THE U N I V E R S I T Y OF M I C H I G A N COLLEGE OF ENGINEERING Department of Naval Architecture and Marine Engineering RESEARCH IN RESISTANCE AND PROPULSION Part II. Streamline Calculations for Singularities Distributed on the Longitudinal Cenverplane Tetsuo Takahei Finn C. Michelsen Hun Chol Kim Nils Salvesen Project Director: R. Bh. Couch ORA Project 04542 under contract with: MARITIME ADMINISTRATION U. S. DEPARTMENT OF COMMERCE CONTRACT NO. MA-2564, TASK 1 WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR February 1963

TABLE OF CONTENTS Page ABSTRACT..................................................... i.ii INTRODUCTIONo.................................................. 1 Ao The Velocity Potential and Velocity Components due to a given Singularity Distribution......................... 6 i) Triangular Draftwise Distribution......... o o,,,,,,, 12 ii) Parabolic Draftwise Distribution................... 12 B. An Approximate Singularity Distribution for Series-60 Water Lines..................................... 17 BIBLIOGRAPHY................ 20 APPENDIX I Computer Program for the Determination of Streamlines Due to a Prescribed Singularity Distribution on the Center Plane....,.............................. 21 A, Cosine Hull.................................... 33 B. Polynomial Hull.o o.........o................... 41 APPENDIX II Hull Forms Obtained from Stream Line Traces..... oo 45 APPENDIX III Effects of Depths of Singularity Distributions on Traced Hulls (Series-60, Polynomial Distribution)~, 59 ii

ABSTRACT A computer program was written for the problem of determining the hull form produced by a given singularity distribution defined on the longitudinal center plane. The program is based on the Runge-.Kutta fourth order method in solving the differential equations of the stream lines. Results indicate the number of subdivisions required.to meet a specified degree of accuracy. The program admits rather general forms of the singularity distribution functions. A computer program was also written for the purpose of determining the two-dimensional singularity distribution for a given shape of the waterlineo Applied to a Series-60 set of lines the three-dimensional effect of truncation has been investigated and found to be substantial, iii

Introduction The linear wave-resistance theory, developed by Michell(l) and Havelock, (2) has until recently found very little direct application in the design of ships. Although it was demonstrated many years ago by Wigley(3) and Weinblum(4) among others that the theory would accurately predict the effect on the wave-resistance of small changes in ship form, and also the location of humps and hollows of the wave-resistance curve, it was soon realized that it would not predict the magnitude of the resistance with sufficient accuracy. Introduction of corrections for viscous effects were helpful but not sufficient, mainly because they did not remove the limitations imposed on the theory by the linearized boundary conditions. The boundaries of concern are the free surface of the sea and the surface of the ship. In regard to the free surface the effect of linearization is probably not too great since it can be shown(5) that the linear velocity potential is exact to within a second order term. The hull surface is a different matter, however. When Michell formulated his wave-resistance theory in 1898 he made the assumption that approximate conditions on the hull surface were to be satisfied on the longitudinal plane of symmetry of the ship. The theory proposed was therefore exact only for a ship of zero beam. For geosim ship forms of finite beam it predicted the wave-resistance to be proportional to the square of the beam. When Havelock introduced the concept of representing ship forms by means of a distribution of singularities over the hull surface he did, at least in principle, provide us with an integral expression for the -1 -

-2 -velocity potential that would, for a given form,satisfy the hull boundary conditions exactly. The evaluation of these integrals, being defined on a rather complex surface, were impractical, if not impossible to solve. By locating the singularities on the longitudinal plane of symmetry and assuming that the strength of the source distribution was proportional to the product of the velocity of the ship and the slope of the hull surface with respect to the direction of motion, Havelock arrived at an expression that was shown to be identical to that of Michello It appeared therefore, that he had not, in reality, been able to remove any of the limitations of the linearized boundary conditions. For many years one very important feature of Havelock's expression for the velocity potential seems to have been overlooked until Inui(6) discussed and made full use of it in much of his extremely important work on wave-resistance. Fully realizing that a source distribution on the center plane did not represent a form as assumed by Havelock, he proceeded to trace the closing streamlines for a given distribution assuming the free surface to be rigid. This assumption can be shown to be not too serious. The resulting hull forms differed considerably from those of Havelock's theory, especially in regard to the bottom configuration. The waterlines near the free surface were also appreciably different in particular towards the ends. It is important to recognize that the hull forms produced from the streamline traces satisfy the hull boundary conditions to a high order of approximation. The relationship between singularity distribution and hull form is one to one, i.eo for a given distribution there exists a unique closing

-3 -stream surface. In the past it has been customary to determine the singularity distribution from the hull form, whereas, in tracing the streamlines, we are determining the hull form from a given singularity distribution. We shall therefore refer the latter procedure as the inverse method. The problem of determining the source distribution on the center plane for a given hull form,that would satisfy the hull boundary conditions to the same order of approximation as that obtained by the inverse method,is extremely difficult. It may indeed be impossible to solve because of the beam draft ratios normally used for conventional ship forms. The conditions to be met are further complicated by the presence of flat bottoms. Since the part of the hull close to the free surface gives rise to the greater portion of the wave-making resistance there is general belief, however, that the boundary conditions being violated near the bottom of a ship is not too serious. Experimental results 78) have verified that this is indeed so. It has been brought to our attention that Dr. Pien of DoT.MoB. is currently working on a method by which he is locating the singularity distribution on four vertical planes forming a rhombus in the plane view. In this way he is apparently able to represent more coventional forms. The publication of his results is expected to bring to light a great many important findings. Other avenues of approach may also be available. Inui has,for instance,proposed an additional line sink source distribution along the keel lineo Although, as stated above, the inverse method cannot be expected to represent conventional ship forms when applied to a center plane

distribution of singularities, it does provide us with an accurate tool by which we can investigate many wave-making resistance phenomena. Theoretical calculations coupled with experimental results will also make it possible to investigate effects of viscosity and eddies in a rational manner. This must be kept in mind when considering the great amount of work going into the study of hull forms far removed from those of final interest, namely ship form of more conventional dimensions. In the world only very few forms have so far been traced. Most of these are shown in the Appendix II. There are two good reasons for this, Firstly, the numerical work of the streamline calculations are lengthy and very involved. Only the very high speed,large computers can bring the cost of calculations down to a reasonable figure. Until the University of Michigan started to trace streamlines using an IBM 7090 digital computer, similar work had, as far as we know, previously been performed on much smaller computers only in Germany and Japan. To emphasize the importance of high speed computers it should be pointed out that the determination of the streamlines for the so-called cosine hull form took two hours in Japan, whereas the same calculations can be performed in three minutes on the IBM 7090. In addition to reducing the required computing time a greater accuracy of results has also been obtainedo Figure 1 shows the maximum beam of traced hull form as a function of the number of station used in the calculations. On the basis of this figure we are now using 100 stations (for L/2) for our stream line calculationso The details of the computer program is given in the Appendix Io Secondly, at present, the calculation of the wave-making resistance from the linear theory is complex even for simple singularity

EFFECT OF NO. OF STATIONS FOR USING RUNGE - KUTTA STREAMLINE TRACE ROUTINE dy v (x,yz) Ox.14 -m( ) - a cos (-) O o UM Data Max Half Beam cJ 113 ImL X Japanese Data. 2 (Estimated). 3%.12 0 20 40 60 80 100 120 140 160 180 200 NO. OF STATIONS FOR RUNGE-KUTTA ROUTINE Figure 1

-6 -distributions. Much work is being done on the simplifications of these calculations and we hope results will be forthcoming soon, In the meantime fundamental research on the simple hull forms should be continued. The fact that such research is extremely fruitful is readily shown by the experimentally verified concept of wave interference effects of large bulbs. In developing the waveless hull form from theory Inui has lead the way to a new approach to the study of ship resistance characteristics. The horse has been put before the carriage so that theoretical predictions are made before they are experimentally verified, thus making it possible to perform hull form studies in a rational manner. In the following sections we are reporting on research dealing with the determination of streamlines of hull forms for simple singularity distributions on the center plane completed at the U of M to date. Much of this is still only in the form of equations which remains to be computed. The computer program has been run for the cosine hull forms, however, and is checked against results of calculations made in Japan. The existing program will admit very general singularity distributions and additional forms wil.l be de-ermiur,_d as they become part of our overall study of wave-making resistance characteristic of ship hulls. A, The Velocity Potential and Velocity Components due to a given Singularity Distribution. For a given hull form the linear wave-resistance theory assumes that the hull can be represented by a distribution of singularities. This distribution can be divided into three parts. (i) A distribution of sources in the plane of symmetry of the ship of strength given by the slope of the hull surface.

-7 -(ii) An image system, with respect to the static free surface, of source distribution as given by (i). (iii) A distribution of singularities distributed above the freesurface to satisfy the free-surface conditions. The systems (i) and (i-ii) combined represent the double model of the given hull form moving in an infinite fluid region. The assumption of the linear theory that the source strength at a point in the plane of symmetry of the ship is given by the slope of hull form at that point is only exact in the limit as the beam approaches zero. For a hypothetical wall sided model of infinite draft the problem of determining the closed stream lines of the source distribution representing the hull, neglecting the free surface effects, is reduced to a two-dimensional case. It can be solved either by the method of conformal mapping or by solving a first degree integral equation. (6) In the case of finite draft, i.e., a three dimensional hull form, the'direct problem of obtaining the singularity distribution which represents a given hull form becomes extremely difficult even if the free surface effects are neglected. Its solution involves integral equations of the second order. No simple method of correcting for the finite beam exists. In an attempt to satisfy the hull boundary conditions exactly, Inui(6) and Eggers(9) considered therefore the inverse problem, namely, the determination of the hull form produced by a given singularity distribution. The complete inverse problem, which includes the contribution from the singularity system (iii), becomes a very difficult one to solve.

