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A study of the hydrodynamic performance of Voith-Schneider propelled escort tugs Ratcliff, Arlon J.T. 2003

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A STUDY OF THE HYDRODYNAMIC P E R F O R M A N C E OF VOITH - SCHNEIDER PROPELLED ESCORT TUGS by A R L O N J . T . R A T C L I F F B.Sc, University of Alberta, 1999 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in The Faculty of Graduate Studies Department of Mechanical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November 2003 © Arlon J. T. Ratcliff, 2003 Library Authorization In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Name of Author (please print) Date Title of Thesis: Degree: ABSTRACT The hydrodynamic performance of a typical modern escort tug, as it relates to indirectly applied ship handling force, is examined through an experimental and theoretical study. Experimental results are presented from a series of tow tank tests performed on a 1:18 scale model of the Voith-Schneider Propelled (VSP) escort tug, AJAX. Steady state flow pat-terns close to the hull are investigated and possible areas of focus for improvement are highlighted. Using this information, three alternative skeg configurations are proposed and tested. A semi-empirical numerical model, based on current and previous experimental data, is developed as a standard basis of comparison for the alternative configurations. The numerical model also allows prediction of performance for hypothetical tug designs. A sen-sitivity analysis is performed using the numerical model to determine areas of focus for design optimization. No significant difference in performance was found between the alternative and baseline skeg configurations, indicating more comprehensive design changes are required to effec-tively improve escort performance. However, an alternative configuration, consisting of a skeg similar to the baseline skeg with a gap at mid- chord, achieved this comparable per-formance with 14% less lateral area, which may be of benefit to other aspects of tug performance. The results of the flow visualization study and sensitivity analysis indicate location of the skeg has a greater influence on performance than shape. Stability and tow point location are also relatively important. However, no single design parameter was identified for focus. Instead, design optimization requires a balance of several parameters. The numerical model can be used for this purpose. The numerical model produces performance estimates with an uncertainty of 15% of the maximum steering force. This uncertainty is adequate for initial estimates at the early stages of escort tug development. ii TABLE OF CONTENTS Abstract ii Table of Contents iii List of Tables v List of Figures V 1 Acknowledgements viii Introduction 1 1.1 Background 1 Evolution Of Escort Tugs 1 Types Of Escort Tugs 2 1.2 Previous Work 3 1.3 Current Work 5 Motivation 5 Scope • 5 Experimental Work 6 Numerical Model 6 Document Conventions 6 Experimental Work 7 2.1 Equipment 7 Tug Model 7 Hull Only Configuration (HO) 8 Baseline Configuration (SK) 8 End Plate Configuration (EP) 11 Twin, Parallel Configuration (TP) 12 Twin, Series Configuration (TS) 13 Planar Motion Mechanism (PMM) 14 Instrumentation 14 Carriage 14 Tow Tank 14 2.2 Method 15 Setup 15 Calibration 15 Testing 16 Data Collection 17 Data Reduction 17 Experimental Uncertainty 18 2.3 Results 19 Forces 19 Centre of Lateral Resistance 20 Numerical Model 27 3.1 Input 27 Hull Parameters 30 Skeg Parameters 30 Stability Parameters 30 iii Propulsion Parameters 31 Hull Coefficients 31 Skeg Coefficients 34 3.2 Calculations 39 Assumptions 39 Hydrodynamic Forces 40 Propulsion Forces and Stability 41 Towline Forces 43 Uncertainty 43 3.3 Output 44 Visual Representation 44 Plots 44 Output Summary 44 Analysis 47 4.1 Discussion of Experimental Results 47 Steering Force 47 Towline Force 48 Overall Performance 48 4.2 Discussion of Performance Prediction Program 51 Validation 51 Sensitivity Analysis 55 Conclusions 57 5.1 Escort Tug Design 57 Flow Patterns Around Hull and Skeg 57 Performance Comparison of Skeg Configurations 57 Sensitivity of Design Parameters 58 Recommendations for Design of Future Escort Tugs 58 5.2 Performance Prediction Program 58 Accuracy of the Program 59 Performance of the Program 59 Recommendations for Improvements to the Program 59 Nomenclature 60 References 63 Appendix A - Figures 66 Appendix B - Tables 73 Appendix C - Calculations 76 iv LIST OF TABLES Table Title Page Table 2.1: Particulars of AJAX Tug and Model 9 Table 2.2: Comparison of Change in Centre of Lateral Resistance 20 Table 4.1: Comparison of Maximum Steering Force 48 Table 4.2: Largest Influences on Steering Force 55 Table 4.3: Largest Influences on Heel Angle 55 Table 6.1: Description of Symbols and Variables 60 Table B . l : Test Matrix 74 V. LIST OF FIGURES Figure Title Page Figure 1.1: Direct and Indirect Escorting Modes 2 Figure 1.2: Azimuthing Stern Drive (ASD) Escort Tug 3 Figure 1.3: Voith-Schneider Propelled (VSP) Escort Tug 3 Figure 1.4: VSP Escort Tug AJAX 4 Figure 2.1: Photo of Test Apparatus 7 Figure 2.2: Lines of the AJAX Hull 8 Figure 2.3: Photo of Experimental Model 9 Figure 2.4: Photographs of Model Configurations 10 Figure 2.5: End Plate Configuration 11 Figure 2.6: Span-wise Flow 11 Figure 2.7: Twin, Parallel Configuration 12 Figure 2.8: Flow Separation Around Chine 12 Figure 2.9: Twin, Series Configuration 13 Figure 2.10: Flow Separation on Skeg 13 Figure 2.11: Underwater Camera Position and Orientation 14 Figure 2.12: Preliminary Comparison of Steering and Braking Coefficients 21 Figure 2.13: Preliminary Comparison of Non-Dimensional Towline Force 23 Figure 2.14: Comparison of Change in Longitudinal Centre of Lateral Resistance . . . . 25 Figure 3.1: Numerical Model Input Sheet 28 Figure 3.2: Effects of Normalizing Function on Sway Coefficients 32 Figure 3.3: Effective Heel Angle 33 Figure 3.4: Calculated Lift Curve Control Points 35 Figure 3.5: Effect of Skeg Presence on Flow Around Hull 37 Figure 3.6: Effect of Hull Presence on Flow Around Skeg 38 Figure 3.7: Composition of Spreadsheet Calculations 39 Figure 3.8: Static Equilibrium of Tug Forces 41 Figure 3.9: Visual Representation of Stability Modification Factor 42 Figure 3.10: Model Output of Towline Force 45 Figure 4.1: Towline Force Envelope for 6 kn Case, 0 to 40 Degrees Drift Angle 49 Figure 4.2: Towline Force Envelope for 8 kn Case, 0 to 40 Degrees Drift Angle 49 Figure 4.3: Towline Force Envelope for 10 kn Case, 0 to 35 Degrees Drift Angle.... 50 Figure 4.4: Towline Force Envelope for 12 kn Case, 0 to 35 Degrees Drift Angle.... 50 Figure 4.5: Performance Prediction Validation, 6 kn 53 Figure 4.6: Performance Prediction Validation, 8 kn . . . . 53 Figure 4.7: Performance Prediction Validation, 10 kn 54 Figure 4.8: Performance Prediction Validation, 12 kn 54 Figure A. 1: Illustration of Terms Used 67 Figure A.2: Yarn Tuft Distribution 69 Figure A.3: Sample of Data Acquisition Output 70 Figure C. 1: Lift and Drag Curve for NAC A 0012 Section from Reference [20] 77 Figure C.2: Lift and Drag Curves for NACA 4415 Section from Reference [26] 78 Figure C.3: Lift and Drag Curves for Clark Y Section from Reference [27] 79 vi Figure C.4: Lift and Drag Curves for Clark Y Section from Reference [28] 80 Figure C.5: Linear Approximation of Longitudinal Centre of Lateral Resistance 81 Figure C.6: Dependency of Stall Point on Aspect Ratio 82 Figure C.7: Control Point Spacing Relative to Lift Curve Slope 83 Figure C.8: Linear Approximation of Lift Curve After Stall 84 Figure C.9: Lift Coefficient at 50 Degrees with Respect to Lift Coefficient at Stall... 85 Figure CIO: Calculated Lift Curve For NACA 0012 Section 86 Figure C l 1: Effects of Modification Terms on Calculated Lift Curve 87 Figure C.12: Dependency of Maximum Drag Coefficient on Aspect Ratio 89 vii ACKNOWLEDGEMENTS I would like to thank my supervisor, Dr. S.M. Calisal for the instruction, advice, and flexibility he has provided. I would also like to express my sincere appreciation to Mr. David Molyneux and Mr. Robert G. Allan for providing me with a project on which to work and for patiently taking time to guide me. The model used in this project was fabricated and tested at the Institute for Marine Dynamics in St. John's, Newfoundland and Labrador. The assistance provided by everyone at the IMD is gratefully acknowledged, particularly that of Blair Parsons, Chris Meadus, and Austin Bugden who helped to make testing an enjoyable experience. Finally, I would like to thank my colleagues and friends, Peter Ostafichuk and Chad Larson, for all of the many tidbits that added up to save me a lot of grief. v i i i INTRODUCTION As tankers are built larger and are required to deliver goods faster, a greater demand is placed on the performance of the tugs which handle them. This is especially apparent in long, narrow passages where ship system failure or human error can lead to grounding and devastating environmental impact. A specialized class of tug, the escort tug, has been developed to assist tankers through these hazardous waterways. This thesis presents a study performed to determine areas of possible improvement in escort tug design. 1.1 Background Tug performance typically refers to the capability of a tug to move or alter the heading of a ship [1]. At operating speeds below 6 or 7 kn, this is achieved by simply increasing the engine power. At higher speeds, however, stability becomes an important concern and any increase in engine power is offset by an even greater increase in drag [2]. Evolution Of Escort Tugs One solution to achieve better performance at higher speeds is to increase the size of the tug. A wider beam will increase stability and a longer waterline length will reduce form drag on the hull, allowing more engine power for ship handling [3]. However, practical constraints such as cost and manoeuvrability limit the power and size of the tug. Another solution which is used, often in addition to increased power and size, is to tow the tug behind the tanker. This allows the tug to keep pace with the tanker while using its engine power for ship handling rather than propulsion [ 1 ]. In this configuration, the tug can apply a force to the tanker either by directly opposing the pull of the towline or by indirectly steering the tug at an angle to the towline. When directly opposing the towline, the maximum force which can be applied is equal to the maximum static thrust of the tug's drive unit plus the drag of the tug. When acting indi-rectly, the tug's hull can generate a hydrodynamic force greater than the static thrust, resulting in a higher towline force [4]. This concept is illustrated in Figure 1.1. 1 Tanker Direct Mode \ _/ Force Indirect Mode Propeller Force Figure 1.1: Direct and Indirect Escorting Modes In many cases, conventional tugs are used in this manner. However, certain factors may dictate the need for a specialized escort tug. Often, the passage through which the tanker must travel, the speed and size of the tanker, and the weather conditions require a tug of a minimum size and ship handling capacity [5]. To suit this need, larger conventional tugs can be fitted with a protruding keel, or skeg, which increases the hydrodynamic force for relatively little cost [6]. One drawback of retrofitting a skeg to a conventional tug is a detrimental effect on the manoeuvrability of the vessel. The addition of a skeg can limit the tug's turning radius and ability to move laterally through the water, or cause directional instability [7]. Because of this handicap, tugs with omnidirectional drives are usually more suited to escort operations. The thrust can be oriented in any direction and there is no rudder to inhibit the flow, resulting in much more efficient manoeuvring. Types Of Escort Tugs The two types of omnidirectional drives in use are horizontal and vertical axis. The hori-zontal axis drives are similar to conventional propellers with the exception that they are driven by a vertical drive shaft about which they can rotate 360 degrees. The vertical axis drives consist of vertically oriented lifting surfaces, or blades, rotating about a central axis. As the blades move, they are continuously oriented to produce a maximum (horizontal) lift-ing force in the desired direction. 