DAVID W. TAYLOR NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER Bethesda, Maryland 20084 DTNSRDC-83/007 r STREET eh REPAIR Adn re nerulstieeaa/ eer: Ire, ie Laue | Wo GS ion t | ERI j WS a wma ceed ON THE THEORY OF THREE-DIMENSIONAL FLOW ABOUT A SHIP MOVING WITH CONSTANT SPEED IN A REGULAR SEA by Francis Noblesse ©) S| DEPARTMENT OF OCEAN ENGINEERING READING ROOM APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED SHIP PERFORMANCE DEPARTMENT RESEARCH AND DEVELOPMENT REPORT April 1983 DTNSRDC-83/007 ON THE THEORY OF THREE-DIMENSIONAL FLOW ABOUT A SHIP MOVING WITH CONSTANT SPEED IN A REGULAR SEA NDW-DTNSRDC 5602/29 (2-80) (supersedes 3960/44) MAJOR DTNSRDC ORGANIZATIONAL COMPONENTS OFFICER-IN-CHARGE CARDEROCK SYSTEMS DEVELOPMENT DEPARTMENT SHIP PERFORMANCE DEPARTMENT 15 STRUCTURES DEPARTMENT SHIP ACOUSTICS DEPARTMENT SHIP MATERIALS ENGINEERING DEPARTMENT COMMANDER TECHNICAL DI Bae 00 OFFICER-IN-CHARGE ANNAPOLIS AVIATION AND SURFACE EFFECTS DEPARTMENT COMPUTATION, MATHEMATICS AND LOGISTICS DEA TMENT PROPULSION AND AUXILIARY SYSTEMS DEPARTMENT CENTRAL INSTRUMENTATION DEPARTMENT GPO 866 987 NDW-DTNSRDC 5602/21 (2-80) iii i ll INR I Ih 0 0301 ooy, corm nesomc Tye UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) READ INSTRUCTIONS REPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM 2. GOVT ACCESSION NO.| 3. RECIPIENT'S CATALOG NUMBER 1. REPORT NUMBER DINSRDC-8 3/007 TITLE (and Subtitle) ON THE THEORY OF THREE-DIMENSIONAL FLOW ABOUT A SHIP MOVING WITH CONSTANT SPEED IN A REGULAR SEA 5. TYPE OF REPORT & PERIOD COVERED 4. - Final 6. PERFORMING ORG. REPORT NUMBER 8. CONTRACT OR GRANT NUMBER(s) AUTHOR(s) 7 Francis Noblesse NT, PROJECT, TASK I(T NUMBERS PERFORMING ORGANIZATION NAME AND ADDRESS David W. Taylor Naval Ship Research and Development Center Bethesda, Maryland 20084 11. CONTROLLING OFFICE NAME AND ADDRESS 9. (See reverse side) 12. REPORT DATE April 1983 - NUMBER OF PAGES 34 - SECURITY CLASS. (of this report) UNCLASSIFIED Aen DON NG AIG 14. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) 1Sa. DECLASSI! SCHEDUL 16. DISTRIBUTION STATEMENT (of this Report) APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED 9 « - ’ F + Bey r. Ea] jt wd = * * = . 3-8 SKOTTAMEGNY A Sanna a GaTASS 348 (eR Hox Rege a AE i? sOHOTT ANT CNRGR SVETARET) EO) Mid at ap Dt ont :- ine At * 2 . w Fa ee ee ee nee a Se ica emia y epi lier iw tals en en (penn Marte ei Nt r soratigym whan gl anime gtr ire Seem Ly roe Pats ar Si Ri nde ad Ec lan eta a ar ABSTRACT The study presents a new integral identity for the velocity potential of three-dimensional flow about a ship moving with constant speed in regular waves. This integral identity is valid outside, inside, and exactly on the surface of the ship, and is equivalent to the set of three classical identities valid strictly outside, inside, and on the ship's surface, respectively. For the usual problem of ship motions in a regular sea, the integral identity obtained in this study yields an integro- differential equation for determining the velocity potential on the ship's surface. A recurrence relation for solving the pro- posed new integro-differential equation is presented. ADMINISTRATIVE INFORMATION The research reported here was performed under the Numerical Ship Hydrodynamics Program at the David W. Taylor Naval Ship Research and Development Center (DTINSRDC). This program is jointly supported by the Office of Naval Research under Program Element 61153N, Task Area RRO140302, and by the Independent Research Program at DINSRDC under Program Element 61152N, Task Area ZRO230101, and using Work Unit 1542-018. 1. INTRODUCTION Motions of a ship advancing with constant average speed in regular waves are predicted theoretically by using approximate theories based on the slenderness of ship forms. These theories are the strip theory, most useful in the short-wavelength regime, and the complementary low-frequency slender-body theory; these complementary slender-ship approximations have recently been united and extended in a unified slender-ship theory valid for all frequencies. A detailed mathematical presentation and historical account, including extensive references to the relevant literature, of these slender-ship theories may be found in Nomar Agreement between strip-theory predictions and experimental measurements has been found, in a large number of cases, to be sufficient for many practical purposes. This, and the relative mathematical and computational simplicity of the strip theory, have made that theory the most widely used method for predicting ship motions. Indeed, the theory, with the improvements of the recently proposed unified slender- ship theory, seems likely to continue to provide a very useful and practical tool in the future, even if significant improvements in computer performance are made and calculations based on