advertisement

III.4 Second-Order Theory of Oscillating Cylinders in a Regular Steep Wave A. Papanikolaou and H. Nowacki Technical lJniversity af Berlin Berlin. Germany Thirteenth Symposium on NAVAL HYDRODYNAMICS OctoberO - 10, 1980 Sasakawa Hall, Tokyo Office of Naval Research National Academy of Sciences Shipbuilding Research Association of Japan III-4 Second-Order Theory of Oscillating Cylinders in a Regular Steep Wave A. Papanikolaou and H. Nowacki Technical University of Berlin Betlin, Germany (n) l ABSTRACT r-J< The nonlinear two-dimensional hydrodynamic problem of a cylinder with arbitrary cross-section shape performing finite ampli- tude oscillations in the free surface of a regular steep wave j_s treated on the basis of nonlinear system dynamicsr potential and perturbation theory. A complete second order transfer model is developed from the assumptions of a second-order incident wave, a quadratic dynamic model of the system, and a second-order perturbation expansion of the hydrodynamic nonlinearities in potential flow. The given nonlinear wave-body flow system 1s decomposed into six second-order subsystems to which a perturbation expansion in several small parameters is applied. This results in a set of linear boundary value subproblems of uniform type, which can be solved by close-fitting methods. On this basis second-order expresslons for the hydrodynamic pressures, forces, and moments are obtained and introduced in the equations of the body motions to obtain the motion response. Numerlcal examples illustrate the soIution procedure and several physical aspects of the second-order mode1. NOUENCLATURE jt (t) (t) ,n g c(t) (r, h , (n) i ? l_ r44 j srrmbo,l Description a wave amplitude b maximum half-beam w A -(1) -(2) j'k 'jtik j area of cylinder cross section reference quantity for motion i, eg. l2l waterline beam, Fig. 3 k + k 1 III-4.1 generalized restoring force coefficient of order n for (i,k), eq. (106) hydrodynamic force or moment amplitude of order n in direction j caused by problem k or by interaction between probIems i and k, as applicable, eqs. (90) to (921 , rt = 1 or 2 hydrodynamic force or moment of order n in direction j, eq. (88) , with ,F (') H (t) . = (.r) (n) (n) -nr (,)= . (2t rik 6) =v , u acceleration of gravity Green's function for order n, eq. (s3) water depth hcrizontal hydrodynamic force of order n lmaginary unit (space), e.9. eq. (54) unit vector in positive x-di-rection moment of inertia (ro11), eq. (ee) imagi-nary unit (time), e.g. eq. (55) unit vector in positive y-dlrection wave numberr €9. (58) unit vector in positive ro11 axis directlon free sqrface inhomog. for sec.order problem (i,k), (451 ,(4gl Symbol Description ,ftll) aJ( M iu ( t') ,{") + n ."1.") llK o-x-y 5-;-y p (n) (r,i, t) nl') (r,i) , nji' (*,tr i ,.(1)_ _(2) K ', f-ik i s (t) So t V vn x, (t) "j")v :{n,:,ll) o1 Yc - Y (x, ^R v-R(n) t) , B 6 d 'k I 1.r, + (n) aI< ^" body p (n) uik lnhomogeneities ilt.4 -2 incident wave phase angle, eq. (57), 6r-o to. regular sinewave (t-0) in posltive x-direction perturbation parameter for rnctctent wave , eq. (21 Eff::::ii:;: 3;:"Ui"- for perturbation parameter motj.on in direction i, for (2) eq. perturbation parameter in general couplex variable source Point, eq. (54) ' incident wave length, eg. (5g) di mensionles s hydrodynamic coefricieni-6f *;;;.. ,,,, lppils. caused by (i,k), ee. (1Og) di."r=iorrless hydrodynamic mass coefficieni or caused by {i,k;, .n. 6.J..*.,, (1Og) frequency parameter, eq. (5g) fluid density body mass density, eq. (99) velocity potential, eg. (1:y space potential functions orders (1) or (2), eq. (13)of decomlrcsed space of orders (1) or potentials (2), eqs .116) , (171 1{ I ot,Problems appli_ !i,k), as(79) caDrer €![S. (67) to position vector l-n o-x-Y-sYstem, eq. (g1) draft, Fig.3 velocity vector of body motion normal component of body velo_ city, eq. (9) body.motion in direction 1, (111.,_for sway S?: neave (i=3), and roll-l:.=2), (i= ) body motion amplitudes of or_ oer n, eg. (141 vertical coordinate of body center of gravity horizontal inertia force of order n, eg. (100) free surface elevatj_on profile vertical inertia force of or_ der n, eg. (1OO) variable, field poj_nt, :".P1:T. (s4) eq. (n) "7 hydrodynamic pressure ampli_ t.ude of order n caused by problem, k or by interaction -boundary ror, tirst (k) or second_order (ar,<) problems, eqs. (30) (44), , (47) , (48) arc length on cylinder contour unit tangent vector, Fig. 3 wetted cyllnder contour at ,time t, Fig. 3 wetted cylinder contour at time t=0, Fig. 3 between 3 ana pos. -1"?+. x-direction, Fig. 3 section area coefficient Phase angle of order Dr 9€D€_ ral, eg. (93) to unit normal vector, positive outward, Fig. 3 directional cosine components of order n for directlln il-eqs. (21) to (24) " inertiaf right-handed Carte_ sran coordj-nate system, Fig.3 body-fixed right-handed Carte_ sran coordinate system, Fig.3 hydrodynamic pressure of order n time T (n) c generalized body coefficlent of order n mass (i,k) , for eq. (106) cylinder mass per un j_t length, eq. - (99) hydrodynamic rolI moment of order n moment of inerti_a of order n, eg. (100) PB @(x,y;t) oJ1)t*,vl "0ffr)rx,y1 t,j^'),*,"1 ^dli)t "''r frequency of incident wave and of first-order motio" ;;;;";;; body boundary differential ope_ rator, eq. (19) Laplace operator free surface differential ope_ rator, eq. (19) radiation differential opera_ tor, eq. (2ol Eapiltonrs nabla operator OJ B{} A{ F(n R(n v{ ] 2r) 2r) { } { } } Subscript Conventions : The paper contains several multiply sub_ scripted guantities rn symbol expressions of the general form .i t tjl) , where some or gf- the subscripts may be present. These .11 symbols denote: a) Superscript n_j-n_ parentheses the order n to of the quantity inrefers ttre per_ gfrq otre that .i.o 4l shii[r for reoil r x.mdilni hry ir sl!rE ilrN eridl: lhaGiG @ooul.f aEE I co-.d ,f .d ucull mnrfu I PI!!.$ilI paeEtl @ !imqrr *rl i [email protected]{t, surfro toal qu"td tcd dq try ILq EEoer ran I nrnl seamld -tr-aild dercti I l) )Lnt, r8) E rlr , turbatLoh expansion. bl I€ft subscript m denotes the dependence , upon the motions of first (m=1 ) and second order (m=2) of the followlng quantities: potentials, eq. (17), associated disturbances, e.9. eqs. (441 to (49) , pressures, eq. (65), and forces, eq. (87). cl l,eft subscript j (j = 2, 3, 4l is used with direction cosines, eqs. (21) Lo l24l , or \dith forces, eqs. (90) to (92). It denotes the reference direction j. For the forces both subscripts m and j may occur, eq. (87). dl Right subscripts i,k (i,k=0,2,3,4,71 relate to the cause of the subscripted quantity. First-order quantities have a single subscript k, second-order quantlties require at least two subscripts i, k; second-order steady-state quantities carry the additional subscript o. Other Conventions: Lengths are made dimensionless by b or la l, ancrleebv maximum wave slor:e or with 'rELpu.i t6 i raaian, time hv ,-t, and mass per unit length by p bz. INTRODUCTION B) I E 16), 4nd nse ope- Pe- a- o r- The past decade has witnessed a rapid growth of j-nterest in the nonlj-near aspects of ship motj-ons. It is well established that linear theory succeeds extremely well in predicting many important phenomena in ship motions within the accuracy required for desj-gn purposes; but it has also been recognized all along that the range of validity of the linear model is restricted by its fundamental assumptions of small w_avq heights and small motion amplitudes t t ] rfrere has been growing experimental evidence in recent years, accompanied by basic theoretical results, suggesting that nonlinearities 1n waves and ship motions are also a matter of important practical conseguence. This holds for the prediction of any large amplitude motj-on effectsr p&rticularly extreme values, and has received much attention in connectlon with large amplitude roll- (capsizing), hydrodynamic j-mpacts (slamming, section flare effects), wave-induced bending moments and other nonlinear phenomena. The present paper addresses itself to one speclfic aspect of nonlinear ship motions, that is, the problem of an infinitely long cylinder oscillating in the free surface in shray, heave and roII in response to a regular incident beam $rave of finite amplitude (on deep water or water of limited depth). Ihis case is an important reference poj-nt for solving the much more general problem of ship motions in an i.rregular seaway, although a comprehensive nonlinear theory for the latter purpose remains to be developed. III.4-3 The current two-dimensional hydrodynamic problem is approached by means of nonlinear system dynamics, perturbation theory and potential flow methods. This implies that several, but not aII, of the nonlinearities present in the physical problem can be taken into account and that thb results may be valid for moderate wave heights and moderate motion amplitudes, in other words, for nonlj-nearlties of such degree that the second-order perturbation approxi-mation will remain sufficiently realistic. Specifically the approach will a11ow for nonlinearities in the - I'ree surface boundary condition for incident and motion generated waves (by including second-order effects and satisfying this condition in the true free surface - in a perturbation sense), - Body boundary condition (by satisfying it over the rea1Iy wetted regime, approxJ-mating the shape of the latter by a Taylor expansion around the rest position of the body in the spirit of perturbation theory) . A11 derived responses such as pressures, forces, motions are also evaluated to the second order. However, nonlinearities of viscous orlgin (Iike separation at sharp corners) are dlsregarded, although subsequent, empirically based corrections of the system model remain feasible. Much of the previous work on the subject of this paper has been devoted to the incident wave and forced oscillation subproblems. Numerous higher-order wave theoiies have been developed by Stokes [2] t Levi-Civitd, Sielbreia and others. Papanikolaou has reviewed several of these theories, including a third order solution of hj-s own, in [:I . For the present purpose a Stokes second-order wave wiII be sufficient to represent the j-ncident wave flow. In regard to the nonllnear forced motion problem Ogilvie [4] derived the secondorder steady force on a submerged circular cylinder, which could be achieved without explicitly solving the second-order boundary value problem. This was done later by Parissis [ 5] for the circular cylinder heav-ing in the free surface and by C.M. f,ee [5] for Lewis forms. Lee's solution to the secondorder set of boundary value problems is based on multipole expansions and conformal mapping. Potash [7] , working with close-fit techniques instead, extended the problem to sway, heave, and rol1, including their coup1ing, and.to erbitrary secti-on shapes. Papanikolaou [8,9J ,on the basis of a similar approach, reexamined the foregolng results for the heaving cylinder and removed some j-nconsistencies in the analytical expressions and numerical results. Fina11y, Masumoto [tO] developed an approach to Lhe complete second-order oscillation problem using multipole expansions in analogy to C.M.Leers treatment of the forced oscillation case. Several approximate solutions to the second-order forced motion problem have been derived without soLving the corresponding boundary vaLue problem completely and rlgo- rously. C.H. Kim hll dealt with forced heaving of triangular cyli-nders by an iteration which departed from a zero-frequency solutlon. Grim llZl derived approximate solutions based ol low-frequency assumptions for larqe amplitudes of roll. Salvesen's -[t:] approacf, in to nonlinear heave (and pitch) is also based on low frequency approximations. Sdding [ta] used Greents theorem to derive the second-order force on an oscillatlng cylinder from first-ord-er potentials of tfie problem. Yamashita [tS] developed an approximate solutlon for "thj-n" oscillatlng cylinders with results up to the third orderseveral authors, €.9. [to-ta] , in contrast to the perturbation school of thought, have pursued direct time-domain solutions of the complete nonllnear problem, using initial value formulations and numerical integration schemes. These methods do not depend on linearizing assumptions and are attractive, in fact, sometimes perhaps the on1y available recourse, for dealing with very large motion amplitudes and waves. However, aside from some unresolved questions regarding the treatment of the radiation conditj-on 1n the complete nonl-inear case there are also some practical limitations: Computer time requirements tend to be heavy, and validation and generalization of results are difficult to perform for lack of frequency- dependent submodels. Experimental results for several aspects of the nonllnear problem have been presented by Vugts [tS] .- ,^ Tasai and Kote-rayama [20i , Yamashita t1 5J . The agneement between indi-vidual test results and secondorder theories is encouraging, although the available evidence is far from systematic and complete. The research reported in this paper has the alm of defining a complete second-order modeL for a cylinder of arbitrary shape oscillating in the free surface in three degrees of freedom. This model combines a quadratic dynamic system moclel- with a secondorder hydrodynamic model. The incident beam wave 1s thus a second-order regular wave, that is, a "steep" rvave in the sense of Stokes. The system model is subclivj-ded into several separate, but interacting flow systems:Incldent wave, forced motion, and diffraction are three primary subsystems familiar from linear theory. In second-order theory it is not sufficient lo model these individual flows to the second order, but it is al-so necessary to account for the mutual- second-order interactions in these f1ows. This requires introdueing three further second-order subsystems. A perturbation expansion j-n five sma11 parameters is performed on the nonLinear system in order to recognize the "smallness parameters" characterlstic of each subprobIem. The expansion yie3-ds a corresponding set of.20 linear boundary value problems. The great majority of these can be regarded as radiation problems and represented by Freclholm integral equations of the second kind. This permits uniform numerical treat- I[t-4-4 I ment of all these subproblems by close-fittj-ng methods. The standardized hydrodynamic coefficients obtained from the submodels are assembled into the complete transfer model via the equations of motion to solve for the motion response and other responses. The paper describes the soluti-on procedure in detai1. Some numerical examples are g'iven to illustrate the physical properties of the transfer model and its components. 2. i t quadr a mol of fr of fr stanl flow tage set ( flow amena FORMULATION will of 1r tion 2. 1 Dynamic System ltodel- The nonLinear system of an incident wave and a body oscillating in response to this wave on the free surface of a fl-uid domain may be clescribecl by two sets of assumptions, cal-l-ed the dlmamic system model and the hydrodynamic modeL. The former is related to the general- dlmamic behavior of the system in terms of its input and output, the latter consists of the hydrodynamic characteristics assumed for the fl-ow system. The two models must' of course, be chosen in accordance wittr eactr ottrer. Motions NONLI NEAR WAVE/SHI Forces P SYSTEM !'ig. 1: Dlmanic System Model The dynamic system model of the nonlinear wave-ship systen (Fig. 1) is characterized by the relation between its input signal i(t) antl its output signaLs oi(t),j=1, ...,J. In the foll-wing we are interested ln the steady-state betravior of the system, that is, in the time-periodic response of the ship to excitation by a regular steep wave. Initial- transient phenomena of the response are disregarded. Por this purpose we assume a nonLinear reLationship between input and outPut of the form: N o.(r) , = x a1"){ttr)}n J J (1) ,.=6 and i.n particular a quactratic time response model (N=2)..,_-fhis modeL is introduced l:lere (as in earl-ier work by r.ee [Zt] and ottrers) as an a priori working trlpotlresis, chiefly because it can be demonstrated. to be comPatible with the second-order hydrodynamic model to be deveIopedl by perturbation methods. intre sionl chari pertr IIe fu I .fit- lamic [s are odel Dr the [e patr de- Ito Lhe nt eto Ld asodel is rof utput, c cha- It Is a particular ProPertY of this qoadratic model that the response oi (t) to aL ronochromatic harmonlc input si9n61 i(t) of frequency o will contain harmonic terms of frequencies t: and 2rrr , as well as a constant "d.c." shift term. In vie\,{ of Lhe complexity of the total flow process it is now'of practlcal advantage to decompose the system model into a set of separate, but interacting nonlinear flow subsystems, each individually better amenable to hydrodynamic analysis. This riI1 be done much in analogy to the concepts of incident flow, forced motion anil iliffraction f1ow, familiar from linear theory. At the same time it is convenient to introduce a system of several small dimensionless parameters e1 suitable for defining characteristic "smallness ratios" in the perturbation expansions of the subprobl-ems. lle introduce: .o em. = Ztr it la,l= k I ., =1":to-' sen lu,ul = .o (xry-1 r (1)r l*i .:- = -Ul , 1 = 2' 3, (2) of the motion amplitudes 1n each degree of freedom. Thus we have available a set of physically relevant smallness parameters for the major subsystems of the flow mode1. The use of five small parameters does not imply that they are meant to be lndependent of each other, but only that their physical interdependence need not be considered until after the subsystem flow problems are solved. In fact, eo and a7 are both dependent on wave height. and e7(e6) and ei are physically linked by body dimensions. However, at least two physically independent smallness assumptions can be made, for examp1e, the traditj-ona1 "small wave steepness" and "smaI1 motion amplitude" assumptions. To this extent our approach parallels New: man's in IZZ) who used three perturbatlon parameters in a thin shi-p oscillation problem to characterize the orders of beamlength ratio (thinness), wave steepness, and motion amplitudes. The decomposj-tion of the dynamic system model S into a set of second-order subsystem models iS, assuming for the time being that all- ek are of equal order of magnitude, can now be expressed in terms of the characteristic sma11 paramelers present 1n each subsystem: I 4 S(x,y;t;eO,er,e.) + L ,S(xry;t;eO) + 2s(x'yiL;er) + 3S(x,y;t;er) + 45(x,Y;t;e.eO) with .* -> = amPlitude of regular beam wave (a, in general complex) I = wave length w k = 2r = wave number vb = r{'1 - 5=dimensionless frequency parameter onliactet sigj=1, ted kb here hers) efty rolnPa- ric me- Further: = maximum half-beam of section = 2, 3, 4 for sway, heave. roll, respectiveLy *11) = complex first-order amplitude in rdr-rect]-on L b,l_ = reference quantity for motion i bZ = b3 = b = maximum half-beam be = 1 rad = unit reference angle The parameter e0 is a measure of wave steepness in the incident flow, e7 is characteristic of the magnitude of the diffraction ftow, and the ei define the smallness b i TII.4.5 I- * o(.;) A ) = (8c'-Bc-+9) /'8"4 = s/c , "1 s = sinh (kh), c = cosh (kh) he ,Ponse 65(x,y;t;eOer) CO of (1) * = (vb){co{ r + efr ct)} -1 + orel) eep Eeen 55(*,y;E;eie7) (3) + and physically for a second-order wave on water depth h [ 3] : stem, POse = FrG, 2 : SECoND-0RDEn DYilAMlc sYsTEfl HoDEL Fig. 2 illustrates the mutual interactlons of the six basic nonlinear subsystems: Nonlinear Nonlinear Nonlinear Nonlinear US: Nonlinear US: Nonlinear ,S: ,S: ,S: nS: i-ncident wave di-f f raction forced motlon interaction of ., S and ,S interaction of ,S and ,S interaction of ., S and ,S 1 The second-order model- of (3) differs form linear theory in two ways: The famil-1ar basic flow systems 1s, 25,. 35 have to be extended to second-order leve1, and their second-order interactions have to be taken into account, which is done by the subsystem 45, 53, 65. Each subsystem iS corresponds to a nonlinear boundary falue problem of potential f1ow, which by perturbaof tion methods can be reduced to classesIinear boundary value subproblems is (n) . Some of these classes are further Eubilivided into erements-oi types is{n) ana is{p), that is, linear subproblems-to be derived in cletail from the hydroilynamic model. Fig. 2 shows the complete scheme of nonlinear flow subsystems iS and their subprob1em classes and elemefits, connected by so1id lines, whereas the dashed l-ines with arrows indicate how the nonlinear subsystems +s interact with each other via their lineal boundary value subproblems. * = *2+ x cos *4 - Y sin xn (4) y=x3+!cosxn*isinxn The motion is assumed to have existed long enough for alL transient effects to have decayed. Further, we assume inviscid, irrota- tional flow, which ensures the existence of a velocity potential- 6(x,y,t) satisfying Laplace's equati-on for an incompressible f1uid. Combining the ki-nematic and dynami-c boundary conditions on the free surface y=y(x,t), extending to infinity-on both sides of the body, one obtains [8] : + Ott(x,Y(1;t);t) o +o o. ) - o2o 2(o -xExytyxxx q v 20 (s o o -o2o y vv xyxy If the fluid has a horizontal bottom at y=-h, then 2.2 Boundary Condltions A cylinder of arbitrary cross section in or just below the free liquid surface in response to a beam, regular, steep wave of amplitude aw and frequency to. (5) =0 The kinematic body boundary condition implies that the normal velocity of the fluid on equals that of the body Vrr(Fi9. 