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Experimental and computational studies of a ducted tip propeller Straver, Michelle Corinne 2002

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EXPERIMENTAL AND COMPUTATIONAL STUDIES OF A DUCTED TIP PROPELLER by Michelle Corinne Straver B.Sc , Queen's University, 2000 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF M A S T E R OF APPLIED SCIENCE in THE F A C U L T Y OF G R A D U A T E STUDIES D E P A R T M E N T OF M E C H A N I C A L ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH C O L U M B I A August 2002 © Michelle Corinne Straver, 2002 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree t h a t permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of M in icq I frn<^ n Mf>'/u The U n i v e r s i t y of B r i t i s h Columbia Vancouver, Canada Date 0,0. AutjU^ QOOZ-ABSTRACT When a fluid flows past a lifting surface, tip vortices are created. On marine propellers, these tip vortices reduce the efficiency of the blades and can have a number of other undesirable effects if cavitation occurs. Several propeller modifications have been proposed to suppress tip vortex roll-up, reduce the strength of the vortices, and prevent tip vortex cavitation from occurring. One of these modifications involves attaching a flow-through duct to the tip of each propeller blade. Past research on these ducted tips has shown promising results. In this study, experiments were conducted to compare the performance of a conventional propeller with that of a ducted tip propeller. There were two primary objectives: to determine the effect of the duct on propeller efficiency, and to determine how cavitation behaviour changes with the addition of the duct. This research differs from previous work in that the experiments were conducted in a controlled environment, allowing for more accurate measurements with fewer uncontrollable external factors that could affect the data. The results showed a small decrease in efficiency over a large range of test conditions. However, the general trends were in agreement with the trends seen in sea trials conducted by Hordnes and Green (1998), in that the presence of the duct was most advantageous at high advance ratios. The addition of the duct dramatically changed the cavitation behaviour. The flow patterns indicated that the tip vortex was substantially diffused with the duct. Evidence also suggested that the tip vortex cavitation inception index was reduced. These findings are in agreement with past research. ii TABLE OF CONTENTS ABSTRACT ii TABLE OF CONTENTS iii LIST OF TABLES v LIST OF FIGURES vi LIST OF SYMBOLS ix ACKNOWLEDGEMENTS x CHAPTER 1 - Introduction 1 1.1 Tip Vortices 1 1.2 Previous Efforts to Reduce Tip Vortex Cavitation 2 1.2.1 The Ducted Tip 4 1.3 Previous Computational Studies 5 1.4 Research Objectives 6 1.5 Presentation of Data 6 1.6 Structure of Thesis 8 CHAPTER 2 - Experimental Apparatus and Setup 9 2.1 The Propeller.. .' . 9 2.2 Experimental Instrumentation and Techniques 10 2.2.1 Tow Tank 11 2.2.2 Cavitation Tunnel 12 CHAPTER 3 - Propeller Hydrodynamic Performance 15 3.1 Data Correction 15 3.2 Conventional Propeller Tests 16 3.3 Ducted Tip Propeller Tests 18 3.4 Assessment of Errors 31 3.4.1 Instrumentation Error Analysis 31 3.4.2 Discussion of Unquantified Errors 32 CHAPTER 4 - Propeller Cavitation Performance 34 4.1 Conventional Propeller 34 4.2 Ducted Tip Propeller 36 CHAPTER 5 - Conclusions and Recommendations for Future Work 40 5.1 Conclusions 40 5.2 Recommendations for Future Work 40 REFERENCES.. 42 i i i APPENDIX A - Comparison Between Conventional Propeller and Wageningen B4-52 Propeller 44 APPENDIX B - Propeller Performance Plots 47 APPENDIX C - Comparative Plots of KT, 10KQ, and rj 49 APPENDIX D - Effective Angle of Attack as a Function of r/R 51 APPENDIX E - Comparative Plots of KT, WKQ, and 77 as a Function of Effective Angle of Attack 56 APPENDIX F - Correction Plots 58 APPENDIX G - Data 64 APPENDIX H - Test for Dependence on Reynolds Number (Cavitation Tunnel)... 70 APPENDIX I - Propeller Performance Plots (Cavitation Tunnel) 72 APPENDIX J - Correction Plots (Cavitation Tunnel) 74 APPENDIX K - Data (Cavitation Tunnel) 77 APPENDIX L - Computational Efforts 81 APPENDIX M - Information on CFD-ACE(U)+ Rotating Systems 94 iv LIST OF TABLES Table 3.1 Summary of changes to propeller performance after installation of ducted tips 20 Table 3.2 Uncertainty estimates for some variables 31 v LIST OF FIGURES Figure 1.1 Induced flow around a lifting surface 1 Figure 1.2 Tip vortex cavitation from a propeller 1 Figure 1.3 A typical propeller performance plot 7 Figure 2.1 The ducted tip propeller 10 Figure 2.2 The duct 10 Figure 2.3 The IMD tow tank 11 Figure 2.4 The IMD cavitation tunnel 13 Figure 3.1 Comparison of the conventional propeller at 8 rps with the Wageningen B4-52 propeller at PID=\.0 17 Figure 3.2 Propeller performance at 8 rps for P/D=0.6, 1.0, and 1.4 18 Figure 3.3 Comparative plots of KT, IQKQ, and f] at 8 rps for varying pitch ratios. .. 20 Figure 3.4 Blade force and velocity diagram 22 Figure 3.5 The changes in pressure and velocity at the propeller disk, according to the momentum theory 24 Figure 3.6 Effective angle of attack at 8 rps as a function of rlR for varying advance ratios 25 Figure 3.7 Comparative plots of K T , 10KQ, and 7] at 8 rps for varying pitch ratios as a function of a. 28 Figure 4.1 The progression of cavitation on the conventional propeller 35 Figure 4.2 The progression of cavitation on the ducted tip propeller 38 Figure A.1 Comparison of the conventional propeller at 8 rps and 14 rps with the Wageningen B4-52 propeller at P/D=0.6 44 Figure A.2 Comparison of the conventional propeller at 14 rps with the Wageningen B4-52 propeller at P/D=1.0 45 vi Figure A.3 Comparison of the conventional propeller at 8 rps and 14 rps with Wageningen B4-52 propeller at P/D=IA 46 Figure B . l Propeller performance at 14 rps for P/D=0.6, 1.0, and 1.4 47 Figure C . l Comparative plots of Kr, IOKQ, and JJ at 14 rps for varying pitch ratios 49 Figure D. l Effective angle of attack at 14 rps as a function of rlR for varying advance ratios. A l l lines are for the ducted tip propeller 50 Figure D.2 Effective angle of attack at 8 rps as a function of rlR for varying advance ratios. A l l lines are for the conventional propeller 52 Figure D.3 Effective angle of attack at 14 rps as a function of rlR for varying advance ratios. A l l lines are for the conventional propeller 54 Figure E . l Comparative plots of KT, IOKQ, and 77 at 14 rps for varying pitch ratios as a function of a. 56 Figure F . l Thrust correction factors for the ducted tip propeller 58 Figure F.2 Torque correction factors for the ducted tip propeller 59 Figure F.3 Thrust correction factors for the conventional propeller 61 Figure F.4 Torque correction factors for the conventional propeller 62 Figure H . l Comparative plots of KT, IOKQ, and 77 at P/D = 0.6 for varying rotation speeds 70 Figure 1.1 Propeller performance at 14 rps for P/D=0.6, 1.0, and 1.4 72 Figure J . l Thrust correction factors for the ducted tip propeller 74 Figure J.2 Torque correction factors for the ducted tip propeller 75 Figure L . l The starting grid on the suction side of the propeller blade 83 Figure L.2 The grid on the suction side of the propeller blade 84 Figure L.3 The flow domain 85 Figure L.4 The flow development region 87 vii Figure L.5 This section links the upstream flow development region to the areas connected with the propeller 88 Figure L.6 The flow domainfrom blade to flow channel wall, as viewed from the suction side and from the pressure side 89 Figure L.7 The flow domain surrounding the blade, as viewed from the suction side and from the pressure side 90 Figure L.8 This section links the areas connected with the propeller to the downstream flow region 91 Figure L.9 Boundary condition types 92 viii LIST OF SYMBOLS a axial induced flow component (as fraction of freestream velocity) at location of propeller a' rotational induced flow component (as fraction of rotational velocity) A0 expanded area ratio b axial induced flow component (as fraction of freestream velocity) far downstream of propeller COJ chord length at 70% of propeller radius D propeller diameter (m) / advance ratio KQ torque coefficient KT thrust coefficient n rotation speed (revolutions per second, rps) np number of data points PID pitch-to-diameter ratio p freestream pressure (Pa) pv vapour pressure (Pa) Q torque (N-m) r local radius (m) rms root mean squared rv radius of Rankine vortex (m) Re Reynolds number T thrust (N) V freestream velocity (m/s) xm mean a angle of attack (°) P inverse tangent of ratio of axial (freestream) velocity to rotational velocity (°) <f> face pitch angle (°) r circulation (m2/s) TJ efficiency v kinematic viscosity (m2/s) p fluid density (kg/m3) <rcav cavitation number Ocav,i cavitation inception index Odev standard deviation ax standard error for variable x co rotation speed (rad/s) ix A C K N O W L E D G E M E N T S I would like to express my gratitude to my supervisors, Dr. Sheldon Green and Dr. Carl Ollivier-Gooch for their support, guidance, and endless patience. Thanks to Doug Yuen of the U B C Mechanical Engineering machine shop for his help with the propeller modifications. I would also like to acknowledge the staff at IMD for their varying contributions, especially Dr. Pengfei Liu, Kent Brett, Andrew Kuczora, Craig Kirby, Greg Janes, David Molyneux, and Erik Johnston. Many thanks to Dr. Neil Bose of the Memorial University of Newfoundland for his technical expertise. Finally, I would like to thank my friends and colleagues at UBC, especially Hildur Ingvarsdottir for her knowledge, support, and encouragement during the most trying times. Funding for this research was provided by the Natural Sciences and Engineering Research Council of Canada. x CHAPTER 1 - Introduction 1.1 Tip Vortices On any lifting surface, there is by definition a higher pressure on the lower side of the surface and a lower pressure on the upper side. This pressure gradient induces flow around the tip from the lower side to the upper side (Figure 1.1). low pressure *) + + + + + high pressure Figure 1.1 Induced flow around a lifting surface. As the fluid flows past the surface's trailing edge, the circulatory motion results in the formation of what are termed tip vortices. The application of interest in this study is the marine propeller. The motion of the tips of the propeller blades causes the tip vortices to form a helical pattern (Figure 1.2). Figure 1.2 Tip vortex cavitation from a propeller. (Source: Munson et al. 1998) 1 Tip vortices on propellers cause two undesirable effects: loss of efficiency and cavitation. The efficiency decrease is caused by the induced flow component associated with the production of thrust; this is equivalent to downwash on an airfoil. Downwash reduces the effective angle of attack of the foil (or blade), decreases the lift (or thrust), and increases the drag, all of which work to reduce efficiency. In addition, a strong tip vortex can cause cavitation. The circumferential flow of the vortex induces a pressure gradient through the cross-section of the vortex, with the pressure being lowest in the core. If the pressure in the core falls below the vapour pressure of water, the water effectively boils, or cavitates, forming vapour bubbles. The bubbles eventually move into a region of higher pressure, where they collapse. This produces a number of undesirable effects, including noise (which is of particular interest to fishing boats and naval craft), vibration, and, if bubbles continually collapse on a solid surface, surface erosion. 1.2 Previous Efforts to Reduce Tip Vortex Cavitation A great deal of past research has been directed at suppressing the formation of tip vortices and preventing tip vortex cavitation for both aerodynamic applications and marine propeller applications. Only the latter will be discussed here. One of the most common and oldest devices used to delay tip vortex cavitation is the Kort Nozzle. The nozzle, which is aligned with the propeller axis, fits around the propeller with a very small clearance so that tip vortices are unable to develop. This technique is especially useful for heavily loaded propellers, in which cases it improves 2 propulsive efficiency. The main disadvantage of the Kort Nozzle is an increase in drag due to the extra wetted area contributed by the nozzle (Hordnes and Green 1998). Platzer and Souders (1979) discuss the techniques used to alleviate the effects of tip vortices prior to 1980. One such concept that is of particular interest to the present study is Crump's work on bulbous tips installed on propeller blades. Crump (1948) tested several bulb geometries, and showed that the bulbous tips.effectively delayed tip vortex cavitation, most likely by modifying the core size of the vortices. The effects on efficiency were varied. Several other techniques have been explored since the Platzer and Souders (1979) report. Itoh et al. (1987) tested "bladelets" on propeller blade tips (equivalent to winglets commonly used on aircraft) and found that the cavitation inception index was reduced, and the efficiency was slightly increased. However, Goodman and Breslin (1980) conducted a similar study and found a decrease in efficiency. In both cases, the bladelet interfered with the vortex roll-up process. The contradiction in performance data suggests that the effectiveness of the bladelets is quite sensitive to the bladelet geometry. Arakeri et al. (1985) found improved cavitation behaviour (with no significant performance penalty) by drilling small holes in the tip and leading edge areas of the blade, reducing the pressure gradient between the pressure and suction surfaces. This technique was found to be particularly effective at low advance ratios. Chahine et al. (1993) injected a drag-reducing polymer solution into the tip vortex region of a propeller. This was effective in delaying tip vortex cavitation, but was fairly sensitive to the locations of the injection ports. The reduction in cavitation inception 3 index was attributed to the thickening of the vortex core caused by the viscoelastic properties of the polymer solution. 1.2.1 The Ducted Tip Green et al. (1988) were first to explore the idea of a ducted tip geometry for hydrofoils, with promising results. This geometry consisted of a flow-through duct attached to the tip of the foil, approximately aligned with the chord. The diameter of the duct was on the order of 20% of the chord length. Since then, the ducted tip has been the object of several more studies. Green and Duan (1995) found that tip vortex cavitation could be significantly delayed (with no significant performance penalty) by using a ducted tip hydrofoil in place of a conventional one. Additionally, they found an increase in the lift/drag ratio at high angles of attack. Hordnes and Green (1998) conducted sea trials on a ducted tip propeller, and found that the addition of the duct delayed tip vortex cavitation by nearly 50% with no performance penalty. In fact, efficiency actually improved under some operating conditions. The ducted tip delays tip vortex cavitation by physically increasing the diameter of the vortex, and by injecting fluid into the vortex core. For an unducted foil (or propeller blade), the vortex diameter is on the order of the boundary layer thickness. When the duct is added, the fluid moving around the tip is forced to flow around the duct, such that the vortex diameter is comparable to the duct diameter. The Rankine vortex model is a two-dimensional model of an irrotational vortex with solid body rotation in the core. Although it does not perfectly predict the actual tip vortex, it can predict the correct trends. The model states the following for the pressure in the vortex core (Green 1995): pr 2 An r The value for the vortex radius rv in the above equation is significantly larger for the ducted tip case, and therefore pc is also greater. Hence, the pressure in the core of the vortex is less likely to fall below the vapour pressure of water and cavitate. 