Multiple Wave Spectra. Richard May Team Lead, Aqwa development

Transcription

1 Multiple Wave Spectra Richard May Team Lead, Aqwa development

2 . Introduction Ocean waves with different frequencies and directions are very difficult to model mathematically Various simplified theories & spectral models of ocean waves small amplitude linear Airy wave higher order Stokes wave long crested irregular waves multiple directional short crested irregular waves

3 Waves induce forces on offshore structures: the wave exciting forces at wave frequency non linear wave forces: low frequency drift force & sum frequency forces, due to the instantaneous wetted hull surface varying, impact, slamming forces. Long crested wave case does not necessarily lead to conservative results, the interaction effect between waves from different directions may be important Renaud et al

4 . Ocean Waves Modeling in Aqwa 3.0 and before 4.0 and thereafter Airy wave Second order Stokes wave same ormulated spectrum User define spectrum Imported wave elevation IWHT Main + cross swell Spreading waves D carpet waves same same same Spectral group Any combination 4.0 in Librium and Drift 4.5 in er one sub dirn in Line & aut 4

5 Structure of wave spectral group Wave Spectral Group Single spectrum Main + cross swell D Carpet Spread Sea Imported WHT sub dirn sub dirns Up to 4 sub dirns 7 sub dirns sub dirn Up to 5 IWHTs Up to 0 Spectra, 4 sub directions * IWHTs: Imported files of wave elevation time history 5

6 Definition of spectral group in Aqwa suite 3SPGR 3SGM UDE + PSMZ + IWHTs 3CURR ISOW 3WID * AME.UDE - LOG CRESTED * SEED 005 3SPD UDE UDE II * AME. LOG CRESTED -ORMULATTED * SEED 006 3PSMZ * AME 3. IMPORTED WAVE ELEVATIO * SEED 007 3IWHT MTWAVE.WHT 3IWHT MTWAVE.WHT 3ODR ED3 Spectral group & name One current/wind setting Wave spectrum definition Seed definition see Aqwa reference for default values Ignore nd force of this sub directional spectrum 6

7 Definition of spectral group in Aqwa Workbench 7

8 8 Wave representation: Linear superposition no interaction 3. Multi Directional Wave Effects on Loads sin cos, m m w d t Y K X K i j m e a t X st order wave exciting force: Linear superposition w d t i j m e a t Morison drag force: Linear superposition of fluid particle velocities S C t i j m r r r d d U U e X V a t U t U t U C t w d,

9 Second order wave forces t d d m n a a { P cos[ t m n j k jkmn jkmn jkmn jkmn Quadruple summations Interaction between frequencies sum and difference and between directions Drift damping Kim et al, 997 Q P Q sin[ cos[ sin[ t t t ] ] ] ] }, B d d w j; m, n x j; m n m n j j; m, n { cos m cos k n j,, K C j j gj C gj sin m m sinn } n 9

10 Mean Drift force for multiple directional wave case d d w a m n j a jn { P jn cos jn Q jn sin jn } Mean Drift force for single directional wave case w j a j P jj Triple summations including directional coupling, In phase and out of phase components, Sensitive to wavelet random phases. 0

11 irst order wave exciting force orce spectrum in er m m m S S d Difference frequency second order wave force v4.5,,, },, {8 mn mn mn mn n m n m Q P D d D S S S d d Total wave force S S S

12 Second order force coefficients g S 0 S 0 WL [ X r [ r d dt n dl ] n ] n ds ds Water line integral Bernoulli Acceleration Momentum S 0 d dt n ds nd order potential without the 5 th term and using complex values for unit wave amplitude P P jkmn jkmn, Q, jkmn Q jkmn 4 g 4 S0 S0 WL g [ X WL [ X ' r ' ' r ' ' r d dt '* r n dl [ 4 ' d dt ] n ds n dl [ 4 '* S0 ] n ds ' S0 ' ' [ M ' s [ M ' X s ] n ds ' g '* X ], ] n ds '* g ].

