THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING CALCULATIONS OF SHIP- HULL FORMS WITH ELECTRONIC DIGITAL COMPUTERS Tet suo Takahei May, 1962 IP-564

ABSTRACT The author has calculated the hull forms mathematically induced by the following source distribution m(, O) = fl() f2(0) fl(5) = a1 sin (2 O) - 1< < f2(W) = 1 - t< < < t Two models were calculated; one is a fine model C-101 (al = 0.4 t = 0.1), the other is a full model C-201 (ac = o06, t = 0.l)o These models (L = 2.500m) were subjected to the tank experiments at the twoing tank of the Tokyo University. ii

I, INTRODUCTION A very important problem in the study of wave-resistance theory is that of obtaining a hull form which satisfies the hull boundary conditions exactly.(l) To reduce the error arrising from the usual approximate hull boundary conditions of the linear waveresistance theory in the case of a ship of finite beam, such a hull form becomes indeed a requirement of that theory. Furthermore, since the wave-resistance theory is based on the assumption of an ideal fluid, a surface satisfying the exact hull boundary conditions also provides a more correct form for the separation of the effects of viscosity on the wave-making characteristics of the hullo In addition it should lead to a more correct understanding of the relationship between hull form, wave-resistance and wave pattern. 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. (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 free-surface to satisfy the free-surface conditions. -1

-2The systems (i) and (iii) 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. No simple method of correcting for the finite beam exists. In the attempt to satisfy the hull boundary conditions exactly one is therefore forced to consider the inverse problem namely, for a given source distribution what is the hull form that this distribution represents. (2) The complete inverse problem, which includes the contribution from the singularity system (iii),becomes a very difficult one to solve. The secondary effect of system (iii) is relatively small at small Froude numbers, however, and in this paper only the source distribution representing a double model moving at constant speed in an infinite fluid will be considered. 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. ( 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

-3integral equations of the second order. The inverse problem is considerably simpler, however, and this approach was therefore used by Inui and Eggers, ( 2) This paper describes the solution of the inverse problem by means of electronic digital computers as applied to two different source distributions.

II. SOURCE DISTRIBUTION AND HULL FORM Let us assume the singularity distribution to be of the form m(g, ) = fl(f) f2() (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 consideredthe region of source distribution is given by -l< < 1 t < 5t (2) where xt Z7 L 2L the length of the ship being equal to 2L and the draft of the ship at bow 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 X fm(5)d~ d = 0 -t -1 Due to symmetry with respect to the LoWoLo it follows that m(n- ) = m(9,90) If the ship is stationary, and if the uniform flow is in the negative t direction -4

-5m(s) < 0< O > 0 m(a) < 09, < 0 For fore-and-aft symmetry, m(~) = -m(S,9) For a source distribution of this type, the location of stagnation points at bow and stern if and (a becomeif - 1+ = -1- C a Ship length is in reality 2 + 2E, but E is normally small and is neglected. Drafts at bow and stern becomes t + e by similar analogy, but e' again can be neglected, The velocity potential due to a source distribution as given by (1) is: t 1 9(Xfy4z) = - 1 n( )d d - x,41T_ 1 S(x') 2 _t y2 + (z - L)2 -t -1 and by definition the velocity components become {uv},w = grad 0 The velocity components can be substituted into the differential equation for the streamlines dx _ dy _ dz u v Vw

-6and 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 O0, 0) = 0 gives the stagnation point at the bow. The value of e 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 coverge 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 - it - fl(()(x - )d d ^ J j [(X _ ( )2+ y2+ (z._ )2]3/2 1 t1 fl(i) Y z-t z+t - + f ( 1) ) (t - ) di (x (x- +y2 + y2 r r' ~~~~~~~~~~-1 ~(5) t 1 fl(0) y d4 d_ - "Av= ~ [(x-_ )2+y2+ (z_-)2]3/2 ry 7-t z+t..

