‘s Technical Report Sponsored by NAVAL FACILITIES ENGINEERING COMMAND August 1974 Robert 6. Wee graphic tet CIVIL ENGINEERING LABORATORY Woods Hole Os NAVAL CONSTRUCTION BATTALION CENTERNoodSs Hote, Port Hueneme, CA 93043 ite eater te A SURVEY OF TECHNIQUES FOR THE ANALYSIS AND DESIGN OF SUBMERGED MOORING SYSTEMS By N. D. Albertsen eee a eee : Approved for public release; distribution unlimited. cis it rY ve {rv Technical Report Sponsored by NAVAL FACILITIES ENGINEERING COMMAND August 1974 rt G. Walden Rolie Oceanographic inais 2 CIVIL ENGINEERING LABORATORY Woods Hole NAVAL CONSTRUCTION BATTALION CENTERNooss Hote, Port Hueneme, CA 93043 A SURVEY OF TECHNIQUES FOR THE ANALYSIS AND DESIGN OF SUBMERGED MOORING SYSTEMS By N. D. Albertsen 4 Geapod es : esp Approved for public release; distribution unlimited. cis, a ey, — § ae s oi Dy Yow lle UTM DWT IV) A4 ITA UDAN SAV AP PAV tps VHOTP RA SPIO eer a cee FO ebay Nay OM MAY vo YUNA LAV AM, “SURE ND srbayh ae ~ j mes er a ——— ———) . : oe 7 oe, Ht SE oe a ems 4 AOE MMI THT RON AUG HANI Ley ia pares Ae | : ny, VE OWI TORII C505 SAL EN TS A ave (LAA, She vee PA OF Ke uh ae Liutiveeda’s vittiery Peal A edeneta a Unclassified SECURITY CLASSIFICATION nF THIS PAGE (When Data Entered) REPORT DOCUNENTATION PAGE 1. REPORT NUMBER 2. GOVT ACCESSION NO| 3. RECIPIENT'S CATALOG NUMBER 4. TITLE (end Subtitle) 5. TYPE OF REPORT & PERIOD COVERED Final; Sep 1972—May 1974 6. PERFORMING ORG. REPORT NUMBER A Survey of Techniyues for the Analysis and Design of Submerged Mooring Systems 7. AUTHOR(s; 8. CONTRACT OR GRANT NUMBER/Ss) N. D. Albertsen 9. PERFORMING ORGANI: ATION NAME AND ADORESS CIVIL ENGINEEKING LABORATORY Naval Construction Rattalion Center Port Hueneme, California 93043 11. CONTROLLING OFF!c ct NaME AND ADORESS 10. PROGRAM ELEMENT, PROJECT, TASK AREA & WORK UNIT NUMBERS 62755N YF53.535.004.01.005 12. REPORT DATE August 1974 13. NUMBER OF PAGES 34 SECURITY CLASS. (of this report) Naval Facilities En: incering Command Alexandria, VA 2°?%32 14. MONITORING AGENC\ NAME & ADDCRESS(1/ different from Controlling Office) 15. Unclassified WSa. DECL Spagice DOWNGRADING 16. DISTRIBUTION STATEMENT (ol this Report) Approved for public release; distribution unlimited. 17. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report) —] 18. SUPPLEMENTARY NOYES 19. KEY WORDS (Continus «+4 reverse side if necessary and identify by block number) Submerged mooriiys, cables, computer simulations, cable strumming, static analysis, dynamic analysis, vortex shedding. 20. ABSTRACT (Continue ws reverse side If necessary and identify by block number) Techniques \xeful for the design and analysis of moored cable systems are presented. Included are compilations of information describing computer programs for steady-state and dynamic analyses and method for the analysis and suppression of cable strumming. 1! was found that the steady-state analysis of complex redundant structures and the \iynamic analysis of simple structures are now possible. However, experimental data aye required to validate the analysis procedures and related computer FORM . 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Vo aricnt teeahrnmhenin EET: ARO Serene 0 aN a anna ty 8 alti ge quniie 10 bad ise ttt ' i Suber ny hifrvelon ota worviuna’ > seat git et angie ats ties, topic o i dayth Ab beally (gotonilnarad. | rehorbtag A al aeiea aivrobett wisest gt erle, waligen hy oo Mh etpuaatit oO Aaa gore’ one aay ben ii Wa soit hres wely yi aN aynivongits belli au ad ey ae ts ron binned eke WPA TY ae ta ewan 0 a vino ehh batt tn ca ‘hetedt tptonsanir out a ® mibajile> cre ioibesgin, eh lisieen to Rorteneneney a asin a a ‘Rode ‘ath ment 7 mm: bisaaar assumptions associated with the structure and with hydrodynamic loading criteria and because of errors inherent in the computational techniques and machines. Therefore, to provide full confidence in a program it should be validated by comparing pre- dicted results with precise experimental data. How- ever, because little experimental data exists to validate programs, only a few of the programs for the simple mooring types have been validated. Assumptions. The assumptions vary from pro- gram to program but basically center on the method used to model the structure and on the simplifica- tions used in the calculation and application of the hydrodynamic loading forces. Remarks. A brief summary of program charac- teristics and capabilities are presented, and the availa- bility of program listings is indicated. Survey Results All the programs presented in Table 1 are operational. It should be mentioned that some of the dynamic programs use a steady-state program to establish a starting point for dynamic analysis. The entries in Table 1 based on Reference 16 by Patton and on References 37 and 38 by Goodman and Sargent are examples of static programs which are extracted from a dynamic program. DYNAMIC ANALYSIS In this section computer programs for the dynamic analysis of moored cable systems are pre- sented. These programs try to simulate the dynamic response of various moored structures under the action of time-dependent forces other than those pro- duced by vortex shedding. Table 2 presents the pro- grams and attempts to update and complement