3): 0r(xr-h;t) shape is osciltating S(t): wi th 0 n = + n ={ (nV)Q(x,y;t) sino, -coso } , 0n = sino Vn- = n V(x,v;t) s ={coso,sinc} coso 0 Vn = sino 1 = tx/ax = i=avlat=i:* :r.2 - x4 (y cos 0 v cos0 v xn + i sin x4) in{* .o" xn - i stn sin c(t) = ?yl0s = y cos x,q + i-sin cos o(t) = Fig. 3: Coordinate Ex,/0s = x cos x 4 - y sr"n xn x4 thL.s ar llE L br leqnatll ,-3 EeI IJ dary rt aectaq asguirto a', batam the sq | (x,t;l rith { o tie fq mif6a tfte ts gest tl are of, firstDitude vergeo clMY Il lEtese' lthysic cial. c sectio to er.i a1thf, rios yetI &1 lt riU g frequ fact t lnteut Gorret teoti,r lnsitl ry cd tlat I secd exf,)r'e! where the prime denotes the derivative with Systems Cartesian coordinate systems are enployed (Fig. 3): The right-handed system o-x-y is fixed in space so that o-x corresponds to the undisturbed fluid surface and y is positive upward; the right-handed system 6-x-y is fixed in the moving body anil coincides with o-x-y when the body is in its equilibrium position. The displ-acements of sway, heave and ro11 are denoted by x2, x3 and x4, respectively. Due to these motions a point (f,y) moving with the body has the following coordinates measured in the stationary system: Two respect to arc length s, the dot indicates a time derivative, o(t) is the angle between unit tangent vector 3 and the positive xdirection, and ri the unlt normal vector which is positive out$/ard. At large dj-stance from the body a suitable Sommerfeld radiation condition is imposed. Physically thj-s corresponds to the fact that the incident wave and the motion generated by the ship are the only disturbances present. Mathematically this condltion ensures uniqueness of the solution potentials. In determining the potentials the motions of the body xi (t) will be assumed to be known. The normalized potentials and corresponding hydrodynamic forces determined on a o (-'?;l , ! i Sis G tials, cooed are o{ i presstr protrtI tifiei that I ard r fit-4-6 / placements xi(t) maY be.correspondingly expanded into i perturbation series in terms Lt tn. small plrameters e1, characterizing itre magnitude of the motions j-n each direc- aasumPtlon l,ili be substitutecl into the of motion l-ater to derive the acbody motions in a given wave. tion i, eq. urbation Tn order to reduce the nonlinear boun."to" problem defined in- the- preceding to I set of linear subProblems, Ye ttat tne potentiat o can be expanded a power series in terms of five perturt" n iarameters..k (k=0,- 2, 3:,1:_ Jl,Yn r."6"a order inKaccordance with (3): !,;t; e*) = t* ,1" (t2l 22 (14) .-jkt'tt r e:r xjl) =r Kr x=o n=o Neglecting trivial terms: 2 jnurt - -:?) l rat) x.t (t) = , *.(") "or "=1^i where we have introduced the abbreviated notation to be used from now on: t(o) = o , i,k = (ruestions concerning the convergence of firegoing expansion, or particularly its iotm convergence, must be left open for roment. Cuirent calculation results -999: that the second-order potentials ai[' the arAar ^f maonitude to to the ^-..r..^1^h+ order -G equivalent of -oflTagnilude. mag0 the that potentlals t-order 1[,''"o con-de of the ei, e1 will govern the should e-of the 6*p.n'iiott. The series of fot suffilientlY smal1 values " but how this l-imits the e farameters, secondi"ii t.rg. oi validity of thisout' Sper theory has yet to be found eautioir is i-n place for non-vertical in tire waterline with regard tiot "frap"s uniqueness of solutions ' and existence in our exPerience to date no sehave arisen practical difficulties 7 According to the quadratic response (1) a regular steep wave of frequency ut 1'produce-physical output effects with ies , ana Zo . AlLowing for this i" tt separati-on of variables forof the iiars " in itz) bY including terms esponding frequencies, expanding the poi-ais for-sma1i perturbations about the itions at rest, and treating the boundasrrow [8] conditlons .."otairrgivl-o""-."" the relevant potentlal terms up to the J-oraet are iircluded in the simplified ,y;t;e*) =i.*rlt' rl*"'* -'i ot e'+ z x.(r) "i'-' i oli)r*'v'tl r ot.ll rff i'* o, 2, 3, 4, (21 .l ,.ll'* *1"' , kto, ,', *!1' - .i?) lrar) This change of notation is eguivalent toall consider will initially ="Vi"S thit we problems and their responses ny&rodynamic way, namely for e i=1 ' Howii-, . t6t*.tizea -ti ieintroiluced later in solbe will "rr"i, of motions and determi"i"g'tL.^"quations ning the actual resPonse Bv substituting these perturbation exana (i4), and-corresponding p..rsioi=-iir) its Lnes for the wave profile y=y(x,t) and and equation taplace Lhe into aeiivatives, in secthe boundary conditions fgrmulated reduce the iion z .2, i-:t is possible [8] to boundary qirr.r, nonllnear time-ilependent to a set of onl-v ;;i;; pi-ure* rot o (x,v,t) boundary value problinear =""".-h"""ndent tlms tor- * (n) 1x,y). The nonlinear problem contains boundary- conditlons on free and ,""i"g boundariei, the linear subproblems the i"""f"" only fixed boundaries, namely of the liquid and body ""ai=t"tf"d-positions rhis is in the spirit of the per"rif.."". where the conditions at the method t"if"ii"" approxtrue positions of these surfaces are imateh by Taylor series expansions about the positions at rest. The linear boundary value problems resulting from this perturbation development we are deicribed in the followi-ng section' restrict ourselves to the deep water caseand iio* tt"r" on. Details of the derivation iEqaraj-ng the conseque1gg.s of limlted water aeptn can be found in [231 (13) olfl)t*'vl e-2jort + orel) expressj-on omits some trj-vial Potenls as welL as some time-indePendent seterms whose hydrodynamic effects of fourth order. By analogous reasonj-ng one obtains exi-ons of equivalent form for the-wave file y=y(x,t) and for all physical quanies t-o be derived from the potential ' forces t is, for example the pressures, dj-smoment on the body. The unknown tlt-4-7 2.4 Boundary Value Problems itn order to obtain a formulation for potenthe boundary value problems for theunknown the of independent is tials which motion amptitudes it is convenj-ent to norof the dismalize thL potentials in terms velocity oii".m.trt.v6Iocity and angular Lomponents of the body or the exciting wave' are for ;h;;. applicable. These components(14) and potentials, from lr',"-ilrll-order the incictent wave velocitY: { I a 2 v1r-1 =i =r n= I ti L v.atnl vln) = ? I { -jnot n=1 (n) n,rl xl"', (n) (r) v. e =- UJK -r 'a, i= 2, 3, iA ";"w r L = o, (1s) 4 1 ! 7 .! measures the distance of The phase angle 6w"crest d{ from the origin at the incident wave time zero. In terms of these velocity components the first order potentlals may be expressed by introducing the normalized potentials tf,") a" follows: '* ol')* ol') odl)r",vv = ,* tl", .il r.j.g. 4: Geometry of the BoundaryVal-ue Problems The time-dependent.direction cosines of the unit normal vector fr, aetinea in(8) , ana (f1) , Bdy be approximated by means of a perturbation expansion about the body position at rest and, thus expressed in terms of body-f ixed coordinates : for k= 0, 2, 3, 4, 7 The second-order potentlals may be split up into two sets of normalized terms, those due to flrst-order displacem"ents and veloclties (l-eft subscript equal to one), and those resulting from second=order veloclties (1eft subscript two): . (2) . (2) . (2) * . (21 titk 9ik 9ik = tQik t 29ik ' (21 tQix o.s *11) for i, k= 0, 2, 3, 29ii for i=k= 2, 3, 4 ' (2) 4,7' \;(2j v.L(2) 2'LL ^o. d (1G) a ,.1n=ot = i .In) "-inrot . ,jl' , (til "l",tli' , ' Nl2) a n(?) o.1 For the sake of brevity we introduce the following differential 6perators ( [ 6l lal ): Free-Surface Differential Operator Ftvl i F(x,y) i =( ry - vr' ), (x,y) e sr Body-Surface Differential B { r(x,y) } .(7'rx-f-Fy), R(v){ F(x,Y)} =Rer(F* * JvF }, , r.to' , "1" = *l').n(1) k;2) .":22' + o.25 *;"'r"11) t t,jlir"(]) (r' + o.2s1*j1)12r"j?) (;' .n(o)={_;,"rl={;. Lo tf*"'*vv-) (18) Operator (x,v) e so (19) Radiation Condition Differential rator I I I k=I with nlo) = a . i'= 2,3,4 (2r\ , Ope- ,*,r, d* (24 I L (22) ) , i--2 , i=3 { l23l (2) (1) (21 12r Q) ' j,n2r = itot L")r' = ino2 = it (0) .,; tzal (2) f-in 'Lra itz r =Lo, i=4 A similar expansion is performed for the dirgction cosines of the unit tangent vector s. Co1lectinq terms whose conimon factor is .. .3-1 ln=1 , 2; i, k=0 , 2, 3, 4, 7l the following well-posed linear we'obtain"-jnot boundary value problems, generally of mixed forn (third kind, Robin problem), for the unknown potential functiirns fi(n| [z:] : I I : I I HI 1 1 I I I First-Order Boundarv Value Problems ( The boundaries trated in Fig. iT' So, SR and S" are illus- ,l fl( trr-4-8 otl' with right-hand sides a) i = k = 2, 3, 4 O, (x,y) e DUDi for k=0, (xry)e D, fot k*O l2s) k=0 (x,y) esousi,for F(v){?11)} = O, tx,y) esi ,'fot klO (26 ) (27 ) stt{1)r= rl" , (x,y) eso , k{o tiu 3.t t ) O , ,*ry)eoSB ( Y* --) 128) [email protected] K y ' nt"rttll) t= o , t*,vl iR , klo sr. (2) 1 129) right-hand sides tk(1) (0) kn'-' (1) ,tli) = u(zt , k= 2, 3, x=r 144) =ltI'), I rxs .v?jil . ril,,, k=4 [,rrj]l '*,' * rr'tl',' * tl',t,ll], (4s) (1 k:r m 4 = k-1 in Eq. (43) ient to evaluate ing potentials For this reason b) i* sfrnmetry conditions t-*,rt = (-r)*-'tJ1) ljl) tk=z,3,ai^xlo,7 146'.l, (30) - attjl)l ,7 rkk [,;l: , k=2 (x,v), !1".r1Iflt+ EN 1< K, (31) a, 2, 3,4, 7. right-hand sides Second-Order Boundary Va1ue Problems 1. Potentials caused uy t{2) , ee. , 3, 4z o, tli' = 0, (x,y) e rra"rqfilfl)t = 0r arrtlfl)r =.n K +ie- ,tliI R(4,)!tli) ) = 0r ( ,tff)t-*,yr k E-SB ( y+ f (avr {,tli, ,= at,tjf) r= ilg- ,tlil, R(,!v) t rt ,ill) jf t-*,r1 ) = r= ,'ji', ,'ji', y1 = 12 ,41 -1,(l) *Q3*" - -'v(1) Y93y= , (4?') 11=(1,4) ,t--21 (2) (35) -6) (x,y)E1st"R 0. (xry) tik (34) so ( 36 ) ( 37 ) t ,r=3 )k=0,7 lrik I t[])r,t=nJ L = (-1 f-'rtli)(*,y) 3, 4 = l (33) 2. Potentials caused uv *j.1 'rr,l' ' , 2, 3, 4, 7z o,?li' (2) (32) D (x,y) e Sa o), (x,y)e = 0, (x,y) k)=(2,3) -!(1) *Q2*" - -1,(1) !Q2r", 117) , , (x,v) e s (x,y)e s F o (3e) (40) 0, (x,y) eS", ( y+ --) (41) Or,*rrrd* " (4tll, tll) * o ,2tj1)t(1) 'y(1)'v(1) 9k** Ii (42t . ?l"tlll + , in Eq. (43),k/O,7 (49) (50) The first and seconcl order potentials for k=0, 7 (incident wave and diffraction problems) are in general nonsymmetric about the potentials can f-axis. But the first order (k odd) and antibe spl-it up into symmetric symmetric (k even) parts. Then, for k=0, 7r the first-order problem must be solved twice for each k, modifying eq. (31) to apply to both the odd and the even case. This requlis in four complex first-order potentials U rl'' L r-D',tjl)r*, y) (21 m = i+k L1 (38) eD - f -ik (48) (43) III.4 .9 (k=0, 7). Accordingly.four second-order po- .t.,gPt?il: obtained, tential Eential- functions fi? {ft (ft=0, {t=0, 7) 71 are (a:), ;;";;.;i";-i,itn replacins eq. tsol',rtr, by relations for the odd and even cases. It may be of interest to note i-n passing that previously pubLished resuLts for the forceh oscillalion case (r,ee [6] , Potash t Z] ) regarding the second-order free sur- face inhomogenous terms, corresponding to our eqs. (45) and (49) (both complex), are not consistent with each other nor with our results. However, there is fu1l agreement with Lee's later (corrected) problem formutation [21] . For more detait!, see [8] , I z:] 3. PROBLEM SOLUTION 3.1 Boundary Value Problems Type In the preceding section a set of linear boundary value problems was introduced by eqs. l?.tr\, to (31) for the first-order potentials $,!'' and by eqs. (32) to (50) for the secondlorder potentials O'Ji). aff potential functions are in general*Complex (with respect to time) according to eqs. (13) to (17). Referring back to ?ig. 2 for