 21.93 s, or an average velocity of about 8.3 km/h. A An object moving in a fluid experiences lift (L) and drag United States Navy Los Angeles class submarine is said to (D). As an example, Fig. 9 shows the L=D ratios as a func- have a top speed in excess of 56 km/h. However a dolphin tion of attack angle for the NACA64(1)-412 airfoil and using its body and tail fin/flippers as airfoils/propulsors change in lift and drag coefficients at different attack can swim at about 40 km/h (we generally call both angles.29) Clearly for different attack angles, the L=D ratio airfoil/hydrofoil as airfoil for simplicity). In 1936, the can be easily up to 10 or even 100, and notably the variation famous British biologist James Gray created a great stir by in drag is quite small at attack angles between 0 and 4.As calculating the drag power of a dolphin swimming around shown later, the fish body and tail may use similar advantage 37 km/h in seawater.17) Surprisingly, his calculations to produce off-c.g. lift for ‘‘circulating horsepower’’ without showed the dolphin needed seven times more muscle power incurring much extra power for drag. 14 Trans. Japan Soc. Aero. Space Sci. Vol. 52, No. 175

Fig. 11. Swordﬁsh tail ﬁn like airplane wing.12)

Fig. 12. Schematic track of swimming ﬁsh body.30)

Fig. 10. The sail boat zigzags to the destination.

For the preparation of later explaining the ‘‘circulating horsepower’’ to recover some energy using lift force as a commonplace matter, firstly let us look the motion of sail boat. It is interesting to see this well-known example that a sail boat bound for a target area in the direction against the wind can extract some energy even from the adverse wind. Sail and hull function for propelling the sailing boat Fig. 13. Schematic of lift/drag forces of fish body/tail fin as kidnapped in this case as an elegant synergy of aerodynamics and airfoils. hydrodynamics. The sail can be made and imagined as a nimble deformable airfoil with adjusted attack angle against wind as shown in Fig. 10 (likewise later for fish tail and the typical mode30) (Fig. 12), the body and tail fin work like body). The moving direction of the boat is accordingly a set of kidnapped nimble deformable airfoils (Fig. 13). The adjusted to have an appropriate angle with wind, too. The light thin/high tail fin generates only relatively small drag dominant lift generated by the sail can resolve into two main together with large lift. The body with most of the total mass components: thrust T and side force P. Since the lift is experiences a relatively much larger drag than the tail fin. generally much higher than the side drag, the drag can be The body can be viewed as another larger airfoil like the tail ignored for simplicity. Thrust propels the boat forward, but obtains much larger lift. When a fish swims in water, while the very high sidewise water resistance largely pre- the deformable body/tail airfoils are adjusted to the best vents the boat from moving sideways. To reach the destina- attack angles to use the body/tail lift to overcome the tion, the sail must alternately swing side-to-side to produce a respective drag of body and tail and increase propul- zigzag path (Fig. 10). Moreover, except for thinking the sion efficiency ingeniously. We call this phenomenon boomerang of Australian aborigines, by the way, more sur- ‘‘kidnapped airfoils.’’ prisingly, even pure drag power can be partly recovered, just 2.3. Circulating horsepower effect of swordfish

think the simple unsymmetrical models b or c moving Because the lift due to the forward-enlarged thin/high

toward the left in air/water instead of a symmetrical .? fish body with forward-biased dorsal fin does not generally 2.2. Kidnapped airfoils concept of fish body and tail fin act on the body center of mass during swimming, the body The swordfish fins are important tools in swordfish swim- may recover some rotational power of lift moving in the lift ming and maneuvering. The shape of the tail fin is like the direction, and thus dramatically increasing propulsion effi- wing of a typical airplane (Fig. 11). When a fish swims in ciency. The forward-biased dorsal fin is similar to the canard May 2009 H.-J. LEE et al.: Propulsion Strategy Analysis of High-Speed Swordfish 15

of power ½ð1 þ 8Þ7 ¼ 2, but obtains 8 Nm/s of propulsion power to counteract the body drag in order to move forward. Therefore, surprisingly, the propulsion eﬃciency is 400%. Furthermore, if we assume that the L=D ratio of the wing- like tail ﬁn airfoil as 20:1, similarly, we can acquire:

W_ tail ¼ Ftail Vtail ¼ 1N 1 ¼ 1N m/s ð4Þ

W_ body ¼ Fbody Vbody ¼ 20 N 0:8 ¼ 16 N m/s ð5Þ

W_ lift ¼ Lbody Vlift ¼ 200 N 0:07 ¼ 14 N m/s ð6Þ In this case, the fish tail fin consumes 1 Nm/s of power and the body consumes 16 Nm/s, but the fish obtains Fig. 14. Schematic of kidnapped airfoils for fish body and tail fin with 14 Nm/s of power for energy recycling. It actually con- simulated power source. sumes 3 Nm/s of power, but obtains 16 Nm/s of propulsion power to counteract the body drag of water (W_ tail þ winglet of some airplanes with inherently unstable lift, thus W_ body W_ lift ¼ 1 þ 16 14 ¼ 3 Nm/s). Therefore, in- providing both risk and chances of energy recovery. The fish credibly the propulsion efficiency is 533% in this case. body and tail fin can use associated muscles to sweep From above analyses, we can see that by paying a through water as a kidnapped airfoil system. In Fig. 14, relatively small amount tail fin drag power, the efficiency- the muscle as an energy source can be simulated as stretched shaped tail fin can play a crucial role somewhat like an or compressed springs. For the convenience, we may tempo- improvised solid wall for the fish body to push against rarily assume a right angle between the fish body and tail fin in order to move the body forward. The fish body and tail as shown. work like two nimble deformable airfoils tightly linked to The spring pulling force Ftail resists the drag Dtail as the use their lift for each other to form the so-called kidnapped tail fin goes forward by consuming relatively small energy; airfoils. Moreover, they use the sensitive rostrum and the tail fin with high lift Ltail can be imagined as a solid wall lateral-line sensors to detect the upcoming/ambient water that the fish body pushes against when moving forward. The pressure and attain the best attack angle to capture the rota- fish tail may go forward at a specified speed Vtail as shown in tional power of lift with the forward-enlarged thin/high Fig. 14. At the same time, Fbody resists the drag Dbody of the body and the forward-biased dorsal fin. This strategy enhan- fish body and generates the lift Lbody by the fish body. From ces propulsion efficiency greatly to easily exceed 500%, Fig. 13, due to the larger amplitude of the tail fin sweep, the and elegantly solves the Gray paradox. When mammals speed of the fish body Vbody is relatively smaller than the returned to the sea more than 5 million years ago, they swam speed of the tail fin Vtail (say, 80%). Also, clearly the swing- vertically with the forward-biased flippers evolving from ing speed of the fish body Vlift is usually much smaller than forearms; this natural forwardness may be the source of the speed of fish tail Vtail (say, 7% as representative velocity their long secret. Furthermore, even the seemingly not- for power recovery). so-streamlined mahi-mahi with similarly distinct forward- Base on reported experimental data even to 100:1, we can enlarged thin/high body and forward-biased dorsal fin can safely assume that the lift-to-drag ratio of fish body and swim at 93 km/h, faster than other seemingly more stream- tail fin is 10:1. For convenience and not to distract from lined fishes such as the 45 km/h barracuda, tiger shark, the long digits, we purposely use simple numbers to explain pointed-nose shark, and even the much more muscular the concept of circulating horsepower. As an example, 80 km/h blue-fin tuna.31) Therefore, the mahi-mahi and when Dtail ¼ 1 N, then Ltail ¼ Dbody ¼ 10 N, and Lbody ¼ Pacific sailfish with partly/fully retractable dorsal fin 16) 100 N. Using Videler’s assumption Vtail:Vbody = 1:0.8, (Figs. 4 and 5) are the best living models for demonstrating Vlift can be assumed to be 0.07 Vtail. Further, denoting their superior swimming strategy of kidnapped airfoils and the power of the tail fin as W_ tail, the power of the fish circulating horsepower. body spring as W_ body, and W_ lift as the lift power of the fish body: 3. Review of Generalized Efficiency Equations for Jet Propulsion System and Associated Revelations for W_ tail ¼ Ftail Vtail ¼ 1N 1 m/s ¼ 1N m/s ð1Þ Fish Tail/Body Propulsion Evolution Trend W_ body ¼ Fbody Vbody ¼ 10 N 0:8 m/s ¼ 8N m/s ð2Þ 3.1. Lagrangian Reynolds transport equation W_ ¼ L V ¼ 100 N 0:07 m/s ¼ 7N m/s ð3Þ lift body lift This section uses the advanced jet propulsion concepts of Now, we can see that when the fish consumes 1 Nm/s Lee et al.32) and discusses swordfish propulsion. First, from in tail fin power and 8 Nm/s in fish body power, the Fig. 15 and appendix deriving lagrangian Reynolds trans- fish obtains 7 Nm/s for energy recycling by the body lift port equation (LRTE),33) the highly natural/efficient LRTE force. In total, the fish actually consumes only 2 Nm/s is presented as: 16 Trans. Japan Soc. Aero. Space Sci. Vol. 52, No. 175