-8 -The secondary effect of system (iii) is relatively small at small Froude numbers, however, and only the singularity distribution representing a double model moving at constant speed in an infinite fluid needs to be considered. Let the singularity distribution to be of the simple form m(S,X) = fl() ~ f2(W) (1) where m(S, ) is the strength of a distributed source (flow per unit area per unit time) divided by the ship's speed V. Because a double model has to be considered' the region of source distribution is given by - 1 < < 1< - t < ~ < t (2) where X' ZT L 2L the length of the ship being equal to 2L and the draft of the ship atbow and stern, TF = tL/2. Since it is required that the hull form -be closed, the total source strength must be equal to zero, thus t 1 Jm(~,) d~ dd = 0 -t -1 Due to symmetry with respect to the LoW.L. it follows that For fore-and-aft symmetry,

-9 -For a source distribution of this type, the location of stagnation points at bow and stern if and ~a become: f = 1 + E a = -1- C Ship length is in reality 2 + 2e, but E is normally small and is neglected. Drafts at bow and stern becomes t + &' by similar analogy, but E' can also be neglected. The velocity potential due to a source distribution m(~, ) in a uniform flow is t 1 O(x"y.z) = - 1. j mm(~, )d~ d x -t -1 $T(x - )2 + y2 + (z - 2 - x and by definition the velocity components become +t 1 -t -1 +t 1 v = + 1 m(5,() dY d d -t -1 +t 1 az 47 R3 It -1 where R2 = (x - ~)2 + y2 + (z - )2 The velocity components can be substituted into the differential equation for the streamlines given by

-10 -dx dy dz U V W and a stream line may thus -be obtained by means of successive integration using the Runge-Kutta method. Four or five suitably chosen streamlines will normally define a hull form with sufficient accuracy. As for the initial values of the integral, u(l + e, O, 0) = 0 gives the stagnation point at the bow, The value of C is determined by equating u to zero. It is preferable to start at x = 1, and since the flow around the bow is essentially two-dimensional the initial value of y may be determined from the two-dimensional case as outlined later. The initial value of z is optional within the depth of the singularity distribution. Actually the streamlines converge closely toward the midship and a slight error in the starting value of y makes little difference in the tracing of the streamlines. When the singularity distribution is uniform draftwise, f2(~) = 1 (-t < < _ t) the velocity components become t 1 fl(i) (x - ) di dC R3 1 fl(O) (x - O) z- t z + td (x - ) 2 Y rt rt d -1 t 1 - v fl(~) y dS d~ R3 -t -1

-11 -r fl(O) Y z - t z + t (x ) 2+ y2 rt rt -1 t 1 - 4rw =3- R3 - 4irw -/ fl(j) (z - ~) dS d~ -t -1 1 - f1(i) ( 1 ) dS where rt = [(x- )2 + y2 + (z t)2 ]1/2 r-t = [(x_ )2 + ya+ (z + t)2 ] For two dimensional flow, the streamline is given by 1 Y = 2r, fl (f) tan-1 ( Y) dI (5) Zt L-2 I/ — x TF TM Figure 2. Uniform draftwise distribution.

-12 -For non-uniform draftwise rectangular source distribution the velocity components for two special cases become i. Triangular draftwise distribution m(S,) = ml(O). m2(W) where = m2(t) = 1 - 1 +1 u = - 1 + r4t - 2r d -t r2 ( +1 v ml() (rt + rt - 2ro) d (6) -1 +1 2 1 (z + ro) w 4,t J -1 1(W) ge (z - t + rt) (z + t + r_t) 2 2 2 2 where ro = (x - +) y + z 2 2 2 r =(x- _) + y r2= (x- )2 + y2 ii. Parabolic draftwise distribution m(t, ) = ml(O). m2(W) where m2( ) =1 - (t) +1 1-1 r2 +1 1 Z - t + rt + 4t2 m1(5)(x ) logez + t + r 4~rt t -lt + '-t +1 (7) v 12 ml() 2 [(z + t). rt - (z - t) rt] dS 4[t2 -l r

-13 -Figure 3. Triangular draftwise source distribution. Figur 4. Parabolic draftwise source distribut~I Figure 4. Parabolic draftwise source distribution.

-14 -+ 1 ml(i) y loge z - t + rt 4-tt -! z + t + rt +1 w =4t2 J 2 ml(t) (rt - r_t + z log z + t + r-t ) d~ 4t2 -l z - t + rt In an attempt to reduce the draft of the mathematical hull forms amidships it was decided to investigate the effect of a variation of depth of singularity distribution over the length of the hull. The distribution was again assumed to be of the simple form m(E;,) = fl(f) o f2(W) with fl(5) = aljii + a2 2 + a312I l + a44 + a5~4|I (8) and f2() = 1 - (C) By definition m(5,5) = 0 |1 > 1 o= i > t(~) In general h will also be a function of ~o For source distribution given in general by Equation 8, the velocity component can be expressed in the following form.

u = -1 + 1J 2 m() (x ) log t() + rt d 4 - -1 z + t() + rt +1 + m(( - [(2 -h2) (z - t) +4 h -1 h r 2 2 + - )(z + t) - (z - t) ]d rt r2 rt r v _1 + 1 y log z - t + rt dS 4 -1 h z + t + r-t +1 (9) + 1 1 m1(5)y [z2 - h2(z - t) -. mlr2 rt + z + t h2)(z + t) z - t ] d rt r2 rt r-t +1 w 4 h1 ml2 () [rt - r-t + z log z +t+t ] d -1 z - t + r t] +1 +1 J ml() [1 t) [ ] 4-1 h rt r_+

-16 -Three special cases of singularity distribution are as follows:,. - ~~fiJ Figure 5 Figure 6 c'Ccj h ' CONSTANT, Figure 7

-17 -A computer program for the determination of streamlines due to a singularity distribution as given by Equations (8) has been completed. Trial runs of the program did not produce usable streamline traces, however, since too small initial vlaues lead to difficulties caused by the logarithmic singularity of the integrands of Equations (9). Results will become available as this work is being continued. Singularity distributions other than those mentioned above will also be tried. The purpose of the complete streamline study is twofold: 1. To provides a rational basis on which to investigate effects of hull form changes and the effect of large appendages such as bulbs. 2. To determine singularity distributions which represent hull forms of conventional proportions. The development of singularity distributions must be closely related to our ability to evaluate the wave-resistance from theory. It is for this reason impractical at present to consider distributions that are not defined in a rectangular region on the centerline plane. An additional limitation is intorduced when it is assumed that the distribution function m(5, ) is given by Equation (1). This assumption puts a very serious restriction on the admissible form of m(, ). A distribution as shown in Figure 8, which may produce a flat bottomed hull cannot be represented by the simple relationship of Equation (1). Because of progress being made at the University of Michigan on the problem of wave(10) resistance evaluation from theory, we hope that we shall be able to consider the distribution, as shown in Figure 8, in the near future. B. An Approximate Singularity Distribution for Series-60 Water Lines. From the discussion above it is clear that with the restrictions imposed by Equation (1), it is impossible to find a particular distribution

-18 -function that will produce streamlines which correspond closely to a particular set of Series-60 lines. To arrive at an estimate of the size and Figure 8 location of optimum bow bulbs, however, it is important to know which distribution function might possibly produce a L.W.L approximately equal to that of the Series-60 hull form. It was therefore decided to try to obtain the best possible fit to the load waterlines of a Series-60 set of lines of block coefficient CB =.60 using a fifth degree polynomial for fl(5) of Equation (1). Appendix IV shows a computer program written for this purpose. Inui(6) has tabulated the integral of the velocity components for the two-dimensional case. Inui only considered symmetric distribution functions. It was believed sufficient to do so also for the Series-60 lines, the reason being that the shape of the fore-body would be of primary importance as far as bow bulbs were concerned. In regard to the draft-wise distribution of singularities only the uniform case, with the distribution terminated at various depths, have been evaluated

-19 -at the University of Michigan to date. As the depth of singularity distribution is decreased, so is, as a result, the beam of the LoWoL. If the LoWoL shape is to be preserved, the strength of the singularities of the two-dimensional case for that waterline configuration must therefore be increased as the depth of singularity distribution is reduced to a finite dimension, Based on results published by Inui(6) this increase in strength was estimated to be approximately equal to 20 per cent at the midship for the case of the Series-60 lines. Calculations revealed that this amount was not large enough. A number of depths of singularity distributions were then run on the computer to determine the relationship between depth and L.W.L. beam. The results are shown in Appendix III. It is noted that the variation of maximum beam with respect to depth of singularity distribution is rather large. This leads us to believe that early investigations(2) of the effect of draft on the wavemaking resistance resulted in erroneous conclusions. Such investigations were based on the Michell approximation of the hull boundary conditions and significant variations in beam could therefore not be avoided, as shown by the results of streamline calculations published in this report. It is also observed that the draft amidships is approximately twice that at the fore-foot. Since we do not as yet have a program which ~will produce a flat bottom streamlines it was of interest to determine experimentally the effect on wave-making resistance of removing of the major portion of the hull form below the depth of the singularity distributiono In general it can be said that the changes in the wave-making resistance were found to be small. Details of results will be presented in a seperate reporto(8)