13 4. Extended ewman s Approximation Store Pjkmn, Qjkmn, Pjkmn, Qjkmn database, Interpolating P, Q, P, Q of all wavelets at each time step, jkmn jkmn jkmn Quadruple summation of all nd order force components jkmn is numerically prohibitive ewman s approximation for single directional waves P ' jk [ P jj P kk ] Extended ewman s approximation for multiple directional waves ' Pjkmn [ Pjn Pkm ], ' Q jkmn [ Q jn Qkm ]. 3

14 4. ewman s Approximation ω ω =... j... j 4

15 4. Extended ewman s Approximation ω ω =... j... j 5

16 Employing extended ewman s approximation { c [ m j n k { s [ m j n k d m d n d m d n c s P km P km s c Q Q km km ]} ]} where c a cos t, s a sin t Require directional coupling mean drift force coefficients database, Obtain P, of actual relative direction at each time step, km Q km Quadruple summation reduced to triple summation. Less hard disk and memory requirement, more efficient 6

17 Validation of extended ewman s approximation 7 Rectangular box in deep water, Internal lid, o 5 th term m=n: ewman s approximation m 80 n P jkmm Q jkmm Good for small 0.E+5 0.E+0 -.E+5 -.E+5-3.E+5-4.E+5-5.E+5 ' P jkmm ull QT ewman Approximation -6.E E+5 0.E+0 -.E+5 -.E+5-3.E+5-4.E+5-5.E+5-6.E E+5 0.E+0 -.E+5 -.E+5-3.E+5-4.E+5-5.E Surge 6.E+4 4.E+4.E+4 0.E+0 -.E+4-4.E+4-6.E rad / s ' Q jkmm -8.E E+4 4.0E+4 0.0E+0-4.0E+4-8.0E+4 -.E rad / s 8.0E+4 4.0E+4 0.0E+0-4.0E+4-8.0E+4 -.E+5 -.6E+5 -.0E rad / s

18 nd order surge force m 80, n E+5.0E+5 5.0E+4 0.0E+0-5.0E+4 -.0E+5 -.5E+5 ull QT ewman Approximation -.0E E+5 5.0E+4 0.0E+0-5.0E+4 -.0E+5 -.5E+5 -.0E E+5 0.0E+0 -.0E+5 ' P jkmn -.0E E+5 0.0E+0 -.0E+5 -.0E+5-3.0E E+4 0.0E+0-5.0E+4 -.0E+5 -.5E+5 -.0E+5 -.5E+5-3.0E E+5 0.0E+0 -.0E+5 -.0E+5 ' Q jkmn -3.0E rad / s 0.05 rad / s 0. rad / s ' ' P jkmn, Q jkmn same order Good for small 8

19 Directional coupling effect on nd order surge force 6.E+5 5.E+5 4.E+5 3.E+5.E+5.E+5 ull QT ewman Approximation 0.E E+5.5E+5.0E+5.5E+5.0E+5 5.0E E E+5.5E+5.0E+5.5E+5.0E+5 5.0E+4 0.0E E+5.5E+5.0E+5.5E+5.0E+5 5.0E E ' P jkmn Q ' jkmn m rad / s Extended ewman s approximation P jkmn Q jkmn Enable to estimate, to include directional coupling effect As good as the original ewman s approximation for multiple directions Easy accomplishment 9

20 5. Effect of multi directional spectrum on a moored LG carrier Length between perpendiculars: 74m Draft: m Soft mooring system: Longitudinal mooring stiffness: 8k/m Transverse mooring stiffness: 54k/m Rotational mooring stiffness: 4.58E6km 3500m^3 storage capacity LG carrier by courtesy of SBM Roll damping: 4.0e5 kms/rad Water depth: 5m Spread sea: JOSWAP spectrum: Hs=m, Tp=0s, 0 spreading form: cos,

