Identification of hydrodynamic forces developed by flapping fins in a watercraft propulsion flow field

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1 University o New Orleans ScholarWorks@UNO University o New Orleans Theses and Dissertations Dissertations and Theses Fall Identiication o hydrodynamic orces developed by lapping ins in a watercrat propulsion low ield Erdem Aktosun University o New Orleans, erdemaktosun@gmail.com Follow this and additional works at: Part o the Aerodynamics and Fluid Mechanics Commons, Controls and Control Theory Commons, Navigation, Guidance, Control and Dynamics Commons, Propulsion and Power Commons, Signal Processing Commons, and the Systems Engineering and Multidisciplinary Design Optimization Commons Recommended Citation Aktosun, Erdem, "Identiication o hydrodynamic orces developed by lapping ins in a watercrat propulsion low ield" 014. University o New Orleans Theses and Dissertations This Thesis-Restricted is brought to you or ree and open access by the Dissertations and Theses at ScholarWorks@UNO. It has been accepted or inclusion in University o New Orleans Theses and Dissertations by an authorized administrator o ScholarWorks@UNO. The author is solely responsible or ensuring compliance with copyright. For more inormation, please contact scholarworks@uno.edu.

2 Identiication o hydrodynamic orces developed by lapping ins in a watercrat propulsion low ield A Thesis Submitted to the Graduate Faculty o the University o New Orleans in partial ulillment o the requirements or the degree o Master o Science in Engineering By Erdem Aktosun B.Sc Yildiz Technical University, 010 December, 014

3 Acknowledgments First o all, I would like to acknowledge Dr. Nikolas Xiros or always having the patience and knowledge to help guide me to complete this thesis on time. I would also like to acknowledge the ollowing people or their role in helping me conquer my eat o this thesis: Dr. Kostas A. Belibassakis rom National Technical University o Athens or helping me with getting CFD lapping ins data. PHD student Vasileios Tsarsitalidis rom National Technical University o Athens or helping and explaining concept and previous work about lapping ins. PHD student Mustaa Arda rom Trakya University with helping or MATLAB coding concept. My amily and riends or always supporting me through my academic and lie challenges. ii

4 Table o Contents List o Figures... v List o Tables... vii Abstract... viii CHAPTER Introduction General Overview Purpose o the Study... 5 CHAPTER... 7 Review o Literature Previous Studies The Investigation o Aerohydrobionts The investigation o the lapping in theory Kinematics o biomimetic in thrusters Dynamics o biomimetic in thrusters Free Surace Eects The outline o CFD code by NTUA Harvesting power and energy by lapping ins... 3 CHAPTER Methodology Data Analysis or Heave Force Describing Function or Heave Force Data Analysis or Surge Force CHAPTER Results Comparison Transer Function Model and CFD Data or Heave Force Data points o Surge orce CHAPTER Conclusions The Contribution o the Study... 8 iii

5 5. Future Work Bibliography Appendix Appendix A: CFD older tree made spectrum analysis Appendix B: One o the CFD output data Appendix C: MATLAB codes Appendix D: Spectrum plots or ish15ar4 and ish15ar Appendix E: Comparison plots or ish15ar4 and ish15ar Vita iv

6 List o Figures Figure 1: MIT laboratory robot "Robotuna" [1]... 1 Figure : Considered laws o unsteady motion: A: combined translational-rotational oscillations, B: purely translational oscillations, C: purely rotational oscillations, and D: advancing wave-type deormations [1]... 3 Figure 3: Vortex structures, orming behind the in, perorming heaving oscillations: A: near ree surace, B:Near solid lat ground and C: In unbounded luid [1]... 4 Figure 4: Cutter on underwater lapping ins [1] Figure 5: Russian ship with a in wave energy extraction system [1] Figure 6: Japanese ship with a in wave energy extraction system [1] Figure 7: Description o motion o the lapping in during CFD experiment [46] Figure 8: Flapping ins simulation model in CFD environment Figure 9: a Ship hull equipped with a horizontal lapping wing located below the keel, orward the mid-ship section. b Same hull with a vertical lapping wing located below the keel, at midship [49]... 1 Figure 10: Schematics o a low energy harvesting system based upon a lapping in [50]... 4 Figure 11: Analog signal [51]... 7 Figure 1: Discrete time signal - low sampling rate [51]... 8 Figure 13: Discrete-time signal - high sampling rate [51]... 8 Figure 14: Discretization o the time signal [51] Figure 15: Dividing o the signal into the two new signals [51] Figure 16: Rectangular Fin outline or s/c =, 4, 6 respectively [46] Figure 17: Fish-like in outline or s/c=4 and s = 15 0, 30 0, 45 0 respectively [46] Figure 18: Cross section area or NACA 001 [5] Figure 19: FFT low chart Figure 0: Heave orce spectrum - requency domain pattern or one type o data Figure 1: Heave orce spectrum - requency domain pattern or one type o data Figure : Heave orce spectrum - requency domain pattern or one type o data Figure 3: Heave orce spectrum - requency domain pattern or one type o data Figure 4: Heave orce spectrum - requency domain pattern or one type o data Figure 5: Heave orce spectrum - requency domain pattern or one type o data Figure 6: Describing unction heave orce system Figure 7: Surge orce spectrum - requency domain pattern or one type o data Figure 8: Surge orce spectrum - requency domain pattern or one type o data Figure 9: Surge orce spectrum - requency domain pattern or one type o data Figure 30: Surge orce spectrum - requency domain pattern or one type o data Figure 31: Surge orce spectrum - requency domain pattern or one type o data Figure 3: Surge orce spectrum - requency domain pattern or one type o data Figure 33: One o surge orce in time domain or irst data v

7 Figure 34: FFT analysis spectrum - requency or irst data series Figure 35: Nonlinear surge orce system Figure 36: Comparison between CFD data and heave orce transer unction model Figure 37: Comparison between CFD data and heave orce transer unction model Figure 38: Comparison between CFD data and heave orce transer unction model Figure 39: Comparison between CFD data and heave orce transer unction model Figure 40: Comparison between CFD data and heave orce transer unction model... 7 Figure 41: Comparison between CFD data and heave orce transer unction model Figure 4: Magnitudes o HF,F Figure 43: Phases o HF,F Figure 44: Magnitudes o H-F,-F Figure 45: Phases o H-F,-F Figure 46: Magnitudes o H-F,F Figure 47: Phases o H-F,F Figure 48: Magnitudes o HF,-F Figure 49: Phases o HF,-F vi

8 List o Tables Table 1: Aerohydrodynamic characteristics o various systems [1] Table : Speed characteristic o some biological and technical systems [3] Table 3: ish15ar4, Heave Amplitude = 1m Table 4: ish15ar4, Heave Amplitude = 1.5 m Table 5: ish15ar4, Heave Amplitude = m Table 6: ish15ar6, Heave Amplitude = 1 m Table 7: ish15ar6, Heave Amplitude = 1.5 m Table 8: ish15ar6, Heave Amplitude = m Table 9: Optimization result ish15ar4, Heave Amplitude = 1m Table 10: Optimization result ish15ar4, Heave Amplitude = 1.5 m Table 11: Optimization result ish15ar4, Heave Amplitude = m Table 1: Optimization result ish15ar6, Heave Amplitude = 1 m Table 13: Optimization result ish15ar6, Heave Amplitude = 1.5 m Table 14: Optimization result ish15ar6, Heave Amplitude = m vii

9 Abstract In this work, the data analysis o oscillating lapping ins is conducted or mathematical model. Data points o heave and surge orce obtained by the CFD Computational Fluid Dynamics or dierent geometrical kinds o lapping ins. The in undergoes a combination o vertical and angular oscillatory motion, while travelling at constant orward speed. The surge thrust and heave lit are generated by the combined motion o the lapping ins, especially due to the carrier vehicle s heave and pitch motion will be investigated to acquire system identiication with CFD data available while the in pitching motion is selected as a unction o in vertical motion and it is imposed by an external mechanism. The data series applied to model unsteady liting low around the system will be employed to develop an optimization algorithm to establish an approximation transer unction model or heave orce and obtain a predicting black box system with nonlinear theory or surge orce with in motion control synthesis. Flapping ins, lapping in thruster, biomimetic ship propulsion, energy rom waves, unconventional propulsion viii