BIBLIOGRAPHY 1. J. H. Michell, "The Wave Resistance of a Ship," Philosophical Magazine, Vol 45, (1898), pp. 106-122. 20 T. H. Havelock, "The Calculation of Wave Resistance," Proceedings of the Royal Society, London, England. Series A, Volo 144, p. 514, 1934. 3. W. C. S, Wigley, "Ship Wave Resistance - A Comparison of Mathematical Theory with Experimental Results," Transactions of the Institution of Naval Architects, vol 68 ppo 124-137, (1926) 4. Go PO Weinblum, "Untersuchungen uber den Wellenmiderstand Velliger Schiffefomen," Jahrbuck Schiffbau Gesellschaft, vol 35 pp. 164-192, (1934). 5. H. Lamb, Hydrodynamics, Sixth Edition, Dover Publications, N. Y., p. 417. 6. T. Inui, "A New Theory of Wave-Making Resistance Based on the Exact Conditions if the Surface of Ships," JSNA Japan vol 93, 1953. 7. T. Inui "Wave-Resistance of Ships," Transactions of Society of Naval Architects and Marine Engineering, New York, 1962. 8. Resistance Characteristics of Mathematical Hull Forms. University of Michigan ORA Report (To be Published) 9. K. Eggers and Wo Wetterling, "Uber die Ermittung der Schiffsahnlichen Umstromungskorper vargegebe Quell-Senken-Derteilungen mit Hilfe electronischer Rechenmashinen, Schiffstechnik Bd. 4 (1957),Hift 24. 10. F. CO Michelsen, Wave Resistance Solution of Michell's Integral for Polynomial Ship Forms, Microfilms, Inco Ann Arbor, Michigan, 1960. (Thesis) 11. P. Thomsen and K. Eggers, "Zur Berechnung von Umstromungskorpern zu Ouell - Lenken - Verseilungen" Schiffsticknik Bdo 7 -1960-Hift 38. 12. T. Takahei, "Calculations of Ship Hull Terms With Electronic Digital Computers" The University of Michigan, Department of Naval Architecture and Marine Engineering. -20 -

APPENDIX I COMPUTER PROGRAM FOR THE DETERMINATION OF STREAMLINES DUE TO A PRESCRIBED SINGULARITY DISTRIBUTION ON THE CENTER PLANE The program is written for the IBM 7090 digital computer in the language of MAD (Michigan Algorithm Decoder). It traces the streamlines, defined by the system of differential, equations dx dy dz U(X,Y,Z) V(xyz) w(xy, z) by means of the Runge-Kutta fourth order method. As mentioned in Section A of this report the starting point selected in tracing a particular streamline is taken at x = 1l0o The starting value of z is arbitrary as long as it is less or equal to the draft at the fore foot plus some small quantity 'o To insure that the straced streamline belong to the hull surface or is located in the region slightly outside the hull surface the starting value of y is estimated under the assumption that the flow near the bow is not very much different as that described by the two-dimensional case. For two dimensional flow the streamline is given by Equation (5) i.e. 1 y(x) fl(i) tan -1 di 2t -1 x - The y coordinate at x = 1 can be evaluated approximately by assuming that fl(t) = fl(l) = c; o < x < 1 Thus 1 1 y(l) = cot 11 d =V cot 1 1 di o Y oY -21 -

-22 -= tan -1 y + Y log (1 + - ) 2 T For y < < 1 it follows that (2! - 1) 2 log (-) c 2 from which y (1) e (c -1) As a starting value of y we therefore use y(1) > e-(c-1) 16 4 normally y(l) = 10- to 10 -It is not too serious if the initial value of y is slightly too large since the streamlines will converge toward amidships. On the other hand a too small initial value of y will lead to great difficulties, probably because the streamline is then inside the hull surface and therefore turns sharply normal to the center plane. Such a sharp turn can be described with the aid of the Runge-Kutta method only if a large number of subdivisions are being used. Whether the initial value of y actually used in the computations is satisfactory can most readily be judged from a plot of the velocity components such as shown in Figure I-lo Runge-Kutta step size must also be giveno Approximately 100 stations along the half-length of the hull seem to be satisfactory as explained earlier. In the present program, the Runge-Kutta step size is automatically refined until a predetermined accuracy is obtained.

