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Yaw Effects on Model-Scale Ship Flows

J.Longo, F.Stern (University of Iowa, USA)

ABSTRACT

Yaw effects on model-scale ship flow is documented through towing-tank experiments for a 3.048 m Series 60 CB=0.6 model ship. The data includes: photographs and video; resistance, side force, and yaw moment; sinkage, trim, and heel angle; wave profiles along the hull and wave elevations; and mean-velocity and pressure fields for numerous crossplanes from the bow to the near wake. Detailed descriptions are provided of the experimental equipment, procedures, and uncertainty analysis. Comparison of results for low (0.16) and high (0.316) Froude number with those from an earlier study for the without-yaw condition enables identification of the salient yaw-and wave-induced effects. The forces, yaw moment, and displacements increase significantly with increasing yaw angle. The wave pattern is asymmetric with increased/decreased Kelvin angle, wave lengths, and amplitudes on the windward/leeward sides, respectively. Close to the hull, the differences are confined to the bow, whereas away from the hull the differences are throughout the measurement region, i.e., nearfield. The mean-velocity and pressure fields are dominated by strong crossflow effects, including forebody and afterbody keel and bilge vortices. The results should be useful for computational fluid dynamics validation and are available along with the geometry and conditions.

NOMENCLATURE

αK, β, η

Kelvin (wave envelope), yaw, heel angle

AP, FP

After and forward perpendicular

B

Ship model beam

CB

Block coefficient [=∇/(LPPBT)]

CM

Moment coefficient

CS

Sideforce coefficient

CT

Total-resistance coefficient

δ

Boundary-layer thickness

dFP, m, AP

Displacements at FP, midship, and AP

Fr

Froude number

Fx, y

Resistance and sideforce, respectively

g

Gravitational constant

H

Total head

LPP

Model length between perpendiculars

λt, λd

Transverse and diverging wavelengths

Mz

Yaw moment

v

Kinematic viscosity

p

Mean pressure

θ

Diverging-wave angle

ρ

Density

Re

Reynolds number (=UoLPP/v)

S

Hull surface area

σ, τ

Sinkage and trim, respectively

T

Ship model draft

u, v, w

Mean velocities in ship coordinate system

Uo

Carriage speed

ωx

Axial vorticity

ζ, ζx

Wave elevation and axial wave slope

Ship model displacement

INTRODUCTION

Ship-flow patterns depend on hull geometry and operating conditions such as speed, maneuvering, and ambient seas. Ships that operate in the straight-ahead condition in calm water generate symmetric flow patterns with respect to the hull centerplane. This is the design condition. For slender, medium-speed ships, these patterns have limited wave breaking and wave- and body-induced vortices and are well documented, but only partially documented for other hull forms that have some of the above features. Furthermore, high-speed and full-form ships in the design condition can exhibit features such as wave breaking and wave- and body-induced vortices that are observed in what are called off-design conditions for a ship. Other examples of off-design features that occur for design conditions include flow over an asymmetric stern, a yacht that requires a sideforce to counteract the load on the sails, or the straight-ahead motion of a SWATH or catamaran where