port analysis for deriving the momentum equation in Fig. 16.32) When F is the net force of the jet propulsion system, this force is assumed to be forward positive.

F ¼ Fext,surf PiAi þ PeAe MðtÞgðtÞ DðmomentumÞ ¼ system ð8Þ Dt

Where Fext,surf is the external force component acting on the external hard surface (excluding inlet/outlet) of the (a) t jet propulsion system in the direction along the centerline. Also, PiAi and PeAe are the pressure forces acting on the inlet/outlet ports, respectively. MðtÞ is the time-varying mass in the control volume, and gðtÞ is the instantaneous gravitationalZZZ acceleration. D F ¼ ðV v Þd8 Dt CV rel SM ZZZ D D @ ¼ V M þ V M v d8 Dt CV CV Dt @t rel (b) t+∆t CVcomoving ZZ

Fig. 15. Jet engine system. vrelv~rel dA~ CScomoving ZZZ u @ i ¼ MV_ v d8ðm_ v m_ v Þð9Þ CV @t rel e e i i Pi , Ai CVcomoving where SM represents the jet system mass. Note that the term involving material derivative of system mass M naturally

g(t) VCV disappears in the above processing due to the conserva- tion of mass. Further, VCV is the control volume velocity (forward positive) and is the mass density; m_ i and m_ e

Pe , Ae are the inlet and outlet mass ﬂow rates, and vi and ve denote the inlet and outlet v , respectively. υe rel Equation (9) can be written as the momentum equation Fig. 16. Schematics of general jet propulsion system. for a jet propulsion system.32)

Fext,surf þ PeAe PiAi MðtÞgðtÞþðm_ eve m_ iviÞ ZZZ ZZ ZZZ DðBsystÞ @ @ ¼ d8þ v~ dA~ ð7Þ ¼ MðtÞV_CV vreld8ð10Þ Dt @t rel @t CVcomoving CScomoving CVcomoving D 3.3. Generalized total kinetic power for jet propulsion Whereas is material derivative, v~ is the velocity rel- Dt rel system ative to the jet propulsion system CV and backward positive. Here, we present the generalized total kinetic power Letting be the velocity relative to the inertial coordinate equation derivation for a jet propulsion system using the system,32) CV and CS are the comoving control volume similar highly eﬃcient interweaved procedure. The gener- and comoving control surface of the system, respectively, alized total kinetic power (TKP) of the jet propulsion and A~ is the traditional area vector. system (JPS) includes both the kinetic power Ek,syst and 3.2. Momentum equation for jet propulsion system the rate of work W_ done to the surroundings. TKP can be Lee invokes the interweaved lagrangian Reynolds trans- processed as:32) ZZZ DðE Þ D 1 k,syst þ W_ ¼ ðV v Þ2d8þW_ Dt Dt 2 CV rel ZZZSM ZZZ ZZZ D 1 D D 1 ¼ ðV Þ2d8 V v d8þ ðv Þ2d8þW_ Dt 2 CV Dt CV rel Dt 2 rel SM SM SM May 2009 H.-J. LEE et al.: Propulsion Strategy Analysis of High-Speed Swordﬁsh 17 ZZZ ZZZ 1 D ¼ V ½V_ MðtÞ þ V 2 d8V_ v d8 CV CV 2 CV Dt CV rel ZZZ SMZZZ SM D D 1 V v d8þ ðv Þ2d8þW_ CV Dt rel Dt 2 rel SM SM 2 3 ZZZ ZZZ 6 @ 7 ¼ V 4V_ MðtÞ v d85 V_ v d8 CV CV @t rel CV rel CVcomoving CVcomoving ZZZ @ 1 1 þ ðv Þ2d8V ðm_ v m_ v Þþ ðm_ v 2 m_ v 2ÞþW_ ð11Þ @t 2 rel CV e e i i 2 e e i i CVcomoving

