Numerical Analysis of Transom stern high speed ship
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SEAKEEPING OF HIGH SPEED SHIPS WITH TRANSOM STERN AND THE VALIDATION METHOD WITH UNSTEADY WAVES AROUND SHIPS A Dissertation by Muniyandy ELANGOVAN Submitted in Partial Fulﬁlment of the Requirements for the Degree of Doctor of Engineering Graduate School of Engineering December 2011 JAPAN
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Abstract Improvement in seakeeping qualities of the ship hull through numerical computation isalways a demand from shipping industry to have the better hull to reduce the resistance whichis related with the engine power requirement. So far, plenty of eﬀort has been made to improvethe estimation accuracy of the seakeeping qualities that are the hydrodynamic forces, motions,wave ﬁeld around a ship and added wave resistance. Nowadays, Rankine Panel Method is popularto carry out this numerical analysis. In this thesis, the Rankine panel method in the frequencydomain is developed and applied to several kinds of ships including the high speed transomstern for the purpose of conﬁrming its eﬃciency. The Neumann-Kelvin and Double-body ﬂowformulations are examined as a basis ﬂow. As numerical estimation methods become higher grade, more accurate and detailed exper-imental data for their validation is required. An unsteady wave pattern is being used as one ofthe methods for that purpose. To capture the wave pattern, Ohkusu, RIAM, Kyushu University,Japan has developed a method to measure the waves and analysis the added wave resistanceby means of unsteady waves. In this thesis, the analysis method is improved by including theinteraction eﬀect of steady wave and the incident wave in the original method. A ModiﬁedWigley hull is analyzed numerically for the unsteady waves and compared with experimentaldata obtained by the present method. This interaction eﬀect has been observed remarkably inthe comparison. In addition to the unsteady wave ﬁeld, hydrodynamics forces, motions andadded wave resistance are also compared with experimental data. To treat the high speed vessel which has a transom, a new boundary condition has beenintroduced. This condition has been derived from the experimental observation which conﬁrmsthat the transom stern part is completely dry at the high forward speed. From this point ofview, the boundary condition is formulated at the transom stern just behind stern to implementin the potential theory panel code. This condition corresponding to the Kutta condition inthe lifting body theory. High speed monohull and trimaran are taken for the analysis, and thecomputed numerical results are compared with experimental data. Inﬂuence of a transom sternis observed in hydrodynamic forces and moments, ship motions, unsteady waves and addedwave resistance. It is concluded that the new transom boundary condition can capture thehydrodynamics phenomena around the transom and this can improve the estimation accuracyof the seakeeping qualities in numerical computation for this kind of vessel. ii
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Acknowledgements The completion of this thesis has been facilitated by several persons. I would like to thankall of them for their help and cooperation during this research. First, I would like to express my deepest gratitude to my academic supervisor Prof. Hidet-sugu Iwashita, who has given me an opportunity to do research under his supervision. Histechnical advice and guidance have signiﬁcantly contributed to the success of this research andalso thank him for his valuable time to explain the critical research point which was raised duringthe research period. He has helped me and my family in terms of advice and ﬁnancial supportwithout which will be diﬃcult to complete this research. Myself and my family members willremember in our lifetime and thankful to him. It is my great pleasure to thank Prof Yasuaki Doi, Prof Hironori Yasukawa and Prof. HidemiMutsuda for evaluating this thesis and for their valuable suggestions. I take this opportunity tothank Prof. Mikio Takaki, and we had a departmental party in the ﬁrst year which is memorablein my life. I would like to thank students from Airworthiness and seakeeping for vehicles laboratory,for their direct and indirect support. I also thanks to Mr. Tanabe and Mr. Ito for their supportin the ﬁnal stage of thesis preparation. I very much acknowledge the help of the staﬀ, Facultyof Engineering Department and Graduate School of Engineering. My deepest thanks go to my father, mother, wife, two daughters and father in-law whoselove, aﬀection and encouragement during the period of course had been main back born to keepme with more conﬁdent and motivation towards the completion of this research. My stay and studies in Japan have been supported in-terms of scholarship of the JapaneseGovernment, Ministry of Education, Science, Sports and Culture, for which I am very thankful. I would like to thank ”www.google.com” making the search part and translation of Japanesedocument very easy and fast. My last thanks but not least, will be for the citizens of Japan and especially Saijo city oﬃceand people for their great cooperation for my family stay and me during our stay. I had a fewopportunities to participate in some of the Japanese cultural program in Japan, which madeunforgettable in my life and love their traditional culture. Therefore, once again, I am thankfulto the people in Japan. Muniyandy Elangovan December, 2011 Higashi-Hiroshima Shi iii
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DedicationTo All Citizens of Japan and My Family Members iv
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NomenclatureAcronyms φj Radiation PotentialBEM Boundary Element Method Ψ Total Velocity PotentialBVP Boundary Value Problem ρ Fluid DensityCFD Computational Fluid Dynamics σ Source StrengthEUT Enhanced Uniﬁed Theory τ Reduced FrequencyGFM Green Function Method ξj Motion in j-th DirectionHSST High Speed Strip Theory ζ Free Surface ElevationLES Large-Eddy Simulation ζ7 Diﬀraction waveRANS Reynolds Averaged Navier-Stokes ζj Radiation wave in jth DirectionRPM Rankine Panel Method ζs Steady Wave ElevationTSC Transom Stern Condition Mathematical SymbolsGreek Symbols Two Dimmensional Laplacian with re- spect to x and yα Displacement Vector GML Longitudinal Metacentric Heightχ Encounter Angle of Incident Waves GMT Transverse Metacentric Heightλ Wave Length mj jth Component of m Vectorω0 Wave Circular Frequency nj jth Component of n Vectorωe Encounter Circular Frequency n Unit Normal VectorΦ Double body ﬂow potential A Wave Amplitudeφ Unsteady velocity Potential Aij Added Mass acting in i-th direction dueφ0 Incident Wave Potential to j-th Motionφ7 Scattering Potential v
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viBij Damping Coeﬃcient acting in i-th di- Miscellaneous rection due to j-th Motion V Steady Velocity VectorCb Block Coeﬃcient B BreadthCij Matrix of Restoring Coeﬃcients d DraughtD/Dt Substantial Derivatives g Gravitational AccelerationEj Exciting Force in j-th Direction H Kochin FunctionFj Steady Force in j-th Direction k Moments of Inertia and Centrifugal Mo-Fn Froude Number mentG Green Function k1 , k2 Elementary WaveH Wave Height L Length Between PerpendicularsK Wave Number NF Number of Elements on Free SurfaceK0 Steady Wave Number NH Number of Elements on Hull SurfaceKe Encounter Wave Number NT Total Number of ElementsMij Mass Matrix Associated with Body NF A No. of Elements on Transom Surfacep Unsteady Pressure P (x, y, z) Field Pointps Steady Pressure Q(x, y, z) Source PointU Forward Speed of the Ship RAW Added Wave Resistancexg , zg Coordinates of the Center of Gravity SC Transom Surface SF Free Surface SH Hull Surface Sw Waterline Area xw Center of Water Line Area