2 For escort tugs, the horizontal axis drives are typically found aft of amidships, as shown in Figure 1.2. These are known as Azimuthing Stern Drive (ASD) tugs. The vertical axis drives are typically found forward of amidships, otherwise called a "tractor" configuration, as shown in Figure 1.3. These vessels are known as Voith-Schneider Propelled (VSP) tugs. While VSP tugs usually operate with the drive units forward, during indirect escort opera-tions the tugs run backwards, or skeg-first. As a result, for rough comparisons, VSP and ASD tugs operating indirectly can be considered similar [8]. Figure 1.2: Azimuthing Stern Drive (ASD) Escort Tug Figure 1.3: Voith-Schneider Propelled (VSP) Escort Tug 1.2 Previous Work Since 1990, much work has been done to improve the design of escort tugs. These improve-ments have largely been a result of public pressure and government regulations after the Exxon Valdez oil spill off the coast of Alaska and the Mercantile Marica grounding off the coast of Bergen, Norway in 1989 [1,6]. Hutchison et al. [1], Scalzo [2], Jagannathan et al. [5], and Laible et al. [9] performed an extensive study using model test results from Maritime Research Institute Netherlands (MARIN) and full scale trials to evaluate tug designs and to create a tug - tanker computer simulation. One of the main conclusions of the study was that tractor tugs are generally 3 better suited to escort duty than stern drive tugs. In terms of numerical modelling of per-formance, the study showed little evidence of interaction between the flow around the hull and the flow induced by the propeller. The study also provided a basic outline for the numerical model of this thesis project. The structure and interface for the numerical model are based on that of the relatively simple sailboat model created by Martin and Beck [10]. The model uses the empirical for-mulas developed from the extensive experimental work of Gerritsma et al. on a series of sailboat hulls [ 11 ]. As more escort tug test data becomes available, it can be accommodated in a similar manner. The test data that was used for the skeg comparison in this project was obtained from the experiments of Molyneux and Earle [12]. This data includes tests of the hull with propeller cage and without propeller and skeg. The tug model used for these experiments was also used for this thesis project. Additional experimental data, obtained subsequent to the start of this thesis project from Molyneux [13], contained data from tests of the bare hull. These experiments were performed as part of an extensive development program with Allan [3, 8, 14], and Amundsen [6] whose design work incorporated the experimental findings and calculations of Sturmhofel et al. [4]. The result of this work is the 38 m escort tug AJAX shown in Figure 1.4. The results of the experiments were also used in the development of a numerical tug-tanker simulation developed by Waclawek et al. [15]. The numerical model developed in this thesis project can be incorporated into this numerical simulation to predict dynamic tug performance during escort manoeuvres. Figure 1.4: V S P Escort Tug A J A X A similar simulation has been developed analytically by Birmingham et al. [16] with an ongoing experimental program at the University of Newcastle upon Tyne, England. It is hoped that the results of this program can be incorporated with those of Allan et al. 4 1.3 Current Work This thesis deals mainly with the indirect escort operation of VSP tugs. The main purpose of the study was to find a new skeg configuration which, if retrofitted to an existing escort tug, would enable the tug to achieve higher towline forces and therefore manoeuvre a given tanker more effectively. The effects of other geometric parameters on tug performance were also studied. To accomplish these goals, flow patterns around a model of an existing tug were examined, 3 alternative skeg models were fabricated and tested, and a numerical model was devel-oped. The numerical model also has the capability of predicting performance for hypothetical tugs, enabling it to be used as a design tool. Motivation The desire for higher towline forces stems mainly from the desire to reduce costs and increase safety. A tug that can achieve greater forces by incorporating a different skeg con-figuration can be built smaller and with less power, thus saving construction and operating costs. Additionally, for a given towline force, the required drift angle would be smaller and the towline would pull less in the lateral direction, resulting in a more stable situation. Part of this thesis project is a visual study of the localized flow around a typical escort tug hull. This study was performed to provide insight into the phenomena observed in previous model tests and to lead to a better theoretical understanding of the behaviour of escort tugs in operation. The flow visualization was also used as a guide for the design of 3 alternative skeg config-urations. The intention of the alternative skeg designs was to overcome some of the inefficiencies identified while studying the flow patterns. Model tests were performed to determine how these skegs compare to the existing configuration. As a means of meaningfully comparing the tested configurations, a numerical model was developed. The numerical model was constructed in such a way that it can be used as a design tool to predict the performance of hypothetical tugs, thus reducing design time and cost. Scope The scope of the experimental study is limited to the effects of the alternative skegs on steady state hydrodynamic force. Model testing was limited to the shape and position of the forward appendages on the model of the AJAX in calm water with no propulsion units present. The numerical model was developed as an aid to the experimental analysis. The output is limited to predicted towline force produced by the tug in calm water under steady state con-ditions for the typical range of operating conditions. There is no consideration of tanker position or velocity. 5 Experimental Work A series of experiments was performed on a 1:18 scale model of an existing escort tug with five different skeg configurations. Video recordings of tests were made to identify flow pat-terns and loads and positions of the model were recorded over a range of operational parameters to assess the performance of each configuration. Numerical Model A numerical model was developed to evaluate the experimental results on a standard basis. The model uses preexisting test data and the results from the experimental work described above to determine the effects of separate tug components and their interactions, resulting in a full scale estimate of performance. Additionally, performance of theoretical escort tugs can be calculated, allowing the numer-ical model to be used to predict tug performance in the early stages of design. Document Conventions The different operating modes of the VSP tugs, as well as differing tug and tanker headings, can lead to some confusion with respect to the meaning of certain relative terms, such as forward and aft. For this document, the terms forward and aft are defined with respect to the indirect VSP escorting mode, when the tug is running stern- first. The terms and sym-bols used in this document are defined in the Nomenclature Section and illustrated in Figure A. 1 in Appendix A. 6 EXPERIMENTAL WORK The experiments were performed in two stages: flow visualization and force measurement. The flow visualization study was performed first to determine areas of potential perform-ance improvement and the results were used to design the models used for the force measurement stage. A l l experimental work was performed in the Ice Tank at the Institute for Marine Dynamics (IMD) in St. John's, Newfoundland. A photograph of the model being tested is shown in Figure 2.1. Figure 2.1: Photo of Test Apparatus 2.1 Equipment The equipment used for the experiments consisted of a model tug with five different skeg configurations, a towing tank with carriage, a Planar Motion Mechanism (PMM) to mount the model to the carriage, and a camera mount for the flow visualization tests. Tug Model The tug model is a 1:18 scale model of the 38 m escort tug AJAX designed by Robert Allan Ltd. of Vancouver, BC. The lines are shown in Figure 2.2 and the particulars are presented in Table 2.1. For these tests, no propulsion unit was installed, but a propeller cage was present. Extended bulwarks were also installed to prevent flooding of the hull at large heel angles. Due to the large drift angles and extensive turbulence throughout the tests, no turbulence stimulators were attached. A photo of the model is shown in Figure 2.3 and a detailed description of the model fabrication can be found in Reference [17]. Five different configurations were created by fitting the hull with either one of four sets of skegs, or no skeg. Al l four sets of skegs have N A C A 0012 cross sections, the same depth 7 below the waterline, and characteristic areas of 0.100 m ± 0.002 m. The roots were cut to fit the keel of the tug and the tips were cut square. Photographs of the configurations are shown in Figure 2.4. Hull Only Configuration (HO) The flow visualization experiments included tests of the hull with the propeller cage, but without any skegs. For these tests, the mounting holes for the skegs were sealed with tape. This arrangement was chosen so the flow around the hull could be visualized and compared with that of the hull and skeg to determine the influence of the skeg. The characteristic area of the hull is defined as the lateral projected area, which is 0.387 m . Baseline Configuration (SK) The Baseline skeg was one of three pre-existing skegs built for the model hull. It was chosen as a control for this project because it had existing mounting holes for an endplate and a sufficient amount of existing test data with which to compare new data. The characteristic area of the skeg configuration is defined as the lateral projected area, which is 0.098 m2. Figure 2.2: Lines of the A J A X Hull 8 Model Ship Waterline Length m 2.12 38.2 Wateriine Beam m 0.79 14.2 Depth of Keel m 0.21 3.8 Maximum Depth m 0.38 6.8 Wetted Surface Area m 2.05 651 Displacement kg;t 213 1276 Transverse Metacentric Height m 0.15 2.64 Table 2.1: Particulars of A J A X Tug and Model Figure 2.3: Photo of Experimental Model 9 (c) Twin, Parallel Configuration (TP) (d) Twin, Series Configuration (TS) Figure 2.4: Photographs of Model Configurations 10 End Plate Configuration (EP) The End Plate configuration is the same as the Baseline configuration, with an end plate attached. The end plate has a height which is 2.5 times the thickness of the tip of the skeg, as shown in Figure 2.5. Figure 2.5: End Plate Configuration This arrangement was chosen as an attempt to reduce the amount of span-wise flow which occurs at large heel angles, as indicated in Figure 2.6. Figure 2.6: Span-wise Flow The characteristic area of the skeg configuration is the same as that of the Baseline config-uration, 0.098 m 2 . 11 Twin, Parallel Configuration (TP) The Twin, Parallel configuration consists of two identical skegs attached to the hull such that the longitudinal position of the mid-chord is at the same location as that of the Baseline configuration and spaced laterally such that the outboard surface at the root is tangent to the chine on the hull, as shown in Figure 2.7. The placement results in a gap of approximately 2/3 chord between the centrelines of the skegs. 2/3 c Figure 2.7: Twin, Parallel Configuration This arrangement was chosen as an attempt to reduce the effects of the flow separation which occurs around the chine at high drift angles, as indicated in Figure 2.8. Figure 2.8: Flow Separation Around Chine The characteristic area of the skeg configuration is defined as the sum of the lateral areas of each skeg, which is 0.101 m 2 . 12 Twin, Series Configuration (TS) The Twin, Series configuration consists of two skegs attached along the centreline of the hull such that the leading edge of the forward skeg and the trailing edge of the aft skeg are at the same positions as the leading and trailing edges of the Baseline skeg, respectively, as show in Figure 2.9. The placement results in a gap of approximately 1/3 chord between the skegs. Figure 2.9: Twin, Series Configuration This arrangement was chosen as an attempt to reattach separated flow on the downstream side of the skeg by transferring momentum from the upstream side, similar to a slotted air-plane wing. The separated flow is indicated by the forward pointing tufts in Figure 2.10. Figure 2.10: Flow Separation on Skeg The characteristic area of the skeg configuration is defined as the sum of the lateral areas of each skeg, which is 0.098 m2. 13 Planar Motion Mechanism (PMM) The PMM was used to mount the model to the carriage. It held the model at a set drift angle and restricted surge and sway while allowing heave, pitch, and roll. The towing force of the carriage was applied at the elevation of the tow point to properly model roll behaviour. A tether was attached to the upstream gunwale to prevent excessive heel. A more detailed description of the PMM can be found in Reference [17]. The PMM also housed the instrumentation used to measure the magnitude of the forces in the restricted directions of motion and the displacement of the motions in the remaining degrees of freedom. Instrumentation The load cells used to measure the surge and sway forces were Interface Sealed Superbeam Load Cell models SSB-AJ-100 (100 lbf rating) and SSB-AJ-500 (500 lbf rating), respec-tively. The error for both models is ±0.05% full scale. The data acquisition system consisted of a VMS and Windows NT based system using IOtech DaqBoards, each with 256 channel capability at 100 kHz aggregate. Carriage The carriage traversed the length of the tow tank on two rails. It housed the drive unit, con-trol panel, and data acquisition system for the experiments. It also held the PMM at the required height above the water. For the flow visualization tests, an underwater frame was connected by a vertical post to the aft end of the carriage. Two underwater cameras were mounted to the frame as shown in Figure 2.11. Port Camera Center Camera Figure 2.11: Underwater Camera Position and Orientation Tow Tank The tow tank used for these experiments was the Ice Tank at the IMD. The Ice Tank is 90 m long, 12 m wide, and 3 m deep. It holds a solution which can be used to simulate a sheet of arctic ice. The solution is mostly water and has a density of999.9 kg/m3 and a dynamic vis-cosity of 1.62 x 10"6 m2/s between 3 °C and 6 °C, the temperature range at which testing was conducted. 14 2.2 Method The tests were performed according to IMD test procedures which are summarized below. A more detailed description can be found in Reference [17]. Setup For the flow visualization tests, yarn tufts were taped to the hull and skeg, as illustrated in Figures 2.6,2.8, and 2.10. The tufts were attached to the entire skeg and the forward half of the hull. Figure A.2 in Appendix A shows the size and distribution of the tufts. Before each set of tests, the model was ballasted to provide the proper displacement and transverse stability. The displacement was calculated based on the full scale volumetric dis-placement, Vy-5, and the model scale, Xm, thus accounting for fluid density: It (2.1) where volumetric displacement is a function of mass displacement, A: V = — The transverse metacentric height, GMt, was set by adjusting the ballast laterally until the roll period satisfied the equation of motion for the model: B (2-2) Roll Period = C J(GMm))/\n where C = 0.44 for the tug. Calibration After ballasting and mounting the model to the PMM, and at the start of each test day, the load cells were calibrated. Pull tests were performed by applying known loads simultane-ously in both the surge and sway directions while measuring the surge force, forward sway force, and aft sway force. A calibration matrix was generated to account for crosstalk in the PMM: Surge ni| m 2 V Ypull Sway measured m 3 m 4 Xpull applied where mx A represent the straight line best fits, found using the least squares method. This gives: Surge ni| rrtj -1 X Surge Sway actual m 3 m 4 Sway (2.3) measured for determining the surge and total sway forces. 