In Fig. 15, the kinetic power output to the environment by the system mass can be expressed as: ZZZ

W_ ¼ðFext,surfÞVCV þ PeAeðve VCV ÞþPiAiðVCV viÞþ gðtÞðVCV vrelÞd8 CVcomoving 2 3 ZZZ 6 7 ¼ðFext,surfÞVCV þ PeAeðve VCV ÞþPiAiðVCV viÞþ4VCV gðtÞMðtÞgðtÞ vreld85 ð12Þ CVcomoving where PeAeðve VCV Þ is the kinetic power done to the environment by the outlet pressure force in addition to corresponding inlet term. Substituting Eq. (10) for the right hand bracketed terms of Eq. (11), and substituting Eq. (12) for W_ in Eq. (11), after some algebraic manipulations and canceling, the TKP produced by the jet propulsion system is obtained as:32) ZZZ ZZZ DðE Þ k,syst þ W_ ¼ P A v P A v gðtÞ v d8V_ v d8 Dt e e e i i i rel CV rel CVcomoving CVcomoving ZZZ @ 1 1 þ ðv Þ2d8þ ðm_ v 2 m_ v 2Þð13Þ @t 2 rel 2 e e i i CVcomoving

Note TKP is invariant to any non-accelerating observer with different velocities as it should be. 3.4. Generalized propulsion efficiency for JPS and associated revelations for fish tail/body propulsion evolution While deriving the generalized efficiency equations for JPS, it is important to note the kinetic power ejected from the ve- 1 hicle due to losing mass. If the existing kinetic energy (not newly produced) of the losing mass in CV and m_ ðV v Þ2 is 2 i CV i added to the generalized Eq. (13) of TKP, we obtain the following generalized available propulsion power (APP) equation for JPS. ZZZ 1 @ 1 APP ¼ TKP ðV v Þ2 ðd8Þ þ m_ ðV v Þ2 2 CV rel @t 2 i CV i CVcomoving ZZZ ZZZ ZZZ @ 1 1 ¼ P A v P A v gðtÞ v d8V_ v d8þ ðv Þ2d8þ ðm_ v 2 m_ v 2Þ e e e i i i rel CV rel @t 2 rel 2 e e i i CVcomoving CVcomoving CVcomoving ZZZ 1 @ 1 ðV v Þ2 ðd8Þ þ m_ ðV v Þ2 ð14Þ 2 CV rel @t 2 i CV i CVcomoving

Further, the generalized thrust power (TP) is obtained by subtracting the kinetic powers gone with the exhaust gas ﬂow and done to the environment by the inlet/outlet pressure forces from APP. Then, the generalized TP delivered to the vehicle can be written as:32)

1 2 TP ¼ APP PeAeðve VCV ÞPiAiðVCV viÞ m_ eðve VCV Þ ZZZ 2 ZZZ ZZZ @ 1 1 ¼ P A v P A v gðtÞ v d8V_ v d8þ ðv Þ2d8þ ðm_ v 2 m_ v 2Þ e e e i i i rel CV rel @t 2 rel 2 e e i i CVcomoving CVcomoving CVcomoving 18 Trans. Japan Soc. Aero. Space Sci. Vol. 52, No. 175 ZZZ 1 @ 1 1 ðV v Þ2 d8þ m_ ðV v Þ2 P A ðv V ÞP A ðV v Þ m_ ðv V Þ2 2 CV rel @t 2 i CV i e e e CV i i CV i 2 e e CV CVcomoving ZZZ ZZZ ZZZ @ 1 ¼ V ½ðP A P A Þþðm_ v m_ v Þ gðtÞ v d8V_ v d8þ ðv Þ2d8 CV e e i i e e i i rel CV rel @t 2 rel CVcomoving CVcomoving CVcomoving ZZZ 1 @ 1 ðV v Þ2 ðd8Þ ðm_ m_ ÞV 2 ð15Þ 2 CV rel @t 2 e i CV CVcomoving Finally, the generalized propulsive eﬃciency for JPS can be expressed in the short form as: TP ¼ P APP