1.3 Previous Computational Studies A small amount of computational work related to the present study has been done in recent years. Hsiao and Pauley (1999) simulated the flow around a rotating marine propeller using the incompressible Reynolds-averaged Navier-Stokes equations and the Baldwin-Barth turbulence model. Comparison with experimental data showed that the model predicted the general flow patterns fairly well, but slightly under-predicted the tip vortex strength. Hsiao and Pauley (1999) also explored (numerically) a modified tip geometry, where the thickness of the blade cross-section was increased near the tip. They found that this geometry resulted in a weaker tip vortex with no associated performance penalty. Ingvarsdottir et al. (2002) studied the flow around a ducted tip hydrofoil and found good agreement between the model and available experimental data. They illustrated several points of interest. First, the sectional lift was higher for the ducted tip case than for the rounded tip, which suggests that the ducted tip improves the lift-to-drag ratio (and therefore efficiency). Also, as expected, the vorticity from the ducted tip foil was shed in the shape of the duct, as opposed to the more concentrated circular vortex shed by the rounded tip foil. Finally, the minimum pressure of the tip vortex was lower for the rounded tip case, indicating that tip vortex cavitation would be less likely to occur with the ducted tip geometry. 5 1.4 Research Objectives This research was divided into two major components: one experimental, and one computational. The experimental research objectives were to determine the effect of the duct on propeller performance and on cavitation behaviour by conducting tests in controlled environments. The computational efforts were directed toward simulating the flow around a rotating propeller; however, this aspect of the research met with limited success. 1.5 Presentation of Data The data from the experiments conducted in this study will be presented in accordance with the standard methods for propeller open-water testing. The measured variables were: propeller diameter D (m), fluid density p (kg/m3), freestream velocity V (m/s), rotation speed n (revolutions per second (rps)), thrust T (N), and torque Q (N-m). From these, the following dimensionless parameters could be calculated: • r v advance ratio J = — nD T thrust coefficient K 7 pn2D4 torque coefficient KN = —%—T Q pn2D5 JKT VT efficiency TJ 2KKQ 2mQ The thrust and torque coefficients and the efficiency are normally plotted against advance ratio. A typical propeller performance plot is shown in Figure 1.3. 6 1.0 0.8 o F ~ 0 .6 o < 0.4 0 . 2 0 —. ^ j i I 1 0 . 1 0 0 . 0 8 0 . 0 6 o 0 . 0 4 0 . 0 2 0 0 0 . 2 0 . 4 0 . 6 0 . 8 1.0 J = V A / D n Figure 13 A typical propeller performance plot. (Source: Lewis, 1998) 1.2 Reynolds number is based on the chord length of the blade at 70% of the radial distance, and is given, by: :0 j ^V2+(0.7rniDf c 0 7 nD^J2 +(0.7TT)2 Re = v A measure of cavitation is given by the cavitation number acav, which is a non-dimensionalized indication of the difference between freestream pressure p (N/m2) and the vapour pressure of water pv. Based on the propeller rotation rate, the cavitation number is: P-Pv l/pn2D2 The freestream pressure is taken at the level of the propeller hub. The point at which tip vortex cavitation first becomes visible in the form of bubbles on the blade tips is defined as the cavitation inception index, ocav,i- This value is dependent not only on fluid density, pressure, and velocity, but also on factors such as the 7 dissolved air content of the water. Dissolved nuclei (air or otherwise) act to induce cavitation earlier than what one would expect based on density, pressure, and velocity measurements. 1.6 Structure of Thesis This thesis is divided into five chapters. The following chapter (Chapter 2) describes the apparatus and setup used for the experiments. The experimental results related to propeller hydrodynamic performance are presented and discussed in Chapter 3, while Chapter 4 contains the experimental results related to propeller cavitation behaviour. Finally, Chapter' 5 contains conclusions of the research as well as recommendations for future work. These chapters are followed by a number of appendices containing graphs and data not presented in the main body of the thesis, as well as appendices pertaining to computational efforts. 8 CHAPTER 2 - Experimental Apparatus and Setup This chapter presents information related to the experimental part of this study. It includes a description of the conventional propeller and of the modifications required to transform it into the ducted tip propeller, as well as a description of the test facilities. 2.1 The Propeller For the conventional propeller experiments, a four-bladed, right-hand screw propeller with an expanded area ratio (A0) of 0.52 and a diameter of 0.20 m was used. The pitch-to-diameter ratio PID (also referred to simply as the pitch ratio) could be adjusted to 0.6, 1.0, or 1.4 by rotating and re-setting the blades. After each angle adjustment, the blades were faired to minimize the chance of flow separation near the hub. The same propeller was modified (Figure 2.1) for the ducted tip experiments. First, approximately 10% of the total radius (3/8 in.) was removed from the tip of each blade. A copper tube with an outer diameter of 3/8 in. and a wall thickness of 1/16 in. was bent so that the inside radius matched the radius of curvature of the exposed chord of the cut blades. Four 1-1/8 in. lengths of the tube were cut. One piece was soldered to each blade tip such that the tube was aligned with the exposed chord, with one end lined up with the trailing edge of the blade. Body filler was used to create a smooth transition between the ducts and the blades. Finally, the leading edges of the ducts were rounded and the trailing edges were tapered in an effort to streamline the body and minimize the wake (Figure 2.2). The duct geometry was chosen to approximate the geometry used in previous studies by Green and Duan (1995) and Hordnes and Green (1998). 9 Figure 2.1 The ducted tip propeller. Figure 2.2 The duct. 2.2 Experimental Instrumentation and Techniques A l l of the experiments in this study were conducted at the Institute for Marine Dynamics (IMD), National Research Council (NRC), in St. John's, Newfoundland. The tow tank was used for measuring torque and thrust to evaluate the propeller's hydrodynamic performance before and after the addition of the duct, while the cavitation tunnel was used primarily for qualitative cavitation observations. Torque and thrust 10 measurements were taken in the cavitation tunnel as well, but were not used extensively in the analysis due to the data's questionable reliability. 2.2.1 Tow Tank The IMD tow tank (Figure 2.3) is 200 m in length, 12 m in width and 7 m in depth. Models are towed through the water by a carriage that spans the width of the tank. The maximum attainable speed is 10.0 m/s. The tow tank is capable of wind and current generation, although these features were not utilized for the present study. The instrumentation used for the experiments included a tachometer on the motor shaft for rotation speed measurements and a Kempf & Remmers open water propeller dynamometer for torque and thrust measurements. The IMD standard data acquisition system was used. The sampling rate was 50 Hz with a low pass filter of 10 Hz. Figure 2.3 The IMD tow tank. The test procedure followed the IMD standard test methods for propeller open-water tests (Murdey, 2000). Some of the important points are highlighted below. 11 The propeller shaft was set horizontally at an immersion depth of 1.5 propeller diameters. A check was made to ensure that air was not being drawn into the propeller by operating the propeller at the highest rotation speed expected to be used for the tests. Before beginning the tests, the "zero" carriage speed measurement was found by replacing the propeller with a "dummy hub" having the same weight as the propeller and operating at the test rotation rate with the carriage stationary. The resulting measurement for carriage speed was used to correct each speed measured throughout the test. The procedure was repeated for each rotation rate. The friction in the torque bearings was also measured by operating the dummy hub at a range of rotation speeds (both below and above the speeds to be used in the tests) with the carriage stationary. This was done at the beginning and end of each day of testing, and the average values for the day were used to correct the measured torque values. For the performance tests, the propeller was operated at two different rotation speeds for each of the three pitch ratios (0.6, 1.0, and 1.4). The first rotation speed (8 rps) represented a median value estimated for the propeller in practical use in model self-propulsion tests, while the second (14 rps) was the highest practical value. 2.2.2 Cavitation Tunnel The IMD cavitation tunnel (Figure 2.4) is a vertical plane, closed recirculation tunnel with a 0.5 m square test section measuring 2.2 m in length. The maximum velocity attainable in the test section is 12 m/s. The absolute pressure can be adjusted from 5.7 kPa to 200 kPa. 12 Figure 2.4 The IMD cavitation tunnel. A sealed strain gauge shaft dynamometer was used to measure thrust and torque for the conventional propeller experiments. During the testing process, the dynamometer required a number of repairs. Shortly before the ducted tip propeller experiments were scheduled to begin, the dynamometer was broken again and deemed irreparable, so the dynamometer used for the tow tank tests was used instead. Some concerns were raised about the fact that two sets of data were taken using two different dynamometers and subsequently compared. A repeatability test was done by performing some of the ducted tip propeller tests that were previously done in the tow tank, and the results showed some level of disagreement. The issue was resolved by using the data from the tow tank for the primary analysis. Graphs and data related to the tests done in the cavitation tunnel can be found in Appendices H through K. A de-aeration system was used to lower the freestream nuclei content of the water. 13 Stroboscopic illumination was used to facilitate qualitative cavitation observations. The strobe light was set at a frequency that made the propeller appear stationary (or rotating slowly) so that details of the cavitation could be observed. 14 CHAPTER 3 - Propeller Hydrodynamic Performance A l l of the experimental results are presented and discussed in this chapter. The conventional propeller will be discussed in relation to a standard propeller for which performance data is readily available. This will be followed by a discussion of the ducted tip propeller as it compares to the conventional propeller, and an assessment of the errors associated with the experiments. 3.1 Data Correction The measured thrust and torque values were not zeroed; that is, they did not account for factors such as friction. It was assumed that the correction factor would be a simple offset of the measured values, independent of the propeller rotation speed. This is a valid assumption only if the thrust and torque coefficients are independent of Reynolds number, as is normally the case except at very low Reynolds numbers. The following procedure was carried out to correct the thrust data: first, the measured thrust values were plotted against the square of the rotation speed for several values of advance ratio. Since T varies only with n (given the aforementioned assumptions), this resulted in a straight line. The data were extrapolated to zero rotation speed, and the corresponding value for thrust was subsequently subtracted from the measured data to give the new, corrected value. A similar procedure was used to correct the measured torque values. A l l plots generated for these adjustments are shown in Appendix F. For the tests done in the tow tank, no rotation speed dependence existed; the data for 8 rps and for 14 rps agreed well within experimental error. A more in-depth study of Reynolds number was performed in the cavitation tunnel (Appendix H), and again, no 15 obvious trends emerged. This validates the assumption that KT and KQ were independent of Reynolds number for the tests performed. 3.2 Conventional Propeller Tests The data for the conventional propeller were compared to the performance plots of the Wageningen B-Series propeller having the same pitch ratio, expanded area ratio, and number of blades (called the Wageningen B4-52 propeller). Figure 3.1 shows the performance plots for the conventional experimental propeller at P/D=1.0 and 8 rps along with the equivalent plots for the B4-52 propeller. The plots for all other test conditions show the same trends (Appendix A). In all cases, the torque and thrust coefficients of the experimental propeller are significantly higher than for the B4-52 propeller. The efficiency of the B4-52 is greater than that for the experimental propeller, up until a point near the peak, where the efficiency of the B4-52 falls off steeply. These discrepancies are most pronounced at the higher pitch ratios. 16 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 J Figure 3.1 Comparison of the conventional propeller at 8 rps with the Wageningen B4-52 propeller at PID= 1.0. The triangles represent KT, the circles represent 10KQ, and the squares represent 77. Solid lines with solid symbols represent the performance characteristics of the conventional propeller, and dotted lines with open symbols represent that of the Wageningen B4-52 propeller. It is important to note that close agreement between the experimental propeller and the B4-52 propeller was not expected. Factors describing the geometry of the propeller, including rake, skew, and blade section shape, are not accounted for in the comparison. These factors have a significant impact on propeller performance. Comparison with the B4-52 serves to illustrate that the performance parameters are of the expected order of magnitude and follow the expected trends. In all cases, when pitch ratio increased, torque, thrust, and peak efficiency also increased, and the peak efficiency occurred at a higher advance ratio. This behaviour is to be expected for any conventional marine propeller. 17 3.3 Ducted Tip Propeller Tests Figure 3.2 shows the performance of the ducted tip propeller as compared to the conventional propeller. Only the 8 rps case is shown for each pitch ratio as the results for 14 rps are qualitatively similar. (Appendix B contains the equivalent plots for 14 rps.) P/D = 0.6 0.7 -j 0.6 Figure 3.2a 18 P/D = 1.0 1.0 0.8 0.6 6 o ^ 0.4 0.2 0.0 • i i i i i i i .. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Figure 3.2b P/D = 1.4 0.0 0.2 0.4 0.6 0.8 1.0 J 1.2 1.4 1.6 1.8 2.0 Figure 3.2c Figure 3.2 Propeller performance at 8 rps for P/D=0.6, 1.0, and 1.4. The triangles represent KT, the circles represent IOKQ, and the squares represent JJ. Solid lines with solid symbols represent the ducted tip propeller, and dotted lines with open symbols represent the conventional propeller. 19 Table 3.1 summarizes the changes to the propeller performance after the installation of the ducted tips at the advance ratios corresponding to the peak efficiency. P/D J 10KQ V 0.6 0.50 +5.2% +33% -21% 1.0 1.0 +13% +20% -6.2% 1.4 1.5 -10% -0.29% -9.9% Table 3.1 Summary of changes to propeller performance after installation of the ducted tips. Figures 3.3a, b, and c illustrate the trends in KT, \0KQ, and 77 respectively with changes in pitch ratio for 8 rps. Equivalent plots for 14 rps can be found in Appendix C. 0.8 Figure 3.3a 20 Figure 3.3b 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 ft A* \ • Jr A • T 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 J Figure 3.3c Figure 3.3 Comparative plots of KT, \0KQ, and n at 8 rps for varying pitch ratios. The triangles represent P/D=0.6, the circles represent P/D=1.0, and the squares represent P/D=l.4. Solid lines with solid symbols represent the ducted tip propeller, and dotted lines with open symbols represent the conventional propeller. 21 The trends for the lower two pitch ratios (0.6 and 1.0) are similar to those of the sea trials conducted with a ducted tip propeller (Hordnes and Green 1998), which had a pitch ratio of 0.8. The results of the sea trials showed that installation of the ducted tips causes a decrease in both Kj and KQ at low advance ratios and an increase at high advance ratios. The efficiency trends were similar in that the ducted tips were most beneficial (or least detrimental) at high advance ratios. To better compare the data between the various pitch ratios, the independent axis was changed from the advance ratio to the effective angle of attack. The effective angle of attack was calculated in such a way that it accounted for blade geometry, blade motion, and the induced flow caused by the acceleration of the local flow at the propeller. Figure 3.4 shows the force and velocity diagram for a blade section. Figure 3.4 Blade force and velocity diagram. (Source: Lewis 1988) 22 The geometric angle of attack is the angle of incidence at the blade section, not accounting for the induced flow. It is given by: a = <p-pl The angle 0 represents the face pitch angle of the propeller. This angle was measured on the original propeller at several radial positions along the blade. For each desired pitch ratio, the angle of rotation necessary to achieve the required pitch angle at r//?=0.7 was determined, and this angle of rotation was added to the original angle measurements. The angle /?is dependent on the inflow velocity and the propeller's angular velocity. f V ^ P = tan" In Figure 3.4, aV is the induced flow in the axial direction, while a'cor is the induced flow caused by changes in angular momentum at the propeller location. For these calculations, a' was taken to be zero since it is expected to be small in comparison to a. The momentum theory (illustrated in Figure 3.5 and explained in detail in Lewis (1988)) was used to estimate the magnitude of a. 23 Vflt'-b) RACE COLUMN PRESSURE p, T PRESSURE p ( PROPELLER DISC. AREA A Q •INCREASE IN PRESSURE AT SCREW DISC INCREASE IN PRESSURE ! 4 PRESSURE p, Figure 3.5 The changes in pressure and velocity at the propeller disk, according to the momentum theory. (Source: Lewis 1998) According to this theory, a can be determined using the following equation: T = pAaV2(l + a)b The expression bV is the total acceleration of the race column, downstream of the propeller. It can be assumed that half of the total acceleration occurs at the location of the propeller, thus setting b equal to 2a. T = pAaV2(\ + a)2a After determining the value of a, the effective angle of attack a could be adjusted to account for induced axial flow by replacing ft in the angle of attack equation with R\. /?, = tan - i f y q + a ) 2mir /?, = tan "'((1+a) tan/?) 24 As indicated earlier, the effective angle of attack varies with the radial distance from the propeller hub. This relationship is shown in Figure 3.6 for each pitch ratio at various advance ratios. A l l data shown are for the ducted tip propeller at 8 rps. (Additional plots can be found in Appendix D.) The data for the conventional propeller are essentially the same as for the ducted tip propeller over most of the domain. The results suggest that flow separation on the blade is most likely to occur fairly close to the hub (approximately 20% of the distance along the blade) at low advance ratios, and, with the exception of P/D=0.6, at the tip of the blade for higher advance ratios. P/D = 0.6 r/R Figure 3.6a 25 P/D = 1.0 15 r/R Figure 3.6b P/D = 1.4 r/R Figure 3.6c Figure 3.6 Effective angle of attack at 8 rps as a function of r/R for varying advance ratios. All lines are for the ducted tip propeller. 2 6 In order to give an indication of the overall propeller performance, the effective angle of attack was averaged from r/R=0.5 to r/R=l.O, representing the part of the propeller that does the most work. Figure 3.7 illustrates KT, IOKQ, and TJ as a function of this averaged effective angle of attack for 8 rps. The plots for 14 rps are shown in Appendix E. In all cases, KT and KQ increased with increasing angle of attack, as expected. For P/D=0.6 and P/D=1.0, the addition of the ducted tip decreased KT at low a, and increased KT at high a. (Although, for P/D=0.6 the results were within experimental error.) At P/D=l .4, the ducted tip decreased KT over the entire testing range, but the greatest drop occurred at low advance ratios. This drop in thrust may have been caused by stall conditions on the duct. The addition of the duct increased the value of KQ for P/D=0.6 and P/D=l .0. Interestingly, over the majority of the test conditions, the increase in KQ was not dependent on the angle of attack. At PID=1.4, KQ decreased, especially at low values of a. The reason for this inconsistency is unknown. In order to maximize propeller efficiency, it is desirable to have a high thrust coefficient and a low torque coefficient. Hence, from the thrust point of view, the duct is best used at low angles of attack, achieved by operating at low loading conditions (i.e. at high advance ratios), especially at the lower pitch ratios. Conversely, it is favourable to use the duct at a high pitch ratio in order to minimize torque. This suggests that there is some (unknown) pitch ratio and angle of attack for which the duct is most beneficial. Figure 3.7c indicates that this is indeed likely to be the case. There was a drop in peak efficiency at all pitch ratios after the installation of the ducted tip. However, the drop was 27 greatest at P/Z)=0.6, and smallest at P/D=l .0. At very high angles of attack, the efficiency improved with the ducted tip for all pitch ratios. 0.8 a (degrees) Figure 3.7b 28 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 tor*T)WL \ iTf ftt i*n m* ^ ^ ,„. - r - , , \ 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 a (degrees) 1.6 1.8 2.0 Figure 3.7c Figure 3.7 Comparative plots of KT, IOKQ, and 77 at 8 rps for varying pitch ratios as a function of a. The triangles represent P/D=0.6, the circles represent PID=\.0, and the squares represent P/D=1.4. Solid lines with solid symbols represent the ducted tip propeller, and dotted lines with open symbols represent the conventional propeller. Figure 3.7c also indicates that the peak efficiency occurs at a small negative angle of attack for all pitch ratios. The position of peak efficiency was not significantly changed by the presence of the duct. These results are not surprising. Propellers are designed to operate at zero angle of attack, such that all thrust is generated by the camber of the blades. The reason for this is to minimize the magnitude of leading edge negative pressure peaks and hence reduce the chance of cavitation. Additionally, propellers are normally operated at an advance ratio slightly lower than the advance ratio that would give maximum efficiency. At this position, the angle of attack would be slightly higher than for the point of maximum efficiency, or approximately zero. 29 The performance penalty associated with the addition of the ducted tips was somewhat disappointing, but not entirely unexpected. Although the sea trials (Hordnes and Green 1998) showed an improvement in efficiency, this was not the case for other tip devices used in the past. Crump (1948) tested bulbous tips (having a geometry somewhat similar to the ducted tips) on two different types of propellers. For one propeller, it was found that efficiency increased slightly at high advance ratios after bulb installation, and that there was no change at low advance ratios. The trends were opposite for the other propeller. These results indicate that the change in propeller performance with the addition of a blade tip fitting may be very sensitive to the geometry of both the blade itself and of the tip fitting. It is important to note that the duct geometry used in this study was not optimized. Although efforts were made to streamline the duct, there is room for improvement. It is worth noting that in many of the cavitation videos (discussed in the following chapter), sheet cavitation (i.e. flow separation) was observed on the suction side of the duct. It is likely that flow separation was also occurring in the non-cavitating flows tested in the tow tank, which could have been a major contributor to the loss of efficiency. An improved geometry could increase the overall effectiveness of the duct by eliminating (or at least delaying) flow separation. It is possible that an optimized geometry would actually improve the propeller performance with the addition of the duct, as was seen in the sea trials. 30 3.4 Assessment of Errors 3.4.1 Instrumentation Error Analysis For each measured variable (V, n, T, and Q), the data acquisition system provided the mean (xm), maximum, minimum, and root mean squared (rms) statistics for each test run. The corresponding standard deviations could then be calculated according to the following equation: 0DEV=4 rmsl -*» This equation represents the standard deviation of the data. The standard error, which is a better representation of the deviation of the true mean of the variable, is determined by dividing the standard deviation by the square root of the number of data points: 4np Some variables did not have associated statistics. Each of these variables was assigned an estimated value of uncertainty, shown in Table 3.2. These values were used for the standard error for the variables. Variable Uncertainty local radius r 0.001 m propeller radius R 0.001 m propeller diameter D 0.002 m blade angle <p 3° Table 3.2 Uncertainty estimates for some variables. The standard errors for the calculated values (J, KT, KQ, JJ, and a) were determined using the following formula for error propagation: " ( dy Y V '•=' l d x ' J 31 where y is a function of several variables JC, with standard errors at. For example, advance ratio is given by: nD - _ • • At one point, V = 1.607 ± 0.00336 m/s, n = 7.992 ± 0.000607 rps, and D = 0.200 ± 0.002 m. Therefore, 1.607 / = • 7.992-0.200 / =1.005 The corresponding standard error is: A2 CX, + fa/ Y ra/ Y va« y + A2 + V n2D + \ 2 2 1 7.992-0.200 Oj =0.0021 -x 0.00336 + -1.607 \ 2 7.9922-0.200 -x 0.000607 + -1.607 7.992-0.2002 x 0.002 The error bars shown on the graphs in the preceding sections are equal to 1.96<7of the corresponding variable, representing a 95% confidence interval. 3.4.2 Discussion of Unqualified Errors Some errors could not be quantified and were therefore not included in the numerical uncertainty analysis. However, they still should be mentioned. These errors are primarily due to geometric inconsistencies of the propeller. The most significant source of geometric inconsistency was introduced when modifying the propeller for the ducted tip tests. It was impossible to ensure that all of the 32 ducts were precisely the same length and shape, and were all mounted in exactly the same position relative to the blade. Additionally, keeping the blades tightly in position throughout each test run was an issue. This caused a small amount of blade shifting, thereby changing the pitch angle slightly. These errors, although not entirely insignificant, are not expected to have greatly influenced either the quantitative or qualitative results. 33 CHAPTER 4 - Propeller Cavitation Performance This chapter compares the cavitation behaviour of the conventional and ducted tip propellers qualitatively. 4.1 Conventional Propeller Figure 4.1 illustrates the qualitative cavitation behaviour for the conventional propeller. A l l of the photographs are for a pitch ratio of 1.4 as this was the only setting for which visual information was available. The same trends would likely be observed at the other pitch ratios. The degree of cavitation increased with a reduction in advance ratio or cavitation number. At relatively high advance ratios, cavitation appeared as a helix emanating from the tip of the trailing edge of each blade (Figure 4.1a). (The appearance of the helix initialization could not be determined.) Decreasing the advance ratio or cavitation number caused the helix to become larger in diameter and less defined, and bubbles began cover the tip of the suction side of the blade (Figure 4.1b). Further reduction of / or acav induced this bubble cavitation to spread down the blade, while the trailing vortices appeared much more diffuse (Figure 4.1c). These observations are consistent with the findings of Gawn and Burrill (1957) and Hordnes and Green (1998). In many cases where there was a large degree of blade coverage, it was difficult to determine whether the cavitation remained in the form of bubble cavitation or if it underwent a transformation to sheet cavitation (induced by flow separation). Sharma et al. (1990) consistently observed that leading edge suction side sheet cavitation accompanied tip vortex cavitation. Interestingly, the trajectory of the tip vortices did not appear to be 34 dependent on the physical parameters that varied, such as P/D, J, or ocav. This observation is in agreement with that of Arndt et al. (1991). Figure 4.1c Figure 4.1 The progression of cavitation on the conventional propeller. All pictures are for P/D-1.4. 4.2 Ducted Tip Propeller The addition of the ducted tip dramatically changed the cavitation behaviour of the propeller in several ways. No visual information was available to give an indication of the appearance of cavitation at inception, but a fair amount of information was obtained regarding the progression of cavitation. At relatively low advance ratios or high cavitation numbers (but still within the range for cavitation), there appeared to be two separate streams of cavitating flow. The first was in the form of separated flow on the suction side of the duct (i.e. sheet cavitation). The second stream appeared as a jet exiting the duct. The two streams interfered with each other slightly past the trailing edge of the duct, and at the location of this interference, they terminated in what appeared to be a diffuse vapour cloud (Figure 4.2a). These observations are consistent with those of Hordnes and Green (1998), and are in contrast to the sharply defined helices seen with the conventional propeller. 