15 From inspection, the crosstalk affected the forward and aft sway measurements differently. Because known loads were not applied independently to the forward and aft load cells, they could not be calibrated directly. For the case with no applied surge load, the approximation: Fwd Sway a c t u a l « Fwd Sway m e a s u r e d was used. Then a correction factor was added to determine the actual forward sway with non-zero surge loads: Fwd Sway a c t u a l = Fwd Sway m e a s u r e d + m 5 x Surgeac tua l (2.4) where m 5 represents the straight line best fit for the measured forward sway with respect to Xpull. The aft sway force was then found using: Aft Sway a c t u a l - Total Swaya c t u a, - Forward Sway a c t u a l (2.5) Testing Tests were performed over the typical range of operating parameters of an escort tug. The speeds corresponded to nominal full scale speeds of 4, 6, 8, 10, and 12 kn and were scaled using the Froude number: Fr = - M = (2.6) For simplicity in this document, the speeds are referred to by the nominal full scale values. The drift angles ranged from 15 degrees to starboard to 45 degrees to port. The full test matrix is shown in Table B.l in Appendix B. The length of the tow tank permitted the testing of multiple speeds in a single run. For drift angles of 30 degrees or less, one run was performed at 4, 6, and 8 kn speeds, then another run was performed at 10 and 12 kn speeds. For drift angles 35 degrees or more, longer sam-pling times were required and the first run was divided into a run at the 4 kn speed and another at the 6 and 8 kn speeds. An 11 minute minimum cycle time was used for each run, allowing approximately 6 minutes for settling between runs. Extra runs were performed for the flow visualization tests. For these runs, only one speed was used but the drift angle was incremented by one degree every 3 seconds throughout the run. The range of drift angles for these tests enveloped the stall angle of the hull or skeg. To determine the uncertainty of the measured data due to randomness of environmental variables and error in drift angle setting, an additional set of tests was performed. For these tests, the TS configuration model was alternately run at a 15 degree drift angle with 4, 6, and 8 kn speeds and a 30 degree drift angle with 10 and 12 kn speeds. Ten cycles were per-formed to determine the repeatability of the test data. 16 Data Collection For each run, the following data was collected: • Surge force • Forward sway force • Aft sway force • Drift angle • Heel angle • Carriage speed The force data, measured by load cells, was automatically zeroed at the beginning of each run. The data was plotted and average values were obtained from the data acquisition system by manually selecting a time frame for each speed of the run. For proprietary rea-sons, the raw data are not included in this report. A sample of the output from the Data Acquisition System is shown in Figure A.3 in Appendix A. Data Reduction For initial comparison of the different skeg configurations, the measured forces, F, were non-dimensionalized with respect to speed, U, and lateral area, A: C = —^— (2.7) and the longitudinal Centre of Lateral Resistance, CLRx, defined as the location about which there is zero yaw moment, was non-dimensionalized with respect to the distance from the forward end of the waterline: CLRX = f (2.8) This definition is illustrated in Figure A. 1 in Appendix A. Although the forces were non-dimensionalized with respect to speed, variations were found between some coefficients at different speeds. This is thought to be a result of free surface effects and all calculations were performed for each speed to compensate. For each configuration, the error due to misalignment and drift angle measurement was cor-rected by setting the drift angle to zero where zero sway force occurred. Best fit lines were determined for sway force with respect to drift angle, from -15 to +15 degrees, for each speed. The roots of these equations were determined and averaged over all speeds to deter-mine the drift angle offset. Coefficients corresponding to the nominal drift angles {0, 5, 10, •••$max} were then found by third order spline interpolation. The procedure described above was applied to the results from this set of tests as well as results from previous tests of the hull without skeg from Reference [12]. For each speed 17 and drift angle, the force coefficients for the hull without skeg were subtracted from the coefficients for the hull with skegs to determine the coefficients for the skegs only: r _ ( ^ )hull + skeg ~ (C A)hull n Qx C ske g _ A ^skeg The effect of friction on the skeg was then removed by subtracting a coefficient of friction. Because the Reynolds Number varied greatly throughout the tests and there is no way to determine whether the flow was laminar, turbulent, or separated at any given point, the coefficient of friction was approximated by turbulent flow over a flat plate [18]: C / ^ = 2 x £ p (2-10) where the 2 represents the two sides of the skeg and the Reynolds number is based on the chord length of the skeg. The CLRx coefficients for the hull without skeg were subtracted from those of the hull with skeg to determine the change in CLRx. The resulting coefficients are products of the force due to the skegs and the effects of the interaction between the hull and skegs, but not the force due to the hull itself. The coefficients are a measure of the "efficiency" of the skeg configurations, in terms of characteristic area and speed. Hence, a negative coefficient means the force is smaller than it would have been for an unappended hull of equivalent lateral area, not a negative force; larger coefficients at a lower speed means more efficiency at a lower speed, not more force. The coefficients for surge and sway were then redefined in terms of steering and braking directions, relative to the tanker, for more intuitive comparison. Experimental Uncertainty The uncertainty for the experimental measurements was determined from the accuracy of the instrumentation and from the repeatability tests. Although the model was not rebal-lasted and remounted for the repeatability tests, any error due to ballasting will be very small and have very little effect on performance and error due to misaligned mounting is corrected in the calculations, as described above. The accuracy of the instrumentation was determined from the calibration tests discussed in Section 2.2. The root mean square (rms) deviation of the measured forces from the applied forces was determined at the 95% confidence level: - n - | ' ^ i£(*,.-**,)2 (2.11) _i '=i where z = 1.96 for 95% confidence, n is the number of measurements, x is the measured force, and x* is the applied force. This calibration uncertainty was non-dimensionalized using Equation (2.7) and the maximum coefficient uncertainty was obtained for surge and sway at each speed. The maximum calibration uncertainty for longitudinal centre of lateral resistance was calculated through simple error propagation. 18 The uncertainty of the repeatability tests was determined from the sample standard devia-tion of the mean measured values for each speed at the 95% confidence level: -.1/2 Zn , .2 ,•=!<*/-*») ^repeat Z 1 (2.12) where x is the non-dimensionalized measured force and x m is the mean of these values. The total uncertainty for the non-dimensionalized measured forces is given by: ^ total = hleas + Repeat ( 2 J 3 ) for surge, sway, and centre of lateral resistance at each speed. A simple error propagation gives the uncertainty in the non-dimensionalized forces for the skeg without the hull. The uncertainty in the values for the hull without skeg are ignored here since these values are subtracted from all configurations. The uncertainty in towline force is the vector addition of the uncertainties in surge and sway. 2.3 Results The initial comparison of the test results is discussed below. This is a simple comparison which is useful as an initial study but does not take into account certain factors such as pro-pulsion and stability and therefore does not fully represent the performance of the skeg configurations. Additionally, the characteristic areas chosen for the configurations are based on geometry and may not be representative of the effective areas. A more compre-hensive analysis is described in Section 4.1. For the flow visualization tests, the presence of the underwater carriage and yarn tufts made the force measurements unreliable and useful for reference only. Additionally, technical problems with the data acquisition system resulted in no measurement of data for several runs. However, a large amount of qualitative information was obtained from the recorded videotape. This information was used to design the models for the force measurement stage of the project, as discussed in Section 2.1. The results of the force measurement tests for the 4 kn case were omitted because many of the values are too small with respect to the uncertainty and are therefore unreliable. Forces Polar plots of non-dimensional steering and braking forces are shown in Figure 2.12. These plots can be compared in several ways. Of importance to this thesis project are the compar-isons of the maximum steering, braking, and towline (combined steering and braking) forces. The towline force is represented by the vector from the origin to a given point. Plots of non-dimensional towline force for each configuration are shown in Figure 2.13. The dis-crepancy between different speeds is illustrated here. The maximum uncertainty in non-dimensional towline force is ±0.058 with 95% confidence. 19 With respect to producing the largest steering force, regardless of the braking force, it appears the EP configuration has the highest steering coefficient (Cs) for the 12 kn case and the SK configuration has the highest Cs for the 10, 8, and 6 kn cases. With respect to producing the largest braking force, regardless of the steering force, the acquired data is limited with respect to drift angle and the assumption must be made that the tug would not operate above the maximum drift angles presented, 35 degrees at 12 and 10 kn and 40 degrees at 8 and 6 kn. With this assumption, the TS configuration has the highest braking coefficient (Cb) for the 12 and 10 kn cases, the EP configuration has the highest Cb for the 8 kn case and the SK configuration has the highest Cb for the 6 kn case. With respect to producing the largest towline force, the TS configuration has the highest coefficient for the 12 kn case, the EP configuration has the highest coefficient for the 10 kn case, and the SK configuration has the highest coefficient for the 8 and 6 kn cases. Overall, these initial results indicate the SK configuration has the best performance at lower speeds and the EP configuration has the best performance at higher speeds. As mentioned earlier, however, these results do not take into account major factors affecting tug perform-ance. Therefore, the performance of the TP configuration is not as deficient as it appears here. A more comprehensive analysis is described in Section 4.1. Centre of Lateral Resistance The change in the longitudinal centre of lateral resistance (CLRx) is shown in Figure 2.14, where a positive number indicates the CLRx is located farther forward relative to the hull without a skeg. The maximum uncertainty is ±0.017%LW L with 95% confidence It is difficult to determine which configuration offers the best performance, in terms of CLRx position, since the maximum values occur at small drift angles, where it makes little difference. Therefore the configurations have been compared at each speed using the cumu-lative effects of all test points presented. The results of this comparison show that the EP configuration moves the CLRx further forward than the other configurations, in general, for all speeds presented. The changes in CLRx are summarized in Table 2.2 with maximum values shown in bold type. - fr^Vfl^t f ft a* n * apt C u m u l a t i v e C h a n g e iri L o n g i t u d i n a l CLF? (% Lwj j | uonTiguraiion • •* y • , z 6 k n 8 k n 10 kn 12 kn S K 1.188 1.035 0.909 0.803 EP 1.239 1.095 0.962 0.868 TP 1.137 1.012 0.859 0.765 * \ T S -<M • 1.211 1.091 0.956 0.843 Table 2.2: Comparison of Change in Centre of Lateral Resistance Very little variation in vertical centre of lateral resistance (CLRz) from the different con-figurations was found. This small variation is made even more insignificant when propulsion is considered and therefore it is not discussed in this section. The vertical centre of lateral resistance is accounted for in the numerical model discussed in Chapter 3. 20 Towline Coefficients for Skegs - 6 kn Steering F o r c e Coefficient, C s 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 -0.2 H 1 1 1 1 1 1 1 1 1 (a) 6 k n Towline Coefficients for Skegs - 8 kn Steering F o r c e Coefficient, C s 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 -0.2 i 1 1 1 1 1 1 1 1 (b) 8 kn Figure 2.12: Preliminary Comparison of Steering and Braking Coefficients 21 Towline Coefficients for Skegs -10 kn Steering F o r c e Coefficient, C s 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 -0.2 i 1 1 1 1 1 1 1 i c (c) 10 kn Towline Coefficients for Skegs -12 kn Steering F o r c e Coefficient, C s 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 -0.2 i ' 1 1 1 1 1 1 (d) 12 kn Figure 2.12: Preliminary Comparison of Steering and Braking Coefficients (Continued) 22 Towline Coefficients for Baseline Configuration (a) Baseline Configuration Towline Coefficients for End Plate Configuration (b) End Plate Configuration Figure 2.13: Preliminary Comparison of Non-Dimensional Towline Force 23 Towline Coefficients for Twin, Parallel Configuration 2.0 c .2 o (c) Twin, Parallel Configuration Towline Coefficients for Twin, Series Configuration 2.0 (d) Twin, Series Configuration Figure 2.13: Preliminary Comparison of Non-Dimensional Towline Force (Continued) 24 0.25 0.20 0.15 x tt _i o c n —i O 2- 0.10 T> re jjj 0.05 o u. 0.00 Change in CLRx - 6 kn 10 20 30 Drift Angle (Degrees) 40 • Baseline • End Plate •Twin, Parallel -Twin, Series (a) 6 k n x OC _l o re —i 0.25 0.20 0.15 O 2- 0.10 •o re o u. 0.05 0.00 Change in CLRx - 8 kn 10 20 30 Drift Angle (Degrees) 40 -Baseline •End Plate -Twin, Parallel -Twin, Series (b) 8 kn Figure 2.14: Comparison of Change in Longitudinal Centre of Lateral Resistance 25 Change in CLRx -10 kn 0.25 o Li. 0.00 J . 