HYDRODYNAMIC ANALYSIS OF A RECTANGULAR FLOATING BREAKWATER IN REGULAR WAVES

Transcription

1 Turkey Offsore Energy Conference, 213 HYDRODYNAMIC ANALYSIS OF A RECTANGULAR FLOATING BREAKWATER IN REGULAR WAVES Hayriye Pelivan and Ömer Gören

2 Contents Introduction Literature summary Problem definition Te matcing tecnique Matematical model of te problem Solution procedure for eave motion Wave forces and moments (diffraction) problem Results and comparison Field of application and future work 2/36

3 Introduction Motivation Establis a simple matematical basis for rectangular cylinder ydrodynamics To develope a code for mooring problem of rectangular floating breakwaters To ave a matematical tool for plunging rectangular cylinders intended to extract wave energy In tis study Wy analytical metod? Zeng, Y. H., Sen, Y. M., You, Y.G., Wu, B. J. Rong, L. (25). Hydrodynamic properties of two vertical truncated cylinders in waves. Ocean Engineering, 32, Wy rectangular section? Hales, L. Z., (1981). Floating Breakwaters: state of te art literature review. Tecnical Report no US Army Engineer Waterways Experiment Station. Vicksburg, Mississippi, USA. Koftis, T., Prinos, P., On te ydrodynamic efficiency of floating breakwaters. Aristotle University of Tessaloniki, Greece. 3/36

4 Literature summary Kim (1969) Bai (1975) Sabuncu (1978) Sabuncu, Calışal (1981) 4/36

5 Te neccesity of te study Tis metod Makes te solution more simple Needs sorter computing time Converges very quickly Sabuncu, T., Gören, Ö. (1981), (1983) Sabuncu, T., Çalışal, S. (1981), (1989) Pek, N. (1989) Gören, Ö., Söylemez, M. (24) 5/36

6 For te solution; Separation of variables Matcing tecnique Potential and linear teory Te flow is irrotational Viscosity and surface tension neglected Free surface condition and kinematic condition on te body are eliminated. Te particle kinematics and te wave pressure are derived from te velocity potential. 6/36

7 Problem definition Te problem will be divided into two independent situation: Radiation Problem (Heave-Sway-Roll) + Diffraction Problem (Wave forces-moment) = Hydrodynamic analysis (Coupling effects are neglected) 7/36

8 Radiation problem Heave Sway Roll 8/36

9 Te boundary value problem(bvp)-heave it V Voe k x z g ( z d) tt z ( z ) z ( x a, x a, z d) x e it V ( ve ) a x a z z i t ( x, z; t) Re V d( x, z) e Te velocity potential is in dimensionless form ( xz, ) due to te Vd term and will be used in tis form. 9/36

10 Te velocity potentials in region I, II & III In region I Te potentials tat are gained by using te separation of variables metod A nx cos nx sin nz B x nz a sin I ( x, z) n cos A Bn cos 2 na n 1 2 na n1 cos I ( xz, ) expanded into Fourier series ( x, z) ( x, z) ( x, z) I I I p 1 I x z z x p 2d 2 2 (, ) ( ) (Sabuncu, T.) 1/36

11 In region II & III ( x, z) ve ( x, z) II III expanded into Eigen functions. ikx kmx e e ( x, z) C Z ( z) C Z ( z) II ik a m k m ma e m1 e ikx kmx e e ( x, z) D Z ( z) D Z ( z) III ik a m k m ma e m1 e 11/36

12 Matcing conditions Pressure Velocity ( x, z) ( x, z) x a, z I II ( x, z) ( x, z) x a, z I III I ( x, z) II ( x, z) x x ( x a, z d) I ( x, z) III ( x, z) x x ( x a, z d) 12/36

13 1st Equation system/set 2 n z A B ( a, z)cos dz n n I 2 n z 2 n z A B C Z ( z)cos dz C Z ( z)cos dz n n m m m1 2 a A B C L n 3d d m m m n 2 ( 1) A B C L n 1, d ( n ) n n m mn m 13/36