10 CHAPTER 1 Introduction 1.1 General Overview The idea o biomimetic lapping ins based on observations o ish and cetaceans swimming marine mammals, e.g whale. The present work o biomimetic ins has given much inormation about kinematics. For example, it shows us how these animals use their lapping tails and ins to produce propulsive and maneuvering orces. In order to understand how these creatures generate orces, the in theory needs to be combined with modern luid mechanics. The ish tails have high aspect ratio similarly to ins as shown Fig.1. To sum up, these ins have been investigated by developing kinematics and dynamics model and by calculating its hydrodynamic response. Figure 1: MIT laboratory robot "Robotuna" [1] In general, a basic dierence between a biomimetic thruster and a conventional propeller is that the ormer absorbs its energy by two independent motions: the heaving motion and the 1

11 pitching in motion, while or the propeller there is only rotational power eeding. In realistic sea conditions, the ship undergoes a moderate or higher-amplitude oscillatory motion due to waves, and the vertical ship motion could be exploited or providing one o the modes o combined/complex oscillatory motion o a biomimetic propulsion system. At the same time, due to waves, wind and other reasons, ship propulsion energy demand in rough sea is usually increased well above the corresponding value in calm water or the same speed, especially in the case o bow/quartering seas. Many experimental and theoretical analyses have been perormed or lapping ins and ins. The result o these recent research and development eorts illustrates that the lapping ins systems at optimum conditions can manage high thrust levels [,3,4]. Furthermore, the requirements or environment and inter-governmental regulations are getting stricter or ocean vehicles. According to Kyoto Treaty, the reduction o pollution and environmental impact o especially ocean vehicles is the great key to eect o global warming and climate change. For example, the environmental pollution caused by cargo ships all over the world has been considered as one o the great actors [5,6] contributing to adverse impact on environment due to bad uel use o vessels. Another proo about environmental problems is coming rom satellite s data [7]. It suggests that the main maritime-waterborne o the transportation has permanently spots with high concentration o pollutants induced by the engine o vessels. As aorementioned, the study o generation o thrust by lapping ins has been increased signiicantly in recent years due to quest or clean energy generation by regulations. The principles o lapping ins perormance analysis is based on luid mechanics especially hydrodynamics o in theory. In order to analyze thrust perormance o lapping ins, it is getting

12 approximated as ish-like motion which is combination o harmonic heave and pitch motion as shown in Fig.. Figure : Considered laws o unsteady motion: A: combined translational-rotational oscillations, B: purely translational oscillations, C: purely rotational oscillations, and D: advancing wave-type deormations [1] The idea is that ish have their own thrust mechanism and they propel themselves in water very eiciently due to rhythmic motion o their tail [8]. Propulsion is generated by means o the combined motion o the tail. Thrust is manageable due to the act that the combined translational and rotational motions o the in. 3

13 Figure 3: Vortex structures, orming behind the in, perorming heaving oscillations: A: near ree surace, B:Near solid lat ground and C: In unbounded luid [1] Flapping ins o a thin plate is approached as ish tails in steady orward motion. They are thought o oscillating with combinations o harmonic heave and pitch motion during calculation o their thrust perormance. However, the hydrodynamics o oscillating ins or thin plates has been mostly studied in experimental way because there is diiculty o analyzing vortex shedding induced by boundary layer separation and estimation o nonlinear dynamics o the large vortices generated by ree layers as shown in Fig.3 [8]. Some may raise the question why many scientists have been interested in this ield so ar. The answer is the useul advantages o lapping in system [1] below; Can be considered as environmentally riendly Relatively low-requency systems Suiciently high eiciency systems 4

14 Multi-unctional being capable o operating in dierent regimes o motion Can provide static thrust Possess more acceptable cavitation characteristics than conventional propellers Can provide high maneuverability 1. Purpose o the Study In this study, the data analysis o lapping in oscillation which is combination o harmonic heave and pitch motion is implemented or post-processing data rom CFD code in order to perorm system identiication or heave and surge orce. The data results rom CFD include 6 degrees o reedom outputs which are orces and moments. These orces are surge, sway and heave orces and the moments are roll, pitch and yaw moments. In this work, especially heave and surge orce will be analyzed to develop system identiication between inputs which is the combined motions o lapping in due to the orward motion o ship in waves and outputs which are heave and surge orce. Note that surge orce is playing very important role to acquire thrust rom lapping in and heave orce is very important to get lit action. First o all, CFD sotware is very slow to make analysis. I one parameter is changed or calculation, time needs to be dedicated. In addition, CFD results are not a closed orm relationship. Thereore, the relationship between orce and requency, velocity or amplitude is unknown. As known, orce is a unction o the requency motion, the orward velocity o the vehicle and the amplitude o motion o the lapping ins. For instance, once one parameter is changed, the eect o change o this parameter is unknown on orce due to unknown relationship. Furthermore, this parameter could be optimized or better operations. To sum up, the objective o this thesis is to deine a system identiication based computer algorithm through a set o equations which come rom 5

15 control system theory and can be used to provide a prediction o relationships based on inputs and outputs. 6

16 CHAPTER Review o Literature This chapter includes a review o literature describing previous studies perormed and the undamental inormation o lapping ins..1 Previous Studies The irst study was implemented by Leonardo da Vinci to explain and apply the mechanism o thrust generation by a lapping in in Based on studies o Leonardo da Vinci, at the end o the 19 th century and the beginning o the 0 th century, many studies were emerged in this ield to develop light vehicles using lapping ins [9,10]. That time studies about bionics were limited due to having inadequate scientiic and engineering background. The irst explanation o the physics o lapping ins is given by Knoller and Betz [1] in 1909 and 191 respectively. Both o them reached the conclusion that longitudinal thrust orce and vertical lit orce is induced by lapping in oscillations. However, an extensive research was done about lapping in by the end o the 19 th century [1]. The irst experimental study was implemented by Katzmayr [11] to veriy Knoller Betz s work in 19. Prandtl studied development o unsteady motion o a in in incompressible low and he concluded that vortices are shed rom a sharp trailing edge [1] in 19. Birnbaum contributed a linearized solution or Prandtl s ormulation and provided the resultant thrust orce generated by a lapping in [1]. Also urther studies o this unsteady airoil theory were conducted by Wagner, Kussner and Glauert. Keldysh and Lavrentiev provided a ormulation or thrust generated by a harmonically oscillating lat plate. The solution was obtained with conormal mapping method [1] in

17 Golubev developed a lapping in theory, dierent rom Prandtl s model, based on the discrete Karman orm o the wake arrangement. Using the momentum theorem, he obtained an integral equation whose solution allowed him to obtain the aerodynamic characteristics, including the thrust o the lapping in in the 1940s [1]. Polonsky and Bratt analyzed low visualization experiments to veriy von Karman and Burgers observations. They illustrated the existence o dierent types o vortex structures behind the oscillating airoil [1]. Since 1960s, the investigations in this ield have been intensiied. Many researchers have been conducting investigations about the aero-hydrodynamics o lapping ins especially development o more extensive mathematical model [1]. The irst symposium regarding bionics was held in Dayton, Ohio in 1960 [1]. Since that day, there has been considerable progress in lapping ins. 8