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Twenty-First Symposium on NAVAL HYDRODYNAMICS Yaw Effects on Model-Scale Ship Flows J.Longo, F.Stern (University of Iowa, USA) ABSTRACT Yaw effects on model-scale ship flow is documented through towing-tank experiments for a 3.048 m Series 60 CB=0.6 model ship. The data includes: photographs and video; resistance, side force, and yaw moment; sinkage, trim, and heel angle; wave profiles along the hull and wave elevations; and mean-velocity and pressure fields for numerous crossplanes from the bow to the near wake. Detailed descriptions are provided of the experimental equipment, procedures, and uncertainty analysis. Comparison of results for low (0.16) and high (0.316) Froude number with those from an earlier study for the without-yaw condition enables identification of the salient yaw-and wave-induced effects. The forces, yaw moment, and displacements increase significantly with increasing yaw angle. The wave pattern is asymmetric with increased/decreased Kelvin angle, wave lengths, and amplitudes on the windward/leeward sides, respectively. Close to the hull, the differences are confined to the bow, whereas away from the hull the differences are throughout the measurement region, i.e., nearfield. The mean-velocity and pressure fields are dominated by strong crossflow effects, including forebody and afterbody keel and bilge vortices. The results should be useful for computational fluid dynamics validation and are available along with the geometry and conditions. NOMENCLATURE αK, β, η Kelvin (wave envelope), yaw, heel angle AP, FP After and forward perpendicular B Ship model beam CB Block coefficient [=∇/(LPPBT)] CM Moment coefficient CS Sideforce coefficient CT Total-resistance coefficient δ Boundary-layer thickness dFP, m, AP Displacements at FP, midship, and AP Fr Froude number Fx, y Resistance and sideforce, respectively g Gravitational constant H Total head LPP Model length between perpendiculars λt, λd Transverse and diverging wavelengths Mz Yaw moment v Kinematic viscosity p Mean pressure θ Diverging-wave angle ρ Density Re Reynolds number (=UoLPP/v) S Hull surface area σ, τ Sinkage and trim, respectively T Ship model draft u, v, w Mean velocities in ship coordinate system Uo Carriage speed ωx Axial vorticity ζ, ζx Wave elevation and axial wave slope ∇ Ship model displacement INTRODUCTION Ship-flow patterns depend on hull geometry and operating conditions such as speed, maneuvering, and ambient seas. Ships that operate in the straight-ahead condition in calm water generate symmetric flow patterns with respect to the hull centerplane. This is the design condition. For slender, medium-speed ships, these patterns have limited wave breaking and wave- and body-induced vortices and are well documented, but only partially documented for other hull forms that have some of the above features. Furthermore, high-speed and full-form ships in the design condition can exhibit features such as wave breaking and wave- and body-induced vortices that are observed in what are called off-design conditions for a ship. Other examples of off-design features that occur for design conditions include flow over an asymmetric stern, a yacht that requires a sideforce to counteract the load on the sails, or the straight-ahead motion of a SWATH or catamaran where

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Twenty-First Symposium on NAVAL HYDRODYNAMICS interaction of the hulls creates lifting effects. There are limited data that document off-design characteristics, and lack of understanding as to the underlying physics of the flow features. Off-design conditions frequently occur such as when ships maneuver or advance through a cross current. These situations produce the off-design features defined above and spray and bubble entrainment which are also not well documented. A summary of documented surface-ship studies is provided in (1), including an evaluation of their usefulness for computational fluid dynamics validation. Herein, the straight-ahead condition is defined as the zero-yaw, i.e., β=0° condition. The other cases, such as maneuvering, cross current, or lift cases (yacht, SWATH, or catamaran), are defined as yaw or β≠0° conditions. It is shown that even for a slender ship, the yaw condition displays many flow features that are common in off-design ship flows. Admittedly, the yaw case is an approximation to the off-design condition, i.e., nominal-wake measurements in the propeller plane of a high-speed FF 1052 combatant (2) display significant differences between zero-yaw tests in a straight basin, yaw tests in a straight basin, and yaw tests in a maneuvering basin (rotating arm). Some of these differences, i.e., axial-velocity contours, were unexpected. Nonetheless, the yaw case has many features of interest that are shared by both design and off-design conditions, it can be performed in a towing tank, and it builds on previous experimental work. This paper is concerned with documentation of the yaw case. The goals for the present work are to (i) explore and identify the important flow features associated with a yawed body and (ii) document the flow features in sufficient detail for validation of RANS CFD codes. The emphasis is on ‘explore' because the range of yaw-induced changes to the flow field was unknown. The yaw project is part of an Iowa Institute of Hydraulic Research (IIHR) study concerning free-surface effects on boundary-layer and wake flow. The research and discussions of wave-boundary layer and wake interactions herein, rely upon and compliment previous fundamental studies with flat plates, (3)–(6) and previous work at the IIHR for the 3.048 m Series 60 CB=0.6 in the zero-yaw condition, (7) and (8), which is precursory for the present study. Also of relevance is the recent Series 60 CB=0.6 bow study (9). The yaw study is related to the basic topics of three-dimensional separation (10), wave breaking (11), and vortex-free surface interaction (12) each of which represents a major field of study. Other related work with application to ship flows are provided in (13). In this study, model-scale experiments are performed in a towing tank with a 3.048 m Series 60 CB=0.6 ship hull. Visualization of the flow is performed with photographs and video for a range of Froude number (Fr) and β. Resistance, sideforce, yaw moment, sinkage, trim, and heel angle are measured for a range of Fr and β. Additionally, wave profiles along the hull and wave elevations are measured for a range of yaw angles and high (0.316) and low (0.16) Fr. For β=10°, detailed mean-velocity and pressure fields are measured at ten axial stations for Fr=0.16 and 0.316 and at several stations in the wake of a breaking wave for Fr=0.316. Finally, and in accordance with established standards and guidelines, an uncertainty analysis is performed for each experiment. The organization of this paper is as follows. First, a brief description of the experimental facilities and equipment, procedures and conditions, and uncertainty analysis is given. Then, the results are presented with regard to the essential yaw- and wave-induced effects observed in the experiments. This paper is based on Ph.D. thesis research and the results are extensive. Many discussions are abbreviated, however, the complete results are provided in (13). Finally, conclusions from this study are given with recommendations for future work. OVERVIEW OF THE EXPERIMENTS Facilities and equipment A cartesian-coordinate system (Figures 1 and 2) is used that is fixed to the model. The x-axis is coincident with the model centerline, and the y- and z-axes are directed toward the starboard side of the model and upward, respectively, with the origin at the intersection of the waterplane and the FP. Yaw angle is the angle between the model centerline and the tank centerline. For the orientation shown in Figures 1 and 2, the port side is the windward side and the starboard side is the leeward side. Alternately, if the yaw condition approximates a turning ship, the port and starboard sides correspond to the outboard and inboard sides of the turn, respectively. Towing tank The IIHR towing tank is 100 m long and 3.05 m wide and deep. Wave dampeners near the free surface on the sidewalls allow for twelve-minute intervals between carriage runs. The carriage is cable driven by a 15-horsepower motor. On board the carriage, a cabinet holds the computer and other instrumentation. Ship model and geometry The ship model for the experiments is a 1:40 scale 3.048 m Series 60 CB=0.6 (Figure 1). The Series 60 CB=0.6 is a single-propeller merchant-type ship, a standard for ship-hydrodynamic research, and in particular, was chosen with three other hull types as a representative hull form for the CEP (14). The lines of

