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 Final Report RESEARCH IN RESISTANCE AND PROPULSION Part IV. Studies on Wave Interference of Mathematical Hull Forms with Large Bow Bulbs Tetsuo Takahei Finn C, Michelsen Project Director~ R. B. Couch ORA Project 04542 under contract witho MARITIME ADMINISTRATION U. So DEPARTMENT OF COMMERCE CONTRACT NO. MA-2564, TASK 1 WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR April 1.963

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS v ABSTRACT vii I. INTRODUCTION 1 II. THEORY OF WAVE-INTERFERENCE 3 III. EXPERIMENTAL RESULTS 6 REFERENCES 27 APPENDIX I. AMPLITUDE FUNCTIONS 29 APPENDIX II. MEASURED WAVE PROFILES AND THEIR ANALYSES 37 iii

LIST OF ILLUSTRATIONS Table Page I. Particulars of Variations of C-101 Model (U of M No. 938) 7 II. Particulars of Variations of C-201 Model (U of M No. 945) 8 III. Bulb Data for C-101 and C-201 10 IV. Test Conditions for C-201 (12-ft LWL) and Bulb Data 11 Figure 1. Details of F1 bulb connection to hull C-101 and CFK-101. 12 2. Details of F2 3ulb connection to hull C-101 and CFK-101. 13 3. Total model resistance of C-101 with rocker bottom (UM 938) and CFK-101 with flat bottom; 12-ft models. 14 4. Total model resistance of CFK-101, CFK-lOlFl, and CFK-lOlF2. 15 5. Cr for cosine from C-101 with rocker bottom and C-lOlF1 16 6. (a) Cr for C-101, CFK-101, CFK-lOlFl, and CFK-lOlF2. 17 6. (b) Cr for C-101, CFK-101, CFK-lOlFl, and CFK-lOlF2. 18 7. EHP for C-101 and C-lOlF1. 19 8. EHP for C-101, CFK-101, CFK-lOlFl, and CFK-lOlF2. 20 9. Total model resistance for C-201 and C-201F2. 21 10. Total model resistance of C-201 and CFK-201; 12-ft models. 22 11. Cr for C-201 and C-201F2. 23 12. Cr for C-201 and CFK-201 24 135 EHP for C-201 and C-201F2. 25 14, EHP for C-201 and CFK-2O01. 26 v

LIST OF ILLUSTRATIONS (Concluded) Figure Page I-1. Amplitude functions vs. speed for hull C-101 and two bulbs, spherical and conical. 30 I-2. Amplitude functions vs. direction angles for hull C-101 and two bulbs, spherical and conical. 30 I-3. Amplitude functions vs. speed for hull C-201 and a bulb (F-201F2). 32 I-4. Amplitude functions vs. direction angles for hull C-201 and a bulb (C-201F2). 32 I-5. Doublet distribution of "conical bulb" investigated. 33 I-6. Wave profile by sphere. 35 I-7. Wave profiles by conical bulb. 35 II-1o Comparison between measured and calculated wave profiles for C-201 and C-201F2. 42 II-2. Comparisons between measured and calculated wave profiles for C-101 and C-lOlF1. 46 II-3. Effect due to modifying the bottom flat for CFK-lOlFl. 51 II-4. Comparison between CFK-101 and CFK-lOlF2. 53 vi

ABSTRACT A comprehensive experimental and theoretical study has been made of the wave-resistance characteristics of the cosine hull forms associated with bow bulbs. These hull forms have previously been investigated by Inui in Japan. The current study, which is based on larger models, confirms, in general, results obtained by him. In addition new conditions of hull and bulb shapes have been investigated. vii

I. INTRODUCTION A ship hull form, being normally of a rather complex shape, will generally produce a wave pattern which can be broken down into several distinct systems such as bow, stern and shoulder waves. All these systems will be in interference with one another, and the success of a particular hull design can usually be judged by how well the designer has been able to produce a favorable interference (wave cancellation). Over the years a great number of emperical rules have been presented to the naval architects, rules which proclaim to provide them with the means of obtaining optimum hull forms. It is to be realized, however, that the theory of wave-resistance provides the only method by which wave cancellation can be studied in a rational manner. In addition to the wave interference that exists between the various wave systems produced by the parent hull form itself, further interference can be effected by the introduction of appendages. The bow bulb is the only appendage form that has so far found general usage in practical hull designs. Experimentally, the benefit of bow bulbs was well established by D. W. Taylor. On a theoretical basis, their effects were first studied by Weinblum in Germany and Wigley in England. By applying the linear wave-resistance theory to the problem of wave cancellation, these investigators were able to essentially verify D. W. Taylor's predictions. They did not proceed far enough in their investigations, however, and it was left to Inui and his Japanese colleagues to produce what is now being generally referred to as a "waveless " hull and bulb combination. It will become clear from the formal formulation of the problem, which is reproduced in outline in this report, that complete and exact wave cancellation is not achieved by bulbs. For a given speed) it is possible, however, to reduce the wave-resistance for the hull and bulb combination to a small fraction of that of the hull alone.* Thus the term "waveless" is appropriate for all practical purposes as long as one is referring to a particular speed. It is fortunate that a bulb also proves to be beneficial over a substantial speed range extending both above and below the design speed. This is clearly evidenced by the results of this report. The failure of the early investigators to realize a "waveless" hull form by means of bulb appendages was apparently due to a lack of computational facility which was necessary to compute the traces of streamlines for the determination of hull forms and also the analysis of wave patterns. *For maximum wave cancellations it is necessary to use both bow and stern bulbs. The bow bulb is by far the most effective and only wave cancellation effects of bow bulbs are included in this report..1