a three-dimensional theory become more practical. Notwithstanding its many merits, the strip theory evidently has limitation, and in some cases there is a need for potentially more accurate calculations based on a fully three-dimensional theory. For instance, three-dimensional calculations would be useful for predicting the pressure distribution on a complex bow shape, such as one equipped with a bulb or a sonar dome. At present, fully three-dimensional calculations represent a difficult task, and indeed only a very limited number of numerical results have been obtained by only a few authors: Chane," Guevel and Boueene and Inglis and meieas Further- more, these sets of numerical results are not entirely consistent: while agree- ment is good for some hydrodynamic coefficients, discrepancies are very large for other coefficients. This lack of consistency suggests that the accuracy of three- dimensional calculations may be difficult to control, as is the case for the problem of wave resistance. Three-dimensional calculation methods are based either on numerical solution of an integral equation for the velocity potential or on related assumed distributions of singularities (sources or/and dipoles) on the ship surface. Different integral *A complete listing of references is given on page 2/7. equations can thus be formulated, and these can be solved in several ways, in parti- cular by using an iterative solution procedure or inverting a matrix of influence coefficients. The performance of a three-dimensional calculation method (measured in terms of accuracy control, computing times, and complexity of implementation) must obviously depend, to a large extent, upon the form and mathematical properties of the integral equation and upon the solution procedure selected as the basis of the calculation method. It thus may be useful to consider various alternative integral equations and solution procedures. The object of this study is to present a new integro-differential equation and a related recurrence relation for determining the velocity potential. The results given in this study generalize those obtained previously for the particular problems of wave radiation and diffraction at zero forward spenila” ship wave resistance, and potential flow about a body in an unbounded raltatteleas The integro-differential equation, defined by Equations (5.1)-(5.4), is an equa- tion for the velocity potential $, rather than for the density of a related distri- bution of sources or dipoles. This equation involves both a waterline integral (i.e., a line integral around the intersection curve between the mean hull surface and the mean sea plane), as in the problem of ship wave wesdigizance.," and a water- plane integral (i.e., an integral of the Green function over the portion of the mean sea plane inside the mean hull surface), as in the problem of radiation and diffrac-— tion at zero forward epee” The highly singular dipole terms (x) dG(E,x)/dn and (&) 9G (E,x) /3x in the hull and waterline integrals, respectively, in the classical integro-differential equation defined by Equations (4.10c), (4.8), and (4.9), take the forms [6 (x)-o (€) ]9G(E, x) /dn and [6 (x)-$ (E) ]9G(E, x) /9x, respectively, in the modified integro-differential equation obtained in this study. These modified dipole terms are nonsingular, i.e., remain finite, as the integration point x approaches any field point @ where the hull is smooth (i.e., has a tangent plane). A recurrence relation is proposed for solving the integro-differential Equation (5.1) iteratively. This recurrence relation is defined by Equation (5.7), where the initial (zeroth) approximation is taken as the nonhomogeneous term ve) in the integro-differential Equation (5.1). In the particular case of potential flow about an ellipsoid (with arbitrary beam-to-length and draft-to-length ratios) in trans- latory motion, along any direction, in an unbounded fluid, the first approximation, given by Equation (5.6), actually is exact, as is proved in Reference 11; this first approximation was also shown to provide a good approximation to the exact potential for arbitrary translatory motions in an unbounded fluid of a cylinder in the shape of an ogive with arbitrary thickness ratio. The plan of the study is as follows. The basic potential-flow problem of the three-dimensional theory of flow about a ship moving with constant speed in regular waves is briefly formulated in Section 2; a more detailed formulation of the problem may be found elsewhere, for instance in Reference 1. The basic equations satisfied by the Green function associated with the free-surface boundary condition (2.5) are given in Section 3. Specifically, the Green function, G, satisfies Equations (3.3a and b) or (3.4a and b), depending upon whether the singularity is fully submerged (T<0O) or exactly at the mean