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Twenty-First Symposium on NAVAL HYDRODYNAMICS the present model conform to the standard offsets. The stern geometry is based on the original methodical series (15). The model is constructed of fiber-reinforced Plexiglas and epoxy resin. It is equipped with a stern tube aperture and propeller hub, however, all experiments are for the without-propeller condition. A set of grids is applied to the model for the wave-profile measurements and replaced with station lines from the FP to the AP every 0.1× Lpp for the mean-flow measurements. To initiate transition to turbulent flow, a row of cylindrical studs of 1.6 mm height and 3.2 mm diameter are fixed with 9.5 mm spacing on the model at x=0.05. The size and spacing of the studs is in accordance with standard practices, however, for the yaw cases, there is some question about the effectiveness of the port-side studs because the boundary layer (δ) is very thin on that side. Instrumentation A three-channel (two force and one moment) load cell is fixed at the midpoint of the model to measure the total resistance, sideforce, and yaw moment. The towing height is above the waterline, z/T=0.24. Dedicated signal conditioners are used at each channel for filtering and amplifying the output from the load cell. Sinkage, trim, and heel angle are determined from measurements of the linear displacements at the FP, midship, and AP with rectilinear potentiometers. The range of each potentiometer is 10 cm. Global-wave elevations are measured with two capacitance-wire probes that have digital interfaces developed at the IIHR for high-resolution, low-noise, wave-elevation measurements (16). The local-wave elevations are measured with four manual point gauges. These devices have a measurement range of about 20 cm. Mean-velocity and pressure measurements are taken with two five-hole pitot probes and a static-pressure tube. Each probe orifice is ported via vinyl tubing to a differential pressure transducer with a 0.32 psi range. The transducers are electrically connected to individual signal conditioners equipped with 1-Hz low-pass filters. A Modulynx precision-positioning system is used in the local-wave and mean-velocity and pressure measurements for movement of the traverses in the yz-plane. The Modulynx communicates through the computer serial port by a RS-232 cable. Data-acquisition systems Data is sampled with an IBM PC-XT compatible microcomputer and a multi-channel analog-digital translation board. An eight-channel board is used for the forces, yaw moment, displacements, and global-wave elevation tests. A sixteen-channel board is used for the mean-velocity and pressure measurements. All data is sampled through a low-pass filter in order to remove the effect of carriage vibration noise. Calibration procedures The load cell is statically calibrated with weights and a moment arm. The potentiometers are calibrated by placing the unit horizontal and extracting the core to its midpoint. Then, the core is moved in increments throughout the measurement range using a ruler as a reference for the changes in position. The capacitance-wire probes are statically calibrated. Initially, the probes are submerged at their midpoints and then manually moved up and down throughout the measurement range. The ten differential pressure transducers are statically calibrated with two water tanks. One is moved up and down with an automated traverse while the other is fixed. The calibration setup and procedures for the five-hole probes are very similar to those used in (7). The details of the calibrations are reported in (13). Procedures and conditions Three types of measurements are made: (i) forces, yaw moment, and displacements, (ii) wave profiles and elevations, and (iii) mean-velocity and pressure fields. All tests are done with the static model at its design waterline. For the forces, yaw moment, and displacements, the model is in the free condition, i.e., it is allowed to heave, pitch, and roll. For all other measurements, the model is fixed to the carriage, i.e., restrained from moving in any direction relative to the carriage. A summary of the type (ii) and (iii) experiments is shown in Figure 3. Forces, yaw moment, and displacements Resistance, sideforce, yaw moment, sinkage, trim, and heel angle are measured for five yaw angles, β=0°, 2.5°, 5°, 7.5°, and 10° and a range of Fr, Fr=0.1– 0.35 (Uo=0.55–1.91 m/s). The carriage is started and after the initial acceleration and necessary delay for steady-flow conditions, data are acquired on four channels (speed, total resistance, sideforce, and yaw moment; or speed, FP, midship, and AP displacements) at a rate of 100 Hz and 200 samples per channel. The total sampling period is eight seconds per carriage run. Wave profiles Wave profiles on the hull are recorded for two Fr, Fr=0.16 and 0.316 (0.87 and 1.73 m/s, respectively) at three yaw angles, β=0°, 5°, and 10°. For each combination of Fr and yaw angle, twenty-nine measurements are made on port and starboard sides of the model except for β=0° where symmetry of the flow allows measurements on one side, only. The measurements are made by marking the top of the wave profile at every axial station with a fine-tipped wax pencil. The position of the mark is converted to a wave height with the grid on the hull surface. The wave