the surveys presented in References 1 through 3. Terminology of Table 2 Each of the descriptive terms found in Table 2 are briefly discussed below. Program Dimensionality. As with the steady-state programs, the dynamic programs may be one-, two- or three-dimensional. The two programs in - Table 2 that are one-dimensional examine the behav- ior of a mooring system along the cable axis. The remaining programs are concerned with tensions and displacement of the system in a plane or in space. The authors of several of the two-dimensional programs feel that extension to the third dimension would not require large amounts of reprogramming. To simulate the ocean environment, particularly near the surface, three-dimensional analysis is many times a necessity because of nonplanar wind, wave, and current forces. Mooring Type. Again the reader is referred to Figure 1 for definitions of the terms used to describe mooring-system types. Most of the programs listed in Table 2 are for the single-point mooring. Solution Method. A variety of solution methods are used in the dynamic programs. Reference 3 gives an excellent summary of the methods with their merits and demerits. System Excitation. The environmental factors which produce the dynamic system response are sum- marized here. The excitation sources are related to wave, wind, or current. System Damping. The forces generated by mooring systems which oppose the dynamic response of the systems are called damping forces. Damping is generally hydrodynamic- or cable-related but may be assumed to be a function of the geometry or mass of the system. An excellent presentation of the Voight and Maxwell linear-damping models for cables is given in Reference 40. Properties of Cables. The assumptions associ- ated with the cables are summarized here. In most cases all forces and mass are assumed to be concen- trated at discrete points along the cable. The cables are assumed either to be completely flexible and elastic or to be made up of a series of springs or rigid segments interconnected with springs. Status of Program. Information on whether the program Is operational or proprietary Is given. nhs die aA Awe iei maygest ad ace unt viene chs ath remangetg steteyfaliasy | wf BsTTayOry ont aT sLacronam neni be * -Gwl ano anyleatt jytlirnnniysnsbohid wold Shain, wlhsieuts..: inp aeeaR ike: oti ives i a ee iqerseey gentry i) —— PS Puan reclaim wane qrminiticvn bee Dili Gp awl Rah tito (hin hha peng) aaenaegt ni baie uy & quant » wot! 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L = cable length (ft) T = cable tension (Ib) virtual mass of cable (slug/ft) = mass of cable + mass of equivalent volume of water it has also been shown that the approximate frequency of vortex shedding from relatively short* cylinders and cables perpendicular to flow may be characterized by the Strouhal Equation [103, 104]. The Strouhal relation is f, = SV,/d (2) where f, = Strouhal frequency (Hz) S = Strouhal number = 0.2 when 2x 10% ran, Yo el =e" i) Peetdes anil) + iy? ovedw sieenteonage say salle srr 7 ea Se a en “Pendle phe papa emnal | met pat erm yeeesl) a: let en ‘alitibvwn wit nagtert rai: gle ey a? hives ‘gry secant tied i (Pr warrenty alten. 3h as a (5) CATA ote) ‘allio Yo iain enytie oe Ul ; warner at hierae Pl bari t yn ay. psu aeai & nay 4 Ay aan Aiea acre t Ab be See aie RS au aguas y a ae a ee pros Viveay ATO, met Ce: roan Rvaeherre) phtisn VOR in gn bey cane wil ghiver: enh eer mnitnape sil Ve aneiynaiibewe. cat whhidaty Ys rupee wi) ie En volte gn na hare ad Mess salynaiy “Petia: we tal trseiyerdtiysill neonates Lat (miroir t Pee wee THE GW CMM) e sive ayty yyy , Oh é . * * y f te cat ano ic silted inva ie sae rm hen be Bel wt sake usd Leavin ae iB. et hd ar full Pea NRO. Neel h ARE a Geto yeh ih grate vabilina ssilr teangenge ah, ‘ tres Kypabriw phe vet we at won ihe Rte sigh tf iatieatait hy bis netiehalthecg peek «amt Pe a Plena aah sR ea “ VPREY 1 YTS os ie TELE : bi Wei hag ~ heodkigndd. cag surest Hnmieilhe demmelie:, € SIE wl sd fila 1 nil CAEP WLAD Rake Abra tipo: hare i % D Ath pe sided 4 aS ress: gaily oN aquyhen ave: Rh aD Senay hos bediaila oo ' ert i no heres pune ehs qeaitarateteo. rah "benny stint arginnaneg MMePM nt He retorhooagindt ‘Teanga, ait 9 TIO a ohana neneailiy ae teehee Weer) = thi lentvcabaii an} dled alana stay 7 beaboseng muse tres abiivad ‘ah liemeriidi ast ore ot ante xen phy ‘SE 10 Kips ra ih Piities Chir wy PAT NEY Ee iene Bilas ‘eas a paliiouey een] ay ‘qPhaeneiggvity, Hi ce ‘eryire WEAR Ae ee Sa amas Siberia: sin gues cies! aa De wh a eonisy AeA ¥ vesneendtag iui Nine Un erly worry Tia edie potiady ie ie Pere Lp: ees Cammy phage ada shia wilt feclesereee insta tiie T4 RoyeRt pA Hleliga. eke way FF ani sds caelaaas ibs WK tO LS anotwet) voll ux) elds! wee: de9 Arey a ‘olgpeaeh ’ i F : a i 4 yh i \ Oo. mmrnrentarion: PE Wal” We AL eae a Me Teta lt ts wh att Vi. ORRioe au pero ; ; ae Ye merny ti | Wiraersiely Pat bh peyy Hireacalisree “464 wernt Drs i Wen wel baat hut a ate Sy Gi tel el ge me eeee Hep HERDING j ; cabs Veetibeagginitcrs, ‘eadevia tS Hinenyneidy ccf ielit Tint ene Oink" seston TUNE Sac, sti iain ws hue sa yistelesnry Sidi iba belch wes shite bay Pistia CONCLUSIONS AND RECOMMENDATIONS 1. The steady-state analysis of complicated, multileg, redundant, submerged mooring systems is now possible. 