1 1 : r -0 10 20 30 40 Drift Angle (Degrees) Baseline • End Plate A Twin, Parallel 0 Twin, Series (c) 10 kn Change in CLRx -12 kn 0.25 (d) 12 kn Figure 2.14: Comparison of Change in Longitudinal Centre of Lateral Resistance 26 NUMERICAL MODEL The numerical model was developed to produce an estimate of full scale tug performance based on geometry and propulsion parameters. It takes into account hull stability, yawing moment, and interaction between components to provide a basis of comparison for the dif-ferent skeg configurations. The numerical model can also be used by designers as a basic performance prediction pro-gram (PPP). The program can be used to estimate tug performance at the initial design stage. The model consists of a single spreadsheet composed in Microsoft Excel 2000. It was lim-ited to a spreadsheet to allow for easy portability, expansion, and adjustment, at the cost of versatility and precision. The spreadsheet consists of separate input, output, and calculation sections. As a numerical model, experimental data are used directly to calculate full scale performance and the range of output data is equal to that of the input data. As a PPP, experimental data from various sources are indirectly used in semi-empirical formulae and extrapolated to the entire oper-ating range of the tug. 3.1 Input The input section of the spreadsheet consists of one sheet for manual input of geometric, propulsion, and environmental parameters and one sheet which contains the coefficients for the tug. The manual input sheet is shown in Figure 3.1. The parameters are defined in the Nomenclature Section 27 VSP Escort Tug Performance Prediction Model SK Lwl Waterline Length 38.2 125.3 i 1 • m ft T Draft 3.8 12.5 < 1 l» HH9I ft AL Lateral Area 1351 n f If Bwl Waterline Beam If 7 7 14.2 m 46.6 ft A Displacement 1243 1224 * ! • t LT CB Block Coefficient 0.59 Tmax Depth Below WL «| • K I 22.3 ft be Effective Span 12.9 ft c, Chord at Root 10.2 m 33.4 ft X Taper Ratio i l l» AL Lateral Area 345 ft* AR Aspect Ratio A Leading Edge Sweep Angle h Endplate Height <l • 0.0 m 0.0 ft S T / GMt Transverse Stability 8.7 i | • Sponson Rare Angle <l » FB Minimum Freeboard 8.0 * ill » FBr Residual Freeboard (Heeled) 0.0 ft <\ • m 1>max Maximum Heel Angle • Constrain Propeller Position PPx 9.9 32.4 <| ft LP i | Blade Length • USUI 8.9 ft Static Bollard Pull BPmax IJ|^ CL3B 110.2 i ! T TOWING STAPLE TPx Long. Towing Staple Pos. 18.7 ill • ft TPz Vert. Towing Staple Pos. ill • 4.4 rn 14.5 ft I^max Towline Interference Angle 110 • i l - • deg Constrain p Water Density ill 1026 kg/m"3 V Water Viscosity 1.2E-6 i l |» mA2/s Figure 3.1: Numerical Model Input Sheet 2 8 iin iiwi" iiim H ira r~T2"k"rv~l 96.3 124.6 160.6 193.7 35 35 35 35 6 10 15 18 •ni.inunagiai 153.9 138.7 125.6 115.1 Body Plan -30.0 -20.0 -10.0 0.0 10.0 20.0 30.0 rmra 15.0 10.0 5.0 0.0 Z -5.0 -10.0 L -15.0 Figure 3.1: Numerical Model Input Sheet (Continued) 29 Hull Parameters The following parameters are required to define the magnitude and location of the hydro-dynamic force on the hull and the location of the towline force: • Waterline Length • Waterline Beam • Draft • Displacement • Longitudinal Tow Point Location • Vertical Tow Point Location • Towline Interference Angle Additionally, the Underwater Lateral Area, which is calculated by integrating over control points based on the AJAX hull, can be tweaked slightly. This is done by adjusting the hor-izontal spread of the control points. The Block Coefficient is shown for reference Skeg Parameters The following parameters are required to define the magnitude and location of the hydro-dynamic force on the skeg: • Depth (below waterline) • Chord (at root) • Taper Ratio • Sweep Angle (of leading edge) • Endplate Height Additionally, there is an option to remove the skeg entirely. This can be used to design a tug without a skeg or to illustrate the effect of the skeg. The Effective Span, Lateral Area, and Aspect Ratio are shown for reference. Stability Parameters The following parameters are required to define the maximum allowable heel angle for the tug: • Transverse Metacentric Height • Sponson Flare Angle • Minimum Freeboard (upright) • Residual Freeboard (when heeled) Additionally, there is an option to constrain the output values to the calculated Maximum Heel Angle, which is shown for reference. 30 Propulsion Parameters The following parameters are required to define the magnitude and location of the propul-sion force: • Longitudinal Position • Blade Length • Static Bollard Pull Additionally, there is an option to remove the propulsion entirely to simulate unpropelled model experiments. Hull Coefficients The coefficients for the hull are entered as a single table containing coefficients for surge force, sway force, longitudinal centre of lateral resistance, and vertical centre of lateral resistance at drift angles from 0 to 90 degrees, in 5 degree increments. The coefficients were obtained from an average of the experimental values for the bare hull from Reference [13] and include corrections for friction and free surface effects. Friction is corrected by including a friction coefficient, Cf, according to the ITTC '57 Cor-rection Line: Cfhuii = 0.0004 + - (3.1) f ' h u " (log(/?e*)-2)2 where Re* = 0.7ReL W L for the tug's hull [19]. This coefficient is then subtracted from the drag coefficient. The free surface effects, which cause variations in sway coefficients and vertical CLR between different speeds, are corrected using functions of the Froude Number. The normal-izing functions were determined by setting the power of the Froude Number such that the variation of coefficients between different speeds was minimized over the full range of drift angles. The effects of this procedure are illustrated in Figure 3.2. The normalized sway coefficient is defined by: and the normalized vertical CLR position is defined by: CLR-z C L R : ' = - & 2 T + Z <3-3) where z = 0.55 is the vertical geometric centroid of the underwater lateral area of the upright AJAX hull, located at 55% of the hull depth, measured up from the baseline. The vertical CLR is therefore normalized with respect to the centroid rather than the baseline. The non-dimensional vertical centre of lateral resistance, CLRZ, is the horizontal axis about which there is a side force but no resultant moment. It can be determined by equating the 31 1.00 0.80 o 0.60 0.40 0.20 0.00 Cy - Bare Hull 10 20 30 40 50 60 70 Yaw Angle (Degrees) •12 kn - B - 1 0 k n -8kn -6kn 80 90 (a) Calculated Coefficients 2.00 0.00 Normalized Cy - Bare Hull 10 20 30 40 50 60 Yaw Angle (Degrees) 70 80 •12 kn - B - 1 0 k n •8kn -6kn 90 (b) Normalized Coefficients Figure 3.2: Effects of Normalizing Function on Sway Coefficients heeling moment, H M , to the righting moment, R M . The heeling moment is the moment caused by the heeling force or sway force, F y , applied at the tow point, TP Z : HM = F • (TP -CLRZ • Thull) = RM Thus: CLRZ = T p RM z Fy Thutl (3.4) (3.5) 32 where the righting moment is defined by: RM=A • g • GZ (3.6) and the GZ curve is approximated by: GZ=3.02xlO~5(|>2 + 2.60xlO~3<|>e (3.7) for angles of <|>e in degrees. This equation is the least squares best fit polynomial to the right-ing arm values calculated using the FastShip 6.1 Inclined Hydrostatics module. The effective heel angle, <()e, is determined from the height of the effective bow wave above the waterline, h e: •« = atn(^ ) (3.8) Using the assumption that CLRz is located at the geometric centroid of the underwater lat-eral hull area, that location being above the baseline a distance equal to 55% of the distance between the baseline and effective bow wave crest, as illustrated in Figure 3.3, the height of the effective bow wave can be determined: CLRz ' Thull 0.55 hull (3.9) This method results in a interdependence of vertical CLR and effective heel angle, requir-ing an iterative solution. Figure 3.3: Effective Heel Angle 33 Skeg Coefficients The coefficients for the skeg are entered as tables containing coefficients for surge force, sway force, longitudinal centre of lateral resistance, and vertical centre of lateral resistance at drift angles from 0 to 90 degrees, in 5 degree increments. For the numerical model, the coefficients are entered as a set of tables with one table for each speed and configuration. For the performance prediction program, the coefficients are entered in a single table. For simplicity, longitudinal CLR in terms of skeg chord length, CLRX, is assumed identical for all speeds and configurations. It varies with drift angle according to: CLRX = 0.3 + ( 3 . 1 0 ) which is a linear approximation of the test results from Reference [20], as described in Appendix C. CLRz is approximated at 50% of the span. These approximations introduce little error with respect to full scale tug forces. For the numerical model, the surge and sway coefficients used are the ones calculated in Section 2.2. For the performance prediction program, the surge and sway coefficients are derived from lift and drag coefficients, which are calculated from semi-empirical formulae. Modification factors are applied to the lift coefficients to simulate the interference effects of the skeg and hull. Lift Coefficients The lift coefficients are calculated in various ways, depending on the drift angle. Ranges of drift angles are defined by control points, as shown in Figure 3.4. For the first segment, from 0 degrees to just before stall, the lift coefficients, C L , are deter-mined according to Whicker and Fehlner [21]: CL = (dCj\ p = o where (3 is in radians and: 2n-AR. B = 0 2 + cosA ARi (3.11) (3.12) c / 4 . cos4 A + 4 c /4 and for a square tip, the crossflow drag coefficient, C D c , is defined in terms of the taper ratio, X, as: 0.1 + 1.6X (3.13) 34 Lift Curve Control Points and Segments c o "5 £ Q) O u 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 One Before Stall (s Two Before Stall (s-2) Stall Point (s) Modified Stall Point (s') • One After Stall (s+1) 10 20 30 40 50 60 Drift Angle (Degrees) 70 80 90 Figure 3.4: Calculated Lift Curve Control Points The effective aspect ratio, AR e , used here is determined from the aspect ratio, AR, and the end plate height, h e p , according to Hoerner [22]: .2-ARe = 2+1.9(if)-0.5(4f) •AR (3.14) where the first term inside the brackets, 2, represents the mirroring effect of the hull bottom and the aspect ratio, AR, is defined in terms of the span, b, and lateral area, A L : AR 2A, (3.15) Equation (3.11) predicts C L well in the region before stall but is not valid after stall. Stall angle is predicted by setting the empirical equation: CLs = 1.65 cos0-75 P (3.16) equal to Equation (3.11) and solving for p\ Equation (3.16) was developed from inspection of maximum stall angles for various fins, as described in Appendix C. Because the stall angle, and therefore the slope of the lift curve, varies with aspect ratio, a relation between horizontal control point spacing and aspect ratio is required. Equation (3.17) was developed such that the control points around stall are spaced sufficiently close 35 together for higher aspect ratios when the curve is steeper, and sufficiently far apart for lower aspect ratios when the curve is less steep: _5_ Me This concept is described further in Appendix C. The control point before Ps_ j is defined as the nearest whole degree which is at least 0.5 degrees smaller. P,±1 = P s±-7^= (3.17) Before stall, C L is predicted well by Equation (3.11). The portion of the curve immediately after stall, however, is very unpredictable. Therefore, a linear approximation is used from the control point immediately after stall, Ps+i, to the local minimum, Pmjn. The slope of this line is set equal to the slope immediately before stall by setting C L immediately after stall equal to C L immediately before stall: C L , s + i = CL,s-\ (3-18) and extrapolating from Ps to Pmjn. This is described further in Appendix C. The value of the local minimum, Pmjn, has a lift coefficient equal to: Cr m i „ = ihr (3-19) L, min | Q v ' Equation (3.19) was developed from inspection of lift curves for various fins, as described in Appendix C. Due to the uncertainty of the coefficients in the region of stall, and the asymptotic behav-iour of Equation (3.11) for low aspect ratios, the maximum lift coefficient at stall was conservatively modified. Around stall, C L is described by a smooth, second degree polyno-mial which is fit to the two control points immediately before stall and the control point immediately after. The local maximum, Pmax, occurs at a drift angle midway between Pmin and 50 degrees. The lift coefficient for this point is a linear interpolation of the measured data from Refer-ence [20], multiplied by the lift coefficient scaling factor, kL. The scaling factor is defined by the ratio of the calculated lift coefficient at 50 degrees to the measured lift coefficient at 50 degrees: h = r Cl'5° ( 3 2 0 ) Where the calculated lift coefficient, C L 5 0 , is defined by the empirical relation: CL.5O = iy (3-21) which was developed from inspection of lift curves for various fins, as described in Appen-dix C. All lift coefficients between these control points are determined by linear interpolation. 