THE UNIVERSITY OF M I C H I G A N COLLEGE OF ENGINEERING Department of Naval Architecture and Marine Engineering RESEARCH IN RESISTANCE AND PROPULSION A Program for Long-Range Research on Ship Resistance and Propulsion F. C. Michelsen Project Director: R. B. 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

ACKNOWLEDGMENTS Although my name appears as the author of this report my role has in many respects been merely that of an editor. Much of the material presented is the result of discussions with my colleagues, notably Dr. Tetsuo Takehei, Hun Chol Kim, James L. Moss and Nils Salvesen. Our former colleague, Louis Landweber contributed Appendix II, "Wave Resistance of Ships," and Appendix III, "Ship Resistance in Rectangular Channels." In addition he also worked on some of the parts of Section E "Improvement of Ship Model Test Procedures." And, of course, R. B. Couch furnished not only overall guidance but much sound advice as well. Finally, W. Taylor Potter, Robert Taylor, E. Scott Dillon, and others of our professional contacts at the Maritime Administration deserve special recognition for their helpful suggestions. Finn C. Michelsen Ann Arbor January, 1963. ii

TABLE OF CONTENTS Page ACKNOWLEDGMENTS................... ii LIST OF FIGURES..................... iv PREFACEoooo..ooo..... o...........................oo.o.o v Io PROPOSED PROGRAM.................................. 1 Ao Introduction...................................... 1 B. Hull Form Research,...........................OOOO 2 C. Ship Propulsion Research........................ 6 D, Research on the Effects of Ship Motions on Resistance and Propulsion.......................... 10 E. Improvement of Ship Model Test Procedures..o.oooooo 12 II. PERSONNEL & FACILITY NEEDS.......................... 29 A. Introduction...................................... 29 B. Personnel,, 30 Co Facilities..................................... 34 III. FIVE YEAR RESEARCH PROGRAM SUMMARY. eoooo.....o........ 44 APPENDICES I The Determination of the Wave-Resistance of a Model from the Wave Pattern................................ 53 II Wave Resistance of Ships...o........................ 67 III Ship Resistance in Rectangular Channels..,.......... 77 IV Short Range Proposals to the Maritime Administration. 84 REFERENCESo....OOOO.OOOOOOOOOOO.OO................. 49 0 a0 00...0....

LIST OF FIGURES Figure Page 1 Typical Wave Profile Obtained 12 ft. Behind the Model in the HSVA Model Basin............................................. 19 2 Relationships Between Measured and Calculated Drag Coefficient s..................................................... 19 3 Flume Experiments on Interactions Between a Wave System and a Boundary Layer........................... 22 I-1 Definition of Control Planes and Coordinates................... 52 iv

PREFACE This report has two main objectives. The first is to propose a five-year program of research in the field of ship resistance and propulsion, with emphasis on developments of importance to Uo SO merchant shipping. The second is to suggest what additional personnel and facilities would be needed to accomplish the proposed program. These recommendations are presented in partial fulfillment of The University of Michigan's obligations under The Maritime Administrations' university program. Maritime's university program was established in response to recommendations of the Maritime Research Advisory Committee. That group's final report(22) asked the Maritime Administration to encourage education through emphasis on contract research in those universities having particular interest in the marine field. The University of Michigan was one of four selected, and was asked to work in the area of ship resistance and propulsion. We are pleased to have this responsibility but would emphasize at this point that most of our recommendations relative to research projects, personnel, or facilities are made without respect to locationo There is no point in denying our natural desire to see the bulk of these research projects carried out at Michigan insofar as present or possible future facilities will permit; but, since we can make no claim to objectivity in this matter, we leave it to the Maritime Administration to reach its own conclusions. We would, however, strongly recommend that the nation's universities be given particular consideration in decisions affecting the assignment of this work. Our reasons for this are well expressed by the following quotation (23) v

The vigor and well-being of our universities is essential for the continued leadership of U. S. science. Opportunities for development of effective careers in research and in the production of superior young scientists must be made so attractive that the majority of our best scientists will desire university careers. The principal talent of the nation flows through the universities and it is here that students can be most readily attracted to fields of marine science through close association with professors engaged both in research and teachingo In our society there are no other posts where so small a necleus of talent can produce such great improvements in succeeding generations. Appropriate research facilities must be available at the universities and under their control. and, since the Maritime Administration's research budget is too small to finance all the proposals that come it's way, we should like to recall two of the principal recommendations of the Maritime Research Advisory Committee (22) About 15 percent of the Maritime Administration research budget should be devoted to basic research in such maritime related areas as hydrodynamics, atmosphere - ocean interactions, structures and materiels. The Maritime Administration should promote maritime related education through: a. Emphasis on university research b. Fellowships for advanced study c. Assistance in the construction of university laboratorieso The text of this report contains what we believe to be strong evidence in support of the above recommendations as well as a carefully considered program of related projects in the field of ship resistance and propulsion. vi

CHAPTER I PROPOSED PROGRAM A. Introduction In his recent paper dealing with the problems of research in the field of sea transportation, Professor E. V. Lewis(l) gave this breakdown of the steps involved in carrying out research with the ultimate goal of providing better and more complete information to the design engineer: 1. Identify, define and clarify the problems. 2. Evaluate present status of knowledge and identify significant gaps therein. 3. Plan and carry out exploratory projects aimed at possible new solutions to problems or filling in the gaps. 4. Follow up with more detailed studies to broaden and generalize results and make information and design techniques available to industry. We believe this outline to be an excellent one and have kept it in mind in developing our recommendations relative to long term research needs in the field of ship resistance and propulsion. In the proposed research programs that follow, we have tried to define the problems (Step 1), to evaluate present levels of knowledge (Step 2), and to provide plans for initial initial investigations (first part of Step 3). Thus, we believe we have carefully prepared for the final, important steps leading -1

-2to the eventual determination and dissemination of practical knowledge. We are particularly anxious to stress the necessity of carrying each particular project all the way through the fourth step, making the results of research available to design engineers in a form that is readily understandable. A fairly large part of our proposals are concerned with research directed towards improvement of model test technique and analysis. We believe this work is particularly important at this time because recent theoretical developments place us at the threshold of important breakthroughs in our understanding of ship and model hydrodynamic relationships. B. Hull Form Research The realization of a so-called "waveless ship form," by the introduction of large bulbs, is the most stimulating event in many years in the field of ship hydrodynamics. Most fundamental mathematical studies of the concept are the product of Dr. Inui and his assistants at the University of Tokyo. This group has also performed an impressive amount of experiment testing with basic hull forms. The test results, in general, verify theoretical predictions. During the past year, the staff at The University of Michigan has translatedallof them:-re important Japanese publications relating to the Inui bulb.(27) It has also built and tested two 12-ft. models of forms previously tested to a smaller scale in Japan. In addition, bulbs for conventional ship forms have been designed and tested. This work has provided basic knowledge of the mechanism of wave cancellation and of the details of computational work involved, a knowledge that will be

-3of great value to the continued research on the problem of reducing the wave-making resistance of ships. Most of this work was carried out under the direction of Professor Tetsuo Takahei of Ibaraki University. Professor Takahei was Dr. Inui's collaborator in the development of the Inui bulb; he was brought to Michigan for two years, his principal support being furnished by The University of Michigan. A separate research proposal has recently been submitted to the Maritime Administration covering some immediate tasks that The University of Michigan wishes to undertake in further pursuit of minimizing wave-making resistance of conventional ship forms by means of large bow bulbs. The details of this proposal is given in Appendix IV. It should be pointed out that results of the proposed investigations are not expected to be final. Dr. Inui's paper(11) is an indication of the extensive amount of research that will be needed to answer all the questionson the relationships among hull geometry, wave patterns and resistance, and it must be expected that continuing research will be needed over a period of years. We feel it unwise to prepare a detailed description of such future activities at this time, however, since substantial progress can be assured only if efforts are based on a careful study of results as they are obtained from both domestic and foreign sources. To provide for continuity of research, the Maritime Administration should therefore earmark funds on a long term basis for the study of the hull form relationship to wave making resistance. The number of investigators actively engaged in research in the field of theoretical wave resistance is rather small. Cooperation and communication is complicated by the fact that these individuals are widely

-4scattered geographically. Thus, there is a real need for an international seminar on the topic of theoretical wave resistance. The purpose of the seminar should be an evaluation of recent advances and the discussion of guide lines for future research. The University of Michigan has recently sent out inquiries to prospective participants all over the world to evaluate degree of interest and desired scope of such a seminar. Tentative plans are for the seminar to be held in Ann Arbor during August 1963. Expenses are to be shared by The University of Michigan and the Office of Naval Research. The knowledge to be gained will undoubtedly be of great value to the Maritime Administration's research program. We are currently working on a limited study on the effect of variations of section shapes of Series-60 hull forms with regard to resistance and propulsion. This preliminary study is being undertaken by a separate Maritime Administration Task. Results of tests will be reported elsewhere. It can be said, however, that three completed tests indicate that extreme V-shaped sections produce a substantial increase in both E.H.P and S.H.P. Although two more tests to be made, may bear out this fact we feel that this study should be continued. Moderate Vshaped sections may lead to much improved performance. Furthermore it may become evident that the sectional area curves should be different from those of the Series-60 parent forms. This point ought to be investigated, possibly with the aid of theoretical wave-resistance theory. The original Series-60 family of hull forms covered a range of block coefficients up to and including CB = 0.80. Many of the large tankers and ore-carriers being built today have -block coefficients of considerably higher value. There is a definite lack of information on

-5resistance characteristics of these ships in the literature. It is therefore proposed that a limited family of full hull forms be developed and tested for both E.H.P and S.H.P versus speed. The parent form could possibly be developed from the Series-60 lines, but this should not be a requirement. Careful studies of the flow conditions around the aft body of the models must be undertaken as part of the investigation of full hull forms. In a recent paper D. J. Doust (28) describes the development of trawler hull forms based on statistical methods of analysis of the N.P.L resistance and propulsion data. The resistance qualities of these forms have been expressed in equational terms, dependent on certain non-dimensional parameters of their hull shape and dimensions. By minimizing this equation, new combinations of parameters have been derived which give superior performance relative to all previous results. Predictions of optimum hull forms have apparently been verified -by full scale results. It is noteworthy that the evolution of the trawler form based on experience did not lead to an optimum choice of design parameters, in spite of extensive model testing. This leads one to wonder whether in the past new designs have not been based too heavily on existing ships. The regression equations used by Doust are not necessarily the only equations of this type suitable for the analysis of ship performance. Several similar methods are already available at the University of Michigan Computing Center. These have already proven to be very successful in the analysis of complex engineering problems, and it is therefore proposed that the Maritime Administration sponsor a computer project for the purpose of the statistical determination of optimum hull form parameters of commercial ship forms.

-6C. Ship Propulsion Research Although it has always been recognized that an interplay exists between the flow to a ship's propeller and the flow around the ship itself, it has been customary to more or less separate the field of ship propulsion from that of ship resistance. In the case of slow speed cargo ships such a separation is probably not too serious, since it is often sufficient to obtain a nominal wake at the propeller disc from the resistance test model, and then design, from propeller charts or by more refined methods (such as the circulation theory) an optimum propeller to suit this wake. Frequently, an average wake obtained from a windmilling propeller is all that the designers have been willing to pay for, and in many cases not that much. The development of highspeed ships has changed such simple design procedures, however, and it has become more common to provide the propeller designer with a complete wake survey, determining the wake condition at all points on the propeller disc. To measure velocities in the wakes, a five-hole pitot tube was developed at DTMB several years ago. With this instrument, pressures in the flow are measured by means of U-tube manometers. The dynamic response of such an instrument is so low that one reading, at the most, is obtained from a single run down the basin. More than one run per reading is often necessary in order to reach equilibrium conditions in the U-tube. Owing to modern electronics technology, major improvements can be made in the design of the five-hole pitot tube. In an instrument, now under construction at The University of Michigan, it is planned to use electronic pressure transducers instead of U-tube manometers. The response of these

-7transducers is sufficiently high that it will enable the wake survey to be made around a complete circle, of constant radius from the shaft line, during one towing tank run. By digitizing the transducer outputs directly onto magnetic tapes and developing a computer program a harmonic analysis can be made with great facility. The purpose of the instrument is to provide a simple technique whereby the wake survey becomes a matter of routine. In the future it should therefore be included as part of the testing of new model forms. The construction of the five-hole pitot tube is already partially supported by the Maritime Administration. It is proposed that this support be extended to cover electronic components and programming of calculations. In the discussion above, only the steady state values of wake velocity components are considered. Except in cases of unusual forms, the opinion is generally held that the wake is essentially steady. From recent measurements it appears that a revision of this opinion is in order. These measurements have indicated that predominant time-dependent wake variations may exist around a frequency of approximately 20 cycles per second. Two distinctly different causes may give rise to such variations. Several flow studies have shown the formation of large vortices around the aft body of models, originating at about the aft shoulders. These vortices are undoubtedly unstable and will therefore produce a time-variable flow at the propeller. It is interesting to note that Dr. Hogben and Dr. Gadd(10) mention that such vortices may always be present, and that they may, indeed, partially account for the discrepency between experimental and theoretical values of energy transfer of ship waves,

-8Although vortices of the type described cause variations in the wake flow, it is difficult to believe that the predominant frequency of these variations will be as high as 20 cycles/second. It is therefore reasonable to suspect that the principal cause of non-steadiness of the wake flow is to be found in instability flow phenomena of the boundary layer. Measurements recently performed elsewhere point to what may be called, for lack of a better name, a ragged edge of the boundary layer. In a particular case, this raggedness appears to be located at about eight-tenths of the radius of the propeller, resulting in significant thrust and torque variations at that location. Sufficient evidence is not available at this time to make it possible to predict the occurrence and nature of this type of boundary layer instability. Detailed studies of the flow around the aft body of the models should therefore be undertaken as soon as possible. A complete description of the wake flow will be of immense value to the propeller designer. Non-steady propeller theory has not as yet advanced to a point where it can accurately predict the magnitude of alternating torque, thrust, and transverse forces. Application of quasisteady techniques in some cases produce calculated values close to those measured, but there is serious doubt that such methods are fully reliable, especially where the wake has sharp peaks smaller in width than the propeller blades. It is, in fact, even doubtful that Sears' non-steady airfoil theory applies to this type of flow conditions. Professor Frank M. Lewis is now installing, in the MIT towing tank, a special water tunnel designed for the study of forces on a foil subjected to a sudden gust.