2. The dynamic analysis of simple mooring systems, such as the single-point mooring and the bi-mooring is now possible. 3. Precise validation data for the steady-state and dynamic programs are needed to quantify the errors associated with the various techniques and to help select correct hydrodynamic loading criteria and added mass and damping coefficients used in the computer programs. 4. Analysis and design procedures need to be developed to reliably predict, describe, and suppress strumming in long cables in an ocean environment. 5. Experimental data are needed to aid in the devel- opment of a reliable tool to predict and describe cable strumming and in the development of practical strumming suppressors. REFERENCES 1. M. J. Casarella and M. Parsons. “Cable systems under hydrodynamic loading,” Marine Technology Society Journal, vol. 4, no. 4, July/Aug. 1970, pp. 27-44. 2. Hydrospace-Challenger, Inc. HCI TR 4450 0001: An inventory of current mathematical models of scientific data-gathering moors, by D. B. Dillon. Rockville, Md., Feb. 1973. 3. Cathodic University of America. Report 73-1: A survey of analytical methods for dynamic simulation of cable-body systems, by Y. I. Choo and M. J. Casarella. Washington, D. C., Mar. 1973. 4. Naval Research Laboratory. Report 7296: The analysis of internally redundant structural cable arrays, by R. A. Skop and G. J. O’Hara. Washington, D. C., Sep. 1971. 5. R. A. Skop and G. J. O'Hara. ‘““The method of imaginary reactions — a new technique for analyzing structural cable systems,” Marine Technology Society Journal, vol. 4, no. 1, Jan/Feb. 1970, pp. 21-30. 6. R. A. Skop, “The method of imaginary reactions: applications to N-point moors,” Preprints of the Sixth Annual Conference, June/July 1970, vol. 1. Washington, D. C., Marine Technology Society, 1970, pp. 1-22. 7. Naval Research Laboratory. NRL Report 6894: The static configuration of a tri-moored subsurface, buoy-cable array acted on by current-induced forces, by R. A. Skop and R. E. Kaplan. Washington, D. C., 14 May 1969. 8 NRL Report 7640: A Fortran IV program for computing the static deflections of structural arrays, by R. A. Skop and J. Mark. Washington, D. C., Aug. 1973. 9. Woods Hole Oceanographic Institution. Report WHOI-73-85: mooring for design and prediction, by N. N. Panicker. Woods Hole, Mass., Nov. 1973. Computer simulation of a tripod‘ 10. Bell Telephone Laboratories. Technical Report No. 14. The analysis of mooring systems and rigid body dynamics for suspended structures, by J. M. Gormally. Whippany, N. J., Dec. 1966. 11. Massachusetts Institute of Technology, Instru- mentation Laboratory. Report E-2321: A stress and stability analysis of the pilot project oceanic telescope, by H. F. Little. Cambridge, Mass., Aug. 1968. 12. Naval Underwater Systems Center. Technical Memorandum No. SA2302-323-71: A finite element model of a submerged line array moored in uniform current fields, by J. D. Wilcox. New London, Conn., July 1971. 13. J. W. Bendendender. ‘“Three-dimensional boundary value problems for flexible cables,” paper presented at Second Annual Offshore Technology Conference, Houston, Tex., April 1970. (OTC Paper No. 1281). 14. S. Bell, General Electric Co., personal communi- cation: Outline of GE-RESD capability in static and dynamic analysis of mooring systems. Philadelphia, Pa., Jan. 1973. 15. Woods Hole Oceanographic Institution, Report WHOI 68-79: Tension and geometry of single point moored surface buoy system, by W. D. Martin. Woods Hole, Mass., Dec. 1968. hi , enoitaany’ ULaTER Gar 7 hulivhe ait” qo A | ry | avi. INOW OP srdiawallgge aa " ofr te sinner CW OTe Vintiens! assthiwo hewn Mande ones vevaban ai elaieta! aw snuvrynideaW j $5-1, 4 OA Poet 1st speineyoda.) vibes iment 5 aehuece boreediiny 4 ty eer Catcar ower aun oft ane) ba ubnbhstan yo ae horas pene ‘aban yauad Af cotyaidtes pilqat 33 bae quel A yd POOL wpe, #1 mirgory VY Arized A (het hoded (ube Lo imran ter auetreehinb onde eld edna: Io jo 0 changin dwelt Aten quite A od aia ANET gees noga” seoerite) siviqeiyennysG pol abaRiy ice ‘hoger 810 eoiliale wtihie tea Wy roading? 4 Oe yd eter tte ‘Ayia mat 'y Heaney ieted eo, sae ott nol nod Sioladiet ubroter nde)’ sevitells't ih Bit Winger Waa eeensieg? .geviviodiion Nah abe rhea, ‘yt. A ofl gel, porns nuit bol sqhne wel afman’ s Ae sat we Nu A ery db vitinierventy ewes Miaohondan' he. suiiaianen Redoute Were-araue A SEE gad Jrntetindad ‘nt eiadin ginese duno aelq wir W meuhoak ye Wepre pk ant oben’ | -vkati,t Wt sania ; al unknitee T roansly a viel erie | lava’ = Tras Tint A yi ers ay? aL i ae imnbermenciat anole Hi tareatn De er be a aes Potecite yd severthinies sd walt) eo ‘pa atand seeelegy : Ped vant lenuitvemitoakT". .vhnehashawt WotR cepeg “enliieo olkhikell wat ari sathy vba Waohonton!. aeotelit) fenced, braved, ve hotoaina TH OPO) OEE yh, aE oteeett sonst: ; LLRST 4 inten Lakes yar oe letona, HG oe) ee Cea Ae) Yo sartiud dleriorta Evel at ed 4hodi tis Dhue at, wi sliefed ed heluistiod ings tale ares wert acntay pl he oe: weet ibe yon sy eta # f Ire phorie Ft AEST Lea meget |e bbe ory phonv? .miacth 41.9 vet opps OOM ee Pai Lai Sika {Beet 990) abel hott a ey ‘de irie ae xitlan duup my baboon oe ' oa. best onde aan ae corminuryh: buns sean ad 101 on Git OY Bre eampindlans ania wh a beta pyr ‘yoithanad | sbenaat vt wilt bee gh es a | eetrTg( at lok sadn sieNB yh Shraretets vel: “ine: ihe “toil oxy at tie 0 bain in we sdiiieshs ‘Set Vian) oe Lear gle TT a tn Te aaeaene: ‘aldinel nega’: we baw tls adlonnval ‘wnt "yaattwal 4 ryebonerlel Af Vet. gat i ao ra yi ty POO Geen HE NOT aie: rgiaeltinit9 vd Yeh, eiataoly Lead smeeenrl aint | aayrs "he seman oe yet” eae gota gage gay Ab TOD ee ill Aah rb§ neh, Aen Asi Wiles feats sutivanle * ibn vot ages Tess sition vl: ai wlth Bion: a yo) ‘yo anne vaginal a fucabroas femme Aa i aN iste Ldaetodowen . balay" 16. Naval Underwater System Center. Technical Report 4331: The response of cable-moored axi- symmetric buoys to ocean-wave excitation, by K. T. Patton. New London, Conn., June 1972. 17. D. B. Dillon. Hydrospace-Challenger Inc. cable programs, personal communication: Hydrospace- Challenger Inc., Rockville, Md., Jan. 1973. 18. S. M. Gay. “Computer analysis and design of undersea cable systems,”’ Undersea Technology, vol. 9; no. 10, Oct. 1968, pp. 43-49. 19. Naval Ship Research and Development Center. Report 3692: Effects of Nonplanar current profiles on the configuration of moored cable systems, by H. T. Wang. Bethesda, Md., Oct. 1971. 20. H. T. Wang and B. L. Wester. “Current profiles which give rise to nonunique solutions of moored cable systems,” paper presented at the Fourth Annual Offshore Technology Conference, Houston, Tex., May 1972. (OTC Paper No. 1538.) 21. John McChandless, Naval Air Center, Sonar Sensor Division. Personal communica- tion: Moor motion analysis at the Naval Air Develop- ment Center. Warminster, Pa., Dec. 1972. Development 22. D. Hubbard, Naval Ordnance Laboratory, White Oak. Personal communication: Moor motion analysis at the Naval Ordnance Laboratory. Silver Spring, Md., Jan. 1973. 23. M. Schreiber, Lockheed Missile and Space Co. Personal communication: Computer programs for the analysis of moored cable systems. Sunnyvale, Calif., Mar, 1973. 24. Lockheed Missile and Space Co. Unpublished document: Ocean systems structural analysis com- puter capabilities, by H. H. Eiriz. Sunnyvale, Calif., Feb. 1973. 25. Stanford University, Department of Petroleum Engineering, School of Earth Sciences. Report No. : The design and analysis of a submerged, buoyant, anchored pipeline for transporting natural gas through the deep ocean, by G. H. Savage. Stanford, Calif., Aug. 1970. (NSF Contract No. GH-61 and ONR Contract No. NO0014-67-A-D158-004) 26. University of New Hampshire, Engineering Design and Analysis Laboratory. Technical Report No. 104: Project Seaspider — Pacific; analysis of the motion of a tri-moored buoy with neutrally buoyant legs, by G. N. Sniffin and G. H. Savage. Durham, N. H., Feb. 1968. : 27. John Mariano and A. Shashaty, Bell Labora- tories, Cable Placing Group. Personal communication: Moor motion analysis at Bell Laboratories. Whippany, N. J., Dec. 1972. 28. Naval Underwater Systems Center. Technical Report 4379: Nondimensional steady-state cable con- figurations, by G. T. Griffin. New London, Conn., Aug. 1972. 29. Texas A and M University. COE Report No. 157: Three dimensional response of deep water mooring lines in steady current flow; mechanics of cable mooring systems, vol. I., by R. F. Dominguez and G. E. Owens. College Station, Tex., Dec. 1972. 30. . COE Report No. 158: Bi and quad cable arrays systems summary data report; Mechanics of cable mooring systems, vol. II, by R. F. Dominguez. College Station, Tex., Feb. 1973. Zl, COE Report No. 159: Parametric evaluations of bi and quad cable array systems; Mechanics of cable mooring systems, vol. III, by R. F. Dominguez and R. deCastongrene. College Station, Tex., Feb. 1973. 32. . COE Report No. 160: A computer program for analyzing the steady-state response of bi and quad cable arrays; Mechanics of cable mooring systems, vol. IV, by R. F. Dominguez and N. M. Muska. Col- lege Station, Tex., Feb. 1973. SBE COE Report No. 161: The response of a tri-moored cable array with an included deformable cylindrical systems, vol. V, by R. F. Dominguez and G. G. Greer. College Station, Tex., Feb. 1973. member; Mechanics of cable mooring 34. for analyzing the steady-state configuration of a tri- COE Report No. 162: A computer program moored array with included rigid and deformable members; Mechanics of cable mooring systems, vol. VI, by R. F. Dominguez, G. G. Greer, and A. C, Liau. College Station, Tex., Jan. 1973. ppbosalga? istiteqmatt: Ws lo: erheyeinls it “Preqnh levitihe'T scryeteds,1 ateylun A, nr A ype dneee ud oe liersecon Hive youl hievworrenes! iy # we mobi ber a een ee ee ae oa aienierse ' r ” » BAR dat sitsdiiet “Fest Nipchiaeh ania adel 38 ~nivteoinamynos inmene4, ouriad gusisat@ wrblsc3d atdhsenn eanqitv earned toa to. alkaline gitar wooly OVI wemmleyE: qtRwreb oy lesiorbesT zee haved’ US Bt pe | | i m9 didn ste yew Lingkanibio OTne Hgyodl a ' AOD web de eT art erect CTS? WA AK wie? Os ers h i) *7 i qi dy aati)! qenlaeien Jew weroubign yribw tient coho OPN oietlent comtemeav? (Gl) a pen pia nog GORD ovecroalielt Me dom ralew oaks, te ad io Pe Le) brag Soot st yd