G.10 G BD ~~~~~~~~~~~~~~D.15 H. 20 H (a) Traced streamlines on cosine hull C-201..0.2.4.6.8.9 1.0.05 (a) Traced streamlines on cosine hull C-201. Figure I-1

~~~~~~~~~~~~~~~~.7 \ t~~I.0 I.zR.6.3.4. 3 1 1 I 1.4 ~ 990.9925 -995.9975 1.0.990.9925.995.9975 1.0 BoW EtD DETAlLs 0.1.2,3.4.5.6.7.8.9 1.0 (b) Velocity components, streamline AB. Figure I-1 (Continued)

= Ic UC' +.7 \.5.2.4.I,,, 15.3,,, lo.990.9925.995.9975 1.0.990.9925.995.9915 1.0 Bow ENDt) TsTLs Wcr.1.2.3.4.5.6.7.8.9 \.o (c) Velocity components, streamline CD. Figure I-1 (Continued)

U E.8 \J.8 U EF -7 VE.3.6.Y.6 W E P 990 o.99a5.995.9975 1.0.990.9525.995.9975.0 BOaV ENA D 'ET~iL5 w E r VEF o.1.2.3.4.5.9.7.8.5 1.0 (d) Velocity components, streamline EF. Figure 1-1 (Continued)

U Ca~~~~~~G 0~~~~..8.8.2 +..990.9925.995.3515 1.0.990 9925 995.39T 1.0 BoW ERID DEdAILS 0~~~~~~~~~~~~~~~~~~~~~~~~ 0.1.2 3.4.5.6.7.8.9 1.0 (e) Velocity components, streamline GH. Figure I-i (Concluded)

-28 -Since calculations have been restricted to source distributions which are symmetric with respect to midships only the streamline traces of the fore-body have had to be determined. In the following the computational procedure is given in more detail for the case of a singularity distribution described by ml(t) = a1 cos (za 0); m2(W) = 1 The depth of distribution is given as t = to - b1 cos (a O) Input Data (1) Initial values of x,y and z (starting point) (2) al and a in source al cos (a ) (3) to and b1 in bottom trunkation t = to - b1 cos (a ) (4) Maximum number of print out line (5) Accuracy of y i,e. 10 (6) Stations in x,i.e., xo - - - - -xn and n At the station Xn along the hull, following substations are set up. \o~~ ad. 1i~~n STERN BOW $Po i at which G, V Ir, A_ t _ J)s

-29 -xi = -lo0 x2 = 0 4 where yo = y(l) x3 = Xn - Yo x4 = Xn - 4o x =Xn - 2yo x6 = xn + 2yo x7 = xn + 4yo X8 = + Yo In order to obtain a good accuracy of computed values of velocity components the region O< x < 1 is divided into two subregions, thus for Xn <.95, u = -10 - [DUIo(xl,x3) + DUojx,x5) + DUL(x5,xn) +DU (xn x6) + DU (x6 x8) + DU (x8 x9)] v = - [DV. (x,x3) + DV.(x3,x5) + DV.(x5,x6) + DV.(x6,x8) +DV. (x8,x9) ] w = - [DW. (xl,x3) + DW.o (x3,x5) + DW (x5,x6) + DWo(x6 x8) +DWo (x8,x9)] whereas for Xn > o95 u = -u' -1.0 - [DU(x1,x2) + DU(x2,x3) + DU.(x3,x5) + DU.(x5,xn) ]

-30 -= -' -[DV(xx 1 2) + DV(x2,x3) + DV(x3,x) + DV(x4,n) w = -w -[DVo(x1,x2) + DW(x2,x3) + DWj(x3,x4) + DW.(x4,xn)] where u' = DUo(xn,x6) + DU (x x ) + DU8(x x V = DVo(xn,x7) + DV.(x7,x8) + DV~(x8,x9) w' = DWo(xn,x7) + DW.(x7,x8) + DWe(x8 x9) for jx1n -lol > 10o-6 u' = v = w' = 0 for Ixn - 1oI0 < 10-6 The functions DU(A,B) are treated as internal functions in the computer program. They are defined as follows: 9 I ~ sia n (.z t DUo(AB) = (B - A) Ii in ( Xi) Ri -t — i R' R-ti i = 1 z + t)] R+ti 9 a y z - t z + t DVo(AB) = (B - A) - sin (a ) [ (] 8in Ri R-t R+t ai 1 1 2;-l 8 R ti R+ti

-31 -where Ii = Gauss weight functions at the proper stations see Table I-I i = 1,2,- - 9- - - - a1 = Amplitude of source strength = phase parameter a = - for c-101 and c-201 2 Si = stations for Gauss integral formula. See Table I-I. x,y,z = coordinate at which velocity is sought t = to - b1 cos (ai): depth of singularity distribution with bottom trunkated b1 cos (a Si) b1 = amplitude of bottom trunkation in source distribution b = 0 for c-101 and c-201 R. = 1 1 ~ ti 2 221/2 [(x - Si) + y ] / [(x - ~i) + y + (z - t) ] Rt [(x - i)2 2 211/2 ti [(x -j) + y + (z + t) ]/ The velocity components u, v and w are calculated at station xn. By Runge-Kutta method, values of y and z at station xn + 1 and at an intermediate station ~(xn - Xn + 1) are calculated and compared. If lY,z(xnl) -, z( X ) I Values of x,y and z and u,v, and w at Xn+l are printed out. If IYNZ(Xn+l) YyZ1(xxnl) > '

-32 -y and z at (xn - xn+l) are calculated and the process is repeated until the absolute values of difference in y and z are smaller than a preset value E, designated "YERR" in the program. Print-out includes x,y,z,u,v,w, and the absolute value of differences in y or z (whichever is larger) when the above limiting condition is satisfied. The computer programs for the cosine and the polynominal hull forms are shown below, printed in the MAD language. Print out for the L.W.L of the cosine hull form has also been included. It is noted that coordinates are given only for x <.9. The reason for this is that for the particular run recorded here the initial value of y was that obtained at x =.9 from a seperate calculation. It should be emphasized that the stream line program can be used to trace any streamline in the fluid region outside the hull and also for the determination of velocity components. The magnitude of the potential wake can therefore be readily evaluated. TABLE I-I STATIONS AND WEIGHT FUNCTIONS FOR GAUSS INTEGRAL FORMULA (When A is at -1.0 and B is at +1.0) i A 9 8 7 6 5 4 3 2 1 B Station -1.0 -.96816 -.83603 -.61337 -.32425.32425.61337.8337.96816 +1.o Weight.08127.18065.26061.31235.33024.31235.26061.18065.08127 Functions

.. 003C3...... 03003...... 003003....... C003003...... 003003...... 003003...... D03003....... 033.......003003...... 30 3003....... 0 3G JOB NO. 003,003 UNIVERSITY OF MICHIGAN EXECUTIVE SYSTEM (MOCEL 0V212) TUESDAY, NOVEMBER 27, 1962 8 1 32.8 H'M TAKAHEI SALVESEN 'REENE S160F 002 002 070 04542 A. Cosine Hull Yerr = 10-5 N = 36 automatically changed to N = 94 (two separate runs). See the last page for "Yerr.". 003003......00300 3.....003003.....003003......003003......003003.......003003......003003.....003003. 003003

$ COMPILE MAD, EXECUTE, DUMP, PRINT OBJECT, PUNCH OBJECT TAKA 003003 11/21/62 8 16 32.8 PM MAD (16 APR 1962 VERSION) PROGRAM LISTING. MARITIME TASK I PROJECT NR. 04542 CALCULATION OF SHIP HULL FORM MATHEMATICALLY PRODUCED BY THE GIVEN SINGULARITY DISTRIBUTION TAKAHEI COSINE SHIP RUNGE-KUTTA STEP SIZE IS AUTOMATICALLY REFINED DIMENSION E(13), F(2), H(9), 1(9), 0(2), X(9999), Y(2) *001 INTEGER CO, CO MAX, K, L, M, N, NI *002 STATEMENT LABEL 0 *003 INTERNAL FUNCTION (A, B) *004 ENTRY TO DU. *005 D = BETA *006 TRANSFER TO ALPHA *007 ENTRY TO DV. *008 O GAMMA *009 TRANSFER rO ALPHA *010 ENTRY TO OW. *011 O = DELTA *012 AlPHA C = O. *013 THROUGH EPSILN, FOR M = 1, 1, M.G.9 *014 I= ((1.+H(10-M)) * B + (1.-H(I0-M)) * A) *.5 *015 XII = XI * ALFA *016 X12 = X - XI 7 X13 = XI2 * X12 + Y(1) * Y(1) *018 G = TNAUT - B1 * COS. (XII) *019 ZGM = Y(2) - G *020 LGP = Y(2) + G *021 Fl = Al * SIN.(XIL) * 3978B7357E-10 *022 R INVS = I(M) / SQRT.(XI3 + ZGM * ZGM) *023 RPRIMI = I(M.) / SQRT.(XI3 + ZGP * ZGP) *024 01FF 1 = R INVS * ZGM - RPRIMI * ZGP DIFF 2 = R INVS - RPRIMI *026 ITGRND = Fl * DIFF 1 / XI3 -~ 0- - TRANSFER TO 0 *028 BETA C = C + ITGRND * X12 *022 TRANSFER TO EPSILN *030 GAMMKA. C = C + ITGRND * Y(l) *0 3L TRANSFER TO EPSILN *032 DE-IA C = C + DIFF2 * F1 *Q3 0 3 3 EPSILN CONTINUE *034 FUNCTION RETURN C * (B - A) *035, END OF FUNCTION *036 EXECUTE SETRKD. (2, Y(l), F(l), 0, X, STEP) *037 H(l) =.9.9166024..*018 H(2) =.83603111 *039 H(3) =.61337143 *Q& H(4) =.32425342 *041 =01 320 1(1) =.08127439 *043