The development of new functional relationships of the wave-resistance integral led to the conclusion that for minimum total wave resistance of a hull and bulb combination, it is necessary to have a basic hull form which by itself produces a simple total wave system. The parent hull need not be one of optimum form to obtain a minimum wave resistance of the hull-bulb combination. Stated in a different way, the design of the hull and the bulb must be considered jointly. Thus, it is not surprising that the first studies on "waveless" hull and bulb combination, performed in Japan, were made with the so-called cosine parent hull form. This form has the advantage that it has no shoulder waves, and furthermore, it can be shown that cancellation of bow waves can be almost completely accomplished by means of a single concentrated doublet (spherical bulb). The experimental work described in this report represents in certain aspects a repetition of tests performed in Japan. It was considered important to verify results obtained by Inui with 2.5 m model length by using the somewhat larger model size of 12 ft. The greater part of the experimental work on the cosine hull forms performed at The University of Michigan constitute an extention of investigations performed in Japan. The most important extention is probably the series of tests with flat bottom models. It has been shown in a previous report (12,p.51) that the depth of the original cosine hulls as obtained from streamline traces is about twice the draft at the bow and stern. The "rocker" bottom produced by these traces have been subject to criticism since it is duly felt that such a bottom can never find any practical application. It was, therefore, essential that tests with flat bottoms be made to determine what effect this feature would have on the overall performance of the hulls and also if a flat bottom would influence the effect of the bow bulb. Results obtained are encouraging, indicating a reduced resistance over certain speed ranges. It should, however, be remembered that both wetted surface and displacement has been materially reduced. Optimum bulbs have been reported by Inui for particular speeds V =.87 forC-101 VL =.90 for C-20. Other bulbs suitable for L VL different speeds have been designed and tested at The University of Michigan during the past year and available results are reported herein. In our particular case, a modification of a bulb to make it more suitable for the flat bottom of the C-101 proved to improve on the results reported by Inui. In another case, a bulb designed for a speed length ratio of 1.1 was shown to be less effective at design speed than the smaller bulb designed for the speed length ratio of.87. These results point to the fact that, although the linear wave resistance theory will predict possible reductions in wave resistance, it hardly eliminiates model testing as the ultimate mean by which the final determination of design parameters are made. 2

II. THEORY OF WAVE-INTERFERENCE Assuming that a ship is moving in the direction of the negative x-axis with the origin located at the bow, the wave pattern of the bow system can for large values of x be defined by (X) () sin[Ko sec2 O(x-xo cos 0 + y sin 8)]dO (1) where AF(G) is the amplitude function determined from hull geometry and Ko - g The subscript F refers to the bow wave system. The corresponding expression for the wave system produced by a bulb located at (xo, 0, f) is;w,,D("XY) -A L BF(0) sin [Kc sec2 e(x-xo cos 0 + y sin e)l]d (2) 2 Superimposing the two wave systems of Equations (1) and (2) we get D(X'Y) - X (AF(O)-BF(O)) Sin [Ko sec2 e(x-x cos a + y sin O)lde 2 (3) The amplitude functions A(0) and B(S) are described in more detail in Appendix I. The contribution to the wave-resistance by the bow wave system has been shown (1) to be given by 2 i_ 2 and it follows directly from Eq. (3) that the wave-resistance due to the bow wavesystem and the bulb wave system becomes 3

Jt w,F+D= PVV2 2[AF(e)(BF(e)]2 CS3 de d(5) 2 _(e 2 Equation (5) -shows that the condition of zero wave-resistance is met if and only if AF(O) = BF(0)o It is indeed difficult to imagine,'from a physical point of view, that two bodies of completely different shapes can produce wave-systems that are of equal amplitude for all values of e. From a practical point of view the equality of amplitude functions mentioned above does not have to be fully satisfied, however. As e0 +I, cos 0 -+ 0 and the factor cos3 0 of the integrand of Equations (4) and (5) shows that the contribution to the wave-resistance of the waves associated with the larger values of 0 is negligible. In a physical sense this means that wave components which propagate in a direction close to being perpendicular to the direction of motion of the ship have a negligible effect on the wave-resistance. Assuming that a hull form is properly represented by a distribution of singularities on the center-plane of symmetry, the wave amplitude function can be completely determined from this distribution. In the case of the cosine ship form, the distribution is such that only bow and stern wave systems are produced. Furthermore it can be shown that the resultant wave system is a sine system (xo = 0 in Eq. (1)). It is noted from Eq. (2) that the wave system due to a doublet located at Fo Po is an inverse sine system and the condition of inverse phase is satisfied for the particular hull form in question. The inverse phase relationship is of the utmost importance to wave cancellation and is indeed implied in writing Eq. (3) as the sum of Eqs. (1) and (2). It was also implied that xo = constant. In general this is not so. In fact, the general form of Eq. (1) is SwF(x,y) ~ i2AF(0) sin [Ko sec2 E(y+XF(cos 0 + y sin E)]dG (6) 2 where [iAF(e)3 = 72(e) + CF2(E) and -=tan't W} I/ " seo

SF(e) and CF(O) are the amplitude functions of the sine and cosine components respectively. It is noted that xF is a function of both 0 and Ko and it follows therefore that it is not possible to write Eq. (5) for the general case of hull forms, assuming the bulbs to be represented by a single doublet. A bulb represented by a distribution of doublets or even a single doublet may still produce a substantial wave-interference, however, provided it is located properly. If the cosine component of the bow wave system is predominent the center of the bulb should be moved aft, for instance, and the optimum position of its center may actually be aft of *the forward perpendicularo The most desirable main hull form for maximum wave cancellation is one that does not have any shoulder waves and creates a bow wave system that calls for the bulb being located within a practical distance of the forward perpendicular. To accurately determine size, form and location of bulbs, full use must be made of the analysis of wave patterns together with theoretical calculations of the amplitude functions and wave profileso The University of Michigan does not as yet have the necessary facilities for complete analysis of wave patterns and neither are all the computer programs ready for use. For the analysis of the results obtained with the cosine hull forms it was possible to use computations made previously in Japan by Inui and Takahei. In this way it was possible immediately to continue important research and thus lay the foundation for continued investigation of wave cancellation effects of bulbs. 5

III. EXPERIMENTAL RESULTS The experimental results reported here pertain for the greater part to C-101, the narrower of the two cosine hull forms tested. (C-101 and C-201.) Particulars of the hulls are given in Tables I and IIo Traced streamlines are shown in Figs. 3 and 4 of Ref. (2) and also in Figo II-1 in Ref. (12). Figures 1 and 2 show the details of C-101 including bulbs and bottom cut-off. The details of C-201 are similar. The source distribution representing the cosine hull forms is given by m(t) = a cos (3k) (7) where the origin is located at the forward perpendicular. The draftwise distribution is uniform. Within the Michell approximation the waterlines are cosine curves with respect to amidships. Because of finite beam the waterlines are not exactly cosine curves however and there is also a slight difference between the forms of the waterlines of C-101 and C-201. Models were made in wood and so designed that bottoms and bulbs could be easily interchanged. Smooth fairing between bulbs and hull was made in wax. In the designation of individual hull bulb combinations subscripts FK refers to flat keel and F1, F2, etc., identify the bow bulbs. Bulb C-1O1F1 was designed at The University of Michigan to replace the original bulb C-lOlF tested in Japan. Comparisons between tests of bare hull and hull with bulbs are based on the same draft of models, in agreement with the procedures used in Japan. The.models were not restrained in trim or heave, howevero In Japan the models are customarily restrained from trimmingo Preliminary tests at The University of Michigan revealed that the effect of trim is negligible when the amount of trim is small, and it was found that this condition was satisfied in all tests made. For turbulence stimulation a trip wire of.036 inO diameter was attached to the model at o05L from the bowo In addition, studs of.032 in. diameter and.020 in. length were spaced.5 in. apart in a circle around the head of the bulb (see Figs. 1 and 2). Injection of dye ascertained that the flow around the bulb behind thle studs was turbulent for even the lowest speeds of interest. No blockage correction has been applied to test results. Such corrections would be very small and besides no correstions were used in the analysis of the data obtained at Tokyo University. Blockage ratio (the ratio of model crosssection to tank cross-section) is approximately 1/160 for the model sizes used in the cosine hull form tests for both the Michigan and the Tokyo tanks. 6