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Twenty-First Symposium on NAVAL HYDRODYNAMICS profiles are recorded after the carriage achieves a steady speed and the wave pattern is steady state. Wave elevations Global wave elevations are measured for high and low Fr and two yaw angles, β=5° and 10°. Wave-elevation data are taken with a capacitance-wire system situated 55 m upstream of the carriage and model. A capacitance-wire probe on the port and starboard side are fixed to wall-mounted traverses and electrically connected to an IBM-compatible computer via a digital interface developed for processing wave-height data. The carriage attains a steady speed 25 m prior to reaching the traverses, and after contacting a switch that is fixed to the tank wall, the sampling period is initiated. A sampling frequency of 100 Hz over 8 seconds is used for each wave cut. The initial wave cuts are taken very near the model (within 2 cm) and subsequent cuts are taken at 5 cm intervals in the ±y-direction up to the tank sidewalls. In the x-direction, data is taken 1 m upstream of the FP to 1 ship length and 3 ship lengths downstream of the AP for low and high Fr, respectively. Wave reflection from the tank sidewalls limits the axial amount of useful data in the wake. Roughly sixteen cuts on each side of the model are recorded at every Fr and yaw angle. Local regions at the bow, stern, and near wake on port and starboard sides of the model could not be measured with the wire probes. These areas are measured with a set of point gauges. The gauges are fixed to the automated traverses and positioned relative to the model. After reaching a steady speed and allowing the wave field to reach steady state, the gauges are adjusted to rest on the free surface. In areas of high free-surface turbulence such as the near wake and at the crest of the port bow wave for Fr=0.316 and β=10°, the probes are adjusted to a level of average wave elevation1. Several runs are made at each x-station in order to measure the gaps between the model and the global wave field. Mean-velocity and pressure field Mean-velocity and pressure fields are measured for high and low Fr (Re=5.3e06 and 2.7e06, respectively) at one yaw angle, β=10° and on both the port and starboard sides of the model. Before every carriage run, a ‘zero-point' file is taken for the zero-velocity condition, i.e., the voltage output from the speed circuit and ten pressure transducers is measured while the carriage is at rest, thus, creating a speed and pressure reference for the pending measurements. The carriage is started and the data acquisition is manually initiated at a designated point at which the model speed is steady and the wave pattern is steady state. Two or four measurements per probe are taken per carriage run for high and low Fr, respectively. A delay of 3.5–4 seconds is used between each measurement position to allow for probe movements and transducer responsiveness. Following each carriage run, the velocity components, pressure coefficients, and total head are plotted. Data is acquired at a rate of 750 Hz and 600 samples are collected per channel giving a sampling period of about 0.9 second per channel. At every station and high and low Fr, approximately 800–1500 points are measured in the yz-plane. The yz-measurement positions are determined from the zero-yaw experiment (7), however, several additional points are measured further from the model where the effects of yaw angle are evident but absent in the zero-yaw experiment. Uncertainty analysis The uncertainty analysis is carried out according to the standards and guidelines of the American Society of Mechanical Engineers ( 17). The detailed methodology and procedures are provided in (18). The error sources are divided into three categories: calibration, data acquisition, and data reduction. The measurement systems are those for the model geometry, data locations, carriage speed, and the various (loadcell, marker/grid, capacitance wire, five-hole probe) data-acquisition systems. Each measurement system has associated variables that are measured and contain bias (fixed) and precision (random) errors. The bias errors for the measurement systems are estimated and the precision errors are determined through a program of repeated measurements. Often, several measured variables are used to derive an experimental result, and the errors from each variable propagate into the final result. Propagation of errors are addressed with data-reduction equations in order to derive final values of bias and precision errors for each result. The bias and precision errors are combined with a root-sum-square method to derive an uncertainty for each result with a 95% confidence level. The uncertainties are listed in Table 1 and are considered reasonable for a towing tank experiment. There tends to be more scatter of the data for low Fr which explains the Fr dependence of the uncertainties. Values for the resistance and flow are comparable to the uncertainties listed in (1). RESULTS AND DISCUSSION Discussion of the results follow with regard to yaw- and wave-induced effects and the features of the flow for both high and low Fr. Initially, an overview 1   Average wave elevation is defined as a point where the probe tip appears to make contact with the free surface approximately 50 % of the carriage-run duration.