1(2) =.18064;16 *644 I (3) =.26`61?)70 *045 1(4) =.31234708 *046 I ( 5 ) =.33023936 *047 THROUGH ZETA, FOR L = 6, 1, L.G.9 *048 H(L) = -H(10 - L) _-49 ZETA I(L) 1 (10 L), ETA READ FORMAT TITLER, E...E(13) *051 KEAD FORMAT W, X, Y(1), Y(2) *052 READ DATA Al, TNAUT, BI, ALFA, N, CO MAX, Y CRIT *053 READ FORMAT R, X(1)...X(N) *054 WHENEVER ALFA.E. Y, ALFA = 1.5707963 *055 PRINT FORMAT TITLE, E...E(13) *056 PRINT COMMENT $4 INPUT VALUES READ WERE$ *057 PRINT COMMENT $4$ *0 58 PRINT RESULTS Al, TNAUT, 81, N, ALFA, CO MAX, Y CRIT, -59 1 X... X(N) 59 PRINT COMMENT $1$ *b6O WHENEVER N.G.28, TRANSFER TO THETA * PRINT COMMENT $4$ *062 THETA PRINT FORMAT St X, Y(1), Y(2),C'..3 CO = I *064 CO MAX = 4 * CO MAX *065 K = 0 *66 Ni1 = N + 1 *067 XtNl) = X *068 XI = -1. *069 X2 =.4 *070 X9 = -Xl *071 THROUGH MU, FOR L = 0, 1, L.G.NI *072 X3 = X - SQRT.(Y(1)) *073 1 X4 = X - 4 t Y(1) *074 X5 = X - 2 * Y(l) -075 X6 = X + 2 * Y(1) *076 X7 = X + 4 * Y(I) *077 X8 = X * 2. - X3 *078 WHENEVER X.LE..95 *079 U = DU.(Xl, X3) + OU.(X3, X5) + DU.(X5, X ) + DU.(X, X6) *080 ~~~~1 ~+ DU.(X6, XE) + DU.(X8, X9) *6:O V = DV.(X1, X3) + DV.(X3, X5) + DV.(X5, X6) + DV.(X6, XS) 081 ~~~~~~~~~~1 ~~+ DV.(X8, X9) *081 = DW.(Xl, X3) + DW.(X3, X5) + DW.(X5, X6) + DW.(X6, X8) *0.52 ~~~~~~~~~~1 ~~+ OW.(X8, X9) *8)2 OTHERWISE -083 U = DU.(Xl, X2) + DU.(X2, X3) + DU.(X3, XS) + DU.(X5, X) "84 V = DV.(XI, X2) + DV.(X2, X3) + DV.(X3, X4) + DV.KX4, X) *085 W = DW.(XI, X2) + DW.(X2, X3) + DW.(X3, X4) + DW.(X4, X) *086 WHENEVER.ABS.(X-1.).G. LE-6 *087 U = U + DU.(X, X6) + DU.(X6, X,) + DU.(XE, X9) *088 V = V + DV.(X, X7) + DV.(X1, XE) + I)V.(X8, X9) *089 W = W + DW.(IX, X7) + DW.(X7, X8) + DW.(X8, X9) *090 END OF CONDITIONAL *091 EL\D OF CONDITIONAL *0.92 U = -1. - U *093 V = -V *094 VHENEVER CO.GE. CO MAX *095 PRINT RESULTS CO, L, Y1, K, UO, XO, YO, ZO, U, X, Y(1), *096 I~ ~ ~vY(2), P *096 TRANSFER TU ETA *097 OR WHENEVER K.E. 0 *098

WHENEVER L.NE. 0 *099 PRINT FORMAT T, X, Y(1), Y(2), U, V, W, Y1 *1s0 WHENEVER L.E. N1 lZ01 PRINT COMMENT $- 'Y ERR' IS THE IMPROVEMENT IN *132 1 Y OBTAINED AT EACH STEP BY RECALCULATION WITH TWICE AS FIN *102 2E A STEP SIZE AS IS INDICATED.$ *102 TRANSFER TO ETA *103 END OF CONDITIONAL *1 04 OTHERWI SE *105 PRINT FORMAT T1, X, Y(l), Y(2), U, V, W *106 END OF CONDITIONAL *107 STEP = X(L + 1) - X *108 UO = U *109 VO = V *110 WO = W *111 XO = X *112 YO = Y(1) *113 ZO = Y(2) it114 OR WHENEVER K.E. 1 *115 L = L - 2 *116 Y1 = Y(1) *117 IOTA STEP = STEP *.5 *118 U = [JO *119 V = VU *120 W = WO *121 X = XO *122 Y(1) = YO *123 Y(2) = ZO *124 [R WHENEVER K.E. 2 *125 P = Y(1) *126 END OF CONDITIONAL *127 KAPPA F = RKDEQ.(O) *128 WHENEVER F.E. 1. *129 F(1) = V / U *130 F(2) = W / U *131 TRANSFER TO KAPPA *132 'R WHE'IEVER K.E. 2 *133 Y1 =.ABS.(Y1 - Y(1)) *134 WHENEVER Y1.G. Y CRIT *135 K = 1 *136 L = L - 1 *137 Y1 = P *138 TRANSFER TO IOTA *i39 END OF CON')ITIONAL *140 K = -1 *141 LAMBDA THROUGH LAMBDA, FOR L = 1, 1, X(L).L.X+STEP.OR. L.G.N1 *142 L = L - 2 *143 rND CF CONDITIONAL *144 CO = Co + 1 *145 MU K = K + 1 *146 VECTOR VALUES 0 = $F10,2E20*$ *147 VECTOR VALUES R = $8F1CG*$ 148 VECTOR VALUES S = SS17,91HCALCULATION OF SHIP HIULL FO[RM MATHE *149 1F:ATICALLY PRODUCED BY THE GIVEN SINGULARITY DIS-TRIBUTION//S91 *149 2,7HTAKAHEI////S7,89HCOORDINATES OF POINTS ON STREAMLINES AND *149 3CORRESPONDING VALUES OF FLOW VELOCITY COMPONENTS////S7,28HINI *149 4TIAL VALUES OF STREAMLINES10,4HX =F8.4,S7,4HY =Fll.8,S',4HL *149 5 =F1 1.8///S 16, lHXS15, HYS16, 1HZS2i, lHUS16, 1HVS16,1HiS 14, 5HY *149 6ERR// *$ *149

VECTOR VALUES T = $F20.7,2E19.9,4E17..4*$ *150 VECTOR VALUES T1 = $F20.7,2E19.9,3E17.4*$ *151 VECTOR VALUES TITLE = $1Hl/lH813C6,C2*$ *152 VECTOR VALUES TITLER = $13C6,C2*$ *153 END OF PROGRAM *154 -—,,

RUN NR. 12 WATER LINE MODEL C-201 INPUT VALUES READ WERE Al =.600000, TNAUT =.100000, BI =.000000, N = 36 ALFA = 1.570796, COMAX = 999, YCRIT = 10.0000GOE-06 X(O)...X(36) 9.000000E-01 8.750000E-01 8.500000E-01 8.250000E-01 8.000000E-D1 7.750000E-01 7.500000E-01 7.2500D0E-1 7.OOOOOOE-1 6.750000E-01 6.500E-01 6.255000E-01 6.500000E-01 6.25003E-01O 6.0000DE-01 5.750000E-01 5.500000OOE-01 5.25 OE-01 5.000000E-01 4.750000E-01 4.500000E-Cl 4.250000E-01 4.OOOOOE-01 3.750000E-01 3.500000E-31 3.253000E-Q1 3.000000E-01 2.750000E-01 2.500GOOE-01 2.250000E-0G1 2.000000E-31 1.750000E-01 1.500300E-01 1.250003E-01 10.000000E-02 f.500000E-02 5.OOOOO E-02 2.500000E-02.000000DOE 00

TAKAHEI INITIAL VALUES OF STREAMLINE X =.90GO =.03025303 Z =.00000000 X Y Z U V W Y ERR,_9000000.302499995E-01.000000000E O0 -.9478E OO.2280E 0:3.0000E O0..8937500.317477487E-01 -.000000000E O0 -.9521E O0.2256E 00.0000_F O0.8890625.328551732E-01 -.000000000E O0 -.9552EO0 _-22_38E O0.0000E O0 ~ 8820312.344954669E-01 -.000000000E O0 -.9593E OD.2210E OG.0000E O0.8150000.361087136E-01 -.0000COOOOE O0 -.9632E 03,2183E OO.0000E O0.6657E-05.8687500.375200383E-01 -.000000000E O0 -.9664E O0.2159E OO.0000E OO.5106E-05.... 86_4._0.~525,385643363E-01 -.000000000E O0 -.9685E OS _:.2!_4.!E O0.000OF- 00.2803E-05.8570312.401120633E-01 -.00000000DE O0 -.9718E 03.2114E O0.0000E OD.6185E-05 ~ 8500000.416353777E-01 -.00COOGOOOE O0 -.9747E OO.