-7t 1 _ r r fl()(z - 5) dS d: - - J [(x - )2 + y + (z- 5)2]3/ -t -1 - fl(S) (r1 r- i -1 ) where r = 2(x )2+y2 (z - t)2 r' = (x- )2+ y2 + (z + t)2 For two dimensional flow, the streamline is given by 1 y = fl() tan-1 ) -1

III. COMPUTING METHOD AND RESULT For the present computations, the functions of singularity distribution in (1) and (2) were chosen as followso fl(S) = al sin (a s) - 1 < < 1 f2(~) = 1 - t < t TABLE I Hull Type A1 t Computor digit, required time C-101 0.4 o.1 HITAC - 301 10-decimal 12 digit, floato pt, 6-1/2 hours C-201 0.6 Oo1 BENDIX G-15D 10-decimal 7 digit, floato pto 4 hours. Two hull forms were computed for the values of parameters as given in Table IL For both hulls, 4 streamlines were traced starting at the following points i) x = 1, z = 0, y = l1 (waterplane) ii) x = 1, z = 0.05, y = ~2 iii) x = 1, z = 0,08, y = ~3 iv) x = 1, y = 0, z = o.1 + C4 (Keel line) The y and z coordinates were evaluated at the following values of x: x = 0.9975, 0,995, 0.9925, 0.99, 09 8, 0o6, 0.4, 0,2, 0 The velocity components were also evaluated at these points. -8

-9The initial values of 1, e2, E3 and e4 are determined according to the integral equation for streamlines of two twodimensional flow. When these values are two small for the computer, however, suitable modified values might be needed~ For the computed cases the following values were used. C-101L 1l = e2 = 3 = o12 x 10, 4 1010 C-201: =e = C = l169 x 10-6, e4 = 145 x 10-6 Since computation methods were different for the two hull forms, each method will be explained separately. 351 Hull Form C-10I It is observed that the integrands of Equations (3) have sharp peaks or steepness in the vecinity of the bow and stern. At these points the streamlines pass close to the singularities. Proper care must be taken in the computations to account for these features. In detail the extreme values were found to be located as followsu" peaks at ~ = x + y, equal to zero at x = x v: peaks at e = x w~ peaks at t = x Numerical integration was performed by the method of Legendre-Gauss, The region of 5 was subdivided into sub-regions of various lengths so that maximum accuracy could be obtained, i.e., coordinates were closely spaced in regions where the integrands displayed sharp variationso Figure 2 shows the sub-division of the region - 1 < i < 1 used in the computations of the velocity component uo

-10To check the final programming a linear distribution of sources (fl(S) = a1i), was tried. Professor Inui of the Tokyo University had independently computed the streamlines for this form previously. A complete agreement up to 3-digits was found to exist, Computation results of Hull Type C-101 are given in Tables II and IIIo 3.2 Hull Type C-201 The integrals to be computed in this case are exactly the same as in the case of C-101, but for the regions of close proximity of the singularities, an analytical treatment was employed., When = 1, sin (T2 )' 1o For 0 << E ~ 1, (3) can be changed to: 1 sin( )(x-) - Oo z+ol x z-Ol z+Ol j (x- + ) 2 -+ y2 r (-)2 1-~ 1-~ 2 2 (x-) + y in ( (x- g)2+y2 +(z - 0.1)2 (01 -z))((x 2+y +(z +0ol )2+(0O.1+ z)) l 1 1 sin(2 -)( ) d = ( ) d r r r r 1-E 1-E r z - 0. + (x )2 + (z - 01)2 ^=1 L nz + Ol +f(x _ )2 + (z + 0O1)2 j= 1 In this computation, E = 10-2 (sin (1-10- ) = 0o999), and -2 - 1 < 1 - 10 was suitably divided to apply numerical integration by Simpson's rule, The result is shown in Tables IV and Vo * Because of this treatment the precedure of calculation for C-201 is not suitable for a more arbitrary distribution of the singularity,