TABLE I PARTICULARS OF VARIATIONS OF C-LO1. MODEL (U OF M NO. 938) C-101 C-1101F1 CFK0.1l CFK-01lFl - CFK-101F2 Model Ship Mo elShiphip Model Ship Model Ship Model Ship M Dimens ions LWL, ft 12.00 600oo 12 00 600oo 12.00 600 12. 00 600oo 12.00 600 LOA, ft 12.00 600oo 12 72 636 12.00 600 12.72 636 12.97 648 B, ft.083 54.2 1.o083 54.2 1.083 54.2 1.083 54.2 10083 54.2 HF, ft 0. 600 30 0.720 36 o 600 30 0.720 36 0o917 45.9 HAs ft 0o 600 30 o 60 0 30 0.600 30 0600 30 0o 600 30 HX, ft 1o030 51.5 L[030 51.5 0.720 36 0.720 36 0.720 36 V, ft3 6911 -- 7.058 -- 5851 -- 5.998 -- 6120 -- A, tons -- 24, 700 - 525,200 -- 20,800 -- 21,400 21860 S, ft2 25.55 6339900 26.678 66,700 235057 57,650 24.185 60, 450 24.739 61,750 V/f and'V design -- -- 0.85 21 kt. - 085 21 kt 1.1 27 kt Coefficients B/LWL 0 0904 o0904 0.o0904 0.0904 0.0904 HX/LWL o. 0859 o.o859 0.060 0o060 0.060 CB 0.515 0.526 0.627 0.642 0.655 CX 0.795 0.795 0.913 0.913 0.913 Cp o.648 o.663 0.697 0.705 0.719 LWL/7V / 6, 30 6.26 6.66 6 60 6.55

TABLE II PARTICULARS OF VARIATIONS OF C-201 MODEL (U OF M NO. 945) C-201 C-201F2 CFK-201 Model Ship Model Ship Model Ship Dimensions LWL, ft 12.00 6oo 12.00 600 12.00 600 LOA, ft 12.00 600 13053 652 12.00 600 B, ft 1.446 72.2 1.446 72.2 1.446 72.2 HF ft o.6o 60 0.902 45.1 o.60 30 A, ft o.6o 30.6o 30 30 60o 30 HX, ft 1.180 59 1.180 59 0.641 32.1 V, ft3 10o 320 -- 0.685 -- 7. 312 -- A, ton -- 35,900 -- 37,100 -- 25,400 S, ft2 28.006 69,500 30.489 76,200 24,850 -- V/I and V, design -- - 0.9 22 kt -__ Coefficients B/LWL 0.1206 0.1206 0.1206 -X/LWL o 0.o984 o. 0o84 0.05355 CB 0.503 0.524 0.657 CX 0. 783 0.783 o0.946 co.643 o.670 o.695 WL/V13 5.52 5.45 6.18 Although the proportions of the cosine hull forms are not suitable for practical ship designs, it was deemed desirable to calculate the resistance for a hypothetical ship of 600 fto length based on the model resultso In so doing the 1947 ATTC friction extrapolator with a ACf =.0004 was used throughout. Japanese data originally incorporated the Hughes extrapolator with a form factor K. For comparison these data have been re-evaluated on the same friction basis as the Michigan datao The values of the residuary resistance coefficients are based on the square of the model lengtho This was done to avoid any ambiguity that might have been introduced if the wetted surface had been used, since the area of this surface varies from model to model. In summary the results of the C-101 series of tests are shown in Figs. 35-8. Some comments in regard to these figures are in order. 8

Ao C-101 Compared to the Japanese data the larger Michigan model exhibits slightly lower values of Cr throughout the speed range. The difference may have resulted from a too small steepness of the ATTC friction curve at low Reynolds numbers. The constant difference in Cr values supports this point of view. The difference is small, however, and the agreement between results can only be judged to be very good. Bo C-101lFl The bulb F1, which is a modification of the bulb F tested in Japan, produced excellent wave cancellation effects at the design speed of V//~ =.87. The modification was made for the purpose of obtaining a better fit with model CFK-101. The cancellation effect of bulb Fl proved to be superior to that of bulb F even when fitted to the original parent form C-101o This can be easily verified from Fig. 5 and also from the wave profiles shown in Figs. II-2(a-e) of Appendix II. Special attention is called to Fig. II-2(a) showing the wave profile at design speed. Details of the bulb designs are as shown in Table III. C. CFK-101 The effect of the flat bottom is to increase Cr as shown in Fig~ 4(a) and (b). When the data are extrapolated to the 600 ft. ship, however, it is found that the total resistance is approximately the same for both C-101 and CFK-lOl1 The decrease in frictional resistance due to a smaller wetted surface appear to have compensated for the increase in residual resistance. Do CFK-lO1F1 Bulb Fl was equally effective as a wave cancelling device in the case of the flat bottom model as when attached to the parent form C-101 (C101F1). Eo CFK-101F2 The F2 bulb was primarily designed for the hull C-101 to minimize the waveresistance at a Fronde number of.26(V/IL = 1ol). To date it has only been tested with the hull CFK-101, however. Although the F2 bulb was efective in reducing the wave resistance the smaller F1 bulb was more so even at V/ = 1.1oo It is too early to give a full explanation for these results. Some plausible explanations may be offered at this time, 9