__20_8_7EO0.0000E O0,6024E-05.8437500.429688714E-01 -.000000000E OO -.9771E OD.2063E O0.0000E OD.q. 643E-05.8390624,439560652E-01 -.000000000E O0 -.9789E OD.2046E O0.0000E OD.2557E-05 ~ 8320312.454197541E-01 -.0000COOCOE O0 -.9814E OO.2019E O0.0000E O0.5658E-05 ~8250000,468610749E-01 -.000000000E O0 -.9837E OD,!993EO0.00_00E00.5528E-05.8187500,481233522E-01 -.000000000E O0 -.9857E O0.1971E O0.0000E OD.4273E-05 ko~,_.8_093_750,499882452E-01 -.000000000E O0 -.9884E OO.1937E O0.O_000E O0.9419E-05 kO ~ 8000000.518159851E-01 -.000000000E O0 -.9910E OD.1903E O0.0000E O0.9159E-05.7937500.530123644E-01 -.000000000E O0 -.9926E O0.1881E OO.0000E 00.3957E-05.7843750.547804318E-01 -.000000000E O0 -.9949E O0.1849E GO.0000E O0.8760E-05.7750C00.565138973E-01 -.000000000E O0 -.9971E 03.1817E O0.0000E OO.8541E-05 ~ 7687500.576489002E-01 -.000000000E O0 -.9984E O~.1796E O0.QOOOE O0.3707E-05 ~ 7593750.593266137E-01 -.0000000DOE O0 -.1DODE O1.1764E OO,0000E O0.8200E-05 ~ 7500000.609719075E-01 -.000000000E O0 -.lO02E O1.1733E O0.0000E 09.8010E-35.7437500.620493919E-01 -.000000000E O0 -.10G3E O1.1713E OO.0000E OO.3433E-35 ~ 7343750.636423044E-01 -.000000000E O0 -.1005ED1.1683E O0.0000E O0.7711E-05.7250C30.652046822E-C1 -.000000000E DO -.10S7EO1.1654EO0.0000E O0.7552E-05.7187500.662279859E-01 -.000000000E O0 -.1008E O1.1634E O0.0000E O0.3289E-05.7093750.677409098E-01 -.000000000E O0 -.1009EO1.1605E O0,0000E OO,7297E-35.7000000.692249425E-01 -.000000000E OO -.1011E O1.1577E O0.0000E O0.7152~-05.693750C.7019697205-01 -.000000000E O0 -.lO12E O1.1558E O0.0000E O0.3120E-05 ~ 6843750.716341108E-01 -.000000000E O0 -.1013E O1.1530E O0.0000E OO.6921E-~5.6750000.7304379425-01 -.00000000OR O0 -.1014E 01.1503EO0.0000E O0.6801E-95 ~ 6687500.739670992E-01 -.000000000E O0 -.lO15E O1.1485E O0.0000E O0.2971E-05.6593750.753321417E-01 -.00000000OR O0 -.1316EO1.1458EO0.0000E O0.6603E-05.6500C00.766709991F-01 -.000000000E O0 -.1017E O1.1432E O0.0000E O0.6492E-05.6437500.775478229E-01 -.000COOODOR O0 -.1018E O1.1414E O0.0000E OD o2841E-05.6343750.788440168E-01 -.000000000E O0 -.lO19EO1.1389E O0.0000E O0.6317E-05.6250000.801151417E-01 -.009000000E O0 -.1020E01.1363E O0.0000E DO....6219E-05 ~ 6187500.809474669E-01 -.OCOOOOO90E O0 -.1321E O1.1346E O0.0000E OD.2723E-~5 ~ 6093750.821776830E-01 -.000000030E O0 -.1022E O1.1321E O0,QOOOE O0 o6063E,05.6000GO0.833838239F-01 -.000000ODOR O0 -.1023E D1.1296E O0.0000E O0.5975E-05.5937500.841/34067E-01 -.000000000E O0 -.1023EO1.1287, E OC.0000E OO,2621E-05 -~ 5843750.853401899E-01 -.00000000OR O0 -.102~E O1.1256E O0.0000E OD.5839E-35.5750CC0.864837766E-01 -.00000000OR O0 -.1025E Ol.1232EDO ~,OOQOE OD.5762~-~5 ~ 5687500.872321732E-C1 -.000000000E O0 -.1026EO1.1216E O0.00035 O0.25295-05

.5593750.883377872E-01 -.000000000E O0 -.1326E O1.1192E O0.0000E O0.5638E-05 ~ 5500C00.894209862E-01 -.000000090E O0 -.1027E O1.1169E O0.0300E 03.5573E-35,5375C00.908332989E-01 -.000000000E O0 -.1028E O1.1138E O0.0000E O0.9779E-05,5250000.922068030E-01 -.000000000E O0 -.1029E 01.1197E O0.0000E O0.9629E-05.5125000.935420811E-01 -.000000000E O0 -.1030E O1.1077E O0.0000E O0.948lE-05.5000000.948396817E-01 -.000000000E O0 -.1031E O1.1047E O0.0000E OD.9354E-05.4875000.961001284E-01 -.000000000E O0 -.1032E 01.1018E O0.0300E OO.~226E-05.4750000.973239064E-01 -.000000000E O0 -.1033E 01.9886E-01.0000E O0.9109E-05.4625000.985114865E-01 -.0000000DOE O0 -.1034E O1.9597E-01.0000E OD.8996E-05.4500000.996633068E-01 -.000000000E O0.lOBrE O1.9311E-01.0000E O0.8839E-D5,4375000.100779779E O0 -.000000000E O0.1035E O1.9028E-01.0000E O0.8781E-05.4250000.101861313E O0 -.000000000E O0.1036E 01.8747E-01.0000E O0.8691E-05 ~ 4125000.102908261E O0 -.000000000E O0 -.1036E O1.8468E-01.0000E OD.8601E-05.4000000.103920989E O0 -.000000000E O0 -.1037E O1.8191E-01.0000E OD.8514E-05 ~ 3875000.104899831E O0 -.000000000E O0 -.1038E O1.7917E-01.0000E O0.8435E-05.3750000.105845101E O0 -.000000000E O0 -.1038E O1.7645E-01.0000E O0.8358E-05.3625000.106757089E O0 -.000000000E O0 -.1039E O1.7374E-01.0000E O0.8283E-05.............................3500000.107636087E O0 -.000000000E O0 -.1039E O1.7105E-01.0000E O0.8214E-05.3375000.108482368E O0.000000000E O0 -.1040E 01. ~6.8.38.E.~9_1... -..0000E...00.814.9E-.05.3250000.109296173E O0.000000000E O0 -.1041E O1.6573E-01 -.0000E O0.8088E-05.3125000.110077739E O0.000000000E O0 -.1041E O1.6309E,01 -.0000E O0.8031E-05.3000000.110827304E O0.000000000E O0 -.1041E O1.6046E-01 -.0000E OD.7974E-05 ' 2875000.111545071E OC,000000000E O0 -.1042E O!....5785E~Pl -.0003E O0.7926E-05.2750000.112231240E O0.000000000E O0 -.1042E 01.5525E-01 -.0000E O0.7875E-05,2625000.112886004E O0.000000000E O0 -.1043E O1 _-526~.~. -.0000E O0 o7831E,05.2500000.113509543E O0.000000000E O0 -.1043E O1.5009E'01 -.0000E OO.7786E-05.2375000.114102021E O0.000000000E O0 -.1043E O1.4752E~0! -.ODDDE 90.7747E-05.2250000.114663593E O0.000000000E O0 -.1044E D1.4497E-01 -.0000E O0.7709E-05,2125000.115194410E O0.000000000E O0 -.1044E01.4242E-01 -.0000E O0.7674E~05.2000000.115694605E GO.0000000DOE O0 -.1044E O1.3988E-01 -.0000E OD.7542E-05...... 18Z5000.116164304E O0.000000000E O0 -.1045E O1.3735E-01 -.000DE O0.76~7E-05 0.1750C'00.116603635E O0.000000000E O0 -'1045E......0~.3483E-01 J. ooodE....66.i579E'b5.1625000.117012694E O0.000000000E O0 -.1345E O1.3231E, Q1 -.0000E O0.7555E-05.1500000.117391SOLE O0.000000000E O0 -.1045E O1.2980E-01 -.0000E OD.7532E-35.... 1375000.117740430E O0.000000000E O0 -.1046E 01 -2730E-0! -.0003E O0.7536E-05.1250000.118059285E O0.000000000E O0 -.1046E D1.2480E-01 -.0000E O0.7487E-05 ~1_!2~000.118348241E O0.000000000E O0 -.1346E O1 -~23~.E_~01 -.0000E O0.,7~59E-05.1000000.118607357E O0.000000000E O0 -.1046E O1.1982E-01 -.0000E OD.7451E-05.0875000.118836701E O0.000000000E O0 -.1046E O1.1733E-01 -.000DE O0.7437E-05.0750000.119036339E O0.000000000E O0 -.1046E O1.1485E-01 -.0000E O0.742~E-05.... 06250.00.119206309E O0.0000000DOE O0 -.1347E O1.1237E~0! -.0000E O0.7412E-05.0500000.119346656E O0.000000000E O0 -.1047E O1.9893E-02 -.0000E 00.7433E-05.0375C00.119457416E 00.000000000E O0 -.1047E O1.....7~8E_~02....r....O ODOE O0.73~5E,05.0250000.119538613E O0.000000000E O0 -.1047E D1.4944E-02 -.0000E O0.7388E-05.0125000.119590275E O0,000000000E O0 -.1347E O1.2472E-02 -.0000E O0.7384E-05 -.0000000.119612403E O0.000000000E O0 -.1047E,01 -.1164E-09 -.0000E OD.7332E-~5 '~ ERR' IS THE IMPqOVEMENT IN Y

..003886......003886......003886.....003886.....003886....003886...... 003886.....003886.....003886... 003886...... 00386 JOB NO. 003886 UNIVERSITY OF MICHIGAN EXECUTIVE SYSTEM (MODEL OV302) THURSDAY, DECEMBER 13, 1962 7 08 30.8 PM KIM - GREENE SOOOF 001 020 160 ORA04542 B. Polynomial Hull..003886...... 03886......003886......003886......003886......003386......003886..... 003886.....003886......003886......003886