S2Or ath Pe by “lake Baniyy be 18 laren ean nity 4?” ota THVT WAD THEY correc 0 Whi aasten elas tA hep hbo win ‘pew te rd Crete ae Ty «ROCA RHC a bi Wb. dOw ee ie: ETL PE TERS yr! Ui a4 ie | ar i ae se | fiat ire Per Get S4e blu $i win: eee | ARS Vara Ube: bien nO Ce WA aplve Anih ) Me yd Dt heey Sierra vie jee nena tian! ins ae pays lay A, bine No Pee Ae wit > TEER hfe tianiNes EL SOE ile te Pe: PORCHES: # yt ee A shen Lipans Oiadeer shee ein od Vwiisilena teil sstiotinae yulencre silica Fer ebicuatualtd Spar wibas tes ofan MS toe gap Ls Sm ed ad WY tow at ee | Hod arte Pe wy99 Se oinogert AdT TOF 2 Poe aD Lee Sliema ‘boatulie ctw initk otats mewn, brriatins lho te ees ceetdini brass erred) ad tara osiunarieeoh wl ety Aree EVR to, 207. welt suelipp Piste Oe Mt al eng Al aM wom 305 rn) 3 40 8 ts Ghia Moy sth ylhast aif wre Hleteh. “yer qitoreihal haw ‘hege batwtom!) dive Noni, Peenien Ni RGN vaeviongige green te Ce Don 2 sisyick pate Pee et road a! FHL d and huihitees yore d i) hh g Renee A) RL ites BY OE On) eT wipriae De) fi Luh ' Vee: aeeiineen | Haresen MT Ra il ity henna Aunt | i, hnoom-aides Mn san edt to dvasns joitiont — pShiqenn?, trae POL OM “sbingealwenet. ea nagenniylt AMO un gittaadtenti lopli ORE ae BM, dt Ww. pyiashs bre aieylune omni fov iablobibs raerabiat) : ; aT ae, eee na sama, ronan “hye rout @ liq weentos unalgnnlt a ua | het eee plein: bianive bi Aw indadi) AM yd oamiezas Die he taeriah ; eye eaten ins. yee yodli® AD We ome ancy 2 ST. Jo, temtwof “yoni gitiqome le Hingigns. simwayti bERK 9g ENR egghnian (flow oinoie 21g oqug aati yhoo lo naylees at Seay, - morewol! aainrolie:) ygiionduel gamut) aearawe (sui wi egal etic ARUP AQ veel HOW wevihysiarat wigenmpinic® ‘lol eign tie oe evince aidiew yeh) HO rragiee yoegney tnHey ood vedere bog yadline bowaro vintdg. atypia worieetil?., ae MO foe cell At hi qd. SireRae eR STU set eM ebtgho. erpamelZe. stdipstont it, eee ae es shiners TCT UD ane? TeemeraD whl otloes¥? Weeder dan Fi niewlicnn ipitnils anniv ENOL Vit OE AED Br wd lind eee sara!) Some ined | oe BE EOELE 0b vyyiloenrll quitiniz anv ye pivscuterandined sin vide ahianlrniebt ab hapwiam: (seruke sid pie A: NBA Aho wel patabentt a ee ores a muianigth Svan te dijon abagisitre ated vapulnsivhstntens lengua NER ake “ohio saneiadl Some A te etal’ OVO" cand, Je. ordeal dine yeshihio" Neh AE bane tmaitow lt DFT ee ioerieds me ates rom messi buprsnnlia, y Lv ie alnjasd ,£T neaO) a6 Wading ad ah Maa @ Te (ENCE nal Of bongh dommes) 2804 aque | seiW Pagmandue In'vievishy mamolihwo Mam, A.W) be Mi witeroviil) jetendly 2 Oh” are: Golit pte Aor Aen ites eet” ene Tenrninen lo zie vincAY eer. ot A we ¢ anes tert hirer ei ty Eeaecyid gree midis th nent +4 Colony s a. Leas win yey eit Yes, eerie an | oe hE RRA | dent. Me udt “ah Hea Raa einai, heels heatey Los itcaeng Wt cys “iro Teather ed | evar) UATE Feet iad LG? AyeGhaee vahnnlnd: tegen ote tiptenghosite AT (EOE pare ovlenpdyepero>. 6 pvinadvom gatende imaitenpeny, be toennton Moony teye eg ; Bre LU “ a a Ai F ut . ile in pl a A eae Onn te heaping maqeq “eeuaidotg 90) Patol 19 Fem if A til wens Hae Me | vate a ory bberila yall pint Hiern Mi el eral (inti ‘gar . NOL, dM, Lit wen ' casi hapal gon, Fines. camille Hi 1 “ky! D, inl pp isa ty ~~ ee @ DO) Hoare li an” Hose ey ATO eet (9) aah’ & dow cc alieatia ted at», aos oshambeth i “ RO eerie ND erMeie ~ RiinOhuMnEbaee abil Sh : qowite weraioiue a a ed ama ian RFRA IA ita jasanwoid 4 bang wk te hi ae pegennenbbtic eT, syvlon P al ed He yal aig ee aalieny 8 iin slmte Stik, gate ab watt el ; j 56. Naval Ship Research and Development Center. NSRDC Report No. 3663: A two-degree-of-freedom model for the two-dimensional dynamic motions of suspended extensible cable systems, by H. T. Wang. Bethesda, Md., Oct. 1971. 57. Naval Air Systems Command. Report No. T72-AV-9602-V: Sonobuoy cable system analysis, by D. R. Sanders and M. L. Collier. Washington, D. C., Tracor Corp., 1972. (Contract NG0019-72-C-0239). Nov. 58. R. A. McLauchlan, et al. “A dynamic analysis of moored and free-floating cable systems,’ paper pre- sented at the Fifth Annual Offshore Technology Con- ference, Houston, Tex., Apr. 1973. (OTC Paper No. 1742) 59. R. D. Taylor. An analytic model to predict the dynamic loading of a cable system in a linearly varying current profile, M. S. thesis, University of Texas, Austin, Tex., May 1973. 60. W. D. Iwan. “Dynamic response of suspended underwater systems,” Journal of the Acoustical Society of America, vol. 51, no. 5, Part 2, 1972, pp. 1688-1696. 61. R. F. Dominguez and C- E. Smith, “Dynamic analysis of cable systems,” Journal of the Structural Division, American Society of Civil Engineers, vol. 98, no. ST8, Aug. 1972, pp. 1817-1834. 62. B. W. Wilson and D. H. Garbaccio. “Dynamics of ship anchor-lines in waves and current,’”’ Journal of the Waterways and Harbors Division, ASCE, vol. 95, no. WW4, Nov. 1969, pp. 449-463. 63. Catholic University of America. Report 72-5: The dynamics of a ship moored by a cable system under sea state excitation, by E. E. Zarnick and M. J. . Casarella. Washington, D. C., July 1972. 64. E. E. Zarnick. The dynamics of a ship moored by a cable system under sea state excitation, Ph.D. thesis, Catholic University of America. Washington, DIG 1g97 2. 