14 2nd Equation system 2 n z A B ( a, z)cos dz n n I 2 n z 2 n z A B D Z ( z)cos dz D Z ( z)cos dz n n m m m1 2 a A B D L n 3d d m m m n 2 ( 1) A B D L n 1, d ( n ) n n m mn m 14/36

15 3rd Equation system d II ( a, z) II ( a, z) x x ik dc Z ( z) dz Z ( z) dz m= d II ( a, z) II ( a, z) m m m m x x k dc Z ( z) dz Z ( z) dz m 1 n n a n n a a N A L B L B L C ik d k 1/2 n tan n n cot n sin n1 2 4a n1 2 kd n n a n n a a N A L B L B L C k d k 1/2 m n tan mn m n cot mn m m sin m n1 2 4a n1 2 m1 kmd 15/36

16 4t Equation system d III ( a, z) III ( a, z) x x ik dd Z ( z) dz Z ( z) dz m= d III ( a, z) III ( a, z) m m m m x x k dd Z ( z) dz Z ( z) dz m 1 n n a n n a a N A L B L B L D ik d k 1/2 n tan n n cot n sin n1 2 4a n1 2 kd n n a n n a a N A L B L B L D k d k 1/2 m n tan mn m n cot mn m m sin m n1 2 4a n1 2 m1 kmd 16/36

17 Complete set of linear equations 2 a A B C L n 3d d m m m n 2 ( 1) An Bn CmLmn 2 m d ( n ) n 1, a A B DmLm 3d d n m n 2 ( 1) A B D L n 1,2... Linear equation system is solved wit a Fortran code wic uses Gauss elimination metod. n n m mn 2 m d ( n ) 1/2 n n a n n a a N n tan n n cot n sin n1 2 4a n1 2 kd A L B L B L C ik d k n n a n n a a N A L B L B L C k d k 1/2 m n tan mn m n cot mn m m sin m n1 2 4a n1 2 m1 kmd n n a n n a a N A L B L B L D ik d k 1/2 n tan n n cot n sin n1 2 4a n1 2 kd 1/2 n n a n n a a Nm An tan Lmn B Lm Bn cot Lmn Dmkmd sin km 2 4a 2 k d 17/36 n1 n1 m1 m

18 During te solution steps 1 Z ( z) Z ( z) dz Z N cos( k z) (m=) 1/2 Z N cos( k z) (m 1) 1/2 m m m N 1 sin(2 kd ) 1 2 2kd N m 1 sin(2 kd) 1 m 2 2kd m 2 N ( 1) k sin( k ) L ( m ve n,1,2...) 1/2 n n 2 2 ( n ) ( k) 1/2 n 2 Nm ( 1) kmsin( km) 2 2 ( km) ( n ) L ( m 1 ve n,1,2...) mn 18/36

19 Added mass and damping coefficients By integrating te pressure F ( a, z; t) nds t C (, ; ) V p V a a x z t t a V b V Pdx a V b V F V V V az bz a d 2d n i A 2 2 An ( 1) tan a a a 3 a a n1 n a n a 19/36

20 Damping Coefficient Added Mass Coefficient Hydrodynamic coefficients results and comparison Nondimensional Wave Frequency Nondimensional Wave Frequency 2/36

21 BVP for sway motion it V Voe i x z g ( z d) tt z z ( z ) it Voe x ( x a, x a, z d) z ( a x a ve z ) i t ( x, z; t) Re V d( x, z) e Te velocity potential is in dimensionless form ( xz, ) due to te Vd term and will be used in tis form. 21/36

22 Damping Coefficient Added Mass Coefficient Hydrodynamic coefficients results and comparison Nondimensional Wave Frequency Nondimensional Wave Frequency 22/36