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List of Figures2.1 Body boundary bondition - coordinate system . . . . . . . . . . . . . . . . . . . 132.2 Transom stern with steady wave . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3 Deﬁnition of problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4 Basis ﬂow approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.1 SPM - Comp. of wave pattern and pers. view for Fn = 0.2, Ke = 5.0, τ = 0.447 . 413.2 SPM - Comp. of wave pattern and pers. view for Fn = 0.2, Ke = 10, τ = 0.633 . 413.3 SPM - Comp. of wave pattern and pers. view for Fn = 0.2, Ke = 20, τ = 0.894 . 423.4 SPM - Comp. of wave pattern and pers. view for Fn = 0.2, Ke = 30, τ = 1.095 . 423.5 SIM - Comp. of wave pattern and pers. view for Fn = 0.2, Ke = 5.0, τ = 0.447 . 433.6 SIM - Comp. of wave pattern and pers. view for Fn = 0.2, Ke = 10, τ = 0.633 . . 433.7 SIM - Comp. of wave pattern and pers. view for Fn = 0.2, Ke = 20, τ = 0.894 . . 443.8 SIM - Comp. of wave pattern and pers. view for Fn = 0.2, Ke = 30, τ = 1.095 . . 443.9 Numerical treatment of transom stern . . . . . . . . . . . . . . . . . . . . . . . . 453.10 Panel shift method for the transom stern problem . . . . . . . . . . . . . . . . . 454.1 Motion free experimental setup diagram . . . . . . . . . . . . . . . . . . . . . . . 484.2 Forced motion experimental setup diagram . . . . . . . . . . . . . . . . . . . . . 494.3 Schematic diagram for the wave measurement by multifold method . . . . . . . . 554.4 Wave propagation with respect to time . . . . . . . . . . . . . . . . . . . . . . . . 565.1 Plans of the modiﬁed Wigley hull . . . . . . . . . . . . . . . . . . . . . . . . . . 625.2 Computation grids for Rankine panel method . . . . . . . . . . . . . . . . . . . 635.3 Eﬀect of number of wave-probe in accuracy of wave pattern analysis (Diﬀraction wave at y/(B/2) = 1.4 for Fn = 0.2, λ/L = 0.5, χ = π) . . . . . . . . . . . . . . . 65 x
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List of Figures xi 5.4 2nd-order term of wave pattern analysis (Diﬀraction wave at y/(B/2) = 1.4 for Fn = 0.2, λ/L = 0.5, χ = π) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.5 Interaction eﬀect between double-body ﬂow and incident wave in wave pattern analysis (Diﬀraction wave at y/(B/2) = 1.4 for Fn = 0.2, λ/L = 0.5, χ = π) . . . 66 5.6 Kelvin wave at Fn = 0.2, y/(B/2) = 1.4 . . . . . . . . . . . . . . . . . . . . . . . 66 5.7 Interaction eﬀect between Kelvin wave and incident wave in wave pattern analysis (Diﬀraction wave at y/(B/2) = 1.4 for Fn = 0.2, χ = π) . . . . . . . . . . . . . . 67 5.8 Kochin functions computed with diﬀraction waves measured at Fn = 0.2, λ/L = 0.5, χ = π, y/(B/2) = 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.9 Steady Kelvin wave pattern of modiﬁed Wigley model (blunt) at Fn = 0.2 . . . . 70 5.10 Added mass and damping coeﬃcients due to forced heave motion at Fn = 0.2 . . 71 5.11 Added mass and damping coeﬃcients due to forced pitch motion at Fn = 0.2 . . 72 5.12 Wave exciting forces and moment at Fn = 0.2, χ = π . . . . . . . . . . . . . . . . 73 5.13 Ship motions at Fn = 0.2, χ = π . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.14 Steady pressure distribution of modiﬁed Wigley model (blunt) at Fn = 0.2 . . . . 74 5.15 Wave pressure (cos component) at Fn = 0.2, λ/L = 0.5, χ = π . . . . . . . . . . 75 5.16 Wave pressure (cos component) at Fn = 0.2, λ/L = 1.3, χ = π . . . . . . . . . . 75 5.17 Unsteady pressure (cos component) at Fn = 0.2, λ/L = 0.5, χ = π . . . . . . . . 76 5.18 Unsteady pressure (cos component) at Fn = 0.2, λ/L = 1.3, χ = π . . . . . . . . 76 5.19 Contour plots of heave radiation wave at Fn = 0.2, KL = 30 . . . . . . . . . . . . 78 5.20 Contour plots of diﬀraction wave at Fn = 0.2, λ/L = 0.5, χ = π . . . . . . . . . . 78 5.21 Contour plots of total wave at Fn = 0.2, λ/L = 0.5, χ = π . . . . . . . . . . . . . 79 5.22 Contour plots of total wave at Fn = 0.2, λ/L = 1.3, χ = π . . . . . . . . . . . . . 79 5.23 Heave radiation waves at y/(B/2) = 1.4 for Fn = 0.2, KL = 30, 35 . . . . . . . . 80 5.24 Diﬀraction waves at y/(B/2) = 1.4 for Fn = 0.2, λ/L = 0.5, 0.7, χ = π . . . . . . 80 5.25 Wave proﬁle for Wigley model at y/(B/2) = 1.4 for Fn = 0.2, λ/L = 0.7, χ = π . 82 5.26 Wave proﬁle for Wigley model at y/(B/2) = 1.4 for Fn = 0.2, λ/L = 0.9, χ = π . 83 5.27 Wave proﬁle for Wigley model at y/(B/2) = 1.4 for Fn = 0.2, λ/L = 1.1, χ = π . 83 5.28 Wave proﬁle for Wigley model at y/(B/2) = 1.4 for Fn = 0.2, λ/L = 1.4, χ = π . 84 5.29 Added wave resistance at Fn = 0.2, χ = π . . . . . . . . . . . . . . . . . . . . . . 84
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List of Figures xii 6.1 Plans of the monohull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.2 Perspective view of the monohull with computation grids . . . . . . . . . . . . . 88 6.3 Computation grids (NH = 1480(74 × 20), NF = 3888(162 × 24), NF A = 297(99 × 3)) . 89 6.4 A snapshot of the transom stern in the motion measurement test . . . . . . . . . 90 6.5 Steady Kelvin wave pattern at Fn = 0.5 . . . . . . . . . . . . . . . . . . . . . . . 91 6.6 Measured steady resistance (total), sinkage and trim . . . . . . . . . . . . . . . . 91 6.7 Added mass and damping coeﬃcients due to forced heave motion at Fn = 0.5 . . 92 6.8 Added mass and damping coeﬃcients due to forced pitch motion at Fn = 0.5 . . 93 6.9 Wave exciting forces and moment at Fn = 0.5, χ = 180degs. . . . . . . . . . . . . 94 6.10 Ship motions at Fn = 0.5, χ = 180degs. . . . . . . . . . . . . . . . . . . . . . . . 94 6.11 Wave pressure on the hull at Fn = 0.5, λ/L = 1.1, χ = 180degs. . . . . . . . . . . 96 6.12 Total unsteady pressure on the hull at Fn = 0.5, λ/L = 1.1, χ = 180degs. . . . . 96 6.13 Wave pressure on the hull at Fn = 0.5, λ/L = 1.1, χ = 180degs. . . . . . . . . . . 97 6.14 Total unsteady pressure on the hull at Fn = 0.5, λ/L = 1.1, χ = 180degs. . . . . 97 6.15 Comparisons of measured and computed wave patterns . . . . . . . . . . . . . . . 98 6.16 Comparisons of measured and computed wave proﬁles along y/(B/2) = 1.52 . . . 98 6.17 Added wave resistance computed by the wave pattern analysis . . . . . . . . . . 101 6.18 Plans of the Trimaran . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.19 Perspective view of the model with computation grids . . . . . . . . . . . . . . . 104 6.20 Computational grids used for Trimaran . . . . . . . . . . . . . . . . . . . . . . . 104 6.21 Steady Kelvin wave pattern at Fn = 0.5 . . . . . . . . . . . . . . . . . . . . . . . 105 6.22 Steady wave view at Fn = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.23 Steady pressure on hull at Fn = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6.24 Added mass and damping coeﬃcients due to forced heave motion at Fn = 0.5 . . 107 6.25 Added mass and damping coeﬃcients due to forced pitch motion at Fn = 0.5 . . 107 6.26 Wave exciting forces and moment at Fn = 0.5, χ = 180degs. . . . . . . . . . . . . 108 6.27 Diﬀraction and Radiation Wave Pattern . . . . . . . . . . . . . . . . . . . . . . . 109 6.28 Perspective view of the model with sinkage and trim . . . . . . . . . . . . . . . . 109
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List of Figures xiii 6.29 Computation grids which include sinkage and trim . . . . . . . . . . . . . . . . . 110 6.30 Ship motions at Fn = 0.5, χ = 180degs. . . . . . . . . . . . . . . . . . . . . . . . 111 6.31 Pressure p/ρ g A at Fn = 0.5, λ/L = 1.1χ = 180degs. (with TSC, sinkage & trim eﬀect) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.32 Comparison of wave pressure on monohull Vs trimaran . . . . . . . . . . . . . . . 112 6.33 Comparison of total unsteady pressure on monohull Vs trimaran . . . . . . . . . 112
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List of Tables5.1 Principal dimensions of the model (Modiﬁed Wigley Hull) . . . . . . . . . . . . . 626.1 Main particulars of monohull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.2 Main particulars of trimaran . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 xiv