-9Pictures of flow around the foil may reveal whether the Kutta conditions are being satisfied at the trailing edge. A similar study is being planned to be carried out in a wind-tunnel at The University of Michigan in the near future. These programs should produce some valuaOle information on two-dimensional non-steady foil theory. After the two-dimensional theory has been formulated, there remains the extension to three-dimensional cases such as the ship propeller. Since 1960, DTMB has been using a stern section separated from the rest of the hull as a means of investigating propeller-induced forces on the hu11.(18) Although the magnitude of the forces acting on the stern section reveals the overall effects of the propeller, little information is provided about the distribution of pressure forces on the hull. The magnitudes of pressure forces and their distribution has a great effect on the elastic response of a ship, and it is important that more information be produced in this area of propulsion research. A few years ago attempts were made at DTNMB to measure propeller induced pressures. (29) The tests did not produce reliable results and were, in general, discouraging. Although modern pressure transducers are far superior to those used then, more immediate demands have prevented a renewed attack on the problem at DTMB. The recent theoretical developments by Dr. Breslin(l9) and others serve to give impetus to further research on this subject. We recommend that the Maritime Administration sponsor additional tests in this area, at DTMB or elsewhere. It is interesting to note that pressure measurements on a full-scale destroyer have been made in Japan.(30)

-10A separate proposal for a study to investigate propeller scale effects has recently been submitted to the Maritime Administration. Details of this is given in Appendix IV. Chapter II discusses added facilities and equipment that would'be needed to carry out the proposals outlined above. These include: a. Five-hole Pitot tube and recording instruments. b. Arrangements for underwater television viewing, for flow studies. c. Propulsion dynamometer capable of measuring time-variable forces acting on model propeller blades. d. Water tunnel for propeller flow pattern and other studies. D. Research On The Effects Of Ship Motion On Resistance And Propulsion St. Denis and Pierson's 1953 paper (20) provided such a stimulus to the study of ship motions that the subject has gained more study and attention since that date than during the entire previous history of naval architecture. The introduction of the concepts of statistical mathematics made it possible, for the first time, to describe rationally typical sea states and the resulting motion of a ship. The deleterious effects of slamming, extreme rolling, or the shipping of green seas are well known. It is therefore not surprising that these new tools were applied principally to the problems of extreme ship motion. The question of maximum loading of the hull girder also received new attention. Although the above mentioned developments have been beneficial, we believe that one economically important area has been badly neglected. We refer to the determination of the influence of sea state on

-11ship's speed within the range of conditions where full power can be maintained. Ship operators are interested in using electronic computers in optimizing the routing of their ships. Clearly, any realistic computer program will demand more information than the stillwater speed and power relationships. The importance of further work in this area is also underscored by Benford's recent report(26) in which he concludes that the lack of quantitative data on sea speed is the missing link in the use of economics to optimize ship design. The problem of providing speed and power relationships for various headings, wave lengths, and wave heights may appear overwhelming. However, there is reasonable evidence (21) that a ship's response is a linear function of wave amplitude, or nearly so, for the greater part of the region wherein it is practical to maintain full power. It would therefore probably be sufficient to test models of standard size in waves of standard height. This would facilitate direct comparison of results. In view of the above, we recommend that the Maritime Administration sponsor comprehensive self-propelled model tests in head-sea conditions. Subsequently the program should be expanded to cover the complete range of ship headings with respect to the waves. In addition to measuring average values of propeller torques and thrusts, we hope that time variations of torque and thrust can also be evaluated experimentally. It is often stated that ship motion studies do not require large models, and that it therefore would be less expensive to build a large model for SHP tests and a smaller one for ship motion tests, rather than using the large model for the whole program. In view of the high costs of models in general, it is doubtful that the two-model procedure

-12will prove to be less expensive in all cases. If self-propulsion in waves is considered, only a large model can be used if scale effects are to be avoided. The large models impose requirements on the tank width, however, since it is known that ship motion is quite susceptible to tank wall effects. A further need in the area of ship motion is the study of the damping effect of the large Inui bulbs. Information on this aspect is of great importance in the overall evaluation of these bulbs. E. Improvement of Ship Model Test Procedures With a single stroke of genius, William Froude laid the foundation of the theory of model testing when he assumed that the total resistance Rt of model or ship could be divided into two parts, i.e., Rt = Rf + Rr, and furthermore, that the frictional resistance Rf is a function of the Reynold's number only, whereas the residual resistance Rr is assumed, in addition, to be the sum of the wavemaking resistance and the eddy resistance. We have long realized that this simple relationship between the forces acting on a ship's hull, even when moving at constant speed on a straight course in smooth water, ignores possible interaction effects. If the forces are expressed in terms of non-dimensional coefficients, it would probably be more correct to write Ct = Cw + Cf + Ce + Cwf + Cwe + Cef where the terms Cwf, Cwe and Cef represent the interaction effects of

-13waves and boundary layer, waves and eddies, and eddies and boundary layer respectively. Research in the field of ship resistance in the past has indeed shown that these secondary interference effects are appreciable. It has, however, been incapable of producting accurate methods by which they can be evaluated. The standardization of model sizes used in individual tanks, together with the prediction of performance of ships of standard lengths, and the introduction of correlation coefficients, such as A Cf = o0004, may very well have blindly compensated for a great deal of the interference effects in most cases, thus providing satisfactory estimates of full-scale performance based on model test results. It has taken a great many years and an immense number of model tests to get this far, the main reason being that -- because of differences in size, technique, or instrumentation -- almost every tank must determine its own correlation factors. Even today, we find it desirable to test one and the same model in the various testing tanks of the world so as to assure a continuation of comparable results. One may logically ask for the reason why interference effects are not well knowno The answer is not hard to find; it can, in fact, be stated in two words: analytical complexity. In the past, the problems of treating both the wavemaking and the frictional. resistance components on a purely theoretical basis have been dealt with quite intensively. These contributions to our knowledge of the fundamentals of ship resistance have been significant indeed. The application of theories to quantitative evaluation of resistance components in the case of practical ship forms, however, has not been too successful. We believe the main

reason for this may be the limitations imposed on theories by the assumptions made in their derivations. In the case of frictional resistance, for instance, we are still applying flat plate theories to the three-dimensional ship's surface. As far as wave making is concerned, the theory is restricted by the linearization of boundary conditions and by the fact that even these are not satisfied at the true boundary surfaces. Introducing limitations so as to avoid mathematical complexities is a feature that is certainly not confined to ship theories alone. In many non-ship cases it has been found possible to evaluate these limitations by means of carefully planned and executed laboratory tests. Until recently, however, it has been impossible to do so in the field of ship resistance. The only quantity that the ship model tester has been able to measure is that of total resistance, and he has not been well equipped with analytical tools to deduce from this measurement the magnitude of individual force components and their interplay. How, then, could the mathematical wave-resistance theory be evaluated when it was not even agreed what part of the total resistance of a model was due to the generation of waves? Within the past year, investigators have undertaken to measure the magnitudes of the separate forces contributing to the resistance of a model. The methods proposed by these investigators offer an opportunity to determine the validity of the basic Froude assumptions of ship-model resistance testing, and possibly to develop improved procedures. One way of analyzing the resistance of a ship is to express it as the sum of the frictional and pressure drags. The latter, which

can be obtained from measurements of the pressure distribution over a hull, has been obtained for a few ship models in the past, and has been (2) measured recently by Townsin at King's College for a Victory ship model. The significant result of all these tests is that the frictional resistance, derived by subtracting the pressure drag from the total, is considerably greater than the flat plate or the extrapolator values usually assumed in the analysis of ship-model data. An alternative analysis expresses ship-model resistance as the sum of the viscous and wavemaking drags. The former, obtained by Wu(3) from a wake survey conducted at The University of Iowa, has yielded a value, for the viscous drag of a Series-60 ship model, about 4 percent greater than the frictional resistance computed from Schoenherr's formula for the resistance of a flat plate. Since the viscous drag is the sum of the frictional resistance and the viscous pressure drag, and 4 percent is probably too small a value to allow for the latter, it follows that a ship model must have a lower frictional resistance than a flat plate. This result contradicts those obtained from the hull-pressure surveys, noted above, but is in agreement with some measurements of Wieghardt(4) on the frictional resistance of cylinders of non-circular sections. The wavemaking part of resistance has been measured by Ward at Webb Institute of Naval Architecture(5) employing a method, based on theory, requiring the measurement of surface-wave profiles. His values for the wavemaking resistance of the ATTC Standard Model are about twothirds of those calculated by Wigley (6) from the Michel ship-wave-resistance integral, with corrections for viscosity. They are also about twothirds of those obtained by Nevitt who subtracts an assumed value for

the frictional resistance from the total. In analyzing his measurements, Ward assumes that the flow is potential, so that his results may be subject to an undetermined error owing to the failure of this hypothesis in the wake region. Assuming that this error is small in comparison to the magnitude of the aforementioned discrepancies, his results would indicate that linearized gravity-wave theory seriously over-estimates the wavemaking resistance and that the viscous drag, at a Froude number v/fLg = 0.28, is 11 percent greater than the frictional resistance given by Schoenherr's flat plate resistance formula. In January of this year Eggers(8) presented a paper in which he proposed to evaluate the wave resistance of the model from two wave profiles located behind the ship, an arbitrary distance apart along sections perpendicular to the direction of motion. During the past summer this principle has been used by Sharma, of the Institut fur Schiffbau, in the evaluation of the wave resistance of a mathematical form derived from a linearly varying source distribution. Some of Mr. Sharma's tests were observed by the author who also had the opportunity to discuss results and many of the questions left open for evaluation in the future. At a Froude number of 0.121, calculations indicate that the wave-resistance obtained from wave profiles was only about 60 percent of that arrived at in the usual manner (subtracting the frictional drag -- as given by Dr. Hughes formula -- together with a form factor K, from the total drag measured). Obviously, such a surprising result led to a series of questions about the validity of theory, accuracy of measurements, magnitude of viscous forces, etc. As far as theoretical formulation is concerned, Eggers' method is probably better

-17suited than Ward's for model testing, since it does not require a wide tank with respect to model size. In Ward's case there is also the question of selecting the point of origin of the coordinate system, a limitation that is removed in Eggers' work, Mr. Sharma has taken profiles along six sections and, by pairing any two of these, he has been able to obtain several independent values of the wave resistance -- from which an average can be taken. Furthermore, he obtains in this way a good evaluation of the accuracy of numerical calculationso It should be mentioned that two independent methods were used in obtaining the wave profiles: stereo photographs and a sonic surface-wave transducer of the type developed at St. Anthony Falls Laboratories, University of (9) Minnesota. Both methods gave comparable results, and since the echo sounder gives the profile directly and is easier to handle, it will probably be preferred for this type of work in the future. A sample wave profile obtained with the echo sounder is shown in Figure 1. In the higher speed ranges, Sharma finds that the discrepency between theory and test varies. The best agreement was reached at a Froude number of 0.34. This corresponds to a speed at which the linear wave-resistance theory generally checks most closely with experimental results. To eliminate the question of the correctness of the frictional resistance formula used in the calculations, some wake measurements were made with pressure probes. Calculations of frictional drag, using the Betz formula, indicate good correlation with Hughes. It is therefore our opinion that the low value of wave resistance predicted by Eggers method could not be explained by a possible insufficient magnitude of the frictional resistance. One would expect that a constant discrepency