36 As the advance ratio was increased, the point of interference occurred slightly further past the trailing edge of the duct, and the vapour cloud extended into longer diffuse helices. This can be seen in Figure 4.2b, and in Figure 4.2c as a more extreme case. The advance ratio was increased by holding the rotation speed constant while increasing the freestream velocity. This resulted in an increased flow rate through the duct, which may have been responsible for forcing the point of interference slightly downstream. A decrease in cavitation number also forced the point of interference downstream and extended the diffuse helices. This may simply have been a result of the increase in the intensity of cavitation associated with the reduction in the freestream pressure operating condition. Increasing the blade pitch ratio resulted in similar behaviour. A possible explanation for this behaviour is that at the higher pitch angle, the point of flow separation near the leading edge of the duct was physically further away from the jet exiting the duct. The two streams therefore interfered with each other further downstream than with the smaller pitch angles. As with the conventional propeller, some bubble cavitation occurred on the suction side of the propeller blades. This bubble cavitation increased with an increase in advance ratio or with a decrease in cavitation number. It is surprising that the cavitation would increase with an increase in advance ratio, as the opposite trend is usually observed. However, this observation may not be reliable for two reasons: first, the angle from which the propeller was observed did not allow for a clear view of the blade surface; second, in many cases, there were bubbles in the freestream flow that may have made the degree of bubble cavitation appear larger than it actually was. There was no apparent 37 correlation between these freestream bubbles and the test conditions - it is more likely that the amount of bubbles was related to the dissolved air content of the water. Figure 4.2a Figure 4.2ft Figure 4.2c Figure 4.2 The progression of cavitation on the ducted tip propeller. A l l pictures are for P/D=1.0. 3 8 Due to insufficient data, a numerical value for the cavitation inception index could not be obtained for either the ducted tip propeller or the conventional propeller. However, the duct is clearly effective in reducing the concentration of the tip vortices. Additionally, one may safely assume that the ducted tip significantly delayed the onset of tip vortex cavitation. The general patterns of cavitation behaviour were very similar to the patterns observed in the sea trials (Hordnes and Green 1998), where it was found that cavitation inception was delayed by a minimum of 47% (based on the shaft rotation speed). 39 CHAPTER 5 - Conclusions and Recommendations for Future Work 5.1 Conclusions A marine propeller was modified by replacing the blade tips with a flow-through duct. Experiments were conducted in controlled environments on the propeller, both before and after modification, to determine the effect of the duct on the open-water performance characteristics and on the characteristics of tip vortex cavitation. The addition of the duct reduced the propeller's efficiency over a large fraction of the test conditions. However, the general trends were in agreement with previous sea trials (Hordnes and Green 1998), which showed an increase in efficiency for some operating conditions. This discrepancy suggests that the performance characteristics are sensitive to the tip geometry, inviting future work on duct optimization. The ducted tip was highly effective in diffusing the tip vortices, and also appeared to delay the onset of tip vortex cavitation. These results indicate that the ducted tip propeller may be of great interest to a number of applications in the marine propulsion industry. Computational efforts were directed toward modelling the flow around the conventional propeller. Although convergence was achieved, the solution was not sensible. The problem seemed to be caused by a fault in the software. 5.2 Recommendations for Future Work More research needs to be done before the ducted tip propeller can be implemented commercially. 40 It may be desirable to determine a numerical value for the cavitation inception index of a ducted tip propeller as it compares to that of a conventional propeller. This would require testing to be done in a cavitation tunnel with continuous video recording beginning from non-cavitating conditions and continuing past the point of inception. The video would also illustrate the flow patterns at cavitation inception, which could be important for duct optimization. More importantly, future work should include optimization of the duct geometry. This research could be either experimental or computational, and would entail exploring different configurations of duct length, inner and outer diameters, shape, and position relative to the blade tip. Any experimental work in this area should be done using a full-scale model propeller in order to minimize the impact of geometric inconsistencies. The computational efforts presented in Appendix L should be extended to successfully simulate the flow around the conventional and ducted tip propellers. The option of using an unstructured grid should be considered. Although an unstructured grid is far less efficient than a structured grid computationally, it is much less demanding of human resources. Depending on the number of simulations required and the computational time required for each, the reduction of required human efforts may be well worth the sacrifice of computational efficiency. Finally, it would be useful to extend the research (both computational and experimental) to propellers of different geometries by varying the diameter, pitch ratio, expanded area ratio, number of blades, and/or blade shape. 41 REFERENCES Arakeri, V . H . , Sharma, S.D., Mani, K. , 1985, " A technique to delay the inception of tip vortex cavitation from marine propellers", ASME Cavitation and Multiphase Flow Forum, pp. 28-30. Arndt, R.E.A., Arakeri, V . H . , and Higuchi, H. , 1991, "Some observations of tip-vortex cavitation", Journal of Fluid Mechanics, Vol . 229, pp. 269-289. Chahine, G.L., Frederick, G.F., and Bateman, R.D., 1993, "Propeller tip vortex cavitation suppression using selective polymer injection", Journal of Fluids Engineering, Vol . 115, pp. 497-503. Crump, S.F., 1948, "The effects of bulbous blade tips on the development of tip vortex cavitation on model marine propellers", Report C-99, David Taylor Naval Ship Research and Development Center. Gawn, R. and Burrill, L. , 1957, "Effect of cavitation on the performance of a series of 16 in. model propellers", Transactions of the Royal Institution of Naval Architects, Vol . 99, pp. 690-728. Goodman, T.R. and Breslin, J.P., 1980, "Feasibility study of the effectiveness of tip sails on propeller performance", Report MA-RD-940-81006, Department of Ocean Engineering, Stevens Institute of Technology. Green, Sheldon I., ed., 1995, Fluid Vortices, Kluwer Academic Publishers, Boston. Green, S.I., Acosta, A.J . , and Akbar, R., 1988, "The influence of tip geometry on trailing vortex rollup and cavitation performance", ASME Cavitation and Multiphase Flow Forum, pp. 76-80. Green, S.I. and Duan, S.Z., 1995, "The ducted tip - a hydrofoil geometry with superior cavitation performance", Journal of Fluids Engineering, Vol . 117, pp. 665-672. Hordnes, I. and Green, S.I., 1998, "Sea trials of the ducted tip propeller", Journal of Fluids Engineering, Vo l . 120, pp. 808-817. Hsiao, C. and Pauley, L .L . , 1999, "Numerical computation of tip vortex flow generated by a marine propeller", Journal of Fluids Engineering, Vol . 121, pp. 638-645. Ingvarsdottir, H. , Ollivier-Gooch, C.F., and Green, S.I., 2002, "Computational study of the flow around a ducted tip hydrofoil", 2002 Joint US ASME-European Fluids Engineering Summer Conference, Montreal, Canada. 42 Itoh, S., 1987, "Study of the propeller with small blades on the blade tips (2 n d report: cavitation characteristics)", Journal of the Society of Naval Architects of Japan, Vol . 161, pp. 82-91. Published in Japanese with English abstract. Lewis, Edward V. , ed., 1988, Principles of Naval Architecture Vol II: Resistance, Propulstion and Vibration, The Society of Naval Architects and Marine Engineers, Jersey City. Munson, B.R., Young, D.F., and Okiishi, T.H., 1998, Fundamentals of Fluid Mechanics, 3 r d ed., John Wiley & Sons, Inc., Toronto. Murdey, D., 2000, "Propeller Open Water Tests", Institute for Marine Dynamics Standard Test Methods #42-8595-S/TM-2, Version 3. Platzer, G.P. and Souders, W.G., 1979, "Tip vortex cavitation delay with application to marine lifting surfaces. A literature survey", Report 79/051, David Taylor Naval Ship Research and Development Center. 43 APPENDIX A - Comparison Between Conventional Propeller and Wageningen B4-52 Propeller Figures A . l , A.2, and A.3 compare the conventional propeller at various operating conditions with the equivalent Wageningen B4-52 propeller. These graphs are associated with Figure 3.1. a/ \\ / *A- *-<: 0 / "'A 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 J Figure A. la 44 Figure A.lb Figure A . l Comparison of the conventional propeller at 8 rps and 14 rps with the Wageningen B4-52 propeller at P/D=0.6. The triangles represent KT, the circles represent 10KQ, and the squares represent 77. Solid lines with solid symbols represent the performance characteristics of the conventional propeller, and dotted lines with open symbols represent that of the Wageningen B4-52 propeller. 0.9 0.8 0.7 0.6 6 0.5 0 *~ 0.4 t— 0.3 0.2 0.1 0.0 LLS^~~4 *"-o n** '< y * A. * - . * \ 1 0.0 0.2 0.4 0.6 0.8 J 1.0 1.2 1.4 Figure A.2 Comparison of the conventional propeller at 14 rps with the Wageningen B4-52 propeller at P/D=1.0. The triangles represent KT, the circles represent \0KQ, and the squares represent n. Solid lines with solid symbols represent the performance characteristics of the conventional propeller, and dotted lines with open symbols represent that of the Wageningen B4-52 propeller. 45 J Figure A.3b Figure A.3 Comparison of the conventional propeller at 8 rps and 14 rps with Wageningen B4-52 propeller at PID=\ .4. The triangles represent KT, the circles represent 10KQ, and the squares represent n. Solid lines with solid symbols represent the performance characteristics of the conventional propeller, and dotted lines with open symbols represent that of the Wageningen B4-52 propeller. 46 APPENDIX B - Propeller Performance Plots The performance plots for the ducted tip and conventional propellers are illustrated Figure B . l for the three different pitch ratios at 14 rps. These are related to the plots in Figure 3.2. P/D = 0.6 o o Figure B.la P/D= 1.0 Figure B.lb 47 P/D = 1.4 Figure B.lc Figure B . l Propeller performance at 14 rps for P/D=0.6,1.0, and 1.4. The triangles represent KT, the circles represent \QKQ, and the squares represent 77. Solid lines with solid symbols represent the ducted tip propeller, and dotted lines with open symbols represent the conventional propeller. 48 APPENDIX C - Comparative Plots of KT, 10% and 77 Figure C . l is equivalent to Figure 3.3, but contains data from the tests conducted at 14 rps instead of at 8 rps. 0.8 -] - . Figure C.la 1.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 J Figure C.lb 49 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 \ r \ - i 0.0 0.2 0.4 0.6 0.8 1.0 J 1.2 1.4 1.6 1.8 2.0 Figure C.lc Figure C. l Comparative plots of /T r, 10/Ye, and 77 at 14 rps for varying pitch ratios. The triangles represent P/D=0.6, the circles represent P/£>= 1.0, and the squares represent PID=\A. Solid lines with solid symbols represent the ducted tip propeller, and dotted lines with open symbols represent the conventional propeller. 50 APPENDIX D - Effective Angle of Attack as a Function of r/R This appendix contains graphs illustrating the effective angle of attack as a function of the radial distance along the propeller blade. These graphs are related to Figure 3.6. P/D = 0.6 Figure D.la r/R P/D= 1.0 r/R Figure D.lb 51 P/D= 1.4 25 -, 20 -r/R Figure D.lc Figure D. l Effective angle of attack at 14 rps as a function of r/R for varying advance ratios. All lines are for the ducted tip propeller. 52 P/D= 1.0 15 , r/R Figure D.2b P/D = 1.4 25 -j 20 r/R Figure D.2c Figure D.2 Effective angle of attack at 8 rps as a function of r/R for varying advance ratios. All lines are for the conventional propeller. 53 P/D = 0.6 P/D = 1.4 25 20 r/R Figure D.3c Figure D.3 Effective angle of attack at 14 rps as a function of r/R for varying advance ratios. All lines are for the conventional propeller. 55 APPENDIX E - Comparative Plots of KT, 10KQ, and 7] as a Function of Effective Angle of Attack The following plots show Kj, IOKQ, and 77 as a function of effective angle of attack for P/D=0.6, 1.0, and 1.4 at 14 rps. The equivalent plots for 8 rps are shown in Figure 3.7. | Gr7-0 6 A' 0 5 0 4 0 0 *W J T n . O -10 -5 0 5 10 15 20 a Figure E.la I IS-•1 A H*t— 1 0 1 0 0 3_, J»T 0 fx*> , 0.0 Figure E.lft 56 a Figure E.lc Figure E . l Comparative plots of KT, 10KQ, and 77 at 14 rps for varying pitch ratios as a function of a. The triangles represent P/£>=0.6, the circles represent P/D =1.0, and the squares represent PID=-\A. Solid lines with solid symbols represent the ducted tip propeller, and dotted lines with open symbols represent the conventional propeller. 57 APPENDIX F - Correction Plots Figures F . l through F.4 contain the thrust and torque correction plots used to establish the factors needed to "zero" the data for both the ducted tip and conventional propellers. This is discussed in greater detail in §3 .1 . P/D = 0.6 J=0.1 T = 0.40n2 + 0.076 J =0.2 7 =0.35n2 + 0.007 J =0.3 7 = 0.30n2-0.008 J =0.4 7 = 0.24n2-0.066 J =0.5 7 = 0.18n2 + 0.026 J =0.6 T = 0.11 n 2 +0.024 200 correction factor: 0.010 ± 0.05 Figure F.la P/D = 1.0 100 150 n2 (rps2) J =0.3 T = 0.69n2-0.33 J =0.4 T =0.64n2-0.11 J =0.6 T =0.53n2 + 0.10 J =0.8 T =0.40n2 + 0.50 J = 1.0 T = 0.26n2-0.19 J =1.2 7 =0.10n2-0.20 200 correction factor: -0.04 ± 0.3 Figure F.lb 58 P/D= 1.4 200 0 50 100 150 200 n 2 Ops2) correction factor: 0.0086 ± 0.38 Figure F.lc Figure F . l Thrust correction factors for the ducted tip propeller. 59 P/D= 1.0 Figure F.2b n 2 (rps2) 200 J =1.2 Q = -0.009n2 + 0.118 J =1.0 Q =-0.013nz + 0.115 J =0.8 O = -0.018n2 + 0.101 J =0.6 Q =-0.021 n 2 + 0.123 J =0.4 O =-0.024n2 +0.103 J =0.3 Q = -0.025n2 +0.115 correction factor: 0.112 ± 0.01 P/D= 1.4 2IP0 J =1.8 Q = -0.010n2 +0.148 J = 1.6 Q =-0.016n2 + 0.100 J = 1.3 Q = -0.025n2 + 0.093 J =1.0 O =-0.032n2 + 0.152 J =0.7 Q = -0.038n2 +0.132 J =0.4 Q =-0.042n2 + 0.129 J =0.1 Q =-0.043n2 + 0.086 I n 2 (rps2) correction factor: 0.120 ±0.027 Figure F.2c Figure F.2 Torque correction factors for the ducted tip propeller. 