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Chapter 1 Introduction1.1 Background of Theoretical Estimations of SeakeepingA ship operated at sea is exposed to forces due to waves, current and winds. These forcesnot only cause the motions of the ship, which can be very annoying for its passengers but alsoaccount for the resistance of the ship or drift the ship away from its course. Resistance due towind is very low when compared to water resistance. Estimations of resistance are important fordeciding the power requirements of any ship. The resistance is decomposed into the resistanceoriginated in the viscosity and the resistance due to the wave making. Before a ship is built, knowing the maximum data about the ship’s performance in calmwater, and in waves can lead to a better ship construction. Therefore, a scale model of theship is built and tested in a basin with and without incoming waves. These model tests arevery expensive, and it is time consuming. Making new computer simulation software, which canpredict well about the ship behavior can replace a model test. With the increase of computertechnology, it has become possible to simulate a ship’s behavior in waves numerically. Thesesimulations are based on mathematical descriptions of the physics of the ship and the sea whichare extremely complex. To understand the gross ﬂuid motion and corresponding interaction withthe ship, one must understand and predict turbulence, wave breaking, water spray, non-linearmotion, slamming, green water on deck, sloshing, acoustics, etc. Viscous ﬂuid motion is governed by the continuity equation and the motion equations of therigid body must be coupled when we treat the ﬁxed-body interaction problem. When we focuson the ocean waves and the interaction between the ship and ocean waves, the gravity eﬀect isdominant and the viscous eﬀect is negligible. Therefore, the ﬂuid can be treated as an ideal ﬂuidand the potential theory can be applied. Potential ﬂow solvers are usually based on boundary 1
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1.1. Background of Theoretical Estimations of Seakeeping 2element methods and need only to be discretized the boundaries of the domain, not the wholedomain. This reduces the load in grid generation and less time computation. Potential ﬂowsolver needs suitable boundary conditions that consist of the body boundary condition, the freesurface boundary condition and the radiation condition to satisfy the physical condition on thefree surface. The well-known strip theory was the ﬁrst numerical method used as a practical designtool for predicting ship hydrodynamic forces and ship motions. This method solves the 2-Dﬂow problem for each strip of the ship and integrates the results over the ship to ﬁnd out thehydrodynamic forces and motions. This problem was solved by U rsell[1]for heaving motion ofa half-immersed circular cylinder. Other extensive works were done afterward by Korvin −Kroukovski[2], T asai[3] and Chapman[4]. The rational foundations for the strip theory wereprovided by W atanabe[5], T asai & T akagi[6] , Salvasan et al. [7] and Gerritsma[8]. Now alsoit is popular in this ﬁeld because of satisfactory performance and computational simplicity. When the forward speed increases, the eﬃciency of the strip theory based on the 2D theorygot reduced, due to a strong 3D eﬀect near the bow part and forward speed inﬂuence from thesteady ﬂow to the unsteady wave ﬁeld. It has been reported by T akaki & Iwashita[9]that theapplicable limitation of the strip theory is around Fn = 0.4 for the typical high speed vessel.The high speed strip theory (HSST), so called 2.5D theory, can be applied eﬀectively for thehigh speed vessels. The theory originated in Chapman[10] and developed subsequently by manyresearchers, Saito & T akagi[11], Y ueng & Kim[12][13], F altinsen[14] , Ohkusu & F altinsen[15]can capture the forward speed eﬀect within the framework of uniform ﬂow approximation in thefree-surface condition. The rational justiﬁcation of strip theory, as a method valid for high frequencies and moderateFroude numbers, was derived from systematic analysis based on the slender-body theory byOgilvie and T uck[16]. This theory was extended to the diﬀraction problem by F altinsen[17]and was further reﬁned by M aruo & Sasaki[18]. The high-frequency restriction in slender shiptheories was removed by the uniﬁed theory framework presented by N ewman [19]. Its extensionto the diﬀraction problem was derived by Sclavounos[20] and applied to the seakeeping of shipsby N ewman and Sclavounos[21] and Sclavounos[22]. The increasing accessibility of computers of high capacity led to the development of three-dimensional theories that removed some of the deﬁciencies of strip theory. The choice of the
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1.1. Background of Theoretical Estimations of Seakeeping 3elementary singularities leads to the classiﬁcation of these methods into the Green functionmethod (GFM) and the Rankine panel method (RPM). In GFM, the wave Green function isapplied only at the ship surface and in the RPM, simple source will be applied to ship surfaceand free surface. The 3-D Green function method has been applied and succeded for the ﬂoating bodieswithout forward speed and extended for the forward speed. In the Green Function method,the unsteady wave source, which satisﬁes the radiation condition and the linearized free surfacecondition based on the uniform ﬂow, is chosen as the elementary singularity. Important devel-opments for its fast and accurate evaluation were made by Iwashita & Ohkusu[23] based onthe single integral formulation derived by Bessho[24]. Iwashita et al.[25][26] have rigorouslyexamined the wave pressure distribution on a blunt VLCC advancing in oblique waves by ap-plying the Green function method and demonstrated that the strip method practically used forthe estimation of ship motions is insuﬃcient for this purpose. They have also showed that a signiﬁcant discrepancy of the wave pressure between numericalresults and experiments still remains at blunt bow part even if the three-dimensional methodis applied. It was decided to include the inﬂuence of the steady ﬁeld in an unsteady wave ﬁeldin diﬀerent boundary condition. Then some improvements have been reported by Iwashita &Bertram[27], where the inﬂuence of the steady ﬂow on the wave pressure is taken into accountthrough the body boundary condition. The problem is often formulated in the frequency domain,which assumes that the body motions are strictly sinusoidal in time. First achievements werereported by Chang[28], Kabayashi[29], Inglis & price[30], Guevel & Bougis[31] and they foundgood agreement with experimental data. The Ranking panel method was proposed by Gadd[32] and Dawson[33] for the steadyproblem and extended by N akos & sclavounos[34] to the unsteady problem. Y asukawa [42]and Iwashita et al.[35] suspected that alternative inﬂuence of the steady ﬂow through thefree surface condition might aﬀect more strongly the local wave pressure, especially at the bowpart. They used for the computation a Rankine panel method based on the double body ﬂowlinearization for steady wave ﬁeld, and its inﬂuence is taken into account in the unsteady problemthrough the free surface condition and the body boundary condition. This formulation will becalled as double body ﬂow formulation. It was considered that the inﬂuence of the steady doublebody ﬂow through the free surface condition is important for estimating hydrodynamic forces