-11Two sets of hull lines, C-101 and C-201, each developed from four streamlines as obtained in the above are given in Figure 30 Also their off sets are given in Tables VI and VIIo

IV. CONCLUSION Hull types C-101 and C-201 were constructed into models of 2.5 meters, and were tested for resistance and wave observation at Tokyo University Towing Tank, Also, to these models, suitable bulbs were fitted at bow and stern, and Waveless Hull Form studies based on the interference cancellation of bulb wave and main hull wave were conducted (3) Finally, the author wishes to acknowledge guidance and assistance given by Professor Inui of Tokyo University and Mr. Fujino of the Technical Bureau of Mitsubishi Zosen Company, special assistance was given for the computation by Japan Electronics Industry Promoting Association and Tokyo Electronic Computor Service Ko Ko The author wishes to express his appreciation. -12

-15Table 2 Coordinates of Calculated Stream-Lines for Model C-101 Unit: L= 20 Square Station L.W.L. Stream-LineA Stream-LineB Keel Line Number Y y z y z z 10 0 10 000 * 000 500 000 800 1.000 9.9 20 / 9120 -025 -025 -508 024 -815 1.030 9.0 1/2 9/2 201 -196 -564 -167 -937 1.174 8.0 1 9 - 362 343 - 634 - 258 1.067 1.306 6.0 2 8 608 *552 *758 362 1.263 1.492 4.0 3 7 -773 -685 -848 -423 1.391 1.619 2.0 4 6. 868 1 7 59 -902 456 1.465' 1.693 0._ 5. *900 I -783.920 467 1.490 1.718 Table 3 Velocity on Calculated Stream-Lines for Medel C-101 Unit: Uniform Flow u0=1 L.W.L. S-L A S-L B K.L. u v u u Wv u W 10 * 0573 * 1001 - 0674 * 1001 ~ 0451. 0984 * 1001 * 1026 * 3405 * 6723 9.9 * 7995 - 1860 - 8107 ~ 1852 * 0448 * 8375 * 1825 * 0958 * 9493 * 2442 9.0 * 9601 1682 - 9632 * 1593 * 0659 * 9676 * 1131 * 1365 * 9762 * 1369 8.0 - 9930 * 1452 - 9930 * 1295 * 0683 * 9924 * 0704 * 1177 * 9927 * 1136 6.0 1.0174 ~ 1032 1.0157 * 0844 - 0549 1.0128 * 0393 * 0803 1.0119 ~ 0794 4.0 1.0286 - 0664 1.0263 - 0521 - 0368 1.0288 * 0235 0512 1.0215 *;08 2.0 1.0343 C326 1.0317 ~ 0251 0184 1.0280 0112 -0251 1.0264 0250 0. 1.0361 -0000 1.0333 -0000 0000 1.0297 ~ 0000 ~ 0000 1.0279 0000 Table 4 Coordinates of Calculated Stream-Lines for Model C-201 Unit: L-=20 Square Station L.W.L. Stream-Line A Stream-Line B Keel Line x --- _ —Number y y z y z z 10 I 0 10 002 " 002 -500 -002.800 1.000 9.9 9 Ao 910 041'041 -509 -038.817 1.040 9.0 /2 91/2 -295 2 83 590 -230 -987 1.256 8.0 1 9'515 4 78 688'340 1.155 1.418 6.0 2 8'8334 -739 853 -470 1.397 1.660 4.0 3 7 1.048 -905 968 -547 1.554 1.848 2.0 4 6 1.164 -994 1.033 -590 1.644 1.936 0. 5 1.206 1.025 1.057 -606 1.677 1.968