60 P/D = 0.6 Figure F.3a J=0.1 7 = 0.40n2-0.36 J=0.2 7=0.35n2-0.46 J =0.3 7 = 0.30n2-0.60 J=0.4 7 = 0.24n2-1.13 J =0.5 7 =0.18n2- 1.20 J =0.6 7 = 0.11n2-1.17 200 correction factor: -0.91 ± 0.43 P/D = 1.0 J =0.3 7 = 0.70n2-1.60 J =0.4 T =0.64n2-1.70 J =0.6 7 = 0.51/72 - 0.96 J =0.8 7 = 0.38n2-1.02 J =1.1 7 =0.16n2-1.83 J = 1.2 7 = 0.08n2-2.22 200 correction factor: -1.55 ± 0.49 Figure F.3b 61 P/D = 1.4 250 " 2 OPS2) correction factor: -2.32 ± 0.75 Figure F.3c Figure F.3 Thrust correction factors for the conventional propeller. 62 P/D = 1.0 E z Figure FAb n2 (rps2) 2©0 J =1.2 Q = -0.007n2 +0.044 J = 1.1 O =-0.009n2 + 0.045 J =0.8 Q =-0.015n2 +0.039 J =0.6 0 =-0.018n2 +0.053 J =0.4 Q =-0.021 n 2 +0.058 J = 0.3 Q = -0.023n2 + 0.075 correction factor: 0.052 ± 0.01 P/D= 1.4 200 J =1.8 0 =-0.011n2 +0.062 J = i.6 Q =-0.018n2 + 0.030 J = 1.3 Q =-0.026n2 + 0.014 J = 1.0 O = -0.033n2-0.002 J =0.7 O =-0.040n2-0.013 J =0.4 Q = -0.046n2-0.030 J =0.1 O =-0.050n2 +0.001 " 2 OPS2) correction factor: 0.009 ± 0.03 Figure F.4c Figure F.4 Torque correction factors for the conventional propeller. 63 8J ea Q O Q © II ft, I* a o u a. a -a 01 3 O t - T - ^ ^ T - T - M t M O J N C M W r - C J ' -o o o o o o o o o o o o o o o o p p p p o o o o o o o o o o o o d d d d d d d d d d d d d d d d 0 0 * - - I - T - I - I - I - I - I - V - I - , - I - . - C \ J o o o o o o o o o o o o o o o o p o o p o o o p p p p p p p p p o o o o o o o o o o o o o o o o o^-m^O)OiT-(Di-onococD<o a ' d d d d d d d d d d d d d d d d ? 0 ( D N 0 3 T t ( ! ) n N O ( M W T - N O S S o O ' - ' - W N n i o ^ ^ i ^ n t > ) ' - 9 o ' o ' o o ' o o ' o o o ' o o o o ' o o p o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o p p p p p p p p o o o o o o o o o o o o o o o o o o o o 1 0 0 1 0 S ' J ' ! f l M O W < 0 0 ) i - O N S W f ( M ! 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CO CM CO oo o> CO o co Tf Tf o CO o CM CD CD Tf CO cn CM co CM ,_ o io CO O) CD CM CD o Tf Tf CO o 00 o 00 CD CO o 5 CO o CO CO Tf o cn r- CM o o CM CM co Tf TJ" CO CO CO CO CO CO Tf o o T- CM co co Tf Tf IO CO CO CO CO CO CO d d d d d d d O d d d d d d d d d d d d d d d d d d d d d d d d • o o o o o o o O o o o o o o o o o o o o o o o o o o o o o o o o O o o o o o o o O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o C o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d <35 Tf 05 CM CO r- CO CO cn CO CM CO CM cn CO 00 00 cn CO CD CM CM Tf r-- CM 00 r~- CO CO TT CM o 00 CO CO CM cn CO r- CD co Tf CO CM o cn CD Tf CM o 00 CO CN CM CM CNJ CM CM CM CM o o o CM CM CM CM CM CM CM CM o o o o o o O o o o o o o o o o o o o o o o o O o o o o o o o o o o d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d ^ CD o T* o ^ o CD Tf Tf CD o CO o CD o Tf 00 CM CO CO o CO CO o CO CO . h- CO -a- CM i— CO CO o CO CD CO o CM 00 CO Tf CM o 00 r- CO CO CO CD Tf CNJ CM CM CM CM CM t— o O o o o CM CM CM CM CM CM T— y— o o o o o d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d o o o o o o o o O o o ,-3 o o o o O o o o o o o o o o o o o o o o o o o o o o o o o o o o D o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d o o o o o CO cn cn o cn CO CM o o Tf CM o CM o o o _^ ^ _ cn cn o o o o CM o CM o co o cn o TT cn Tf o Tf o CO o CO o CO o CO o CO o CO o Tf cn CO o CO o CO o CO o o '— CM CM CM co Tf Tf CO CO CD CD o o i— T— CM CM CO CO CO Tf CO CO CD CD d d d O d d d d d d d d d d d d d d d d d d d d d d d d d d d d o o o o O o o o o o o o o o o o o o o o o o o o o o o o o o o o o O o o o O o o o o o o o o o o o o o o o o o o o o o o o o o o o o D d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d _ 00 00 Tf CO CO o> CD CO CD Tf CO CD CD Tf cn CM ^  CM CM CM ^  CD CO cn CM CD CO CD CM CO o 00 lO CM O) CD co 00 Tf OO Tf cn Tf o co CO o Tf CO CM CO o o cn CO CD CO c o in CO CO CO Tf Tf TT TJ- CO CO CO CM CM •»— T- CD CD CO Tf Tf CO CO CM q q 00 CD CO CO d i d d • d • d • d • d i d 1 d d i O O d • d i d d • 1 i 1 1 i 1 • • d • d i d 1 d i d 1-CU a o c c > © U O o o o o o o o o o o o o o o o o q q q q q q q q o o o o o o o o d d d d d d d d d d d d d d d d CM i - CM CO cn O |-» Tf CM O I CD CO CO CO lO C O . - O C M O C O C M O O C O iocoocococncOT-coi-TfT fT fcOCOCMCMCM'^T -o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o COCDCOCDCDCDCOOf-COCDOOOCDh-C O C D T - C O O J C O C D C M - ^ - ^ C M O C O C O C M C O h ~ C D C 0 C 0 T f T f c 0 C 0 C M T - O 0 > h - C D C 0 C 0 T - ' T - ^ T ^ v ^ - r ^ T ^ ^ ^ r - ^ ^ T ^ d d d d d O O O O O O O O O O O O O O O O O T T C M C U C U C ^ I ^ C T J T T O O C O i n ^ K ^ / r i C M •t-CMCMi-CMCMCMCMCMCMCMCOTfCOTfTf O O O O O O O O O O O O O T - O O c i c i o c i o c S c i c i c i d c D c i c i c i c i c i CM CM CM CM CM CM TfOT - c o r ^ c o o i - c n c o c o r - c M C M f n ^ OTcrjcocq^wcpcocDCMCMcriOTcqjgoo cDTf 'dTf 'dTf ' d co ' r^T^Tr ' r ^dcM -NTf cocor-r--cocococoTfTfcocMCM'>- < 7-' i O ' - i - ' - . - C M t - C M - ' - ' - C M C M - r - T - C M C M O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O c o c 7 > r ^ c o ^ o o c M T f r o m m , - i a - l r v l r > . r ^ ^ c o c 7 > C M T f c n c o T f c o g ° p 3 c M | ^ "1 . .001— CO N , w i o o r ^ c o m c M O c o c O T f c M 0 ; N : i r ; r j o c M cncocD-^CMT-cocDTrcococMf~r~--,io qTtajqcTjiot^cDcqTfCTiOTfcDgcD ^cDcot^cMcp'dricpcri-r-'^io' co co oo CO oo • h-CDCDCOCOTfTfcOCM O O T - T - T - T - T - T - T - T - C M C M C M C M C M C M o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o d o d c i d o o c i c i c i c i c i c i d c i O O l O l O l O l M O S O M D t - O K C I t ^ o r - cocor -cnh- iOTf^oco iococMO q o ^ c x j n q ^ i q c B M O o o i O i - N O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O q q q q q q q o o o o o o o o o dooc2o<D<Dcidoc6cidcDcic6 inc0C00>0)0)(3)0)OSNNC0011D0) cocooococococococnoocooooooocooo oictoiOTbO)OiO)(3)0)0)0)0)0)0)0) i ^ i ^ i ^ r ^ ^ r ^ r ^ i ^ r ^ ' i ^ r ^ r ^ h ^ i ^ t ^ r ^ O T - T - T - T - T - T - C M C M C O C M C M C M C M C M C M o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o c i c 5 o d c 5 c i c 5 c i c i c i c 5 d < D c i c i d O I D T - O I O O I - O I O S S M O W S ' -OTfooi-cDOTfr^.-^coo>cor-cMioo O'-CMTtlof-COCn- ' -CMCOCOCDCOCDi-d d d d d d d d T ^ T ^ T ^ T ^ T ^ T ^ T - ^ c M o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o COCOCOCOh-COOOCOCOCDCOCDCOOOCOr*-0)0>qo)qqc7)05qo50)050)C7>c7)C7> c^crSc^wdcT)codcodcr)cr>cooococo in o ft, a o u ft. u 3 O O ' - T - 7 - T - W r - i - ^ 0 J W i - i - T - i - ^ T - T - T - f A l 0 J 0 J ' - T - ' - 0 J ( D O O O O O O O O O O O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o O O O ^ ^ C M C V J O v J C O C O o o o o o ' o o o o o o o o o ' o ' o ' o ' o o o o o o o o o o 0 O T - T - I - T - T - T - ' - ' - T - ' - I - W N N o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o IT) CO CNJ 00 CM n w i~ ^ ^ • i - « J T " CM CO IO CO I I o o i - c M C v ^ ^ m m t o t o L O ^ o ^ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ' O T - O O O O T - T - I - T - I - I - O I - ' - I - I - T - ' - O I - T - O O O O O O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o i f l i n c O O ' t N ' t ^ l f l t ' - C n n t D W l D C O O C O O C O ' - C O C M S ^ t D t S S ( D ( O r r C \ J ' - C D N l O n O C O ^ W m ( O t C O ( D P J O ) ^ T - ( O W N W C O C O C O 0 0 C O r a c O N N N S N ( D ( D ( D i n i n U ) ^ , ^ , ^ , n C ' 3 C O N O l i - ' -o o o o o p o o o o o o o o o o o o p o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o S W C C J C O C D O C D W i - C O C D i - i D r A j N t , " ^ W C O W ( D O C M l r ) N N C O C O C O C O O O N N C O C D i n L O ' J C O C s J i - T o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o p o p o o o o o o o p p p p p p o p o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o " . ^: ^  ^ ^; . . . . _ _ _ _ _ _ _ _ _ _ _ _ _ o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o ' - C O O O I S ^ S C O I O S W < - N " t a ) L r ) i - < O T - ( D o S • e J - C Q C O C M C N J C N J i - i - O O P 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ° CM O O CO O O) N - t i n rj- n- 3-O O ^ r - ^ T - f J ' - ' - C S J C J p j W r - r - ^ T - r - T - C J W W W ' - i - T - O i -O O O O O O O O O O O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o > a ) ( 0 ^ n i o o w o o o ) 0 ) - n o . ^ O ( 0 ( O i n i - . ( D i i - c o c o o ) t 3 ) 0 o t o t o m o m o m o ^ o i m a i m o t o i o o i f i o m o i f l o i D q q o r - N w w n ^ ^ i r ) i f l . q o ( D s r a ( D 0 ) 0 ) o o ' - ' - c v i w o O O O O ' O O O " O O O O O O O O O O O O T ~ ^ ^ ^ ^ ^ , ~ ~ , - : O i - i - i - « - C \ | P J ' - ' - ' - C \ J C J W C J C \ J o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o O O O O O O O O O O T - ^ T - ^ T - ^ I - ^ T - ' o o o o o o o o o o o o o o o o o o o o o o o o o o o o i - o h - o o m c D O i - o - N N N ( D ( D 1 0 I _ 1 W l f l r - O D C O r - i - C D O < D __ _- _ ' - T f s c n c x j n ^ - n ^ w ^ ^ C v O O W T - i - o c n o ) c o s a ) ( D U . ' t n c v o o o o o o o o o o o o o o o ' - O m S C O O U . - l ' - U . O J t O t D t D N O i n T 0 ) C 0 C 1 O ( M ' T O ( 0 i f l ( D 0 ) c o c o T - m i o ' C ' - c o v o - ' - t D o t T T V V Y 9 9 9 9 9 9 9 9 9 9 7 t t t CO CO CO CM CM 1— '— < o o o o o o o o o o o o o o o o o o o o o o o o o o o o a)cou.-OsnNa)C)corAjco(D^wu.K)coo)WO)(D^ff iO)^corr W W ' - 0 ) ( 0 0 0 ( O W C O ' T O ) t S C O S ' - ( D N O W ( 0 ( O O i - C v P . n ( D ( D ( D l f l l f l l O l f i ^ ^ C O C O W W ' - i - 0 0 0 ) C O C O S l O i n i r ) T t C v W T -o o o o o o o o o o o o o o o p o c p c p c p c p c p c p c p c p c p i n u p u D ^ ^ ^ ^ C O C O C M C M O J ^ - c p c p C M C M i - ^ i - C M C N J C M C M C M C V J ^ C M C N J C M C M C M C M C M C M C M C M i - i - i - C M C O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o i o d o d r ^ o u ^ T t o i o o r ^ O T r ^ o r ^ • - -^ ^ ^ ^ ^ ^ ^ ^ ^ C O C O C O C O C O C O O J C N J O J O J l O o o r - c o o o c n o j ' ^ C M u n c D C D C D C D a D O O O O O O O O O O O O O O O o o o o o o o o o o o o o o o 0 ) 0 ) 0 ) l O ( D ( D N n , n r_ m i n _ri r*» r - 1 - r ^ r r r r d r 1 - - - T - T - W ^ r r W W ' - N N T - r r N n o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o ^ ( O t o o o N i n i n s o s o ) T f o n n i n s ( o n m ^ - , A i - . ^ o r - 0 ) W » i p n r i n o ) n ( p O ) o q w o ) r r T t c p w q ( p g ^ 5 s - r T - T - i - w c o w w w n n c O ' - ' - ' - i - i - T - w c o c o c o r - w w o w o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o O C O U . W i n ( v O N O U . l O P 3 C O W N O W i r ) ' - 0 ) N O C O C \ J I N r t W ( , 3 o i ^ i n c o c M o r ^ i n ^ r o o r ^ i O T j - t - o r ^ ^ i - ^ r o O c o i n ^ - c M o c o t D q q T - w c o t ^ i r ) ( q s N ( D O ) q r - q c \ j q ^ i f l i O ( O N i D O O O O O O O O O O O O O ^ T - ^ T-' r-' r 1 T-' t-' i - ^ C\i W N 0 0 0 0 0 0 T - T — O-^'— T - T - O T - T - T — T - O ' ^ - ' - ' ^ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 0 ) 0 0 ( ^ 0 ) 0 ) 0 ) ^ 0 ) 0 ) 0 ) 0 ) 0 ) 0 ) 0 ) 0 1 0 ) 0 ) 0 ) 0 ) 0 ) 0 ) 0 1 0 ) 0 ) 0 ) 0 ) 0 ) 0 ) p O ) p O ) p O O ) 0 ) 0 ) 0 0 ) 0 ) C T ) 0 ) 0 ) 0 ) 0 ) 0 ) 0 ) 0 ) 0 ) 0 ) 0 ) 0 > 0 ) 0 ) 0 ) 0 ) --^  r--^  r-^  r--^  r-^  r ^ r ^ r ^ r ^ r ^ r ^ r ^ r ^ " r - - * r - - - i r—-" r--^  r-^  r-^  r~-* r-^  r-^  r-^  r-^  W C 0 N N C O O ) O ) ' J C J i n t D t O t D ! D ( D O O O O O O O O r - O O O O O O O N W C 0 C 0 ^ l f l C 0 ^ T f t U - < D ( 0 ( D o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o c o c M ^ - c o r ^ i o ^ t - c o i n ^ i n r - . r ^ o c o W ' t c j Q M n a j ' - o i M f l n O ) O C N J t n c O T - c O C D O ) C M U O h - O C O < D C O O i - i - i - i - i - i - O O O O O O i - O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o o o o C D i - t - t - C M o C M C M o O C M C M C M ' - C O c o c o c o c o c o c o c o c o c o c o c o c o c o c o c o I _ T - l ~ l - ' T - l - T - - f - T - l - t - T - T - T - 1 - 1 - 1 - C N J C N J < N C N l ( O C O T - O C O a > 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ^ 1 - 0 0 p o o o o o o o o o o o o o o o o o o o o o o o o o o 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 _ T - o m O S C D C D 0 ) ' - O ' - N - ' - 0 ) O ' - 0 D T - a D 0 0 N n O ) ' t ' T f 2 o i n o N - o ^ c n c o N r A j ( o o w s o ) T - N - i / ) < D s i r ) N - o t O i - s ' P p o ^ W W ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ L " ' : I " C N j C N 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 9 O O O O O O O O O O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o d d d d d d d o d o d d d d d d d d d d d d d d d d d intoroui(D(Din©tt>cvj©N t N t(»<Dcoswo3NnN ,coo)^roc\i w a ) s c D M - w o o o ( D T r r - o ) N ' f c V ' - s i n r v i c o ( O W Q O N - ' - i n ' -e o s s N N N s < D < D ( D ( O i n w i n w i n ^ ^ ' v t o n c ) w w N ' - ' -o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o O O O O O O O O O O O O O O O O O O O O O T - ^ T - T - T - O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o p o o o o p p p p o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o c D c o N O ^ S D N c n o ^ o j w n n o ^ c o T t T - c D Q L n i n N ' ^ ' W ' - o c o N w n i - o ) N ( D N - N o c o ( D T r c v i o ) S i n w c D N ' t r i - ^ i n i o i f l ^ t ^ ' T ^ ^ c o n n n n c o w i N N N T - T - T - ' - o o o o O o o d o o ' d d d o o d d d o o ' d d d o o o d o o o o p O O O T - T - T - T - T - ^ T - O T - T - O J o o o o o o o o o o o o o o o o o o o o o o o o o o o o d d d d d d d d d d d d d d O O J i - N - O N O i m c O C O W O D N ^ OOOOJCOiriCMCOCOCDCDr-^-OD O i - C N J C \ J « ^ - i r ) i r ) C D C D C D t D ' T , ~ : d d d d d d d d d d d d d p O O O O O O O O O O O O O O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o d d d d d d d d d d d d d d N ' t N ' - C O C M l f i N ^ O C O t O O O O ) O J C O ' ! t T - ( D O J N O J N i - l O C O O o O r ^ r ^ r ^ t O C D l O l O ^ - ^ f C O C v J C v J O O O O O O O O O O O O O O d d d d d d d d d d d d d d O O O O O O O O O O O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o o o d d d d d d d d d d d d d d 0 0 0 0 0 0 0 0 6 0 6 0 0 9 O O r i - T - r - i - i - T - T - r - T - r - r - i - T - i - T - ^ ^ M ^ ^ ^ r T - C M o o o o o o o o o o o o o o o o o o o o o o o o o o o p p p p p p p p p p p p p p p p p p p p p p p p p p p o d d d d d d d d d d d d d d d d d d d d d d d d d d oocooooooococ»cooooc0 ' i—oic3)oc£>ooocoa)cj)a>oia>oo o i n o ^ o ^ o ^ a j ^ o ^ o i r - o ^ o ^ O L n o T r c n ^ r o ^ r o i dddddddddddddddddddooT- :'- :'- :'- :T- :i- : O O O ' - T - T - ^ T - I - T - O ' - ' - O J O O O O O O O O O O O O O O o o o o o o o o o o o o o o O O O i - * - O O O O O ^ O i - 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 O r - C V J C O - ^ - L O C D h - h - C T J O ' - O J C O o o d d d d d d d d - r - ' - ' - ^ ' ' - " ^ ' - i - ' - ' - ' - ' - O J i - ' - r - i - T - T - T - i - T - C V J r - T - i - T - W I N W ' - ' -O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O o d d d d d d d d d d d d d d d d d d d d d d d d d d N ^ C D i - i - O S C O n i n N l D i n c O S O l D r f t D C O ' t T t O O T - l O O C O W C O ( O W © C O O ( D ^ ( f l i - S ( N S W N W N O l N - C O C I ) ' - n W C O ( 0 C 0 L 0 l 0 W ^ ^ N ' W n W W T - i - O O 0 > 0 ) 0 3 S N ( D U ) l f l ^ W C J o o o o o o o o o o o o o o o o o o o o o o o o o o o o . 