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1.1. Background of Theoretical Estimations of Seakeeping 4and local pressures. In RPM, the radiation condition must be satisﬁed numerically, and this numerical radiationcan be solved in diﬀerent method. The typical one is the ﬁnite diﬀerence method originated inDawson[33]. The upward diﬀerential operator is used to evaluate the partial derivative of thevelocity potential on the free surface and the radiation condition which proves non-radiatingwaves in front of a ship is satisﬁed. Sclavounos & N akos[36] introduced the B-Spline functionto express the potential distribution on the free surface and satisﬁed the radiation conditionby adding a non wave condition at the leading edge of the computation domain of the freesurface. They also applied their method to seakeeping problem provided the high reducedfrequency τ (= U we /g) > 0.25 where any disturbed wave due to a ship does not propagateupward. The alternative method is Jensen s[37] method so called collocation method. Thecollocation points on the free surface are shifted just one panel upward to satisfy the radiationcondition numerically. This method has been applied to the seakeeping problem by Bertram[38],and the analytical proof on the numerical radiation condition has been done by Seto[39] for the2D problem. For the investigation of fully non-linear steady kelvin wave ﬁeld and its inﬂuence on theunsteady wave ﬁeld, Bertram[40] employed a desingularized Rankine panel method proposedinitially for steady problem by Jensen et al.[37]. The steady problem is solved so that thefully nonlinear free surface condition is satisﬁed and the inﬂuence terms of the steady wave ﬁeldon the unsteady ﬂow are evaluated by assuming the small amplitude of the incident wave andsmall ship motions. The boundary conditions are satisﬁed on the steady free surface and wettedsurface of the body. Rankine source method is a highly eﬃcient method for seakeeping analysis. There aremany researchers working with RPM to satisfy the radiation condition in the frequency domainin diﬀerent ways. T akagi[41] and Y asukawa [42] have introduced Rayleigh’s viscosity numeri-cally to satisfy the radiation condition numerically and getting the adequate value of Rayleigh’sviscosity is not strait forward. Bertram [43] and Sclavounos et al. [44] has introduced desingu-larized panel method to satisfy the radiation condition, and this is applicable when the reducedfrequency is τ > 0.25. Iwashita et al.[45], Lin et al.[46], T akagi et al.[47] proposed a hybridmethod named combined boundary integral equation method (CBIEM) which is the combina-tion of Rankine Panel Method and Green Function Method. The method makes possible to
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1.1. Background of Theoretical Estimations of Seakeeping 5satisfy the radiation condition accurately by introducing the Green function method in the farﬁeld. Three dimensional computation method was extended to the time domain byLin et al.[50],[51] had developed in Green function based 3D panel method satisfying the linearfree surface condition, and the ship surface is treated nonlinear boundary condition. The freesurface is to be re-meshed according to ship position. This program is called as LAMP (LargeAmplitude Motion Program), and it was made commercially available. Later, free surface hasbeen improved with partially nonlinear and further; there are many series as Lamp-1 to Lamp-4.This software has been extended as a Non-linear Large Amplitude Motions and Loads methodby Shin et al. [52] used for oﬀshore structure application. Using this program, many calculationshave been carried out by W eems et al. [53], [54] for the trimaran and wave piercer. M askew [55] has developed Rankine source based 3D panel method where the three surfacetreated as a fully nonlinear and the ship surface is treated as partially nonlinear. It has beenapplied for Frigate ship with waves. This program also considered lift eﬀect and made availablecommercially. Though it is the ﬁrst program based Rankine source in time domian but there aresome disadvantages in accuracy of the calculation. This has been applied for the high speed anddemonstrated few results for S-60, SAWTH, S175 and Frigate with waves by M askew [56], [57].Beck.R.F [58], [59]. has developed Rankine source based fully nonlinear boundary condition forfree surface as well as ship surface and this method used the desingularized panel source. Thiscode has been applied for Wigley hull for ship motion problems. This has been extended byScorpio et al.[60] for the container ship. N akos et al.[61] have developed the Rankine source based time domain program wherethe double ﬂow is introduced, and it has been applied for Wigley hull with linear free surfacecondition. Later, it has been improved by Kring et al. [62] for the application of contained shipmotion and load estimation where the free surface and body surface have taken as nonlinear.This program gives much higher accuracy without many panels. However, this software hasbeen made commercially for the ship motion and load estimation with the name called SWAN.Adegeest et al[63] has improved in the Rankine based time domain program for the containership in unsteady waves. Bunnik and Hermans [64] & Buunik [65]have developed code based Rankine sourcemethod for the estimation added wave resistance and ship motion. It used the numerical beach
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1.1. Background of Theoretical Estimations of Seakeeping 6to satisfy the radiation condition. However, it cannot be used for blunt ship and high speedvessel. Rankine source based code is developed by Colagrossi et al.[66], [67] with non linearfree surface condition and applied for slender ship body. This method used the desingularizedmethod for free surface condition and numerical beach for radiation condition. It has beenapplied for catamaran and trimaran. This method has been developed and applied for the S-60(Cb=0.8) by Iwashita et al.[68]. T anizawa[69] and Shirokura[70] has developed the fullynonlinear free surface and body boundary condition seakeeping code based on Rankine sourcemethod. This code is applied for the Wigley hull and demonstrated high-accuracy results. Y asukawa[71], [72] has developed nonlinear code for seakeeping problem based on Rankinesource. For the high speed calculation SOR method has been introduced where after the transomis modelled with a dummy grid surface to get the smooth ﬂow. It has been applied for the Wigleyhull and S175 by Y asukawa[73], [74] and validated with experimental results. Green function based time domain code is developed by Kataoka and Iwashita [75], [76]in which the ﬂuid domain is decomposed into near ﬁeld and far ﬁeld, and the space ﬁxed artiﬁcialsurface like side-walls of a towing tank is employed to separate them. The Rankine method andthe Green function method are applied to two domains respectively, and the solutions in bothdomains are combined on the artiﬁcial surface. This method is called hybrid method. It hasbeen applied for the modiﬁed Wigley hull and S-60 to estimate the forces and motions. It hasbeen extended for the diﬀerent kind of vessel by Kataoka & Iwashita [77] ∼ [81] for the S175,S-60 (Cb=0.7, Cb=0.8) and modiﬁed Wigley hull. The added wave resistance and pressure arecalculated and compared with an experimental result for the validation. In practical ﬂuid motion problem, due to the presence of turbulence suggests using theunsteady Reynolds Averaged Navier-Stokes (RANS) equations or Large-Eddy Simulations (LES)to model the ﬂuid motion or seakeeping problem. In recent years, the CFD simulation toolsare applied to seakeeping. In grid-based method, three-dimensional CFD simulations of verticalplane motion in waves are reported by Sato et al.[82] and W eymouth et al.[83]. A non-linear motion for a high speed catamaran vessel, and the ship resistance in 3D was presentedby P anahi et al.[84]. On the other hand, in particle based method, Shibata et al.[85]&[86]computed a shipping water and pressure onto moving ship. Oger et al.[87] SPH method appliedto a ship motion at high Froude number. M utsuda.H. et al.[88] − [90], has validated for freesurface and impact pressure problems.