36 The lift coefficients after 50 degrees are defined by scaled coefficients from Reference [20]: CL, fj > 50 = V CL(meas) (3-22) The interaction effects between the hull and skeg are applied as modification factors, K] and K 2 , which were determined by comparison of the calculated lift coefficients and those derived in Section 2.2, as described in Appendix C: Cucorr) = ^ + K l + K 2 ) C L ( c a l c ) (3.23) The effect of the skeg on the lift coefficient of the hull is modelled by: Ki = 3sm2$-£- (3.24) LWL and the effect of the hull on the lift coefficient of the skeg is modelled by: ii-2b tf2 = - l s i n ^ (3.25) Equation (3.24) represents the effect of the skeg chord length on the lift coefficient of the hull. This is a positive effect since the presence of the skeg alters the flow such that a lower pressure region on the suction side of the hull is created. This phenomenon is illustrated in Figure 3.5. The term increases with drift angle as the skeg becomes more exposed to the flow. (c) No Skeg (d) Low Pressure Region with Skeg Figure 3.5: Effect of Skeg Presence on Flow Around Hull 37 Equation (3.25) represents the speculated effect of the bow shape on the lift coefficient of the skeg, where the taper of the bow is generally proportional to the beam. This is a negative effect since the taper of the bow results in an increased local angle of incidence, which causes increased effective drift and heel angles for the skeg. This phenomenon is illustrated in Figure 3.6. (a) Deflection at 30 Degree Drift Angle (b) Deflection at 60 Degree Drift Angle Figure 3.6: Effect of Hull Presence on Flow Around Skeg It is expected that a more pronounced flow pattern would be observed at higher speeds and drift angles than the ones tested. Additionally, the flat bottom of the hull can shadow a por-tion of the skeg when heeled. The magnitude of the term increases with drift angle as the effect becomes more pronounced. Drag Coefficients Below 45 degrees of drift, little variation in drag coefficients was observed for various skeg geometries. Therefore, for simplicity, the measured coefficients from Reference [20] are used directly for this range. For drift angles above 45 degrees, the drag coefficient is determined by: CD - C A 4 5 ( _ ) + ^ ^ f ^ ( P - 4 5 ) (3.26) + ^ C A 9 0 - C ^ 5 ( w e o i ) s i n [ 2 ( p _ 4 5 ) ] . c o s [ 2 ( [ 3 _ 4 5 ) ] where the second term on the right hand side is a linear interpolation between 45 and 90 degrees, and the third term defines a curvature with a maximum at 90 degrees. C D 9 u is determined by the empirical expression: CD 9 0 = 1.16 + 0.02 ARe (3.27) which was developed from inspection of drag curves for various fins, as described in Appendix C. Since the discrepancy between calculated and measured drag coefficients was small, no interference effects were modelled. 38 3.2 Calculations The calculation section of the spreadsheet consists of distinct components which are imple-mented consecutively, as illustrated in Figure 3.7. The first block is the table of coefficients discussed in Section 3.1. Once the geometry is added to this, the hydrodynamic forces can be determined. These forces are combined with the propulsion parameters in static equilib-rium to determine the heel angle. The longitudinal propulsion force is then incrementally perturbed to determine the full range of possible towline forces. GEOMETRY COEFFICIENTS LATERAL PROPULSION LONGITUDINAL PROPULSION HYDRODYNAMIC FORCES STABILITY TOWLINE FORCES Figure 3.7: Composition of Spreadsheet Calculations Assumptions Some general assumptions were made to accommodate the limited scope of this thesis project, as described below. 1) Hull Coefficients and Parameters All hydrodynamic hull coefficients for the program were based on tests of the model AJAX tug. Furthermore, the calculated hull parameters, and in particular the lateral area and sta-bility, are based on characteristics of the AJAX. It is assumed that all tugs of a similar design will have similar coefficients and characteristics. 2) Propulsion Force It is assumed that the flow field around the propeller and cage is entirely in the direction of thrust and therefore the propeller and cage have negligible effect on drag, and tug velocity and heel have negligible effect of propeller force. The assumption is partially justified by the presence of the hull bottom and bottom plate of the cage which help direct the flow in line with the propeller. Since the propeller force is entered as a static bollard pull, it is assumed that the resistance of the hull and cage on the propeller force is accounted for. The interaction between pro-pellers is ignored for simplicity. However, a more accurate representation could be made in the future using the findings of Brandner et al. [23]. 3) Interaction Effects It is assumed that the skeg and propeller are separated enough as to have negligible effects on the flow around each other. Additionally, the effect of the propeller on the flow around the hull is assumed negligible. 39 The error introduced by these assumptions cannot be determined quantitatively without fur-ther testing. Validation of numerical model output with full scale test results would provide a good qualitative indication of this error. Hydrodynamic Forces The hydrodynamic forces for the hull and skeg are calculated individually, then added together to determine the hydrodynamic forces, and the centre of pressure, for the whole tug. The hydrodynamic forces for the skeg include the effects of interaction between the hull and skeg which are incorporated into the coefficients. The hydrodynamic forces for the hull and skeg do not take into account the heel angle of the tug. Although some error is introduced as a result of this omission, insufficient experimental data was collected to enable an accurate correlation between the two factors. The full scale forces for the hull are determined by: Fy,hull = Q • P • & • Aku^C; • Fr°'45 + Cfjhul, (3-28) and: where: Fx.kuU = Q-9-&-A^(Cx + CftX)hull . (3.29) Cfy = C / Sinp Cfx= Cycosp and C f and C y ' are defined by Equation (3.1) and Equation (3.2), respectively. (3.30) The centre of pressure, in metres, for these forces is located at: CPx.kuiiM = LWL-CLRxhull (3.31) and: CPZthuiiW = Thull[(CLRz'-0.55)Fr2X + 0.55] (3.32) where CLR Z ' is defined by Equation (3.3). CP y is assumed to be amidships. The full scale forces for the skeg are determined by: and: Fx,skeg = { \ - p - ^ - ^ ( C ^ C f J s k e g (3.34) where Cf is defined by Equation (2.10). The centre of pressure is located along the chord, c, aft of the leading edge and along the span, b, below the baseline according to the CLR: C^Wm] - c-CLRx,skeg (3.35) 40 and: ^ . ^ [ m ] = b-CLRzskeg (3.36) CP y is assumed to be at the centreline of the skeg. The hydrodynamic forces for the tug are then added and the centre of pressure is deter-mined, with respect to the tow point, through a weighted average: CP ^ rx, tug CP ^ r x, tug F - F + F y, hull y, skeg (3.37) F x,tug ~- P + F x,hull x,skeg (3.38) Fy, hull' CPX, hull  + Fy, skeg CFX, skeg (3.39) F y, tug Fy, hull' CPX, hull  + Fy> skeg CPX, skeg (3.40) F y, '"g Propulsion Forces and Stability The lateral propulsion force is determined through static equilibrium about the vertical axis through the tow point: F CP F = —* (3 41) y,proP Cos§ CEX y ' where CE X is the longitudinal centre of effort of the propulsion force, as shown in Figure 3.8. 41 Heel angle is determined from the righting moment, RM, and transverse stability, GM t , which is approximately constant for the range of heel angles considered [19]: * " t , . g % A . g < 3 ' 4 2 ) For static equilibrium, the righting moment equals the heeling moment, taken about the lon-gitudinal axis through the tow point, as shown in Figure 3.8: RM = HM = Fy< t o t a l • Heeling Arm (3.43) The stability modification factor, kg, represents the additional stability due to the sponson when heeled. This factor effectively scales the righting arm curve. It is defined by the per-centage increase in cross sectional immersed area due to the sponson when heeled 17 degrees: A + A i _ immersed sponson A A*. immersed where 17 degrees of heel is the angle at which the sponson is fully immersed. This defini-tion is illustrated in Figure 3.9. Inclined Waterline Xi, A: Cross Sectional Immersed Area B: Additional Immersed Area Due to Sponson C: Cross Sectional Exposed Area Figure 3.9: Visual Representation of Stability Modification Factor Combining Equations 3.42 and 3.43 provides an expression for heel angle: _ C P ^ - F -CEZ ksGM,Ag which is combined with Equation (3.41): CE, CP 4> Fy,tug-{CPZ- C O S * - z x X cos(|) CE ksGMtAg (3.46) and solved iteratively for (|>. Due to the limitations of the spreadsheet, the resolution of <() is 1 degree. Equation (3.41) is then solved for lateral propulsion force. The longitudinal propulsion force is independent of all hydrodynamic forces. It is incre-mentally perturbed from the negative maximum to the positive maximum to determine the 42 spectrum of possible towline forces. The total propulsion force, which is the vector addition of the lateral and longitudinal propulsion forces, is limited to the maximum static bollard pull. Towline Forces The magnitude and direction of the total towline force is determined through static equilib-rium of the hydrodynamic and propulsion forces about the tow point, as shown in Figure 3.8. This force is then resolved into steering and braking forces, relative to the tanker. Towline forces which occur at excessive heel and towline angles, as defined by the input parameters, are flagged for reference. Uncertainty The uncertainty in the numerical model results used for comparison of configurations was obtained through a perturbation method described in Reference [24]. By making small changes in C x and C y and observing the corresponding change in results for each speed, the partial derivatives were approximated for the error propagation: where (0Cx and mCy are defined in Section 2.2. The partial derivatives for CLRX and CLR Z were negligible. Using this method, the resulting uncertainties in steering and braking force were found for each speed. The vector sum of these two uncertainties is the uncertainty in towline force. The uncertainty in the performance prediction program output was obtained through a com-bination of error propagation and validation with experimental results. Because no tests were performed with the skeg and without the propeller cage, and because the program does not model the propeller cage, the predicted performance was compared to a simulated tug, where the simulation was based on experimental data. No compatible experimental data with propulsion was found, so the simulated tug includes the skeg but no propulsion. The uncertainty in the simulation of the hull was determined through a comparison of the simulation with the bare hull data from Molyneux [13]. The root mean square (rms) devi-ation of steering and braking forces at each speed was determined using Equation (2.11). The uncertainty in the simulation of the skeg was determined using the perturbation method mentioned above, where the uncertainty in skeg coefficients was determined as discussed in Section 2.2. In this case, however, the uncertainty in the coefficients for the bare hull was included. The total uncertainty for the simulation of the hull with skeg was found using: (3.47) (3.48) for steering and braking force at each speed. 43 The uncertainty in the prediction with respect to the simulation was determined using the rms deviation of steering and braking forces at each speed. The total uncertainty in the pre-dicted results was then found using: ^prediction ~ J ^prediction/simulation ^simulation/ experiment (3.49) for steering and braking force at each speed. 3.3 Output The spreadsheet provides 3 types of output: a basic visual representation of the tug geom-etry, polar plots of towline forces, and a summary table with maximum steering force and relative tug position. Visual Representation A visual representation of the tug is presented in two diagrams on the main input sheet of the spreadsheet, as shown in Figure 3.1. The main purpose of these diagrams is to provide immediate feedback for the geometric parameters entered. This enables the user to easily identify any incongruous parameters. The diagrams also help to illustrate the definitions of the geometric parameters. Plots Four polar plots are created to represent the towline forces for the 6, 8,10, and 12 kn cases. The plots consist of the steering force plotted against the braking force at all drift angles and propulsion forces calculated, as shown in Figure 3.10. Circles around the data points indicate towline interference with structures or equipment on deck. Crosses indicate excessive heel. The vector from the origin to a given point represents the towline force. Data points shown in the upper two quadrants are indicative of an impelling (negative braking) force on the tanker. Data points shown in the lower left quadrant are indicative of forces achieved through direct opposition to the towline. Data points in the lower right quadrant are indic-ative of towline forces achieved through indirect (hydrodynamic) operation. Output Summary Often, escort tug performance is defined simply as the maximum achievable steering force at a given speed. To facilitate this definition, a summary of maximum steering forces is tab-ulated on the main input sheet, as shown in Figure 3.1. The table consists of the maximum achievable steering force as well as the resultant drift, heel, and towline angles for each of the 6, 8,10, and 12 kn cases. The angles are presented for reference and are indicators of the relative position and safety of the tug. They are par-ticularly useful when the options for heel and towline angle constraint are unselected. 44 Achievable Towline Force - 6 kn Steering Force (t) 150 200 250 2 150 CQ 200 250 I • Towline Force x Excessive Heel O Excessive Towline Angle (a) 6 kn, 0 < f3 < 40 degrees Achievable Towline Force - 8 kn Steering Force (t) -250 -200 -150 -100 -50 0 50 100 150 200 250 SS 150 03 200 • 250 I • Towline Force x Excessive Heel O Excessive Towline Angle (b) 8 kn, 0 < P < 40 degrees Figure 3.10: Model Output of Towline Force 45 Achievable Towline Force -10 kn Steering Force (t) 250 200 250 I ! • Towline Force x Excessive Heel O Excessive Towline Angle (c) 10 kn, 0 < (3 < 35 degrees Achievable Towline Force -12 kn Steering Force (t) -250 -200 -150 -100 -50 0 50 100 150 200 250 250 |« Towline Force x Excessive Heel O Excessive Towline Angle (d) 12 kn, 0 < p < 35 degrees Figure 3.10: Model Output of Towline Force (Continued) 46 ANALYSIS The results of this thesis project are discussed here as a performance comparison of the model skeg configurations and a sensitivity analysis of the performance prediction program (PPP). The numerical model is used with experimental results to evaluate the skeg perform-ance in terms of towline force and stability and then to validate the PPP. Several tug parameters are then isolated and their influence on predicted performance is discussed. 