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Twenty-First Symposium on NAVAL HYDRODYNAMICS of ship-wave patterns and definitions of certain parameters are provided. Then, photographs of the wave field at the port bow and the stern and wake for β=0°, 5°, and 10° provide qualitative descriptions of yaw effects. Subsequent discussion of the detailed quantitative results begin with resistance, sideforce, yaw moment, sinkage, trim and heel angle. Next, the wave profiles along the hull and the wave patterns are discussed. Finally, the mean velocity and pressure field results are presented for Fr=0.316. General observations The Kelvin wave pattern (Figure 4c) can be used to explain the features of full- and model-scale (Figure 4a-b) wave patterns. A Kelvin wave pattern results from a point source moving at a speed, Us, across a free surface with gravity waves, which is a potential-flow solution for an infinite fluid with linearized free-surface boundary conditions. The resulting Kelvin wave pattern is enclosed in a delta-shaped envelope with a semiangle (wave-envelope angle) of αK=19°28'. The pattern consists of transverse waves that travel along the line of motion and diverging waves that radiate at an angle, θ=35 ° (diverging wave angle), from the symmetry plane. The transverse waves move at the speed of the disturbance (Ut=Us) and have wavelength: λt=2πFr2 (1) The diverging waves move at a reduced speed (Ud=Uscosθ) and have wavelength: λd=2πFr2cos2θ=λtcos2θ (2) The diverging wave amplitudes are larger and more conspicuous than the transverse wave amplitudes. In the far field, a ship hull can be approximated as a single point source because the global wave pattern exhibits many characteristics that are illustrated in Figure 4c. The near field is considerably more complex as it includes interactions of many point sources, i.e., every point along the hull acts as a source. However, the wave pattern can still be characterized by the same parameters (αK, θ, λt, λd) and some additional flow features. Table 2 summarizes the Kelvin and present (for β=0°, 5°, and 10°) values of these parameters for Fr=0.316. The present study focuses on the near field, which is conveniently referred to by regions. The local region is close to the hull (Figure 4a) and the global region is further from the hull (Figure 4b). The local region is cha racterized by steep waves that overturn and break and create significant white water and bubbles. The important features of the local wave pattern include the bow and stern waves which initiate as crests and the shoulder waves which initiate as troughs. In the global region, the wave pattern is complex due to the interactions of the divergent and transverse wave systems. At the bow, the incident flow creates a stagnant region (decelerated flow) near the bow stem. At the FP of a wedge-shaped bow, a thin film develops aft of the free surface-bow stem juncture and transitions into a bow wave. The thin film can be a site where bow vortices are generated. The hull-free surface contact line is the juncture line between the hull surface and the free surface. Typically, the contact line is a localized region of high free-surface turbulence. Free-surface turbulence is characterized by random or unsteady fluctuations of the free surface and also occurs in the wake and when wave breaking is present. Wave breaking is characterized by white water and free-surface turbulence along a wave front that becomes unstable (this is also observed in the yaw study). Waves that curl, overturn, and entrap air pockets are called plunging breakers. Spilling breakers occur when the crestline of a wave becomes unstable and slides down the wave front. When a wave breaks and impinges on the free surface, air can be entrained into the flow field (bubbles) and vorticity can be introduced into the flow, (wave-induced vorticity). Clearly, at model-scale, there is a comparative absence of wave breaking and white water near the hull and in the wake which is largely not understood. The wake regions of ships or models are characterized by significant free-surface turbulence and unsteady flow. Photographs of the port-side bow and wake are shown in Figure 5 for β=0°, 5°, and 10°. At β=0°, the wave pattern is typical of a slender hull form and follows the general pattern described above, i.e., the wave pattern is enveloped in a Kelvin wedge that sweeps back from the bow. In the figures, the local regions show the bow and fore-shoulder waves and the stern wave system. The global regions show interactions of the bow and fore-shoulder wave and interaction of the wave systems at the stern. The stern flow and wake are symmetric. Although not apparent from the photo, measurements indicate that the flow gradually rises for –0.05≤x≤–0.0027 followed by a sharp increase of a thin film at the bow-stem/free-surface juncture. Similarly, at x=0, measurements of the wave elevation indicate a gradual rise for 0.03≤y≤0.002 followed by a sharp increase of a thin film up to the hull surface. Beads of fluid originate at the bow stem and FP and form along the top of the thin film and appear vortical. Initially, they are steady and laminar. They are carried downstream and transition to unsteady and turbulent and merge with the downstream flow. The flow at the bow has the appearance of an attached spray sheet and bow vortices. Observations of the FP