23 BVP for roll motion e it o j x z g ( z d) tt z ( z ) z ( z d) ( x a, x a, z d) x x ( a x a ve z ) z i t x z t d x z e 2 (, ; ) Re (, ) Te velocity potential is in dimensionless ( xz, ) d will be used in tis form. 2 form due to te term and 23/36

24 For roll motion Particular solution (presently introduced) 3 2 x xz I p ( xz, ) 6d 2d 2 2 For ydrodynamic coefficients M p( r n) ds S r xi ( z d) k 24/36

25 Damping Coefficient Added Mass Coefficient Hydrodynamic coefficients results and comparison Nondimensional Wave Frequeny Nondimensional Wave Frequency 25/36

26 Wave force and moment (Diffraction) analysis x z g ( z d) tt z ( z ) z n i t ( x, z; t) Re c( x, z) e i Te velocity potential is in dimensionless form ( xz, ) due to te c i term and will be used in tis form. 26/36

27 Velocity potentials in regions A nx cos nx sin nz B x nz a sin I ( x, z) n cos A Bn cos 2 na n 1 2 na n1 cos ikx kmx e e ( x, z) C Z ( z) C Z ( z) II ik a m k m ma e m1 e e cos( kz) ikx kmx ikx III ( x, z) D Z( z) m m( ) ika D Z z e kma e m1 e sin( kd) e 27/36

28 Wave force and moment F pnds M p( r n) ds S p t S d III ( a, z) II ( a, z) X dz dz t t d Z a a I ( x, ) dz t d a d III ( a, z) I ( x, ) II ( a, z) M ( z d) dz xdz ( z d) dz t t t a Söylemez, M., Gören, Ö., (23). Diffraction of oblique waves by tick rectangular barriers. Applied Ocean Researc, 25, /36

29 Dimensionless Moment My Dimensionless Force Fx Dimensionless Force Fz Wave force and moment results and comparison Nondimensional Wave Frequency Nondimensional Wave Frequency Nondimensional Wave Frequency 29/36

30 Equation of motion 6 F n it n it n g zds Re i e jds Re i e ( 7) ds M rxn S j1 rxn rxn S S B B B Radiation problem analysis Added mass (acceleration) Damping (velocity) Diffraction problem analysis Wave load and moment Hydrostatic restoring forces and moment Force and moment (displacement) 3/36

31 Equation of motion ( m a ) ( ) e b ( i) e F e 2 i t i t i t x X F ga x 2 a ( m aij )( ) bij ( i ) ( m a ) ( ) e b ( i) e c e F e 2 i t i t i t i t z Z F ga z 2 a ( m aij )( ) bij ( i ) cij ( m a ) ( ) e b ( i) e c e M e 2 i t i t i t i t y M ga 3a ( I a )( ) b ( i) c m 2 y /36

32 RAO-Sway motion RAO- Heave motion Response amplitude operator (RAO) 3 1a a Wave frequency Numerical M. Present Study 1.2 2a a.8.4 Numerical M. Present Study Wave frequency 32/36

33 Results & Discussion For te radiation problem Good agreement wit te experiment results For low frequency values te comparison wit te experimental work is muc more better tan te numerical metods. Also wave force and moment values presented are in good agreement wit te experimental work. Calculation metod and getting te result in sort time are te advantages wen compared to numerical metods. 33/36

34 Field of application Not only for floating breakwaters A design approac for new landscape requirements for social activities, entertainment establisments or living for industrial factories or military facilities Terminals (suc as for container sips, airports etc) Seakeeping caracteristics of rectangular structures (Strip teory) Calculating te wave force and moment of wavemakers 34/36

35 Future work Planned future works as a continuation of te present study are Generating a code for ydrodynamic analysis of moored floating structures by te linear teory approac Nonlinear analysis of te same problem Wave energy conversion in te soreline by considering rectangular cylinder as a source of driving unit on linear generators. 35/36

36 Tank you for your attention 36/36