however. The size of the optimized bulb increases rapidly with the design speed (approximately proportional to the cube of the speed). With the larger bulb more fairing is needed to fit it to the main hull. This fairing may change the wave making characteristics of the main hull sufficiently to eliminate some of the wave cancelling effect of the bulb. Another factor, which is described in Appendix I, is that for the case of the cosine hull forms the condition of equal magnitude of the amplitude functions A(e) and B(e) become rather difficult to satisfy at higher speeds over a significant region of values of O. The amplitude functions of a variety of bulb shapes do not change materially as a function of 0 at various speeds. It follows therefore that it is difficult to accomplish a better wave cancellation by a redesign of the bulb. The course to follow is therefore to design the hull to fit the bulb, and not vice versa. Future studies will be based on this approach. The characteristics of the F2 bulb are as shown in Table III. TABLE III BULB DATA FOR C-101 AND C-201 Percent Bulb Designed at ao/L AV/V AS/S AB/AX f/T Q/L C-lOlFl UM 2.0 2.13 4.41 21.0 4 4 C-1OlF Tokyo Univ. 2.7 3.60 5.80 20.0 4 5 C-101F2 UM 2.8 35.87 6.60 40.5 4.8 5.53 C-201F2 Tokyo Univ. 5.2 4.o6 9.96 22.1 5.4 6.0 ao radius of a sphere circumscribed by the bulb L length of ship AV/V ratio of bulb volume displacement to that of main hull AS/S ratio of bulb wetted surface to that of main hull AB/AX ratio of maximum cross-sectional area of bulb to that of main hull at midship f bulb immersion location of bulb center forward of FP in longitudinal direction Test results on C-201 series is summarized in Figs. 9-14. For these figures the following comments are added: 10

F. C-201 As can be seen in Fig. 11, small differences exist between the results obtained in Japan and at The University of Michigan. G. C-201F2 At The University of Michigan, the model with bulb was towed free to trim and heave at different draft. Trim was found to be negligibly small, being less than 0~2 of a degree. These tests are identified as Test Nos. 2, 3 and 4. No appreciable difference was found due to the change of draft although the tested depths of immersion of the bulbs were varied considerably. The following table shows the test conditions indicated in Figs. 9, 11 and 13. For bulb data see Table III. TABLE IV TEST CONDITIONS FOR C0-201 (12-FT LWL) AND BULB DATA Test Test Conditions Model Draft V, ft3 Wetted Surface, ft2 1 C-201 Designed 10 320 28.oo006 2 C-201F2 Designed 10.683 30.489 3 C-201F2 DWL —O 16 in. 10.523 (1% less) 30.165 4 C-201F2 DWL —0.48 in. 10.203 ( 3 less) 29. 519 H. CFK 201 Figures 12 and 14 show the resistance of this model as compared to 0-201. It is noted from Fig. 12 that above a speed length ratio of 1.2 the Cr for CFK-201 is less than that of C-201. This is surprising, especially since *the coefficients are calculated on the basis of the wetted surface. It appears therefore that the removal of the rocker bottom has had a slight beneficial effect in regard to wave-resistance. This feature needs further investigation from a theoretical point of view. 11

LWL __ l 1 2 31. —t i rENTF~E F. IJOINT NT -101!~-cTO\ 2/ //;/ H) ___ ___ ___I ___F1 BULB PROFRLE 3 I I d 4 2 4 FP (FP)~ ~~~~~~(L LEGEND s,, (.................(FLAT. BOTTOM) Fig. 1. Details of Fl bulb connection to hull C-101 and CFK-lOl.

LWL 4 1235.......................- ( JC-N. CENTER QF~~~~~~~~~TON ~;~.~~ CEn~5nF!C' Vl I I_ CURVATURE i / -101 - - _2x _ _//__ ~~~F2'~BU{~Bi ~ P-I1(FL Fig. 2. Details of F2 bulb connection to hull C- and CFK-Lo1. i ~~~~~~~~~(FLAT BOTTOM) Fig. 2. Details of F2 bulb connection to hull C-101 and CFK-101.

14 - Legend - e C-101 (690F) A CFK-101 (71~F) 12..... 10 6;1; 6~~~~~~~~~~~~/ 4.6.8 1.0 1. 2 1.4 O I...l...... I 1 I 11 I I 3 4 5 6 7 8 Model Speed, fps I l I I I I I 3 4 5 6 7 8 9 10 Reynolds Number x 10-6 Fig. 3. Total model resistance of C-101 with rocker botton (UM 938) and CFK-l101 with flat bottom; 12-ft models. 14

14. / Legend -- {3 CFK-l101 (710F) - CFK-lOlFl (70~F) 12 CFK-lOlF2 (68~F) 10 ---- I I 10 1 1...... 84.6.8 1.0 1.4 I I I I. I I I I 3 4 5 6 7 8 Model Speed, fps 3 4 5 6 7 8 9 10 Reynolds Number x 10-6 Fig. 4. Totalmodel resistance of CFK-lOl, CFK-lO1Fl, and CFK-101F2. 15

6 t / Legend t —O C-101 (U of M Model 938) --- C-101lF (U of M Model 938 Fl) 5 C-io1 ~/C-101F C-lOlF j From Japanese data with 8.2-ft models Froude Number I I r 11 I I I I I 1 5 16,, i i.......__ _..6.7.8.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 Speed-Length Ratio Fig. 5. Cr for cosine from C-101 with rocker bottom and C-1O1F1.

5 - - CFK-lOlFl l - CFK-1l1F2 L e g'end I:/ le-...... 1speed Fig.-(a). C, for C-101 CFK-1a CFK-lO1 1 F2. 2 _/.6.7.8.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6

Legend C-101 10 -- i, CFK_101 O: l~ ~-l' Des Speed,_10- CFK-l0 Speed-Length Ratio Fig. 6(b). C_ for C-101, C CF — CiF ___- 10 F 2. -r 8 r / o a- _ _ _ _ _I_ _

Legend C-101 C-101F1 o O 50 1 50 20 2a o w rI4 / Speed I Speed-Length Hatio.5.6.7.8.9 1.0 1.1 1.2 15 20 25 Ship Speed, knots Fig. 7. EHP for C-101 and C-1O1F1. Ship length = 600 ft. 19

75 Legend C-101 CFK-101O CFK-101F2 o0.... >D~~~~~~~-20' Pq 50 o q o 20 CFK- 10 F1-/ 0 p|S.rI /c 25 Des gngn Speeded forF2 For Fl Speed-Length Ratio.6 1.0 15 20 25 Ship Speed, knots Fig. 8. EHP for C-101, CFK-101, CFK-101F1 and CFK-101F2. Ship length = 600 ft. 20