orlentation, one can now distinguish: - First-order incident wave potential (k=0; n='l ) , solution known. - First-order forced oscillation potentials (k--2, 3, 4i n=l ), three unknown functions. - First-order diffraction potentlal (k=7; n=1), symmetrj-c and antisymmetric parts, two unknown potential functions. - Second-order forced oscillation potentials lj- , k=2, 3, 4; i n=2), total of six unknown potential=k;functions. - Second-order incident wave and diffraction potentials (k=0, 7i n=2). On deep water these potentJ-a1s are trj-vial and need not be cons-idered. - Second-order interaction potentials (i=2, 31 4i k=01 1i n=21, symmetric and antisymmetric parts, basically 12 unknown potentj-aI functions, but these may be reduced back to 6 numerical evaluations by combining the symmetrlc and antisymmetric parts, respectively, of the disturbances resulting from the first-order free wave and tliffraction potentials. - Second-order interaction potential between free wave and diffraction flow (i, k=0, 7 i n=21, this vanishes trivially on deep water. Aside from the lncident \^/ave problem, whose solution to the second order is well known, the boundary value problems for all of the potentials mentioned in the precedlng are of the same type, whose general form is: a6(n) = o, (x,y)e D (x,y)e S" F(r2v){o(n)1= "(t'), (x,y)e so (51) Bto("))= "(t), (n) .o= O, (x,y)e"sr(y+ --1 v R(n2v){o(")}= O, ,*,t,{rl The only essential dj-fference bet\,/een and the second order lies in the the first f?TT=3ff:l'.i:.;,;$:.::uTuf3.I i.tt+r:"3, for the second order. Boundary value problems of type (51) as "radlation problems" according to Sommerfeld, summarizing under this name a1l those problems which may be may be addressed d described by pulsating sources j-n a finite fluid domain. The uniform format of the boundary vaIue problem type describing the linear submodels of the nonlinear flow system has several practical advantages. Questions of existence and unl-queness of solutlons may be discussed in a very general way once and for all. Above all, however, it is of immediate benefit that a sj-ng1e numerical solution approach may be applied to all subproblems under discussion. 3.2 Integral Equation rffili, @ fted tdt @u ffi aII( h d ail tril Uet tEo Method Green's third theorem of potentiaf theory, applied to the 6(n)-boundary value problem (51), yields the following inhomogeneous integral equation of Fredholm type of the second kind (Helmholtz integral equation), tgl : mil mEn @M mud affi lkr 'UnSt p,J[tdl,t !tuba - 2ro(n)(*,r) * s [email protected] . aG(n) (*,y iL,nl .ro(n) tE, n)-----an;--dsQ o . (52) (*,r;E,0)L(ntqnE = tc(n)*(n) (E,n)dso - { "tn) to-t" cryllt ,@@ [ @ffi6 [email protected] Trr @ilufr In order to apply thj-s formulation, suitable Green's functions must be avaitable i.e., functions which are solutions to a boundary value problem similar to (51), but with homogeneous boundary conditions [8] On deep water the Greents function of n'th order is, [ 8] : (t) (z; E) = R"i {Log(z-e) - rog (z-E) ) r - ir (z-L) *2+}-------_u*j 2ne-in-v (z-e) \ c @- trmrmt [email protected] ffi fforr I arflfrl brnr I + (53) 1 n v-K U with space complex variables for field point P (zl and source point Q ( 6 ) : z=x+lY e = E + ln , E = € - in (s4) In the complex time domain the time-dependent Greenis function of nrth order is represented in accordance with eq. (13): [email protected] *{ @rryl III-4-r0 gln) lrr4ral (zrr)e-inurt} = R€.{"(n) ) proper treatment of this effect is even more lmportant than in 11near theory. Proposed cures-to this problem may be analytical (Ufse11 [ 27] , ogitvie-Shin [28] , Papanikotaou [ 8]), semi-analyticat and -numerical (pau1ling ana wood [29] , ohmatsu [:O]), or- purely. numerical (f'altinsen [:t] , papanj-kolaou l. I, 23) ). fn the present context numerical methods, based on interpolation of regular frequency results, were preferred to others up to moderately high frequencies because they were convenient to use and gave reliable results, though not always without substantial expense of computer time [ 2S] But it must be mentioned that purely analytical methods for this purpose have not yet been extended to second-order situations with inhomogeneous free surface boundary conditions. (55) The integral eguation (52) represents Iy a pair of coupled integral eqqa: because the unknown functions q (n) complex. A physical interpretation of fogUrplation is as follows: The poten0(nl (x,y) in a field point p(xly) is of contributions from a double of (unknown) intensity f potenLial produced dipoles arraiged ) E, n '( Se, from a slnglebyl.aygF potential of source intensity -Rtrr, ( E, n ) on S in the event of second-order potentia?l second-order potentia f,urther single lEyer potential of krror., rr.a e- lntensi-ty in{-anci +,, Lrzt r 12) ( t E, r O) ^r on Sp. The : Helmholtz integral inteqral eguatlon equatlon formuformu(52), which was apparently first ininto ship motion theory by potash fl zl ,, 4p is analytilally srrqrJ Lruolry g"rr6.ui tItdll *ore geIIeIctl. [rutc th.r, Is source-sink -weIl-Kno\^rn conventional source-well-kno\^rn [zq) by including boundary value lems of inhomogeneous mixed Lype. In tion, even when applied strictly to a p6oblem, it has shown certain advannumerically, especially for more comsectlon shapes, and with regard to sensitivity to the well-known irregular uency phenomenon [ 8] , lzl) , lzsl - . The discretization of the integral tion system (52) into an algebraic set equatlons is conventlonal in the sense close-flt methods and will not be dishere. Details can be found in [8] , 231 . In contrast to Frank's close-fit the normal derivatives of Greenrs tion in (52) are taken with respect to e point coordinates, which simplifies integral expressions to be evaluated the discrete panels [ 8] Some particular analytical difficulties i. evaluating the integral expressJ_ons ?! (52). 91" of these pertains to Lhe line Gegrlt over Sp, which contains a singulalry where Sp intersects with Ss. This caui numerical difficulties in evaluatlng the r:ntia1 and its derivatives in the viciniof the slngular point. Further the intetion along Sp, which should be extended infinity, muit be truncated, so that some ytical criterion for the truncation Iit is.reguired. Our approach to these quesons is discussed in section 3.3.3. rt is well-known [ 26] that integral ion formulations like (52), or is used Frank, fail to provicte finite, unique, sically meaningful solutions at or near in "irregular" frequencies corresponto eigenvalues of the adjoint lnterior ial boundary value problem. This phen is known as "irregularity problem,'. second-order theory this type oi effect ins at much l-ower frequencies, the first rder irregularity occurring at about quarter the frequency of the correspon'first-order vaLue. This suggests that IrI.4 3.3 Solution Potentials 3.3.1 Incident Wave problem The derivation of second-order potentials for a regular, steep (Stokes) wave may be assumed to be known, e.g. [ 3] . rt will be presented in a special, normalized form in the present context For water fiJz) "r i"ii"ite-a"ptn is trivial and need not be consideied. For a second-order regular, steep $rave propagating in the positive x-direction on deep water (beamwise), the first-order potentiil for the wave with amplitude aw and wave length tr, must satisfy eqs. (251, (26) and (28) and is of the form tJ" = (kb) "-1 .k(Y+jx) This (16) may ( s6 ) be normalized according to (15), *J" =-s ur 1 1.,;.kY uj (kx+6r) (57) where (vb) = (kb) (1 + + otej),eo=kl.,l{sa) '3r with the di-mensi-onless frequency parameter ) -1 (vb) = r,rg'b (59) Note that the wave corresponding to this potential is, in fact, exact to the third order although here nonlinear effects are visible only in the relation between wave frequency and !,rave length, eg. (59) . The phase angle 6, measures the position of the wave crest retative to the origin at time zero. - u 3.3.2 First-order Radiation Problems AIl remaining potential flow problems are rafliEtion problems of the form (51 ) , with L(l)=g ge1 the first order. These probIems can all be solved by a uniform procedure: The integral equation formulation (52) is discretized by introducing N straightline element panels of constant potential value on.the body contour So and, in the spirit of the close-fit method, setting up an algebraic system of equations based on the boundary conclitions at the midpoints of the dj-screte panels. Details of the procedure and the system of influence coefficients and right-hand side terms in the ?Igebraj-c equation system are given in [23] The forced motion first-order potentials (k=2, 3, 4l are determined in a single calculation using synmetry or antisymmetrY, whereas the diffraction problem (k=7) must be solved twice for the svmmetrical and antisymmetrical parts of rjl^) .in (30). once the potentials/'dr1tl ;." known on So, eq. (52) may be used again to calculate this potential at any desired point (x,y) in the fluid domain D. In particular the first-order potentlals can be evaluated on the boundary Sp (free surface) which is required to obtain the lnhomogeneous terms (45) and (491, for the second11{f), problems. "q-": orcter 3.3.3 Second-Order Radiation Problems The general procedure in solving the second-order radiation problems is the same as for the first order, that is, eq. (52) is again discretized and applied to points on the body contour (x,y) e Se. In fact, the problems associated with second-order onset ilo*= "l2l Or=2, 3, 4), €9s. (32) to (37), are comi)Ietely analogous in form to the first-order problems and can be treated accordingly without difficulty. However, the remaining second-order radiation problems, which are caused by flrst-order disturbances proportional to *.( 1). ,rJ1) ti,l=0, 2, 3, 4', 7i, d.o introduce s6me spiicial questions regardlng the evaluation of the right-hand sides, €eS. (441 to (49). These expressions involve first and second partial derivatives of the potential which are to be approximated numerically. The accuracy of these approxlmations must be examined carefully. In our computing experience for a variety of different section shapes it has been found advantageous to transform some of the second deri-vatives with respect to x and y, where required, to expressions with derivatives in the tangential clirection s, eqs. (441 to (49). This has helped, in Befticular, to obtain stable results for . r!