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1.2. Validation Methods of Theoretical Estimations 7 M utsuda.H.etal.[91], the seakeeping performance of a blunt ship in nonlinear waves studiedand validated. As an improvement, CFD has been applied by Suandar.B. et al.[92]. for ﬁshingboat and high speed ferry. Ship motions, pressure distribution on the hull and velocity ﬁeldhave been validated with panel method and experimental results. A practical computational method to analysis the transom stern ﬂow is important in thearea of marine hydrodynamics. Research on the transom stern steady ﬂow was conducted bySaunders[93]. He provided advice that the speed of ventilation occurs at a FT ∼ 4.0 to 5.0(where FT is the Froude number based on the transom draft). More recent observation onnaval combatants put the ventilation in the range of FT ∼ 3.0 to 3.5. V anden − Broeck andT uck[94] describes a potential ﬂow solution with series expansion in FT that is valid for lowspeeds where the transom stagnation point travels vertically upward on the transom as theFT increases. V anden − Broeck[95] oﬀers a second solution applicable to the post ventilationspeeds. For a 3D full ship problem, Subramani[96] extended the fully nonlinear desingularizedpotential code (DELTA) to analyze a body with a transom stern. Additionally, Doctors[97] hasstrived to suitably model the transom hollow for use in linearized potential ﬂow program. ByDoctors et al.[98] non linear eﬀect has been introduced to analyze the transom stern vessel.Because of its advantage, most of the high speed vessel and naval ships are built with transom.It is also important to study hydrodynamic ﬂow behavior at transom and behind the ships.When the ship reaches to a suﬃcient speed, ﬂow leaves at transom and the transom area isexposed to air. The transom ﬂow detaches smoothly from the underside of the transom, anda depression is created on the free surface behind the transom. This creates inﬂuence in thepressure reduction and resistance on the hull. For unsteady ﬂow, many researchers are working on this to get transom ﬂow with diﬀerentapproaches but there is no exact solution is arrived yet. Considering the need and a newboundary condition has been proposed to treat the transom part and details will be coveredinside the thesis.1.2 Validation Methods of Theoretical EstimationsA ship advancing in waves generate waves. The generated waves are classiﬁed into the time-independent and time-dependent waves. The former is the steady wave called Kelvin’s wave and
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1.3. Scope of Present Research 8corresponds to the wave that is generated by the ship advancing in calm water surface. The waveis independent of time when we observe it from the body-ﬁxed coordinate system. The secondone is the unsteady wave that consists of the radiation waves and the diﬀraction wave. Theradiation waves are the wave generated by the ship motions, and the diﬀraction wave is scattedwave of the incident wave by the hull. It is understood that both the steady and unsteady wavesdissipate more energy, which leads to a power loss. The resistance of a ship in wave minus thecalm water resistance is called added wave resistance. Besides this power loss, a transverse forcewill drift the ship from its course, and a rotating moment about the ship’s vertical axis leads toa change in its course. Therefore, it is necessary to consider the above when designing the ship,especially its hull. The asymptotic unsteady wave patterns which are for away from the ship were studiedinitially by Eggers[99]. Ohkusus.M [100] proposed a method for measuring ship generatedunsteady waves and then evaluating the wave amplitude function and the added wave resistance.Iwashita[101] has made systematic work, including the analysis near the cusp (the caustics)wave pattern. That was successful in treating with the surface elevation near the cusp for awave system of shorter wave components with introducing Hogner’s approach [102]. A 2nd orderunsteady wave pattern has been investigated by Ohkusu.M [103]. These unsteady wave patternsphysically show the pressure distributions over the free surface. Therefore, the comparison ofunsteady wave distribution between computed and measured corresponds to the comparison asfor the pressure distributions that can be considered as the local physical value. From thesereasons, it is very valuable to utilize the unsteady waves in order to validate the numericalcomputation methods more precisely. In this paper, the interaction eﬀect between the incident wave and steady wave, which hasnot been considered in the measurement analysis up to now, is investigated in order to makethe experimental unsteady waves more accurate.1.3 Scope of Present ResearchTo the analysis of seakeeping problems, Rankine panel method based mathematical formulationis derived which includes the body boundary condition, the free surface boundary condition andthe radiation condition. Computational code is developed to handle the conventional ship hull.
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1.4. Organization of the Thesis 9As a basis ﬂow, uniform ﬂow approximation and the Neumann-Kelvin ﬂow approximation areconsidered in the formulation and applied for the modiﬁed Wigley hull to study the seakeepingqualities. Improve the measured unsteady wave taking into account of the interaction between theincident wave and the steady wave. A new analysis method of the measured unsteady wave isproposed and the interaction eﬀect related above is conﬁrmed by applying the method for twomodiﬁed Wigley hull models. Additionally, the RPM code is developed in the thesis is validatedthrough the comparisons with experimental data, that is, hydrodynamic forces, motions, addedwave resistance and wave ﬁelds. To treat the transom stern condition, until now there is no proper boundary condition totreat the transom stern by panel method. A new boundary condition is derived to treat thetransom eﬀect mathematically in numerical method. Therefore, the developed code is extendedfor the application of high speed monhull to predict the transom stern eﬀect at the transom.The RPM is applied to the high speed ships with the newly introduced transom stern conditionand validated through the comparison with experiments. Transom eﬀect is compared betweenby numerical method taking into account of transom condition and without transom condition.This method is also applied to a trimaran and the seakeeping data is compared with experimentaldata.1.4 Organization of the ThesisThe general scope of the thesis is divided into ﬁve chapters excluding the ﬁrst chapter is theintroduction which covers the previous research and the conclusion as a last chapter. The thesis is outlined as follows:Chapter 2: This chapter covers the mathematical formulation of a boundary value problem (BVP) for seakeeping analysis. As a ﬁrst step, to formulate BVP, appropriate bound- ary conditions need to be derived for frequency-domain formulation. Derivation of body boundary condition, free surface boundary condition and radiation condition derivation are given. A new boundary condition to deal with transom stern is developed and the details also covered. The ﬁnal section is the formulation of the boundary value problem to
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1.4. Organization of the Thesis 10 solve the potential equation in the panel method. And also the formulation of estimating the hydrodynamic forces, exciting forces and moments, motions and waves are derived.Chapter 3: In panel method, it is very critical to select the suitable formulation to solve the problem numerically. To calculate the potential in RPM, there are two methods available i.e., (i) direct method and (ii) indirect method. Each method has been discussed. As a next step, it is very important to select the appropriate method to satisfy the radiation condition either panel shift method or spline interpolation method. Therefore, attempt has been made to plot wave pattern by both the method and compared with an analytical wave pattern. This chapter also covers explanation about the treatment of the transom boundary conditions in panel method.Chapter 4: In Engineering ﬁeld, any numerical solution must be veriﬁed and compared with experimental results. Therefore, required tests are identiﬁed based on requirements. Mea- suring method for forces, motions and unsteady wave method is discussed towards im- provement in capturing the waves around the hull. Formulation is derived to capture interaction eﬀect of incident wave and steady wave in an unsteady waves.Chapter 5: To investigate the wave interaction eﬀect, modiﬁed Wigley hulls have been numer- ically analyzed and the results are compared with experimental results. Hydrodynamics forces, exciting forces and moments are compared with experimental results. Ship motions and added wave resistance are also compared with experimental results. This chapter is to deal with the validation of incident wave and steady wave interaction eﬀect with the unsteady waves.Chapter 6: High speed monohull has been taken for the analysis. Radiation forces and ex- citing forces are calculated with transom and without transom stern condition to see the eﬀect of a newly introduced condition. All the results are compared with experimental results. Unsteady waves and pressure plots are also validated with an experimental data. Ship motion and added wave resistance are also compared with experimental results. It has been extended for the trimaran application and numerical results are compared with experimental data. This chapter is to deal with transom stern boundary condition with an experimental validation.Chapter 7: In this chapter, the thesis is concluded making clear the obtained results.
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Chapter 2 Mathematical Formulation of Seakeeping2.1 IntroductionDue to the development in computer technology and advanced applied mathematics, engineeringproblems are solved by a numerical method, and being validated by experimental results. Inmarine hydrodynamics, ship/oﬀshore structure’s problems are formulated as a boundary valueproblem where the physical parameters are to be deﬁned well in the form of mathematicalexpression to solve the problem numerically. It is very important that boundary conditions ofreal physics are to be expressed mathematically, which contribute to the prediction accuracy ofthe expected results. In this research, boundary value problems are solved by boundary element method (BEM)also referred as a panel method or boundary integral equation methods. BEM is more suitablefor the marine structure hydrodynamics problem which takes the surface of the ﬂuid domain, andit is the best numerical method when compare to other methods like ﬁnite diﬀerence method,ﬁnite-element method, ﬁnite volume method, etc. in terms of computational time because itsolves for entire domain, which is not required for practical needs. Consider a ship which is ﬂoating on water and the water surface which is in turn touchwith the air is called a free surface. The most exact description of the ﬂow of water is given bythe Navier-Stokes equations, which take into account of the water viscosity. Viscosity in shiphydrodynamics can be important in turbulent areas like, for example, near a rudder, propulsionor sharp edges of hull, but none of these is considered in this research. Near the hull, a smallboundary layer exists in which viscous eﬀects dominate, but this layer does not really aﬀect thelarge-scale interactions of ocean waves and ship motions. Assuming that the ﬂow is irrotational and incompressible, the ﬂow can be described using 11
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2.2. Body Boundary Condition 12potential theory. By deﬁning the ﬂuid velocity by a scalar potential, the velocity ﬁeld of theﬂow can be expressed as the gradient of a scalar function, namely the velocity potential, u(x, t) = Ψ(x, t) (2.1). The continuity equation or conservation of mass reduces to Laplace’s equation because theﬂuid is incompressible, which states that the divergence of the velocity ﬁeld is equal to zero. ·V =0 (2.2)and the velocity potential must be a harmonic function which satisﬁes the Laplace equation. 2 ∂2Ψ ∂2Ψ ∂2Ψ Ψ= + + =0 (2.3) ∂x2 ∂y 2 ∂z 2The conservation of momentum equation can be reduced to Bernouli’s equation. −1 ∂Ψ 1 p − pa 1 2 z= + Ψ· Ψ+ − U (2.4) g ∂t 2 ρ 2where ρ is the ﬂuid density, g is the gravitational constant and pa is the atmospheric pressure.The total velocity potential can be decomposed into steady velocity potential and unsteadyvelocity potential. Ψ (x, y, z; t) = Ψs (x, y, z) + Ψt (x, y, z; t) (2.5)For a real ship in a seaway, the ﬂuid domain is eﬀectively unbounded relative to the scale of theship. For the computational purpose, the ﬂuid domain must be truncated. On the boundariesof the truncated ﬂuid domain, the hull of the ship SH , the free surface SF , the bottom of thewater SB and the control surface (truncation surface) SC are covered. The total surface canbe written as S = SH + SF + SB + SC . In addition to the above boundary condition, it isnecessary to satisfy the radiation mathematically to get unique results. The control surface andthe bottom surface need not be considered for analysis because of the simple source in RPMmethod. We discuss on each boundary in details on further section.2.2 Body Boundary ConditionThe boundary condition on the hull should take into account the interaction between themotion of the hull and the motion of the water at the hull. Just like the water, thehull of the ship cannot be crossed by a ﬂuid particle. The water should therefore havethe same normal velocity as the ship’s hull and the water does not penetrate the hull.