-18would exist over the speed range if this were the case, or -- at the most -- that the difference would be monotonically increasing or decreasing. It appears, however, that the variation between results is an oscillating function of Froude number. The mathematical formulation of Dr. Eggers' work is given in Appendix Io What is known at present is that the flux of energy, owing to free waves, as evaluated by means of a linear theory, is in apparent disagreement with the value arrived at from the towrope pull. This disagreement is not the same as that which exists between the linear Mitchell theory and the measured resistance. Perhaps the linear theory of free waves is not sufficiently accurate in portraying free surface conditions. Any improvement in this regard, however, could hardly be expected to make up for the total difference of results. Is it then possible that some mechanism of exchange of energy exists between waves and boundary layer? Dr. Inui, in his work, introduces a correction factor to account for the attenuation of bow wave amplitude owing to the presence of a boundary layer. He also adds a shelving correction factor that accounts for the fact that the ship runs on top of its own bow wave. Both these factors serve to bring the linear theory into better agreement with experiments. Considering the energy in the free waves, however, one is forced to look for further corrections. We want to point out that a mathematical theory based on potential flow, even accounting for nonlinear boundary conditions, assumes that the ship is 100 percent efficient in generating waves, It is reasonable to respect that this is not true. If the ship is less than fully efficient in producing waves,

-19I MODEL PORT -- -- STARBOARD TIME Figure 1. Typical Wave Profile Obtained 12 Ft. behind the Model in the HSVA Model Basin..20.30.40.0.0 F — Figure 2. Relationships Between Measured and Calculated Drag Coefficients.

-20it simply means that part of the wavemaking resistance will not appear in the form of free waves. This idea is quite distinct from that of Dr. Inui since we are considering here a reduction in wave amplitude even before the waves have been in contact with much or any boundary layer, or have been affected by rigid boundary surfaces, After witnessing the manner in which bow waves are normally generated by a full size ship, it is not difficult to imagine how some wave energy may be dissipated in the breaking crests, The result of Sharma's work is outlined in Figure 2. He is currently continuing his work in this field and we expect that a formal publication will be ready for the proposed seminar on wave resistance to be held at The University of Michigan in August, 1963. The method of analyzing the wave resistance from wave profiles behind the ship opens up a great many possibilities in regard to the evaluation of the effects of design parameters and testing conditions and we hope that some investigations, parallel to Sharma's, can soon be started in this country. One fundamental question that immediately presents itself is that of scale effect. If the efficiency of the model as a wave-maker is less than 100 percent, there is a definite possibility that a new scale effect would exist in relation to wave making. This scale effect would be distinct from that already known to exist because of the difference in the thicknesses of the boundary layers. Dr. Hogben of NPL has also formulated a method for obtaining the wave resistance from wave profiles, Some fundamental questions have been raised by Eggers in regard to the exactness of Hogben's formulation. It is of interest, however, that one of his measurements gave a

-21calculated value of the wave resistance in the order of one-half of the value obtained from the towrope pull. This led Dr. Gadd, also of NPL, to perform some simple experiments in a small flow channel. Assuming that the boundary layer was the cause of the difference between wave profile and towrope wave resistance values, he wished to introduce a boundary layer in a two dimensional wave train without creating a new wave. To accomplish this he used an essentially flat plate strut. The general experimental setup is shown in Figure 3. Dr. Gadd observed that, when the trailing edge coincided with a trough, the wave behind the strut was essentially undiminished; whereas when the trailing edge was at the crest, the wave downstream from the strut was reduced to about one-half its original amplitude. An adverse pressure gradient existed in the second case, which may be of some signifcanceo The results are startling, and this case of fluid flow should be more fully investigated. In a report on their work, Dro Hogben and Dr. Gadd(10) attempt to ascribe the lack of energy transfer of the free waves to energy transfer of eddies. They plan to present further details of the study at the proposed wave-resistance seminar next fallo A fourth procedure for obtaining the wave energy of a ship from its wave pattern is presented by Mr. Wisashi Kajitani in his discussion of Dr. Inui's recent paper on "waveless" hull forms(11) In this, the path of integration is chosen along crest lines, and a fairly complete picture of the wave field is therefore required, We note that a good correlation between calculated wave-resistance values obtained from wave patterns and those evaluated from towrope pull has been obtained in Japan. This fact only serves to emphasize the need for more detailed investigations.

(as) -- -STEP GENERATING WAVES )PLATE TRAILING EDGE IN CREST PLATE,TRAILING EDGE IN TROUGH Figure 3. Flume Experiments on Interactions Between a Wave System and a Boundary Layer.

-23A clear and consistent view of the magnitudes of the various kinds of resistance force components is not yet available from the experiments described above. What is significant is that they indicate the availability of means for measuring these individual components, thus leading to a fuller understanding of their interdependence, The new developments in the field of ship resistance theory have provided us with several independent methods by which resistance forces can be measured. This redundancy should be used to reveal characteristics of the flow around ships and to provide data by which their effects can be evaluated, This suggests the following research projects: 1. Select a ship form for which the wavemaking resistance is readily calculable by potential theory. Calculate the wave resistance versus Froude number not only from the Michell integral but also in accordance with Appendix II "Wave Resistance of Ships," which introduces non-linear corrections to the Michell solution and a more accurate determination of the source distribution on the center plane as the solution of an integral equation. It is a well established fact that the Michell theory, in addition to exaggerating the humps and hollows, overestimates the wave resistance of ships of finite beam. The introduction of non-linear boundary conditions is expected to bring theory and experiments into better overall agreement, although correction factors and an efficiency concept, as mentioned before, may be necessary to explain the apparent lack of wave interference effects.

-24It is possible to include the theoretical effects of changes in sinkage and trim in the calculations of the non-linear case. This would greatly complicate the analysis, however, so we propose to assume no change in the ship's attitude. Calculations based on the linear theory have shown that correlation with theory is improved if the model is left free to adjust its own sinkage, but is restrained from trimming, If sinkage is restrained, the displacement of the model would also change. It is proposed that a similar procedure be used initially in any experimental verification of the non-linear theory. 2. Determine the viscous drag versus Froude number by means of a wake survey in one transverse plane, Analyze the data in accordance with the most accurate of the viscous drag formulas discussed in a recent report by Landweber and Wu.(12) Also measure total resistance. The determination of frictional resistance may seem to run into the fundamental question of the degree of turbulence stimulation, But this is really not the case here since the purpose of this investigation is to determine accurately what the total frictional force is in a given case, regardless of what type of turbulence stimulation is used. One may say that stimulation will be necessary only in so far as it stabilizes the flow and improves the consistency of the measurements. The values of the residuary resistance versus Froude number, obtained as the difference between the total and viscous drag, should be compared with the values of wavemaking resistance obtained from non-linear potential theory and from wave profile analysis in accordance with Eggers and others.

-253. The new theories advanced by Landweber, Wu, and Eggers should make it possible to determine the total resistance of a ship model from the flow conditions at some arbitrary location behind the model. The wave profile analysis should therefore be put to full use in conjunction with the analysis of wake survey. Naturally, we hope that the total resistance obtained by means of the theories mentioned above will be the same as the towrope pull. Sharma's failure to achieve this in the few tests made to date indicates, however, that a great deal of detailed experimental work and analysis remain to be done. The ultimate reward from such research should be improved methods for the determination of the hydrodynamic forces acting on a ship's hull. 4. The wake analysis advanced by Landweber and Wu provides a method by which the change in frictional resistance can be determined directly. Since the thickness of the boundary layer will influence both frictional and wavemaking resistance, it is not sufficient or satisfactory merely to evaluate the frictional drag from the towrope pull. Therefore, it is proposed that the problem of boundary layer stimulation be studied in greater detail, using various -types of devices, and measuring simultaneously both the wake and wave profiles. This study should also include some investigations of scale effects. It is proposed that this be done by means of performing identical test series with geometrically similar models of different sizes.

-26Wavemaking resistance for this series of models should also be computed from theory and compared with wave profile evaluations, Although it may become possible to determine with certainty what the frictional resistance of a given model is, one does not escape the question of turbulence stimulation, Since the degree of turbulence in the boundary layer has a direct influence on the separation of flow around three-dimensional ships, it is important that the flow around models be turbulent. In spite of the large amount of research done to date in the field of turbulence stimulation, no final conclusion has been reached in regard to type and amount of stimulation requiredo Some towing tanks use one trip wire, others use two, Studs of various shapes and location have also been fitted to models. From this fact alone it should be obvious that a standard roughness correction of A Cf = 0o0004 is at best a guess. The name "correlation factor" used by many investigators is probably a better choice than "roughness correction," The opinion held by several reseachers today, among them Dr. Hughes, is that it is preferable to introduce complete turbulence stimulation to the flow around models by mechanical means, such as studs. These devices must, however, be of such design that their own drag can be easily estimated, Three dimensional boundary layer theory has been developed(14) to a point where efforts should be made to calculate the viscous drag of ship forms, The known viscous drag of the model could be used to suggest correction factors for these calculations, The theory may also indicate the nature of the laws of variation of the viscous drag with hull shape and Reynolds number.

-276. During the past year, a successful semi-empirical formula was derived and tested, allowing very good estimates of the blockage effect in The University of Michigan towing basin. We are still not sure that it does not also take into account some scale effects and we therefore suggest that the blockage effect on the wave resistance be studied from a theoretical point of view as indicated in Appendix III, and that the contribution to the frictional resistance, owing to the presence of tank boundaries, be determined from wake surveys. The problem of predicting the resistance of the full scale ship remains. Undoubtedly, the wavemaking resistance depends primarily upon the Froude number and can be obtained approximately from the values for the model. It is important, however, to investigate the effect of Reynolds number of the wavemaking resistance coefficient. The following additional tasks are therefore proposed. 7. If wave resistance calculations based on potential flow with corrections for vixcous effects are found to be in good agreement, repeat the calculations, taking into account viscous effects at the Reynolds number of the ship, assuming a smooth hull. 8. We believe that the Maritime Administration should also sponsor fullscale measurement of wavemaking resistance on an existing hull. Thus, the lines selected for model tests should be that of an existing vessel. Selection of a form on the basis of ease of theoretical wavemaking calculations

-28would then have to be sacrificed. The full size surface wave profiles could be measured by means of stereoaerial photographs. Measuring the ship's total resistance would also be desirable. Since the wave-making would be disturbed by the usual propeller drive, we suggest that some form of air thrust propulsion be employed. Further, we suggest that this full-scale trial be carried out both with a clean, freshly-painted hull and also after the ship has been out of dock for a considerable time. A practical procedure for the above might be first to conduct tests with the ship as is, then to dock it to examine the nature of the roughness, and then to clean and paint. These full-scale results should yield an evaluation of the degree of validity of the Froude law of comparison and of the success of theoretical methods for computing wavemaking resistance. If the effect of Reynolds number on wave-making is found to be significant, the theory might be applied to compute a correction factor -- presumably differing only slightly from unity -- for the extrapolation from model to ship.

CHAPTER II PERSONNEL & FACILITY NEEDS A. Introduction In the conduct of experimental research, laboratory facilities and research personnel are intimately connected, One cannot consider one of these factors without, at the same time, taking the other into account, We fear that the intimate relationship between facilities and personnel has quite often been overlooked with the result that research facilities have been built more as a symbol of stature than as a place for intensive scientific investigations. Research facilities must be considered as tools in the hands of able and competent researchers, The type and size of facilities must, therefore, bear direct relationship to the immediate projects at hand; they must also be, as much as possible, adaptable to future requirements. Simple principles of economics dictate that the tools in the hands of the researchers be of the right type size and of the best design. Size is of the utmost importance, One must be careful to distinguish between overall plant size and size of individual pieces of equipment. The plant size should be geared to the rate of flow of the overall production, whereas the size of the item of equipment in question should be sufficient to perform the tasks at hand properly. If it is foolish to attempt to lift a two-ton load with a one-ton crane, it is also foolish to buy a two-ton creane if loads never exceed one ton, As a matter of expediency, this chapter treats personnel and facility needs in separate sections. In several important areas, however, -29

-30the two must be considered simultaneously. Thus, the section on facilities makes frequent reference to associated personnel considerations. B. Personnel Chapter I proposes a long range research program in the field of resistance and propulsion of ships. The program is ambitious, and it should be. The United States has established itself as a leader in the field of theoretical studies, but as far as follow-up investigations are concerned, it does not enjoy the position it had in the past. The number of research establishments in Europe and Japan, compared to those in this country, underlines this fact. The difference in training of our researchers is possibly the main reason for this state of affirs. Most European Maritime researchers are naval architects, whereas, in the United States they are more likely to be mathematicians or physicists and, as such, improvement of merchant shipping is likely to be anywhere but at the forefront of their minds. This is not to say that our researchers have not made significant contributions to the field of naval architecture, for indeed they have. The main problem today, however, is that pitifully few of their contributions have ever been proven out or converted into practical terms for use in shipyards and design offices. The solution of this problem will require two things: 1. Continued experimental research to verify theories and to investigate forms and parameters important in the actual design of ships, and the presentation of results suitable for use in ship design.