4.1 Discussion of Experimental Results The numerical model allows the experimental results to be compared on a standard basis by taking into account stability and propulsion. The result is a comparison of allowable towline forces rather than actual coefficients; the latter translating into theoretically unlim-ited forces if the above factors are not considered. The experimental results are used in the numerical model in conjunction with the full scale parameters for the AJAX tug, with 100 tonnes of static bollard pull. The output of the model then represents the performance of the AJAX as if it were fitted with each of these configurations. For comparison, the aspect ratios of the skegs were maintained while the size of each skeg was maximized to produce the greatest possible towline forces without causing excessive heel. Excessive heel is defined here as greater than 19 degrees, which is the angle at which deck edge immersion occurs for the given stability parameters. The tow point was then placed at 90% of the distance from the forward end of the waterline to the farthest forward hydrodynamic centre of pressure. Moving the tow point towards the centre of pressure results in greater achievable towline forces, while keeping the tow point forward of the centre of pressure ensures yaw stability in case of tug engine failure. The output of the numerical model is limited to drift angles of 40 degrees for the 6 kn and 8 kn cases and 35 degrees for the 10 kn and 12 kn cases due to limited experimental data. Because of this limitation, the maximum braking force, which is achieved at higher drift angles, cannot be determined. The experimental results are therefore presented here in terms of maximum steering force and overall towline force. Steering Force The maximum allowable steering force is defined as the maximum steering force which could be achieved by a configuration such that no impelling force (negative braking force) is generated. These steering forces, which are subject to the limiting drift angles stated above, are summarized in Table 4.1 with the maximum values shown in bold type. The maximum uncertainty in these values is +1.5 tonnes with 95% confidence. The Baseline configuration, SK, shows the highest overall steering force while the Twin, Parallel configuration, TP, shows the lowest. However, with respect to the uncertainty, no meaningful difference between the SK, EP, and TS configurations is shown. 47 p- Maximum^ Steer ingj l^ejtor ines) - asm 6 k n », «^ 8 kri'f 10 kn 12 kn SK 59.1 105.2 165.6 199.7 -EP, 57.9 104.5 166.2 198.9 -1% T P 47.9 102.5 138.7 193.5 -10% TS 50.0 101.6 159.0 199.9 -6% Table 4.1: Comparison of Maximum Steering Force The End Plate configuration, EP, and the Twin, Series configuration produce approxi-mately the same maximum steering force as the SK configuration, but have 6% and 14% less lateral area, respectively. Therefore, the EP configuration is a feasible concept and, despite the slightly lower maximum steering force, the TS configuration may also be a fea-sible concept. Although the preliminary experimental results indicate maximum steering coefficients are reached at about a 30 degree drift angle, the numerical model output suggests that for some cases, the steering forces are still increasing at the maximum drift angles presented. There-fore, a definite conclusion cannot be drawn with respect to maximum steering forces without further testing. This idea is further discussed below. Towline Force The towline forces for the configurations are presented as plots of braking force versus steering force for a given speed, as shown in Figures 4.1,4.2,4.3, and 4.4. The lines repre-sent the envelope of towline forces which the theoretical tug could achieve within the range of drift angles specified above. The vector from the origin to any given point within the envelope represents the achievable towline force. Forces shown in the upper two quadrants are indicative of an impelling (negative braking) force on the tanker which is not standard operating practise but is shown here for reference. Forces shown in the lower left quadrant are indicative of forces achieved through hull drag and direct towline pull, as opposed to indirect hydrodynamic force. The abrupt turn at the end of the top portion of the curves in Figures 4.1 and 4.2 illustrate the increasing steering force with respect to drift angle mentioned above. Figures 4.3 and 4.4 have a slightly rounded turn, as well as a sharp hook at the bottom of the curve, indicat-ing a maximum or near maximum steering force with respect to drift angle. The plots show that the SK configuration works the best in all cases except for the 12 kn case where the EP configuration shows a slight advantage. The difference in performance between configurations is more apparent at lower speeds. At higher speeds, however, there is no meaningful difference between the SK, EP, and TS configurations. Overall Performance The low performance of the TP configuration is likely due to its higher aspect ratio. Although high aspect ratio wings typically have a higher maximum lift coefficient, they also tend to stall sooner than lower aspect ratio wings. In this case, the lower stall angle likely had more of an effect than the higher lift coefficient. Additionally, the higher aspect 48 -40 -100 _ -60 0) o -20 o Li. O) 20 c !Z (0 CQ 60 100 -20 Towline Force - 6 kn Steering Force (t) 0 20 40 60 80 100 ^^^^ •SK EP TP TS Figure 4.1: Towline Force Envelope for 6 kn Case, 0 to 40 Degrees Drift Angle Towline Force - 8 kn -20 -100 -60 8 -20 C 12 2 03 20 60 100 140 20 Steering Force (t) 40 60 80 •SK - EP TP 100 TS 120 140 Figure 4.2: Towline Force Envelope for 8 kn Case, 0 to 40 Degrees Drift Angle 4 9 Towline Force -10 kn Steering Force (t) 180 SK EP TP TS Figure 4.3: Towline Force Envelope for 10 kn Case, 0 to 35 Degrees Drift Angle Towline Force -12 kn Steering Force (t) -20 0 20 40 60 80 100 120 140 160 180 200 220 -100 i 1 1 220 SK EP TP TS Figure 4.4: Towline Force Envelope for 12 kn Case, 0 to 35 Degrees Drift Angle 50 ratio resulted in a longer skeg which lowered the vertical centre of pressure, causing more heel and therefore limiting the projected area. One other contributing factor is the longitu-dinal centre of pressure being located farther aft relative to the SK and EP configurations, likely as a result of the shorter chord having a smaller positive effect on the lift at the for-ward end of the hull. The longitudinal position of the centre of pressure is proportional to the lateral propulsion force required to maintain stability, thereby affecting the towline force as described in Section 3.2. One possible solution to these problems is to base the design of the TP configuration on the aspect ratio of the baseline configuration rather than the depth. Another solution would be to place the skegs farther forward. However, this would require them to be placed farther inboard, due to the shape of the hull, and may result in worse performance. The longitudinal centre of pressure for the TS configuration is also located aft of that of the SK configuration, thereby adversely affecting its performance. A smaller gap between the skegs may help to alleviate this problem. The lower performance of the EP configuration at the lower speeds is likely a result of the smaller lateral area mentioned above. At lower speeds, there is less heel and therefore less span-wise flow over the skeg. In this situation, the end plate has less of an effect and the smaller lateral area becomes more important. As mentioned above, however, there is only a slight decrease in performance relative to the SK configuration. 4.2 D i s c u s s i o n of Performance Predict ion Program The performance prediction program can be used as a design tool to calculate the perform-ance of hypothetical escort tugs at the early stages of development. The program requires several basic input parameters relating to geometry, stability, and propulsion. The output consists of plots of towline force for speeds of 6,8,10, and 12 knots, as well as a summary of maximum achievable steering forces and the heel, drift, and tow-line angles at which they occur. To achieve this, experimental data from various sources are indirectly used in semi-empirical formulae and extrapolated to the entire operating range of a full scale tug. The program was validated at the model scale with respect to experimental results. Subject to the limitations of these experiments, the program shows good agreement. The program was also used to perform a sensitivity analysis in order to determine which design parameters have the most impact on tug performance. This information is useful for focusing the design of new tugs as well as for the development of new concepts. Validation The simulation aspect of the program was validated by simulating the experiments with a bare hull and no propulsion, conducted by Molyneux [13], and comparing the results. 51 The performance prediction aspect of the program could not be validated using experimen-tal data directly because no experiments were performed with a skeg that excluded the propeller cage, and the propeller cage cannot be modelled with this program. The predic-tion performance was therefore validated by modelling the AJAX at model scale with calculated skeg coefficients and no propulsion and comparing the results to a simulation using experimentally determined skeg coefficients. The overall uncertainty in predicted performance is then a combination of the prediction uncertainty with respect to simulation and the simulation uncertainty with respect to experiments, plus the experimental uncertainty. No compatible experimental results with propulsion could be obtained for validation. How-ever, the propulsion calculations are relatively straightforward and little discrepancy is anticipated. No experimental results at full scale could be obtained for validation so the cal-culation of scale effects discussed in Section 3.1 must be assumed adequate. Some of the experimental results used for validation are limited to maximum drift angles in the range of 35 to 40 degrees. Therefore, the calculated uncertainties are only valid for this range. Normal escort operation, however, seldom exceeds this range. The validation shows reasonable overall agreement, as illustrated in Figures 4.5, 4.6, 4.7, and 4.8. The discrepancies between the Bare Hull simulation and the experimental values are largely a result of the averaging of hull coefficients over the speed range for the numerical model. These discrepancies are minimized by taking into account free surface effects, as discussed in Section 3.1. The resulting maximum uncertainty in towline force for the sim-ulation is ±14 N with 95% confidence. The discrepancies between the performance prediction of the hull with the SK configura-tion and the simulated values are largely a result of the calculation of one set of skeg coefficients for the whole speed range. As expected, these discrepancies are larger at the extremes of 6 and 12 kn. The resulting maximum uncertainty in towline force for the per-formance prediction of the hull with skeg is ±42 N with 95% confidence. The results of this validation were incorporated in the uncertainty analysis discussed in Sec-tion 3.2. The results of the analysis indicate reliable full scale performance estimates for typical escort operation can be obtained within 15% of the maximum steering force value, with 95% confidence, for small deviations from the AJAX design. Although this error seems fairly large, it is adequate for quick, initial estimates at the early stages of escort tug development [25]. The major contribution to this uncertainty comes from poor prediction of skeg coefficients at the extreme speeds. A module which calculates a separate set of skeg coefficients for each speed would greatly reduce this uncertainty. A separate set of hull coefficients for each speed would further reduce the uncertainty. However, this would require more testing to determine the hull coefficients for large drift angles at high speeds. 52 6 k n U O) c 2 co 100 10 20 Steering Force (N) 30 40 50 60 70 O Bare Hull Experiment • SK Simulation Bare Hull Simulation SK Prediction Figure 4.5: Performance Prediction Validation, 6 kn 8 k n B O Q O : Z 40 g 80 ? 120 2 CQ 160 200 20 Steering Force (N) 40 60 80 -o 100 120 O Bare Hull Experiment • SK Simulation Bare Hull Simulation SK Prediction Figure 4.6: Performance Prediction Validation, 8 kn 53 10 kn Steering Force (N) 160 O Bare Hull Experiment • SK Simulation Bare Hull Simulation - SK Prediction Figure 4.7: Performance Prediction Validation, 10 kn 12 kn Steering Force (N) 0 40 80 120 160 200 240 280 0 -j 1 1 1 1 1 1 1 240 J O Bare Hull Experiment • SK Simulation Bare Hull Simulation - SK Prediction Figure 4.8: Performance Prediction Validation, 12 kn 54 Sensitivity Analysis For reference, the program was used to determine how certain input parameters affected the estimate of performance. A sensitivity analysis was performed whereby these parameters were individually perturbed and the changes in tug performance, relative to the simulated AJAX, were recorded and compared. The parameters were altered by 10% with respect to either relative position or magnitude and the changes in performance are reported as a percentage difference from the original values. All changes were made with the attempt of either increasing steering force or decreasing heel angle. The results for the largest influences on steering force are summa-rized in Table 4.2 and the results for the largest influences on heel angle are summarized in Table 4.3. Longitudinal Skeg Position Change in Steering Force | Static Bollard Pull 5% Skeg Span 3% Skeg Chord 2% Waterline Length 1% End Plate Presence 1% Table 4.2: Largest Influences on Steering Force | Altered Parameter • ChangeHn^Hgef.Angle ,»| Transverse Stability •7% Vertical Tow Point Position -3% Skeg Chord 8% Skeg Span 11% Longitudinal Skeg Position. 12% Table 4.3: Largest Influences on Heel Angle The parameter with the largest influence on performance is the longitudinal skeg position. The skeg was moved forward a distance equal to 10% of the waterline length, resulting in a 6% increase in steering force. However, this change also resulted in a 12% increase in heel angle, which offsets the increase in steering force. The skeg is also typically placed as far forward as possible so it may be very difficult to apply this change practically. A 10% increase in static bollard pull showed a 5% increase in steering force with negligible difference in heel angle. This is a relatively basic design change with a large influence on performance. However, engine power is typically limited by cost constraints which may be difficult to overcome. A 10% increase in skeg span also showed an increase in performance. However, the increase of 3% in steering force is offset by an 11% increase in heel angle. Similar results are shown with a 10% increase in skeg chord which resulted in a 2% increase in steering force, offset by an 8% increase in heel angle. 