sea surface (T=0), respectively. Equations (3.3a and b) for a fully submerged source are well known. However, Equations (3.4a and b), cor- responding to a flux across the mean sea surface, are proper in the limiting case when the singularity is exactly at the mean sea surface. Equations (3.3a and b) and (3.4a and b) generalize the corresponding equations obtained previously for the par- ticular cases of ship wave resistance and of wave radiation and diffraction at zero forward pasion Equations (3.3a and b) and (3.4a and b) are used in Section 4 for obtaining basic integral identities satisfied by the velocity potential. The three classical identities (4.10a, b, and c)--valid strictly outside, inside, and on the ship surface, respectively--are obtained first. However, the main new result of Section 4 is identity (4.13). This identity is valid outside, inside, and exactly on the hull surface, and indeed is equivalent to the set of the three usual identities (4.10a, b, and c). The integral identity (4.13) yields an integro-differential equation for determining the velocity potential on the surface of a ship moving at constant ppeed in regular waves. This equation is examined in Section 5. Finally, an approach to the numerical evaluation of the iterative approximations defined in Section 5 is presented in Section 6. 2. THE BASIC POTENTIAL-FLOW PROBLEM The basic potential-flow problem of the linearized theory of ship motions in a regular sea is briefly formulated in this section. The sea is assumed to be of infinite depth and horizontal extent. Water is regarded as homogeneous and incom- pressible, with density p. Viscosity effects are ignored, and irrotational flow is assumed. Surface tension, wavebreaking, spray formation at the ship bow, and non- linearities in the sea-surface boundary condition are neglected. A moving system of coordinates (X,Y,Z) in steady translation with the mean forward velocity U of the ship is defined. Specifically, the mean (undisturbed) sea surface and the center- plane of the ship in its mean position are taken as the planes Z = O and Y = 0, respectively; the Z axis is directed vertically upwards, and the X axis is directed toward the ship bow. In the above-defined translating system of coordinates, the linearized sea- surface boundary condition takes the form 2 2 2 2 _ = a _ = Dodl [gd,+(Ud,-9,,) J? (Ud,-3,,)P /p - gQ° on Z = O (Di) where g is the acceleration of gravity, T is the time, ® = 0° (X,T) is the velocity potential, P~ = P~(X,Y,T) and Q~ = Q°(X%,Y,T) correspond to distributions of pressure and flux, respectively, at the sea surface (we have Q° = O for all practical applica- tions, and P~ = 0 except for surface-effect ships), and the notation 35> dy. 3p is meant for the differential operators 0/0Z, 3/0X, 3/dT, respectively. The present study is concerned with flows simple-harmonic in time, with radiant frequency | where wW is the frequency of encounter. However, such free-surface flows are not completely (or uniquely) determined unless an appropriate "radiation condi- tion" is imposed, as is well known and is discussed by Stoker, for instance. A convenient alternative approach, employed previously in idleineRe and eile oe to the use of such a "radiation condition" consists in defining a time-harmonic flow as the limit, as the small positive auxiliary parameter 0 vanishes, of a flow defined by a velocity potential of the form 6° (X,T) = Red(X)exp[(o-iw)T] (on)) where Re represents the real part of the function on the right side. The eventual sea-surface distribution of pressure P°(X,Y,T) and flux Q°(X,Y,T) similarly are assumed to be of the form P (X;Y,T) = Re P(X, Y)exp| (o-iw)T] (Qs va) Q(X, Y,T) Re Q(X, Y)exp[(o-iw)T] (2.2b) In this alternative approach, one is then faced with an initial-value problem, with the obvious initial conditions @° = 0 and 36°/dT = O for T = -~. Use of Equations (2.2) and (2.2a and b) in Equation (2.1) then yields the sea-surface boundary condition [g2,-(w- 403, tio)“ 1@ = i(w-iUd,+io)P/p - gQ on Z = 0 (2.3) > ~o for the "spatial component" 0(X) of the actual potential © (X,T). Nondimensional variables are defined in terms of 1/w as reference time, the ship length L as reference length, and the acceleration of gravity g as reference If Z yf 2 acceleration, from which the reference velocity (gL) » potential (gL) iy and pressure pgL can be formed. The nondimensional variables > > 1/2 1/2 2am, 22h, 6 CO al@y 2, p= Bloat, a= Gn” (2.4) are then defined. In terms of these nondimensional variables, the sea-surface boundary condition (2.3) can be shown to take the form [9 -(£-4F3 tie) 714 = i(f-iFo +ie)p -qonz=0 (@25)) where £ is the frequency parameter, F is the Froude number, and € is the time- growth parameter defined as f = ajay? (@62)) 2S Gn (2. 