(.t on 56. Reoarclidq the numerical evaluation of the terfr ,L!?l on Sp some further problems arise. ThA 'f*prop.i integral over sp i-n (521 involving this term in the lntegrand requires integration to infinity, but in practice integration must be truncated at a "sufficient" distance "x- " from the body. fll.4 This problem is also familiar from time-domaln and finite element formulations of the present problem llll , lzzl , [::] , where to in addition it is fundamentally difficult It meet the radiation condition at infinity. can be shown by theory that dpg.lo the harmonic asymptotic behavior of ,if, l,, rim I.r1.') t"l = o at< (60) lI ( r]D. ffi([ '!f ,{" "{r, lx1+- Nevertheless an independent criterion must rq! be used to measure the truncation error. In [email protected] the present context the following indirect [email protected]@ procedure was used: The first-order damping [email protected] coefficient was first calculated from near [email protected] field quantlties (by pressure integration) ffi[r and then compared to results derlved from [email protected] far field potentials (via radiated wave amplitudes), extending the range of [email protected] ffielfl tion on Sp step by step until suffj-cient agreement was reached asymptotically. This drfi defined the truncation point x- . In prac_Wr tice, x- was found to depend on frequency ( v b) and body shape. Dil;"i A particular difficulty exists at tlre intersection between S^ and Sp wfrere ,r{fl) and 1r{f) are slngul.r? a==r^ing the sr-ngularity to be integrable, which cannot be taken for granted for any section shape, we treat this problem numerically by close1y approaching, but sti1l exempting the pole in the integrations. However, the fundamen! tal analytical problem, particularly for nonvertical sections, remalns unresolved desmli[@ pite John's valuable basic work lzaj troM il ,{ 3.4 Pressures Forces 3.4. 1 Pressures Moment *p ' ,(I [email protected] The hydrodynamic pressure P (xryrt), measured relative to atmospheric pressure Ieve1, according to Bernoulli's equation lii:rnrfflll rfu is P(x,y;t) 1illlI = - P9Y - P0t(x,Y;t) 1 - 2t' lvoi2 (61) Using the abbreviations from eqs. (13) to 117 ) (2\ o(1) = I *l',, ,0,', =.r. r . ix (62) ' I'k .(2)-: .(2) i= 2, 31 4, k= , O, 2, 3,4,7 2q' = | z+ii' Q ," ora.i-r, the hydrodynamic pressure on S (t) up to the second order: { nt P(;(st,f t=);t)= - psi - {psr*j 1) t1\(21 + - jpo0"t*,y)- ]e-'i,dt '*- - toe(x]-' -o( 2) o.25i x(l12) - 2)po(.0(2)+ ,1"' *"ojt'', - o.s; ,^ri'*1" - o.sip.,(;' *2(1) Y *3(1) (2) {ps(x o3 G;'- i;' r*j1)rrj"r e-2iu'lt G;' *ri't*j1) I oj1) + + -L2 + ( o]fl uPuu, * M-{[ - o.2s;l *f,lt 12t * o.25ptlOj1) l2 - *'*1" - (xx'+yy-) /t \ + y , xi\L I'+(xy--yx ) * 't', * 0.5 j pro ,i'*1" lTjr) * o.5jprrr ti'*)t\ rljr) /l I (21 2Pi i = -L (21 ' p(Ir29ii il (53) t;e*) = p(o) rvl .n ,1" ,i,t,.-Jot t k n.l*'.'* ,nll) r*'Yl f, + ,r!?) I (2t = ,nli' rostatics 9g . . (1) =l0o9t+ (2) Eeg:ond ,nli' = - tt o (72) 7 .25 pt, ,l:''. ,l:'') (73) 4. Second Order (n=0, 1=1, m=1; quasi-hydrostatics of second order) al i=k=2, 3, (2) lPoii 4 rll,nli, . ,11, all,, p,(i'to'ali' * r"to'ti:', *1" *l fo.r, on tl*lt'l' i=4 = - 0.25 pt + L. , it4 o s *j3), m='l (1) *3 , K=J (2') 2P oLL pg rn=1; order) oi1) is r"(o'rll', *1" 2r b)i ,,jil 4 (21 z, p^rOii' - o.2s pt Oin :!)'* ,o(in(o)o(l) * ojl)o{l' . ol:'r,11', s) i=k=0, (65) Order (n=2, I=1r a) i=k=2, 3, + i<k k=0 o.s pt lrik o *1'), k=4 (67) -0 , kl 3, 4 t (70) , pgi t=0 + sl L=2, 3, 4i k=0, 7 = 2j p,r oli) - o. s o r oll'}o{l) d) - 0.sj st Order .(2\ _ o.s ot olr)0.(1) ln Kn 9i1 (7 (65) (n=0 p ptor 4 oil'}o{l)r + 0.5i p,(i'to'r,ll' * r.(0)ol1))x(r) "-2 Gmparing (64) with (63). yields the 1 pressure terms p(nl, dropping fiactors er,.k, X. Zeroth Order zl (6e) ,*1" * {k'to'ril'* *"to)oll)l*ll)t jtot .r"' I;l,llll: ltjll' ,ri*l't,L=4 oll)of l'}r + 0.5 jpr.,i (. n to'0,1,1' * ,"(o'r,ll' ,l*'r'* convention: (68) 714 i< k b) i,k=2 tPit< !=4 pg x3(21 r r=J , !* 3, (2) _2 ^p ,njll t;'7r + ]'Ie. . r l-011 (6,4) 2, 3, 4, k= 0, 2, 3, 4,'l the left subscript is defined by the = , \ lhese expressi-ons were,derived aftgr ding the potentials 6(1) 3n6 6(2) laylor series about the eguilibrium of the body contour S- so that all ls and theii derivativSs in (53) to this position. ES. (53) represents a complete hydrodytransfer model for the pressures on rretted body contour S (t) . This transfer corresponds to the dynamic system mol3l and can be decomposed into terms of t orders (and frequencj-es) in analo(13): 3- (1)2 25 pg V *i-'-, .[: * + Itr-4-13 = -0.5 k=2 l7 4) 1=3 r ori*!1',1=4 L; 3, 4; ,Ll 3, i< k 4 ,( oll'} ol:'} . *l:'all" -0.5j pr,r{(.n(o)6(1) +(.n(0);(t) *."K 'lN K (7s) . r"to'0,11)r *jl)* to'tl:', *,1"1 06t c,l i=2, 3, 4i k=0, ,r:?l ' 7 oll'} 6ll) - ollta[l)r *1" otr(rn(o) ol:) * r=(o'a,ll" = -0.5 e( -..., ,r(t;e*) =.F(o)+ i.* rf(l)e-J(!t + -(2.\t -2.1' ot + -(2) * itik ,l*tttk ,l*ttt* jtoik (77') d) i,k=0, 7; i< k ,"j?l = -0.5 ,( ol:)Tll' . ,ll'*,11" s) i=k=0, ,rjil = where, in accordance with (65): -o.2s (78) - (2) rl2l r(2', t Jz'il J^ii - j1-ii _ (21 ( l2l f(21 + 'j-oii jl-oii i2toii The left subscriPts denote force and components, resPectivelY: 7 e(+l:)oll'} . 'l:'Tl:') (7e) denote rn these expression" in(ol tt'a i=*(o) the directibn cosines-of fr and 5 at time t=g with in(o) from eq. Q3l and {x' , t=2 t=:,"'o'={i-: Ltiv'- Yx') , 3 .4. 2 Forces, moment u f)=z' J( ) =t 1=:, vu (88) Li=4, collectinq ter$s whose common factor is --l .,T ;-inry' (1, m, 11=e,722i t=2, 3, 4; tlao,-f 3, 4, 7) the inclividual components are defined by the fotlowing expressions: (n=0, I=0, in=0) 1 . Ilydrostatics (80) ''=4 Ivloment P in (61) Integration of the pressures (t) leads to S contour body over the ietted the body Lte nyaroaynamic foices acting on and tire noirent about the origin o ittl =-.r e i a", ilrtt = - I n,;tlt* ) a', ? = (r) (87) ."(o) =-tp l.rr(o)d" Jg 2. First-Order HYlrodYnamics ln=1 [=9, m=1) (81) (x-xz'Y-*3) s of Leibniz' integratlon-rule the to an iitegration over s(t) may be reduced at rest' i"teg.af over the wetted contour s , i"a a few additional terms of second or- By means a8i tzl t (S(t) , lztl ; .n'-'xl-'ds .n(1)x(1) -]7r,0, s I 'r, : eu) = i i(s;t; ^ rr{1)rr,(o)a" so t.-:_________-,/ E*) ds * IR .(o)_-.._* I* = R* i,,+ .rr(o) i.._ (g3) Rj = - 0.25 pst"jl)* i *1"-'.,:. o(t)li ti.l;r"-r:,t - 0.25 psl*rl"* i ,1" - -j ,r-'o(" l1 (t-);1 k=4 182) 3. Second-Orcler Eydrodynamics ln=2 , L=1 , m=1 ) (0) !?).r,(o)a. --lrPii'; laK so - o.s ^I rl" i.(1)*j1),a. .l2t (84) Ll- - ' i or k=4 where the subscripts denote ( )* = ( )l(;,f)=(o.5g,o) ( )- = ( )lt;,tt=(-o.sB,o) * (Bs) i"".?lIr,*,r,, i=k=4 rtl" 4. Second-Order Quasi-HYdr Ti=O, The transfer modet for the hydrodynamic is deforces and moment to the second order(63) pressures rivea by substituting the(21) to (24) into and normal vectors fiom is of anaexpression resulting iai f . ihe logous form to (54): (o) *12)* t.,,1?) - _ l,p\ '"'(=ni"'x)-'t zz ,-------J + 0ls.n;;'x;-'-l l=ffi=Tl j'i?l = -r:,"i?l .n(o)a" -rrrri2,l ,,,(o)a" IIr-4-14 o-so.r G_ (2)xo4 (2) * nl":n(1);(1)u" -^,. ..P(0) (ino2 o, + 0.25.n j?'t,.j')f )u= ) i or k=4 i=k=4 + = A (n) sin ( n,,:t + 6 from (14) with xi(t) + joR t(2) ill';J,I,:=e (t) M_pB 7AA= Il(x'+y') ) To determine the unknown motions x. (t), €{I. (14), of the body in a given incideilt mve the equations of motion must be taken into consideration. These equations serve to determine, in a second-order sense, the actual motion amplitudes, hence the paraEters ei (L=2, 3, 4), and further any other erplicit hydrodynamic quantities of interest. Force components relate to the inertial eordinate system o-X-y, the moment is taken rith respect to the origin d of the coordioates fixed in the body. Equating hydrodyuanic pressure forces (moments), subscri_pt F ' to inertia forces (moments), subscript E. , in the equations of motion, we obtain: i"=o n= I ,_ .R =? .R .-inr,rr "]r) (roo) i ,(n) tR e--jnort * .-l2l tRo' -rr1, "'''R = Substituting (100) and (86) to (88) lnto (94) and separating terms of different order (and frequency) the following sets of equations of motion are obtained for motion components of matching order (and frequency): 1. Hydrostatics (n=0) r x n )ds (95) + x(gj (99) 2 ,R = t. *1"' .- jnurt (94) MR It ="1 " (n) ) s(r) A The expressions in (98) can be developed into a perturbatj-on expansion to the second order with respect to ei, using xi (t) from (14) and a power series expansion-of the trigonometric functions. The resulting expressions are of the form t-5 Body Motions -/pids,MD=-Ip( ' s(t) an -1 y^=M-IIydnl, uA n=u F xn) and _) _n A(n) = lu(') l=(*.3{u(')} + r,?{u(')} )o.s . (n) 6i"' = arc ts( E": {u("1 i/ r*.{.(n)) (93) Fp=FR,rr= x, +tsi)I"sin ru1"xrv"."" (e2) fre expressions r{2) una fji) represenr tfre contributions from the additional integrals (83) to (85) which approximate the effects of the actual $retted contour S (t) deviating from So. Thg lime-complex expressions for pressT-eP.p(n), lsz) to (73), and forces g(nl , eqs. "q=. (SO1 , O1'), can readily be conrgrped to real notation for any quantltv ampritude and phase ias a(nI , . It* ='x'nrrn - + v)dm (e6) E(o) = o (101) ;l:l :;r'' 2. Hydrodynamics of Eirst Order (n=1) E(t) = L,(1) = x(1)'1 Rl k=0 k v(1) ,Ju =" k=0 = "l"f-.11),ei M(1) =', = ,l,J l( k=o "_(1) 3. Hydrodynamics of. Segond OJder (102) (p=2) rith il. Mn, ir g. fi^, from (81), v from (10) and A = body cioss sectional area. The.hydrodynamlc pressure forces +En _t were deriv94, in 3.4 except for the !$l .i or ei factors fattors or *,n,, respectivLly. respectiv6l specEr-very. "l.),forces (moment are deThe inertiS s (moment) fined by: !t =x*t *r"J,fi*=tu* oo2I.=''ri; - x4yccos xn + xiyosln o = x{g + oo oo H(2) v(2) = ', 1 (97) M(2) xn) o)_ _ x3 - x4ycsln xn - xf,yocos x4) I "ll)= =i,k=Q rk (98) III-4-15 ul-2)= ^ rl( ' '<=u = ', ,1.2)= i,k=Q rk ,l,l "l'' f* *1" ,I', (r03) 4. Quasi-Hydrostatics of Second_Order(n=2) Exciting forces and moment are made dimen* sionless by J s(?l= ,1') o = i 'i=6--"it o 1 ,:2) =. ;. ^v(?l= o v or.K +X r ' X=U (2) fi1") 1 (104) ol- t II1) 7 .-. M(2'.t ; M(2) = = M(2) o RoI , k=o oik t - ,,2 wi [fi !(n) ak 2pcl . d* tu(n) xik * l il:) ].i< 1(n) 9k 2pd. d*nt.r 'utL' , ,= =-0.5o / t.("1,,(o)a= K 1 rrt" l, ,l=2 illi' fljl)r., tr",.o , i=4 , i=2 ,3 (114) , i=4 drift components (of second ora.r) *1?) result from the algebraic set of equatioH* (104).,f,[e hgqizontal drift force is derive from H1'' (xl!'remains indeterminate) and may be"compaY6d to Maruo,s results [34] from a momentum theory. From (86) and (87): _(21 ,;-'= ] ;(2) ;-;J -= r,i=o'r'k Hoik with _12) no H:" = o z=en[2)"12 tr(zt "o = ( 11 o. spg I arr The nondi-mensional verti.cal drift is 47 ;(1) = *t?)r-1 f, E e.e. = oJ crj 1=2 k=0 1 k ,viil (115 with Ij3' = -:3' l.,l-' = -(2) xo3 e7-r j'o'' , (108) = trlaJern-1) , i=2,3,n,-. -1 The eo ,n tf "rf .rt-1)-1 ti and, finally, (a, i=2 ,3 . (n) nik (n) and E The dinensionless hydrodynamic mass and damping in (105) are defined as ( th \ ^x;{i tii' =u,* tu 0.5(k_i) .33 = 1, d.*=(kb) ^22 = ,33 = 0.5 A, ^" ttu tu --^-a tmnO=O .5t nn ^24 = ^42 =-O.5 Ic, " 44 = 0. 5cM -tutu m.k erse3 (to6) = 0, cik = o, 3r. = I =r!r, ]11,,-1 t(n) ru(rr) f-1"' *i = tt (n) ) . II"' (l0s) uit (f rom (85) i ttll', I,l") . tsjn) 4 o(n) uik (l) k k=2 lk are obtained by matrix inversion ) ;r" ;ft='*jt .? 1 , L=4' '7= The solutions to ('105), which is of the form lji'. ftjl') + r lll ) a1;, .}jl'r = ffjn),i= 2, 3, wlth: f= a U-2 !(2) r(1) m.. mik' IK = m.'