I I I! I I 28 Legend 26 - C-201 (Design draft, water temp. 700F) O- -~ - C-201F2 (Design draft,* water temp. 710F) 24 -- C-201F2 (Design draft less 0.16 in., water te. 71~ t 9 —-- C-201F2 (Design draft less 0o.48 in., water te~p. 710F) 22 t *Design draft bulb center depth = 7.8 in. Model LWL = 12 ft o 20 t 18 co 16 10 __o 4 12 -.6.8 1.0 1.2 1.4 Speed-Length Ratio 4 6 7 8 10 I Ir A I I II Reynolds Number x 10Fig. 9. Total model resistance for C-201 and C-201F2. 21

28 26 24 22 Legend 20 - C-201 (Water temp. 680F) CFK-201 (Water temp. 700F) 18 - o 16 - m 14 O 12 2 4 5 6 7 8..... L,I, I, I, I,. Model Speed, fps.6.8 1.0 1.2 1.4 I I I I I l I I I Speed-Length Ratio 4 6 7 0 Reynolds Number (x 106) Fig. 10. Total model resistance of C-201 (rocker bottom) and CFK-201 (flat bottom); 12-ft models. 22

12 - Legend - -& — C-201 (Design draft) -A —-- C-20LF2 (Design draft; bulb center depth = 7.8 in.) 10 - C-201F2 (Design draft less 0.16 in.) — a- C-201F2 (Design draft less 0.48 in.) C-2017 C 201F2 J Japanese Data with 8.2-ft model 0 6. 0 IV~~~~~~~~~-.6.7.8.9 1.0 1.1 1.2 1..4 Speed-Length Ratio Fig. 11. Cr for C-201 and C-201F2.

12 11 10 Legend C-201 8 7 6 3 2 1 Foude Number 15.20.25.30.35.40.6.7.8.9 1.0 1.1 1.2 1.3 1.4 Fig. 12. Cr for C —201 and CFK-201.

Thousand Effective Horsepower Imi R)~~~~~~~~~C \J1 IF.0 \ 0 01 I ~~~~~~~~ H O'i d I M~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~r CD 0~~~a 0 5 O~~~~~~~~~~~~~~~~~~~ PI 0 U1 ~ ~ ~ ~ ~ ~ ~ ~ ~0 CD P~r td (D R)~~~~~~~~~( PI ~ ro 1. 0 n1 ro n x F CD Cf)~~~~~~~~~~~~~C O (D Ct vl~~~~~~~~~~~~0 0 Percent Reduction =s'~~~in EHP Due to Bulb I' / OP ct~~~~~~~~~~~~~~~~~~~~ ~ ~~~~~~~~~~~~~~~~~Percent Reduction in EHP Due to Bulb

75,000 Legend C-201 CFK-201 0 P) h 50,000 C H 10 15 20 25 30 Ship Speed, kt Fig 14. EHP for C-201 and CFK-2Ol. Ship length = 600 ft. 26

REFERENCES *1. "Wave Profile Measurements on the Wave-Making Characteristics of the Bulbous Bow," by T. Inui, T. Takahei, and M. Kumano, Journal of Society of Naval Architects of Japan, Vol. 108, 1960o *2. "A Study on the Waveless Bow," Part I, by T. Takahei, Journal of Society of Naval Architects of Japan, Vol. 108, 1960. 3. "A Study on the Waveless Bow," Part II, by T. Takahei, Journal of Society of Naval Architects of Japan, Vol. 109, 1961. *4. "A Study on the Waveless Stern," Part I, by M. Kumano, Journal of Society of Naval Architects of Japan, Vol. 108, 1960. 5. "A Study on the Waveless Stern," Part II, by M. Kumano, Journal of Society of Naval Architects of Japan, Vol. 109, 1961. 6. "A Study on the Waveless Stern," by M. Kumano, Part III, Journal of Society of Naval Architects of Japan, Vol. 110, 1961. 7. "The Wave-Cancelling Effects of Waveless Bulb on the High-Speed Passenger Coaster M/S KURENAI MARU; Part I: The Model Resistance and Propulsion Experiments, by T. Inui and T. Takahei; Part II: The Full Scale Experiment, by M. Shigemitsu and K. Kai; Part III: Photogrammetrical Observation of Ship Waves," by T. Inui and T. Takahei; Journal of Society of Naval Architects of Japan, Vol. 110, 1961. **8. "A Study on the Large Bulbous Bow of High-Speed Displacement Ships; Part I: Resistance Tests in Still Water," by S. Takezawa, Journal of Society of Naval Architects of Japan, Vol. 110, 1961. **9. "A Study on the Large Bulbous Bow of High-Speed Displacement Ships; Part II: Performance in Waves," by S. Takezawa, Journal of Society of Naval Architects of Japan, Vol. 111, 1962. 10. "Fishing Boat of the Waveless Hull Form" (in English), by N. Yokoyama, Journal of Society of Naval Architects of Japan, Vol. 110, 1961. 11. "Wave-Making Resistance of Ships" by T. Inui, T SNAME, Nov. 1962. *Translated into English at The University of Michigan, Dec. 1961. **Translated into English at The University of Michigan, Part I: May 1962; Part II: July, 1962. 27

12. Takahei, Michelsen, Kim and Salvesen. Research in Resistance and Propulsion, Part II. Streamline Calculation for Singularities Distributed on the Longitudinal Centerplane, University of Michigan Office of Research Administration Report, 04542-2-F, February, 1963. 28

APPENDIX I AMPLITUDE FUNCTIONS The singularity distributions of the "cosine ship" form has the property that both bow and stern free travelling wave systems consists only of the sine wave component throughout the whole speed range, i.e., in Eq. (1), Xo = O. The amplitude function has been shown elsewhere (2, transl. p, 52) to be given by all V(KoT,0G)sec2G AF(G) = E K~ L (KoL sec G)2-2 (I-1) where V(KoT2G) = 1 -exp (-KoT sec2G) with a1 = 0.4 for C-l01 and a1 = o.6 for C-201 AF(G) increases with speed. This is shown in Fig. I-1 for C-101 for a few values of G (C-101 identified by U-shape frame lines). For G varying from zero to about 35~, Eq. (1) refers to the transverse wave component at any location, whereas for G > 35~> divergent wave components is referred to. In the same figure, the curves identified as the V-shape frame lines show the values of the amplitude function for a main hull form generated by singularities whose lengthwise distribution function ml(S) is the same as that for the C-101. The draftwise distribution function m2(S) is changed from a uniform to a linear function, however, i.e., from m2( ) = 1 to m2( ) = 1:- /t. Compared to a U-shape frame line, a V-shape frame line has smaller component 29