6b) 22 see? (2.6c) The basic potential-flow problem of the linearized theory of ship motions in a regular sea may now be stated. As is well known, the problem consists in solving the Laplace equation TW) = 0 and (2.7) subject to the boundary conditions specified below. The solution domain d in Equa- tion (2.7) is the domain exterior to the ship hull and bounded upwards by the mean sea surface 6. On the mean sea surface 0, the boundary condition (2.5) must be satisfied: [9 -(£-4F9 tie) "]¢ = i(£-iF0 +ie)p - q ong (2.8) > > where we generally have p = 0 = q. The potential (x) vanishes as | |x| |> co at least > as fast as 1/||x||; that is, we have > = 0(1/||x|]) as ||x||> (2.9) Finally, on the mean position of the ship hull surface h the potential must satisfy the usual Neumann condition do/dn given on h (2, iL0)) a . where 36/dn = Voden is the derivative of » in the direction of the unit normal vector -}- n to h, taken to be pointing inside the fluid. The precise form taken by the expression for 0$/dn on h in the usual "radiation" and "diffraction" problems may be : 1 F found in Newman, for instance. A classical technique for solving a potential-flow problem such as that defined by Equations (2.7) through (2.10), in the general case of an arbitrary ship form, consists in formulating an integral equation for the potential based on the use of a Green function satisfying all the equations of the problem except the "hull boun- dary condition," which is to be satisfied by means of the integral equation. The required Green function is defined in the following section. 3. THE GREEN FUNCTION +> The Green function, G(&,x), associated with the sea-surface boundary condition (2.5) satisfies the equations Ve € = VE DOG) ne So Guile) [8,- (£-1F8 tie) “I< iO) Gt ES. 0 (Gist) where 6( ) is the usual Dirac "delta function," and V_ represents the differential operator (0,9, ,9,)- Physically, ihe Gpeee function G(E,x) is the "spatial compo- nent" of the velocity potential ReG(&,x)exp[(e/f-i)t] of the flow created at the field point E(E,n, <0) by a moving source of pulsating strength Re exp[(e/f-i)t] located at point x(x,y,2<0). In the limiting case, z = 0, the source at point x evidently is no longer fully submerged, so that this physical interpretation of the Green function becomes ambiguous. A complementary physical interpretation for this limiting case is that the pulsating flow created at point x(x,y,2Z=0) stems from a flux across the plane z = O of the mean sea surface. In the limit z = 0, the Green >> function G(&,x) must then satisfy the equations V2 = 0 inz< 0 (3.2a) [3,-(£-4F8 tic) “6 = §€=)SG@=9) oa = = 0 (3.2b) as may be seen from the sea-surface boundary condition (2.5). Equations (3.2a and b), justified above on physical grounds, can be justified mathematically in the man- ae for the particular problems of wave radiation and diffrac-— ner shown in Noblesse tion at zero mean forward speed (F=0) and of steady flow about a ship advancing at constant speed in calm water (f=0). The Green function G(E,x) actually is a function of the four variables &-x, N-y, Ctz, and (ae and thus is invariant under the substitutions € +> - x, n+- y, 6+*z. By performing these changes of variables in Equations (3.la and b) and (3.2a and b), it then may be seen that the Green function G(E,x) also satisfies the following equations 2 Toe = OCLa NGO Gewte a 210 (nse) for € < 0 [9 -(£+iF3 tic) “Jc 20240 (3.3b) 2 VWGse Otma2a< @ (3.4a) for C= 0 [9 -(£+iFd tie)"]G = -6(x-£)6(y-n) on z = 0 (Guan) where V is the differential operator (200 p08) Equations (3.3a and b) and (3.4a and b) will be used in the next section for obtaining integral identities satisfied by the velocity potential. A well known expression for the Green function, in terms of a double integral, can be obtained by using a double Fourier transformation of Equations (3.la and b) with respect to the horizontal coordinates € and n. This Fourier representation of the Green function is given by ime) = Ke AGay MeO? + IGG oe ss Die pone 2a | i { gp ple Ur) ilo- Dug] G5) i (uw+v") - (f£-Futie) The "Cartesian Fourier integral representation" (3.5) can also be expressed in the form of a "polar Fourier representation" by performing the change of variables H = A cos® and v = A sinO, which express the Cartesian Fourier variable i and v in terms of the polar variables X and 8. These equivalent double-integral representa- tions were first obtained by Hae letndag and Bream,” ead later by Eemeolka,~° Strezensil,- Raaciae,~— Herallac, and WWehhemecn,“— and are therefore well known. 