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2.2. Body Boundary Condition 13 iωt Z r = r + αe Z ∂Ψ = V SH · n on SH (2.6) ∂n YWhere the normal vector n is from hull to- Ywards the ﬂuid domain. This condition must rbe simpliﬁed to the mean wetted surface from ran instantaneous surface. Let us consider a iωt SH αethree-dimensional object in a ﬂuid with a free SHsurface. The object, for instance, a ship, sail Instantaneous Wetted Surface SHthrough an incident wave ﬁeld with a velocity SH Mean Wetted SurfaceU (t) in the negative x direction; this is equiv- Figure 2.1: Body boundary bondition - coordinatealent to an object with zero speed in current systemU (t) in positive x direction. The object is free to translate in threedirections and to rotate around the three axes. Therefore, six motions are taken to representthe body boundary condition. To derive the mathematical equation of the moving body, thereare two coordinate systems are followed. The ﬁrst coordinate is the space coordinate systemwhich can be represented as, r = (x, y, z) (2.7)The second coordinate system is body ﬁxed coordinate system is represented as r = (x, y, z) (2.8)Bring the relation between teh Space and body ﬁxed coordinate system, the equation shall bewritten as r = r − αeiωt (2.9) Where α is a displacement vector of a point on the body and ω is a circular frequency ofoscillations. α = ξj ; (j=1,2..6 surge, sway, heave, roll, pitch, yaw). α = iξ1 + jξ2 + kξ3 + (iξ4 +jξ5 + kξ6 ) × r. The r coordinates are ﬁxed with respect to a body which is deﬁned by theequation F (x, y, z) = 0, then the potential ﬂow kinematic boundary condition to be satisﬁed on DF (r)the surface is 0 = . Dt
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2.2. Body Boundary Condition 14 The velocity potential is decomposed into a steady mean potential and unsteady perturba-tion potential. The Velocity of the ﬂuid is represented by the vector Φ = V (r) + eiωt φ(r)and V is the steady ﬂow ﬁeld due to forward motion of the body(negative x direction ) i.e.,V (r) = U (−x + ϕ) and φ(r) is the potential of the oscillating velocity vector. The coordinatesystem chosen such that undisturbed free surface coincides with the plane z=0 and the centreof gravity of the object is on the z axis, with z pointing upwards. In this formulation, SH is themean wetted surface and S H is the instantaneous wetted surface, Fig. 2.1The boundary condition on the body is DF (r) ∂ 0= = + Φ. F (r) (2.10) Dt ∂t DF (r) ∂F (r) 0 = = + V (r) + eiωt φ(r) . F (r) on S H (2.11) Dt ∂tThe above equation can be written as ∂F ∂x ∂F ∂y ∂F ∂z 0 = + + + V (r) + eiωt φ(r) . ∂x ∂t ∂y ∂t ∂z ∂t ∂F ∂x ∂F ∂y ∂F ∂z ∂F ∂x ∂F ∂y ∂F ∂z i + + +j + + ∂x ∂x ∂y ∂x ∂z ∂x ∂x ∂y ∂y ∂y ∂z ∂y ∂F ∂x ∂F ∂y ∂F ∂z +k + + on S H (2.12) ∂x ∂z ∂y ∂z ∂z ∂zIn the above equation, equation can be simplied using the below relation as iωt ∂ α . F (r) − jeiωt ∂ α . F (r) F (r) = F (r) − ie ∂x ∂y −ke iωt ∂ α . F (r) ∂z ∂ ∂ ∂ (r ) = i ∂x + j ∂y + k ∂z Now the equation can be simpliﬁed as ∂F ∂x ∂F ∂y ∂F ∂z 0 = + + + (V + eiωt φ). ∂x ∂t ∂y ∂t ∂z ∂t ∂α ∂α F (r) − ieiωt . F (r) − jeiωt . F (r) ∂x ∂y ∂α − keiωt . F (r) on S H (2.13) ∂z
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2.2. Body Boundary Condition 15Now the equatin must be simplied. Let us see each term individually ∂F ∂x ∂F ∂y ∂F ∂z First Term : + + = −iωeiωt α. F (r) (2.14) ∂x ∂t ∂y ∂t ∂z ∂tNow the equation can be rewritten as 0 = −iωeiωt α. F (r) + (V + eiωt φ). ∂α ∂α F (r) − ieiωt . F (r) − jeiωt . F (r) ∂x ∂y ∂α − keiωt . F (r) on S H (2.15) ∂z The above formulation is satisﬁed on the instantaneous wetted surface (Exact body surfaceF (x , y, z , t). This can be linearized under the assumption that unsteady displacement amplitudesmall. So Taylor’s expansion can be applied for both steady and unsteady velocity ﬁeld on themean wetted surface. Taylor’s expansion for V and φ V (r) = V (r)mean + (α. )V (r) eiωt + 0(α2 ) (2.16) mean φ = φ(r)mean + (α. ) φ(r) eiωt + 0(α2 ) (2.17) meanSubstituting the above formation in boundary condition ( 2.15 ), we can obtain the equationon mean wetted surface 0 = −iωeiωt α. F (r) + V (r) + eiωt φ + eiωt (α. )V (r) . ∂α ∂α F (r) − i eiωt . F (r) − j eiωt . F (r) ∂x ∂y ∂α − k eiωt . F (r) on SH (2.18) ∂z ∂α ∂α Last Term : F (r) − i eiωt . F (r) + j eiωt . F (r) ∂x ∂y ∂α +k eiωt . F (r) (2.19) ∂zBoundary Condition by Steady State Term for: V V (r). F (r) = 0; or V . F = 0
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2.2. Body Boundary Condition 16Boundary Condition by Oscillatory Function for: φ eiωt φ. F (r) = iωeiωt α. F (r) − eiωt (α. )V . F r + (V )mean . mean ∂α ∂α i eiωt . F (r) + j eiωt . F (r) ∂x ∂y ∂α + k eiωt . F (r) on SH (2.20) ∂zAfter removing the time part out of this equation, the equation becomes φ. F (r) = iω (α. F (r)) − (α. )V . F (r) + (V )mean . mean ∂α ∂α i . F (r) + j . F (r) ∂x ∂y ∂α +k . F (r) on SH (2.21) ∂z or φ. F (r) = iω α. F (r) − (V . )α − (α. )V . F (r) on SH (2.22) meanAll the terms are small, of the same order as α or φ. Thus to this order of approximation itis no longer necessary to distinguish between the actual position and of the body and its meanposition, or between the co-ordinates r and r φ· F = {iωα · F − (α · )V · F + (V · ) α} · F (2.23)Vector Identity: (V . )A − (A. )V = × (A × V ) − A .V + V .A (2.24) .V = 0 for Incompressibility;V. F =0 from the Steady Potential Condition;so the unsteady condition shall be written from the equation ( 2.23 ) as φ. F = iωα. F + × (α × V ) . F on SH (2.25) ∂since: F ⇒ vector normal to the body surface and n. = ∂nBoundary Condition for φ on the Body is : ∂φ = iωα + × (α × V ) .n on SH (2.26) ∂n