-312. Stronger training of naval architects in the fields of mathematics, physics and applied mechanics so that they can fully appreciate potentially useful developments in theoretical aspects of hydrodynamicso Training in the field of experimental research is of equal importance. Both require increased emphasis on graduate training. Part 2, above, has long been overlooked. As a result, the process of translating new scientific knowledge into a form suitable for the design of ships is seriously inhibited by a lack of competent personnel, Since the Maritime Administration is directly connected with the welfare of the Uo SO merchant marine, it can no longer afford to overlook our country's critical shortage of naval architects with advanced university training. The original 1936 Act (Section 201-e) specifically, authorized the establishment of scholarships for advanced study, and this was echoed by The Maritime Research Advisory Committee ) "The Maritime Administration should establish a program of 20 to 30 annual unconditional fellowships for advanced study in maritime or related fieldso" The Maritime Research Advisory Committee also understood the important interrelationship of education and research activities as witnessed by the following recommendation: An important consideration in assigning research facilities should be to exploit to the maximum the potential influence of such facilities to arouse the interest of able young researchers, to provide them with opportunities for employment both before and after graduation and to develop the research interest and abilities of senior scientists in maritime fields of need.

-32In placing contracts, attention should be given to furthering education in maritime related fields. Particular consideration should be given to universities, especially those with strong interest in naval architecture and marine engineering. Consideration should also be given to private non-university research or industrial organizations and to the David Taylor Model Basin for its specialized facilities and services. The state of Michigan deserves a great deal of credit for the way in which it instituted and now voluntarily supports the national resource which is this Department of Naval Architecture and Marine Engineering, the country's largest source of trained naval architects and marine architects and marine engineers. The University of Michigan will without doubt continue to support undergraduate and graduate training in the marine field, but the critical needs of a greatly expanded graduate program cannot be met without financial assistance from outside the state Such assistance would include funding for unconditional fellowships for graduate study and -- if we are to attract topnotch young people -- there should be scholarships for undergraduates as wello We do not mean to imply that the Maritime Administration, alone, should furnish such support nor that Maritime's educational assistance should be confined to The University of Michigan. We do feel, however, that the common interests of the Maritime Administration and The University of Michigan are uncommonly strong. Therefore, educational funding, of the type suggested here, seems uniquely prudent and natural.

-33Personnel needs, of course, have two facets. The first -- production of better educated naval architects -- has been stressed in the preceding paragraphs. The second relates to the more immediate problem of finding properly trained researchers to assist in carrying out the research efforts now being planned under Maritime's university program, We believe the Maritime Administration could be of greatest assistance here if it would make special provisions to assure long term support to the various cooperating universities. We say this because qualified men are in short supply and we find it difficult to bring them into our employ without at least some promise for the future. Obviously, too, good researchers demand reasonably generous support for facilities. We believe our own personnel needs could be met if we had assurances of long term funding for both salaries and facilities, coupled with the ability to offer scholarships and fellowships to promising students. The other universities would presumably be in the same position. Assurances of long term support have important secondary effects in that they free the researcher from the unattractive burden of preparing large numbers of proposals. This has implications, not only in facilitating the technical work, but in making the positions attractive to topflight technical people. In summary of this section, we would suggest that the Maritime Administration 1) confer with representatives of the interested universities to learn what might best be done to stimulate greater interest in the marine field through the provision of scholarships and fellowships; 2) following the first step, procure funds for annual allocation to the universities for their individual decisions and distribution to students;

-343) grant future university contracts on long term bases (3 to 5 years). C. Facilities 1. Model Basin: Resistance and propulsion play the most important parts in the study of ship hydrodynamics; they were, in fact, for many years the only aspects of significance to ship model researchers. Resistance and propulsion will always be important simply because they are so closely associated with economics. During the last 10 years we have also witnessed concentrated efforts in the fields of ship motion and maneuvering. It is unfortunate that these fields are considered, by some, to be separate from that of resistance and propulsion, Such a divorce can have only detrimental effects on the overall success of ship design, Cause and effect of hydrodynamics are intimately related by the laws of physics, and all of them must be considered without prejudice in the design of a ship. In the end, any design is a compromise, and the neglect of one or more design factors can lead only to an inferior end product. It is therefore important that laboratories engaged in ship hydrodynamic research be equipped to investigate all aspects necessary to the successful design of a ship as a complete entity from the hydrodynamic point of view. If this can be accepted as a criterion for the specification of a resistance and propulsion research laboratory, it follows that there exists in the United States today only one such laboratory, namely the David Taylor Model Basin, These facts were at the forefront of the late Admiral E. Lo Cochrane's mind when, in 1959, he recommended that the Maritime Administration consider the establishment of a new model basinj(22) His

coincided with a proposal from The University of Michigan asking for Maritime Administration sponsorship of a large new ship hydrodynamics laboratory at the Michigan campus. The Maritime Administration studied the possibility and found considerable favor on the part of U. So shipyard management. Spokesmen for DTMB and Stevens Institute, on the contrary, argued against the proposal. Any decision of the matter was thereupon indefinitely postponed. Following the Maritime Administration's postponement of a decision relative to establishing a new model basin, The University of Michigan has done its utmost to upgrade its original (1903) model basin, which is second in this country only to DTMB in allowable model size (up to 16 feet in length). New tracks were installed, a new towing carriage was purchased, and a new wax model and propeller shop was established. Sophisticated instrumentation was bought or produced within University shops, and key personnel were retained on a full-time basis. Although the Maritime Administration encouraged this with research contracts, including funds for calibrating the new equipment, the State of Michigan furnished by far the major share of the required financial support. Recently also the University has allocated funds for the construction of a wave-maker. We have operated the upgraded model basin for a year, now, and have determined that our facilities are suitable for a great many kinds of tests. We have also found that scale effect limitations preclude dependable measurements in several highly important categories of research, particularly where self-propulsion is involved. In addition, the large number of organizations, both public and private, that have

-36used our tank -- coupled with the knowledge that a large percentage of private U. So testing and research is still. done abroad —; lead us again to the conclusion that this country needs another model, basin capable of handling models of 20- to 24-foot length, As regards model basin operating costs, our experience to date leads us to believe that a well managed university facility in this country could be operated at a cost level. comparable to levels at European model basins, Commercial clients at the Michigan tank. assure us that the cost difference between our tank and those abroad is apparently disappearing~ The proposed basin should be available for regular commercial, testing as well as long range research. We believe such a combination of purposes is both necessary and desirable. We think it is necessary because it is the best way to keep overhead costs within reason~ We think it is desirable because there is no hard and fast line between the two kinds of activity and also because their unnecessary separation tends to cause sterility in both, We believe the proposed m.odel basin should be located at a large university that has a strong interest in the marine field. From. a university's point of view, such. a facility would greatly strengthen its ability to attract outstanding staff and students, From the Mari.. time Administration's point of view, a university location would mean minimum costs as well as those extra benefits inherent in the stimulation of education (as already discussed in the Introduction). These conclusions are entirely in harmony with the following recommendation of the Maritime Research Advisory Committee (

-37The Maritime Administration should give favorable consideration to establishing five-year programs of sustained grants-in-aid to those universities which are particularly interested and well qualified to carry out maritime research. In addition, we would cll attention to the following advantages inherent in large universities: a) A university is already in possession of extensive related research facilities such as general hydrodynamics laboratories, computers, wind tunnels, libraries and electronic equipment. b) A university provides a better environment for basic research and stimulates researchers to look for solutions outside their own special field. It also provides a ready availability of expert consultants in related fields. c) The advantages of the university town makes it easier to attract highly qualified research personnel, d) Fixed costs such as taxes and maintenance are minimized. e) Auniversity provides a better atmosphere and provides better conditions for serving many interests, including that of international cooperation, than does a fully government sponsored laboratory or a privately owned research organization. We believe we are safe in saying that no single government agency, educational institution, or industrial organization is willing

-38to finance the construction of a large new model basin in this country. Many such groups, however, would benefit from the availability of such a laboratory. Therefore, we propose that the Maritime Administration establish a cooperative program involving the many interested parties. A partnership between the Maritime Administration and a large university, with supplementary support from industry, seems to be the only practical solution to the problem of initial financing. Obviously, such a large central facility should be made available to industry for commercial testing and research; it should also be available to qualified researchers from other institutions. We suggest that the Maritime Administration discuss this proposal with the interested universities in order to reach its own conclusions as to the best course of action, The next few paragraphs present our carefully considered conclusions in regard to the size and capabilities of the proposed new hydrodynamics laboratory, To carry out standard testing of self-propelled models, without excessive scale problems or tank wall effects, we consider the following model basin dimensions to be an absolute minimum. Length: 600 ft. Width: 30 fto Depth: 15 fto Max. carriage speed: 30 ft/seco A wave maker installed at one end should be capable of generating waves of 35-fto length and 24-inch height and also of a random unidirectional sea,

-39Although a towing basin of dimensions as given above may prove quite satisfactory for much of the work at hand, it does not provide a good foundation for future expansion. The most serious limitation is its cross section. Much research, such as wave pattern analysis, will require wider basins and, for testing submerged bodies, a greater depth is desirable. We therefore, recommend that the following dimensions be considered for the first stage in the construction of a moderately large towing basin: Length: 800 ft. Width: 40 fto Depth: 20 fto Max. carriage speed 50-60 ft/seco The relatively high carriage speed would provide flexibility of design so that special tests requiring high velocities could be undertaken. We realize that a length of 800 ft. is not sufficient to permit the use of the higher speeds, since it will take from 500-600 ft. to reach the 60 ft/seco upper limit. The ultimate length should therefore be about 1200 ft. Thus, provisions must be made for a lengthening at a later date. This extension could be built as a ship motion basin of the same type as that now in use at the Netherlands Ship Model Basin. The towing carriage should preferably be equipped for programmed acceleration. Electronic components, needed to accomplish this, are already on the market. Controlled acceleration would give researchers a much needed tool for the systematic study of transient and non-steady phenomena.

-402, Water Tunnel, In a model study of propeller performance there is one major facet that cannot be investigated in the model tank, namely that of cavitationo To study cavitation one needs a water tunnel. This is one of the most important tools of the laboratory engaged in research in resistance and propulsion, because it provides a means of isolating parameters governing the performance of the propeller. One such parameter is the effect of non-uniform flow to the propeller. Many of the water tunnels in operation today can vary the ve.l.ocity distribution in the plane of the propeller. To obtain a three-dimensional. flow pattern, a model of the stern section of the ship can also be located in the tunnel ahead of the propeller. Nevertheless, we doubt that the flow conditions in a tunnel can ever be made to correspond fully to those behind a ship. Therefore, in regard to propeller research, the water tunnel should be used to study the effect of selected flow patterns, regardless of whether these patterns can be found behind any particular ship. In addition to propeller research there are a great many hydrodynamics problems that can be investigated most easily in a water tunnel, among them, we would include boundary layer studies in connection with pliant coatings and other surface treatments, The test section of the proposed water tunnel should measure approximately 36" x 36"~ This would permit placing dummy bodies in from of the propeller so as to simulate wake distributions and the effects of the propeller pressure field. Maximum velocity of water in the test section should be 20 ft/sec, with the pressure being variable from a high. vacuum to about 1.5 lbs. per sq4 inch measured at t"he pressure regulation container.

