Linear Seakeeping High Sea State Applicability

Proceedings of the 16 th International Ship Stability Workshop, 5-7 June 2017, Belgrade, Serbia 1 Linear Seakeeping High Sea State Applicability Tiothy C. Sith, Naval Surface Warfare Center Carderock Division, tiothy.c.sith1@navy.il Kevin M. Silva, Naval Surface Warfare Center Carderock Division, kevin..silva1@navy.il ABSTRACT The sall otion assuption of codes is well known. The validity of this assuption is investigated by coparisons with a body exact non- code over a range of significant wave height. A etric based on relative otion is proposed to quantify the validity of the assuption indicate up to what point is appropriate. Keywords: Linear Seakeeping; nuerical siulations. 1. INTRODUCTION Despite the advent of relatively coputationally fast non-linear tie doain seakeeping progras, there is still soe use for linear strip-theory seakeeping progras. Frequency doain progras can produce seakeeping predictions for any speeds, relative wave headings, seaways in seconds of coputation. This is especially useful for including seakeeping in early design analysis of alternatives calculating ission operability. Tie histories based on linear response aplitude operators (RAOs) are also fast to copute provide representative otions for ship syste design/evaluation. The ain assuptions of linear strip-theory seakeeping codes are well known. The first is that calculations are prefored about the ean undisturbed waterline. Hydrostatics, radiation, diffraction, incident wave forces are all calculated on the suberged portion of the hull at the ean undisturbed waterline. This is also stated as a wall sided sall otion assuptions. These descriptions explain in a physical sense what using the ean undisturbed waterline to define the suberged hull actually eans. Wall sided indicates that the hydrostatic properties are not changing as the ship oves. Sall otion indicates that the suberged geoetry used for radiation, diffraction, incident wave force calculations can be considered constant. O Dea Walden (1985) exained with respect to bow flare wave steepness. The other ain assuption of linear striptheory seakeeping relates to the independence of the two diensional strips. The strips are assued to be independent but in actuality flow fro one will influence flow fro strips further aft. As a result low speed strip-theory is liited to Froude nubers less than 0.3-0.35. Higher speed striptheories have been forulated. This paper does not address the validity of using low speed strip-theory above Froude nubers of 0.3-0.35. Lastly, as a direct result of having a constant suberged volue, the equations of otion can be solved for a unit wave height linearly scaled to higher wave heights. This is ost obviously seen with heavily daped heave pitch otion. However, roll has non-linear daping ost progras have soe iterative or coputational schee to account for this do not scale roll linearly with wave height. However, seakeeping predictions in very sall waves, where assuptions are valid, are not very useful. Fortunately, the assuptions can be stretched produce useful results at wave heights of interest. This paper discusses a etric to identify when the linear seakeeping assuptions are ore than stretched but broken. 2. COMPARISON APPROACH The validity of linear scaling of results will be deterined by coparing linear strip theory results with non-linear tie doain results for the sae hull for, loading condition, seaways. Heave,

2 pitch, roll root ean square (RMS) values will be copared for a range of wave heights. The coparison will be ade as a ratio of the RMS otion value at a given wave height over the RMS otion value of the lowest wave height considered. For linear response, thatt ratio is a straight line when plotted against wave height. Non-linear response will deviatee fro that line. Motions will be calculated at 15 knots (Fr=0.2062) for headings fro head to following seas in 30 deg increents. While this is not a coplete atrix, it avoids higher Froude nubers provides enough headings forr a preliinary evaluation. The wave heights considered range fro 3.25 to 12 in 1.75 increent. This is fro id-sea State 5 to id-sea State 8 following STANAG 4194 (NATO 1983). The wave heights correspondss to 0.5 to 1.84 ties the draft. A 14 second odal period is used forr all the wave heights, so the steepness increases with wave height. The waves are long-crested. The spectra shape is Bretschneider. The hull for used for this study is a generic naval cobatant that has been widely studied inn the public doain (DTMB odel 5415) (Longo Stern, 2005). See Figure 1 for a view of geoetry Table 1 loading condition details at full scale. Figure 1: Geoetry of odel 5415 Table 1: Full scale principle diensions of o DTMB 5415.. Paraeter Length between Perpendiculars Bea Draft, baseline Tri (+bow down) Displaceent LCG (aft FP) KG GM Roll Gyradius Pitch Gyradius Yaw Gyradius Units tonnes 142.0 18.87 6.51 0.00 9381.88 72.14 7.86 1.63 7.05 35.5 35.5 This hull for is a traditional onohull with a sall aount of o flare forward. As ost of the hulll is wall sided the expectation is thatt linear strip theory should be b appropriate at a uch higher wave height than hulll fors with ore variation. The siulation tools used for this study are Navy Ship Motion Progra (SMP95) (Conrad, 2005; Meyers a Batis, 1985; Meyers et al., 1981) Large Aplitude Motion Progra (LAMP) (Lin et al., 1990, 1994). SMP95 is a linear strip- theory seakeeping code first developed in 1981 thatt uses the Salvensen-Tuck-Faltinsen strip-theory (Salvensen et al., 1970) with odified forward- speed ters. It uses Frank s close-fit ethod (Frank, 1967) to calculate e radiation forces. Rolll daping is estiated fro appendage geoetry using Ikeda-Tanaka-Hieno (1978) Kato (1958) epirical forulae. Non-linear rolll daping is included by an iterative process to atch the calculated response with roll angle associated withh the roll daping estiate used to calculate the response. SMP95 calculates otions, velocities, accelerationsa s at center off gravity defined points, as well as, relative otion between points on the hull the incident wave. LAMP is a tie doain ship otion wave loads siulation code that was developed by Science Applications International Corporation (SAIC) beginning in 19911 to copleent linear frequency doain codes. LAMP calculates threee diensional wave-body w hydrodynaics using a potential flow approach. The basic hydrodynaic calculations include non-linear Froude Krylov forces non-linear hydrostatics as well as linear potential floww calculations. Roll daping, appendage lift a other viscous vortical forces are estiated using epricial forulae /or tuned coefficient odels. LAMP can calculate cobined seakeeping aneuvering, includes rigid elastic bea odels for coputing hydrodynaic loads. LAMP calculates otions, velocities, accelerations at center of gravity defined points, as well as relativee otion betweenn points on the hull the incident wave. 3. COMPARISON RESULTS The results are non-diensionalized by dividing by the value associated with the 3.25