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2.2. Body Boundary Condition 17(i) Zero Speed CaseThe derived body boundary condition shall be written as ∂φ = iωα + × (α × V ) .n on SH (2.27) ∂nThe forward velocity is considered zero. So the second term will be neglected from equation (2.27 ). So the boundary condition shall be, ∂φ = iωα.n on SH (2.28) ∂nSubstituting the α in the above equation ∂φ = iω iξ1 + jξ2 + kξ3 + (iξ4 + jξ5 + kξ6 ) × r .n on SH (2.29) ∂nVector relations and the normal vectors are written as . . (ξ × r) n = ξ (n × r) n = n1 , n 2 , n 3 (2.30) n×r =n , n , n 4 5 6Applying the above relation in equation ( 2.29 ), the equation shall be written as 3 6 ∂φ ∂n = iω ξj nj + iω . ξ j (n × r)j on SH (2.31) j=1 j=4To consider the six degree of motion, boundary condition shall be written as 6 ∂φ = iω ξj n j on SH (2.32) ∂n j=1(ii) With Forward Speeda) Translatory Motion (j = 1 · · · 3): α = ξ1 i + ξ2 j + ξ3 k ≡ ANow the derived boundary condition shall be applied for the translatory motion as ∂φ = iωα + × (α × V ) .n on SH (2.33) ∂nLet us rewrite the vector identity and the motions are considered only three direction. × (A × V ) = (V · )A + V ( · V )A − V ( · A) − (A · )V (2.34)
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2.2. Body Boundary Condition 18Considering the translatory motion, the below mentioned relation can be used ·A=0 ·V =0 Now in the vector identity, applying the above relations, the last term n (A · ) V can bemodiﬁed using given equation as ∂vi ∂ ∂φ ∂vj = = (1 ≤ i, j ≤ 3) (2.35) ∂xj ∂xj ∂xi ∂xi n (A · ) V = A (n · )V (2.36)Now applying above relation in the body boundary condition, the ﬁnal equation for translatorymotion with forward speed shall be written as ∂φ = iωα · F+ × (α × V ) · n ∂n = iωα · n − (n · )V · A = iωn − (n · )V A (2.37)b) Rotational Motion (j = 4 · · · 6): α = ξ4 i + ξ5 j + ξ6 k × r ≡ B × rBody boundary condition shall be written as × (A × V ) = (V · )A + V ( · V )A − V ( · A) = (A · )V (2.38)Let us apply only the rotation motion in the condition as iωn (B × r) = −iωB (n × r) = iωB (r × n) (2.39)The above equation can be applied to the vector identity asn· × {(B × r) × V } = n {(V · ) (B × r) + (B × r) ( ·V)−V [ · (B × r)] − [(B × r) · } ] V(2.40)Using the below mentioned mathematical relation ∂u ∂v ∂w (B × r) + + =0 ∂x ∂y ∂z (2.41) ∂ ∂ ∂ V + + (B × r) = 0 ∂x ∂y ∂zThe vector identity term can be simpliﬁed as n (V · ) (B × r) = n (B × V ) = −B (n × V ) (2.42) n (B × r) · V = (B × r) · (n · ) V = − (n · ) × r B = r × (n · ) B
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2.3. Free Surface Boundary Condition 19Applying to the boundary condition equation ( 2.26 ) as ∂φ = iωα + × (α × V ) · n ∂n = iω (r × n) − (n × ) − r × (n · )V B (2.43)To simplify the above equation, tensor relation can be used ∂vj ∂ xi + vj = (xi · vj ) ∂xi ∂xi ∂ ∂vk (xj · vk ) = xj (1 ≤ i, j, k ≤ 3) (2.44) ∂xi ∂xiVector idetity which can be used (n × ) + r × (n · ) V = (n · ) (r × V ) (2.45)Applying the above mentioned relations, the ﬁnal equation for body boundary condition shallbe written as ∂φj = (iωnj + U mj ) ξj (j = 1 · · · 6) (2.46) ∂nThe full form of m and n vectors are written as n = n1 i + n2 j + n3 k r × n = n4 i + n5 j + n6 k − (n · )V = m1 i + m2 j + m3 k − (n · ) (r × V ) = m4 i + m5 j + m6 k (2.47)2.3 Free Surface Boundary ConditionIn the free surface, two boundary conditions are to be satisﬁed. The ﬁrst one is (a) kinematic-freesurface boundary condition, and the second one is (b) dynamic free surface boundary condition.The kinematic-free surface boundary condition states that the normal velocity of the ﬂuid surface& air surface must be equal, and the water surface does not penetrate the air surface. The ﬂuidparticle on the ﬂuid surface remains at the free surface. The exact free surface shall be writtenas z = ζ (x, y; t)
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2.3. Free Surface Boundary Condition 20 Equation (2.5) is the total velocity potential. Now this can decomposed into steady waveand unsteady wave ﬁeld as Ψ (x, y, z, t) = U ΦS (x, y, z) + φ (x, y, z) eiωe tIn the abve equation, steady ﬂow can be further expanded as Ψ (x, y, z, t) = U [Φ (x, y, z) + ϕ (x, y, z)] + φ (x, y, z) eiωe t (2.48)Where Φ = −x + ΦD The unsteady velocity potential equation shall be written as, 6 gA φ= (φ0 + φ7 ) + iωe ξj φ j (2.49) ω0 j=1where φ0 = ieKz−iK(x cos χ+y sin χ) Φ means the double body ﬂow, ϕ the steady wave ﬁeld and φ the unsteady wave ﬁeld whichconsists of the incident wave velocity potential, the radiation potentials φj (j = 1 ∼ 6) andthe scattering potential φ7 that represents the disturbance of the incident waves by the ﬁxedship. The radiation potentials represent the velocity potentials of a rigid body motion with unitamplitude, in the absence of the incident waves. The total velocity potential must be appliedover the ﬂuid domian and potential equation has been derived based Green’s theorem as follows, ∂ ∂φj (Q) φj (P ) = − φj (Q) − G(P, Q)dS (2.50) SH ∂n ∂n 1where G(P, Q) = and r = (x − x)2 + (y − y)2 + (z − z)2 , P (x, y, z) is the ﬁeld point 4πrand Q(x, y, z) is the source point. Now let us derive the free surface boundary condition. Kinematic Free surface Boundary Condition: The substantial derivative D/Dt of a functionexpress that the rate of change with time of the function, if we follow a ﬂuid particle in the freesurface. This can be applied in our exact free surface as, D ∂ (z − ζ) = + Ψ· z − ζ (x, y; t) = 0 on z = ζ (x, y; t) (2.51) Dt ∂tD/Dt means the substantial derivative and is deﬁned as a two-dimensional Laplacian withrespect to x and y on the free surface. z = ζ (x, y; t) is the wave elevation around a ship and