-41The propeller dynamometer should meet the following specifications: Maximum RPM 4000 Maximum thrust + 600 lbs. Maximum torque + 75 ft. lbs. We anticipate that a 24" x 24" test section could be supplied in the future so that maximum water speed could be increased to 40 ft/sec,, making it possible to test supercavitating propellers. In addition, it would be desirable to obtain equipment for propeller tests with inclined shafts and for counter-rotating propellers. For testing of bodies and profiles, a 6 - component balance should also be eventually included, We expect the tunnel would be equipped with means for the adjustment of the velocity distribution at the test section. The remaining paragraphs of this section deal. with the facilities and equipment which will be needed at The University of Michigan if we are to carry out the research program described in Chapter I" The proposed new model basin and water tunnel. would, of course, rank first and second in the list of desired facilities. However, the program we propose does not hinge on the availability of these facilities -- at least for the first few years. We do, however, consider them. to be essential to much of the research that is forseeable beyond that proposed here, And, we might add, the sooner the new facilities become available, the greater will be their economic benefit. 3o Work Shop and Tools- Tools and space must be provided for efficient production of ship models and model propellers, and also for manufacturing special instrumentation, Many of the required machine tools are of

-42standard type and are available at the University. Special mode.l-building tools must, however, be supplied to the ship hydrodynamics laboratory. Our wax cutting machine has served a useful purpose up until, now and will probably still. do so if financial conditions prevent the procurement of a new machine. It would be desirable however, to replace it with a machine designed for cutting hard wax, such as that now used and able to cut wood models as well. At The University of Michigan, model propellers are now made almost entirely by hand. As long as the number of propellers (and their physical. size) is small, this probably the cheapest method. Whenever this situation changes, however, we believe it would be of considerable advantage to purchase a propeller cutting machine. 4~ Propeller Dynamometer: Until a realistic propeller theory for nonuniform, non-steady flow is brought forth, experimentation will remain the only means available for the determination of propeller performance in time-varying types of flowo And even when an analytical method becomes available, experimentation would hardly become superfluous., I! n a broad sense, the performance of a propeller includes much more than the efficiency. Considerations of thrust and torque variations, together with transverse forces and moments, are often. of the greatest importance in the design of propellers for modern highspeed ships~. n response to recent design requirements, ship hydrodynamic laboratories have attempted to measure and evaluate these time-variable forces (15 7) The instruments used for these measurements have all been elaborate in design and construction; the most successful and have been those of the Netherlands

-43Ship Model Basin, the Hamburg Ship Model Basin, and DTMB. Fortunately, the electronics industry is continuously producing better measuring equipment, and it should soon be possible to simplify existing designs materially. In case of a propeller, the ideal solution would be to locate the dynamometer in the hub, and we feel that such a dynamometer can be built. The availability of such a device would allow the immediate start of a study of the fundamental aspects of non-steady propeller phenomena in conjunction with the study of non-steady flow conditions. In view of the above considerations, we recommend that the Maritime Administration sponsor, at The University of Michigan, the design and construction of a hub-enclosed propeller dynamometero The instrument should be capable of measuring all forces acting on the propeller, and their time-dependent variations. 5. Hot Film Anemometer: Ro L. Townsin(13) has recently developed a hot film anemometer by which he has been able to determine regions of laminar flow without introducting any disturbance to the flow itself. This anemometer may prove capable of determining areas of flow separation around the run of a model. It should therefore be studied more closely and, if proven successful, be included as a research instrument. 6. Television Camera: We have made successful use of a borrowed television camera as a means of studying flow around a model. We are convinced that such an item of equipment would be of great benefit, particularly in analyzing the flow of water to propellers. We would therefore recommend that the Maritime Administration support the purchase of a suitable television camera, together with the necessary strut and control devices. The Netherland Model Basin and the DTMB have already developed such systems.

CHAPTER III FIVE YEAR RESEARCH PROGRAM SUMMARY In the preceeding sections we have described and elaborated on areas of research on ship resistance and propulsion which we believe require immediate attention, to be followed by continuous research efforts for several years to come. It is realized that in so doing we have been discussing much research which cannot be conducted with the present facilities at the University of Michigan. In this section we will. therefore outline specific tasks which we can perform without any capital expenditures for new major facilities. It is expected, however, that funds will have to be provided for certain instrumentation which will. be needed for the specific investigations proposed. Rather than arranging the separate tasks described below in the order of priority, they have been listed in accordance with the sequence in which they are discussed in this report. A. Hull Form Research a) A study of the relationship between hull form and wave-making characteristics. Both mathmatical and coventional hull forms are to be investigated. The work. on. large bulbs now underway should also be continued. Time: Continuous throughout the five years. b) Systematic evalutaion of the effect of changes of section shape on resistance and propulsion, primarily as applied to the Series60 forms. As an extension, an investigation of the effect of changes of the sectional area curve will be undertaken, based on

-45the linear wave-resistance theory and on model tests. Time: First through third year. c) The study of a family of hull forms of fullness greater than CB =.80 to determine resistance and propulsion characteristics of present day large slow speed commercial ship in a systematic manner. Time: First through third year. d) Application of regression equations to existing tank test data for the purpose of a statistical determination of optimum hull form perameters for commercial ships. Time: First through second year. B. Ship Propulsion Research a) The complete development of the five-hole pitot tube fitted with electronic pressure transducers and mechanical drive for the survey of the wake field in the plane of the propeller. The introduction of electronic recorders and analysers to provide a harmonic analysis of the wake flow, and the application of this instrumentation to the determination of flow conditions behind the hull forms tested under project Ib above, Time: First through third year. b) A detailed study of time variable flow conditions in the wake field of various ship forms to determine parameters governing this phenomena. Time: Second through fifth year. c) Experimental determination of propeller induced pressures acting on the hull. Results to be checked against prediction of

-46existing theories. Time: Second through forth year. d) Development of a propulsion dynamometer to be installed in the hub of the model propeller. The experimental determination of time variations of torgue and thrust in flow fields of known characteristics including time and space variations of velocity components. The comparison of results with theoretically predicted values. Time: Second through fourth year. C. Research on the Effects of Ship Motion on Resistance and Propulsion a) Self-propulsion tests of models in moderate head seas to determine average changes of propulsion characteristics. Time: Second through fifth year. b) Measurements of time variations of thrust and torgue of selfpropelled models in waves, Such measurements will be correlated with ship motion data to determine relationships between variations of Dropellerforces and motion amplitudes at the stern. Time: Third through fifth year. c) Testing of models fitted with large bulbs in head seas to study effect of these bulbs on ship motion and resistance. Self propulsion tests will be included. Time: Second through fifth year. D. Research to Improve Ship Model Testing Procedures a) The development of higher order corrections to the linear wavemaking resistance theory. Evaluation of these simple mathematical

ship forms in an attempt to bring theory in better agreement with experimental results. Time: Second through fourth year. b) Determine the viscous drag by means of wake survey in one transverse plane. Analysis of data in accordance with a recent paper by Landweber and Wu. Mathematical hull forms, for which the wave-making resistance can be readily evaluated,will be used primarily. Time: Second through fifth year. c) Evaluation of wave-making resistance from wave profile measurements by means of several methods recently proposed by Eggers, Ward and others. Investigation of second order corrections necessary to bring theory and experiments in better agreement. The effect of the boundary layer and eddies on the wave-making resistance will be studied very closely, and also the possibility of a scale effect on the generation of ship waves. Time: Second through fifth year. d) Investigation of boundary layer stimulation by means of wake surveys accompanied by calculation of viscous drags. In addition hot film anemometers will be used to determine laminar flow regions and flow seperation at the stern. Models of different sizes are to be tested to provide data for evaluation of scale effects. Time: Third through fifth year. e) Application of three-dimentional boundary layer theory to ship hull forms. Comparison with wake survey measurements. Time: Fourth through fifth year~

-48f) Blockage effect study incorporating theoretical evaluation of increase in wave-resistance of a ship moving in a channel and the evaluation of the change of frictional resistance due to blockage from wave surveys. Time: Third through fifth year. g) On the basis of information obtained on the dependence of wavemaking resistance upon the Reynolds number and scale effects,attempt will be made to calculate the wave-resistance of a full size ship from available theories. Time: Fifth year. h) Planning and execution of full scale trials to verify predictions derived from projects D: a, b, c, d and e. Time: Fourth and fifth year. It must be realized that the time schedule is tentative and that individual projects may be shifted within the five year period. Furthermore it should be recognized that it will not always be possible to fully adhere to time allowed in each case. Progress made at the University of Michigan and elsewhere may change the picture considerably. It is therefore important to add the following project~ Maintain a continuing survey of world wide research activities in related fields and, when necessary, modify or argument the proposed program. Report to be submitted to the Maritime Administration yearly.

REFERENCES 1. E. V. Lewis, "Research Toward More Efficient Transportation by Sea," SNAME Transactions, 1961. 2. R. L. Townsin, "Frictional and Pressure Resistance of a Victory Model," N.E.C. Inst., Transactions, Vol. 78, 1961-62. 3. J. Wu, "The Separation of Viscous from Wave-Making Drag of Ship Forms," Journal of Ship Research, June 1962. 4. K, Wieghardt, "On the Turbulent Flow along a Cylinder and a Prism," Proc. of 8th ITTC, Madrid, 1957. 5. L. W. Ward, A Method for the Direct Experimental Determination of Ship Wave Resistance, Ph.oD Dissertation, Stevens Institute of Technology, May 1962. 6. W. D. S. Wigley, "Ship Wave Resistance. A Comparison of Mathematical Theory with Experimental Results, Part 1," T.IoNoAq, Vol. 68, 1926. 7. C. Ridgely-Nevitt, Resistance Tests at the Webb Towing Tank of the ATTC Standard Model, Webb Institute of Naval Architecture, April 1960o 8. K. Eggers, Uber die Ermittlung des Wellenwiderstandes eines Schiffsmodells durch Analyse seines Wellensystems, Schiffstechnik 9 No. 46 S. 79-84 April 1962. 9. John M.Kill, The Sonic Surface-Wave Transducer, St. Anthony Falls Laboratory Technical Paper No. 23, Series B, July 19590 10. N. Hogben and G. E. Gadd, An Appraisal of the Ship Resistance Problem in the Light of Measurements of the Wave Pattern, Draft for M.P.L. Ship Division Report. 11. Takao Inui, "Wave-Making Resistance of Ships," SNAME Transactions, Vol. 70, 19620. 12. L. Landweber and J. Wu, "The Viscous Drag of Submerged and Floating Bodies," IIHR Report, March 1962. 13. R. L. Townsin, "Turbulence Detection. Results from the Use of an Unobstructive Technique during Ship Model Testing," INA Transactions, Vol. 102, 1960. 14. J. C. Cooke and Mo Go Hall, "Boundary Layers in Three Dimensions," RAE Report No. Aero 2635, February 1960o -49

-5015. JO D. Van Manen and Iro R. Wereldsma, "Dynamic Measurements on Propeller Models," Int Shipb. Prog., Vol. 6 Me.63, November 1959. 16. J. K. Krohn, Numerical and Experimental Investigations of the Dependence of Transverse Force and Bending Moment Fluctuations on the Blade Area Ratio of Five-Bladed Ship Propellers, Fourth symposium on Naval Hydrodynamics, August 1962. 17. J. B. Hadler, Wo Kopko and P. Vo Ruscus, Correlation of Model and Full-Scale Propeller Alternating Thrust Forces on a Sumerged Body, Fourth Symposium on Naval Hydrodynamics, August 1962. 18. G. R. Stuntz, Jr,, PO Co Pien, Wo B. Hinterthan and N. Lo Ficken, "Series 60 - The Effect of Afterbody Shape upon Resistance, Power, Wake Distribution and Propeller Excited Vibratory Forces, SNAME Transactions, 1960o 19. J. P. Breslin, Review and Extension of Theory for Near-Field Propeller Induced Vibratory Effects, Fourth Symposium on Naval Hydrodynamics, August 1962. 20. M. St. Denis and Wo J. Pierson, Jr. "On the Motions of Ships in Confused Seas," SNAME Transactions, 1963. 21. H. J. So Canham, D. E. Cartwright, G. J. Goodrich, and N. Hogben, "Seakeeping Trials on O.W.S. Weather Reporter," I.N.A. Transactions, 1962. 22. Proposed Program for Maritime Administration Research, Report of the Maritime Research Advisory Committee, NAS-NRC, to the Maritime Administration9 1960o 23. Oceanography 1960 to 1970, Report by the Committee on Oceanography, NAS -NRC 24, Ship Propulsion and Resistance Studies, Proposal to Maritime Administration, Project ORA-63-491-Fl, November 1962. 25. B. Vo Korwin-Kroukovsky, "Theory of Seakeeping," SNAME, 1961. 26. Harry Benford, General Cargo Ship Economics and Preliminary Design, Office of Research Administration, The University of Michigan, 1962. 27. Three Recent Papers by Japanese Authors on the Effect of Bulbs on Wave-Making Resistance of Ships, The University of Michigan, Department of Naval Architecture and Marine Engineering, December- 1961 (translation)o

-5128~ D. J. Doust, "Optimized Trawler Forms", T.o ECI, December 1962. 29. A. J. Tachmindji and M. CO Dickerson, "The Measurement of Oscillating Pressures in the Vicinity of Propellers," DTMB Report No.1130, April 1957. 30. Daiichi Nakase, Akio Kaneda, Katsuo Otaka and Kenji Fujita, "On the Effect of the Propeller Surface Force on Ship's Hull," Journal of Japan Society of Naval Architects, Vol 107, 1960.