1 2 APP PeAeðve VCV ÞPiAiðVCV viÞ m_ eðve VCV Þ ¼ 2 ð16Þ APP The above discussion of jet propulsion system has signiﬁcant revelation for even hydrodynamic ﬁsh propulsion. In nearly still water with a large streamtube inlet, vi is generally approaching but slightly larger than VCV . According to the explan- 32) ations following Eq. (29) in previous work, when ½PiAiðVCV viÞ becomes positive (becomes input power), its absolute value should be added to the denominator of P 1 ½APP P A ðV v Þ P A ðv V Þ m_ ðv V Þ2 i i CV i e e e CV 2 e e CV P ¼ ð17Þ ½APP PiAiðVCV viÞ

Further, in Eq. (17), for given available propulsion power energy transforms are still locked in the engine and are and inlet port pressure power ½APP PiAiðVCV viÞ, still possibly available. Conversely, the relatively small together with mass flow rate m_ e (m_ e ¼ m_ i for incompressible heat energy part generated by water viscosity due to tail flow), assuming nearly constant outlet ambient water beating/body-moving is primarily dissipated in nearly pressure, when the absolute outlet velocity ðve VCV Þ is incompressible water and is usually unavailable locally. smaller, propulsive efficiency of the system is higher. On the Moreover, the kinetic energy part imparted to water by other hand, for the most significant thrust term in Eq. (10) viscosity still stands a chance of becoming circulating m_ eðve viÞ ¼ m_ eðve VCV Þ, using 1 ð10Þ¼10 ð1Þ¼ horsepower depending on the individual propulsion 10 as a simple example, for the same thrust of m_ eðve strategy. Therefore, water viscosity and other possible 34,35) VCV Þ¼10, substituting these 1 and 10 values into the last causes will more or less affect the overall propulsion 1 1 efficiency of swimming fish. term of Eq. (17) m_ ðv V Þ2, we obtain 10 2 e e CV 2 1 4. Conclusion ð1Þ2 ¼ 5 and 1 ð10Þ2 ¼ 50. Substituting 5 and 50 into 2 Eq. (17) gives a higher value of P for 5. The tail fin of swordfish evolved to be a light thin/high This demonstrates why a thin/high crescent tail fin and crescent for pushing a large amount of water backward with forward-enlarged thin/high body together with forward- a small velocity difference. Together with the forward- biased dorsal fin can achieve a much better propulsion effi- enlarged thin/high body and forward-biased dorsal fin ciency by pushing a large amount of water as well-known enclosing sizable muscles, the swordfish can thus achieve (say, large m_ e ¼ 10 in our example), with smaller velocity unimaginably high propulsion efficiency and maximum difference [ðve VCV Þ¼1 in our example]. Incidentally, speeds of 130 km/h as marine speed champion, and speed it is interesting to note that just like the wing designs of is one of the most important factors in this diner/dinner club glider, albatross and frigatebird, for the same energy playing incessant sea games of life and death.36,37) efficiency reason earlier slender turbojet engines for jet In this paper, the novel concepts of ‘‘kidnapped airfoils’’ airplanes are generally replaced by the massive turbofan and ‘‘circulating horsepower’’ are presented to illustrate how engines (thinking e.g., Boeing 7x7) with high by-pass the high-speed swordfish can swim so efficiently in a rhyth- ratios in recent decades. Likewise, this is why oarsmen mic manner by catching the body lift power aided by the should row synchronically with reasonably large oars to evolved dorsal fin to compensate most of the water resist- win races. Moreover, it is worthwhile noting the somehow ance power. This superior swimming strategy can thus different viscous effects between the jet engine working enhance its propulsion efficiency greatly to easily exceed with compressible fluid and fish tail beating/body-moving an astonishing 500% as demonstrated numerically by vivid in open water. Inside a jet engine, all the viscosity-affected no-focus-blurring neat-digit model. Sailfish and mahi-mahi May 2009 H.