65. Naval Civil Engineering Laboratory. Technical Note N-1053: Dynamic line force and oscillatory motion of a vessel being winched down in ocean — two computer solutions, by C. L. Liu and J. A. Drelicharz. Port Hueneme, Calif., Oct. 1969. 66. National Data Buoy Center. National Ocean Survey. Unnumbered report: A new technique for the solution of the equations of cable dynamics for the NDBC buoy-cable dynamics simulation, by A. Trampus. Rockville, Md., Dec. 1972. 67. R. Blumberg and C. D. Osborn. “Computer-aided design of submerged ocean buoy systems,” paper presented at the First Annual Offshore Technology Conference. Houston, Tex., May 1969. (OTC Paper No. 1133) 68. C. S. Draper Laboratory. Report in preparation: Mooring dynamics experiment: determination of a verified dynamic model of the W.H.O.I. intermediate mooring, by N. Chhabra, et al. Cambridge, Mass., 1974. 69. J. R. Dale and R. A. Holler. “Spurious signals ” from cable-suspended sonar systems,” Journal of Hydronautics, vol. 3, no. 2, Apr. 1969, pp. 83-87. 70. A. W. Marris. “A review on vortex streets, period wakes, and induced vibration phenomena,” Journal of Basic Engineering, vol. 86, series D, no. 2, June 1964, pp. 185-196. 71. M. V. Morkovin. “Flow around circular cylinder — a kaleidoscope of challenging fluid phenomena,” paper presented at American Society of Mechanical Engineers Symposium on Fully Separated Flow. Philadelphia, Pa., May 1964. 72. I. Dyer, Massachusetts Institute of Technology. Personal communication: Dynamic response of deep mooring systems. Cambridge, Mass., June 1973. 73. 1. L. Ver. ‘Power balance and statistical energy analysis,”’ Proceedings of Intersociety Energy Conver- sion Engineering Conference, Boston, Mass., Society 1971, pp. of Automotive Engineers, Inc., 1041-1047. Aug. 74. The Shock and Vibration Information Center. Monograph SVM-1: Random noise and vibration in space vehicles, by R. H. Lyon. Superintendent of Documents, U.S. Government Printing Office, Washington, D. C., 1967. 75. R. H. Lyon. “Statistical analysis of power injec- tion and response in structures and rooms,” The Journal of the Acoustical Society of America, vol. 45, no. 3, June 1967, pp. 545-565. Ai iw Neal a. fl "0 on, or i Bae ae, i Ls iy . e 5 Pic i : ee th gaa t, : . - ta i en } 7 y : ny , . on } va L tf Py a TK? bane) 98 oak wal (aroizett an ally 203 supiadony won A; how, bevvaelermenns ert ae . 7” To] exmanyh wlilns Jo anoljaupy wid We modmalie oad emblems “gen aldgoeyn wed age TRE 9) bm) itv Rivesdh anqemneT bobis =aityrn0:Y sang (0 bnw igredieniah ah s rot relies Suphadiia ‘Lert nee aie! paqeg “verrsueye youd W920 Hrogristencdane Vy ngied, egmlondvaT mate Vian db aL a Lreo-secmny Taye TO) .2d0T Ya eT inal cea) (ret met ponsinnidee Megs, eee yee Rb 4 ) HOMenionaiab RAYS avery h griivmdh Seiboner tay SONY ada Ye lobews strneiyd toitiray ee: ytd te te wreledl vd onion . } wrens dengie sucipuq’ sellalt A bee oh oH Le lo Ihre, Uatoteve Yetths bibiragemAcitéin | fod WSF ny Foes A oh ier Edw pie veh a A et Aikatoa ih, pollo Dawhtl bay aadper C7. Pact OH law mo rtempign “bao RTS agp ARN DATS eh, eT Ah we vA wore“ oral ,f hole %e ee ners, Te hal ie hiner ha ls 1 Da esha wy catormersdg tilt pel byesit teoatrnetirel? be pea ireaioneds Ag wert Laie? wile ea vnnienyye . ae Nahe ny Warn bod | sinweas vag Molawilyal te Sretiant, ex evevleogpayt ral ou ok Bs 18 Sass RUE io Read Aen Layo EROL onil kek aaheebein® chau igenienthe Aires Monaro in sonmbgd quent eh hd lbh Peng. Youd Nrokyore Le eRvil tatopeen th: !” of, wede . eg | jay pers a “ HO De a ‘Gt Tae LES Weir a ele “AeA ark. htnigeed ey he SOOT ROY vi) Seater Ne neissieaial wojenky Liew ead bt is 7 use Matas, abaceattity auf tne” ot vieskparctiy ue ani hay ast my - tet hirinitieque ie eae | + el eee soir 2a? Gmbh. omnia: 2.0 ane SARE oh A peat ee in) Yoweq tn eneeglanys ender py | en at AT Vermuner Beer: veyed wi: sales bike pile om toy joer, TW Wtdiony? hwireiet ae ieud eae tke AR meh an | let Ji ies t ae ei dal o. ‘WHerovTOL. \aheagly 2 jth pe . babnp gous ‘Ves. -Seintoomeay svi Ativan sie te Suheng ’ norbe oi tornanpabeeins ‘A tn eroureuns divmuriyd. | ana v i “a wmarcraige: okitirg ‘ binarne a a pre e ‘pallies oh ebTTAOOy. Star. to a ; pol et te ve ghions ieee AY {i 't3 ie aqey, “yenisin ye wha | “igi ‘equlnaitas't nonay() eile ; avs al VEO) grep my va r avid neyo: \) Bil x " i\ ht ; - : de tileng o3 bboy, aby tianty eh, WeaRoe oti, Api “althes, te 1 Ras ETOH gall eae i mg DO lh GN Beat Tiere t thai? Ne STO ah Wi ieee ev cote nay Tents Wes Toe aa CL ae asusbinit’s cA iy eg ey) ; a r lo Tey way "Seat Diskiy seem em ene aeaalanaa SO tee aA dteiall ene dull bagi ey RENE YG MORE: ses OT rsa i Ai: Vi erievownnntd aitty qrisieys aiihes' oe Wal ee ey ile, i ee 1 AY, tate ale en 1.3 ed yundaniiages wae | Me | ey the le “d baron miele % tex itrtaay ait eine C2, OI Oe Gee He vosbaigiay | WyetehlaNy, ewer de be wapeenainty: iter oriD TOP viok vm’ AE peaeron aie ges sre ri ay {i lita tence, ail ynaianll, bhie bus ey fy hh Pari vy mee loreny, Ae tak phe oe jemnaatya: | Baw ot Me AA einen pont oa) verkunalticn 76. Y. Nakamura. “Vortex excitation of a circular cylinder treated as a binary flutter,’ Reports of Research Institute of Applied Mechanics, vol. XVIII, no. 59, 1969. (Kyushu University, Kushu, Japan,) pp. 217-234. 