18 . The Investigation o Aerohydrobionts Indeed, the irst explanation about thrust generation by lapping ins is released in 191 [1]. Ater that, the irst study o analysis has been done in and ater 194 [13,14]. During 1960s, bionics had been emerged in science as a separate ield which covers studies o the basics o lapping-in propulsion [15]. Many publications in biology and biomechanics aimed to develop air or water vehicles with lapping ins as well. The major purpose was the explanation o the background o the eicient propulsion o ish, cetacean, birds, and insects. To sum up, ater understanding o the phenomenon o ish eicient propulsion, the desire o those studies is to implement and apply that background to air or ocean vehicles with the principles o hydromechanics mathematical model. As mentioned earlier, many researchers were interested in lapping in propulsors coming rom early observation o ish, insects and cetaceans who all utilize oscillating in mechanisms or thrust generation. It is useul to introduce the coeicient K which represents aero-hydrodynamic mechanisms; NL K mu Where; N is the power o the system, m is the mass o system, U 0 is the swimming or lying speed, L 0 is the maximum length o the object. The table below shows K coeicient or various creatures and engineering system [1] 0 0 9

19 Table 1: Aerohydrodynamic characteristics o various systems [1] Biological or technical system Swimming in nature Coeicient o aerohyrodynamic perection K kws/ton Relative speed U 0 /L 0 s -1 Cateceans High-speed ish Dolphins Swimming in technology Submersibles Flying in nature Birds and insects Flying in technology Jet airplanes As can be seen rom the table, dolphins have the best hydrodynamic perection in comparison with the other systems. However, birds and insects have the highest relative speeds. Many studies appeared about properties o insect light, ish and cetaceans in 190s. The research was about the structure o the ins, light modes, kinematics and deormations o the ins, measurements o dierent characteristics, and simulations o the light o insects [1]. Maxworthy [16], Ellington [17,18], Freymuth [19], Liu [0,1] made research about aerodynamics o insect light. They realized that unsteady low properties are very important or aerodynamics o insects. 10

20 Another study conducted by Schmidt-Nielsen [] showing that it is more economical or an animal to ly rather than to move on land. Table illustrates the comparison between relative speeds o motion or living creatures and man-made objects [3]. Table : Speed characteristic o some biological and technical systems [3] Biological or technical system Maximum speed body lengths/s Human being 4 Cheetah 18 Supersonic aircrat 75 Starling a type o bird 10 Pershin [4] and Kozlov [5] have studied about ish and cetaceans to look into the propulsive characteristics o these creatures. They systematized and analyzed bio-hydrodynamic o properties o swimming. According to many studies about bio-hydrodynamics, there is a correlation between the method o thrust generation, speed o displacement and typical regimes o swimming or the creature. According to observations, low-speed marine creatures utilize wave-type propulsors, like a whole part o body contributing or thrust generation. Thereore, body lexibility is proportional to speed, i.e as increasing speed causes rise in lexibility o body. High-speed sea creatures can be considered by a system that has three main unctional parts: a hull, a stem, and a in. In ollows, that stem-in subsystem has degrees o reedom. The displacement o in can be considered as heaving-pitching oscillations [1]. According to Pershin [4] and Kozlov [5], the analysis o statistical data o hydrobionts marine creatures show that there is relationship between their geometric and kinematic 11

21 parameters. The most important parameters are the dimensional speed, requency, and amplitude o oscillation. Many papers addressed kinematics o hydrobionts [4,6,7,8]. They showed that the body, stem and in have a relation with the swimming velocity and average period o oscillation. Furthermore, the type o motion made by marine creature such as dolphins has been investigated in literature. According to some papers, the motion o hydrobionts is sinusoidal type. However, analysis o swimming motion based on underwater ilms o Cousteau, has shown that during uniorm translatory motion the oscillatory motion o the stem and the in is close to a harmonic motion. The oscillations o ins are optimized in dolphins to avoid low separation. Thus, kinematic parameters o motion which are heave and pitch oscillations having have phase shit between them. The pitch oscillations are ollowing the heave oscillation by an angle close to ψ = π/ [4]. To sum up, the main parameters o hydrobionts that can be as similar to human-made devices can be [1]; Dimensions, shape and platorm o the in Amplitude o the in oscillations Frequency o oscillations Mass-rigidity characteristics o the in variable chordwise or spanwise Large-in deormations 1

22 .3 The investigation o the lapping in theory The mathematical model o lapping in theory is based on linearized inviscid twodimensional -D low. Along with rise o computer technology, more complex 3-D in nonlinear inviscid and viscous approaches were introduced. Keldysh and Lavrentiev implemented the linear theory to obtain thrust orce acting on thin oscillating plate. They used conormal mapping to solve the problem [1]. Furthermore, Garrick provided a solution about same problem by using Theodorsen s linearized solution or incompressible unsteady low past a lat plate. Meticulous investigations o lapping ins propulsors were done by Khaskind, Sedov and Nekrasov to make perormance analysis or heaving, pitching and combined oscillations. The orces acting on the oscillating thin plate were ound by solving a singular integral equation [1]. Another analysis o combined heaving-pitching oscillating in has been done by Gorelov to look into the maximum thrust orce achievement. He reached the conclusion that the maximum thrust orce is managed when there is a phase angle ψ=π/ between heave and pitch motions. Gorelov was the person to develop the propulsive characteristics o a lapping in in nonlinear ormulation. He obtained a solution by the method o discrete vortices and compared his result with suction orce obtained by linear theory. Instead o the method o discrete vortices, panel method which is a nonlinear ormulation is used to investigate perormance o the propulsive characteristics o lapping ins due to its eiciency in comparison with the Navier - Stokes solver [9,30,31,3]. It was used to study the 13

23 eect o in thickness, oscillation amplitude, phase shit, pitch axis location, and Strouhal number [1]. In addition to these studies, many investigations have been conducted about mathematical modeling o low past lapping in systems, two-dimensional low models, three-dimensional low models and experimental investigations o lapping ins. Taking account o all these studies, the application o lapping ins have been used in ship and oshore industry. Recently, many projects are being conducted to improve water vehicles which utilize lapping ins. Indeed, one o the irst tests o ull-scale vehicles with lapping in propulsors was implemented by Grebeshov in 1970s. He conducted the experiment a ull-size cutter on hydro ins. This cutter used hydro ins not only as liting but also as propulsive components as shown Fig.4. Figure 4: Cutter on underwater lapping ins [1] The major work is recently going on in American, Canadian, British and German scientiic institutes. This work is published by Muller [33]. In addition, we have Russian and Japanese scientiic work about lapping ins development [34,35]. 14

24 Figure 5: Russian ship with a in wave energy extraction system [1] Figure 6: Japanese ship with a in wave energy extraction system [1] Last but not least, other work about lapping ins development can be ound in Res. [36,37,38,39,40,41,4,43,44,45]. Among these works, some o them are about biomechanics aspects related to the structure o ins o birds and insects and the mechanics and dynamics o their light. The others are suggested aero-hydro-dynamics o lapping ins and to the practical application o lapping ins vehicles as shown Fig.5 and Fig.6. 15

25 .4 Kinematics o biomimetic in thrusters Figure 7: Description o motion o the lapping in during CFD experiment [46] In this section, it is indicated that the classical mechanisms which describes the motion o lapping ins without consideration o the causes o motion applied by NTUA National Technical University o Athens team Dr. Kostas A. Belibassakis and PHD student Vasileios Tsarsitalidis. As can be seen in Fig.7, the reerence coordinate system is considered or lapping ins. Thereore, the motion o body with lapping ins can be described in accordance to this coordinate system such as translational surge, heave and sway motions and oscillatory roll, pitch and yaw motions. As shown in Fig.7 under constant parallel velocity, the lapping in coordinate system is experiencing a combination o both heave motion and pitch motion due to the waves. In this case, two dierent basic requencies have been considered or the motion o lapping in, one o them is relative requency [47]; 1 1 And the other requency is lapping in pitching requency;. 16