-J. LEE et al.: Propulsion Strategy Analysis of High-Speed Swordfish 19 with partly/fully retractable dorsal fins are the best non- Visualization of Flow around the Caudal Peduncle and Finlets of amputated living models to demonstrate their superior the Chub Mackerel Scomber Japonicus, J. Exp. Biol., 204 (2001), pp. 2251–2263. swimming strategy. Notable for straight high-speed gliding, 22) Lauder, G. V. and Tytell, E. D.: Hydrodynamics of Undulatory Pro- dorsal fin is fully retracted. Meanwhile, the foregoing strat- pulsion, Fish Physiology, Vol. 23, Shadwick, R. E. and Lauder, egy solves the perplexity of Gray paradox lasting for more G. V. (eds.), Academic Press, San Diego, Calif., 2003, pp. 425–468. than 70 years. When cheetah and antelope can only run 23) Wilga, C. D. and Lauder, G. V.: Hydrodynamic Function of the Shark’s Tail, Nature, 430 (2004), p. 850. about 80–95 km/h as land speed champions, then we know 24) Vogel, S.: Life in Moving Fluids: The Physical Biology of Flow, 2nd how marvelous the swordfish propulsion efficiency is to ap- ed., Princeton University Press, Princeton, 1996, pp. 81–377. proach the superme realm as the old saying ‘‘shippass, water 25) Nauen, J. C. and Lauder, G. V.: Hydrodynamics of Caudal Fin Loco- still’’ ( ). Other flying/swimming creatures may motion by Chub Mackerel, Scomber Japonicus (Scombridae), J. Exp. Biol., 205 (2002), pp. 1709–1724. use similar strategy for energy recycling to increase their 26) Van Dam, C. P.: Efficiency Characteristics of Crescent-Shaped Wings critical survivability and provide important revelations for and Caudal Fins, Nature, 325 (1987), pp. 435–437. futuristic biomimic torpedoes, cars, ships, airplanes, etc. 27) Lauder, G. V. and Standen, E. M.: Hydrodynamic Function of Dorsal They can constitute the charming topics for possible future and Anal Fins in Brook Trout (Salvelinus fontinalis), J. Exp. Biol., 210 (2007), pp. 325–339. studies. 28) Lauder, G. V. and Drucker, E. G.: Locomotor Function of the Dorsal While taking a panoramic view of the swordfish’s propul- Fin in Rainbow Trout: Kinematic Patterns and Hydrodynamic Forces, sion strategy, we cannot help but sigh sensationally for its J. Exp. Biol., 208 (2005), pp. 4479–4494. amazing synergy of force/beauty and our luck of unveiling 29) Munson, B. R., Young, D. F. and Okiishi, T. H.: Fundamentals of Fluid Mechanics, John Wiley, New York, 2006, p. 543. the crucial survival secret of innumerable long-evolving 30) Taggart, R.: Marine Propulsion: Principles and Evolution, Gulf fishes. Publishing Company, Houston, Texas, 1969. 31) Block, B. and Stevens, E.: Tuna: Physiology, Ecology, and Evolution, References Fish Physiology, Vol. 19 Academic Press, San Diego, Calif., 2006. 32) Lee, H. J. and Chang, C. L.: Deriving the Generalized Power and 1) McKenzie, D. J., Farrell, A. P. and Brauner, C.: Primitive Fishes, Fish Efficiency Equations for Jet Propulsion System, JSME Int. J., 44,4 Physiology, Vol. 26, Academic Press, San Diego, Calif., 2006. (2001), pp. 658–667. 2) Perry, S. F. and Tufts, B.: Fish Respiration, Academic Press, San Die- 33) Lee, H. J. and Lee, H. W.: Deriving the Generalized Rocket Kinetic go, Calif., 1998, pp. 113–184. Power Equations and Associated Propulsion Indexes, JSME Int. J., 3) Zhuang, C. G.: Encyclopedia of General Scientific: Fish Wonderment, 42, 1 (1999), pp. 127–136. ACME Cultural Enterprise Co., Ltd., Taipei, 1987. 34) Chiang, J. S., Chung, S. H. and Lee, H. J.: Unveiling the Crucial Role 4) http://zh.wikipedia.org/wiki/%E6%97%97%E9%AD%9A of Impact Energy Loss for So-called Incompressible Fluid Flow, 5) Nakamura, I.: Billfishes of the World an Annotated and Illustrated J. Mar. Sci. Tech., 12, 1 (2004), pp. 45–52. Catalogue of Marlins, Sailfishes, Spearfishes and Swordfishes 35) Lee, H. J., et al.: The Life and Death of Euler, Bernoulli, Navier- Known to Date, FAO Fisheries Synopsis, FAO Species Catalogue, Stokes Equations and Associated CFD for So-called Incompressible Vol. 5, No. 125, 1985. http://www.youtube.com/watch?v= Fluid Flow, anticipated to appear. LEd3XvYgFVE&feature=related 36) Goadby, P.: Billfishing: The Quest for Marlin, Swordfish, Spearfish & 6) http://tuna-taiwan.myweb.hinet.net Sailfish, International Marine, Maine, 1996. 