3 significant wave height. The non-diensio onal wave height ranges fro 1.0 to 3.69. If the otions scale linearly with wave height, they should follow the sae range. Figures 2 to t 8 show the coparison of non-linear (LAMP) linear (SMP95) seakeeping predictions. In I Figures 2 8, the roll data are not presented due to values being very sall in head following seas. LAMP heave pitch results are very close to the line over the entire wave height range. The differences are ost notable at head (0 deg), bow (30deg), following (180 deg) seas above non-diensional wave height 2.5 (1.25 ties draft) ). Pitch behavior in bea seas is not linear as values are sall. LAMP SMP95 are very close in diensional values as well. Figure 4: Coparison of non-linear at Fr=0.21 bow seas (60 deg). Figure 5: Coparison of non-linear at Fr=0.21 bea seas (90 deg). Figure 2: Coparison of non-linear at Fr=0.21 head seas (0 deg). Figure 6: Coparison of non-linear at Fr=0.21 quartering seas (120 deg). Figure 3: Coparison of non-linear at Fr=0.21 bow seas (30 deg).

4 Figure 7: Coparison of non-linear at Fr=0.21 quartering seas (150 deg). Figure 8: Coparison of non-linear at Fr=0.21 following seas (180 deg). Both LAMP SMP95 roll results vary fro linear behavior as expected. LAMP results have noticeable curvature as the wave height increases. SMP95 results are fairly linear with a different slope than 1:1 with wave height. LAMP roll results are alost twice the SMP95 roll values in diensioanl values. This is explained as difference in roll daping odels appendage suite. 4. APPLICABILITY METRIC Grigoropoulos et al. (2003) indicates strip theory is appropriate for displaceent onohulls under Fn=0.3. However, the RoPax ferry did not perfor as well as expected. While the geoetry is vertical above the waterline, the below waterline shape has significant taper. DTMB odel 5415 has a relatively large bilge with nearly vertical sides at the waterline along ost of the length. A typical oil tanker has vertical sides for ost of it s length depth. Hull H for considerations lead to a etric that quantifies q the validity of linear seakeeping based on changes of waterplane areaa relative otion. An inforal etric is that is appropriate if the t relative otion is less than half the draft; essentially to thee top of the turn of the bilge. The rationale beingg this is the wall sidedd portion of thee hull the concernn is otion relative to wave, not absolute otions. This etric is soewhat vague v in ters of relative otion statistic, e.g., RMS, 1/100 th highest, point location. Following Meyers et al. (1981) applying the Rayleigh distribution, the probability of relativee otion, σ, exceeding half the draft (critical distance D) can be foundd by P = e D 2 2σ 2 (1) The probability where the linear non-linear results diverege becoess the liitt of linear seakeeping applicability. Even so, this is soewhat subjective in ters of location of pointss at which to evaluate relative otion selection of critical distance, e.g., half h the draft.. This study proposes using points at 0.25LBP 0.75LBP, centerline, baseline to evaluatee relative otion with respect to incidentt wave. The quarter length points bracket parallel iddle body locations whilee representing soe of the t fore aft geoetry changes. Thee critical distance is the average of the distance fro the ean waterline to where the station becoess decidedly non-vertical. This definition accoodates different hulll geoetries fro RoPax ferry to oil tanker. For this case, the critical distance is half the draft (3.251). Also, note that the probability changes with speed heading, so soe iniu probability should be selected as thee liit of applicability. Figures 9 to 122 show the probability off the SMP95 relative otionn exceeding half the draft for casess that showed a differencee between LAMP SMP95. The forward pointt is the liiting point as wave heading ove aft of bea, the forward point line oves towards the aft point line becoes coincident. Looking at head seas, Figure 2 Figure 9, LAMP heave a pitch aree approxiately 2% lesss than linear value near non-diensional wave height