10 More recently, one-fold integral representations (involving the exponential integral in the integrand) have been obtained and used by Inglis and meee" and Guevel and Boieae.” These single-integral representations are modifications of the double-integral Fourier representation in terms of the polar coordinates (i,@). Single-integral representations associated with the Cartesian Fourier representation (3.5), in the manner shown in Reference 12 for the particular problem of ship wave resistance (f=0, F#0), have not been obtained to the author's knowledge. However, such single-integral representations are considerably more complex than the corre- sponding integral representations for the ship wave resistance problem and the series representations obtained in Reference 13 for the particular case of wave radiation and diffraction at zero forward speed (F=0, £#0). For the practical purpose of numerically evaluating the velocity potential defined by a surface (or line) distri- bution of singularities (sources or dipoles) with known strength, it may actually be preferable to use a double-integral Fourier representation, such as that given by Equation (3.5), together with an interchange in the order of integration between the Fourier variables (u,v) and the space variables (x,y,z), as is shown in Section 6. il 4. FUNDAMENTAL INTEGRAL IDENTITIES In this section, basic integral identities for the velocity potential are obtained by applying a classical Green identity to the potential ¢ = (x) and the > previously defined Green function G = G(E,x). The Green identity is { cv?e-cv?o av = [ @2c/02-c09/22) axay d~ oO + | ccoe/n-gac/anyaa + | (¢dG/dn-Gd/dn)da (4.1) h h (oe) where d~ is the finite domain bounded by the ship hull surface h, the mean sea plane z = 0, and some arbitrary, but sufficiently large, exterior surface h,, surrounding the ship surface h, as is shown in Figure 1; furthermore, o is the portion of the plane z = 0 between the intersection curves c and €,, of the plane z — 0) with the ship surface h and the exterior surface h,, respectively. On the surfaces h and h, we have 0¢/dn = Voen and 0G/dn = VGen where n is the unit outward normal vector to h or h,, as is shown in Figure 1. Finally, dv and da represent the differential elements of volume and area at the integration point x of the domain d~ and the sur- faces h or h,, respectively, and dxdy is the differential element of area of the mean sea surface O°. Let the integrand $dG/dz - Gdd/dz of the sea-surface integral in Equation (4.1) be expressed in the form $[9,-(E+iFo tie) “Ic - G[3,-(£-1F9 tie) “14 + 2iF (f+ic)3d(Go)/dox + F73 (G3/9x-$9G/ dx) /dx. Furthermore, we may use the relation 12 Figure 1 - Definition Sketch [ [2i(£+ie) 9 (Ch) /dx+Fd (G9$/3x-$9G/ 9x) /3x]dxdy , oO = f [2i (f+ic) GO+F (GI$/3x-$9G/ 9x) Jdy c af J [2i(f+ie) Gd+F (GI$/Idx-h3G/ dx) Jdy (4.2) c co where the curves c and c,, are oriented clockwise and counterclockwise, respectively, as is shown in Figure 1. The Green identity (4.1) can then be expressed in the form il3) i inne = | $[9,-(£+iFd +ie)”]Gdxdy Gl Om s a { aia < i G[8,-(£-iF)_ tic)” ]daxdy d fe} + { (Gd$/3n-$96/3n) da h i 2H [24 (E+ie) Gh+F (63/3x-99C/ 3x) Idy + 1, (4.3) Cc where the term I, is given by the integrals iL = | (¢dG/dn-Gdo/dn)da + | [2i(f£+ie) GO+F (Gd>/dx-hdG/dx) ]dy h c co (oo) We have G = O(1/r) and » = O(1/r) as r = (6242-472) 1/2 = ©, so that the term I, vanishes as the large surrounding surface h, is made ever larger. The term I, can then be ignored if the finite domain d~ and the finite region 0° of the mean sea plane are replaced by the unbounded mean flow domain d and the unbounded mean sea surface Oo outside the mean hull surface h and its intersection curve c with the plane z = 0; nespectivelly.: By expressing the potential » in the integrands of the two integrals on the left side of Equation (4.3) in the form > = $, + ($-$,), where > = $ (xx) as was defined previously, and >, represents the potential at the field point E, iLoGoog Wo, = o(é), we may obtain i oV"cdv - i $(3,-(EHiF8 +ie)"]Gdxdy = Ch, + C7 (4.4) d 14 where C and C° are defined as 2 Ge | ein = i [9 ,-(E+iF2 tie)” ]Gdxdy d oO ($-,)V-Gav | ($-,) [3,-(£+iF3 tie)” ]Gaxay (4.5) Oo It may be seen from Equations (3.3) and (3.4) that we have C~ = 0 if o - Oo = $ (x) > > > - 0(§&) > 0 as x > €, that is if the potential is continuous everywhere in the solu- tion domain d and on its boundary 0 + h+c, as is assumed here. Use of Equation (4.4), with C” = 0, in Equation (4.3) then yields Ch, | ai-éie < \ G[d,-(£-4F9 tie)” ]ddxdy oO + { (Gdo/dn-bdG/dn)da h + c F | [24 (E+ic)Gd+F(G9$/3x-$9G/9x) Idy (ay > Let BGs ie) represent the unit vector tangent to the curve c oriented in the clockwise direction, as is shown in Figure 1. On the mean waterline c, we have dy = t d&, where d& is the differential element of arc length of c. Furthermore, have 0o/dx = Voed, where i (1,0,0) is the unit positive vector along the x axis. then yields 36/3x = [ndd/In+tdo/d2+(nxt)do/dd]*a =n o0/on + t,26/82 - nt ad/ad, we This > where (n, Sil il ») are the components of the unit Pees normal vector n to sate hull surface h, 3/32 is the derivative of » in the direction of the tangent paaaes t to c, and 36/dd is