APPENDICES

APPENDIX I THE DETERMINATION OF THE WAVE-RESISTANCE OF A MODEL FROM THE WAVE PATTERN (Eggers' Method) A. The Wave Resistance for Steady Motion Derived from the Flow of Energy in the Wave Motion The ship is assumed to move at constant velocity c in a channel of constant width b and constant depth h. Considering the fluid occupied by the region between two fixed planes A and B it can be said that the rate at which work is being done on the fluid is equal to energy transport out of the region plus the time rate of change of energy within the region. It will be assumed that plane A is located so far ahead of the ship that the velocity there is equal to zero (no waves). Only flow conditions at plane B need therefore be considered.'4. - -17- 2 I7-~ l' ___ ___ 7_____________7 _______________ 7 XO XI Figure I-i. Definition of Control Planes and Coordinates. I~~~~-3

Owing to the constant speed of the ship, it is clear that the energy between the planes B and A (see Figure I.1) at the time to + 6t is the same as the energy between the planes B and A at time to. It follows that the energy in the fluid between B and B' represents the increase in energy of the fluid in the region R over a small time interval 5t. The kinetic energy of the region between the planes B and B' is given by 6T f 0' ds = P f (V)v(1) 2 an 2 where 60/6n is differentiation with respect to the inward normal. Limiting our attention to the first integral, Equation (1) can be written as b b 2 2 o 6T = P f{ct (oz)zo dy - f dy f [(OX)x=x (00X)xxl]l dz} 2 - _ _b -h 2 2 (2) The contributions from the bottom and the side walls are clearly equal to zero. Expanding (00x) in a Taylor series it follows that (0xX) x=xl (00x) + cst(00x) and Equation (2) becomes b b 2 2 0 2 6T = t i (z)z_= + f dy f [(Ox) + XX]X dz} (3) 2b b -h =X 2 2 The free surface condition is given by z = - K-o xx where K0 -

Equation (3) can therefore be written as b b 6T = y t2 ft { dy + 0XX2 dz - K f (df xx)z_~ dy} 2 -h = Kob 2 (3a) The potential energy of the fluid between the planes B and B', measured with respect to the undisturbed fluid, is given by b b 6V = pgc6t f zdz f dy = pc t f f2dy (4) o b 2 b 2 b To the usual linear approximation, the free surface is obtained from g Substituting into Equation (4) it follows that b pc 2 2 sV =- t f (OX) dy (5) 2Ko b z=o 2 From Bernoulli's equation the pressure is given by P = - pq - pgz -cpPx The work done on the plane B during the time interval bt is therefore b 2 o bwp f dy f pubtdz b -h b b 2 01 2 o = - f dy f - pq2utdz - dy f pgzuStdz (6) b -h 2 b -h 2 2 b 2 o 2 + cpSt f dy f (0x) dz b -h

-56The energy density per unit volume is defined by 1 2 e =- pq + pgz 2 Thus the energy flux into the region R can be expressed as b b 2 o 2 o E = bt f dy f pq2udz + 6t f dy f pgzudz (7) _b -h.b -h 2 2 The work done by the ship is, by definition, 6Ws = Rtc6t (8) and the total energy balance becomes 6Wp + 5Ws + 3E = 6T + bV Substituting Equations (3a), (5), (6), (7) and (8) one obtains an expression for the resistance as follows b b 2 2 2 o Rt = f f [(Ox) - Oxx] dy - P f dy f [(Ox)2- xx] dz 2Kob Z=o 2b -h X=Xo 2 2 (9) Returning to Equation (1), the use of the right hand volume integral will lead to a slightly different expression for the model resistance. Letting dv = c6tdydz b 2 0 2 T = 6t f dy f (I) d b -h

and b op 2 2 2 T - 6Wp - 6E = t f dy [02 + - 0] dz 2 b -h Z 2 Substituting into (8) b b 2 2 o Rt = g 2dy + P f dy f [O2 + 2z - x dz (10a) 2 2 b -h 2 2 or b b o 2 2 2 R p 2 2 2 dy fdz (b) t 2Ko + fz x] (d (O b) o b b -h 2 2 Assume the velocity potential far aft of the model to be given by 00 O(x,yz) c g 1 {Cuncoscwnx + Pnsinwonx } n=ln cosh Kn(h+z) no, b cos -(- - y) (11) cosh Knh b 2 + g cCOSco X- _oSinoX} cosh uo(h+z) 20oc cosh woh where 2 2 Kn (n b (12) b It is noted that the boundary conditions at y = + - and z = -h are satisfied by this potential. At the free surface ~xx + Kogz = 0; z = o

which gives 2n - KoKn tanh (Knh) = 0 but 2 2 = - (b) or n=\ K2 (nt) > n = 0,1,2,... so that Kn - KoKn tanh (Knh) = 0 (13) In case Equation (13) has no real root for a given n it will be required that an = Pn = 0. The velocity components are given as follows 00 -u = -x = g E {n-a sin wnX + Bn cos hnX} c 1l cosh Kn(h+z) cos n(b y) cosh Knh b 2 + g {-<(o sin cox + Bo cos woX} cosh w0o(h+z) cosh woh 00 -y = g {n cos fnx + sn sin 2nx} nb Y C n=l b cosh Kn(h+z) sin n(b y) cosh Knh b 2 Introducing the following functions An(x) = A-n(x) = 2(n cos Wnx + Pn sin cnx) Bn(x) = B-n(X) = L(Pn cos Dnx - an sin cnx) (14) 2 the derivatives of the velocity potential can be written x = c 0 Bn(x) cosh Kn(h) cos n(b - y)

-5900 g Z An(x) nit cosh Kn(z+h) sin n - (15) Y n=-oo bwn cosh Knh b 2 z g An(x) K sinh Kn(z+h) cos n(b _ y c n=- xon sinh Knh b 2 The free surface, given by g z=o can now be written 00 5 = ] Bn(x) cos -( ) (16) n=-.oo b 2 and also 00 n~~xn =x l An(X) wn sin n-,b y) (17) Due to orthogonality, b nit b mr b f cos b 2 - y) cos b (2 y) dy 0; m ~ n b 2 =; m =n and similarly for the sine functions. Substitution of (15) and (16) into (10a) therefore gives Zo 2)0b2 0b 2 cosh2Kn(z+h) ~R = Pb 00 Bn(x) 2bc2 Z {An(x)[ (b) cosh2Knh dz n =-o n=-o -h O sinh2 Kn(h+z) K2 cosh Knh dz - Bn(x) 2 Kn(z+h) z} (18) -h Csh Kn h

-60The last two integrals can be evaluated, thus 0 cosh2 s(h)d sinh 2Knh + 2Knh -h 0 2 sinh 2Kn - 2Knh S sinh Kn(z+h)dz =sinh 2Kn -h 4Kn Furthermore 22 tanh (Knh) = g Kn g Kn and since 2 1 cosh2 x - 1 sinh 2 x ctgh x it follows that g Kn cosh2 (Kh) = sinh (2Knh) 2g'n2 Equation (18) can therefore be written pgb [2()( (sinh 2Knh + 2Knh)Cn RB -- Y n=[B-o - (sinh 2Knh) K 2 n=R 2 sinh(2Knh) + 2Knh n, 2 sinh(2Knh) - 2Knh An(x)(nh bKn sinh 2Knh(19) or, with 2 2 On pgb {A2(x) + Bn(X)2 sinh(2Knh) + 2Knh (n)2} (20) ~n= 1n~-o f\[ sinh Knh From Equation (16) and (17) b 2J t~~y nio b f 5(x,y) cos b-( - y) dy = b Bn(x)

-61b 2 f x(x,y) sin nT(b - y) dy = b An(x)cn b b 2 2 It follows therefore that b 00 2 2 R -g P [{ I (x,y) cos n -,(b y) dy2 4bn= b b2 2 b + + { IX(xy) sin - - _ y) dy}i (21) 2 [2 - sinh(2Knh) + 2Knh (Dn)2 ] sinh 2Knh Kn Defining the quantities An and cn by the relationships An(x) - A-" sin(cwnx + Ej) (22) Bn(x) = A cos(cnx + c*) (23) where 1 2 AX 4 \/ot- + 2 * n tg En Pn a substitution of Equation (23) into (16) leads to 00'(x,y) A cos(Lnx + (n) cos b (2 - Y) 00 X, + nT nj T Z A* cos (nx + cy +a n Y) n n IAn sin(XUx A+ sinn + n) -( y)

-62The last summation is equal to zero. Hence 00 5(x,y) = E An cos(onx + en + n,- ni y) (24) From (12) 2 (n)2 2 It is therefore advantageous to introduce the variable En such that ni _ Kn sin 8n; cn - Kn cos en (25) b Equation (24) now becomes 00 c * b ~(x,y) = Z An cos(Knx cos an - KnY sin en + En + 2 Kn sin en) n=2 (26) If the angle ~n is the angle between a line and the positive x-axis, Equation (26) represents a separation of the free surface into waves of length equal to 2c/Kn. The group velocity of such a wave is 1 sinh(2Knh) + 2Knh n 2 sinh(2Knh Vph where Vph =- \g tanh (Knh) (28) Because the waves are being generated by a model moving at a constant speed c along the x-axis, the crest of a wave propagating in the direction ~n has a velocity c along the x-axis. This condition is given by vnh = c cos En (350)

-63The flux of energy in the x-direction is similarly given by CE = U cos en = 1 c Cos en sinh(2Knh) + 2Knh (31) 2 sinh(2Knh) It is noted from (22) and (23) that {An(x) + Bn(x)} =(An)2 From (20) it then follows that the wave resistance can be written R = pgb A ( E (32) 2 n C B. Experimental Determination of the Wave-Resistance from Wave Profiles If 5(x,y) and 5x(x,y) were known along some line perpendicular to the direction of motion, it should be possible to determine the wave resistance from Equation (21). To avoid the difficulty of the determination of the wave slope, however, it will be necessary to make use of two parallel sections a distance 2A apart. The free surface ~(x,y) is by (14) and (16) 00 t(xy) = T1 7' (Pn cos wnx - an sin Cnx) cos n(b. y) (33) -2 -b2 Defining the free surface along the sections as x = xs + A and x = xs - A by ~+(y) and S-(y) respectively then 00 1() 2 {in cos wn(x + A) sin n(X + A)} cos n(-y) noo (34)

-64and 1 00 (y) 21 {7 (n COS n(Xs ) CAn sin -An(Xs - A)) cos (b - y) 2 b ( n=o-2 (35) Hence 00 5+(Y) + ~-(Y) = Z {Pn cos cnXs o cos Wn A - Cn sin cnXs o cos nA} n =-.o x cos - y) (36) 00 S+(Y) - -(Y) = Z {-n sin wnXs sin nnA - czn cos wnxs sin tnac} n= (oob x cos n(b y) (37) It is noted, however, that 00 +(y) + 5 (y) = 2 E cos nA Bn(Xs) Cos -(- Y) (38) n2= —cobO and 5+(y) - c-(y) = -2 Z sin UrA An(xs) cos (b- _ Y) (39) n=-oo from which it follows that b 2 Bn(Xs) (~+(y) + - (y))cos ~_(b - y) dy 2b cos cnA b b 2 2 b A(x) (=+(y) - — (y))cos nt(b y) dy ns 2b sin nA b b 2 By defin___n An -An(Xs) + Bn(Xs) 1= (n + cn) An = ~

Hence b 2 2 12 ~ ~ fi 2 2b 2 _nA ( S (~+ + ~-) cos n(b _ Y) dy)2 -n + =n =b2 cos2 +nC b 2 y) dy) b 2 + 2 2 f (+ - b) cos n( y) dy)_ b sin2 wnA b 2 provided sin wnA ~ cos onA A 0 Introducing the identity cos n(b2 -_g) = cos cos n Y + sin n sin n b2 2 b b b one can finally write b n=oi n2 n2 R = [Co2n A {TnIc+ + (1 - Tn)Is+} + in2n { TnI + (1 - Tn) Is}1 2 (41) s2 inh 2Knh + 2Knh (41) 2 sinh 2Knh K2} n where b 2 In ( + =-) cos n-b y dy (42) c+ b b In = (I+ + ~-) sin ny_ y dy (43) si b (45

-66Tn =1 n even = 0 n odd It is noted that values of In and In are obtained from a harmonic analysis of (5+ + 5m)o Substitution of these values into Equation (41) together with the appropriate values of wn and Kn, as given by Equations (12) and (13) (in addition to A'and h) provides a numerical expression for the resistance of the model. Because the fluid is assumed to be inviscid this resistance is taken to be the wave resistance.