. ].K = E = k=0,7 kb) ' 47 a (2) ilt ikt from (86) , (8 i=2 k=0 (t (,o) I 2 pebd. e] :- 2,3 BI2) =ts I The first set of equations, ('101), concerns hydrostatic effects due to zero1utl order pressures, corresponding to the law of Archimedes. The set of eqs. (102) consists of the first-order differential equations for sway and roII (coupled), and for heave. The unknown parameters eill--2, 3, 4) , defined in eq. (2), can be derived from the solution of (102). Eqs. (103), with ei substltuted, comprise second-order equations of motion for coupled sway-roll and for heave, and are used to find second-order rnotion amplitudes t(2) (frequency 20r ). In (104), fina1ly, time-independent second-order effects, called quasi-hydrostatic, are present and can be determined using the s i from the solution of (102). T!!p yields the so-ca11ed afift components *:!t, where the sway drift x:4t is equated to"2ero, by virtue of an aS5umed external force balancing the drift force. The dynannic equations (102) and (103) can be written in dimensionless form as follows: *!r11"' ,1") 1 IrI-4-16 the roll drift ("hee1") l:1' = -::r)irl.,tr-1 = {(klarl) . -r'I 4 ^ i=2 .r'* ,iljil - 0.25(vb) ., in Ijtr;ttl, {o.s d A ) -t 7 r k=0 lt17 l. NUMERTCAL RESULTg A computer program [35] has been developed to numerj.cally evaluate the boundary value problems described in the precedlng sections and to calculate the physical quantj-ties derived from the resulting integial equations. The results presented in the following were generally obtained with N=25 discrete panels on the body surface So and about 50 discrete elements in the free surface Sp (second-order). The free surface ilisturbances for the second order on SF were luated up to where the potentials reached 1imit, usually no more than 9 _asymptotic If-beams away from the body. The frequenrange was 1O-5s (vb) 2.5 in steps of = =0.05. The size of the program is 140 K words a CYBER 175 computer. The program calcullates in one pass all pertineit iydrodynaa (vb) ric quantities (potentials, pressirres, for, moments, and motions, where applicable) any standard problem case of either t order (k=0, 2, 3, 4, 7) or second or(i_2,3, 4, k=0t 2,3, 4, 7). Compilatime is around 1 0 seconds per standard lem case. A complete evaluation up to second order, comprising 13 standard lem cases after suitable rearranqements, about 1 minute of execution time per ncy. The results presented in Figs. 5 to 36 _ back in of the piper) with few 6xceptions in only to second-order quantities. are based on first-order results, which t be included in the present paper. Nor space permit a discussj_on of the ,'irrear frequency" phenomenon. Earlier publiions by papanikolaou [A] , [zz) , lzs] be consulted for details on these is, including numerous first-order results different seqtign shapes over a large uency range [25J less as indicated in the figures. The ned heave amplitudes have been-standaril to. correspond to e3=1 o. xJl)=5. ,"lve phase anglesoare ilways pl5tted with ancrement of 350 Fig. 5 for the triangle of B/T=0.8 ancl . r/2 (finite flare) demonstrates encoungly good agreement between theory aqfl, riment for the hydrodynamic torce- VJj, some allowance, for the phase an-Y+Fh The 51)4'. appreciable phase snift, which Dre abrupt in the measurements, occurs frequencies near where the force has a , an observation made here for the r although in the linear case simieffects are familiar from several_ other . The caLcul-ated results.for the steate second-order force V"{3) in Fig. 5 are not in agreement with yamashitars mea_ surements regarding the absolute levet aI_ though both show a similar flat tendency; we are uncertain of his definition of this quantity. In Fig. 6a fine Lewis form for B/T=0.g and B=0.5 (sectj_on coefficient) is investigated, Fig. 7 shows the ellipse for the same B/T. Hydrodynamic and steady state secondorder forces show excellent agreement with measurements- The phase angles agree better for the ellipse; for the Lewis form the agreement improves for decreasing motion am_ plitudes, proportlonal to e, as one must ex_ pect. fn Fig. 8 for the ellipse at B/T=1.4 all results are 1n very good agreement. Figs. 9 to 1 1 are related to wider sec_ tions (B/T=2.0) of different fulIness. The triangle shows the strongest, the U-shape the weakest nonlinear effects. This appears reasonable because the nonlinearities should be responsive to how rapidly the section shape changes near the waterli_ne. Eor the circle in Fig. 10 the overall agreement in al1 results with data from expeiiments appears extraordlnary. The minor deviations that do exist increase with e, but remain acceptable even at e=0.6. The U-shape (Fig. 11) shows some, but not much greater scatter. It is of interest to compare the steady_ state forces in Figs. 9 to 12 in the limit of (vb)+0. The tri_angle with large positive flare in the water:Iine produces i net posi_ tive steady lift force, circle and U-sirape have zero flare angles and a vanishing ,Zro_ frequency steady lift force, and the bulbous form with negatlve flare causes a sma11 ne_ gative steady 1ift. At finite frequencies the steady-state vertlcal force miy become positlve or negative. The results for the bulb (Fig. 12) shoul-d be vlewed with caution due to the very large e-values in the experiments. The magnitude of the nonlinear effects is rather smal-l, the scatter in the measurements con_ siderabl-e, and comparisons with the theory for e up to 1.9'17 are of questionable valu6. In Figs. 13 to 16, for a Lewls form (B/I=2.0, B=Q.94), further comparlsons of forced motion results for pure sway, heave, and roll_ as well as coupled sway-rb11 notions (with reference to standaidized para_ meters eir ek=1) are presented relative to data from potashrs second-order theory [7J (Simil-ar comparisons with theories Uv-oifrer authors are also found in [B] , [S] ', lzZl ). Fig. 13 relatgq,to the pure sway forced motion.problem -(ZS)5t in Fig. 2) . fne agreement with Potash-in the vertj.cal forces, hy_ drodynamic and steady-state parts, is exceilent. These can be calcul_ated from first-order potentials exclug[vely. The second-order horizontal_ forces frlj), winlcin involve second_ order potentials, differ appreciably from Potash. This seems due to a devlation in his second-order problem formulation Ig] and the presence of irregular frequency effects in his results .Fgr heaving (proUrerq.3sjl)1 in Fig. 14 - hydrodynamic the force VtzJ-ae[enas on tII-4-17 second-order potential-s, but the agreement remains reasonabl-e, in part because the heaving irregutarities arg^grilder. fn rol_l_the hydrodynaing (Fig. 15, problem 3Slfl -ielatively mic nonlinear effects aie weak for this section shape. Fig. 16 illustrates the second-order effects in coupled forced sway and roll motlons (problem 3slfr)y upon the vertical force component. ThiS'emcompasses the second-order effects due to first-order disturbances resuLting from coupled *iay and ro11 motions. Comparisons with Potash's results are problematic because it is not clear whether he was dealing with the same flow subsygfgm. (He ma have included the problems 3S)22, ana 3Slfi simultaneously) . Figs. 17 to 19 present the steady-state forces and moments actj-ng on a fixed body 1n a wave. These results stgqr. from the steady-state part of problem 5slzl Thesefnvestigations, and the ones that fo1Iow, were performed systematically for three section shapes, the triangle (B=0.5), the circle ($=r/4), and the rectangle (B=1.o). Forces, moments etc. are nondi_mensionalized with respect to wave amplitude I uro I , as customary in the literature. Fig. 17"c94gerns the horlzontal_ steadystate force Hn61). rn the limit of (vb )*oo our results for all section shapes approach unity in agreement with Manro,s [:a] analytically derived result frpm,qrgmentum considerations. The quantitlz fi.,61, ,plotted in dashed lines, expresses th6-considerabl_e contribution by the wetted part above water (rest integrals in eqs. (83) , (84) ) upon the total steady-state force. For ( vb ) * this term tends to the limit of 2 sin o_in agreement with [36] . The contribution Hade by the underwater part is smaller and negative, the sum of the two yields the net force. The corresponding vertical steady-state forces (Eig. 'l8) are negative for circle and rectangle (sinkage force), and positive for the triangle (lj-ft). The zero frequency limit of this force is zero for the wall-sided shapes^F,,ri r/2) and equal to 1 for the triangle {trj[it + ct9 o, = n / 4=11 . "tS Fig. 19, are The steady-state fioments, negative, that is, heeling in the direction of the lncident wave. In order to determine the lnfluence of the motions upon the hydrodynamic secondorder contributions, it 1s necessary to solve the equations of motion to the first order (eq. (105), n=1) to start wlth. The results are reguired to solve for the sma1l parameters ei (i=2, 3, 4) in terms of the initially asGumed e7 (e^), which measures the relative slze of ttrE incident wave, usi-ng eq. (114). Ihe first-order exci-tation forces for a circle in a sine wave (6*=91 are shown in Fig. 20. Figs. 21 to 23 present the corrgpponding first-order motion amplitudes *+', (i=2,3, 4), where the results for sway-and roll stem, of course, from their coupled equati-ons of motion. Near resonance all amplitudes are considerable because hydrodynamic damping is smal1, above all in ro1l, unless empirical corrections are made to allow for viscous damping. On the basis of the amplitude results we may now assign some bounds to the ei and limit the wave heights accordingly viE e7 or w€ may assume some wave height, hence'e7, determine the corresponding values of ii, and avoid frequencies where ei exceeds a specified limit, especially near resonance. In Figs. 24 to 28 some first results are presented descrj.bing the second-order forces and moments which result from the second-order interactions between motion poten tials and the combined J,qpident w4yp and dif fraction potentials (4Stzt and 55tz,subproblems combined). These results tike into account the small parameters et=f(vb ) of the motions, deduced from the lifrear response ana1ysis (egs. (105), (114)). The curves exhibit some more or less pronounced resonant peaks. These stem mainly from the behavior of the ei nedr resonance, modi_fied in part according to the basic frequency dependence of the second-order hydrodynamic forces. Th resul-ts presented here are stitl a function of the initially assumed parameter e7. By solving all standardized subfrobl of the system and associating them with the pertinent small parameters .i.uk, it is pos sibLe to assemble all contriEutions of second-order for the body freely oscillating in the wave. The steady-state part of this summati-on corresponds to the so-catled drif tj-ng forces. Fig. 29 compares horizontal drifting forces for three basic section shapes. The results are credible, except resonance. Comparisons based on Maru.ors famil-iar formulas [ga] give similar answers. The asymptotic limit of this force for ( vb)r - shoulcl be one, as in Fig. 17 for the fixecl body, because the motion amplit go to zero at high frequencies. Vertical drifting forces, eq. (1161 , are shown in Fig. 30, drifting moments, eg. (117) , in Fig. 31. Rectangle and triang have positive verticat drift (lift) for mos frequencies. The drifting moments are negatj-ve, they tend to heel toward the wave. Masumotors results [tO] show similar tenden cies. It remains to solve the equations of motlon of the seeond order, eq. (105), n=2. The excitation forces of second order are obtained from eg. (111) by summation of a1I second-order hydrodynamic terms, which are caused by the motions and velocities of fi order. The verti.cal second-order excitation forces for a semj--circle are plotted in Fig. 32. According to eqs. (i07)r (108), n= the second-order hyd.rodynamic mass and damping coefficients for heaving have been dimensionless with the displaced fluid mass of the semi-circle, as usual. The curves in Fig. 32 are obtained by contractj_ng the fre quency scale of the corresponding first-order curves, which are familiar, to one qua ter so that the first-order results for shifted to ( v b) =l . The motion responses of the heavi-ng c cular cylinder to the first and second ( v b) =4 .are are presented in F19. 