AFRW.8.. L La 1.8 Ox _ _S_- S PHERICAL BULB N' " - -— N CONICAL BULB 1.6 \~ \\'. 5 MAF - MAIN HULL 1 \4 \ "\44 " "" \\ \20' \\ \' \\ N \ ~,, \ "\ "N,50RE ) N- 60 0,~~~~~6 N NV/ N 1oi~~. (!U"-SAP).0o. - Fig. I-i. Amplitude functions vs. speed for hull C-101 and two bulbs, spherical and conical. 2.0 F= 0.250 K.L 1/ V/6 O. 87 A(0L -' 1, 1.0 SPHERICAL BULB Bs(0 | - - - - - -- - - JIAIMI~ALLVSI~ATFR1 LIFRANE LINE) - (DERN HE L (E) SHAFs FRAE- LIN).20,I tAF (e)-B6(O}cm/~ MN HUI1 AF WITH BC WI ~ k \ 10 20 30 40 50 60 R0 80 C-101 and two bulbs, spherical and conical.

in the transverse wave system. The same fact is observed from the upper figure of Fig. I-2. It is also noted that the ratio of the divergent wave component to the transverse component is larger for shallow draft vessels* than for deep draft vessels. Figs. I-3 and I-4 show the amplitude functions for C-201. Observations similar to those above can also be made in this case. The free wave pattern to the rear of a doublet with the axis in the direction of advance is described by Eq. (2) in the main text, where the amplitude function is given by MK2 ) = sec4G exp (-Kof sec2e) (I-2) The negative x-axis is in direction of advance, f is the bulb immersion, and M is the strength of the doublet. For an axisymmetrical three dimensional flow in an infinite media M = 2TaaV corresponds to a sphere of radius ao0 It is, therefore, referred to as "sphere bulbo" Figures I-1 and I-3 show the variations of the amplitude function of a sphere bulb with respect to speed changes. From these figures it is noted that the condition of approximately equal magnitudes of AF(G) and BF(O) over a significant range of G can only be attained. over a narrow speed range since the variations in the amplitudes of these two functions with respect to change in speed are opposite. A distribution of doublets has therefore been investigated for the purpose of comparison with the above-mentioned *The V-shaped source distribution will produce smaller maximum draft amidships than the uniform draftwise source distribution (U-shape frame line). 31.

C-20/ F2 BULB -- MAIN HULL L 1.2 - L 0.8o.6DESIGN1C0.2 / SPEED, I I / I.12.14.16.18.20.22.24.26.26.30.32.34.36'i 1 1 J I I J I I I I X.6.8 1.o0 1.2 I.4. Fig. I-3. Amplitude functions vs. speed for hull C-201 and a bulb (F-201F2). C-201 F2 F= o.267 K.L=14 =1O.4.o c20.11.0 L BF(e)lo, CULBI e), ~"/L:O. 03' iL:O. O5 10 i15 188 /8 o0 20 30 40 50 60 70T 80 0 (DEGREE) Fig. I-4. Amplitude functions vs. direction angles for hull C-201 and a bulb (C-201F2). 32

concentrated point doublet. If the doublets are distributed in the vertical direction, the phase of the free travelling wave will be unaltered as compared to that of a point doublet. Therefore, it might be possible to select the following doublet distribution to meet the wave cancellation requirement; 1~(G = 0o t; -f0 < 5 < -ft (I-3) fo-ft This distribution is shown in Fig. I-5. z -ftV Fig. I-5. Doublet distribution for "conical bulb" investigated. The amplitude function of such a distribution,which may be called a "conical bulb," is given byo LBFC(G)]l = K~ %seo2 0 e-Kofosec2_ eosec e-Kofosec (-4) IBFC(G) Irv -e +' K(fo-ft'sec20 (I-4) For ft = 0, a linear distribution function extending from the surface down to the depth fo is obtained, for which we geto Ko osec2 -e KKffsec2G 1 - e-Kofosec2-5) _ BFC(G) 2 e fsecG BFTI~e~lp %eV Kofosec To compare the characteristics of the amplitude functions of Eqso (I-2) and (I-5), let us consider the following example. 5533

KoL = 16, Fronde No. 0.25 or V//L = 0.84 For the conical bulb [io = 2va2V is determined so that its amplitude function at g = 0 is equal to that of a sphere bulb. ao is the diameter of a cylinder corresponding to a doublet in two-dimensional flow. Under the specified design conditions the bulb sizes become a. Spherical bulbo f/L = 0.04; ao/L = 0o024 b. Conical bulbo f/L = 0 0.05; ao/L = 0.021 at fo/L = 0o05 Amplitude functions are shown in Figs~ I-1 and I-2. Wave profiles corresponding to various speeds are shown in Figs. I-6 and I-7o From these figures it may appear that the amplitude function of the conical source distribution can be made to be in better agreement with the hullI amplitude function than that of a sphere bulb, especially at large values of G. We must remember, however, that the linearized theory is being used and that this theory l'ooses accuracy rapidly for large values of G owing to the linearization of the free surface conditions, For this reason conclusions reached on the basis of purely mathematical considerations of the linearized theory may not be strictly valid in this region. An investigation of the wave steepness in the region of large G reveals that the assumption of a small ratio of wave height to wave length is not well. satistied9 and this fact must be taken into consideration in evaluating the significance of a good fit between the hull and bulb amplitude functionso Furthermore, as mentioned before, the weight function cos35 in the resistance integral reduces substantially the effect of any difference that may exist between the amplitude functions in the region of large 9 values. 34

.5 /0 AP I3 5 6 7 8 q P k 6 7 8 L KL 10 1 L oL / [~~~~~~~~~ 1'2%6 IAP.z - 6'7 8"L KoLqrI= I L=12 2. 6 7 8,, 2 3 6' P 3-' 4 AP — I KU - 1 L ~ ~ ~ ~ ~ / KU=P /4 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~01'5 APl, —-'"r'"~ 7 3' —' —' —-'~ t: 5~ 2 7 (1 \9 1.0 AAP 3 4- 7 KoL=16'oL="16 I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~':'* APfo1=6r Ko5 6 7 8 rr ~,,~5" L~~~L 1 4 5 6 8 9;~~~~~~~~4 i. —'" — 18..%.1 -, Ko - KoL=16, c e~~~~~~~~~~~~~',~ -,.3Ii~~ s,o-....6.. L,Ko 01. — 7/ AP 3 S 8 L t~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Ir ~~~~~~~~~~~~~KoL-21-7 L ~~KoL=22~66I.4. Ko L& 2, r,/I tts, Fig. I-6. Wave profile by sphere,* f/L 0.04; aO/L 0.024. Fig. I-7. Wave profiles (free wave) by oic1bub f,- L/.o; ao/L 0.021.