-14Table 5 Velocity on Calculated Stream-Lines for Model C —201 Unit: Uniform Flow u0 =1 L.W.L. S-L A S-L B K.L. x.. U v U v W U w w 10 *367 -155 -390 *150 *051 *430 *150 103.162 -857 9.9 -743 -258 -756 *258.058 -791 -252 -128 -875 -212 9.0 -947.230 *951 *212'094 -958'135'179 *963 *169 8.0 989 -192 990 -165 -093.990 087 -148 *991.138 6.0 1.022 *137 1.020 *108 *072 1.018 -050 -099 1.015 -124 4.0 1.037 082 1.034 063 *046 1.031 030 063 1.029 061 2.0 1.044 040 1.041 -030 -023 1.038 -014 031 1.035 -030 0. 1.047 001 1.044 001 * 00 8 1.040'004 011 1.037 01 1 Table 6 Off-Set for Model C-101 Unit L-20 Half Breadth y..\............. - Height of LW.L. 1 2 3 4 4} 5 534 6 61 7 Keel Line 0 0.24 0.48 0.72 0.96 1.08 1.20 1.32 1.46 1.58 1.68 0 10 1.000 4 9Y *'105 105 * 104 101 086 -032 / 9 * 203 202 *200 188 166 120 1.174 [ Y 9%/ 284 *284 280 268 232 186 *100 1 9 * 366 366 - 355 * 337 298 * 254 180 1.306 1 84 * 498 - 497 487 461 410 374 300 196 2 8 * 608 * 607 * 592 * 562 508 466 * 408 316 * 160 1.4 92 214 6 * 697 * 692 674 * 642 586 * 546 * 488 404 - 280 * 016 3 7 773 767'746 712 657 614 560 *484 376 204 1.619 4 6 868 -861 840 804 745 702' 648 574 475 344 088 1.693 5 -900 -893 -868 832 774 *729 -678.606 *512 -388 -184 1.718 Table 7 Off - Set for Model C-201 IUnit: L=-20 Ha f Breadth y..\\.......... —- Height \ L.W.L1 2 3 4 4 % 5 5 6 6~1 7 7 % 8 of Ke~l 0 0.24 048 0.72 0.96 1.08 1.20 1.32 1.46 1.58 1.68 1.80 1.92 0 10 1.000 4 94 166 ~166'163 157 -122 -089 -i 9~ -298'296 *290 *277 *236 *194'108 1 2S h 9 - 408 405'394 -378'334 290 221 092 1 9 515 *510 *498 474'421 *378 312 *209 1.418 1~ 8 * 686 *681 *669.637'573 -526'462 *382 261 2 8 * 834 *830 *815 -773'703.656 597'526 -433.296 1.660 2~ 7' * 955 * 949'933'893 - 826. 784.725 -653 -562 * 447'282 3 7 1.047 1.040 1.02C0'976'908'868 814 -739 -652 -544'412 -212 1.848 4 6 1.161 1.129 1.121 1.085 1.018'972 -917 -852 -773 -674 - 548 -390-120 1.936 5 1.206 1.194 1.170 1.125 1.060 1.012 - 962. 88 816 - 20 -6 -1 -232 1.968

15z;tTF C =T Im(a.,) o)7 Figure 1. X9 X7 Xl x1o Xs Xe X I u,. I -(5 03- m'o I -I —a Figure 2.

-16C -101 B/L=0.090 TF/L: 0.050 TM/L =0.0859 C - 201 B/L0. 121 TF/L 0.050 TM / L0.0984 Figure 3.

BIBLIOGRAPHY 1o Inui, To "Wave-making Resistance by Correct Hull Boundary Condition," Japan TSNA No. 93 (1955)o 2, Eggers, W,: Uber die Ermittelung derSchiffsahulichen Umstronlungs-Korper Vargegebener Quell-SinkenVerteilungen mit Hilfe dectromischer Rechenmachinen, Schiffstechnik Bd, 4 (1957) Heft 24, 284 - 288~ 3. Takahei, To "Study on Waveless Bow" (No, 1). -17