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Twenty-First Symposium on NAVAL HYDRODYNAMICS at Fr=0.16 show that the incident flow initiates on the bow similarly as Fr=0.316 with a thin film and attached spray sheet but on a much smaller scale. In the photograph, the bow wave grows in amplitude and fullness with increasing x but does not break. There is very little free-surface turbulence in the wave pattern except along the hull-free surface contact line downstream of the bow wave and at the stern and wake where there is moderate unsteadiness. At β=5°, the wave field is altered. On the port side, the Kelvin wedge appears to be displaced further from the hull centerplane, [i.e., αK(5°)>αK(0°)]. At the stern and wake, the wave field is noticeably asymmetric with respect to the hull centerplane due to the diverging stern wave which is roughly aligned with the models longitudinal axis, i.e., the x-axis in Figure 1. Locally at the bow, the wave amplitude and steepness is increased. In the global regions, interaction of the wave systems give the wave pattern an asymmetric appearance. The flow initiates on the FP similarly as for β=0° with an attached spray sheet and bow vortices, although, the amplitude and thickness of the thin film is increased. Again, observation at the low speed confirms a similar flow pattern at the FP but on a smaller scale. In the photograph, wave breaking appears to be imminent which is verified by the free-surface turbulence at the crestline along the wave front. Free-surface turbulence also appears in the wake of the bow wave, along the hull-free surface contact line near the midbody, and in the wake. Observation shows no air entrainment into the flow but increased unsteadiness at the bow and in the wake. At β=10°, the wave field is altered in extreme. On the port side, further displacement of the Kelvin wedge from the model is displayed, [i.e., αK(10°)>αK(5°)>αK(0°)]. In the wake, the diverging stern wave is roughly aligned with the models longitudinal axis. Locally at the port bow, the bow and fore-shoulder wave systems appear to merge. In the global regions, interaction of the wave systems increase the asymmetry of the wave pattern. The flow upstream of the FP initiates on the bow similarly as β=0° and 5°, however, observation shows that the thin film rises more steeply than for the previous cases and is thicker. Observations of the flow over the bow at Fr=0.16 are consistent with those for β=0° and 5°. In the photograph, the bow wave increases in size and rises steeply on the upstream side with increasing x and then falls away from the hull. At the crest, the spray sheet becomes detached and evolves into a plunging-type breaker close to the hull and then a spilling-type breaker as the wave is further displaced from the hull. The overhead side view shows large disturbances including white water, bubbles, and air entrainment into the flow field similarly as is evident in Figure 4b but clearly less than at full-scale in Figure 4a. The wake is characterized by significant free-surface turbulence. At the hull-free surface contact line, the level of turbulence is consistent with β=0° and 5°. The unsteadiness of the wave pattern is increased especially in the wake of the port bow wave and at the stern. Forces, yaw moment, and displacements Resistance, sideforce, yaw moment, sinkage, trim and heel angle results are presented in Figure 6 and the equations are provided below. (3) (4) (5) (6) (7) (8) Resistance and sideforce are the drag and lift, respectively, in the ship coordinate system, and yaw moment is the counter-clockwise moment (top view, Figure 2) in the xy-plane resulting from the yawed orientation. Sinkage quantifies the upward or downward deflection (heave) of the hull and is positive for downward deflection. Trim quantifies the relative deflections of the FP and AP (pitch) about the midbody and is negative for the bow-down orientation. Heel quantifies the motion of the hull about its longitudinal axis (roll) and is positive for counterclockwise rotation as seen by looking upstream. At β=0°, results for resistance are compared with those in (7). For the same model and towing conditions, the quantitative and qualitative character of the results are very similar. At low Fr (0.1≤Fr≤0.2), CT is low, small oscillations are present, and the resistance decreases. Wave making is minimal and the resistance is mainly viscous. For medium Fr (0.2≤Fr≤0.3), CT sharply increases due to increased wave resistance. At high Fr (Fr >0.3), the resistance plateaus and then increases again at Fr=0.34. Through the Fr range, the wave pattern continuously changes especially at higher Fr. The interference of the diverging and transverse wave systems accounts for the Fr dependence of CT. These are the ‘humps and hollows' in the resistance curve. As expected, results for sideforce and yaw moment are zero when β=0°. The