Takahei has tested hull forms fitted with various bulbs in Japan. Some of these bulbs correspond to the cases of so-called conical bulbs. The top of the bulbs were raised up to the waterline, with the shape of the bulb becoming approximately a cone fitted to the main bulb. The test results were rather disappointing, the resistance being considerably greater than predicted by theory, and especially so in the low speed range. The blunt shape of the stem disturbed the water surface sufficiently so that the boundary conditions of the linearized theory were seriously violated. Favorable wave cancellation could and was achieved with the single doublet, or slight modifications thereof, when the top of the bulb was kept at a substantial distance below the water surface, however. For these configurations the linear wave-resistance theory may prove to be valid to a sufficient degree of accuracy. Taking all these facts into account, we believe that the way to achieve the waveless bow is to try to fit the most suitable ship hull form to a sphere bulb, or one modified slightly from it, rather than trying to find a suitable bulb shape for a given ship form. 36

APPENDIX II MEASURED WAVE PROFILES AND THEIR ANALYSES Wave pattern measurements and their analyses have come to play a very important part in the studies of wave-interference phenomena (2,3,4,5,6). The analyses of model and ship waves materially supplements the present linear wave-resistance theory. In particular it can be used in various ways as a means of introducing corrections to the theory due to viscous and nonlinear effects. The most complete record of the wave-patterns can be obtained by means of stereo-pictures. A more expedient method is offered by the acoustic transducer although it only produces wave profiles along predetermined path lines, None of these types of equipment are presently available at The University of Michigan. It was therefore necessary to limit the wave pattern analysis to the traces of wave profiles along the model sides as read off from pictures. The measured profiles are compared with the profiles calculated from the singularity distribution function representing the model hull form. For the numerical calculationstables of wave profile functions prepared in Japan have been used. These tables are not complete and for further analysis of wave patterns extended tables will be greatly needed. Wave-resistance is caused by the fact that waves are continuously being generated by the ship and that some part of this wave system is propagating away from the ship. This part is referred to as the free travelling wave system. At some distance away from the ship only the free travelling wave system persists~ It is therefore the sole contributor to the wave-resistance and 37

thus the object for the wave analysis. The second part of the wave system is referred to as a local disturbance since it attains its maximum value at the ship's boundary and decays exponentially with the distance from it. Although this system is of no direct interest to us it must be included in the wave profile calculations if the profile is to be evaluated along the ship's surface. Figures II-l(a) through (d) show the wave profiles of C-201 and C201F2. The wave profiles obtained at The University of Michigan are compared with those obtained at Tokyo University with a smaller 8.2-ft model. Wave profiles near both ends of either model were traced with less accuracy than amidships because of curvature of hull surface and splashing water. Taking this fact into account, the agreement between test results is very good. As shown on Fig. II-l(b), the composite wave of the model fitted with the F2 bulb becomes entirely flat at the design speed along the model except near the bow and stern where the local disturbance affects the free-wave profile. This is a direct visual verification of the success of the cancellation of transverse waves by the effect of bulb. Theoretically calculated wave profiles of the main bulb coincide with those measured values although some phase lag between them is noticed. Remembering the discrepancy between measured and calculated values of the wave-making resistance, the confirmed agreement between wave profiles is rather startling. A careful look at Figs. II-1 and II-2 reveals that the measured bow wave profile leads the calculated bow wave profile in phase by approximately 5-6* of ship length in the speed range ofFronde number less than 0,4. This

phase shift is due to the finite draft and does not occur in the case of infinite draft. The reason for this may be due to the neglect of secondary corrections to the perturbation velocity components on the ship's surface. Since the correct phase relationship between bulb and hull waves is of the utmost importance to the concept of wave cancellation, it is essential that the phase lag mentioned above be correctly determined from the model tests. According to theory the center of the bulbs should have been located exactly at the FP of the C-101 and C-201 models. Actually the centers of the bulbs were located a few percent of the length of the models ahead of FP, as indicated in Table III,* to insure a proper phase relationship and thus satisfactory wave cancellation. The difference between the wave profiles of the model with bulb and that of the bare hull is expected to be equal to the wave profile of the bulb itself. To investigate if this relationship is properly predicted by the linear theory, the measured differences in wave profiles were compared to the wave profile calculated for a single point doublet, The strength of the doublet was determined from the condition that the volume of an equivalent sphere in a uniform stream was equal to two-thirds of the total increase in model displacement due to bulb and accompanying fillets. Depth of immersion was assumed to be the same as for the bulb. The longitudinal location of the doublet was determined by shifting the calculated wave profile until the fit between it and the measured difference profile was at an optimum. This location was found to be about 5 percent of model length forward of FP in *See page 10 of this report~ 39

the case of C-201 and the F2 bulb, which we shall call the effective center of the bulb. Figures II-2(a) through II-2(e) show the wave profiles obtained from C-101 and C-lOlFlo Figure II-2(a) is the plot of the wave profile measured at the design conditions at which the bow wave is found to be effectively cancelled. The bulb wave profile was calculated on the assumption that the effective center was located 2.5 percent of model length ahead of FP. The plots of wave profiles clearly show that the bulb wave system can be replaced by a doublet wave system except in the vicinity of the bow and in particular at high speeds. The discrepancies in this region may be partially attributed to a lack of accuracy of measurements but is most likely the result of an interference effect unaccounted for. Wave analysis has also been applied to CFK-1Ol. A comparison between the wave profiles measured before and after the installation of the flat bottom shows that the change in keel shape does not affect the wave profile materiallyo Similar results are reported elsewhere (11). The portion of the original form that was cut off in making the model flat bottomed can be approximately represented by a line distribution of singularities defined as followso ( ) = -.*3.; 0 = 0; = -.06L with -.8/ < 8 <.8i It is noted that this distribution provides sinks in the fore-body and sources in the aft-body~ Calculated wave profiles which include the effects 40

of the singularity distribution above are shown in Figs. II-3(a) and II-3(b) as chain lines together with the measured difference profiles. There remains the problem of actually tracing the streamlines with the singularity distribution q(~) added to the regular distribution of the cosine hu:ll form. This will be done as a part of a systematic study of the problem of generating ships with flat bottomso 41

Legend C-201 Bare Hull F2 Bulb 5 r vM = 4.91 fps Measured: U of M Data - F = 0.250 Japanese Data 4 - V/f = o.84 Calculated: 4 KoL = 16 (for bare hull) Free wave 2 -uTotal wave 3 22 ~ H 1 4U/) 00 -1 - -2 Hull Wave -2 I I I I I 1 I 1 i I I AP 9 8 7 6 5 4 3 2 1 FP 2 2 I profiles for C-201 and C-201F2. co -2 -3 - Bulb Wave (C-201F2 less C-~201) (a) Fig. II-1. Comparisons between measured and calculated wave profiles for C-201 and C-201F2.