77. M. Funakawa. “Excitation mechanism of an elastically supported circular cylinder in a flowing fluid,” Bulletin of Journal of Society of Mechanical Engineers, vol. 35, no. 270, 1969, p. 303. 8G. S* Draper Laboratory. Report in preparation: Lumped parameter representation of Karman vortex excitation, by D. G. Shepard. Cambridge, Mass., 1973. 79. G. D. Silvo. ‘‘Self-controlled vibration of a cylinder in fluid stream,” Journal of Engineering Mechanics Division, ASCE, vol. 95, no. EM2, Apr. 1969., pp. 347-361. 80. H. Halle. “Proposed analytical model for the cross-flow-induced vibrations of a circular cylinder,” “in Proceedings of Conference on Flow-Induced Vibrations on Reactor System Components, Argonne National Laboratory, Argonne, Ill., May 1970. (Argonne Laboratory Report ANL-7685). 81. R. T. Hartlen and I. G. Currie. “Lift oscillator model of Engineering Mechanics Division, ASCE, vol. 96, no. EMS, Oct. 1970, pp. 577-591. vortex-induced vibration,” Journal of 82. I. G. Currie, R. T. Hartlen, and W. W. Martin. “The response of circular cylinders to vortex shed- ding,’ paper presented at Symposium on Flow- induced Structural Vibration, Karlsruhe, Germany, Aug. 1972. 83. S. Kawamura. “A design criterion — wind loads of high steel stacks,” Proceedings of the Third International Conference on Wind Effects on Build- ings and Structures, Tokyo, Japan, 1971, pp. 707-716. 84. R. A. Skop and O. M. Griffin. model of determining flow induced vibrations of off- “An hueristic shore structures,” paper presented at the Fifth Annual Offshore Technology Conference, Houston, Tex., Apr. 1973. (OTC Paper No. 1843). 85. O. M. Griffin, R. A. Skop, and G. H. Koopman. “Measurement of the response of bluff cylinders to flow-induced vortex shedding,” paper presented at the Fifth Annual Offshore Technology Conference, Houston, Tex., Apr. 1973. (OTC Paper No. 1814). 86. R. A. Skop and O. M. Griffin. ““A model for the vortex-excited resonant response of bluff cylinders,” Journal of Sound and Vibration, vol. 27, no. 2, March 1973, pp. 225-233. 87. W. D. Iwan and R. Belvins. Unpublished Manu- script. A model for vortex induced oscillation of structures, California Institute of Technology, Pasadena, Calif., Dec. 1973. 88. J. E. Fromm and F. H. Harlow. “Numerical solu- tion of the problem of vortex street development,” The Physics of Fluids, vol. 6, no. 7, July 1963, pp. 975-982. 89. D. C. Thoman and A. A. Szewezyk. “Time- dependent viscous flow over a circular cylinder,” High-Speed Computing in Fluid Dynamics — The Physics of Fluids Supplement II, 1969, pp. II-76 to 11-86. 90. T. Sarpkaya. ‘“‘An analytical study of separated flow about circular cylinders,’ Journal of Basic Engineering, Dec. 1968, pp. 511-520. 91. T. Sarpkaya, Naval Postgraduate School. Personal communication: Flow field studies at the Naval Post- graduate School. Monterey, Calif., Mar. 1973. 92. A. D. K. Laird. “Eddy formation behind circular cylinders,” Journal of the Hydraulics Division, ASCE, vol. 97, no. HY6, June 1971, pp. 763-775. 93. A. D. K. Laird, University of California at Berkeley. Personal communication: Recent work on the vortex shedding phenomenon. Berkeley, Calit., Mar. 1973. 94. S. K. Jordan and J. E. Fromm. “Oscillatory drag lift and torque on a circular cylinder in a uniform flow,’’ The Physics of Fluids, vol. 15, no. 3, Mar. 1973, pp. 371-376. 95. R. R. Clements. “An invisced model of two- dimensional vortex shedding,’ Journal of Fluid Mechanics, vol. 57, part 2, 1973, pp. 321-337. “ haieneyOu WOO bra qo A rT welts Me o ae OP erabitilicy Told NO oaniodiion sit Vn jerome) ta ha trrayrrdy oepary onilisite Sai poy: tpatniewelt Serio) yaotindesT sro lana ‘ivy mr (PLES pl eget STOTT oe tnt Harwell rv reef wel) 1) leben, A" iid? 9 bna ‘godt A ph. reer Wald vo DanKCyeSy eraeneedy beaver Ripe Meh fan 8D dae \sneodigeonchy tne h ov hee Des Tpawevitpes Legere ak tt nah beadallinyld oir OW bee wel ee Yo roiutins besuliat spre 0h Whore) A ayiae Vy ctor tas] to GOL? ms rye eae | 7 7UPoUTIe ETRE ned oly tinal t alas Leseesehidl nebo ht hd Bive pevpeets ThartiedsuM We ‘Unplaor Ver, a Yo & Ia modiany balloano>aa) ui ts ane | ‘folie adiahe # ul yobrlign telestig | (FOE oy AOR WEES . noi eiaqont ni Plevepth gvters Mo teny mart te HOOKER 4 dou hse his en: 2 fe | ot ate u" “Sore Wwoh Poovey cedacee Po aldo orl Toone En AiN ae Ne taertwes | * Cer: bial qe FOOL viut..e ond toh aul be POT OEE AL SRA ua hac deer eA 4 caoate ; “ts “sree rt" Wrsvurenk Aud Sree eine DAD A Weed liodn dats oy boron” 5 ™ yohuplbss TA od. tel, etl OOO: ep ab ope vibvitilyss Pea ip sein denlly ry Sm eae wee WT on YN Loge bésubwh walt HHO: Save eo tele” Meh OP OTD dey AAO ren iare hi Ae ante. aie PUBS Bere, 2 ATR Ra HATE ae ule AM anonetaenyt"siet Pip vee oa . LEMBT AULA rn OT ede a eee ee ee pabtpees. "Pats eet Sy towed! “ahatilive veh Gry: wigehittan,” woah fablLiutes a uated a él dine ‘news see a Bh OE Seen NRL Ry iat ‘\roinetdly boonihutiqearahe : , A Th Me EN Ae! ey nesiyivie Nonna SOMITE JonTloe ordi atlemutt Levu armairpeadl cil NR i nape Hh ttn? DAS eh eaihaia lel? wad iWeHaey wee TED COR RSS eke lehae out ETrade MM NM Lire rot i : ae bike etn asa Badhey