55 A 10% increase in waterline length showed a marginal increase in steering force. However, this modest increase would not likely justify the cost of the additional length. Another small increase in steering force was achieved through the presence of an end plate on the skeg. Although the influence is relatively small, the cost of implementing the change would also be small. Neither one of these changes had a significant effect on heel angle and the size of the end plate had a negligible effect on both heel angle and steering force. Heel angle was influenced the most by longitudinal skeg position, skeg span, and skeg chord, as mentioned above. However, this influence is detrimental if these parameters are changed to benefit steering force. The parameter with the largest positive influence on heel angle is transverse metacentric height. This parameter was increased 10%, resulting in a 7% decrease in heel angle. How-ever, this parameter is typically maximized with respect to many constraints and further increases can be very difficult to achieve. A 10% decrease in the height of the tow point, with respect to the waterline, showed a 3% decrease in heel angle. However, this parameter may also be very difficult to change due to its dependency on such major design parameters as deck height. Other parameters investigated were draft, waterline beam, longitudinal tow point location, longitudinal propeller position, and propeller blade length. Changes of 10% in the relative position or magnitude of these parameters showed no significant influence on performance. Because of the interrelation of the factors influencing steering force and heel angle, as well as practical design considerations, it is very difficult to isolate individual parameters for design optimization. Instead, designers must alter multiple components simultaneously to achieve optimized results. An example is increasing the skeg span to achieve greater steer-ing force while increasing the transverse stability (possibly by placing ballast in the longer skeg) to reduce the increase in heel angle. This sensitivity analysis provides a starting point for the optimization process and the performance prediction program provides a tool. 56 CONCLUSIONS The results of the analysis for this thesis project are summarized below. Using this infor-mation, recommendations have been made for both future escort tug designs and for improvements in the performance prediction model. 5.1 Escort Tug Design The flow patterns around the forward end of the hull were video taped to determine the effect of a skeg on tug performance. This information was used to design 3 alternative skeg configurations, which were then tested in an attempt to improve the performance of the baseline tug. A numerical model was used to analyze the test results using a standard basis of comparison by including effects of propulsion and maximizing skeg area, subject to sta-bility constraints. A sensitivity analysis was then performed using the numerical model to determine which factors have the greatest influence on tug design. Flow Patterns Around Hull and Skeg The study of flow patterns around the forward end of the hull produced several interesting qualitative observations. The major findings are summarized here: I) The presence of the skeg has a large beneficial impact on the flow around the hull. In particular, the low pressure region on the suction side of the skeg reinforces the low pressure region on the suction side of the hull. II) The presence of the hull has a large adverse impact on the flow around the skeg. In particular, the shape of the hull is such that the flow is deflected along the span of the skeg and, in some cases, even forward with respect to the skeg chord. III) The flow in some areas can be extremely turbulent, even at small drift angles. Accurate computational flow simulation would therefore be extremely difficult to achieve. Performance Comparison of Skeg Configurations Although the geometry of the different configurations varied substantially from the base-line configuration, no meaningful difference in performance was found. However, two of the configurations produced very similar performance results with less lateral area and may therefore be worth further investigation: IV) The Twin, Series configuration produced similar results with 14% less lateral area. This smaller area, however, resulted in lower performance at lower speeds. V) The End Plate configuration, which varies the least from the baseline configuration, produced similar results with 6% less lateral area. This smaller area, however, resulted in lower performance at lower speeds. VI) The Twin, Parallel configuration performed undesirably and should only be considered further if a change in hull shape allows the skegs to be placed further forward. 57 It should be noted here that these experiments were performed over a limited range of drift angles and without propulsion. Sensitivity of Design Parameters The sensitivity analysis performed using the numerical model resulted in the isolation of a few design parameters which most affect the predicted performance: VII) Forward adjustment of the skeg position by 10% of the waterline length resulted in a 6% increase in steering force, but a 12% increase in heel angle. VIII) A 10% increase in static bollard pull resulted in a 5% increase in steering force with negligible change in heel angle. IX) A 10% increase in transverse metacentric height resulted in a 7% decrease in heel angle. X) A 10% decrease in the height of the tow point above the waterline resulted in a 3% decrease in heel angle. Recommendations for Design of Future Escort Tugs The following recommendations are made for future design of escort tugs: XI) Design optimization should be focused on placing the skeg as far forward as possible, while increasing transverse stability and lowering tow point height to compensate for increased heeling moment. XII) The power of the propulsion unit should be maximized, subject to spacial and budgetary constraints. XIII) The lateral area of the skeg should be maximized, subject to stability constraints. XIV) Alternative skeg configurations should be investigated to achieve greater performance. Based on the results of this thesis project, very large changes are required to achieve these results. One suggestion, which has received some attention, is a skeg with moveable control surfaces. Another suggestion is a skeg which can be rotated about the vertical axis. 5.2 Performance Prediction Program A numerical model was developed to predict the performance of hypothetical escort tugs for use as a preliminary design tool. The model consists of a single spreadsheet with mod-ular components and is based on experimental data which can be updated as it becomes available. 58 Accuracy of the Program The output of the model was compared with known data to calculate the uncertainty in the results. The following statements summarize the calculated uncertainty: XV) Reliable full scale performance estimates can be obtained within 15% of the maximum steering force value, with 95% confidence. This uncertainty is adequate for initial estimates at the early stages of development XVI) This uncertainty is valid for small deviations from the AJAX design, for the range in drift angle from 0 to 40 degrees. XVII) The major contribution to this uncertainty comes from poor prediction of skeg coefficients at the 6 and 12 kn speeds. Performance of the Program The model is designed such that it consists of only one file and can be modified by anyone who is familiar with spreadsheets. The program generally performs well but has a few lim-itations. The major features and limitations are summarized here: XVIII) Immediate visual feedback provides indication of any incongruities in geometric parameters. XIX) Immediate summary of maximum steering forces allows easy comparison between scenarios. XX) Towline forces which occur at excessive heel or towline angles are flagged for easy reference. XXI) The file is very large and computation can be slow on slower computers. XXII) Iterative calculations are limited in resolution. XXIII) Some equations are very complex due to consolidation of multiple equations in a single cell. This can make modification of the model difficult. Recommendations for Improvements to the Program Due to time constraints, the scope of the model was limited and many recommendations can be made. Some of the more cost effective suggestions are listed here: XXIV) More test data should be integrated to improve the accuracy of the input coefficients. Additionally, coefficients should be provided for each speed. XXV) More testing should be done to investigate the interaction between the hull, skeg, propeller, and propeller cage. XXVI) The results should be validated further to improve the accuracy and investigate scale effects. XXVII) An attempt should be made to increase computation speed, either by removing some features from the current model or by creating a stand-alone program. 59 N O M E N C L A T U R E Symbol Description A, A L Area, lateral area AR, ARe Skeg aspect ratio, effective aspect ratio P Drift angle in degrees b Skeg span; as subscript denotes braking direction (positive aft relative to tanker) B, B W L Hull beam, hull beam on the waterline BP Static bollard pull c Skeg chord length C, Cx, Cy, Cb, Cs Non-dimensionalized force coefficient/Subscript denotes direc-tion C D Drag coefficient C D c Cross-flow drag coefficient Cf Coefficient of friction C L Lift coefficient CE, CEx, CEz Centre of effort of the propulsion force CLR, CLRx, CLRz Non-dimensionalized centre of lateral resistance CP, CPx, CPz Centre of hydrodynamic pressure A Mass displacement of the hull V, Vm, Vfs Volumetric displacement of the hull, displacement of model, dis-placement of tug 0, (|)e Heel angle, effective heel angle F, Fx, Fy, Fb, Fs Force. Subscript denotes direction FB Freeboard or minimum distance from waterline to top of bul-wark. FBr Residual freeboard or freeboard when heeled. Fr Froude number Table 6.1: Description of Symbols and Variables 60 Symbol Description S Acceleration of gravity, 9.81 m/s GM t Transverse metacentric height GZ Transverse righting arm T| Angle between towline and centreline of tug h e Effective bow wave height h e p End plate height HM Moment causing heel k s Added stability factor due to sponson flare A. Skeg taper ratio A™ Model scale factor A, A c / 4 Sweep angle at leading edge, sweep angle at quarter chord LpMM' W L Length of PMM, Hull length on the waterline m j . . . m 5 Calibration factors v Kinematic viscosity of water PMM, PMMx Planar motion mechanism, distance of PMM from forward end of waterline PP Centroid of propulsion unit p Density of water Re, Re*, Rec Reynolds number, Re based on 70% of L W L , Re based on chord length RM Righting moment s As subscript denotes steering direction (positive starboard rela-tive to tanker) T, Tmax Keel depth below waterline, maximum depth below waterline TP Location of towing staple U, Free stream flow velocity co Uncertainty in value Table 6.1: Description of Symbols and Variables (Continued) 61 Symbol Description Q Angle between towline and centreline of tanker X As subscript denotes longitudinal direction (positive aft) Xpull Known applied force in the surge direction Angle between outboard surface of sponson and the vertical y As subscript denotes lateral direction (positive port) Y M Moment causing yaw Ypull Known applied force in the sway direction z As subscript denotes vertical direction (positive up) Table 6.1: Description of Symbols and Variables (Continued) 62 REFERENCES [I] Hutchison, B.L., D.L. Gray, and S. Jagannathan, "New Insights into Voith Schneider Tractor Tug Capability", Marine Technology, Vol. 30, No. 4, Oct. 1993, pp. 233-242 [2] Scalzo, S.T., "Experience With the Design and Operation of Escort Tugs", RINA Inter-national Conference on Escort Tugs, London, England, 1993 [3] Allan, R.G., "The Evolution of Escort Tug Technology: Fulfilling a Promise", SNAME Annual Meeting, Oct. 2000 [4] Sturmhofel, U., and J-E. Bartels, "Basic Requirements for Safe Escort Vessels - Theo-retical Consideration and Model Measurements", RINA International Conference on Escort Tugs, London, England, 1993 [5] Jagannathan, S., D.L. Gray, T. Mathai, and J. deJong, "Tanker Escort: Requirements, Assessments, and Validation", SNAME Transactions, Vol. 103, 1995, pp.255-293 [6] Amundsen, C.J. "Optimised Escort Tug for Norsk Hydro's Sture Terminal", 16th Inter-national Tug and Salvage Convention, Jersey, Channel Islands, May 2000 [7] Jansen, J., "Damen ASD Tugs - an Ongoing Development", 16th International Tug and Salvage Convention, Jersey, Channel Islands, May 2000 [8] Allan, R.G. "Developments in Escort Tug Technology", 15th International Tug and Salvage Convention, Cape Town, South Africa, Jun. 1998 [9] Laible, D.H., and D.L. Gray, "Escort Tug Computer Simulation - Standards and Advances", 16th International Tug and Salvage Convention, Jersey, Channel Islands, May 2000 [10] Martin, D.E., and R.F. Beck, "PCSAIL, A Velocity Prediction Program for a Home Computer", The 15th Chesapeake Sailing Yacht Symposium, Annapolis, U.S.A., Jan. 2001 [II] Gerritsma, J., R. Onnink, and A. Versluis, "Geometry, Resistance and Stability of the Delft Systematic Yacht Hull Series", International Shipbuilding Progress, No. 328, Dec. 1981 [12] Molyneux, D., and G. Earle, "A Comparison of Forces Generated by a Hull and Three Different Skegs for an Escort Tug Design", Institute for Marine Dynamics, Technical Report TR-2000-08, Feb. 2001 [13] Molyneux, D., Personal communication, Dec. 2002 63 [14] Allan, R.G., J-E. Bartels, and D. Molyneux, "The Development of a New Generation of High-Performance Escort Tug", 16th International Tug and Salvage Convention, Jer-sey, Channel Islands, May 2000 [15] Waclawek, P., and D. Molyneux, "Predicting the Performance of a Tug and Tanker During Escort Operations Using Computer Simulations and Model Tests", SNAME Annual Meeting, Oct. 2000 [16] Birmingham, R., and J. Grimwade, "Tanker Stopping Distances and the Performance of Escort Tugs", Royal Institution of Naval Architects, Technical Paper W238, 1999 [17] Molyneux, W.D. "Model Test Procedure for Escort Tugs at the Institute for Marine Dynamics: Voith-Schneider Propellers", Institute for Marine Dynamics, Technical Report TR-1999-09, Oct. 1999 [18] White F.M., Fluid Mechanics, Third Edition, McGraw-Hill, New York, 1994 [19] Larsson, L., and R. Eliasson, Principles of Yacht Design, Second Edition, McGraw-Hill, Great Britain, 2000 [20] Critzos, C.C., H.H. Heyson, and R.W. Boswinkle, "Aerodynamic Characteristics of NACA 0012 Airfoil Section at Angles of Attack from 0 to 180 Degrees", NACA, Technical Note 3361, Jan. 1955 [21] Whicker, L.F., and L.F. Fehlner, "Free Stream Characteristics of a Family of Low Aspect Ratio, All Movable Control Surfaces for Application to Ship Design", David Taylor Model Basin, Technical Report 933, Revised Dec. 1958 [22] Hoerner, S.F., and H.V. Borst, ed, Fluid Dynamic Lift, Published by Mrs. L.A. Hoerner, Brick Town NJ, 1975 [23] Brandner, P., and M. Renilson, "Interaction Between Two Closely Spaced Azimuth-ing Thrusters", Journal of Ship Research, Vol.42, No. 1, Mar. 1998, pp. 15-32 [24] Holman, J.P., Experimental Methods for Engineers, Seventh Edition, McGraw-Hill, New York, 2001 [25] R.G. Allan, Personal Communication, Nov. 2001 [26] Ostowari, C , and D. Naik, "Post Stall Studies of Untwisted Varying Aspect Ratio Blades with an NACA 4415 Airfoil Section - Part I", Wind Engineering, Vol.8, No.3,1984 [27] Knight, M., and C. Wenzinger, "Wind Tunnel Tests on a Series of Wing Models Through a Large Angle of Attack Range: Part I - Force Tests", NACA, Technical Report 317,1930 64 [28] Zimmerman, C.H., "Characteristics of Clark Y Airfoils of Small Aspect Ratios", NA CA, Technical Report 431,1933 [29] Abbott, I.H., and A.E. von Doenhoff, Theory of Wing Sections, Including a Summary of Airfoil Data, Dover Publications, New York, 1959 [30] Anderson, R.F.; "Determination of the Characteristics of Tapered Wings", NACA, Technical Report 572, 1937 65 APPENDIX A FIGURES 66 Tanker (a) Top View Figure A.1: Illustration of Terms Used 67 Bid Plate Leading Edge Sweep Angle (A) (d) Skeg Figure A .1: Illustration of Terms Used (Continued) 68 (e) Planar Motion Mechanism (PMM) Figure A.1: Illustration of Terms Used (Continued) 1%" 1V2" Figure A.2: Yarn Tuft Distribution 69 (a) Sheet 1 Figure A.3: Sample of Data Acquisition Output 70 4 (b) Sheet 2 Figure A.3: Sample of Data Acquisition Output (Continued) 71 (c) Sheet 3 Figure A.3: Sample of Data Acquisition Output (Continued) 72 APPENDIX B TABLES 7 3 Drift Angle 2 • • -15 0 20 25 30 35 40 45 60 No Skeg • . i * f NS 4 • • • • • • 9 to 30 6 • • • • • • 11 to 32 8 • • • • • • 15 to 19, 30 to 35 10 • • • • • • 16 to 20, 36 to 43 « v 1 2 • • • • • 17 to 21, 38 to 44 ' Baseline SK 4 • • • • • • 9 to 14 6 • • • • • • 9 to 17 8 • • • • • 9 to 17 10 • • • • • 14 to 20 12 • • • • • 16 to 21 (a) Flow Visualization Tests •m : ~ Drift Ancle* ^ I -15 0 15 20 25 30 35 40 50 55 Baseline > S K 4 6 8 -, 10 12 End Plate EP 4 6 8 10 12 Twin, Parallel ' T P • 4 6 8 10 . 12 Twin, Series ' ' T S 4 6 8 10 12 (b) Force Measurement Tests Table B.1: Test Matrix 74 Speed1 — 11 j 11II MM - 15 ' Twin, Series T S 4 x10 6 x10 " 8 x10 10* x10 12 x10 (c) Uncertainty Analysis Tests Table B.1: Test Matrix (Continued) Notes: 1) Nominal full scale speed, in knots 2) Nominal drift angle, in degrees 75 APPENDIX C CALCULATIONS 76 The data used in the following empirical relations was obtained from References [20], [26], [27], and [28] and is shown in Figures C l , C.2, C.3, and CA, respectively. Lift and Drag Curves for NACA 0012 Section 10 20 30 40 50 60 A n g l e of Attack (Degrees) 70 - Lift Curve Drag Curve 80 2.5 90 Figure C .1: Lift and Drag Curve for N A C A 0012 Section from Reference [20] 77 tO -•>' 1 *C ,C 60 -C< CORPUS* Fjgu- . i „r Aspect P i ' . . i>) lh« A r r u d y n a ^ - c ) riC.izats o f Uic 4 4 1 J Mrfoil at " 0 ii % 10* (a) Lift Curves 78 CL C, /. 0 a • J LL Aspect ratio 4 • - e I h — — — _ _ _ „ 1, x - V /> a. /> f A ( S y 7/ f \\ , \ : r * / N \ I i \ \\ ft A YA H T f— i SO* 50° 40° a ¥iaim.t MoscpIaaB kings. A«pecs raHo effect. CIsrVY. Circular Hps. frtacli Chord Figure C.3: Lift and Drag Curves for Clark Y Section from Reference [27] 79 ce, degrees FIOTOE Z—Variations of lilt and drag coefficients with angle of attack. Redan* gal&rUps. 0*yavr Figure C.4: Lift and Drag Curves for Clark Y Section from Reference [28] 80 C.1 Empirical Relations for Predicting Skeg CLR The longitudinal centre of lateral resistance is estimated using a linear approximation of the measured coefficients from Reference [20], as shown in Figure C.5. Chord - Wise Centre of Lateral Resistance "E o £ o & _ l o 0.6 0.5 0.4 0.3 0.2 0.1 0.0 10 20 30 40 50 60 70 Angle of Attack (Degrees) 80 NACA 0012 • Linear Approximation 90 Figure C.5: Linear Approximation of Longitudinal Centre of Lateral Resistance Equation (3.10) is the equation of this line: (3.10) The error introduced by this approximation is negligible with respect to full scale tug force. 81 C.2 Empir ica l Relat ions for Predict ing Skeg Lift Coeff ic ients Maximum Lift Coefficient The maximum lift coefficient and stall angle are predicted using trends observed from the experiments performed for this project and from References [20], [26], [27], and [28]. A plot of maximum lift coefficients for NACA 0012, NACA 4415, and Clark Y wing sections is shown in Figure C.6. Stall Points for Various Wing Sections and Aspect Ratios 2.0 c o o 1.5 4 = 1.0 0.5 0.0 A- -A Increasing AR 5 10 15 20 25 30 35 Angle of Attack (Degrees) Figure C.6: Dependency of Stall Point on Aspect Ratio A curve was fit to each set of data using Equation (3.16): CL s = «• cos°-75p (3.16) where n = 1.65 for the NACA 0012 section. Equating Equation (3.16) and Equation (3.11) gives the predicted stall point. 82 Control Point Spacing Because of the variation in the lift curve slope with respect to aspect ratio, an equation was required to set the spacing of the control points. Equation (3.17) was developed such that the control points around stall are spaced sufficiently close together for higher aspect ratios when the curve is steeper, and sufficiently far apart for lower aspect ratios when the curve is less steep, as illustrated in Figure C.7. _5_ Me P, ± 1 = P ,±-^= (3.17) 2.0 Control Point Spacing at Stall for Various Aspect Ratios AR = 20 AR = 2 5 10 15 20 25 30 Angle of Attack (Degrees) 35 Figure C.7: Control Point Spacing Relative to Lift Curve Slope 83 Curve Slope After Stall The portion of the curve immediately after stall has very unpredictable characteristics, as illustrated in Figures C. 1, C.2, C.3, and C.4. Therefore, a linear approximation is used with the slope of the downward curve set equal to the slope of the upward curve immediately before stall, as shown in Figure C.8. This is done by setting the lift coefficient of the control point immediately after stall equal to that of the control point immediately before stall: CL.,+ I = CL,S-I (3-18) Although this over-predicts the lift coefficients immediately after stall, the error tends to be smaller for the lower aspect ratio skegs typically considered for this application. Linear Approximation After Stall o n Stall Point -1 o 1.5 -/ 1 \ Q> // \ \ o E Q> 1.0 Lift C< 0.5 -/ / Local Minimum i i i \ \ 0 10 20 30 40 Angle of Attack (Degrees) Figure C.8: Linear Approximation of Lift Curve After Stall Local Minimum The value of the local minimum after stall is determined by a fraction of the maximum lift coefficient. The ratio of maximum lift coefficient to the local minimum coefficient appears to be consistent for a given wing section of aspect ratio greater than 1, as illustrated in Fig-ures C.2, C.3, and C.4. The data from Reference [20] was used to determine a ratio of 1.9 for the NACA 0012 section, which gives Equation (3.19): CL,min=JJ (3-19) This point is shown in Figure C.8. 84 Stall Point Modification Overall, the portion of the calculated lift curve around stall shows close agreement with measured data. However, no data could be found for comparison of low aspect ratio NACA 0012 sections at the stall point and beyond. Because of this unknown, and the asymptotic behaviour of Equation (3.11) for low aspect ratios, the maximum lift coefficient at stall was conservatively modified. Around stall, CL is described by a smooth, second degree polyno-mial which is fit to the two control points immediately before stall and the control point immediately after. This curve is shown in Figure 3.4. Lift Coefficient Scaling Factor A local maximum, or second stall, occurs after the local minimum, as shown in Figures C. 1, C.2, C.3, and C.4. From inspection of these figures, this maximum occurs approximately mid-way between the local minimum and 50 degrees. The coefficients from this second stall point to 90 degrees are determined by a fraction of the corresponding coefficients from Reference [20]. This fraction is referred to as the lift coefficient scaling factor, kL, and is defined by the ratio of the calculated lift coefficient at 50 degrees to the measured lift coefficient at 50 degrees: h = r Cl'5° <3-2°) U Z , , 50(meas) The calculated lift coefficient at 50 degrees, C L 5 0 , is defined in the same manner as the local minimum lift coefficient: CL, so = % (3-21) where 1.5 is the average ratio of lift coefficient at stall to lift coefficient at 50 degrees for the data shown in Figures C l , C.2, C.3, and C.4. These ratios are shown in Figure C.9. Ratio of Maximum Lift Coefficient to Lift Coefficient at 50 Degrees 2.0 -r o LO _r 1.5 • u X 1.0 (0 E 0.5 -I o 0.0 -0 oo _ 9j ° 8 H TT" 10 20 A s p e c t Ratio 30 40 A NACA 0012 • NACA 4415 O Clark Y •Average Figure C.9: Lift Coefficient at 50 Degrees with Respect to Lift Coefficient at Stall 85 Calculated Lift Curve The calculated lift curve for an infinite aspect ratio is shown in Figure C.10. Lift and Drag Curves for NACA 0012 Section 1.8 -i 1.6 -Angle of Attack (Degrees) — Measured Calculated Figure C.10: Calculated Lift Curve For N A C A 0012 Section Interaction Effects The effects of the interaction of the hull and skeg are included in the skeg coefficients in the form of the modification terms Kj and K 2 : CUcorr) = ^ + K x + K 2 ) C L ( c a l c ) (3.23) Ki is the effect of the skeg on the lift coefficient of the hull, modelled by: Kx = 3sm2^-^- (3.24) From observation of the flow visualization data, it seems that the skeg has a noticeable effect on the flow around the hull in the region of the skeg only. Therefore the effect of the skeg on the lift coefficient of the hull is defined in terms of the length of the skeg, c, with respect to the length of the hull, L W L - This effect is small at low drift angles and would have a maximum at 90 degrees. Therefore, a sine function is used. The exponent, 2, is used to moderate the effect at lower drift angles to suit the measured data. The scaling factor, 3, is used to adjust the lift curve to suit the measured data near the stall angle. K 2 is the effect of the hull on the lift coefficient of the skeg, modelled by: K2 = -lsin4P^- (3.25) 86 From observation of the flow visualization data, it seems that the hull has a more noticeable effect on the flow around the skeg closer to the root of the skeg. This effect is likely due to the flat bottom of the hull shadowing the skeg. Therefore the effect of the hull on the lift coefficient of the skeg is defined in terms of the width of the hull, B, with respect to the depth of the skeg, b. This effect is small at low heel angles and larger at higher heel angles. Additionally, for a given heel angle, the effect is small at low drift angles and would have a maximum at 90 degrees. Therefore, a sine function is used. The exponent, 4, is used to moderate the effects at lower drift angles, which correspond to lower heel angles, and to enhance the effects at higher drift angles, which correspond to higher heel angles. The scal-ing factor, -1, is used to adjust the lift curve to suit the measured data after stall. The effects of modification terms K] and K 2 are shown in Figure C l 1. The experimental values shown for Skeg A, Skeg B, and Skeg EP were obtained from this thesis project and References [12] and [13]. While the calculated lift coefficients do not agree very well with the experimental values past 50 degrees, they have little influence on the overall steering force at this point. Additionally, during escort manoeuvres at higher drift angles, steering force is much less important than braking force. Furthermore, the experimental values were obtained using combinations of the sources mentioned above and should therefore be used as a guideline only. For drift angles below 50 degrees, the modification terms appear to work well. Corrected Lift Coefficient Curve for Skeg B 10 20 30 40 50 60 Yaw Angle (Deg) 70 80 A Skeg B Data Correction Function - Uncorrected Curve -Corrected Curve 90 (a) Skeg B Figure C.11: Effects of Modification Terms on Calculated Lift Curve 87 Corrected Lift Coefficient Curve for Skeg A 1.6 Y a w A n g l e (Deg) • Skeg A Data - - Uncorrected Curve — Correction Function - ~ (b) S k e g A Corrected Lift Coefficient Curve for EP Skeg 1.6 Y a w A n g l e (Deg) A Skeg EP Data - - Uncorrected Curve Correction Function — (c) EP Configuration Figure C.11: Effects of Modification Terms on Calculated Lift Curve 88 C .3 Empir ica l Relat ions for Predict ing Skeg Drag Coeff ic ients From inspection of Figures C l , C.2, C.3, and C.4, the drag coefficient curve seems con-sistent for all aspect ratios below 30 degrees of attack angle. Above this angle of attack, the drag coefficient varies consistently with respect to aspect ratio. A comparison of the measured data from this project with Reference [20] showed similar drag coefficients until 45 degrees so, for simplicity, the data from Reference [20] was used directly from zero to 45 degrees. Above 45 degrees, the drag coefficients were approximated using a linear interpolation between 45 and 90 degrees, with a curvature added such that a maximum drag coefficient occurs at 90 degrees: Co - C O . 4 5 , ^ ^ C A 9 0 9"0CD445S""'" l(|i-45) (3.26) ^"'^-As"'^' s i °[ . 2 <P- 4 5 )] ' c°s[2(P-45)] This maximum drag coefficient at 90 degrees is determined with respect to aspect ratio: CD 90 = 1.16 + 0.02-ARe (3.27) Equation (3.27) was derived from a trend found in the data shown in Figures C. 1, C.2, C.3, and C.4. This trend is illustrated in Figure C.12. Maximum Drag Coefficient for Various Wing Sections and Aspect Ratios 0.5 -0.0 \ 1 1 : 0 10 20 30 40 Aspect Ratio Figure C.12: Dependency of Maximum Drag Coefficient on Aspect Ratio 89 

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