0 9 9 9 9 0 9 9 9 9 0 T T T 7 CO CO CO CM CM CM r~ -l o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a , - 9 9 9 9 9 9 9 9 9 9 9 9 o o o o o o o o o o o o o o C O P J W W O O t D N i n O N l f l N W o > m c o o > c o t ^ r ^ - T - r ^ c o c o ^ - o > o c o c o c j T - o i m o j c n i n T - s o g t D T - O J T - C M T - T - ^ C ^ C \ J C \ J T - C M C ^ C \ J C \ J C \ I C M C M O J O J ^ O O O O O O O O O O O O O O O O O O O O O ' - ' - I - ' - O O 0 0 6 6 0 0 6 6 6 6 6 6 6 6 6 0 0 0 0 6 0 0 6 0 0 6 6 i - t D W ( D Q o u ) S ( O c o t o o i a i m o ) c o i n w w s N - c D ^ ^ [ J ; ^ t u ^ ^ ^ T j - ^ T r ^ ^ e o c o c o c o c o c j c j c \ i c \ i c j ' - ' ^ i - i - n ~ ^ • CMCMCMCMCMCMi-O^CNJCMCOCO^ o o o o o o o o o o o o o o d d d d d d d d d d d d d d a o u Cu. s o S O U IT - i - i - ^ T - ^ c M r - o j i - i - T - T - i - T - T - T - c g c M ^ r c N j T j - i n c v i u D i j D O O O O O O O O O O O O O O O O O O O O O T - I - T - T - O O 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 • CM o J £ _ • O CM 0 3 ICNi-OCbO^COT - c n r^LriCOi-C»r^ 0 - i-T-T-^CMCMCMC\Ji~T-CMCMC\Ji-CMCMCMCMCMCMCMCM^CMCMC0 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O o d d d d d d d d d d d d d d d d d d d d d d d d d d O O ( D ( 0 ( 0 I D 0 ) ( D C & a ) i n c 0 C 0 I N O 0 ) r l 0 l ) 0 O C 0 0 ) 0 ) 0 1 O O C 0 o c o w n T - o i N i n r a T - o i s i f l ^ w c n a j i o w w o i s i n n w o N 0 0 1 - c \ i n n ^ i n < D N s r a c n o » - ^ i N c o ^ i o i n i o s a 3 a ) 0 0 d d d d c i d d d d d d d d r r - V ^  ^ r-' ^  ^ ^ ^ r-' T - ' c\i c\i o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o d d d d d d d d d d d d d d d d d d d d d d d d d d f O ' - t - w ^ ^ ^ ^ t D o a N m c o i n i o i n w i n w i n t n w w w u i i n ' j O O O O O O O O O O O O O O O O O O O O O O O O O O O 0 0 ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 r ^ c o c o c o c o c o c o c o c o c o c o c o c o c o c o o o o o c o c o c o o o o o o o o o o o o o o o (DCOLOCOLOO — ,- — ^ ,^^-,^ C M c q i n p p c o O g ^ ^ S 9 g S 5 ^ W ^ C O C M t - a ) C 0 r ^ L O ^ f C O i - f O i - i - C M C M C O C M C M C O r t i - C O C O L O O O O O O O O O O O O O O O o o o o o o o o o o o o o o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 O 0 ) r i - f M » C 0 0 ) ( 0 C 0 i - 0 ) O O OCMlfiCO'-COCDCJ)CMlf)COOCOCO o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 0 0 0 0 0 0 0 0 0 0 0 0 ) 0 0 o o o o o o o o o o o o o o c o c o c o c o c o c o c o c o c o c o c o c o c o c o t--i - W ^ C O C O O C \ | T f l f i N C O C \ i n N C O O . C O N T - 0 O O r - O ' - W C v l W r A l C v l W C O n n W - O C O t ' - W p q p p o o o o o o o o o o o o o o ^ - o d d d d d d d d d d d d d d d d d d d d T - O ^ N O W O ^ O O W N f A l l O C O C O W C O ' t M O O T - i— i— C M C M C M C M C M C O C O C M CO C M CO C O CO C O C O CO L O O O O O O O O O O O O O O O O O O O O O O d d d d d d d d d d d d d d d d d d d d d p-o n s c T - m f A i o w o T - a K M i - o t t o o i O ' -o c o r o ( o c o f f i i f i i - ( D i - c D r o c o t n i D . o » - c v j c \ i r ; d d d d d d d d d d d d d d o d d d d ° O N C N C D O l O ^ O ^ O J ^ O C O C B ' t O W C O C O C j J ? o ( D n o ) ( D c \ i m r - ( o a ) T - _ o ) o i i O N C D c o t , t s ' O O r - T - w n c O ' T ^ ' t l O l O t O ( D ( D ( D C D ( D L O O T : d d d d d d d d d d d d d d d d d d d d 0 o __ 0 o _ W i - C O C O O O C O C O N C O S C O C O N C O C O C O N N N S o o o o o o o o o o o o o o o o o o o o p p p p o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o N C D T t C O ' - W N C O N O J - l C l N C O O O . C O ' - n N n ^ ^ ^ n c o o j c v i ' - ' - o a ) 0 ) c o s ( D U ) t n c \ i o o o o o o o o o o o o o o o o o o o o " - - O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O d d d d d d d d d d d d d d d d d d d d d c o o ^ c o n W N C v j o i o ^ n T - w w W ' - o c M W i n N O T - c O ' f O l N C O C O l O W C D O C M ^ C D N N S l f l C O C 3 ' f ^ C v C O C \ I O J C \ l ' - i - i - 0 0 0 ) C O N t D l O ' 3 - t O W d d d d d d d d d d d d d d d d d d d d d _• o _ S ( D ( D ( D O l / ) ( 0 1 f l ( D t D O N N O ) ( O l O ( D S n O O O O O T - O O O O O I - O O O - I - I - O O I - O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o N 7 - N O r - 0 5 ' < f O ) < D l O S n c O ( O l O i - C O n O ( 0 N c o c o c o m c o i - c o - o w c o i n T - N n r o ^ O ) ^ ! : o o o o o o o o o o o o o o o o o o o 0 c o c o c o i n c D c D i n i n c o c D W i n c D i f i i o i n i n i n t n T r T i -O O O O O O O O O i - O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d 0 6" C \ | l / ) N i - C O O C O m a ) O N O . C M ' - N ( O l f l ( D i - C O o o t - t - c \ i r ) c o c o c o 5 5 3 - O i n 5 - ^ M - M ' _ ) r -p p o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o O O N _ W ' - C D C O O O U . t C O K I O O ) ' - ' - T - C \ j o O ' - w n ^ i n c N c o a i O T - o j c o t w t t i N c o c \ i i o ^ c D o c o i n c o i o c o c \ i i f i ^ c \ i c o o s c \ i o c j ) i n o - i - C M a i c o c o c o c o ^ n i r j ^ L o ^ i n N i n N L O c D N O O O O O O O O O O O O O O O O O O O O O d d d d d d d d d d d d d d d d d d d d d O o ^ O O ^ C O C O ^ L O ^ U ^ C D h - r - - h - C O C O - T - i - O J O O O O O O I O U ^ U O O L O L O L O I O L O L O L O I O O O C O O T - W C O ^ l f i W O N C O C O C n O T - C N i n ^ l f i t D S C O t g o o o o o o o o o o o o o o o o o o o o ^ ( D O O W C O < D N O ) C \ I C O N C O O ) C O C O O O ) O C O S C O L O CO CO C O C O ^ C O C O C O C O C O C O ^ f C O ^ C O CO § o o o o o o o o o o o o o o o o o o o o rn er\ . is_. —_. «-f- rft fT. fo ro — rT\ rr\ r\i • O O O O O O O O O O O O O O O O O O O O O d d d d d d d d d d d d d d d d d d d d d o o° ^—' *J_ 1^ * V V UJ \Ji \±J mi t^ J VJ? c o r o r - n o T - ^ ^ ^ n r o t u w c D n W ' t C v W i n N s a c q N t D i f i ^ q w q i i i c q x ) d co C M co co r"*- L O r*" r*- co j^- co C M L O J*-- o o> co i*"-c o w T - M s a ) N o s w n - ) c o o n n c o ( D ( D - ) c \ i o i ^ p ^ T - ^ c q r ^ L q o p c q ^ p p o p p ^ c q - r ^ ^ apcocpcpcoiYr^ o D o © T - C v ^ n n f r - n T - ' t t W ' - c o c o o T - T - r - i -C O C 0 C N C M C V C \ J C \ I C \ J C \ J C \ J C \ J C M a J C \ i W p p p p p p p o o p o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 0 ( X l ( O T - 0 ) N T - T t C O ^ C 0 0 3 ( D C O W ' - a ) C O C \ | r -( D N 0 ) r - C 0 0 3 W C \ I C \ | T - ( 0 t O t D r - n C \ J ' - ( D O ( D C D ( D N i n ^ ^ C 0 W r - 0 ) 0 0 N W T t C M O C 0 L 0 O 0 ) 0 ) O O r A l t D ^ l f l C 0 n S ( D N ( D N N O e D 0 ) M n N 3 C O C O C O W W C \ I C \ I C O C \ I C \ I C \ I C \ I C \ I C \ i n C \ I W W p p p p o o p p o p o o o o o o o o o o o d d d d d d d d d d d d d d d d d d d d d O 0 0 T t W C 0 C 0 S i n S N ^ C 0 ^ ( D C \ l - ) N 0 ) C n C 0 N ( o o c f l o m t o m c o i f l O T - ^ t D c O ' - T - t D ^ ^ r o o w ^ ^ c o q N q n q q s n q ^ q u ) 0 ) c o N q c o o p ' c p ' c p ' c O C O ' S N S N C D ' c D ^ L O t o r - o ) n r j ) ( D c \ j . ) c o o a ) W ( o i f l o o j n ^ c O ' t r -s i o i n i n m i f i i n ^ t D i n o c D t D O i c D i o c D s c o o ) d d d d d d d d d d - r ^ d d d ^ ^ d d i - ^ d O J C O T T T - C O U ^ O L O O C O I ^ L O C O C N O I ^ C M O L O C O O ^ q s q q c q i f i ^ f f l q i n i f i q q q w i n i n i n q q o o h-c o o ( 0 ( o ^ T - f £ i c D w t ( _ n c n c o c D n n - m w r ^ c d c x i r ^ ' d c o T - ' c d L r i c v J c o u o Lfi iowLniowio^^^nnnwc\ i i -T- < : n " • ^ W t N © W O S O ^ C » O N ^ M M F F I N T L . - L . N N ( S r - q w q c o ^ c q N W t S S S S ° © w ^ e O W N i n c O l f i ( D O ) N W T f t C O C \ J C O C O T - W O ) C O ( D ^ W T f C O C O ^ C V J ^ ^ f f l l O l f l C O i n ^ l O C D O J C O d d d d d d d d d d d d d o T ^ T ^ d d - r ^ d O O O O O o ^ O O o C M o ^ f r ^ L O C M C O - r f O h - L O C O C q ^ L O r ^ C O L f t ^ l ^ H o i i - s c o i n w s N n i n t D n o Q S T t ' - t - . - . -N c o o D s d c O r - ' a D ^ w o d u i r ^ N c ^ o L X ^ L O L O l X 5 L n u n l ^ T t ^ T t c o c ^ c o o J C ^ J T - ^ - 0 , t J t r ^ i ^ p - O L o p w ^ c q c O L O ^ S ^ ^ g ^ g S ^ > D C O C 0 S N W N W C D T - ^ i r ) Q C \ | i - U ) C O C \ J C O t D N O O W T - 3 5 l / ) l f i O ( O N S C O C O S N N S O ) W o o o o o o o o o o o o o o o o o o o o d d d d d d d d d d d d d d d d d d d d L O C N O ^ C O C O I ^ U ^ O O O L O O C O r - ' t f r l v . O O C M O 0 ^ l f l N ( 0 0 ) 0 ) O W O ' t C \ l t r - ( 0 0 ) l f l O ( , 0 0 ) 0 O O O O O O O T - T - T - ^ - - T - ^ T - ^ T - C M ' - ' - C \ J d d d d d d d d d d d d d d d d d d d d d o o n s T - f f l C v r - n w o i M n o n i - c o c M r - w o a 3 o o _ T - s n o ) ^ r - N t b o ( D r A i a } ' J O _ O O T - L O n n O N ( O N i n ( D C D N ( D T t ( D i r ) C O ( D ( D O CO CD CO T - O *3-'T- C D n M n n r - 0 ) M n C v T - 0 3 N q q o j ^ i O N c o q r - o i n q c q q T - o ^ t D a j c j ) O C M L O O O T ^ ^ I O C O O C M C O O c o c o c o w c o n c \ i n c o n n n n c o o c \ i c \ j w c \ i w 0 o o o o o o o o o o o o o o o o o o o o O O O O O O O O O O O O O O O O O O O O O d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d c C O ^ O C O U ^ L n L O ^ U O L O L O ^ L O ^ O ^ - c t T t - r r - r j -o o o o o o o o o o o o o o o o o o o o O O O p O O O O O O O O O O 0 ) C 3 ) O O O O r--^  r-I r*-^  r-^  r—^  r*-^  r—^  r-^  r*-^  i—^  r-^  r*-^  r-^  r-^  r--^  v—^  r-^  r-^  r-^  r-^  N C O C O C O C O C O C O C O N S N N N N N N C O N N N N O O O O O O O O O O I O O C O C O O C O C O C O C O C O C O C O O C O C O C O C O C O C O C O C O C O C O <u "5 e_ o u, C« "3 c _o "•C c o > a o U o 0 O 0 0 O ' - T - T - r - T - T - 1 - W ' - T - T - ( M C M C \ j n o o o o o o o o o o o o o o o o o o o o p p p o p o p p p p p p o o o o o o o o d d d d d d d d d d d d d d d d d d d d o m o c o c O ' - c o n m c o o w c o s O CO O r»- Tt —- N n O) ^  O) t ' — — — ^ ^ i n m c D q o T - T - w n c o N - ' t i n i o d d d d d d d d d d d d d d d d d . i- LO i- O O t- OJ O Tt CD Tt O O O O O O O I - O O O O O - ^ O O — - I -o o o o o o o o o o o o o o o o o o o o o p p p p p o p o p p o o o o o o o o o o o d d d d d d d d d d d d d d d d d d d d d O Q O © ( D i r j l O C O r O i - S l O S f A | O N C O T - C O C O N -O O T - W O J C O C O N " ^ d d d d d d d d d d i d d d d d d d d d d O O O O O O O O O O O O T - O O O O O — - — -o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o p p p p p O O O O ' o o o o o o N C D L D W N T - M t C 0 i - N ' N 0 1 0 J t © 0 ) O C n 0 0 C D U l l O I ^ U ^ N ^ ^ C O C N J C V I ^ O O l Q C O N C D C D N ^ r O C X J d o d d d d o o o o o o o o o o o o o d o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o d d d d d d d d d d d d d d d d d d d d d o o o o p p o d d d d d d d d d d d d d d d d d d d d d o o o o o o o o o o o o ^ - o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o d d d d d d d d d d d d d d d d d d d d W C V J C O t r ) C O ^ O . T - ^ C O C \ J W ^ f f i l / > N ^ W C O C r ) o o r a r ^ c D ^ i - i ^ ^ o c o w o o ^ C T J i n i - r ^ C A j O i -p p p p p p U ^ L O L O T t T t C T ) C O ( N C N J C \ | T - T — O O d d d d d d d d d d d d d d d d d d d d o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o d d d d d d d d d d d d d d d d d d d d d C D O O i - i n w c o c o c o c n m r o a j N C D t o i o n N c o o ) N c o N i n o j c n N o o i N L n i - N i r i c B i r i ' - N W N i -C D < 0 C 0 t 0 t 0 L 0 L 0 L 0 T t T t T t T t C T ) C 0 C \ J C N J C \ J - i - - ^ O O o o o o o o o o o o o o o o o o o o o o p p p p p p o o o o o o o o o o o o o o d d d d d d d d d d d d d d d d d d d d O T - 0 ) 0 ) O O r - m 0 ) N N ( 0 N N O O N 0 ) 0 5 C 0 o w n ^ in i 0 0 0 0 0 0 0 0 0 O O O O O O — - T - O O - — - i - O O - i - C M C N C M C M o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o d d d d d d d d d d d d d d d d d d d d d O O i — • ^ C 0 - > - O O O O O O ' - T - T - T - T - T - - I - C \ I C \ J O O O O O O L O L O L O O I O L O U O L O L O L O L O L O L O L O L O O T - O J C ^ C O i n i r . C D N C O C O O ) O i - O J C O N b l f ) C D N C O d d d d d d d d d d d d ^ ^ T ^ T ^ i ^ - r ^ T - ' T - ^ I - : o o o o o o o o o o o o o o o o o o o o ^ W C O W i n C V O ^ W C V J l O C O C D i - Q C O i - l O C N I O T - O N W i - C n i O C y C 0 C 0 0 3 C 0 O i r A J ( D i - C 0 ' - C 3 ) T t c d c d c d m c d c M O J o J c M o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o d d d d d d d d d d d d d d d d d d d d d . . _ . T t (N o Lf> CO T t _ _ _ _ _ __ N N i f i q o | i n n c D i f l n r ; ( o w N w c o c g N i - ^ N c p c n c p d c p c p c o — • — - T - O O O O O O O O O T - O O O — • — • • — — • o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o d d d d d d d d d d d d d d d d d d d d O 0 ) C D i - O C 0 ^ T - N f A | C 0 < \ C 0 i - - O f M O i W - r - r - O C O N t D - v t O i - O O D S l O T l - C M O COCOCO-OCOCMCMCMCMCMCMCM o o o o o o o o o o o o o o o o o o o o o C O U 5 n i r , N t l / ) C O O ) Q C D W l O W W Q N ' O Q N k O C O c p c p d c p c o c d c d i ^ o o o o o o o o o o o o o o o o o o o o CO CO - - - - - CD CM o ^ a a a ) i o w i 3 ) ^ i - N c o o ) ' t o © c v i N C \ i , r i J _ ) ( D © _ _ C D I X ) l O I X ) N - N w C 0 C 0 n W f A l - T - ( D ^ o o o o o o o o o o o o o o o o o o o o o CD O CO O T t CO , n LO i - O O CO CD S T - c v J o c o u ^ i n O T c o i n O T c o o r o i o ^ CO CM CM CM i - CM CM o o o o o o o o o o o o o o o o o o o o d d d d d d d T f ^ q . q ^ f ' t q q ^ q q q q q s q T - i - g o | N N N i - J c o ' 6 _ c o o i u i d _ w c 6 c o ' f f i ' c D O C D O C O C O L O U ^ T t T t T t C O C O C M C X I — - T - ^ ^ T i O O O O O O O O O O O O O O O O O O O O O O - - - - T - C 0 C 0 n C 0 C 0 C 0 C 0 i - C M C 0 W C 0 t O O O O O O O O O O O O O O O O O O O p p p p p o p p p p p p p o p o o o o d d d d d d d d d d d d d d d d d d d d O T - C O O ) O T - C \ I C O G S ( O N N S T - T - S O O C O O C O C n O ) C O C M C O C O O ) i r ) i - N C , C n C D C M N ^ O W q q w q i n N q q ^ q i f i q q q T - q T t q q q O O O O O O d i - T ^ i - ^ T - ^ T ^ T - I i - ' c M C M C M C v j O O O - i - i - W O J - ' - W O J C M C O i - O C T t C D C D C O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O d d d d d d d d d d d d d d d d d d d d d o c » c \ i c \ i i n c \ i Q O ) O o r o o - T - n w w W ' - c o ^ O M D f - O C O ' - O ^ M D ^ ' M n - - - • - - • •- — LO CO - - - - - -i— v_ x - . i v i - u v i ^ u - N - C O C O t D - J W O C O O C M L O c q o ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ P ^ ^ ^ - : d d d O —^  T-^ T-^ C M C M C M C M C M I CO CO CO Tt Tt Tt Tt LO O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O d d d d d d d d d d d d d d d d d d d d O O O O O O O O O O O O O O O O O O O O p p p p p p p p o p o o o o o o o o o o cocdc6cdoocoa_c6a .cx .co o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o d d d d d d d d d d d d d d d d d d d d d CO LO CD CD LO CD r-*- CO Tt T t ^t LO Tt Tt LO Tt LO LO TJ" LO CO cnrocnroQfflffiffiroffifflfficoo ) 0 ) 0 ) 0 ) 0)OiO ) f f l p o o p p c ^ p o ^ p p c x p o ^ c n p p p p p o i p o c o n c o o n c o n o n w n n c o n c o c o n c o c o n APPENDIX H - Test for Dependence on Reynolds Number (Cavitation Tunnel) This appendix illustrates the performance characteristics of the ducted tip propeller various rotation speeds. The graphs show that there is no apparent dependence of KT, IOKQ, or rj on Reynolds number. 