the derivative of » in the direction of the unit vector a x t, which is tangent to h and pointing downwards. NS) Equation (4.6) can then be expressed in the form co(E) = we) - 134) (4.7) . . re where C is given by Equation (4.5), W(&) is the potential defined as ve) = { cv? ov x { G[9,-(£-4F) tic)" ]ddxdy (0) +{ Gad/onda ro | Gn, t ,9¢/ dnd (4.8) h c = and L(&;?) is the linear transform of > defined as L(E;4) = | odG/dnda - 2i(f+ie)F { Oe h c A A ( [98G/9x-G(t ,96/d2-n t.99/dd) 1 ds (4.9) c Use of Equations (3.3a and b) and (3.4a and b) in expression (4.5) for C then shows that we have C = 1 if the field point é is strictly outside the hull surface h, ind or on 0, whereas we have C = O if id is strictly inside the ship surface h. It can also be seen from Equations (3.3) and (3.4) that we have C = 1/2 if the point g is exactly on the hull surface h or on its intersection c with the plane z = 0, at least for points ze where the hull h + c is smooth; more generally, the value of 4mC (or 27C) at a point e of h (or c) is equal to the angle at which d (or 0) is = viewed from the point €. We thus have 16 o(@) im Gs @ oho @ (4.10a) 0 = vé) = L(E;4) for é in d, + Ca h-ec (4.10b) (E)/2 exactly on h+c (4.10c) where d, and oO, represent the domain and the portion of the plane z = 0, respec-— tively, strictly inside the ship surface h, as is shown in Figure l. The value of the constant C on the left side of Equation (4.7) is discontinuous across the ship hull surface h; C being equal to 1 outside h and to O inside, as is explicitly indicated in Equations (4.10a, b, and c). This discontinuity in the value of C evidently is accompanied by a corresponding discontinuity on the right side of Equation (4.7). Specifically, the latter discontinuity stems from the dipole- distribution integrals Je ¢0G/dnda and iL OPIEy onset in the potential L(E34) > defined by Equation (4.9). An identity valid for any point €--outside, inside, or exactly on the ship surface h--can be obtained by eliminating the discontinuity in the value of C in Equation (4.7). This can be done by adding the term C.o, on both sides of Equation (4.7), with Cc. given by Ca { VY Gdv = | [3,-(EHiF9 tie)? ]¢dxdy (aD) d 4 O. a at { V-Gav =| Be/aadady +f 9G/3nda d : OF h i i Wy yields CG. € (f+iF) tie) “Gdxdy + | dG/dnda aL OF h i Furthermore, by using the relation j [DACHARE) DC PaO ay 6 = I [24 (E+ie)6-F9G/8x]t al On Cc 1 we may obtain the following alternative expression for C,: C.= (EHie)- | Gdxdy - 2i(erieyr [ Gt ag O. c 1 + 7 ey ore el + I dG/dnda @ h (4.12) By adding the term C.o,, on the left and right sides of Equation (4.7), with C, given by Equation (4.11) on the left side and Equation (4.12) on the right side, we may obtain [1-w(E) ]o(E) = w(E) - LCE) where w(é) is the "waterplane integral" defined as nO) = (ey? i aCe, dbutly On 1 18 (4.13) (4.14) 2 = . . and L (€3$) is the linear transform of @ defined as L* (E34) -{ ($-$,)9G/8nda ~ 2i(ftie)F | G($-9,) tae h (e 2 +F \ [(O-$,,) 9G/dx-G(t 99/d8-n t 3/3d) Jt aL (4.15) c in which we have $ = (x) and 9, = (E) as was defined previously. In obtaining Equation (4.13), the relation C + C. = 1 was used. This relation can be obtained by using Equations (3.3a and b) and (3.4a and b) in Equations (4.5) and (4.11), which yield cic, = f V-cdv - { [9,-(£+iF9 +ie)”]caxdy z<0 =0 Identity (4.13) is valid for any point z, whether outside, inside, or exactly on the ship surface h. This identity thus is essentially equivalent to the set of the three classical identities (4.10a, b, and c), which are exclusively valid for e outside, inside, and on the hull surface h, respectively. Identities (4.10a, b, and c) and (4.13) correspond to the case of an open hull surface piercing the sea surface. For a closed, fully submerged surface h, the waterplane integral w(é) defined by Equation (4.14) and the integrals around the mean waterline c in Equation (4.8) for the potential ve) and in Equations (4.9) and (4.15) for the potentials L(é3) and L’ (E39) are evidently not present. Two other important particular cases of identities (4.10a, b, and c) and (4.13) are obtained in the limiting cases when the Froude number F vanishes, corresponding to wave radiation and diffraction by a body with zero mean forward speed, and when the frequency parameter f vanishes, corresponding to steady flow about a ship advancing in calm water. 