26 In the simple harmonic thrust producing case, relative requency and lapping in pitching requency is equal to each other [47]. 1 And 1 The surge motion o the lapping in is shown as below x t Ut And t indicates time here. The vertical motion becomes; h t h0 sin t Where; h 0 is amplitude o vertical oscillation o the lapping in. Simultaneously, the in undergoes a pitch oscillatory motion at a possibly dierent requency yet as it is mentioned beore in the simple harmonic thrust producing case motions o the requencies are equal. Thereore, pitch oscillatory motion becomes [47]; 0 sin t t m Where θ m is mean angle o attack, θ 0 is amplitude o pitch oscillation o the lapping in and ψ is phase angle between the two movements..5 Dynamics o biomimetic in thrusters The phase dierence ψ between the two oscillatory motions is very important as ar as the eiciency o the thrust development by the lapping system is concerned. As it is discussed beore in the simple harmonic thrust producing case where: 1, it usually takes value ψ = With the pivot point or the angular motion o the in located around the 1/3 chord length rom the leading edge, a minimization o the required torque or pitching is achieved as shown Fig.8 [47]. 17

27 Figure 8: Flapping ins simulation model in CFD environment For lapping systems steadily advancing in unbounded liquid the main low parameter controlling the unsteady lit production mechanism is the Strouhal number [46]; St h / U, 0 Note that the Reynolds number has a secondary role aecting viscous drag corrections. As a result o the simultaneous heaving and pitching motions o the biomimetic in the instantaneous angle o attack is given by [47]: 1 1 t H t t tan U dh/ dt t For relatively low amplitudes o purely harmonic motion and optimum phase dierence ψ=90 o, the angle o attack becomes; 18

28 1 t U h0 0cos t which is equivalently achieved by setting the pitch angle θt proportional to θ H t and thus t w tan 1 U 1 0 wh 0 /U dh/ dt Where; w is termed the pitch control parameter ater [47], usually taking values in 0<w<1, which is amenable to optimization. Decreasing the value o w, the maximum angle o attack is reduced and the in operates at lighter loads. On the contrary, by increasing the above parameter the in loading becomes higher and so is the chance o leading edge separation that would lead to signiicant dynamic stall eects. Following [48], we exploit the above relation, as an active pitch control rule o the lapping-in thruster in the general polychromatic case, based on the time history o vertical motion. In this case, the instantaneous angle o attack is [48] t 1 w tan 1 U 1 dh/ dt.6 Free Surace Eects In the case o the biomimetic system under the calm or wavy ree surace, additional parameters enriches the above set, as the Froude number; 1/ F U /gl 19

29 where L denotes the characteristic ship length and g is gravitational acceleration, as well as various requency parameters associated with the incoming wave, like μ = ω L/g and τ = ωu/g, τ has distinguish subcritical τ <1 / 4 rom supercritical τ >1 / 4 condition [49]..7 The outline o CFD code by NTUA The hydrodynamic analysis o ship and lapping ins are employed by NTUA team Dr. Kostas A. Belibassakis and PHD student Vasileios Tsarsitalidis. First, the hydrodynamic analysis o ship is computed by developing algorithm in computational luid dynamics environment. The lapping in is considered as appendage o a speciic ship and by ollowing the hydrodynamic analysis o the ship, the hydrodynamic analysis o the lapping in is employed as well. The combined translational and pitch motion o the lapping in is induced by the ship 49. Both vertical and horizontal o arrangement o lapping in is analyzed in this study 46,48,49 as shown in Fig.9. The pitching motion o the in about its pivot axis is selected properly in order to produce thrust, with signiicant reduction o reduction o responses and generation o anti-rolling moment by the vertical in, useul or ship stabilization 49. 0

30 Figure 9: a Ship hull equipped with a horizontal lapping wing located below the keel, orward the mid-ship section. b Same hull with a vertical lapping wing located below the keel, at mid-ship [49] Ship hydrodynamic analysis has been applied by using linear theory using a Rankine source-sink ormulation and ship motions are calculated considering the additional orces and moments because o unsteady propulsion systems. Standard linearized sea-keeping analysis is used to achieve the motions and responses o ship and lapping in. The motion equations o ship the coupled equation o heave and pitch motion o the ship can be calculated regarding the mass o ship, added mass and damping coeicients 49. 1

31 A simpliied liting line model is applied or hydrodynamic analysis o lapping in to obtain expressions o the lapping in orces 49. The generated lapping orces are considered as two parts. One part is depending on the oscillatory ship amplitudes and the other part is dependent on the incoming wave potential. The irst part produces modiications o the hydrodynamic coeicients o the system the other part adds on Froude-Krylov and diraction orces in the right hand side 49. The horizontal arrangement o the lapping in thruster and vertical lapping in in quartering and beam waves can be seen in detail 49. To sum up, the analysis o lapping ins being appendage o ocean vehicles can be seen in detail 46,47,48,49.

32 .8 Harvesting power and energy by lapping ins Besides energy harvesting rom application o propellers, lapping ins have a capability o generating energy as well. We are getting energy rom lapping ins induced by vortices, reesurace waves and uniorm currents. In the irst case, at least two dierent methods are considered to generate energy rom the lapping ins; the constructive mode and the destructive mode. In the constructive mode, the vortices created by the in and the incoming vortex are in the same phase and reinorce each other [50]. The destructive mode, on the other hand, is characterized by a phase dierence o approximately between the in-generated vortices and the incoming ones [50]. Related to these studies, generating lapping in energy rom ree-surace waves has also been experienced. According to some studies, submerged ins have the ability to propel itsel orward in sea conditions once it is right under a ree surace. According to some investigations, the coupling o dierent motion modes and external activation is required in order to generate energy rom lapping ins. For instance, one o modes o lapping in is acted as a periodic motion. Hydro-dynamically, this produces periodic variations in the liting and drag orces as well as pitching- rolling moments in an incoming low. These timevarying orces/moments in their turn can trigger other modes rom which power extraction is achieved through attached generators [50]. 3

33 Figure 10: Schematics o a low energy harvesting system based upon a lapping in [50] Two-dimensional in is integrated with a damper c in incoming low U as shown in Fig.10. ρ is the luid density. Here, chord length is 1m. The in has a combined motion which is heave and pitch as described beore. Both motions are deined as harmonic motion. Heave motion has already been described as h t h0 sin t And also pitch motion has been deined as below; t 0 cos t Inertia o the in is ignored. The external moment is needed to trigger the pitching mode as M e = -M. M is deined here as hydrodynamic pitching moment. The power input into the system becomes; P i M M e The power input is obtained by the damper c and it can be expressed as; P O ch 4

34 The mean power input becomes; P i T t0 T t 0 1 Pdt i And the mean power output becomes; P 0 T t0 T t 0 1 P o dt Where, T is deined period o the system. Finally, net power o the system becomes; P P o P i In addition, the eiciency o system is analyzed with respect to net power o the system. The power harvesting eiciency becomes P 1/ U Y P Where; Y P is calculated as dierence between the highest vertical position reached by the leading and trailing edges and the lowest vertical position [50]. 5

35 CHAPTER 3 Methodology Flapping ins motion was previously describing the combination o heave and pitch motion in chapter. All outputs which are orces and moments have been analyzed in CFD code based on the input which is the motion o lapping ins. Thereore, we have discussed how to describe physics behind lapping in motion. In addition, in order to deine system identiication, the correlation between input motion o in and output heave and surge orce needs to be implemented. To do this, surge orce x and heave orce y depending on time needs to be transormed in requency domain. Hence, Fourier Transorm, as well as the method used to transorm both orces in requency domain will be discussed in this section. Furthermore, an optimization method will be developed in MATLAB in order to establish approximating transer unction model or heave orce. Nonlinear model is implemented in order to ind data points o surge orce signals. 6