0.40 -i . J Figure H.la 0.45 -i 0.0 0.2 0.4 0.6 0.8 1.0 J Figure H.16 70 71 APPENDIX I - Propeller Performance Plots (Cavitation Tunnel) The performance plots generated from the tests conducted in the cavitation tunnel are shown in Figure L l . The results are for the ducted tip and conventional propellers at 14 rps. This data was not used in the analysis due to its questionable reliability. P/D =0.6 0.6 0.5 0.4 a _ 0.3 o 0.2 0.1 0.0 . — . u. \ ft \ <Trp_A—A^.^ \ \ "A. " ' A - _ -A • A . . ^~~*^0 V "A.. • - _ ' - 0 . \ — , -v. Figure 1.1a 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 J P/D = 1.0 0.8 0.7 0.6 0.5 p-a _ 0.4 o 0.3 _ 0.2 0.1 0.0 o _ _ » - — « ^ 0 V "o X bx \ ET A^ •o "•o • • y I'- A . A. . A "A A ^ ^ ^ j A ~ A * A A A ' f t ^ '"6 J \ A. \ . '. \ T V 0.0 0.2 0.4 0.6 0.8 1.0 1.2 J 1.4 Figure 1.1b 7 2 P/D= 1.4 1.6 Figure Lie Figure L l Propeller performance at 14 rps for P/D=0.6, 1.0, and 1.4. The triangles representor* the circles represent IOKQ, and the squares represent n. Solid lines with solid symbols represent the ducted tip propeller, and dotted lines with open symbols represent the conventional propeller. The data for the conventional propeller have not been corrected for friction because testing was only done at 30 rps, making it impossible to determine the appropriate correction factors. 7 3 APPENDEX J - Correction Plots (Cavitation Tunnel) Figures F . l through F.4 contain the thrust and torque correction plots used to establish the factors needed to "zero" the data for both the ducted tip and conventional propellers. They were generated from tests conducted in the cavitation tunnel, in contrast to the plots from the tow tank tests shown in Appendix F. P/D = 0 . 6 200 -i . n2 (rps2) correction factor: -15.9 Figure J.la P/D = 1.0 0 200 400 600 800 n2 (rps2) correction factor: -16.4 Figure 3.1b , 7 4 P/D= 1.4 500 400 300 200 100 200 400 n 2 (rps2) J =0.2 7 = 0.67n2 + 35.7 J =0.5 7 = 0.47/72 + 61.4 J =0.7 7 = 0.35/72 + 69.9 J =0.9 7 = 0.20n2 + 78.7 J = 1.1 7 = 0.03/72+ 95.0 600 correction factor: -68.1 Figure J . l c Figure J . l Thrust correction factors for the ducted tip propeller. P/D = 0.6 n 2 (rps2) J =0.8 Q =-0.002n2 +0.190 J =0.7 Q =-0.004n2 +0.180 J =0.6 Q =-0.005n2 + 0.183 J =0.5 Q =-0.007/72 + 0.197 correction factor: 0.190 Figure J.2a 75 P/D = 1.0 E 2 o Figure J.2b n 2 (rps2) = 1.3 O = -0.006n2 +0.156 \J = 1.2 Q =-0.01 On2 + 0.339 •J = 1.0 O = -0.015n2 + 0.593 J =0.8 Q = -0.020n2 +0.844 = 0.5 O =-0.025n 2 +0.906 correction factor: 0.568 P/D= 1.4 J = 1.1 O =-0.01 On 2-3.76 J =0.9 O =-0.017n 2-3.04 J =0.7 Q =-0.023n 2-2.45 J =0.5 O =-0.029n 2-1.98 J = 0.2 Q = -0.036/72 - 1.05 n 2 (rps2) Figure J.2c Figure J.2 Torque correction factors for the ducted tip propeller. correction factor: -2.50 7 6 APPENDIX K -- Data (Cavitation Tunnel) PID = 0.6 Ducted Tip Propeller n V T 1"corr Q Qcorr J K T KQ •n 5.9 0.00 8.8 24.69 -0.347 -0.537 0.000 0.449 0.0491 0.000 5.9 0.93 -18.8 -2.91 0.143 -0.047 0.784 -0.053 0.0043 -1.543 5.9 1.54 -32.6 -16.72 0.420 0.230 1.308 -0.304 -0.0209 3.019 5.9 2.19 -74.9 -59.04 1.296 1.106 1.857 -1.071 -0.1009 3.138 5.9 2.51 -96.0 -80.12 1.735 1.545 2.126 -1.456 -0.1411 3.492 5.9 2.82 -117.8 -101.95 2.167 1.977 2.391 -1.856 -0.1808 3.906 5.9 3.14 -141.1 -125.19 2.635 2.445 2.660 -2.277 -0.2235 4.314 5.9 3.44 -173.8 -157.89 3.295 3.105 2.915 -2.870 -0.2836 4.694 7.9 0.00 19.2 35.11 -0.617 -0.807 0.000 0.359 0.0414 0.000 7.9 0.73 -5.0 10.87 -0.199 -0.389 0.461 0.111 0.0199 0.409 7.9 0.83 -7.6 8.29 -0.150 -0.340 0.530 0.085 0.0174 0.409 7.9 0.94 -9.8 6.09 -0.106 -0.296 0.598 0.062 0.0152 0.390 7.9 1.04 -12.5 3.44 -0.064 -0.254 0.660 0.035 0.0130 0.283 7.9 1.14 -14.5 1.44 -0.020 -0.210 0.721 0.015 0.0108 0.156 7.9 1.25 -17.1 -1.21 0.039 -0.151 0.797 -0.012 0.0078 -0.202 7.9 1.36 -19.6 -3.68 0.083 -0.107 0.865 -0.038 0.0055 -0.944 9.9 0.00 32.9 48.85 -0.931 -1.121 0.000 0.318 0.0367 0.000 9.9 0.72 7.5 23.38 -0.511 -0.701 0.363 0.152 0.0230 0.383 9.9 0.84 4.4 20.35 -0.459 -0.649 0.427 0.133 0.0212 0.425 9.9 0.94 2.0 17.94 -0.417 -0.607 0.478 0.117 0.0199 0.447 9.9 1.15 -2.5 13.45 -0.328 -0.518 0.583 0.088 0.0170 0.479 9.9 1.26 -5.0 10.95 -0.282 -0.472 0.637 0.071 0.0154 0.468 9.9 1.46 -9.7 6.24 -0.197 -0.387 0.742 0.041 0.0127 0.379 9.9 1.67 -19.1 -3.19 -0.020 -0.210 0.845 -0.021 0.0069 -0.406 9.9 1.88 -33.7 -17.85 0.260 0.070 0.953 -0.116 -0.0023 7.677 9.9 2.09 -48.2 -32.32 0.554 0.364 1.058 -0.210 -0.0119 2.977 11.9 0.00 48.9 64.83 -1.314 -1.504 0.000 0.293 0.0342 0.000 11.9 1.04 14.9 30.83 -0.770 -0.960 0.440 0.140 0.0218 0.447 11.9 1.25 9.6 25.46 -0.687 -0.877 0.530 0.115 0.0200 0.487 11.9 1.46 4.8 20.72 -0.607 -0.797 0.619 0.094 0.0181 0.509 11.9 1.67 -4.6 11.26 -0.417 -0.607 0.705 0.051 0.0138 0.414 11.9 1.88 -18.6 -2.73 -0.167 -0.357 0.794 -0.012 0.0081 -0.192 11.9 2.09 -33.1 -17.25 0.105 -0.085 0.883 -0.078 0.0019 -5.688 13.9 0.00 68.2 84.15 -1.796 -1.986 0.000 0.279 0.0331 0.001 13.9 1.25 28.2 44.15 -1.175 -1.365 0.452 0.146 0.0227 0.463 13.9 1.46 23.0 38.89 -1.081 -1.271 0.528 0.129 0.0212 0.512 13.9 1.67 12.8 28.68 -0.901 -1.091 0.605 0.095 0.0182 0.504 13.9 1.88 -1.6 14.33 -0.631 -0.821 0.680 0.047 0.0137 0.376 13.9 2.09 -15.3 0.60 -0.383 -0.573 0.755 0.002 0.0095 0.025 77 n V T Tcorr Q Qcorr J K T KQ Tl 15.9 0.02 91.0 106.87 -2.377 -2.567 0.008 0.270 0.0327 0.010 15.9 0.71 65.5 81.39 -2.010 -2.200 0.225 0.206 0.0280 0.264 15.9 0.83 62.2 78.09 -1.952 -2.142 0.262 0.198 0.0272 0.302 15.9 0.92 59.0 74.88 -1.902 -2.092 0.292 0.189 0.0266 0.331 15.9 1.03 55.9 71.77 -1.859 -2.049 0.326 0.182 0.0261 0.361 15.9 1.13 53.2 69.11 -1.822 -2.012 0.357 0.175 0.0256 0.389 15.9 1.23 50.0 65.93 -1.775 -1.965 0.391 0.167 0.0250 0.415 15.9 1.35 47.4 63.27 -1.733 -1.923 0.426 0.160 0.0245 0.444 15.9 1.45 44.5 60.39 -1.684 -1.874 0.459 0.153 0.0238 0.468 15.9 1.55 41.7 57.61 -1.639 -1.829 0.490 0.146 0.0233 0.489 15.9 1.65 34.1 50.00 -1.508 -1.698 0.521 0.127 0.0216 0.486 15.9 1.76 26.0 41.90 -1.372 -1.562 0.557 0.106 0.0199 0.473 15.9 1.86 18.6 34.46 -1.238 -1.428 0.589 0.087 0.0182 0.450 15.9 1.96 11.4 27.34 -1.112 -1.302 0.621 0.069 0.0166 0.413 15.9 2.07 3.9 19.77 -0.974 -1.164 0.655 0.050 0.0148 0.353 15.9 2.17 -2.2 13.74 -0.864 -1.054 0.686 0.035 0.0134 0.283 15.9 2.29 -10.3 5.55 -0.708 -0.898 0.724 0.014 0.0114 0.142 15.9 2.38 -16.6 -0.70 -0.589 -0.779 0.754 -0.002 0.0099 -0.021 15.9 2.49 -23.6 -7.66 -0.461 -0.651 0.787 -0.019 0.0083 -0.293 29.9 0.70 332.4 348.31 -8.440 -8.630 0.118 0.249 0.0310 0.151 29.9 1.25 307.7 323.61 -8.063 -8.253 0.210 0.231 0.0296 0.261 29.9 1.46 298.9 314.79 -7.924 -8.114 0.245 0.225 0.0291 0.302 29.9 1.76 274.1 290.01 -7.528 -7.718 0.295 0.207 0.0277 0.351 29.9 2.07 241.8 257.73 -7.024 -7.214 0.348 0.184 0.0259 0.394 29.9 2.40 211.6 227.47 -6.524 -6.714 0.403 0.162 0.0241 0.432 29.9 2.71 185.7 201.64 -6.076 -6.266 0.456 0.144 0.0225 0.464 29.9 3.03 160.3 176.20 -5.629 -5.819 0.508 0.126 0.0209 0.487 29.9 3.33 130.3 146.15 -5.093 -5.283 0.560 0.104 0.0190 0.491 29.9 3.66 95.9 111.83 -4.484 -4.674 0.615 0.080 0.0168 0.466 29.9 3.87 71.7 87.61 -4:045 -4.235 0.649 0.063 0.0152 0.426 29.9 4.08 46.6 62,48 -3.587 -3.777 0.685 0.045 0.0135 0.359 29.9 4.38 6.8 22.71 -2.866 -3.056 0.736 0.016 0.0110 0.173 29.9 4.59 -20.6 -4.73 -2.375 -2.565 0.772 -0.003 0.0092 -0.045 29.9 4.80 -48.8 -32.90 -1.877 -2.067 0.806 -0.023 0.0074 -0.406 Conventional Propeller n V T T"corr Q Qcorr J K T KQ r| 31.2 0.90 272.6 N/A -7.070 N/A 0.145 0.175 0.0227 0.178 31.1 1.20 257.3 N/A -6.770 N/A 0.193 0.167 0.0219 0.234 31.2 1.49 241.4 N/A -6.510 N/A 0.238 0.155 0.0209 0.281 31.3 1.81 213.0 N/A -6.180 N/A 0.289 0.136 0.0198 0.317 31.1 2.11 183.6 N/A -5.740 N/A 0.339 0.119 0.0186 0.345 31.3 2.41 162.5 N/A -5.400 N/A 0.385 0.104 0.0173 0.369 31.2 2.70 139.6 N/A -5.020 N/A 0.433 0.090 0.0161 0.383 31.3 3.00 120.4 N/A -4.710 N/A 0.479 0.077 0.0151 0.390 31.1 3.30 88.3 N/A -4.080 N/A 0.531 0.057 0.0132 0.366 31.2 3.61 61.0 N/A -3.700 N/A 0.579 0.039 0.0119 0.303 31.3 4.20 -6.5 N/A -2.360 N/A 0.671 -0.004 0.0075 -0.059 31.3 4.50 -49.9 N/A -1.660 N/A 0.719 -0.032 0.0053 -0.687 31.2 4.80 -95.4 N/A -0.960 N/A 0.769 -0.061 0.0031 -2.432 31.2 5.09 -136.7 N/A -0.053 N/A 0.816 -0.088 0.0002 -66.950 31.4 5.40 -202.8 N/A 0.667 N/A 0.860 -0.129 -0.0021 8.320 41.2 7.50 -440.9 N/A 4.280 N/A 0.910 -0.163 -0.0079 2.984 31.3 5.71 -262.4 N/A 1.760 N/A 0.912 -0.168 -0.0056 4.328 44.7 8.40 -563.3 N/A 4.970 N/A 0.940 -0.177 -0.0078 3.390 38.2 7.20 -419.5 N/A 4.090 N/A 0.942 -0.180 -0.0088 3.076 36.2 6.90 -387.3 N/A 4.010 N/A 0.953 -0.185 -0.0096 2.930 34.9 6.90 -427.7 N/A 4.050 N/A 0.989 -0.220 -0.0104 3.323 32.9 6.60 -399.4 N/A 3.960 N/A 1.003 -0.231 -0.0115 3.220 30.9 6.30 -383.4 N/A 3.750 N/A 1.019 -0.251 -0.0123 3.317 29.2 6.00 -351.2 N/A 2.990 N/A 1.027 -0.258 -0.0110 3.841 78 PID = 1.0 Ducted Tip Propeller n V T Tcorr Q Qcorr J K T KQ 13.9 1.25 107.5 123.93 -4.128 -4.696 0.453 0.411 0.0782 0.378 13.9 1.77 89.6 105.96 -3.650 -4.218 0.641 0.351 0.0703 0.510 13.9 2.09 74.0 90.39 -3.221 -3.789 0.755 0.300 0.0631 0.570 13.9 2.51 55.2 71.58 -2.690 -3.258 0.907 0.237 0.0542 0.631 13.9 3.03 33.4 49.83 -2.037 -2.605 1.096 0.165 0.0434 0.664 13.9 3.45 10.7 27.14 -1.328 -1.896 1.247 0.090 0.0316 0.565 13.9 3.97 -24.5 -8.11 -0.203 -0.771 1.437 -0.027 0.0128 -0.479 13.9 4.39 -57.8 -41.36 0.863 0.295 1.587 -0.137 -0.0049 7.051 13.9 4.70 -84.9 -68.48 1.761 1.193 1.700 -0.227 -0.0199 3.090 13.9 5.01 -116.3 -99.91 2.744 2.176 1.814 -0.331 -0.0362 2.638 13.9 5.33 -150.2 -133.85 3.770 3.202 1.928 -0.443 -0.0533 2.553 13.9 5.75 -202.0 -185.60 5.282 4.714 2.079 -0.615 -0.0784 2.592 13.9 6.06 -246.1 -229.71 6.548 5.980 2.193 -0.761 -0.0995 2.668 30.9 3.26 551.3 567.68 -21.232 -21.800 0.530 0.379 0.0732 0.437 30.9 4.17 473.9 490.33 -19.108 -19.676 0.677 0.328 0.0661 0.534 30.9 4.71 421.2 437.56 -17.602 -18.170 0.765 0.292 0.0610 0.584 30.9 5.37 354.3 370.67 -15.584 -16.152 0.872 0.248 0.0542 0.634 30.9 5.96 290.8 307.18 -13.583 -14.151 0.968 0.205 0.0475 0.666 30.9 6.60 215.7 232.12 -11.260 -11.828 1.071 0.155 0.0397 0.666 30.9 7.22 132.6 149.05 -8.766 -9.334 1.173 0.100 0.0313 0.593 30.9 7.69 63.6 80.01 -6.755 -7.323 1.250 0.053 0.0246 0.433 30.9 8.03 5.2 21.55 -5.030 -5.598 1.305 0.014 0.0188 0.159 30.9 8.46 -81.1 -64.67 -2.483 -3.051 1.375 -0.043 0.0102 -0.923 Conventional Propeller n V T T"corr Q Qcorr J K T KQ •n 31.0 2.64 501.4 N/A -20.256 N/A 0.426 0.327 0.0660 0.336 31.3 2.71 508.5 N/A -20.766 N/A 0.433 0.325 0.0664 0.337 31.1 3.00 477.1 N/A -19.666 N/A 0.482 0.309 0.0637 0.372 31.2 3.31 459.5 N/A -19.166 N/A 0.530 0.296 0.0617 0.405 31.4 3.61 437.8 N/A -18.566 N/A 0.575 0.278 0.0590 0.431 31.0 3.90 396.5 N/A -17.166 N/A 0.629 0.258 0.0559 0.462 31.2 4.20 374.5 N/A -16.546 N/A 0.673 0.241 0.0532 0.485 31.3 4.50 358.9 N/A -15.966 N/A 0.719 0.229 0.0510 0.514 31.3 4:81 329.8 N/A -15.166 N/A 0.768 0.211 0.0485 0.532 31.4 5.10 302.1 N/A -14.326 N/A 0.812 0.192 0.0455 0.545 31.2 5.41 267.0 N/A -13.186 N/A 0.867 0.172 0.0424 0.559 31.1 5.70 228.6 N/A -12.066 N/A 0.916 0.148 0.0391 0.553 31.1 6.00 201.5 N/A -11.166 N/A 0.965 0.130 0.0361 0.554 31.1 6.30 174.1 N/A -9.866 N/A 1.013 0.113 0.0319 0.569 31.0 6.60 140.0 N/A -8.966 N/A 1.065 0.091 0.0292 0.529 31.3 6.90 116.2 N/A -8.386 N/A 1.102 0.074 0.0268 0.486 31.2 7.20 75.5 N/A -7.066 N/A 1.154 0.049 0.0227 0.392 31.2 7.50 38.4 N/A -5.826 N/A 1.202 0:025 0.0187 0.252 31.2 7.80 -7.1 N/A -4.316 N/A 1.250 -0.005 0.0139 -0.066 31.3 8.10 -103.0 N/A -1.146 N/A 1.294 -0.066 0.0037 -3.702 31.3 8.40 -149.3 N/A 0.514 N/A 1.342 -0.095 -0.0016 12.407 31.4 8.70 -209.9 N/A 2.644 N/A 1.385 -0.133 -0.0084 3.501 7 9 P/D = 1.4 Ducted Tip Propeller n V T Tcorr Q Qcorr J K T KQ •n 13.9 0.29 169.4 101.34 -8.130 -5.630 0.103 0.336 0.0938 0.059 13.9 1.35 153.2 85.14 -7.507 -5.007 0.490 0.282 0.0834 0.264 13.9 1.87 138.8 70.73 -7.040 -4.540 0.677 0.234 0.0756 0.334 13.9 2.40 120.7 52.55 -6.428 -3.928 0.870 0.174 0.0654 0.369 13.9 2.93 104.6 36.52 -5.862 -3.362 1.060 0.121 0.0560 0.365 13.9 3.44 84.9 16.80 -5.098 -2.598 1.247 0.056 0.0433 0.255 13.9 3.97 62.1 -5.98 -4.170 -1.670 1.435 -0.020 0.0278 -0.163 13.9 4.39 41.8 -26.28 -3.321 -0.821 1.588 -0.087 0.0137 -1.611 13.9 5.01 7.2 -60.93 -1.846 0.654 1.812 -0.202 -0.0109 5.347 13.9 5.54 -27.5 -95.58 -0.337 2.163 2.003 -0.317 -0.0360 2.803 13.9 6.06 -67.7 -135.78 1.336 3.836 2.193 -0.450 -0.0638 2.459 25.9 0.81 500.4 432.33 -25.660 -23.160 0.158 0.411 0.1107 0.093 25.9 1.25 471.1 403.01 -24.528 -22.028 0.243 0.383 0.1053 0.141 25.9 1.78 435.6 367.49 -23.218 -20.718 0.344 0.349 0.0990 0.193 25.9 2.41 389.6 321.48 -21.538 -19.038 0.467 0.306 0.0909 0.250 25.9 3.03 345.5 277.37 -19.900 -17.400 0.588 0.264 0.0831 0.297 25.9 3.67 297.6 229.51 -17.963 - 1 5 4 6 3 0.710 0.218 0.0739 0.334 25.9 4.19 254.7 186.61 -16.147 -13.647 0.812 0.177 0.0652 0.352 25.9 4.61 218.9 150.75 -14.637 -12.137 0.893 0.143 0.0580 0.351 25.9 5.23 163.7 95.64 -12.407 -9.907 1.013 0.091 0.0473 0.310 25.9 5.86 93.4 25.35 -9.696 -7.196 1.136 0.024 0.0344 0.127 25.9 6.17 53.4 -14.69 -8.077 -5.577 1.196 -0.014 0.0266 -0.100 Conventional Propeller n V T Tcorr Q Qcorr J K T KQ Tl 27.0 3.00 528.2 N/A -32.983 N/A 0.556 0.454 0.1417 0.283 27.0 3.10 520.9 N/A -32.633 N/A 0.574 0.447 0.1402 0.292 26.5 3.00 517.8 N/A -31.903 N/A 0.566 0.462 0.1423 0.292 26.8 3.30 514.8 N/A -31.923 N/A 0.616 0.449 0.1392 0.316 27.0 3.60 502.6 N/A -31.293 N/A 0.667 0.432 0.1344 0.341 27.0 3.90 481.7 N/A -30.163 N/A 0.722 0.414 0.1296 0.367 27.0 4.20 459.7 N/A -29.083 N/A 0.778 0.395 0.1249 0.391 27.0 4.50 437.6 N/A -28.653 N/A 0.833 0.376 0.1231 0.405 27.0 5.10 389.4 N/A -26.583 N/A 0.944 0.335 0.1142 0.440 27.0 5.40 371.6 N/A -25.383 N/A 1.000 0.319 0.1090 0.466 27.0 5.70 352.9 N/A -24.283 N/A 1.056 0.303 0.1043 0.488 27.0 6.30 308.1 N/A -21.883 N/A 1.167 0.265 0.0940 0.523 27.0 6.60 283.5 N/A -20.823 N/A 1.222 0.244 0.0894 0.530 27.0 6.90 260.1 N/A -19.823 N/A 1.278 0.223 0.0851 0.534 27.0 7.20 235.5 N/A -18.653 N/A 1.333 0.202 0.0801 0.536 27.0 7.50 208.5 N/A -17.363 N/A 1.389 0.179 0.0746 0.531 27.0 7.80 180.7 N/A -16.153 N/A 1.444 0.155 0.0694 0.514 27.0 8.10 158.6 N/A -14.983 N/A 1.500 0.136 0.0644 0.505 27.0 8.30 140.2 N/A -14.153 N/A 1.537 0.120 0.0608 0.485 27.0 8.50 122.3 N/A -13.353 N/A 1.574 0.105 0.0574 0.459 27.0 8.70 102.1 N/A -12.423 N/A 1.611 0.088 0.0534 0.422 27.0 8.90 82.7 N/A -11.663 N/A 1.648 0.071 0.0501 0.372 80 APPENDIX L - Computational Efforts Some efforts were directed toward modelling the flow around the propeller using Computational Fluid Dynamics (CFD). The intention of the computational efforts was to model the flow around the conventional and ducted tip propellers and to compare the models with the experimental results described previously. These attempts were unsuccessful in the end due to an apparent glitch in the software; however, they still warrant description. A commercial software package from Computational Fluid Dynamics Research Corporation (CFDRC) was used. The package, called CFD-ACE(U)+, comprises the following parts: CFD-GEOM, a grid generator capable of generating structured, unstructured, and mixed-element meshes; CFD-GUI, an interface for computational model setup; CFD-ACE(U), a general purpose flow physics solver; CFD-VIEW, a visualization tool; and CFD-DTF, a data transfer facility. The propeller was modelled using a structured grid. In CFD-GEOM, grid generation is done using a transfinite interpolation (TFT) technique. The basic element of this type of grid is an edge, which is composed of one or more connecting geometry elements (i.e. lines and curves). The positioning of the nodes on the edges can be controlled by the user. Four sets of edges are combined to form a face, which can then be projected onto a surface (such as the propeller blade) where necessary. The highest-level element is a block, which comprises six sets of faces. The blocks represent a volume in the grid domain. In this study, all blocks represented fluid volumes, although in general they can also be used to represent solid structures. 