nghipe Phe a, iid bonita del nasil ha vl mye 7 Be + ; i es Ge || COE TY ae Ayiptense's aq Rates nhl amet triiaiindlidh IA oie nie Mi laa, deoailbald ruiteaade: fimisanetey”. Henig’ k Clee ag TRY ted Oe diy : ; i ; p 9 sbi Pie nando haan Aes * ya eben) Lae inestirgginy rightly en eh WOR etd 1 sehen Late | eth MUTE Wha | hein aig Sin suiiite Scoala aint ‘use HD eshemaatt une ei. aaa onl n _ 7 ba ad zp Hae eet ‘woreineds ‘ Eyed tnh a ERGY ile — a erste pve ratalh aot)" ener f Cie eo we Me ' a ntiria panies piggreg 7 dati on - ios rere ay" Sa le ‘J ‘ye anak 4 if sie i cl ac he eh, ooh po RDO carbene wollte arabe Aint alirmaeah ™ i mv tilh'D, ‘onthe Te Ho Mamas ni : hy i ‘ , ws. dleata: basedyr ee a a ee HO TNE hut ko terre PO OITOY LUO ROIEE | TEAS ah slo adenine t dy 96. Y.N. Chen. “Wake swing and vortex shedding in a cross flow past a single circular cylinder,” paper presented at the Symposium on Flow-induced Struc- tural Vibrations, Karlsruhe, Germany, IAHR and IUTAM, Aug. 1972. 97. J. S. Sonand T. J. Hanratty. “Numerical solution for the flow around a cylinder at Reynolds numbers of 40, 200, and 500,” Journal of Fluid Mechanics, vol. 35, part 2, 1969, pp. 369-386. 98. }. H. Gerrard. “The stability of unsteady axisymmetric incompressible pipe flow close to a piston, Part 1. Numerical analysis,’ Journal of Fluid Mechanics, vol. 50, part 1, 1971, pp. 625-644. D2). Ealhic separated flows,” paper presented at Symposium on Flow-induced Structural Vibrations, Karlsruhe, Germany, IAHR and IUTAM, Aug. 1972. low frequency components of - 100. J. Shioiri. “Synchronization phenomenon in vortex shedding and its role in vortex induced oscillations of structures,” in Proceedings of the Third International Conference on Wind Effects on Buildings and Structures, Tokyo, Japan, 1971. 101. North American Rockwell Corp. Report SD-68-996: Hydroelastic analysis of a circular cylinder, by B. H. Ujihara. Seal Beach, Calif., Dec. 1968. (Contract NO014-68-C-0223) 102. . Report SD-69-558: Hydroelastic analysis of a circular cylinder/lift on bodies of revolution, by H. D. McLaughlin and B. H. Ujihara. Seal Beach, Calif, Dec. 1969. (Contract NO0014-68-0223) 103. Naval Air Development Center. Report No. NADC-AE-6719: Determination of normal drag coefficients for flexible cables,’ by J. R. Dale. Warminster, Pa., June 1967. 104. Naval Air Development Center. Report No. NADC-AE-6731: Water drag effects of flow induced cable vibrations, by J. R. Dale, J. M. McChandless, and R. A. Holler. Warminster, Pa., Aug. 1967. 105. Naval Mine Defense Laboratory. Interim Report 1-132: Drag and vibration of some wire ropes and fairings,’ by R. E. Kelly and C. N. Goff. Panama City, Florida, Sep. 1967. 106. Naval Ship Research and Development Center. Unpublished manuscript: Vortex induced vibration of bare and ribbon cables. Bethesda, Md., Sep. 1971. Robert Blevins. 107. A. Fabula, Naval Underwater Center. Personal communication: Suppression of cable strumming in towed systems using helical strakes. San Diego, Calif., Feb. 1973. 10 - a fel tee oe t —_ he an i ” Laie ; 7, i 1: ae ig a ye i Wy : / ,7 1 eer 7 i ae ett " if ul! agtas> tara bas ‘ena ae teva 0k Je wonnidn bypuwini ene seqit samme hadeilduqn\l SR. 992 bh waa re A’ Adie nodin Sra wind ee oF aie edo 1 — feowe sain!) eewobntl lawalt hide A sot. te yoni ghdice Yes: woheswque ° veoh tai ew reverts atid OBsith ane eaplerte ahi any Pro NY he ATeT 7 ‘Retaoego ceive, corn to, anti nyeiy,: Ybwsrnw Ta gwlidann ‘ait, hiecaet M a OF mols: welt aqig. alitiaenrqyma ati i tiwl Yo losroot!" eieytnne lesional eae PEIELO ah LOL IT Vena ‘ ki ReeTOgMiba. YonmupaAT web! Fr) miniae gee: 1h bam sev isp | ‘gH’ wcoriendty ATC GaN) MAYO bh tft Paeeenee ae nediiainont rege boawial Kano, mi stay: mh ‘bed eet ant to agribaypantt a Re arvolts” bil WF nm soiree eve ayy wrompil, valetned darned pwthagy dy ey arta rae ial A wali * * " abet sage ¥ , a) sealed wssibsenbca aevenee Aap, wl hel rus Way Ke wpa lh oe ivealenitga , ont took eed ean witty te LESLOD AL ahd Aopen y Pace au neacyns sate? veranteqe trent, wh val gb. lecteioy Ta, ootraoieeet 20 Te we RA Loved aad glen ote TOO) swt wi via OM pogo aD saomiqoinea) aA jyapited Wolh Tankioat he gent von tg See ae ae ee ee Ta SO CE Tt aired Wietointheld Seng tadh ieate gorracie? | hoe WR Gan Aine Bor 7 tri-mooring Note: A General Program is one which can simulate the behavior of the above moorings plus others of additionai complexity. Figure 1. Mooring type terminology. 11 " a : 2 : 0 ‘ : vd Cee bondby cs «men ugeiny wel beget ; Ms ; ipa 1 aermnre mee a ot a” Ne is A ee eT ate ea — sa Obape 2 ony yam pc aha i mes crn Dh een ne pent arma f Ae a A Poa ee Eo) wih yume Sie tenae cbt VAGUE oi) me ey acl PO te etc! Ava ie ees bho Niece seepign Stile vd senemgeltt im s Strumming Drag Coefficient Cy 2.0 2 3518) > 1.84 y » haired streamer or very low strumming force 1.9 [ te 1.7 1.6 1.5 1.4 Reference (smooth round cable) aN 1.3 1.2 strumming force comparable to smooth cylinder 11 twisted pair (pitch = 15d) 1.0 strumming force reduced by more than 50% (Ref. 4) antinode splitter = 7d weathervane fairing > oe) very low strumming force “500 700 900 1,100 1,300 1,500 Reynold’s Number Figure 2. 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