36 3.1 Data Analysis or Heave Force In this section, a brie presentation o Fourier analysis is given. What kind o data do we get out o this model and why do we choose to work these data series. In addition to this, how do we validate our Fourier methodology based on our data series? In general, Fourier series can transorm any periodic signal or unction into harmonic signals or sinusoidal unctions. Thereore, all periodic unctions can be analyzed easier by using Fourier Transorm. There are many reasons to utilize Fast Fourier Transorm FFT. However, the undamental idea is using FFT in many science ields to transorm time-domain signals into requency-domain signals. This approach is very useul to deine parameters o vibrating systems [51]. The use o digital technology is on the rise in various applications because digital signal processing has many advantages in comparison to analog signal processing. In this study, heave and surge orce data series are based upon discrete time signals. Thereore, FFT Fast Fourier Transorm has been applied to all heave orce signals to transorm into requency domain by using MATLAB. Unlike analog type o signals which are essentially continuous time signals, digital technology encodes inormation using discrete-time signals as shown Fig.11 and Fig.1 [51]. Figure 11: Analog signal [51] 7

37 Figure 1: Discrete time signal - low sampling rate [51] I the variable to be represented requires ast transitions, it must be described by using a higher sampling rate as shown in Fig.13 [51]. Figure 13: Discrete-time signal - high sampling rate [51] In this study, the output which is heave and surge orce data series is assumed to consist o periodic unctions. Thereore, any initial transient stage is ignored. For period T t T surge orce can be described as below. y t c0 csin t Where; yt is the unction signal in time domain and c 0 and c are coeicients o series. The output requency is and ψ is phase angle. 8

38 With using Euler s ormula and Fourier integral, the transorm o output unction into requency domain becomes; Y y t e it dt Where ω is angular requency and Yω is a unction o amplitude and phase spectrum o output. This equation is called Fourier Transorm o yt. This analog Fourier Transorm will be needed to ind data points o the system as well in the next section. However, due to the act that our data series o heave and surges orce are composed o discrete signals 564 data, Fast Fourier Transorm methodology needs to be explained briely. Based on the FFT algorithm, the expression o requency domain o heave and surge orces have been obtained in MATLAB or each data series. First o all, the methodology o FFT will be explained and then how FFT algorithm is implemented to surge orce will be clariied. The Fast Fourier Transorm FFT is a very useul algorithm or Discrete Fourier Transorm DFT. By using DFT computation time is decreased rom N to Nlog N where N is number o samples o each surge orce data series. Discretization o the time signal needed or Discrete Fourier Transorm is illustrated in Fig.14 [51]. 9

39 Figure 14: Discretization o the time signal [51] FFT algorithm is based on the act that every discrete Fourier transorm with N samples can be divided into two Fourier transorms, each with N/ samples irst with even samples and second with odd samples as shown in Fig.15 [51]. 30

40 31 Figure 15: Dividing o the signal into the two new signals [51] Fourier Transorm becomes sum o two new smaller Fourier transorms: 1 0 N r N r i r e Y Y N r N r i r N r N r i r e Y e Y / 1 / N r N r N r i r N i N r i r e Y e e Y Where r is sample number. We have two new Fourier transorms in equations above so that they can be deined by real variables [51] 1 0 / N r N r i r e Y A 1 0 / 1 N r N r i r e Y B And the complex variable becomes;

41 W e i N Because equations can be combined between each other, FFT equation which is used or most o digital signal processing can be obtained as below [51]; Y A W B It is better to explain output data series structure beore explaining how Fourier Transorm applied to surge orce and heave orce data series. For each selection o geometry, amplitude o vertical oscillation o the lapping in to chord ratio h/c, phase angle between heave motions and pitch oscillation ψ, the mean angle o attack θ m are created and all iles corresponding to this set are gathered. This typical set has simulations or ive Strouhal numbers ranging rom 0.1 to 0.7 and the amplitude o pitch oscillation rom 5 degree to maximum 75 degree. For each run, the mean angle o attack θ m is 0 degree and phase angle ψ is 90 0 [46]. As shown in Appendix B, 564 output data series are obtained rom CFD. Each column includes iterations, time, surge orce, heave orce, sway orce, roll moment, yaw moment and pitch moment respectively [46]. Strouhal number Str and pitch motion amplitude θ 0 deine each run, its stands or the time steps used and time or the simulated duration. All orces and moments are the mean values divided by water density ρ, pow stands or power also divided by ρ, tra stands or translational and rot stands or rotational., numbers 1,,3 stands or the axes x,y,z and min, max, dev stand or minimum, maximum and standard deviation values Regarding geometrical parameter during CFD experiment, NACA 001 standard in is used or all data series. As ar as an individual lapping in is concerned, the selection o platorm area, in conjunction with horizontal/vertical sweep and twist angles, and generating shapes ranging rom simple orthogonal or trapezoidal-like ins to ish-tail like orms, constitutes 3

42 the set o the most important geometrical parameters. Other important parameters are the in aspect ratio, span-wise distribution o chord, thickness and possibly camber o in sections, as well as the speciic in-sectional orms. Here, some o in geometries regarding rectangular and ish type used or experiment are presented in Fig.16, Fig.17 and Fig.18; Figure 16: Rectangular Fin outline or s/c =, 4, 6 respectively [46] 33

43 Figure 17: Fish-like in outline or s/c=4 and s = 15 0, 30 0, 45 0 respectively [46] Figure 18: Cross section area or NACA 001 [5] As it was mentioned beore two types o ins were used in data structure which is ish like and rectangular ins. The geometric parameters are aspect ratio AR and skewback angle such ish15. For each selection o geometry, heave to chord ratio, phase angle, mean angle o attack and position o pitching axis is created. As known, 6 degrees o reedom 3 orces and 3 moments induced rom the oscillating ins. In this study, surge orce and heave orce are chosen to be analyzed. From the aspect o hydrodynamics and control point o engineering, surge orce is playing undamental role to generate thrust and heave orce is playing important role or lit. 34

44 First o all, FFT Fast Fourier Transorm needs to be calculated and illustrated in some kind o ramework in order to make data analysis or heave orce data series. In MATLAB, an algorithm is developed to transorm heave orce data into the Fourier domain. Due to the act that heave orce time signals are discrete type o signal, Fast Fourier Transorm, FFT is applied or all heave orce signal in order to compute dominant requency. The excitation requency is needed to be computed in order to decide the system is whether linear or nonlinear and what kind o transer unction model can be used to make data itting or all heave orce data series. To do this the ormula as below can be used to calculate the excitation requency which is or heave and pitch motion. StU motion h 0 Where; motion becomes excitation requency or each data, St is Strouhal number, U is low velocity and h 0 is heave motion amplitude that can be calculated as below; h h0 c h/c is heave to chord ratio which is deined or each data and c is chord length that is 1m or all type o ins. Now, or all data FFT analysis results can be seen in Appendix D. The way o obtaining heave orce spectrums can be seen in Fig. 19 as well. 35

45 Figure 19: FFT low chart However, some o them that were identiied as representative o important cases can be seen in log-log plots or heave orce data series in Fig.0, Fig.1, Fig., Fig.3, Fig.4 and Fig.5; 36

46 Figure 0: Heave orce spectrum - requency domain pattern or one type o data Figure 1: Heave orce spectrum - requency domain pattern or one type o data 37

47 Figure : Heave orce spectrum - requency domain pattern or one type o data Figure 3: Heave orce spectrum - requency domain pattern or one type o data 38

48 Figure 4: Heave orce spectrum - requency domain pattern or one type o data Figure 5: Heave orce spectrum - requency domain pattern or one type o data 39

49 All plots generated in MATLAB have been observed to achieve data analysis and deine heave orce system ater Fast Fourier Transorm or all data series. The irst order requency has been become dominant requency o all data spectrum which means the peak requency is equal to the excitation requency which is the requency o motions or almost each data series. Thereore, heave orce system is acting such as linear system. We can collect data points by calculating the value o spectrum with respect to excitation requency on the spectrum plots. All data points have been calculated by developing codes in MATLAB. Now, the heave orce system is ready to develop describing unction to build transer unction between inputs and outputs. 3. Describing Function or Heave Force The data itting concept achieved or heave orce data series is explained in this section. Transer unction will be generated between inputs and outputs or each amplitude to create mathematical model or this system as shown in Fig.6. Figure 6: Describing unction heave orce system Since CFD data points have been gotten, second order transer unction is used or data itting with respect to dierent ish type geometries and dierent operational conditions depending on heave and pitch motion amplitude. For suitable data itting process, the sample second order transer unction as below is used to compare to CFD data series or heave orce system. 40