33. ft is interesting III-4-l8 b- t' : t I tit-- ll i, to note that two resonant peaks are present in the second-order heave amplitudes, one at the resonant frequency of the first-order system which tends to excite the second-order system, and one at the second-order sys1 'temrs own resonant frequency which lies at about one quarter of the first-order resonance. The second-order amplitudes are relatively small compared to the first-order values in this instance. This need not to be so for other degrees of freedom. To obtain an idea of the relative i:rportance of second-order effects in forced heaving motlons we may compare the curves in Fig. 34. They represent the ratio of secondto first-order force amplitudes for a heaving circular cylinder as a function of the amplitude parameter e3. The agreement betr,'reen calculated results and experimental data from Tasai and Koterayama- [ZO] is very good. Only at higher frequencies and amplitudes (e) do some differences develop. Viscous effects as a possible reason for part of these dlfferences at hiqher frequencies are discussea in I Zo] A comparable diagram for heave excitation forces, second- to first-order ratio, for a seml-circular cylinder oscillating in a wave is given in Fig. 35. This ratio depends directly on e7, i.e., the relative size of the incident wave to the body dlmensj-ons. The nonlinear influence on this quantity has a peak at some intermedlate frequency where larger motions are present due to resonance. The time-dependence of the heaving motion of a circular cylinder in a "standard" wave (eZ=1), approximated to the first and second order, is shown in Eig. 36 , together with the second-order steady-state term. The frequency (vb) of 0.25 corresponds to a peak in the second-order effects (Fig. 33). The nonlinear effects are not dramatic, but noticeable. Amplitudes are about 158 greater than in the lj.near analysis and a steady lift effect is present. The result confirms why for a shape like the circl-e linear theory has been so successful in practice whenever heaving motions and waves are reasonably small. That nonlinear effects upon vertical loads (and, of course, local pressures) can be much more substantial in certain frequency ranges will be appreciated from Figs. 34 and 35. Although we have not yet examined motions in other degrees of freedom by the current method, we would also expect significant hydrodynamlc nonlinearities in nearresonant ro1l, coupled with sway, in view of the associated large ro11 amplitudes. 5. CONCLUSIONS By means of an approach based on nonlinear system dynamics and nonlinear hydrodynamic theory it has been possibl-e to develop a complete second-order theoretical model for the motions and hydrodynami-cs of a cylindrical body of arbitrary cross section in a regular, second-order incident hrave. A crucial first step is the decompo- sitj-on of the total second-order flow system into a set of nonl-inear subsyster-s comprising the second-order equivalents of the familiar forced rn-otion, incident r.^rave, and diffraction flow systemsrbut also their mutual second-orcier interaction. Perturbatlon theory, using several small parameters, has then been applied systematically to derive a complete set of first- and second-order linear subsystems of the f1ow. These systems together form the basis for developing a second-order transfer model of the dynamic probl-em. The equations of motion have been derived to the second order with all hydrodynamic couplings present. They include firstorder terms of incident wave frequency o, and second-order terms of frequency 2 o as well as a time-independent expression. A11 system responses are of the same basic form. On the basis of this theoretical model a numerical solution method has been deveIoped using an integral equation formulatlon for a Robin type problem and a close-fit discretization. The fact that aIl problems, with one trivial exception, were of the same boundary value type, that is, radi-ation problems, paved the way for a unified calcuLation procedure for all subproblems. Numerical calculations were performed for a variety of section shapes over a wlde frequency range. These calculations included some samples of each of the major subproblem types. No insurmountable, fundamental obstacles were found in the path of the numerical calculations. In those relatlvely scarce cases (forced motions) where comparison with experimental results was possible the agreement in second-order effects was generally between good and excellent. Eor the heavlng semi-circ1e, for which most experlmental data are avallable, the agreement is outstanding, for sections with flare it is slightl-y worse. The evaluation of drift forces as a second-order phenomenon is possible from subproblems of the system. The results obtained show good agreement with other theories. The response of the system can be evaluated by assembling the results of all first- and second-order subproblems into the equations of motion with small parameter values dssigned to each case on a physical basis. The overall transfer model is based on frequency domain techniques because the linear subproblems, of which the systen consists, all have frequency domain transfer functions. This posslbility evidently has significant practical advantages over tlmedomain solutions. Investl-gations of the type reported are, of course, only a prelude to developing a systematic understanding of nonlinear ship motlons in a nonlinear irregular seaway. We feel, however, that the remaining open questj-ons, despite thei-r great fundamental complexity, show a certain promj-se today of gradually being amenable to higherorder analysis via the frequency domain by rII-4.19 I i.' I sultable extension of bi-spectral analysis and generalized three-dimeirsional flow analysis 1 3. Salvesen, N., "ship Motions in La Waves", R. Timman Mernorial_, De1ft, 1 978. methods. 14. S6ding, H., Forces on Oscillating_Cylinders"Second_Order in Waves", Schiffst nik, vo1. 23, 1976, pp. 205*209. 15. yamashita, S., ,,Calcu1ations of the Hydrodynamic Forces Acting Cylin_ ders oscillating Verticaliy "p""-ffri" irittr-i..g" p1itude"l in.Japanese, Journ. Soc. Nava1Am_ alcl:[email protected], vol. 1 4l;-Jine-iffi 67-76. REFERENCES 1. St. Denls, M., ,,On the Spectrat Technlque for Describing the Seaiay_fnduced I4otions of Ships - - A Review of DLvetop_ ments over the past Two Decades and an 6ut^. l-ine of Problems lrlow i_n Hand',, SNAI{E Symposium S-3 on Seakeeping f SS:'-ng;'T & R ;ture, Glen Cove, Itr.y., June 1974. 2. Stokes, G.G., "On Theory of Oscillatory hTaves", Trans. the Camb. Soc., vo1. 8, r8rt;-;i."iii:ats. 3. Papanikolaou, A., ,,On the Solution _ of- the lrlonlinear problem of Waves of f,irrit" Amplitude on Water of Limited Depth the Method of perturbatlon", in Germln, by Schiffs_ technik, vol. 23, 1976, pp. 63_84. 4. Ogilvie, T.F., "First- and Second_ Order Forces on a Cylinder Submerged- under a I'ree Surf ace,' , .Tegrn. of Fluid Mech., vol . 16, 1963, pp. 451-472. Faltinsen, O., "Numerical Sol-utions of- _17. Transient Nonllnear Free_Surii." Motion outside or j-nside Moving Bodies,,, proc. Sec 18 . l.Iichols, B.D. Hirt, C.W. , "NonIine ar Hydrodynamic Forces, on Floating ", Proc. Sec. Intl. Conf. on Num. Shi Bodi-es H Univ. of CaIiEl Ber 5. Parissis, G., ,,Second_Order poten_ tials- and Forces for Oscillatinq Cyl_inders on a Free Surface,', I,lfT-Rept. ruo. 6O*tO, Dept. of ocean rngineeringl jg66. 6'.Lee, 9:l{:, "The Second*order Theory ^. ., of Heaving Cylinders in a Free Surface,', vol. 12 , 1 968, pp. 394. Vugts, J., ',The Hydrodynamic Coeffi . 19.for cients Swayj_ng, Heaviig n"ifi"g Cy-. finders in a Free Surface", fu ""a O"iit nept. No. 1 94 , 1968 . 20. Tasai, F., Koterayama, W., ,'Nonl1n Aclins on Cylinders .i:_-1y9:"9ynamic-Forces Heaving on the surface of a iluid;;-il;a: I'Io. 77, Res. Inst. of Appl . Mech., Kyushu w, 7. Potash, R.L., ,'second_Order of Oscillating Cylinders',, Journ. of Theory Shio 3es. , vol. 15 , 197 1 , pp. 2g54ZT-. Papanikolaou, A., ,'On the Nonlinear Problem of a Vertically Osci1lating Cylin_ der. of Arbitrary Shape;, in Germa;; Techn. Univ. of Bertin, D 83, g77 . !i-E-:_. , 9. Papanikoloau, A., ,'potential Theory of_ Second Order for Cylinders Oscil_lating ' Vertically", in German, Schiffstechnik, vol . 25,1978, pp. 53-gO:-0. Masumoto, A., ,'On the }tronlinear Hydrodynamic Forces for Oscillating Bodies in-Regular Waves", in Japanese, Soc. of I{a_ va1 Arch. _of of Japan Japan, .l 979 . 11. Kim, C.H., "On the Influence of Non_ 11near Effects upon Hydrodynamic Forces in Forced Heaving Oscillitioni ot in cerman, Sctriffstechnik, vol. Cy1inders,,, i4, 1967, pp. Univ. 12._ Grim, O., ',Hydrodynamic Forces ^ C3yse{ by RoIl Oscillatiois with Large Am_ pIitude", in Germanr Schiffstechnik, vof. 24, 1977 , pp. 'l 43-150:- 1976. 21. Lee, C.M.I "second_Order Theory l onsinusoidal Oscillations of a Cylinder for in ;.5:;:"::'53ii' .1 79-91. , :, ffi i"'o 3 lBuBra-;* 22, Newman, J.N., "A Linearized Theory _ the for Motion of a Thin Ship in negufar Waves " , Journ. of Ship &E., vol . 5] .rrrne 1961, pp.-3-a:5E.- papanikolaou, -23. HydrodynamicA., ',Calcu1ation of Nonlinear Effects in Ship Mo_ tions by Means of Integral Equation Methods (Cl-ose-Fit)',, in cermai, nepd. - i".- lgtl Inst. f. Schiffstechnik, teihn. Univ. of, Berlin, It{arch 1 979. _ _ 2!. Frank, W., "On the Oscillation of Cylinders in or below the Free s"iia." of Deep F1uids", NSRDC, Techn. Note 69, 1967. _ -2_?. papanikolaou, A., ,'Hydrodvnamj_c Coefficients for the Li_near 6sciiiations of Floating Cylinders r' , in Germ.rr, n"pt . llo. 79/7, tnst. f. schiffstechnik;'r;;ir". univ. of-Ber1in, August 1979, and .--' Schifistechnik. vol- . 27, september 19g0. IIr-4-20 26. John, F., "On the Motion of Floating Bodies", parts I and II, Comm. Pure and Appl,. Math., vol. 2, 19491-FFl-l3=ETIand vol. 3, 1950, pp. 45-101. 27. Sayer, p., Ursel1, F., "IntegralEquation Methods for Calculatlng the Virtua1 Mass in Water of Finite Depth,', proc. Sec. Int. Conf. on Num. Shi pp. 1'76-184. 28. Ogilvie, T.F., Shin, y.S., "fntegral Equation Solutions for Time-Dependent Free Surface Problemstr, Journ. Soc. Nav. Arch. Japan, 1978, pp.43T.R., ,,stability and Ship .29. inPaulLing, Motion a Seawayo, Summary Rept. for U.S. Coast Guard, June 1970. 