S COMPILE MAD, EXECUTE, DUMP, PRINT OBJECT, COPIES(2) 000039 09/24/62 9 36 43.2 AM MAD (16 APR 1962 VERSION) PROGRAM LISTING. MARITIME TASK 1 PROJECT NR. 04542 CALCULATION OF SHIP HULL FORM MATHEMATICALLY PRODUCED BY THE GIVEN SINGULARITY DISTRIBUTION POLYNOMIAL SHIP DIMENSION F(2),H(9),I(9,K(13),O(2),P(4),X(999),Y(2),Z( 1) *001 EQUIVALENCE (RTM, XI), (RTP, XI1) *002 BOOLEAN H IS T *003 INTEGER J, L, M, N *004 STATEMENT LABEL D *005 INTERNAL FUNCTION (At B) *006 ENTRY TO DU. *007 D = GAMMA *008 TRANSFER TO ALPHA *009 ENTRY TO DV. *010 O = DELTA *011 TRANSFER TO ALPHA *012 ENTRY TO DW. *013 D = EPSILN *014 ALPHA EXECUTE ZERO.(C, Fl) *015 THROUGH ZETA, FOR M = 1t 1, M.G.9 *016 XI = ((..+H(10-M)) * B + (l.-H(10-M)) * A) m.5 *017 O XI1 =.ABS. XI *018 THROUGH BETA, FOR J = 0, 1, J.G.4 *019 BETA Fl = (F1 + P(J)) * XI1 *020 WHENEVER XI.L.O., Fl = -F1 *021 F1 = E * Fl * I(M) *022 XI2 = X - XI *023 X13 = X12 * XI2 + Y(1) * Y(1) *024 G = TNAUT + Bi * XI * XI *025 ZGM = Y(2) - G *026 ZGP = Y(2) + G *027 RTM = SQRT.(XI3 + ZGM * ZGM) *028 RTP = SQRT.(XI3 + ZGP * ZGP) *029 F2 = ELOG.((ZGM + RTM) / (ZGP + RTP)) *030 WHENEVER H IS T *031 F1 = Fl / (G * G) *032 ITGRND = Fl*(F2+(ZGP*RTM-ZGM*RTP)/XI3) *033 OTHERWISE *034 HI = H1 * HI *035 F1 = Fl / HI *036 ZH2M = Y(2) * Y(2) - H1 *037 ITGRND = Fl*(F2-((ZGP*RTM-ZGM*RTP)*ZH2M+(ZGM*RTM-ZGP*RTP)* *038 1 XI3)/(RTM*RTP*XI3)) *038 END OF CONDITIONAL *039 TRANSFER TO D *040 GAMMA C = C + ITGRND * XI2 *041 TRANSFER TO ZETA *042 DELTA C = C + ITGRND * Y(1) *043 TRANSFER TO ZETA *C44

EPSILN C = C + FL * (RTM - RTP - F2 * Y(2)) *045 WHENEVER H IS T, C = C + Fl*(HI-G*G)*(RTP-RTM)/(2.*RTP*RTM) *046 ZETA CONTINUE *047 FUNCTION RETURN C * (B - A) *048 END OF FUNCTION *049 EXECUTE SETRKD. (2, Y(l), F(1), 0, X, STEP) *050 H(1) =.96816024 *051 H(2) =.83603111 *052 H(3) =.61337143 *053 H(4) =.32425342 *054 H(5) =.0 *055 I(1) =.08127439 *056 1(2) =.18064816 *057 I(3) =.26061070 *058 I(4) =.31234708 *059 1(5) =.33023936 *060 THROUGH ETA, FOR L =6, 1, L.G.9 *061 H(L) = -H(10 - L) *062 ETA I(L) = I(10 - L) *063 READ FORMAT R, P...P(4) *064 READ DATA N *065 READ FORMAT R, X(1)...X(N) *066 E = 1. / (8. * 3.1415927) *067 X 1 = -1. *068 X2 =.4 *069 X9 = -Xl *070 THETA READ FORMAT TITLER, K....K(13) 071 READ DATA TNAUT,B1,H1 *072 READ FORMAT Q, X, Y(l), Y(2) *073 PRINT FORMAT TITLE, K...K(13) *074 Y = Y(1) *075 Z = Y(2) *076 PRINT COMMENT $4 INPUT VALUES READ WERE$ *G77 PRINT COMMENT $4$ *078 PRINT RESULTS TNAUT,Bl,H1,N,P...P(4),X...X(N),X,Y,Z *079 H IS T =.ABS.(HI - TNAUT).L. 1E-5 *080 PRINT COMMENT $1$ *081 WHENEVER N.G.28, TRANSFER TO IOTA *082 PRINT COMMENT $4$ *083 IOTA PRINT FORMAT S, X, Y(l), Y(2) e084 THROUGH LAMBDA, FOR L = Ot 1, L.G.N *085 X = X(L) *086 X3 = X - SQRT.(Y(l), MU) *087 X4 = X - 4 * Y(1) *088 X5 = X - 2 * Y(l) *089 X6 = X + 2 * Y(1) *090 X7 = X + 4 * Y(1) *091 X8 = X * 2. - X3 *092 WHENEVER X.LE..95 *093 U = Dt).(X1, X3) + DU.(X3, XS) + DU.(X5, X ) + DU.IX, X6) *094 1 + DU. (X6, X8) + DU.(X8, X9) *094 V = DV.(X1, X3) + DV.(X3, X5) + DV.(X5, X6) + DV.(X6, X8) *095 1 + UV.(X8, X9) *.095 W = DW.(X1, X3) + DW.{(X3, X5) + DW.(XS, X6) + DW.(X6, X8) *096 1 + DW.(X8, X9) *096 OTHERWISE *097 U = DU.(X1, X2) + DU.(X2, X3) + DU.(X3, X5) + DU.(X5, X) *098 V = DV.(X1, X2) + DV.[X2, X3) + DV.(X3, X4) + DV.(X4, X) *099 W = DW.(X1, X2) + DW.(X2, X3) + DW.(X3, X4) + DW.(X4, X) *100

WHENEVER.ABS.(X-1.).G. LE-6 *101 U = O+ODU.-1XU-XM+ -4DU'. 1X6,' 81 4+ DU.-TX4 -X91J-.1-02 V= V + DV.(X, X7) + DV.(X7, X8) + DV.(X89 X9) *0 W= W + DW.(X, X7) + DW.(X7t X8) + DW.(XB8 9)10 END OF CONDITIONAL.*105 lENflOF-CUaNIYlTr[NA-L *106 -U -1. - U ~~~~~~~~~~~~~~~*107 V-V ~*109 W 2. *W *0 PRINT FORMAT T, Xi Y(1), Y(2)v U, Vs W *110 WHENEVER L.E.N, TRANSFER TO THETA *111 STP -( --- 1) - X *112 KAPPA F = RKDEQ-tO) *1 WHENEVER F'.E.l. *1 F(1) =V IU *115 Fi2) =W /U *116 TRANSFER TO KAPPA *117 -END -OF-C6N'D I T I O-NA L 1 LAMBDA CONTINUE *119 mu PRINT -COMMEN~T $4 ARGUMENT- OF' SQUARE ROOT FUNC-f~N 4A- 12 0 1VE*$ *120 'PRINT"'RE-SUL`TS X, Y, 1, Li, STEP, Y(l), Y(2'), F, U, V, W *121 TRANSFER TO THETA *122 VECTOR VALUES Q = F10,2E20*S *123 VECTOR, VALUES- R = t8F1-O*$ * 124 VECTOR VALUES S = SS17,p91HCALCULATION OF SHIP HULL FORM MATHE *125 IMATICALLY PRODUCED BY THE GIVEN SINGULARITY DISTRIBUTION//S91 *125 217HTAKAHEI///1S7,89HCOORDINATES OF POINTS ON STREAMLINES AND *125 3CORRESPONDING VALUES OF FLOW VELOCITY COMPONENTSt///S7928HINI *125 4TIAL VALUES OF STREAMLINES10,4HX =F8.4,57,4HY =F11.8,57,4HZ *125 5 — =FT11-.-8//-/Sl6,IFIXS18,LHYS18,1HZS22,1HUS18,lHVS18,1HW//*$ *125 VECTOR VALUES T = $Fl9.4,S2,2El9.4,S4,3E19.4*$ *126 VECTOR VALUES- TITLE = $lHl/lH813C6,,C2*$ *127 VECTOR VALUES TITLER = S13C6,C2*$ *128 END OF PROGRAM *129

APPENDIX II HULL FORMS OBTAINED FROM STREAM LINE TRACES Very few hull forms have so far been obtained from the traces of streamlines for given singularity distributions. For convenience we have included the particulars of forms produced elsewhere in Table II-1, and in Figures II-l(a) through II-1(i). It is noted that all hull forms have a singularity distribution of the seperable type, ioe, m(~,) = ml(t). m2(W) Except for hull G-4-V, where m2(Q) is a linear function of -. All hulls have a uniform distribution of singularities with respect to depth i.e. m2(S) = 1 The depth of the singularity distribution is the same as the draft at the F.P. The function ml(e) is shown as a chain line in each figure.

TABLE II - I LIST OF STREAMLINE HULL Source Distribution m(5, 0)=ml(O).m2(W)U Model Lengthwise m () Draftwise m2(W) B/L Tf/L T/L ml( = 1) m~(~ = 1) Figure Remarks S-101 0.4 5 Constant t = 0.1 0.0748 0.050 o.o806 0.4 0.4 II - (a) 8-102 0.4, " t = 0.2 o.o839 0.100 0.1356 o.4 0.4 II - (b) Ref. (6) S-201 o.8 5 " t = 0.1 0.1229 0.050 0.0979 o.8 o.8 II - (c) 8-202 0.8 5 " t = 0.2 0.1454 0.100 0.1562 o.8 o.8 II - (d) #G- o0.8 ~ " t = 0.2 o.1454 0.100 0.1562 0.8 0.8 II - (d) #G-2 1.1705 - 0.78052 " t = 0.1 0.39 -0.39 II - (e) #G-3 1.952~(1 _2) " t = 0.1 0 -3.904 II - (f)) G-4-V o.8 5 1-5 1l, t = 0.2 o.8 0.8 II - (g) Ref (11) C-101 0.4 sin " ~ Constant t = 0.1 0.0904 0.050 0.0859 0.4 0 II - (h) C-201 0.6 sin T I " t = 0.1 0.1208 0 050 o0.09og84 o0.6 o II - (i U = undisturbed flow velocity 1 > > -1 t > _> -t,

-47 -(a) S-101..I T, (b) S-102 Figure II-1

-48 -(c) S-201 (d) S-202 Figure II-1 (Continued)

-49 -(e) G-2 (f) G-3 (g) G-4-V Figure II-1 (Continued)

-50 -B/L-0.090 TF /L 0.050 TM/L -0.0859 (h) C-101 B / L 0. 121 TF /L= 0.050 TM/L=0.0984 (i) C-201 Figure II-1 (Concluded)

APPENDIX III EFFECT OF DEPTHS OF SINGULARITY DISTRIBUTIONS ON TRACED HULLS (SERIES-60, POLYNOMIAL DISTRIBUTION) To investigate the effect of the depth of source distribution on the hull lines, the following six hulls' streamlines (four streamlines each) were traced. The result indicates that the hulls obtained by the uniformly distributed sources at the centerplane have maximum draft approximately twice the depth of the distribution. Therefore, if the bottom has to be cut off for the draft consideration, it is recommended that the effect of this extremely large cut off be first investigated. Fig. III - 1 shows the source distribution used above. Table III - 1 shows the depth of singularity distribution and starting points for Models A-l-l —6. Four streamlines are traced for each hullo Fig. III - 2 shows the body plans of the hulls, A-i-i, A-1-2, A-1-5, A-1-6 and Series-60. Fig. III - 3 shows the load waterlineso The changes in load waterlines due to the changes in source distribution is the maximum at the midship and they taper off parabolically toward the tip~ All waterlines appear to have similar shapes. Fig. III - 4 shows the keel lines. Again all keel lines appear to have the same shapes and the changes in shapes due to those in depth of distribution may for all practical purposes be neglected. -51 -