VM = 5.25 fps F = 0.267 Legend same as Fig. II-l(a) 4 22 2~~~~~~~~~~~~~~~~1 co-1 =X X -2 -Hull Wave 2 AP 9 8 7 6 5 4 3 2 1 FP 0 0 - L Bulb Wave (b) Fig. II-i. (Continued).

VM = 5.68 fps4 3 F = 0.289 Legend same as Fig. II-2(a) 3 v/HL = 0.97 - 2 KoL =12K 12 ~o 1 1 Hull Wave-2 AP 9 8 7 6 5 4 3 2 1FP x a -2L Bulb Wave (C-1lF1 less C1-10 (c) Fig. II-1. (Continued).

VM = 6.21 fps F = 0.316 /v/-2= Legend same as Fig. II-l(a) V/lJ = 1.06 4 - KoL = 10 4 3 3 2t~~~~~~~~~D ~~~~~~~~2 cd Fig. II-1. (Concluded). -1 -2 Hull Wave -2 I I I I I I I I, I AP 9 8 7 6 5 4 3 3 1 FF 2 2 1 0) M -2 2 -3 Bulb Wave (C-201F2 less C-201) (a) Fig. IT-i. (Concluded).

Legend C-101 4 4 V4 M = 4.91 fps Measured: Bare Hull F = 0.250 With F1 Bulb 35 v/v' = o.84 3 K L = 16 Calculated: 2 - 2 (for Bare Hull) Free Wave 2 Total Wave.'O= 1 1 CO3 -1 -1 -2 Hull Wave -2 on, I i! I I I i I I I I AP 9 8 7 6 5 4 3 2 1 FP.11 -1 / 0 0-2 -2 -3 Bulb Wave (C-101F1 less C-101) -3 (a) Fig. II-2. Comparisons between measured and calculated wave profiles for C-101 and C-lOlFl.

VM = 5.25 fps 2 F = 0.267 Legend same as Fig. II-2(a) v/Jr = 0.90 t~ 1 ~K0L =P 14 / 1 CD D"'- --- _ -j -2L Hull Wave -2 I I I i I I I -~ AP 9 8 7 6 5 4 3 2 1 FP 2 4 01 - -21 Bulb Wave (C-10Fl less C-101) -2 (b) Fig. II-2. (Continued).

- vM = 5.68 5 F = 0.289 F = 0.289 Legend is same as Fig. II-l(a) 4 -4 V/TL = 0.97 KoL = 12 3 2. 1 -2 L Hull Wave -2 I I I! I I I I I I I AP 9 8 7 6 5 4 3 2 1 AP 0A - 2 -32 -2 -3 - Bulb Wave (C) Fig. II-2. (Continued).

4 - r VM = 6.21 fps F = 0.316 Legend same as Fig.II-2(a) 3 -:3 V /lJ= 1.06 *d2 KL =10 2- / 4-, -2 L Hull Wave -2 v r I I I I I I I I I \1 AP 9 8 7 6 5 4 3 2 1 FP'~ 1-O I -1 -1 -2 Bulb Wave (C-101F1 less C-101) -2 (d) Fig. 11-2. (Continued).

4r M = 6.96 fps 4 F = 0.354 V/F = 01.19 Legend same as Fig. II-2(a) 2 l KL =8,, / 2 2 / -o 1 -- co -1 - - 1 -2 - Hull Wave 2 I I I I I i I iI I AP 9 8 7 6 4 3 2 1 FP 0 -pd ~ -l-1-1 co -2 Bulb Wave (C-10F1 less C-101) 2 (e) Fig. 11-2. (Concluded).

3 3 3 M = 4.91 fps Legend F = 0.250 2 2 v/ =- 0.84 C-101 rocker bottom KL = 1 6 CFK-lOl flat bottom 1 m -1 LHull Wave.J-1 co Measured 1 - Difference _- Calculated free wave -1 L I J_ -1 3 3 U) vM Di 5.25 fps F = 0.2 2 1 VKoL = 14 -1 Hull Wave -1 1 r Difference 0. 1 I I I = 1 Hull Wav lo (a) Fig. 11-3. Effect due to modifying the bottom flat for CFK-1OL1L. 51

M = 6.21 fps Legend F = 0.316 3 V//i = 1.06 C-101 rocker bottom 2 KoL = 10 - - CFK-101 flat bottom 2 0 0I -1 -1 H ---- Measured 1 Difference -— Calculated free wave,, VM = 6.96 fps F =0.3543 v/ = 1.19 / 2 F KoL = 8 2 -1 L Hull Wave 1 r Difference o 1 1 I I I I I I I I I AP 9 8 7 6 5 4 3 2 1 FP (b) Fig. II-3. (Concluded). 52

3 3 v = 4.91 fps 2 F =.25 2 V/, = 0.84 CFK- 101/ D1 KL L =_16 I / 0o 1 *H3~~ ~ 3 -2 -1 - 2 -2 Bulb Wave (calculated) Pv = 5.25 fps 2, F = 00.267 v/JL = o.90/.1 K~~~~~~~~ 41 0 I-1 AP 9 8 7 6 5 4 3 2 1 FP L - -.1\ ~-3 L 3 v = 5.68 fps 3 F = 0.289 / 2 V/JL= 0.97 2' 2 1 --- I AP 9 7 6 5 3 2 1 FP 12 / Fig. 1-4. Comparison between CFK-lOl and. CFK-1O1F2. 53

4 -F4 v = 6.21 fps CFK-O 1 / F = 0.316/ -3 33 \ v/'L = 1.06 2 KoL = 10 CFK-101F2 2 30 / / -1 -/-1 AP 7 6 FP i 1 (CK-1OF2) - (CFK-101) \: -2 Bulb Wave (calculated) -2 Free Wave Component (calculated) 4 / v = 6.96 fps / 3 F o 354 / 3 3 v/- = 1.19 2 0 2 x 1 AP. 8 7 6 5 4 3 2 1 FP - 1 (b) Fig. II-4. (Concluded). 54

UNIVERSITY OF MICHIGAN 3 9015 03527 2684