7) http://en.wikipedia.org/wiki/Coryphaena hippurus 37) Gibson, C. D.: The Broadbill Swordfishery of the Northwest Atlantic: 8) Jefferson, T. A., Leatherwood, S. and Webber, M. A.: Marine An Economic and Natural History, Ensign Press, Maine, 1998. Mammals of the World, FAO Species Identification Guide, Food and Agriculture Organization of the United Nations, Rome, 1993. Appendix 9) Aleyev, Y. G.: Nekton, Dr. W. Junk b. v. Publishers, The Hague, 1977. 10) Sun, C. L., Wang, S. P. and Yeh, S. Z.: Age and Growth of the Sword- Comparing to the traditional Reynolds transport equation, fish (Xiphias Gladius L.) in the Waters around Taiwan Determined a more direct and natural lagrangian Reynolds transport from Anal-Fin Rays, Fishery Bulletin, 100, 4 (2002), pp. 822–835. 33) 11) Mather, C. O.: Billfish: Marlin, Broadbill, Sailfish, Salaire Publishing equation (LRTE) had been proposed specifically for ana- Co., Sydney, Canada, 1976, p. 148. lyzing jet propulsion. The LRTE is an ingenious analysis 12) http://www.flmnh.ufl.edu/fish/Gallery/Descript/Swordfish/ merging both lagrangian system analysis and Euler’s re- Swordfish.html gional analysis concepts. A brief introduction of LRTE is 13) Maguire, J. J., Sissenwine, M., Csirke, J. and Grainger, R.: The State of the World Highly Migratory, Straddling and Other High Seas Fish given here for the more general jet engine with both inlet Stocks, and Associated Species, FAO Fisheries Technical Paper, No. and outlet areas. The corresponding equation for the rocket 495, 2006. engine can be obtained by closing the inlet port of the jet en- 14) http://www.wretch.cc/blog/aprilyao&article id=10350368 gine. 15) http://a320005.travel-web.com.tw/Show/Style6/News/ c1 News.asp?SItemId=0271030&ProgramNo= Referring to Fig. 15, a jet engine with system mass en- A320005000001&SubjectNo=35234&Page=3 closed in region 1 and region 2 at time t represents the con- 16) Videler, J. J.: Fish Swimming, Chapman & Hall, London, 1993. trol volume in rectilinear motion. After an infinitesimal time 17) Gray, J.: Study in Animal Locomotion IV-the Propulsive Powers of t, the same system moves an infinitesimal distance and ex- the Dolphin, J. Exp. Biol., 10 (1936), pp. 192–199. 18) Hertel, H.: Structure-Form-Movement, Reinhold Publication Co., ists in region 2 and region 3 as shown in Fig. 15. The LRTE New York, 1966, pp. 110–212. approach directly compares the two control volumes sepa- 19) Vogel, S.: Comparative Biomechanics: Life’s Physical World, Prince- rated by an infinitesimal distance. All the observations of ton University Press, Princeton, 2003, pp. 93–297. physical quantity B are made by an observer in an inertial 20) Azuma, A.: The Biokinetics of Flying and Swimming, AIAA Educa- tion Series, 2nd ed., Reston, Virginia, 2006. frame instead of the control volume rider. Let B stand 21) Nauen, J. C. and Lauder, G. V.: Locomotion in Scombrid Fishes: for a certain physical quantity (say, momentum or kinetic 20 Trans. Japan Soc. Aero. Space Sci. Vol. 52, No. 175 energy), and represents its intensive counterpart for unit mass, the following material derivative of B with respect to time can be obtained with non-inertial control volume comoving with the jet engine. Also, note that the complementary volume

½ðB1Þtþt is added and subtracted at the same time to enhance eﬃcient derivation of LRTE as follows DðB Þ ðB Þ ðB Þ syst ¼ lim syst tþt syst t Dt t!0 t ðB þ B Þ ðB þ B Þ ¼ lim 3 2 tþt 1 2 t t!0 t ðB þ B Þ ðB þ B Þ ðB Þ ðB Þ ¼ lim 1 2 tþt 1 2 t þ lim 3 tþt 1 tþt ! ! t 0ZZZ t ZZ t 0 t @ ¼ d8þ v~ dA~ ðA:1Þ @t rel CVcomoving CScomoving Equation (A·1) with control volume comoving with the jet engine is fairly useful for the above work.