5 of 2 with a probability 0.14 for the forward point. The probabilities at bow seas are less for siilar otion differences. Other headings have lower probabilities less difference in otions forr the sae wave height. Roll shows ore non-linearity, but within 10% difference at 2.0 non-diensio onal wave height. The threshold value to exceed, allowable otion difference, relative otion point location are all inter-related acceptability liits cannot be set independently. So taking the relative otion point location as the foward point accepting 2% difference bewteen linear non- linear results sets the threshold probability at 0.14. So for other destroyer-like hull fors, if the probability of a forward relative otion point is less than 0.14, the difference between linear non-linear response is less than 2%. Other relative otion points acceptable differences would have other associated probabilities. Figure 11: Probability of relative otion at bow stern exceeding half thee draft at Fr=0.21 bow seas (60 deg). Figure 12: Probability of relative otion at bow stern exceeding half thee draft at Fr=0.21 following seas (180 deg). Figure 9: Probability of relative otion at bow stern exceeding half the draft at Fr=0.21 head seas (0 deg). Figure 10: Probability of relative otion at bow stern exceeding half the draft at Fr=0.21 bow seas (30 deg). 5. CONCLUSIONS This study proposed a etric to quantify the applicability of o linear strip-theory seakeeping. Motions were calculated for DTMB odel 54155 using SMP95 LAMPP for a range of wave heights, a single speed, d ultiple headings. The otions were copared too see wheree non-linear effects were apparent a d iportant to the root ean square off the otions. A etric based on the probability of the relative otion exceeding a critical distancee was proposed to define the range of applicability of o linear strip-theory seakeeping predictions. This T approach shows proise but needs to be exped to other speeds hulll fors to deterine general applicability. Other statistics such as average of 1/10th highest ay provide ore discriinationd than root ean square

Proceedings of the 16 th International Ship Stability Workshop, 5-7 June 2017, Belgrade, Serbia 6 statistic. Additionally, there ay be soe copleentary etric based on variation in waterplane area that would iprove selection of critical distance. 6. ACKNOWLEDGEMENTS The author would like to thank the any who discussed the idea provided invaluable coputational support. Mr. Robert To Waters provided a rough idea of the relative otion etric. Mr. Andrew Silver first used the probability of keel eergence in a systeatic way in 2016. Mr. Kenneth Wees provided the description of LAMP the input files for the LAMP calculations. DTNSRDC/SPD-0936-01. NATO Military Agency of Stardisation, 1983, STANAG 4194 Ed3, Stardized Wave Wind Environents Shipboard Reporting of Sea Conditions. O Dea, J. F. Walden, D. A., 1985, The Effect of Bow Shape Nonlinearities on the Prediction of Large Aplitude Motions Deck Wetness, 15th Syposiu on Naval Hydrodynaics, Haburg, Gerany, Sep 1984, pp. 163-176. Salvesen N., Tuck E.O., Faltinsen O., 1970, Ship Motions Sea Loads Trans. SNAME 78: 250-287. REFERENCES Conrad, R., 2005, SMP95: Stard Ship Motion Progra User Manual, NSWCCD-50-TR 2005/074. Frank W., 1967, Oscillation of Cylinders in or Below the Free Surface of Deep Fluids DTNSRDC, Rep. No. 2375, Washington, D.C. Grigoropoulos, G., Harries,S.,Daala, D., Birk, L. Heiann, J., 2003, Seakeeping assessent for high-speed onohulls A coparative study, 8thIntl. Marine Design Conf., Athens, 5-8 May. Ikeda, Y., Hieno, Y., Tanaka, N., 1978, Coponents of Roll Daping of Ship at Forward Speed, Journal of Society of Naval Architects of Japan, Vol. 143. Kato, H., 1958, On the Frictional Resistance to the Roll of Ships, Journal of Society of Naval Architects of Japan, Vol. 102. Lin, W. M., Yue, D. K. P., 1990, Nuerical Solutions for Large-Aplitude Ship Motions in the Tie-Doain, Proc. the 18th Syposiu on Naval Hydrodynaics, The University of Michigan, Ann Arbor, Michigan. Lin, W. M., Meinhold, M. J., Salvesen, N. Yue, D. K. P., 1994, Large-Aplitude Motions Waves Loads for Ship Design, Proc. the 20th Syposiu on Naval Hydrodynaics, University of California, Santa Barbara, California. Longo J., Stern F., 2005: Uncertainty Assessent for Towing Tank Tests With Exaple for Surface Cobatant DTMB Model 5415. J. Ship Research. 49 (1) 55-68. Meyers W.G., Baitis A.E., 1985, SMP84: iproveents to capability prediction accuracy of the stard ship otion progra SMP81 DTNSRDC/SPD-0936-04. Meyers, W. G., Applebee, T. R., Baitis, A. E., 1981, User s Manual for the Stard Ship Motion Progra, SMP,