1g 5. INTEGRAL EQUATION AND RELATED ITERATIVE APPROXIMATIONS Identities (4.10a, b, and c) and (4.13) hold for any function ? continuous in the domain d and on its boundary 60 +h+ec. If the function ¢ is taken as the veloc- ity potential of flow about a ship advancing at constant mean speed in a regular sea, then the normal derivative 36/dn of $¢ is given on the hull surface h, and ¢ satisfies the Laplace Equation (2.7) in the mean flow domain d and the sea-surface boundary condition (2.8), with p = 0 = q, at the mean sea surface 06. Identities (4.10c) and (4.13) then yield integro-differential equations for determining the potential $ on the ship surface. Specifically, Equation (4.13) becomes [1-w(E)]6() = ve) - L7(E34) (5.1) > where the waterplane integral w(&) is given by w(é) = (ftie)? { G(E, x) dxdy (5.2) O.: 1 the potential v(é) takes the form ve) = { Gdd/dnda + Fe { sa, tyae/dna2 (5.3) h c 7 $5 . ° and the potential L (&;$) is given by L* (E39) = [ ($-,)0G/anda - 2i(ftie)F | G(d-$,) td h c 2 +F [ ((0-0,26/2x-6(e,,26/0-n, ¢ 36/94) ]t a8 (So4) Cc 20 The potential vé) is known since 0$/dn is given on the ship surface h. The poten- jeieul Ea(Gs0). on the other hand, is evidently not known. An approximate solution of the integro-differential Equation (5.1) may be obtained by seeking a solution of Equation (5.1) of the form (E) = k(E)W(E), where the function k(é) = $(&)/W) is assumed to be slowly varying. Specifically, by adding the term k(E)L7 3) to both sides of Equation (5.1) and multiplying the resulting equation by v(é), we may obtain 6(E){[1-w(E) WE) 44 E3v)} = v2) + o@)u7 E30) - wu 34) (5.5) If the potential > were actually proportional to the potential ~, the term > 2. y o(E)L (E3p) - W@)L (E34) would vanish, and the modified integro-differential Equa- tion (5.5) would yield the solution o(E) = v7@)/{l1-w® we + 17 E;w} (5.6) More generally, the above expression for the potential can be regarded as the (n) first approximation in the sequence of iterative approximations 9 associated with the recurrence relation gD) = vo™ E/tt1-wEy16™ E + 1°E39™)} for n> 0 (5.7) + > ; and the initial (zeroth) approximation 6 é) = W(&). An approach to the numerical evaluation of these iterative approximations is presented in the following section. Pale 6. NUMERICAL EVALUATION OF ITERATIVE APPROXIMATIONS Equation (3.5) for the Green function may be written in the form 4nc(é,x) = -(1/r-1/r*) - (1/m) | dy { duE(U,VsE)E(u,v3x)/D(u,v) (6.1) Eh > where r, r-, E(U,V;x), E(u,v3&), and D(u,v) are defined as r= [(x+6)74(y-n)24(2-0)2y4/? (ua) r= [(x-8)74(y-n) *H(ate)?y1/? (6.1b) BUR) = Seale Ge)! iG (6.1c) E(u,v38) = explcqtev?) 2-4 (euanv)] (6.14) DG, Vie) Gusev)! 2-(EoBuie) 2 (12) The potential (€) defined by Equation (5.3) can then be expressed in the form ve) = ve) + wR) (6.2) where the potentials y and wR correspond to the singular terms 1/r - 1/r~ and the regular term defined by the double integral, respectively, in Equation (6.1) for 3 4 : the Green function. Specifically, the potential ) is given by the hull-surface integral 22 Si 2 w (&) = -(/4T) | (1/r-1/r~)d$/dnda h and the potential yk may be expressed in the form we@) = -(/4n?) | ae | duE(u, v3) A(u, Vv) /D(u, v) where A(u,v) is defined as Ai, Vv) = \ E(u, v;x)0/dnda + Fo | B(U,V3x)n,t 39/andL h c The waterplane integral w(é) defined by Equation (5.2) takes the form a2) = ss Js vfs ay EQS) { exp [i(xutyv) ]dxdy On 1 (6.2a) (6.2b) (6.2c) (6.3) The potential L*(é:¥) defined by Equation (5.4) can then be expressed in the form L (Es) = LZ(E3v) + LZ Esv) where the potentials Le and LR are defined as Zo (6.4) 1g (Es) = -(1/4m) { Lop?4yy- yf) 19 (1 /r-1/27) /anda (Gua) h (oe) co gw = ayer [| av | duE(u,v3€)A“(u,v)/D(u,v) (6.4b) — 0 co with A“ (u,v) given by A“ (u,v) i L(?+0)- yuk) Jon, vs) /dnda h 2icerieye | (YEW DE Cu, vs) ta c + al (PW, 8B/dx-[e 2h /d2—n, ta (PHY) /2d]E}t at (6.4c) c The first iterative approximation o) can then be determined by using Equa- tions (6.2), (6.3), and (6.4) in Equation (5.6). The potential weno 5. a > ib, defined by Equation (5.7), for the second and subsequent iterative approximations 2) Gem), can be expressed in a form almost identical to that given above by Equations (6.4) and (6.4a, b, and c) for the potential L“(E;). The basic computational task common to Equations (6.2b and c), (6.3), and Ss (6.4b and c) consists in evaluating a double Fourier integral, I(&) say, of the form 1(é) = [ fe duexp [ eQue +V 5) ie -i(€u+my) INGu,v)/DG@, v) (6.5) 24 where D(u,v) is given by Equation (6.le), and N(u,v) is defined by a surface or a line integral of the type Hi N(u,v) = [ esplaGi a)! eae) ING@ dad a” (6. 5a) h,o,,¢ If a small positive number is used for the parameter €, the function D(u,v) has no real root, and the integral (6.5) can be evaluated without difficulty in principle. 25 0 suktiun a yd beat at ean in of ” tush Dale nt we! ‘ ; i New , OR, ae A “a woo ; 7 a, elie mT Ph Pie: a ua; Wi? rilaney J hi oh ihe ve we a City #9 eerie HO hiche ‘i Hay 4b. eheib anal twenaaiss® 1 ale f (fe a pieae ane we tot dons ans “0 s92ametey add 102 how at dm: ay . ty: ms ies Bt gifuoteeeh, tyadate besa! ove ‘od ey Re ay tendssan a ha Kay) etven by , AG) } (Py CO AR RRR, "i ALCL e Let Vimde dh a a heecomnaian ae ere Sane ares I Te ree ha j LCe a vaeybar eae (Sirm, Fete te i i us ] \ Fa the fir ar bbe hein aopros atten. 