APPENDIX II WAVE RESISTANCE OF SHIPS Ship wave resistance theory has hardly advanced beyond the Michel integral for the wave-making resistance of thin ships. This formula successfully depicts the nature of the variation of wave-making resistance with Froude number and, because the hull function appears explicitly in the integrand, has been useful in showing the effects of variations in hull form. Because of the restrictive assumptions on which it is based, however, it cannot yield resistance values of the accuracy usually required for ship resistance calculations. Inui has attempted to improve upon the Michel theory by satisfying the boundary condition on the hull surface more accurately. Actually the boundary condition is satisfied more accurately only at very low Froude numbers, and one cannot be certain that there would be any improvement in predictions by this method at moderate and high Froude numbers. An interesting and important alternative to the thin-ship theory is the recently published slender-ship theory of Vossers. This is also a linearized theory, however, and cannot be expected to yield a close approximation to the solution of the exact potential-flow problem. A well-known method for determining the potential flow about a body, when no other boundaries (nor a free surface) are present, leads to an integral equation for an unknown source distribution on the surface of the body. This same procedure has been applied by Havelock to -67

-68formulate the ship wave-resistance problem so as to satisfy the hull surface condition exactly, although the free surface condition remained linearized. The resulting integral equation has been considered to be too complicated for numerical evaluation. The purpose of the present work is to suggest practical means of obtaining more nearly exact solutions of the ship wave resistance problem by satisfying both the nonlinear free surface condition and the hull surface condition to a higher degree of accuracy. A. Formulation of Boundary Value Problem It will be supposed that the ship is at rest in a channel through which a stream of constant velocity U in the negative xdirection is flowing. Take the y-axis positive to port and the z-axis positive upwards, with the origin at the undisturbed level of the free surface at the center section of the ship. Denote the equation of the hull surface by _= +f(xz) L < x < L -H < z < O (1) where L is the length of the ship and H its draft. It will be assumed that the fluid is inviscid and incompressible and that the flow is irrotational. Under the assumed conditions, there exists a velocity potential 0 and a perturbation potential 0 related by = - Ux (2) which satisfy the Laplace equations _- = V = o ()

-69and express the velocity at a point of the fluid in the form s = u - U; u - X, v=, X; X = 60 (4) 6x ax by az where u, v, w are the components of the perturbation velocity. The boundary condition that there is no flow across the hull surface yields the boundary condition ufx - v + wfz = Ufx (5) where f 6 f f ~x ax' ~z -z Denoting the equation of the free surface by z = Z(x,y) (6) we obtain for the boundary condition at this surface (U-u)Zx - vZy + w = O (7) The Bernoulli equation in the present case is p 1 p[(U-u) + v +w ] + gz 2 (8) and the condition p = 0 at the free surface yields gz = uU - q2, q2 2 +v2 +2 (9) 2 Eliminating Z between (7) and (9), we obtain u a~ + — a -= u a (u2 + v2 +- 1 w ) _ + v O(q2)1 (10) ax' aZ ax ~2 2 dx a

-70Thus it is required to find a solution of Laplace's equation which satisfies the linear Equation (5) on the surface of the hull and the nonlinear Equation (10) on the free surface (6). The free surface condition (10) is inconvenient because it is to be applied on a surface of unknown location, and because it is nonlinear, so that superposition techniques are not applicable. The condition may be transferred to the plane z = 0 by means of Taylor expansions, such as 210(xy,z) - a0(x,y,0) + Z a60 (xyO) + 2 40(xPyO) + 6x2 6x2 - x26z 2o 6x26z2 ~(x yIz) = (xy,0) + z A20(x y,0) + xZ2 (,y,0) + which, substituting into (10) and applying (9), give, to terms of second order, U2 + g 0 U[ ( u2 + v2 + 1 w2) _- uo2 ) (11) ax2 az ax 2 g ax2az az2 applied on the plane z = 0. Let 00 denote a solution which satisfies (11) with the right member zero - ioeo. U2 200 + g a = O (12) x2 + g Then, setting uo = a etco., one obtains, to terms of second order, Ua2 - = U[ (u2 + v2 + W2)1 ] a linear equation, applied. on the plane z = 0O This procedure can clearly be continued to obtain linear approximations of higher order of Equation (10).

-71Consider a source of unit strength at the point (9 9 ) within the channel through which the stream of velocity U is flowing in the negative x-direction. It will be assumed that one can obtain the disturbance velocity potential corresponding to this source which satisfies the condition of impermeability at the solid channel boundaries, a linear free surface condition such as (13) or its extension to higher order, and a radiation condition that surface waves are propagated only downstream. This velocity potential, G(x,y,z; 59ng)~9 may be expressed in the form G(x,y,z;S,qa,.) - - + F(xy,z;S,,s) (14) where r =(xx-)2 + (y-_)2 + (y2)2 +1/2 and F(x,y,z;S,r,q) is a regular, harmonic function within the channel which may be considered as the potential of the image system of the source at (an,) in the solid boundaries of the channel and the free surface. B. Source Distribution on Hull Surface Assume that the boundary value problem for a ship form can be satisfied by a distribution of sources of strength m(~,9yr) on the hull surface. The potential at a point (x,y,z) of the surface due to a source element m(\,y, )dS is m( b sJs )G(x,y,z;e, nm)dS which contributes to the normal derivative at (x y9 z)

-72By application of Gauss s flux theorem, it may be shown that the source element m(x,y,z)dS, in the neighborhood of the point at which the component of the velocity normal to the surface is being evaluated, contributes 2irm(x,y,z) to the normal derivative. Thus the hull boundary condition becomes 2im(x,y,z) + f m(,) yG(xyz;ps,,)dS = U n (15) a Fredholm integral equation of the second kind. for m(5, 5)o Since the direction cosines of the normal dlrected outward from the hull surface are fx 1 fz -~ I4 2 f~x — f2, 2 2-2 - + fx+ f 1' + fx f 5 b 71 + f + f 2 \V 1V a= we have aG fxGx - Gy + fzGz an l - f 1 + 2 + fcx +z V or, by (14), G fxx fzF (x)fx - (y) + (5.)fz (16) 6 /1 + + f2 r /1 + f2 + f2 f x z The factor r-3 in the last term of (16) indicates that the integral in (15) is improper, although, as is shown in texts on potential theory, the integral converges. In terms of m( 9,5 ), the velocity potential is 1(xy,z) =- m(f S k)G(xy z;_ t9 )dS (17) 5

-73and the wave-making resistance, given by the Lagally theorem, is R = -47tp fI m(~t9 i)m(xgy z)Fx(xgy z;5,,~)dSdS (18) ss The theory of Fredholm integral equations of the second kind gives assurance that (15) can be solved for m(,~ ~)o Practically, however, because of the complexity of the function F(x~yz;~ 9, 9), the presence of singularities, and the necessity of integrating over a curved surface, a numerical procedure for solving (15) has not yet been developed. A long computing program, which overcomes the two latter difficulties in connection with the potential flow about bodies in an unbounded fluid, has recently been developed. It is suggested that research be continued to shorten the computing program for this case, which would yield means for calculating the last term in (16) and to develop an efficient means of computing the gradient of F(x,y,z;,9,) o C. Source Sink Distribution on Hull Centerplane Assume a source distribution m(S, ) on the centerplane within the hull. The hull boundary condition is now f m(tG) an G(x,y,z;y,oy)dSo U an (19) in which the normal derivative is taken at a point (x,y,z) of the hull surface and the integral extends over the centerplane within the hull. This is a Fredholm integral equation of the first kind with the same kernel (16) as the integral equation (15), but with 0o An iteration formula for solving (19) is 1x mn( ) S (20)x mn+l(X~Z) = mn(xz) + I [U ~ — I m dSo1 (20)

-74where I =f 6 dSo = f [F -a ()]dS (21) s s 7n-s n ir 0 0 2 r". )2"* s i(~-ttj1/2 r -[(x=g)2 + + (z)2i In the numerical evaluation of (20), the integrals in (20) and (21) are replaced by a quadrature formula and one then has a set of linear algebraic equations to solve by iteration. Since the part of the kernel a (1) peaks sharply in the neighborhood of (5 G) = (x,z), it would Ad r be necessary to use very small intervals, or a quadrature formula of very high order to obtain sufficient accuracy. It is desirable, practically, to eliminate the peak in order to permit the use of quadrature formulas of moderate order. This may be accomplished as followso First we note, by Gauss's flux theorem, that if the point (5,~) on the centerplane is held fixed, and the integration extends over the hull surface S, we have f a(p)dS = -4 (22) or, since 6s dS dS dS (23) 0 n [l+f i+f l/2 (25) x z and 1 + fx + f2( a ) = (x-)fx + f(xz) + (z-=)fz] (24) /z ( r Then substituting (23) and (24) into (22) and interchanging the variables (xz) and(5,e), we obtain

-75(XE)fE + f('i9) + (Z-O)ft dSO= 4 (25) o [(x-0 ) + f + (z-2) 1 Let us now write (21) in the form I _f [F K(x,z; 0 S K - (x- =)Fx f(xz) + (z-L)fz so on [1 + fx + fz 2~' [(x)2 + (xz)2 + z)2 (26) Then, noting that (25) may be expressed in the form sf K(~S;x,z)dSo -4t (27) o we have I f [F K(x,z;S,r) - K(ft;x,z)]ds + 4 (28) 1s +n 2 2 + 2 2+f o 1 + fx + fz I + fx + z the integrand of which does not peak in the neighborhood of the point (,n) = (x,z); in fact, the term giving rise to the peak has been annulled at this point. The integral occurring in (20) can be altered in a similar way. We have 6G 6F m(S ()K(x, z; ) S m(~,0) -n dSo - f m(,+) ndS~ - f [m( z95) aF mm(xz)K(xz; z)K(s;) x(,z)'dS x z +[i + f2 + 2f1/2 [lAgain+f+fl/ it is seen that the term giving rise to the peak in this inte gral has been annulledo

-76This is as far as the development will be carried here. The programming of the suggested procedure for solving the integral Equation (19) should be undertaken. It is believed that such a program would be well within the capacity of existing high speed computers. Since the integral Equation (19) is of the first kind, it may not possess an exact solution. Nevertheless, such equations are capable of producing excellent approximations to a solution of a physical problemo In practice, when an iteration formula such as (20) is used to solve an integral equation of the first kind, the error, given by the term in brackets in (20)9 is observed after each iteration, and the iteration is stopped when the error is uniformly sufficiently small over the entire surface or when the error begins to grow and becomes unacceptably large at some point or points with an increasing number of iterations. The above procedure for eliminating the peak in the integrands can also be applied in the integral equation of the second kind (15) to eliminate the singularity at the point (r9.) - (x9y9z)o For numerical evaluation, the principal advantage of the method of a distribution over the centerplane over that of a distribution over the hull surface may be in the greater ease of evaluating the function F(x,yz;,9~y9) and its derivatives on the centerplaneo When m(,)) has been determined from (19)9 the velocity potential is given by ~(x,yz) = f m(,)G(xy,z;~0,~)dSo (30) so and the wave-making resistance is R - ~4-p f f m(x9z)m(~9)Fx(x90;z;;90)dSodSo (31) so So

APPENDIX III SHIP RESISTANCE IN RECTANGULAR CHANNELS A. Introduction The problem is to determine the influence of the finite dimensions of the section of a towing tank on the resistance of a ship model. Although empirical criteria for avoiding this so-called blockage effect are known, these are frequently violated either in connection with tests of a geosim series, or in the deliberate selection of a larger model scale so as to obtain a propeller size large enough for self-propulsion studies. Consequently a procedure for correcting towing tank data for boundary effects is desired. Bo Statement of Problem It will be supposed that a ship model of length L is being towed along the centerline of a rectangular towing tank of depth h and width w, of relative dimensions h > 2 s w>L ( ) -2' at towing speeds U in the range of Froude numbers FL = < 030 (2) 4gLi These limits appear appropriate for the purpose. It is unlikely, for example, that a model longer than 20 feet would normally be tested in a tank 20 feet wide and 10 feet deep. Furthermore, the speed range given by (2) is more than adequate for merchant ship forms. For a 400-foot ship condition (2) corresponds to a top speed of 20 knotso - 7 7 -