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Twenty-First Symposium on NAVAL HYDRODYNAMICS Fr for the detailed measurements are determined from the resistance curve where for Fr=0.16, the resistance is low, and the effects of wavemaking are small, while at Fr=0.316, the resistance has passed through a sharp peak and reached a plateau where the effects of wavemaking are comparatively large. For β>0°, resistance curves increase but are qualitatively similar versus Fr for each β. For fixed Fr, ΔCT is relatively constant for increasing β. The sideforce curves are linear in Fr for low and medium Fr and all β. The slope increases with β. For high Fr, nonlinearity in Fr is indicated with increasing nonlinearity for increasing β. Also, for β=10°, the curve displays wavy characteristics. For fixed Fr, ΔCS increases with increasing β. The yaw-moment curves appear to be roughly linear through the Fr range for β=2.5° and 5° and then nonlinear for increasing β. Resistance increases with increasing β are probably a result of increased viscous and wave-making effects. It will be shown in the following sections that wave breaking and wave- and body-induced vortices are dominant flow features for the yaw case and surely play a role in the towing resistance of the model. Likewise, the wave patterns for the yaw cases will be shown to have greater area and larger amplitudes, and thus, larger components of wave-making resistance. Interestingly, interactions of the separate wave systems in the global regions of the wave pattern for β>0° are probably similar to those for β=0° because the positions of the humps and hollows for increasing β are constant. In the free condition, the changing shear stress and pressure distributions with Fr are responsible for adjusting the models sinkage and trim. Additionally, yaw angle induces a heeling angle toward the windward side. At low Fr small deflection of the hull (sinkage) and downward deflection of the FP in relation to the AP (trim) occurs (19). As Fr increases the bow-down movement is reduced, and at about Fr=0.32, the bow rises upward, the stern sinks downward, and the attitude of the hull is appreciably different than for low Fr. Significant sinkage, trim, and heel are associated with increases in wave resistance and wave elevations. At β=0°, the sinkage is roughly linear in Fr but with oscillations, and σ increases with increasing Fr. The trim decreases slightly for low and medium Fr and then increases sharply in 0.25 ≤Fr≤0.35. Interestingly, τ looks very similar to CT through the Fr range with corresponding sharp increases where the wave resistance grows. As expected, the heel is uniformly zero for β=0°. For β>0° and similarly as the forces and yaw moment, σ, τ, and η increase in magnitude. As per CT, the sinkage curves maintain consistent shape and slope through the range of yaw angles, however, Δσ increases with β. Similarly with σ, trim curves retain consistent shape and slope for all β, and Δτ increases with β. The trim becomes more negative with β which means that increases in yaw angle cause increases in the downward and upward pitch of the bow and stern, respectively. The heel angle is roughly linear in Fr for β=2.5° and then increasingly nonlinear in Fr for the higher yaw angles. Note that for β=10°, Fr=0.32 was the maximum Fr at which the breaking bow wave did not swamp the model. Wave profiles Wave profiles on the hull are measured for high and low Fr and β=0°, 5°, and 10° and are shown in Figure 7. The results are normalized by the velocity head, Several points along the wake centerplane are interpolated from the wave patterns in order to extend the wave profiles downstream of the AP. For β=0° and Fr=0.316, the wave profiles (marker/grid) are in close agreement with those in (7) (35 mm photography). The data in (7) compare favorably with data in the CEP (14). The wave profile along the hull defines the local wave system and the transverse wavelength, λt. The bow and stern waves initiate as crests and the forebody- and afterbody-shoulder waves initiate as troughs. The initial bow crest and forebody trough are larger than the stern crest and afterbody trough, probably due to the relative effects of the expansion and contraction of the hull at the forebody and afterbody, respectively, and the associated pressure field and viscous damping with increasing x. Two transverse wavelengths are apparent on the hull. The first, λt=0.55, is fairly close to the value for a Kelvin source at the same speed, and the second, λt=0.425, is somewhat shorter. The data on the centerplane upstream of the FP show the sharp rise in elevation at the free-surface bow-stem juncture. For β=0° and Fr=0.16, the zero-yaw wave profile is in fair agreement with (7). The bow and stern wave crests are apparent between which there is a broad region along the hull where the profile is negative. The fore- and aft-shoulder wave troughs are very shallow. The initial bow crest and forebody trough rapidly diminish in amplitude with increasing x probably due mainly to viscous damping with increasing x. One transverse wavelength is apparent on the hull, λt=0.13, and is shorter than for the high Fr, λt=0.55, and also shorter than for the Kelvin source at Fr=0.16, The bow wave amplitude exceeds that for the high-Fr case when nondimensionalized by the velocity head. For β=5° and 10°, and Fr=0.316, most resulting changes to the profiles occur upstream of x=0.25. For x<0.25, and the port side, the profiles shift upward significantly with increases in β. The