50 ^ k s z H s s 100 s 00 The purpose is to ind the gain k and zero z along with suitable method or each speciic case depending on the geometry o in and operational circumstances. Here, the poles in the transer unction have been chosen arbitrarily because poles aect gain in this ormula. I dierent poles are deined, the gain will become dierent. The optimization algorithm is developed in order to calculate gains and zeros eectively in MATLAB. An optimization algorithm to make minimum error between approximation transer unction and CFD data points has been developed in MATLAB. Thus, gains and zeros creating small errors have been calculated by using suitable optimization method. The objective unction is deined as below which is calculating errors between transer unction model and CFD data; ObjFunc n i1 ^ 0 log 10 H i log 10 dp i Here gains and zeros are optimization parameters. In MATLAB, the optimization code is created to calculate eectively these parameters or some speciic in geometry and operational points. Unconstrained optimization minimization method is used to minimize the objective unction. In result section, the optimization result, gains and zeros can be seen or two geometries and operational cases. Here the transer unction is compared with ish-like in with skewback angle 15 ish15 with aspect ratio 4 AR4 and skewback angle 15 ish15 with aspect ratio 6 AR6. The operational points 5 degrees, between degrees, between degrees, between degrees pitch amplitudes and 1 m, 1.5 m and m heave amplitudes have been investigated or data itting. 41

51 All pass ilter signal processing can be used to ix the phase o transer unction model. The aim o all pass ilter is to add phase shit delay to the response. An all pass ilter is allowing through all requencies without changing the magnitude o transer unction model. Its magnitude does not dierentiate any requency with all pass ilter but it ixes our transer unction phase with appropriate all pass ilter design. The general transer unction o an all-pass ilter can be seen as below; ^^ H s 1 c n c n 1 s... s n 1 c1... c n s n Where; c are the coeicients here. They depend on the order o the all pass ilter. c must contain one, two, three, or our real elements. For instance, in our case c with two elements generates a second order all pass ilter. Now, the ixed transer unction can be calculated by using an all-pass ilter. The all pass ilter magnitude becomes; The all pass ilter phase becomes; ^^ H i 1 ^^ H i Now, the ixed transer unction model can be calculated based on the inormation. The magnitude o transer unction model that ound by using optimization algorithm does not alter but the ixed phase o transer unction model becomes as below; ^ H i H i H i ^^ 4

52 3.3 Data Analysis or Surge Force In MATLAB, code is developed in order to transorm surge orces x into the requency domain by implementing the FFT algorithm mentioned beore. Due to the act that surge orce time series o discrete type, Fast Fourier Transorm is applied to all surge orce signals in order to determine the dominant requency. Here, all data series is examined in order to deine which the dominant requency that surge orce signals is. Each data series has the same low velocity which is.3 m/s and the chord length o the ins is 1 m. In addition, Strouhal number, heave oscillating amplitude and pitch oscillating amplitude are varying. First o all, the requency o heave and pitch motions has been calculated by using the Strouhal ormula as below; StU motion h 0 Ater calculation o the requency o motion, the FFT algorithm has been applied to all surge orce data series in order to obtain transorm in requency domain and semi-logarithmic scale. Now, observations can be done or all surge orces. For instance, or some o the FFTs o surge orce data analysis, a pattern can be identiied as shown in Fig.7, Fig.8, Fig.9, Fig.30, Fig.31 and Fig. 3 semi-log plots; 43

53 Figure 7: Surge orce spectrum - requency domain pattern or one type o data Figure 8: Surge orce spectrum - requency domain pattern or one type o data 44

54 Figure 9: Surge orce spectrum - requency domain pattern or one type o data Figure 30: Surge orce spectrum - requency domain pattern or one type o data 45

55 Figure 31: Surge orce spectrum - requency domain pattern or one type o data Figure 3: Surge orce spectrum - requency domain pattern or one type o data 46

56 During the data analysis process, each surge orce FFT set is observed in order to see whether there is zero, second order harmonics. However, as can be seen in the analysis above, zero and second harmonics are the dominant in comparison to other harmonics. Per the observation o all output surge data series, motion requency combined heave and pitch oscillation is hal o the dominant second order harmonic requency o the FFT spectrum analysis or surge orces. Also, the spectrum value which is a complex number can be calculated with respect to zero order and second order requency or all data series. Surge orce system seems to be a nonlinear system based on this inormation. Thus, the response s Associated Transorm will be used to make data analysis or surge orce. In addition, in order to validate the FFT algorithm developed in MATLAB, comparison was made between the period o some o surge data in time domain and the requency o surge orce analyzed by FFT code in MATLAB. 1 T Comparison or one data series is given in Fig.33 and Fig

57 Figure 33: One o surge orce in time domain or irst data Figure 34: FFT analysis spectrum - requency or irst data series 48

58 The dominant requency in Fig.34 is 0. [1/s] which corresponds to a 4.54 [s] period and the surge motion period in Fig.33 is close to 4.5 [s]. Using such comparisons, the FFT code can be validated to work. In summary, by using the FFT algorithm in MATLAB, surge orce data series was transormed rom time domain to requency domain. Thus, surge orce can then be observed easily and it is possible to observe whether surge orce data series is linear or nonlinear system. As mentioned earlier, most o surge orce data outputs have double signiicant content o requency in comparison to the requency o motion. Thereore, surge orce will be analyzed as a nonlinear system. Next step will be the explanation o the response s Associated Transorm and how to implement or our surge orce data series to obtain data points. The inputs and the outputs o the lapping in system have been deined beore. It will now be presented how to implement the response s Associated Transorm or input signals and output signals. First o all, the nonlinear data analysis method will be explained with Associated Transorm. In addition to this, how the Associated Transorm is used to compute data points o the surge orce system. Here the surge orce generation system can be considered as a black box along with unknown nonlinear properties o lapping in system. This black box is time invariant which means that the properties o black box do not depend on time [53]. Signal yt is called the system s response and At is called the virtual input signal that combines heave and pitch motion. The nonlinear surge orce system can be illustrated in Fig

59 Figure 35: Nonlinear surge orce system The Associated Transorm is used in order to calculate the data points o the system. The Fourier Transorm values o inputs and outputs have been used or computing transer unctions o the system. Beore explaining how to obtain each data, it is better to provide some undamental inormation about response computation and the Associated Transorm. Associated Transorm is needed in order to calculate data points. The Fourier Transorm o output signal Y will be derived rom Y n 1,, by reducing all but one variables. Then a single variable inverse Fourier Transorm is necessary to compute yt. However, in our process, Fourier Transorm is initially implemented or input and output signals. Then, the data points o the system can be calculated given each point o Fourier Transorm coming rom CFD analysis. The process is to calculate Y is called the associated transorm [54]. The notation below is used to denote association o variables; Y An [ Yn 1,..., ] In this study, n= since second order nonlinear system is considered. Now, the extension to the general case can be made easily. For the Fourier Transorm Y s 1,, the associated transorm can be expressed as below; 50

60 51 ], [ 1, Y A Y Inverse Fourier Transorm can be then expressed as below; , 1, d d e e Y i t t y t i t i And along with setting t 1 =t =t, the equation can be written as it is; ], 1 [ 1 d e d e Y i i t y t i t i Now the variable o integration needs to be changed by writing 1 ], 1 [ 1 d e d e Y i i t y t i t i All in all, once the order o integration is changed, the equation below is obtained; d e d Y i i t y i t ], 1 [ 1 It can be seen rom the equation above, that term in the bracket should be equal to ] [ t y F Y And the proo is complete.