21 Aqwa Line Calculation Soft spring system is defined by additional structural stiffness SST 39 frequencies and 37 directions in [ 80, 80] degrees Use MQT option to calculate the directional coupling mean QT matrices 5% extra CPU time for the directional coupling mean QT calculation *.MQT size of mb compared to.3mb of *.RES

22 Effects on equilibrium position Three treatments of spreading sea: Long crested wave along main heading AB7878PLOG Represented by 7 point Gaussian integration, no directional coupling MQT AB7878POQT 3 Represented by 7 point Gaussian integration, with directional coupling MQT AB7878PMQT 9 DRM 9ILE AL7878PM5.MQT 9CSTR ED9 Input data Iterative procedure

23 Time domain analysis Aqwa Drift 5% CPU increment for 7 sub direction coupling drift force calculation Evident differences between 3 treatment results 3

24 requency domain analysis Aqwa er 4.5 Output total force/response spectrum contributed by all sub spectra Output RAO et al in main sub direction with max. Hs Optionally output RAO in specified sub direction SSPC in deck8 Comparison of the significant translational motions Surge m Sway m Heave m Drift freq. Wave freq. Drift freq. Wave Drift freq. Wave freq. freq. Long crested Spread, no coupling Spread, coupling

25 6. Conclusions Spectral group is introduced to model multi directional waves The directional coupling mean QTs are calculated and used in Aqwa Extended ewman s approximation provides a fast & relatively accurate approach with acceptable hard disk/memory requirements Multiple directional waves and directional coupling QT should be considered for moored offshore structure hydrodynamic analysis. Thanks! 5

26 Azzalini A, Du S., May R. 0 Extended ewman s approximation for the second order drift force in short crested waves, prepared for AV0. Chen X.J., Moan T., u S.X. and Cui W.C. 006 "Second Order Hydroelastic Analysis of a loating Plate in Multidirectional Irregular Waves", International Journal of on Linear Mechanics, Vol.6, pp Kim S., Sclavounos P.D. and ielsen.g. 997 "Slow Drift Responses of Moored Platforms", Proc. of the 8th Int. Conf. on the Behaviour of Offshore Structures BOSS 997, July 997, Delft, The etherland, Vol., pp ewman J "Second Order, Slowly Varying orces on Vessels in Irregular Waves", Proc. of Int. Symp. Dynamics of Marine Vehicles and Structures in Waves, Ed. Bishop RED and Price WG, Mech. Eng. Publications Ltd, London, UK, pp Pinkster J. A. 980 "Low requency Second Order Wave Exciting orces on loating Structures", PhD Thesis, Delft University of Technology. Renaud M., Rezende.C., Chen X.B. and Parigi C.J. 008 "Second Order Wave Loads on a LG Carrier in Multi Directional Irregular Waves", Proc. of 8th Int. Conf. on Hydrodynamics ICHD, October 008, antes, rance, pp Renaud M., Rezende., Waals O., Chen X.B. and van Dijk R. 008 "Second Order Wave Loads on a LG Carrier in Multi Directional Waves", Proc. of the 7th Int. Conf. on Ocean, offshore and Arctic Engineering OMAE008, June 008, Estoril, Portugal, Vol., Paper o. OMAE , pp Rezende.C. and Chen X.B. 00 "Approximation of Second order Low requency Wave Loading in Multi Directional Waves", Proc. of the ASME 9th Int. Conf. on Ocean and Arctic Engineering, June 00, Shanghai, China, Vol.3, Paper o. OMAE00 099, pp Sergent E. and aciri M. 00 "Wave Spreading and Wave Drift Loads for a Standard LG Carrier in Shallow Water", Proc. of the ASME 9th Int. Conf. on Ocean and Arctic Engineering, June 00, Shanghai, China, Vol., Paper o. OMAE , pp Waals O.J. 009 "The Effect of Wave Directionality on Low requency Motions and Mooring orces", Proc. of the 8th Int. Conf. on Ocean, offshore and Arctic Engineering OMAE009, June 009, Honolulu, Hawaii, USA, Vol.4, Paper o. OMAE , pp