Fig. III - 5 cross-plots of Fig. III - 2. Note that the changes amidship are the greatest. This graph may be used for estimating the three-dimensional effect from two-dimensional case and vise versa. Fig. III - 6 is cross-plots of Fig. III - 3. This may be used to estimate the maximum draft for the changes of source distribution. TABLE II - I LIST OF MODELS Source Distribution: m( ) = 9.88 | 2 17 58 113 + 5.96 | + 1.98 1j15 Depth of Source Initial Values of Hull No. Distribution Streamlines UMo Hull to X Y 2/3 to 5/6 to to A - 1 - 1.0775 1.0 11.69 x 10-4.05167 o06458.07751 A - 1 - 2 o1067 1.0 ".07113.08892.10671 A - 1 - 3.070 1.0.04667.05833.07001 A - 1 - 4.060 1.0 04000o.05000 o 6001 A - 1 - 5.050 1,0 t ~.03333 o04166.05001 A - 1 - 6 1.000 1.0. o6667.83333 1.o0001

.8.6 E 4 - MATHEMAT I CAL HULLS E' E / SERIES 60 CB=.60 SOURCE DISTRIBUTION.2 - TYPE A-i I/m(S)=9. 882- 17. 583+ 5. 96 4+ 1. 98k5 0.1.2.3.4.5.6.7.8.9 1.0 x BOW Figure III-1

I A-1-5/- Ser-60 x.3 x=O.02 -A-i-1 -1 A-1-C =.60 TYPE1-A —5 -x.7:~~~x= A-1-i >;( \\ I /l =10.0.4 06.8 10.1.4.1.8.x=0.10 - 7 G=.3.16.18 -.20t x = A-1-6TYPE A-i1-6 m() = 9. 88x=0- 17. 58+5. 964. 98 BEAM — ICAL HULLS FigOff Scalure RIES 60

55 I I I I I I l I MATHEMATICAL HULLS SERIES - 60 Y CB =. 60.20 LWL SOURCE DISTRIBUTION m(E) = 9. 88E2 -17. 58Q3+ 5. 96,4+ 1. 98,5.18 O A-1-1 to =.0775 16 A-1-6 X A-1-2.1067 O A-1-3.070 * A-1-4.060 < A-1-5.050.1!4 A-1-6 1. 00 Series 60 CB =.60 A-1-3.06.04.02 I'_ I I I l I I, 0.1.2.3.4 X.5.6.7.8.9 1.0 _ I I I I I I I STIO. 9 8 7 6 5 4 3 2 1 F.P STATIONS Figure III-3

56.02 MATHEMATICAL HULLS SERIES 60 ~04 -CB -.60 KEEL LINE SOURCE DISTRI BUTI ON A-1-5 05 m(e) - 9. 88E2- 17. 58.3 + 5. 96E4 + 1. 98.5 06 - ~~~~~~~~~~~~~. 06 -1-4. 06 A-1-3.07 *.10 x /.12.14 2. 45 to -85 to 16 0 x.20 1 STIO. 9 8 7 6 5 4 3 2 1 0 STATION Figure III-4

.18 I l l l 2- Dim x=O 0 MATHEMATI'CAL HULLS SERIES 60 CB =.60 ~~~~~~~~~~~~~LWL 2: Dim SOURCE DISTRI BUTION TYPE A-1.14 L m(1;) = 9. 88~2- 17. 5843+ 5. 96 4+ 1. 985 x O 3.12 x 2- Dim K~~~~~~~~~~Fu II I-5 > F10 -A-1-4 A-l-3 06 - A-1- 1-1 A-1-2 x=.7 x=. 7.041 0 02.04.06.08.10.12.14.16.18 DEPTH OF SOURCE AT x = 1. O. to Figure III-5

MATHEMAT I CAL HULLS SERIES 60 CB =.60 KEEL LINE SOURCE DISTRIBUTION TYPE A-1 \a o m(~) = 9. 8802- 17. 58.3+ 5. 96 4+ 1. 98_5 LWL lo \ Station 5 1. 0 A-1-4 A-1-3 A-1-5 I I I A-1-2 0.02.04.06.08.10.12.14.16.18 to Figure III-6

APPENDIX IV COMPUTER PROGRAM FOR FINDING THE SOURCE DISTRIBUTION FOR A GIVEN LOAD WATERLINE SHAPE FOR TWO-DIMENSIONAL CASE The method of finding a general source distribution representing a given hull form is not yet available. For a two-dimensional case, however, the integral equation can be reduced to a set of simultaneous equations, which can be readily solved. Assume a two-dimensional source distribution at the longitudinal centerplane in power series form: N m() = V an n 0 < < 1 n = 0 (IV-1) N m() =V (-1)n ne, -1 <1 n n = 0 where an = coefficients to be determined m(6) = source strength V = uniform velocity N = number of terms in polynomials. The coordinate systems is defined as in Figure IV - 1. The stream function for the singularity distribution defined by Equations (IV - 1) can be written as of those due to the individual terms as: N (x,y) =. n(x=,y) n =1

-60 --1.o (STER) 0 + / 0 (BOW) SrRENGTH OF SOURCE DISTRIBUTION. L W~ Figure IV-1 where 1 'n - G (x,y; 2) nd for n = odd f O (x, y; i) ] f or n = even -1 E- [I OG (x~y; ~) ~ (IV-2) 0n - o (xy; i) ndi] for n = even with G defined as shown in Figure IV-2. From the condition that the dividing streamline at the centerline forms on the hull surface, it follows that 1.0 1 m(i) O (x,y; 5) dS - V y(x) = O (IV-3) -1.0

- r.IH 0 -1 l~~~~~~kc~o

I1 = (91 - G2) - (x - y2) (91 - G2) -2y [1 + x (in r1 - in r 2 2 I2 = (G1 - 2) - x(x 3y ) (G1 + 2 2- o)0) -y [1 + (3x2 _ y2) (in rl + in r2 - z In ro)] I3 = (G1 - G2) - (x4 - 6x2y2 + y4) (G1 - G2) -2y[1/3 + (3x2 _ y2) + 2x(x2- y2) (in rl - in r2)] I4 = (Gi - G2) - x(x4 - lOx2y2 + 5y4) (1l + G2 - 2Oo) -y[1/2 + (3x2 _ y2) + (5x4 10x2y2 + y4) (in rl + in r2 - 2 In ro)] 15 = (91 - G2) - (x2 - y2) (x4 - 14x2y2 + y4) (Gi - G2) -2y[1/5 + 3x- y2) + (5x4 - 1ox2y2 + y4) +x(3x4 - 10ix2Y + 3y4) (in ri - in r2)] with 0o = tan -1 [rad] x G1 = tan -1 x -1 92 = tan -1 Y x+ ro = x2 + y2 ri =f t-y (x ) 2 --- — ~ — -2. — r2 2(x+i2 For x =, x, - - - - - -,xn, the corresponding values of Yn are

-63 -known from the LWL, and an are found by solving n simultaneous equations, The desired solution is found by substituting an into Equa — tion (1). The computer program is written for n = 5. It solves for a in [I] [C] = [Y] and An = Cn 2t (n + 1) and punches out A n

KI M -S2 93F 00-1~ -010 0.00 0-RA04542 5 4 14 K I Sv 293F 001 010 000 ORA04542 $COMPILE MAD, EXECUTE9 DUMP, PRINTOJC R R WATERLINE CALCULATION 2-DIMENSIONA-L R -R TAKAHEI KIM. R 1CU( 20) VECTOR VALUES DIM = 291 95 INTEGER NvJ START f-RAD-~ATA- A ~1I.-t-ktxCL5r VECTOR VA LUES A= $8F10*5*$ AD~~iikA~~XLLL~~~~~X5 1 PRINT FORMAT B, X(1)*..X(5) PRINT COMMENT $4$ PRINT FORMAT B, Y(1)...Y(5) THROUGH ALPHA, FOR J=l1,1,J.G.5 X = -XjLJI Y = Y(J) WHENEVER XoE*0.U THO = 3.1416/2.0 OTHERWIf'E THO = ATAN.(Y/X) END OF CONDLT I ONA L THi = ATAN*(Y/(X-1)).TH2 = A-TAN.(Y/(X+1H Rl ((Y*Y)+((X-l)oP*2))eP*0*5 RO =((Y*Y)+(X*X)).P*O.5 - B(THl-THZID (EL0G.(R1)-EL0G.CR2)) F =(TH1 +TH2- 71*lHn u F =(ELOGO(RlD+ELOG.(R2)-(2*ELOG.(RO))) L = ((X*X)-~(Y*Y)) 1(1,J) B-(L*B)-(2*Y*(l+(X*D))) I (2. J) R- (X)* ( (X*X )- (3*Y*Y)) *F ) - Y*Cl 1+(M*E)) 1(3,J) =B-C(XePo4-(6*G)+Ye'P.4)*B)-(2*Y*(1/3+M+(2*X*L*D))) I(4,J) =B- (X* (X.-P.AIG)(5YP4 *IAY12+ 1+(((5*XsP.4)-(1Q*G)+Y.P.4)*F),) ALPHA -I-(5 9J) - -B —(L* (X*Po 4-14*G)-+Y P.4 Pa)*~ ( ~(15(N~ 1+( (5*XoP.4)-(1O*G)*YoP.4)+(X*( (3*X.P.4) 2-(10*G)+3*Y*P*4))*D))) THROUGH BETA, FOR N1,1,9NoGo5 C=GJRDT *(5,6,1,CO) WHENEVER C=-loo PRINT COMMENT $ DETERMINANT EQUALS ZERO$ BETA PRINT RESULTS XqCO(N) TRANSFER TO START END OF PROGRAM~ $ DATA 0.0 0.2 0.4 0.6 0.8 0.-2 0.-192 0.168 0.128 09072

UNIVE~$1'T'Y OF A, 3 o02 0