81 To maximize computational efficiency, only one propeller blade was modelled. This. was done using cyclic (also known as periodic) boundary conditions, which required that the faces to be matched be either pure translations or pure rotations of each other. This proved to be a challenging constraint. Standard CFD practices were followed throughout the grid generation process as much as possible. These included designing for a high grid density in areas where large velocity and pressure gradients were expected, avoiding large gradients in element size, and minimizing cell skewness. The grid generation procedure began with the blade of the propeller. As an initial attempt, the outline of the blade was divided into five sections: one on each surface at the root, one along the leading edge (approximately), one along the tip, and one along the trailing edge. Each of these sections was specified as an edge set and then combined to form one face on each surface of the blade (Figure L . l ) . 82 Figure L. 1 The starting grid on the suction side of the propeller blade. The pressure side of the blade follows the same structure. This was adequate when an initial coarse grid was used, but became a problem when the grid was refined. In order to achieve adequate boundary layer resolution (which was especially important at the blade tip), the cells surrounding the propeller surface had to be very fine in the direction normal to the surface. This created undesirable large gradients in cell size at the edges of the blade. To resolve this issue, the grid on the blade surface was changed to a "distorted rectangle" shape in the centre of the blade, surrounded on three sides by a loop, as shown in Figure L.2. The loop was further divided into several sections in order to have better control over node placement near the blade edges. 83 Figure L.2 The grid on the suction side of the propeller blade. The pressure side of the blade follows the same structure. After the blade surface was gridded, the flow channel was created. The option of a straight, 90° wedge-shaped flow channel was explored first for its simplicity and ease of construction. The cells generated near the blade were highly skewed in some areas due to the twisted geometry of the blade combined with the requirement that the grids on each side of the wedge be identical. This skewness was not a major problem for the initial coarse grids, but after grid refinement some elements intersected with each other, forming what CFD-ACE(U) calls "negative volumes" and resulting in obvious problems for the solver. After exploring a number of alternative gridding possibilities with the straight flow channel, the idea of using a wedge-shaped helical flow channel was introduced. This technique allowed the gridlines to be better aligned with the curvature of the propeller 84 blade, reducing cell skewness. Although a major improvement over the straight flow channel, this method was not without its problems. Gridding the hub geometry was not a possibility with a helical flow channel. Instead, the propeller was modelled as a blade rotating around a long hub. This is a common practice in propeller modelling, as the hub geometry has a negligible effect on the overall performance. The grid for the entire flow domain is shown in Figure L.3a. The section in the box is enlarged in Figure h.3b. The flow domain is divided into several sections, described in the following paragraphs in order from upstream to downstream. Figure L.3a 85 Figure L.3Z> Figure L.3 The flow domain. The first section is a flow development region, which consists of four identical blocks. Each block is a simple wedge shape (with an inner core removed to represent the hub), gridded with an H-mesh and twisted 180° to form a helix as illustrated in Figure L.4. The pitch of the helix was chosen through a trial-and-error process, where a simple visual technique was used to determine the pitch value that would best fit the curvature of the propeller blade from root to tip. 86 Figure L.4 The flow development region. The second section (Figure L.5) is similar in structure to the first, but it differs in shape. It is a continuation of the flow development region, and simply acts as a link from the first section to the sections directly connected with the blade. 87 Figure L.5 This section links the upstream flow development region to the areas connected with the propeller. The next sections are the most complex, as they are directly connected with the propeller. On each side of the blade, there is an inner block and an outer block. The inner block on the suction side of the propeller (Figure L.6a) was constructed by first projecting the outline of the blade onto the flow channel wall. Inside this projection, an inner loop was drawn so that the grid on this area of the flow channel wall was similar in structure to the grid on the blade itself. Edges were constructed to connect points on the blade to their equivalent points on the flow channel wall. In some cases, these edges were forced to pass through certain points in the grid domain in order to reduce cell skewness and avoid the formation of negative volumes. A similar procedure was used to create the equivalent block on the pressure side of the blade (Figure L.6b). The grid on 88 the flow channel wall in this region is identical to that of the region concerned with the suction side to accommodate the constraints imposed by the cyclic boundary condition. Figure L.6a Figure L.6b Figure L.6 The flow domain from blade to flow channel wall, as viewed from the suction side (a) and from the pressure side (b). The outer blocks surround the inner blocks as illustrated in Figure L.7a (viewed from the suction side) and in Figure L.lb (viewed from the pressure side). 89 Figure L.7a Figure L.7b Figure L.7 The flow domain surrounding the blade, as viewed from the suction side (a) and from the pressure side (b). 90 The next section (Figure L.8) links the blocks connected with the blade to the downstream flow region that follows. It is similar to the upstream link in both form and function. Figure L.8 This section links the areas connected with the propeller to the downstream flow region. The last section is the downstream flow region, where the wake is allowed to develop. It comprises five blocks that are identical to the blocks used for the upstream flow development region. 91 The boundary condition types applied to the grid are as shown in Figure L.9. Additionally, the propeller blade (not shown) is modelled as a rotating surface, and the interior block boundaries are continuous interfaces. constant constant •pressure outlet velocity rotating wall inlet Figure L.9 Boundary condition types. The appropriate parameters of the problem were set, including boundary conditions, initial conditions, and the K-e turbulence model parameters. When CFD-ACE(U) was used to solve the specified problem, convergence was achieved, but the results were not sensible. The suction side of the propeller blade showed positive pressure values, while the pressure side showed negative values. It appeared that the propeller was not rotating, indicating that the problem most likely had to do with the rotating wall boundary condition applied to the inner surface of the flow channel (i.e. the hub) and the blade surfaces. To implement this type of boundary condition, CFD-ACE(U) applies the appropriate velocity profile, determined by a user-specified axis of rotation and angular velocity, directly onto the rotating surface. (More information on this subject is located in Appendix M.) This functioned properly, but the flow conditions applied to the hub and blade surfaces appeared to have no effect on the neighbouring cells, regardless of the magnitude of the angular velocity. CFDRC Customer Support has acknowledged the 92 issue, suggesting that there is a problem with the normal fluxes associated with moving walls that their software developers are looking into. Customer Support also suggested that there may be a way around this problem through the use of moving grids along with arbitrary (mismatched) interfaces, which are supported by CFD-ACE(U). However, no such attempts were undertaken in this study. It is worth noting that a test was performed before going through the efforts of grid construction in order to check that the rotating boundary condition feature was working properly. The test was one of simple Couette flow, where a cylinder rotated inside a larger concentric cylinder, with fluid in between the two. The problem was not detected in this case because all flow was in the direction tangential to the inner cylinder's motion, such that all normal fluxes (the suspected source of the problem) were zero. 93 APPENDIX M - Information on CFD-ACE(U)+ Rotating Systems Rotating Systems Introduction The conservation equations for mass, momentum, and heat presented in the module chapters are for the stationary frames of reference. In many applications, such as turbomachinery problems or flows around propellers, fans, etc., it is more convenient to deal with a rotating sys-tem. One advantage of doing so is that a very complex transient prob-lem may be reduced to a simpler steady-state problem. Features The Rotating Systems feature can be applied to both three-dimen-sional and two-dimensional geometries. Theory Conservation equations for a rotating frame of reference will be pre-sented in this section. Assuming that the system rotates at a constant angular speed, and defining ut as the i-th component of the velocity vector relative to the rotating frame, and wt as the i-th component of the frame velocity vector (ri> = Q x >), we have for the absolute velocity component, v(-U: + W; (9-1) Since the frame velocity does not contribute to the mass balance, the continuity equation for mass remains unchanged. CFD-ACE(U) User Manual 7/23/01 9-1 94 Chapter 9: Rotating Systems Limitations Regarding the momentum equation, it is well known that observers in two systems of reference will not see the same changes when passing from one system to the other. It can be shown that under a steadily rotating system the momentum equation for the i-th component of the relative velocity goes as follows, i ( p « , ) + l(PuiUj) = -1 + ^  + pfi - 2peipppUj -piQfPi-toffy) (9-2) where e ipj is the alternating tensor. The additional terms are known as the Coriolis force and the centrifugal force, respectively. Under a constantly rotating frame of reference, it can also be shown that the transport equation for total enthalpy, H, as presented in the Heat Module, is no longer valid. A new equation has been derived: dpi dp"/ dt dxj _3 dx dp dxUuj dt dX; (9-3) where the quantity / is defined as follows, I=h + - = H - (9-4) Clearly, the RHS of Equation 9-3 is identical to that of the original H-equation, as presented in the Heat Module. Because of this, the quan-tity, /, is called rothalpy and it measures the total energy content in a rotating frame of reference. In CFD-ACE(U), the rothalpy equation is solved if the user specifies a rotating frame. Limitations There are no known limitation. Implementation The Implementation section gives details about how to setup a model for simulation with the CFD-ACE+ package utilizing the rotating sys-tem feature. The Grid Generation section will give tips about what types of grids are allowed and general gridding guidelines. The Solu-tion Setup section describes all of the rotating system related inputs to 9-2 CFD-ACE(U) User Manual 95 Implementation Grid Generation Chapter 9: Rotating Systems the CFD-ACE(U) solver. (Please see chapter 1, "CFD-GUI Over-view" for details on general model setup operations). The Post Pro-cessing section provides tips on what to look for in the solution output. Grid Generation Solution Setup/ Generation The rotating system feature can be applied to 3D or 2D-planar geome-try systems. Furthermore all grid cell types are supported (quad, tri, hex, tet, prism, poly). The CFD-ACE+ Graphical User Interface (CFD-GUI) can be used to provide all of the inputs required for the rotating system feature. This section describes the settings specific to this feature. Please see chap-ter 1, "CFD-GUI Overview" for a discussion of general model set-tings and basic CFD-GUI operation. Problem Type Settings Press the Problem Type [PT] button to show the Problem Type set-tings page in the Control Panel (see "Problem Type" on page 1-23 for general information about this page). Global Under the Global tab activate the "Rotation" feature. You will then be able to select one of two different modes for the rotation feature from a pulldown menu. Frame Mode. Choose the "Frame" mode if you want your system to be calculated in a rotating frame of reference. In the first set of type-in field, provide the x, y, and z coordinates of any point through which the rotation vector of frame passes. Next, input values of Wx, Wy, and Wz for the three components of the rotational speed vector for frame. This mode also allows for the specification of rotating wall boundary condition types as well. Wall Only. Choose the "Wall Only" mode if your model has only rotating wall boundary conditions (and you want the system to be cal-culated in the inertial (steady) frame of reference). Boundary Conditions Press the Boundary Conditions [BC] button to show the Boundary Condition settings page in the Control Panel (see "Boundary Condi-tions" on page 1-33 for general information about this page). Before CFD-ACE(U) User Manual 9-3 96 Chapter 9: Rotating Systems Implementation Post Processing any boundary values can be assigned, one or more boundary condition entities must be made active by picking valid entities from either the Viewer Window or the BC Explorer. Although the computations in a rotating frame are performed using the relative velocities, all of the boundary condition values must be specified in the absolute frame of reference. This applies to all bound-ary condition types but care needs to be taken for all conditions requiring external velocity inputs: Walls, Inlet, and Outlet conditions. For setting up Wall boundaries, any wall that is stationary in the abso-lute frame should be specified as a stationary wall (with zero veloc-ity), and any wall that appears rotating in the absolute frame (for example, blade surfaces) must be specified as a rotating wall. The inputs needed for a rotating wall have been outlined "Rotating Walls" on page 1-24. Note that the flow solver automatically converts all velocity specifica-tions (inlet/outlets and walls) from the absolute frame to the rotating frame using the frame information provided in the problem type set-ting. The solver also will convert any other related quantities (enthalpy to rothalpy) as appropriate. '• V^ S», Post Processing If the computation is performed in the rotational frame, the graphical output will show velocities in the relative coordinates. 9-4 CFD-ACE(U) User Manual 97 

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