30. Ohmatsu, S., "On the Irregular Frequencies in the Theory of Oscillating Bodies in a Free Surface", papers Ship Res. Inst., No. 48. Tokyo, 1975.31. Faltinsen, O., ',A Study of the TwoDimensional Added-Mass and Damping Coefficients by the Erank Close-Fit Method", Norske Veritas Rept. No. 69-10-S, 1969. 32. Bai, K.,yeung, R., "Numerical Solutions to Free-surface Flow problems',, Proc . 1 Oth OUR Symp. , Cambridce },Iass. , ffi 33. Smith, D.A., ',Fi-nite Element Analysis of the Forced Oscillation of Ship Hull Formsr', ltrav. postgr. Sch., Monterey, CaIif. , 1974 34. Maruo, H., "The Drift Floating on Waves', , Journ. of vol. 4, 1960, pp. 35. Papanikolaou, A., "Computer program Version June 1980',, Techn. Univ. of Berlin, June 1980. 36. Kim, C.H., Dalzetl, J.p., "Analytical Investigation of the euadratic Frequency Response for Lateral_ Drifting Force and Moment", Rept. No. SIT-DL-79-9-20G1, Stevens Inst. of Techn., Davidson Lab., May NONLfIIEAR, 1979. m-4-21 .i ooo uoool ": Etl^ olt> !ttr o t> o t> 6rEr61566'1 c i F = = F .i; d r 5> !> rrl!l--E d a !lEl Ell olF I \ tH> Et^T 'C -> ll.!o ot ,8l o o ol -Ev-E-o ot t>a> ': AD ;;:[email protected]@@@ ,l -.. oo TI'I rl-a alb na al ^, NO -t . . { [email protected] o 9 l>oE E> 2 (() ^AALcytr,/ j -t> E a E o e f, .; { d d I[r-1-X2 in+ Eri * ,hil: fl ;Hry ?.- :rt .El !l dr- +l -il6Y116l n Elao-el t>ot>ot> *' a<E ^' ..o{ tr .. [email protected]@@ 3T = = G; d fli[ N s a N{eCl ooo doull gl rl tst !l a oll> oloo> ^ J>a 0 l> 6d b ->ao o l> - = ; d flt- 4 -23 il*fruli s ?x 3r i g il*. j R R' = = : .; 6 d t : ql :: fl?s s,s ?,. : : - i't" ll= il ,=S an El 'F !s! ^. s N! -l E r> r- ', !l 3 9! f*$'ils; [email protected] [email protected] ", ^l [email protected] fllO0006 !l [email protected] :-= [email protected] s Bni g|@ idd "^E t>ti = ) = E t z = fr g. 6 ll-4-24 >t I.EI ' 7 N-"$ fl+ flnPil"l?* "i=-E ?N?O&da 1 I-ll g! -\ OOEE [email protected]@ o .' ?E F E ; ci ,: "t il +/ a !l I il Y -dtri,-t-) ts s,* d $:ep +a + ft t tsl \ $ti ts ;!j?- ^F ;t :jl I = E F F d d rfr-4-25 f i r o r )xt fl+:r+flH flF:tP flP* ))) (_\\ qII >)) > o> tl .!j a> F E F :- = =r F = OF o= o= F J 6i d C; I I a q it .91 H :l flH n+fl Ht flH I I I I = = F E ri C; I Inr-4 -26 r>_ | El I\ + _:tj t:J $.? it7;, -$h ')'^: -\^:^? / )l / aa il r.t 3 rfl lr) =,ji fl+ + : sr i t *p D o t:\ /a z o , F F F F t €J >d ^EO =< = OE OE d d > r Ia> fl+ flHi*lf , fl,l n+ ).Jri \(-i'-a; ));.;o fl I t .- 2: {. >-l' o E t -X P- { (d d fr-4-27 16."ltq ol o dooil ll--T- sl 5 -T il l\ .9t il t4 111 + t a- o F F F F F = F (ii d d qla at o IlFll e #l'N El l)? ll W ; ) \ .) $+ g[i ,N l7 I e_"1-* l; :o 1 e F I F z = F ; d d E IU-4-28 6.1 -dor6HI ooo El N6ts ol <q- algl r>- lr>I E F = = I F 6 d ) , .$ i1 T Jl "11 + -L1 -t- { E I = F ri ,;i d IIr-4-29 d::-*i-rr:-f::.t:1+..rr4fi.:rj+ ilj;!rr:t:i;!i:4ri4trlii:ji*!:g!.#F:j*.!+r&liBg.*., 1 Discussion F. Tasai and W. Koterayama{Kyushu tJniv.) The I i. writers would like to congratulate the Authors on this very interesting and valuable paper and would be most grateful if they be kind enough to comment on a few points. 1. Which boundary condition has the J-arger effect on the second-order force, the free-surface condition or the body-surface one ? 2. The results of the forced heaving tests20 indicate that the first-order forces and the arnplitudes of progressive waves E3 depend on the-amplitude parameter e 3. For example, E: tends to decrease lrith an inerease in e:. can these phenomena be e4plaiqgd,wilh t_lris the.ory. ? - _ 3. rn ria.'17 and 29, the horizontal dtifting f5rces fro(z) are greater than unity at a certain frequencY. BY Maruors theory, the nondimensionalized drifting forces don't become larger than unity. Did you comPare your calculated results with those by Maruo's theory ? H. Maio Uoxonama N. u niv.) In this paper, the auttrors have deveLa general method of computing secondorder hydrodynamic fofges on a cylindrical body oscillating \rrith three degrees of freedom on waves of large amplitude. The computation prograrn seems to be very useful for the assessment of the.Iateral oscillation of a shipTii b-earn seas where the motion amplitude is liabLe to be large so that the non-Iinearity in the motion is no longer negligible. O;1e problenu_wli.ch I wish to point out is the faCt that the theory upon which the computation method is based ii a perturbatton analysis in any case. The boundary;yalue problem is formul-ated on the body srrrface at its average position. It is suspected that the perturbation theory may present some_difficulty -in-6-rder to take account of 'bhe change of shape of the wetted portion ruith time which is liable to become l-arge in the case of rolling. A question is, to what extent, is the perturbation anal.ysis .applicable at J-arge motion amplj.tude ? oped. As a previous pursuer of the same kind .rl problem, I can ful1y appreciate tire pain!.I taking patience the authors must have required during the iourse of this work, particularly, in keeping the orders of magnitude consistent. I congratulate the authors for this excellent milestone work I have to confess that I did not the patience to check every eguation inhav this paper; however, with regird to the passage concerning the irregular frequen_ cies, I have the following iuggestion. This suggestion may only applt if the h(n) = 6(n) - f (,), such that F(n2r){r,(n)} = o, B{h(n)1 = p(n) - $ rt"l, (x,y) e D (x,y) € sF (x,y)e so and the rest of the boundary conditions remain the same as in Equation (51). . ?-i{r.g the new boundary-value problem - n\.., ror rs ]-dentical with the first-orde problem, one of the techniques in t2gJ _ [30] should resolve the problem of indefi nite solution at the iregular frequencie f would think such an appioach wouid save the computer time rnore significantly than the interpol_ation method used in this paPer As to the solution for f (n) orre-can find_the expression in Eguat-ions, (2I,22) of "Surface Waves,' by wehausen a.rd Laiton in whiclr.the expression should !g fe3 _-f ( When L(n) is nondecaying at lxl * - 6rg behave as g)aztttzr , ih"-"oirrii"" i= !iu"n in my earlj-er work [2.1] 1 Authoy's RePly A- Papanikolaou and H. Nowacki (Tech. Univ. Berlin) We would like to thank all- discusse for their valuabl-e commentsProfessors Tasai and Koterayama have C.M. Lee (DINSFDC) * Quote: But it must be mentioned that purely analytical methods for this have not yet been extended to second-orde situations with j-nhomogeneous ffee surfa boundary conditions. The authors have completed the secondorder solution for freely floating twodimensionaL bodies subjeEt to beai waves. -332- I l In Su?:tion?: it is difficult because-the rebody and free suraepl"ds on secti-on three interestins raised tr:'=;;.;'*.I*tt'" rit=t-"it' ansv'er ;;-;r;; a general of the iltit"-i-p"rtance face bound.ry "onai'lio"i upon which shape, frequency p"i"*tttt'-?"d discussed' is subpioulem ill"ir-"taer the body boundary conii-i""v-"iioationsrt"t]=otracl colrdition Bust ii.il"' "ra-ttas" beins of- equal Tr::?1:"' ;;-;;;.;a;d exPansLon 6econd, our peiturbation pathe-sma11 does not show any'eftect of danping coheave linear tnL ili"a".-.r-onA.. consistent with the --ti.n-ii theorv' -sria.ianr we.apprecatit'""t situation different tt'"t-" :;;;'t;;;;"', =;;;;i;";"or results' where uav exist ir, .,'tiy=ing test forces into oi-*"i""t"d orders and fi'. ;il;"ision rto* aitttt"t't. I.ittiu"tror,= do"'-d"P""a on the assumed ;;:;;;;;il" model'..For exevaluition the in orders evaluation model ample, if a secoia:;;a"taclequate to reprenot t-.==*.a, but is effects' show may thig *""r*"d sent the Eg' dependence ;;"i"-;-"false" -"iltt-a"p"-t'd"t'"" mav"f 11-on have -also ;:*;;";, in secondother reasons ""i-""""""ted--f9r viscous uotablv ih"otv' ;;A;; polentiar effects. have been Third, our drifting-forces by potentials i""ilfiAld calcurated r=ot over of second-order Pressures il;;;;;;i;n whereas contour' body wetied i't.-i.i""rly lraruors rheory iilr ;; t"!+ "l.far-fierd our linear ineident'iit" driftins "ott=iderations' tr'"-ii"tizonta:. i"p"iiiL"t ;i;:r; an o,ttott=titined body' Fig'17 force on the- former case ;;i;; io. " r:-xea bgdY:. rn for force the nonilime.r"io"Ir-a;ift u"itv values certain a in the rectangr" t"i*"i"l"J i= due to- a resonant freouency t..,g"l-*tti"tt As stated i-n the PaPer' ;;;j;;";i.;;i";' mtist be viewed the result= ,'"tt't"=ottitt"" considerabl-e the of oe""oi" cautio,, with !.i''iri"a;-:i.i:;.:tl'J*i::"'i:"ffi::;"' i" ti'i= -i:r'l::"'"q"iil:";; fi=;;;-;;pri"ainvesti- situation. rot-i'rtt-iixea uoaies qated, only-tle i"tti"gf" Itl:-" nondj'men?I'"a-'iiiqhtlv srea'terthan :1;;;i ili'rrils one. It must be noted that the-reference drirtins ;;;d i;'-pi"tti"g iiil".rii i;;;;; is o - sysl*ll kinetic energy tude aw- ln reality the. ;:'"'"::*i:"3r'kl1?kinetic energy the of our second-order wave also- includes .whichcomparr1:::-?: second-orde. "o"tiinotiot'=' taken into account in any direct from Maruol:--. fr;";;;-results-derive& some expera-that noLed be ii"i.vl-'-ri-*"v arlo--inaicate the possibilmental results exceedrorces driftins H;:; illi""nttr inc unitY-'^" -'pio't"ssor Maruots remarks remind us of the- perturbaor tn"-iill.-ri*it"tions the nature of in rt-iies tion method.that ihe boundarY.conditions ii;; ;;fi;d or moving boundaries are met on any free I il iill I .pp'o*i*.t"1l-lY.T:Il":.";::;:';fli:=i:Tn, p-p., that-a second-;:"H:":n:ll.EE=iX'aI for moderi=."pI--i"'bL varid llulii"Ir',""ivamPratirdes and moderate wave ate motion are exceeded heights- When.ti!="-fi*its frequency paand shape depends on secEron '=tcond-order extends raireter. The ptl=""[ ti'rllit" to litt"tr -theorv ii'Jl""ii*rt=for second-order hyilrodynamrc comments We are grateful for Dr'L"9:-" the presexperience'with based Jn hi"-lottg ent problem. Iii=-;'"ggestion.to-transform qo 3 homosene" ;ir; :;;;;;:order subPiobrems rirstof ttrat to -i"-"ia"t to-be the ;il ;;ii.';eoi""rt"i to rely able order problems, for methods on the well--knowtt--tt'tfyti"a1 i" alleviating the'iti"soi"titY -l:?:1"* feabe certainly well conceivea ana-iiould exthe this-would-b:-i: sible. However, two boundary value Dense of having to solvefor every secondi"it""a-"r one #:;;;; our under discussion' rn 5"tX!i=I"oi;;i; for schenes pot"fv--ttumerical experience, wqrrea very reliabl-y the same potpo"I-ittve as for the second *"ff for the first ""l"-'"aa"a,that we.have also order. It mayt"iiyiltii semi-analvtical investigat"a ""a the i-1leSr1laritv methods ror aeali'ng- ryit\.lla'13 wolxed quite i'r'itt' ilIilffi rel-, izgil' llley :nrv shift adeouately, too,'"ilrto"gtt other-frequenCtr fi:=i;;;iliaritv !o ?om?. in the ranse or interii;";-fr;i-"t:'ri'lie the second-ordernote ;:i;':p;;ticurairv for important to phei=' ii IiiirS*l--ii""iri i-rr3g1-laritv i1i. ti that the the HeImusing *"tt'-""duced'!Y i=""t"'iii with f6rmuration e;;;;i;; holtz intesrar unknown runction' I;"";";;;;l'ar "i -333 - I i i, j I I I 1' i i I t bv allowing the existeffects and, in i"iti."r"',-f9r waterline' the in il;=;i;';";ri; nomenon I t i t i il t i