61 I the integral is rewritten, another ormula or the association operation is obtained 1 Y 1, 1 1 i Y d Ater providing basic inormation about the Associated Transorm, the theory is implemented or the CFD data series in MATLAB. First o all, the Fourier Transorm is applied to virtual input signals. This unction is deined as virtual ictitious signal due to the act that it consists o heave motion and pitch motion. This combined input signal is created in order to calculate the points o transer unctions o the system eectively. At virtual input signal is deined as below; A t Asin Ft Where; A is deined as virtual amplitude and deined as below; A h 0 0 Where; h 0 is deined as the heave motion amplitude and θ 0 as pitch motion amplitude. Ater deining the virtual ictitious input signal, the Fourier Transorm o the virtual input will be applied or transorm. A A t e it dt Asin Ft e it dt Here, Euler s theorem will be used or sine unction; 5

62 it it e e cos t & e sin t it e i it A A e ift e i ift e it dt A [ i e ift e it dt e ift e it dt] By using Dirac s delta unction property, the virtual input signal in Fourier domain can be obtained as below; e ift dt F A A [ F F] i This equation will be used or computing Fourier Transorm o the output unction with Associated Transorm. Surge orce data points can be calculated based on Associated Transorm. Surge orce signals can be expressed beore as below; Y 1, H 1, A 1 A Data points can be expressed as below; Y 1, H 1, A A 1 53

63 as below; Along with the transer unction points relationship above, surge orce can be expressed 1 Y d i Y, Data points relationship can be substituted into surge orce signal and the new expression between input and output can be achieved along with Associated Transorm. 1 1 Y H, A A d i This equation above shows the relationship between the combined motion o the lapping in and the surge orce obtained by CFD code. be; The input signal which is the motion o lapping ins can be ound in Fourier domain to A A [ F F] i The input signal can be expressed as below; A A [ 1 F 1 i 1 F ] Along with Associated Transorm, the equation can be expressed as below 1 54

64 55 ] [ F F i A A Another input expression becomes; ] [ F F i A A The correlation between the combined motion o the lapping ins and the surge orce signal has been ound beore;, 1 d A A H i Y H represents data points o the surge orce system. Now the input signals derived earlier will be plugged into the relationship between input and output giving; ] [ ] [, 1 d F F i A F F i A H i Y Ater some algebra, the equation becomes;, 8 F F F F F F F F H i A Y And i more algebra is achieved, the equation becomes

65 56 ],,,, [ 8 d F F H d F F H d F F H d F F H i A Y The response surge orce equation has 4 terms; each term will be investigated one by one to provide a simpler expression. The irst term in the equation becomes;..., 8 d F F H i A Y This equation must be solved based on Dirac delta unction property which is shown below; 1 d Thereore, once = F is set or the irst term o response surge orce equation, the equation becomes;..., 8 F F F H i A Y The second term in the equation becomes;..., [... 8 d F F H i A Y

66 57 Here, should be set F to solve the second actor, the equation becomes;..., [... 8 F F H i A Y The third actor in the equation becomes;..., [... 8 d F F H i A Y Here, should be set F to solve the actor, the equation becomes;..., [... 8 F F H i A Y The last actor in the equation becomes;, [... 8 d F F H i A Y Here, should be set -F to solve the actor, the equation becomes; ], [... 8 F F F H i A Y Ater these calculations, once all terms are arranged, surge orce response spectrum becomes;,,,, 8 F F F H F F H F F H F F F H i A Y Now, output signal which is surge orce in Fourier domain can be calculated, once the Dirac delta argument o each term is set to zero. Thereore, the equation becomes;

67 A H F, F F H F, F Y 8i H F, F H F, F F In order to ind the data points o the system or surge orce signals, surge orce signals in Fourier domain needs to be ound by using CFD data. The output signal rom CFD data has been calculated in previous section as below; y t c0 c sin F t Now this output signal will be calculated in order to ind in Fourier domain as calculated beore or input signal in Fourier domain. Y it c0 e dt c sin F t e it dt Where; c 0 and c coeicients in the equation. By using Euler s ormula and Dirac delta properties, we are able to ind output signal rom CFD data. e ift dt F c i Ft i Ft Y c0 e e e i it dt becomes; And then, the mathematical process can be applied to this equation and the equation c i Ft i it Y c0 e e e dt i e i Ft e e i it dt 58

68 59 dt e e e i c dt e e e i c c Y t i Ft i i t i Ft i i 0 Now by using Dirac delta unction property, surge orce signal rom CFD data series in Fourier domain can be computed as below; 0 F e i c F e i c c Y i i By using Associated Transorm, surge orce signal in Fourier domain has been calculated. In addition to this, the surge orce signal calculated rom CFD data series has been used to compute the surge orce signal in Fourier domain. These equations represent the same mathematical phenomenon. Now, the data points o the system can be computed. Surge orce signal in Fourier domain;, 8, 8, 8, 8 F F F H i A F F H i A F F H i A F F F H i A Y The surge orce signal in Fourier domain rom CFD data 0 F e i c F e i c c Y i i 1 F e b F e b b Y i i Here, b 1 and b can be ound rom FFT analysis spectrum previously mentioned by developing code in MATLAB.

69 As can be seen rom the two equations, the coeicients o δ, δ-f and δ+f are the same and data poimts can be calculated rom the coeicients; First o all, or δ-f coeicient, the equation can be written as below; A b H F, F 8i 8bi H F, F A In this equation, the coeicient b or =F is computed or each surge data in MATLAB. In addition, A is deined as the virtual input signal amplitude developed or each data as well in MATLAB. A h 0 Q 0 h 0 is deined as heave motion amplitude and Q 0 as pitch motion amplitude. For δ+f coeicient, the equation can be written as below; A b H F, F 8i 8bi H F, F A The coeicient b or = F and A are computed with the same method or each surge data points in MATLAB. For δ coeicient, the equation can be written as below; 60

70 61 1,, 8 b F F H F F H i A Here, the coeicient b 1 or = 0 is calculated rom FFT data spectrum in MATLAB. It holds that:,, F F H F F H I this property is plugged into the equation, it becomes; 1 8,, A i b F F H F F H Thereore, the real part o H-F,F needs to be set to zero. The imaginary part can be ound by generating code in MATLAB as below; 1 8, Im A b F F H The imaginary part o transer unction HF,-F can be ound as well. Now, all data points o surge orce have been computed. Each data point o the surge orce system can be seen in the chapter 4 in 3-D plots. The mathematical meaning o these results is going to be discussed in the conclusion part.

71 CHAPTER 4 Results This chapter shows and discusses all o the plots and results or lapping ins CFD data. The methodology o the calculations using MATLAB was explained in the previous chapter. First, the FFT spectrum analyses o all heave and surge orce rom CFD data is generated in MATLAB. Then, all o these spectra were observed or suitable method in order to develop mathematical model between inputs the combined motion o the lapping ins and outputs the surge or heave orces o the system. Transer unction model can be built or system identiication or heave orce system because this system is pseudo-linear. Also, the codes generated in MATLAB to compute data points o the surge orce data series were based on nonlinearity. In this chapter, the results are shown and discussed. 6

72 4.1 Comparison Transer Function Model and CFD Data or Heave Force As it was mentioned beore, the comparison between transer unction model and CFD data has been implemented or two types o ish-like ins which are ish15ar4 ish type skewback angle 15 and aspect ratio 4 and ishar6 ish type skewback angle 15 and aspect ratio 6. The transer unction below has been used to generate mathematical model between input and output or two types o ish-like ins. ˆ k s z H s s 100 s 00 Where; k is the gain and z the zero or this type o system. - 8; All gains and zeros or varying pitch motion amplitude are given in the ollowing Table 3 Table 3: ish15ar4, Heave Amplitude = 1m Pitch Amplitude Gain, k Zero, z θ 0 = x x θ x x θ x x θ x x