9 Mite ee eX Chen, ba. met sam Kile ia 1) 4 , i be ate Dh ot Matra a NI. ' wine, “CB AD See Beak hele CSch) ee po nentdont Me ‘a ‘datined by Eawatton oaehd Post Relves chp: eee sha ar ae Tiewrat ay a ce D) ao hat eacpee vot UE, a Lore RE ORRIN Raa, Neat ORR nr: Pai ie a : Sy re rr toe % bers Sao mine git hs S ‘ ¥ ? au rekiag) tee scomaenae! Kae RRS. dente RES 8 MRE A) thy ws y a LET ATi a Bagh ne Hey diy, (ORR MS, 5 ney Ba He he wade cComgulet j y Pas h b A by ; e [xen Nere { R a Nes ; h 4 oN EY ie ANF duce LEA \* PS ah A “Cy ka Git A Paige AP era roo Wo PARMA ade at ean yt * REFERENCES 1. Newman, J.N., "The Theory of Ship Motions,'"' Advances in Applied Mechanics, Vol. 18, pp. 221-283 (1978). 2. Chang, M.S., “Computations of Three-Dimensional Ship Motions with Forward Speed,'' Proc. Second Intl. Conf. on Numerical Ship Hydrodynamics, Berkeley, pp. 124- 1335) (L@7/7))c 3. Guevel, P. and J. Bougis, "Ship Motions with Forward Speed in Infinite Depth," International Shipbuilding Progress, Vol. 29, No. 332, pp. 103-117 (1982). 4. Inglis, R.B. and W.G. Price, "The Hydrodynamic Coefficients of an Ellipsoid Moving in a Free Surface," Journal of Hydronautics, Vol. 14, No. 4, pp. 105-110 (1980). 5. Inglis, R.B. and W.G. Price, "The Influence of Speed Dependent Boundary Conditions in Three-Dimensional Ship Motion Problems," International Shipbuilding Progress, Vol. 28), Now 31 8,mpp.) 22=29) (Gigs) 6. Inglis, R.B. and W.G. Price, "A Three-Dimensional Ship Motion Theory--The Hydrodynamic Coefficients with Forward Speed," Trans. RINA, Vol. 124, pp. 141-157 (1982). 7. Inglis, R.B. and W.G. Price, "A Three-Dimensional Ship Motion Theory: Calculation of Wave Loading and Responses with Forward Speed," Trans. RINA, Vol. 124, pp. 183-192 (1982). 8. Proceedings of the Workshop on Ship Wave-Resistance Computations, Vols. 1 and 2, Edited by K.J. Bai and J.H. McCarthy Jr., David W. Taylor Naval Ship Research and Development Center (13-14 Nov 1979). 9. Noblesse, F., "Integral Identities of Potential Theory of Radiation and Diffraction of Regular Water Waves by a Body," Journal of Engineering Mathematics (in press). 10. Noblesse, F., "A Slender-Ship Theory of Wave Resistance," Journal of Ship Research, Vol. 27, No. 1, pp. 13-33 (1983). 11. Noblésse, F: and G. Triantafyllou, "Explicit Approximations for Calculating Potential Flow about a Body," Journal of Ship Research, Vol. 27, No. 1, pp. 1-12, (1983). 27 12. Noblesse, F., "Alternative Integral Representations for the Green Function of the Theory of Ship Wave Resistance," Journal of Engineering Mathematics, Vol. 15, pp. 241-265 (1981). 13. Noblesse, F., 'The Green Function in the Theory of Radiation and Diffraction of Regular Water Waves by a Body," Journal of Engineering Mathematics, Vol. 16, pp. 137-169 (1982). 14. Stoker, J.J., "Water Waves," Interscience Publishers, New York, 567 p. (1957). 15. Lighthill, M.J., "On Waves Generated in Dispersive Systems by Travelling Forcing Effects, with Applications to the Dynamics of Rotating Fluids," Journal of Fluid Mechanics, Vol. 27, pp. 725-752 (1967). 16. Haskind, M.D., ''The Hydrodynamic Theory of Ship Oscillations in Rolling and Pitching," Prikl. Mat. Mekh., Vol. 10, pp. 33-66 (1946); English Translation: Tech. Res. Bull. No. 1-12, pp. 3-43, Soc. Nav. Archit. Mar. Eng., New York (1953). 17. Brard,R., "Introduction a l'étude théorique du tangage en marche," Bull. Assoc. Tech. Marit. Aeronaut., Vol. 47, pp. 455-479 (1948). 18. Hanaoka, T., "Theoretical Investigation Concerning Ship Motion in Regular Waves,'' Proc. Symposium on the Behaviour of Ships in a Seaway, Wageningen, The Netherlands, pp. 266-283 (1957). 19. Stretenski, L.N., "The Motion of a Vibrator under the Surface of a Fluid," Trudy Moskov. Mat. Obshch., Vol. 3, pp. 3-14 (1954). 20. Eggers, K., "Uber das Wellenbild einer Pulsierenden Storung in Translation," Schiff u. Hafen, Vol. 9, pp. 874-878 (1957). 21. Havelock, T.H., "The Effect of Speed of Advance upon the Damping of Heave andePitehs! “Eranse inst. Naval Arch. Voll. 00s. ppl si —lo >i Glos) 22. Wehausen, J.V. and E.V. Laitone, "Surface Waves," Encyclopedia of Physics, Springer-Verlag, Berlin, Vol. IX, pp. 446-778 (1960). 230 ineldis. Reba and WaiGe) Prices. .CallculationyoL thes Velocity esPotentialotga Translating, Pulsating Source," Trans. Royal Institution of Naval Architects, England (1980). 28 Copies 13 eZ INITIAL DISTRIBUTION CHONR/ 438 Lee NRL 1 Code 2027 1 Code 2627 USNA ekeechwrrb 1 Nav Sys Eng Dept 1 Bhattacheryya 1 Calisal N.VPSGSCOL i! Library, 1 Garrison NCSC 1 Library 1 Higdon NADC NCI.L/Code 131 NAV 3EA SEA 031, R. Johnson SEA 031, G. Kerr SEA 031, C. Kennel SEA O5R, L. Benen SEA O5R, Dilts O5R, N. Kobitz 5EA O5R, J. Schuler {EA 312, P.A. Gale SEA 312, J.W. Kehoe StA 55W, E.N. Comstock SEA 55W, R.G. Keane, Jr. SEA 61433, F. 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