Conditions (1) and (2) may be combined to give a restriction on the Froude number based on depth, F U -'L < 0.42 (3) Thus the radical changes in flow phenomena associated with depth-Froude numbers near unity need not be considered within the scope of this problemo In the indicated ranges of interest the effects of the vertical walls and bottom may be expected to be small, advantage of which should be taken to simplify an otherwise extremely complex problem. C. Proposed Research First consider the case of very low Froude numbers, for which the free surface boundary condition may be taken to be that for a rigid surface. There is no wave making and the principal effect of the walls is to increase the velocity of flow relative to the model, As is well known from wind tunnel practice, the velocity field produced by the presence of the walls may be determined by the method of images in potential theory; see section (b) below. The viscous drag of the body is then assumed to be that associated with the increase in the mean velocity of flow, The induced flow also modifies the pressure gradients at the stern of the model, but these are small, and an examination of their effects has indicated that they are of secondary importance. One would expect then, that, by the method which has been aIplied so successfully in wind tunnels, it would be possible to predict wall effects at low Froude numbers, It appears, however, that the velocity correction, computed by treating the free surface as rigid, is only

-79about one-half of the value observed at a Froude number of about F - 01.5, at which the wave-making resistance is still negligible. This unexpected disagreement has been reported by several, investigators and, as a consequence, empirical formulas for the velocity increase have been based on a crude one-dimensional flow analysis rather than on the sounder method of images. Although the accumulated evidence for the aforementioned disagreement is convincing, it is indirect. It is proposed, then, that research be undertaken to measure the change in velocity of flow about a ship model due to the finite width and depth of a towing tank, at various Froude numbers, beginning from as low a value as possible. For this purpose an accurate set of pressure measurements taken around the girth at midships should suffice, by means of the Bernoulli equation, to give the change in the mean velocityo A possible explanation of the apparent discrepancy between theory and experiment is that, with increasing Froude numbers, appreciable changes in flow about a ship model occur at a much lower Froude number than for wave making resistance. The validity of this hypothesis, which seems to fit the known facts, would be determined by the set of flow measurements proposed above. The velodity about a ship model can also be otained from potential flow, gravity wave theory. For the case of a rectangular channel., the velocity potential which satisfies the linearized boundary-value problem is known, so that, in principle, the velocity of flow about the hull can be computed. This suggests that it would be desirable to select,

-80for the proposed pressure measurements, a form for which flow and waveresistance calculations can conveniently be performed. A successful culmination of the above program would not of itself yield a practical solution. The formulae of gravity wave theory are too complex to yield a direct insight into the effects of hull form, width and depth of channel, and the Froude numbers on. the change in speed or resistance. Since these effects are small, owing to the restrictions of the problem in (1) and (2), it is suggested that attempts be made to simplify the expressions for the velocity potential and the wavemaking resistance, either by means of power series in small. quantities or by asymptotic expansions. If this can be accomplished for a mathematically simple form, such as one generated by a source and a sink near the free surface, the resultant expressions would probably be considerably better suited for displaying the desired effects. 1. Wall Correction at Very Low Froude Numbers Consider a source of strength M in a rectangular channel. of width w and height ho Take coordinate axes (y, z) with origin at the center of the channel. at which a source is situated, Then there is an image system in the walls consisting of sources situated at the points (mw, nh), m, n = 0 + 1, + 2,..., but not both zero. The velocity potential due to these images is =-M EZ Z [x + (y-mw) + (z-nh) 1/2 (1) M -00 - 00 If there is a source M at x -c and a sink -M at x = c in a univorm stream of unit strength the velocity potential owing to the image system is

-81= M 7 7 {[(x-c)2 + (y-mw)2 + (z-nh)2]/2 -[(x+c)2 + (y-mw)2 m n + (z-nh)2]-l/2} (2) The x-component of the velocity at (0, O, O) is then UO = 2Mc Z Z (c2 + m2 w2 + n2 h2)3/2 (3) m n By Taylor's added mass theorm we have V (1 + kl) = 4Mc (4) where kl is the longitudinal added mass coefficient and V the volume of the double model; io.eo, the model and its image in the free surface, Here k1 may be estimated from the value for an equivalent ellipsoid, i.eo, one having the same length and volume as the double body. Equation (3) then becomes Uo = +lk V E (c2 + m2 w2 + n2 h2)3/ () 2-A mn In computing Uo since the double infinite series converges slowly, it will be convenient to terminate it after a finite number of terms and to approximate the remainder by an integral. We obtain 1 + k 2kmn 1 s 2i rdrdG Uo =. -— V [ Z + — f 2t m,n (c2 + m2 w2 + n2 h2)3/2 wh o o (c2+r213!/2 1+ kl kmn1 1 + kl [, kmn _ 1 It m,n (c2 + m2 w2 + n2 h2)3/2 wh -c2 + r2 (6) in which m and n assume all non-negative values such that

-822 22 2 k i1 if m or n = 0 m + n r2 if neither m nor n = 0 (7) An important special case is that in, which w = h, approximately the proportions of most ship model towing tanks. (Depth of towing tank is h/2). For this case a good approximation is obtained by replacing the entire sum by an integral, with ro = 0o64 w, which gives 1 + k1 V UO = w2 J +0lW (8) The the degree of accuracy to be expected from (8) it suffices to select a mean value for kl, k1 = 0O04, and to assume that c = L/2, where L is the length of the model. Then (8) becomes o208 ~ 2o8 V (9) w2L l+ 1.64 (L)2 Example University of Michigan Model 932 has the following characteristics: Volume = 20775 cu, ft. (V = 4105 fto3) L = 14 fto, B = 2o19 fto, H = 0o876 ft. Assume w = 20 ft,, tank depth = 10 ft. From Eq. (9) we obtain U0 = 0.0072 indicating an increase in velocity of about 0o7 percent. Comparison between Michigan and DTMB data for a model of about the same block coefficient gives the following values for TUJc

-83V/fL 0 o38o o.465 0.535 UO 0.0125 0. o05 o0.02 o The computed value U0 = 0.0072, corresponding to zero speed-length ratio, is not inconsistent with the trend indicated by these data.

APPENDIX IV SHORT RANGE PROPOSALS TO THE MARITIME ADMINISTRATION A. Bulbous Bow Study As part of the research program, MA-2564 Task I sponsored by the Maritime Administration at the University of Michigan, investigations of the so-called Inui bulbs have been conducted during the past year. A very important foundation for the continuation of this work has already been laid and it is expected that the University of Michigan will be heavily engaged in this type of research in the future. Since the study of wave-resistance in general, and of bulbs in particular, comes under the category of ship resistance and propulsion, the long range aspect of research in these fields will be carried in greater detail in a proposal for a five year general research program to be submitted to the Maritime Administration by the University of Michigan, in the near future. To avoid any interruption in the work now in progress, however, the Maritime Administration is hereby requested to provide additional funds for the continued investigations of bulbous bows. The reason for this separate request is that Professor Takehei, who has been associated with Inui's work in Japan and who has now been with the University of Michigan for about a year, expects to remain here only until the end of the Summer of 1963. His continued services will undoubtedly have a bearing on our future activities and efforts in connection with wave-resistance research. In a paper, "Wave-Making Resistance of Ships", presented at the November meeting of the Society of Naval Architects and Marine -84

-85Engineers in New York this year, Dr. Inui reports on some interesting research on bulbous bows recently completed in Japan. In particular the results presented in Figure 29 of his paper are referred to, Therein, Dr. Inui shows clearly that the optimum position of the bulb is very much a function of the form of the waterlines of the parent form, More specifically the figures indicate that fine hollow waterlines in the entrance require the optimum bulb to be located close to the FoPo On the basis of this fact Dr. Inui has succeeded in producing a ship form UF3 ( x UF7 ) x Fl which is very close to what could be called a coventional ship-bulb combination. It should be pointed out that the Inui models referred to above have flat keel lines, In the past is has frequently been argued that the reduction in ship resistance achieved with bulbs was not so much due to the bulb itself as to the reduction in angle of entrance that was possible in the bulbous bow designs, Dr, Inui has shown, however, that if a bulb is located close to the F,P,o it can only be fully effective if the waterlines have a certain shape. It can therefore be said that, although the finer lines in the entrance may reduce the wave-making resistance of the parent form, the reduction of entrance angles make the bulb more effective in a given location, The shape of the bulb is not of great significance in this discussion because bulb form is not as important in design as bulb location, The phase relationship between bulb and hull waves must be such that a cancellation of waves takes place, Less than complete cancellation may be all that can be allowed for in a given design due to considerations other than resistance which may dictate that the bulb size

-86be less than optimum. Located properly the bulb will be benificial, however, whereas a bulb located such that its wave is not in inverse phase with respect to the hull wave may prove to be detrimental. no matter to what size it has been built. Up to date most investigations of Inui bulbs have been performed with hull forms generated -by simple sink-source distributions. The choice of such forms for any theoretical evaluation of results is dictated by the present state of the facilities to handle ship wave calculations. Another important factor is the relatively simple wave patterns generated by these forms. The promising results obtained with simplified hull. forms have tempted investigators to try to fit Inui bulbs to real ship forms. The full scale results obtained by the Japanese with the ship "Kurenai Maru" is well knowno Recently a bulb design for an oceanographic vessel has been tested in the University of Michigan tank with very good results. This ship is to be constructed in the near future. In addition a series of bulbs for a destroyer were tested during the last year at U of M, and bulbs have also been fitted to Series-60 models. The tests with the Series-60 models have been made as a preliminary investigation of the effects of Inui bulbous bows to conventional ship forms. The Series-60 forms were not originally designed for bulbous bows and they are therefore most likely not of optimum shape as far as conventional. forms suited for bulbous bows are concerned. To be sure it has been pointed out by Dr. Inui that the bulb should be treated from a theoretical point of view as an appendage to a given ship form. This is not meant to imply, however, that the parent hull form is not important. Indeed it follows from the discussion above that for best results a good match must exist between hull and bulb.

-87The immediate problem at hand is therefore to attempt to develop a practical hull form together with bulb forms suitable for specific speeds. It is proposed that Series-60 provides a basis for such a hull. form and that the University of Michigan be given the task of initiating a study of alterations of Series-60 lines which will be necessary so that an Inui bulb of reasonable size and location may produce a significant reduction in wave-making resistance. This study is intended to be primarily experimental in the initial stages. To obtain a complete set of data from the type of tests considered here, however, it will. be necessary to measure wave patterns and compare these with calculations based upon theory. The instrumentation required for such measurements is presently not available at U of M. Wave pattern analysis will therefore be included in a long range research program. It should be pointed out that the limited research program proposed here is intended as a part of the overall problem of the development of direct methods by which optimum bulb designs can be made. Bo Scale Effect on Propellers & Propulsion When self-propulsion tests were initiated at the University of Michigan a few years ago a careful study indicated that the hull model. size of 14 ft. in length would be free from excessive blockage effects and at the same time bring the propeller size up to above 6 ino in diameter. A limited series of comparative tests with Series-60 models indicated good correlation with DTMB tests of 20 ft. models. A subsequent test of the Maritime Administrations's PID-108 design showed very poor correlation, however. Not only were the overall propulsive efficiencies

-88lower than those obtained by DTMB but both wake and thrust deduction as derived from the propeller open-water characteristics were definitely not consistent with available information, Any reasonable doubts harbored that the University of Michigan results might be due to the special design features of the PD-108 design were almost completely dispelled when similar results were recently obtained from the self-propulsion test of a modified Series-60 form. It is most likely that a scale-effect problem in involved, To investigate this possibility the open-water test of the PD-108 design propeller was rerun at an RPM in the range of the RPM used in the self-propulsion test. The KT and KQ values thus obtained were different than those previously found, and moreover, if these values were applied to the self-propulsion tests the derived values of wake and thrust deduction were almost identical to those reported by DTMB. Propulsion efficiencies were still lows however~ Since some tests appear to be free from scale-effects whereas others are not, it is reasonable to believe that the 14 ft. model size is operating in a transition range in regard to the boundary layer on the propeller bladeso Two modes of attack can be offered leading to a solution of the problem, namely: 1. Turbulence stimulation of the boundary layer on propeller blades. 2. The use of larger models. It is felt that model size cannot be much greater than 1.6 ft. in the University of Michigan tank. It may therefore be necessary to use a combination of both remedies mentioned above.

-89If self-propulsion tests are to be run in the University of Michigan tank in the future there is no doubt that the important problem of propeller scale-effect must be solved. The Maritime Administration is therefore requested to provide financial support for one year's work on this project with the aim of ascertaining the criteria that must be met in self-propulsion testing so that significant scale-effects can be avoided.