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2.3. Free Surface Boundary Condition 21considered to be expressed by the summation of the steady wave ζs and the unsteady wave ζtas follows ζ (x, y; t) = ζs (x, y) + ζt (x, y; t) (2.52)Dynamic Free surface Boundary Condition: The dynamics-free surface boundary condition isthat the pressure on the water surface is equal to the constant atmospheric pressure pa and thiscan be obtained from Bernoulli’s equation. 1 p pa 1 2 Ψt + Ψ· Ψ + gz + = + U (2.53) 2 ρ ρ 2where U is the forward speed of the vessel. Now, shifting the pressure to one side, the equationshall be written as 1 1 p − pa = −ρ Ψt + Ψ· Ψ − U 2 + gz (2.54) 2 2This pressure equation is applied on the exact free surface z = ζ (x, y; t). Equation shall berewritten considering the ﬂuid pressure is equal to atmospheric pressure as, ∂Ψ 1 1 pa − pa = −ρ + Ψ· Ψ − U 2 + gζ (x, y; t) (2.55) ∂t 2 2Keeping the free surface elevation in one side and the free surface is written as 1 ∂Ψ 1 1 ζ (x, y; t) = − + Ψ· Ψ − U2 on z = ζ (x, y; t) (2.56) g ∂t 2 2 Now substituting the dynamic free surface boundary equation ( 2.56 )in the kinematic-freesurface boundary equation eq.(2.51), the free surface boundary condition is written as, ∂ 1 1 1 0= + Ψ· z+ Ψt + Ψ· Ψ − U2 ∂t g 2 2 1 1 = Ψ· z+ Ψtt + Ψ· Ψt + Ψ · Ψt + Ψ· ( Ψ· Ψ) g 2 1 1 = Ψz + Ψtt + 2 Ψ · Ψt + Ψ· ( Ψ· Ψ) on z = ζ (x, y; t) (2.57) g 2Now the exact free surface boundary condition shall be written as 1 Ψtt + 2 Ψ · Ψt + Ψ· ( Ψ· Ψ) + gΨz = 0 on z = ζ (x, y; t) (2.58) 2 Total velocity potential equation (2.48) shall be substituted in the free surface bounndarycondition eq.(2.58) as φtt + 2 (Φ + ϕ + φ) · φt 1 + (Φ + ϕ + φ) · (Φ + ϕ + φ) · (Φ + ϕ + φ) 2 + g (Φz + ϕz + φz ) = 0 (2.59)
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2.3. Free Surface Boundary Condition 22 φtt + 2 Φ · φt + Φ· Φ· (ϕ + φ) 1 + ( Φ· Φ) · (ϕ + φ) + g (ϕz + φz ) 2 1 = −gΦz − ( Φ· Φ) · Φ on z = ζ (x, y; t) (2.60) 2Similarly, total velocity potential shall be applied to the dynamic free surface as 1 1 1 ζ (x, y; t) = − φt + Ψ· Ψ − U2 on z = ζ (x, y; t) (2.61) g 2 2 1 1 1 ζ (x, y; t) = − φt + (Φ + ϕ + φ) · (Φ + ϕ + φ) − U 2 on z = ζ (x, y; t) (2.62) g 2 2 1 1 1 ζ (x, y; t) = − Φ· Φ − U2 − ( Φ· ϕ) − (φt + Φ· φ) on z = ζ (x, y; t) (2.63) 2g g gIn the above wave equation, the ﬁrst part of equation is the wave which is the function of onlyforward speed. The double body potential satisﬁes the Miller condition on z = 0, which do notgenerate waves and the double body ﬂow is satisﬁed on the free surface i.e, ∂Φ/∂z = 0 on z = 0.Considering the above condition, the forward speed wave shall be written as, 1 ζ (x, y) = − Φ· Φ − U 2 on z = ζ (x, y) (2.64) 2gNow the free surface equation eq.(2.63) shall be applied to the forward speed wave z = ζ byapplying Tailor series 1 1 ζ (x, y; t) = ζ − ( Φ· ϕ) − (φt + Φ· φ) g g ∂ 1 + ζ −ζ − Φ· Φ on z = ζ (x, y) (2.65) ∂z 2g ( Φ· ϕ) + (φt + Φ · φ) ζ −ζ = − on z = ζ (x, y) (2.66) g + Φ · ΦzSimilarly the free surface equation equation (2.60) shall be applied on the double body ﬂow freesurface by applying Tailor series φtt + 2 Φ · φt + Φ· Φ· (ϕ + φ) 1 + ( Φ · Φ) · (ϕ + φ) + g (ϕz + φz ) 2 ∂ 1 φt + Φ · (ϕ + φ) − Φ · ( Φ · Φ) + gΦz ∂z 2 g + Φ · Φz 1 = −gΦz − ( Φ · Φ) · Φ on z = ζ (x, y) (2.67) 2
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2.3. Free Surface Boundary Condition 23Presently, the free surface boundary condition is satisﬁed on the double body ﬂow free surface.Now we need to linearize the free surface i.e., z = 0. In this formulation, the steady ﬂowdisturbance potential ϕ is very small when compare to other potential and the higher order canbe neglected. Applying the Tailor series for the above free surface equation with respect to z = 0 φtt + 2 Φ · φt + Φ· Φ· (ϕ + φ) 1 + ( Φ · Φ) · (ϕ + φ) + g (ϕz + φz ) 2 ∂ 1 φt + Φ · (ϕ + φ) − Φ · ( Φ · Φ) + gΦz ∂z 2 g + Φ · Φz ∂ 1 1 − Φ · ( Φ · Φ) + gΦz Φ · Φ − U2 ∂z 2 2g 1 = −gΦz − ( Φ · Φ) · Φ on z = 0 (2.68) 2On the free surface, the double body ﬂow potential ∂Φ/∂z = 0 on z = 0. Considering that wecan bring the summarization as Φ· Φz = 0 on z = 0 (2.69) ∂ Φ· ( Φ· Φ) on z = 0 (2.70) ∂zNow subsituting the above relation and the free surface equation equation (2.68) φtt + 2 Φ · φt + Φ· Φ· (ϕ + φ) 1 + ( Φ· Φ) · (ϕ + φ) + g (ϕz + φz ) 2 1 − Φzz φt + Φ· (ϕ + φ) − Φzz Φ· Φ − U2 2 1 =− ( Φ· Φ) · Φ on z = 0 (2.71) 2This is the linearixed free surface boundary condition which must be statifed on z = 0 and thiscan be separated as a steady part and unsteady part. The steady part potential free surfaceshall be written as 1 Φ· ( Φ· ϕ) + ( Φ· Φ) · ϕ + gϕz − Φzz ( Φ · ϕ) 2 1 1 =− ( Φ· Φ) · Φ− U2 − Φ· Φ Φzz on z = 0 (2.72) 2 2
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2.4. Radiation Condition 24For unsteady velocity potential, the free suface condition shall be written as 1 φtt + 2 Φ · φt + Φ· ( Φ· φ) + ( Φ· Φ) · φ 2 + gφz − Φzz (φt + Φ· φ) = 0 on z = 0 (2.73)The steady wave elevation shall be written as 1 1 1 ζ (x, y) = − Φ· Φ − U2 + Φ· ϕ on z = 0 (2.74) g 2 2The unsteady wave elevation shallbe written as 1 ζ (x, y; t) = − (φt + Φ· φ) on z = 0 (2.75) gIn the free surface boundary condition ( 2.72 ), the boundary condition shall be written ne-glecting the higher order term as 1 1 Φ· ( Φ· ϕ) + ( Φ· Φ) · ϕ + gϕz + ( Φ· Φ) · Φ = 0 on z = 0 (2.76) 2 2The unsteady potential free suface shall be written as 1 φtt + 2 Φ · φt + Φ· ( Φ· φ) + ( Φ· Φ) · φ + gφz = 0 on z = 0 (2.77) 2Taking the account of double body ﬂow velocity potential Φ = U [Φ], the steady disturbancepotential ϕ = U [ϕ] and the unsteady velocity potential φ = φ (x, y, z) eiωe t , steady andunsteady part shall be written. The steady potential free suface shall be written as 1 1 1 ∂ϕ Φ· ( Φ· ϕ)+ ( Φ· Φ)· ϕ+ ( Φ· Φ)· Φ+ = 0 on z = 0 (2.78) K0 2K0 2K0 ∂zThe unsteady potential free suface shall be written as 1 1 ∂φj −Ke φj +2iτ Φ· φj + Φ· ( Φ· φj )+ ( Φ· Φ)· φj + = 0 on z = 0 (2.79) K0 2K0 ∂zwhere K0 = g/U 2 ; Ke = ωe /g and τ = U ωe /g 22.4 Radiation ConditionRadiation condition indicates that the generated waves by the body propagate to inﬁnity. Butit is diﬃcult to express this condition by mathematical equations. This condition is satisﬁednumerically when we introduce the RPM.
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