73 Table 4: ish15ar4, Heave Amplitude = 1.5 m Pitch Amplitude Gain, k Zero, z θ 0 = x x θ x x θ x x θ x x10-09 Table 5: ish15ar4, Heave Amplitude = m Pitch Amplitude Gain, k Zero, z θ 0 = x x θ ex x θ ex x θ ex x10-09 Table 6: ish15ar6, Heave Amplitude = 1 m Pitch Amplitude Gain, k Zero, z θ 0 = x x θ x x θ x x θ x x

74 Table 7: ish15ar6, Heave Amplitude = 1.5 m Pitch Amplitude Gain, k Zero, z θ 0 = x x θ x x θ x x θ x x10-10 Table 8: ish15ar6, Heave Amplitude = m Pitch Amplitude Gain, k Zero, z θ 0 = x x θ x x θ x x θ x x

75 In the methodology section, it was mentioned that unconstrained optimization minimization method is used to perorm data itting or dierent type o geometry and operational points. The optimization results can be seen Table 9-14; Table 9: Optimization result ish15ar4, Heave Amplitude = 1m Pitch Amplitude Obj. unc. Initial Value Obj. unc. Initial Value θ 0 = e θ e θ e θ e Table 10: Optimization result ish15ar4, Heave Amplitude = 1.5 m Pitch Amplitude Obj. unc. Initial Value Obj. unc. Initial Value θ 0 = e θ e θ e θ e Table 11: Optimization result ish15ar4, Heave Amplitude = m Pitch Amplitude Obj. unc. Initial Value Obj. unc. Initial Value θ 0 = e θ e θ e θ e

76 Table 1: Optimization result ish15ar6, Heave Amplitude = 1 m Pitch Amplitude Obj. unc. Initial Value Obj. unc. Initial Value θ 0 = e θ e θ e θ e Table 13: Optimization result ish15ar6, Heave Amplitude = 1.5 m Pitch Amplitude Obj. unc. Initial Value Obj. unc. Initial Value θ 0 = e θ e θ e θ e Table 14: Optimization result ish15ar6, Heave Amplitude = m Pitch Amplitude Obj. unc. Initial Value Obj. unc. Initial Value θ 0 = e θ e θ e θ e

77 Some results o the data itting or heave orce can be seen below in the plots. That ollow the process has been done or one type o lapping in which is ish-like in having 15 degree skew-angle. Two ish like geometries have been investigated ish15ar4 and AR6 rom the CFD data. The transer unction model developed in MATLAB and CFD data can be seen in each plot or every ish-like geometry and operational point. For type ish15ar4 and operational point having 5 degrees pitch amplitude and 1 m heave amplitude, it can be seen in Fig.36 how the transer unction model developed approaches the CFD data. Each point represents one data point with respect to excitation requency and the error can be seen between the mathematical model and CFD data. Figure 36: Comparison between CFD data and heave orce transer unction model 68

78 Fish15AR4 has been investigated with speciic operational points. However, since the operational point is deined by the pitch motion amplitude, varying rom 1 to 13.7 degrees amplitude around 1 degrees pitch motion has been chosen to implement data itting into CFD data, along with 1.5 m heave motion amplitude. The data itting result regarding this speciic operational point as shown in Fig.37. Figure 37: Comparison between CFD data and heave orce transer unction model 69

79 Fish15AR4 has been investigated with speciic operational points. However, since the operational point is deined by the pitch motion amplitude, varying rom 14.4 to 16.6 degrees amplitude around 15 degrees pitch motion has been chosen to implement data itting into CFD data, along with m heave motion amplitude. The data itting result regarding this speciic operational point as shown in Fig.38. Figure 38: Comparison between CFD data and heave orce transer unction model 70

80 Fish15AR6 has been investigated with speciic operational points. However, since the operational point is deined by the pitch motion amplitude, varying rom 0 to 3.7 degrees amplitude around 0 degrees pitch motion has been chosen to implement data itting into CFD data, along with 1 m heave motion amplitude. The data itting result regarding this speciic operational point as shown in Fig.39. Figure 39: Comparison between CFD data and heave orce transer unction model 71

81 Fish15AR6 has been investigated with speciic operational points. However, since the operational point is deined by the pitch motion amplitude, varying rom 14.4 to 16.6 degrees amplitude around 15 degrees pitch motion has been chosen to implement data itting into CFD data, along with 1.5 m heave motion amplitude. The data itting result regarding this speciic operational point as shown in Fig.40. Figure 40: Comparison between CFD data and heave orce transer unction model 7

82 For type ish15ar6 and operational point having 5 degrees pitch amplitude and m heave amplitude, it can be seen in Fig.41 how the transer unction model developed approaches the CFD data. Each point represents one data point with respect to excitation requency and the error can be seen between the mathematical model and CFD data. Figure 41: Comparison between CFD data and heave orce transer unction model 73

83 4. Data points o Surge orce 3D plots have been created or data points o surge orce system. x-y plane is deined as excitation requency [1/s] plane and z axis is deined as magnitude or phase [degrees]. All data points ish15 and ish30 can be seen in plots. Figure 4: Magnitudes o HF,F The magnitude points o HF,F have been plotted to develop approximation model between inputs and outputs as shown in Fig.4. 74

84 Figure 43: Phases o HF,F All phase points o HF,F in degrees can be seen or surge orce with respect to excitation requency as shown in Fig

85 Figure 44: Magnitudes o H-F,-F All magnitude points o H-F,-F can be seen or surge orce with respect to excitation requency as shown in Fig

86 Figure 45: Phases o H-F,-F All phase points o H-F,-F in degrees can be seen or surge orce with respect to excitation requency as shown in Fig

87 Figure 46: Magnitudes o H-F,F All magnitude points o H-F,F can be seen or surge orce with respect to excitation requency as shown in Fig

88 Figure 47: Phases o H-F,F All phase points o H-F,F in degrees can be seen or surge orce with respect to excitation requency as shown in Fig

89 Figure 48: Magnitudes o HF,-F All magnitude points o HF,-F can be seen or surge orce with respect to excitation requency as shown in Fig

90 Figure 49: Phases o HF,-F All phase points o HF,-F in degrees can be seen or surge orce with respect to excitation requency as shown in Fig

91 CHAPTER 5 Conclusions This study provides a closed-orm phenomenological model speciically or heave orce that can produce the useul output signal without using CFD, i the speciic conditions are met. In addition, it is an identiication method o nonlinear system generating e.g. heave orce. Since the data points o surge orce are readily available, the study is open to expand to deine system or nonlinear surge orce generation as well. 5.1 The Contribution o the Study CFD sotware is normally slow, and expensive to buy. Also, the results are not in closed orm. For example, i one parameter s value is changed, CFD run needs to be repeated. As it was pointed out, the result o CFD is not in closed orm relationship which means that the correlation between orce and motion, velocity or amplitude is unknown. Thereore, i one parameter changes, the eect o the changing parameter is unknown. Since the mathematical system built between inputs and outputs especially or heave orce has a closed orm, all these drawbacks are eliminated. For each speciic geometry and operational point, a transer unction is known. This comes with mathematical advantage as well. We can look into all parameters insightully and much more eiciently. In addition, since all data points are known or surge orce response, this study can be expanded to investigate in order to build a mathematical system as nonlinear black box or this variable as well. 8

92 5. Future Work A potential uture study based on this work would include system identiication or surge orce. Due to knowledge o all necessary inormation o surge orce system, it can be expanded to develop nonlinear transer unction describing unction model by using appropriate methodology. For this study, a Volterra model could be a suitable method to obtain mathematical model that properly predicts the hydrodynamic response o lapping ins. Then such model can be used to make some analysis without costly re-execution o the CFD code [55], since surge orce can be predicted by a nonlinear black box system. Volterra theory o nonlinear systems provides a mathematically rigorous approximation technique to describe these unsteady hydrodynamic eects. Another track o uture work might aim to investigate dierent kind o geometries like other ish or rectangular type o ins. In addition to this, the mathematical system can be deined or other degrees o reedom such as sway orce. 83

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97 Appendix Appendix A: CFD older tree made spectrum analysis 88

98 Appendix B: One o the CFD output data 89

99 90