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Twenty-First Symposium on NAVAL HYDRODYNAMICS largest increases in the profiles occur where ζx=0. On the starboard side, the profiles decrease significantly with increases in β. Again, the largest Δζ occur where ζx=0. For x>0.25, the influence of β is fairly small except for decreases and increases in the fore and aft shoulder-wave troughs. On the wake centerplane, ζ increases with increasing yaw angle. On both sides, two transverse wavelengths are apparent on the hull. On the port side, the initial λt increases to 0.6 for β=5 and 10°. On the starboard side, the initial λt decreases to 0.53 and 0.5 for β=5 and 10°, respectively. The subsequent transverse wavelengths are similar as per For β=5° and 10° and Fr=0.16, again most yaw-induced changes to the profiles occur upstream of x=0.25. For x<0.25 on the port side, ζ shifts upward with increases in β. On the starboard side, oscillations are apparent such that the crests and troughs have increased amplitudes. For x>0.25, the influence of β is minimal except for β=10° on the port side which shows reduced crests and troughs and the starboard side which shows increased troughs on the forebody. On the wake centerplane, ζ decreases with increases in yaw angle. On both sides, two transverse wavelengths are a pparent on the hull. On the port side, λt is invariant with increasing β. On the starboard side, λt decreases from 0.13 (β=0°) to 0.1 (β=5° and 10°). Similarly with β=0°, the bow wave amplitudes exceed those for the high-Fr case when nondimensionalized by the velocity head. Wave elevations Next, contours of the wave elevations and axial wave slopes are discussed for Fr=0.316 and β=0°, 5°, and 10° (Figure 8). Figures for β=5° are not shown. The low-Fr data is not discussed because the elevations and gradients are comparatively negligible with those at Fr=0.316. The measurements are made in the tank coordinate (x, y, z) system, but, are presented in the ship coordinate (x, y, z) system by rotating the wave field by the amount of yaw (Figure 3). The axial gradients in the wave patterns are obtained by numerically differentiating ζ. The wave-pattern parameters, αK, θ, and λd are determined from the ζx contours with line drawings as per Figure 4c and presented in Table 2. For β=0° and Fr=0.316, the contours display the overall features of the wave pattern. The diverging systems in ζx give the pattern a wedge-shaped appearance. The wave-envelope angle is 22°, which is close to the Kelvin-source value, and the apex of the wedge is upstream of the FP. The diverging wave angle, θ=53°, is considerably larger than the Kelvin-source value such that λt/λd=2.75, i.e., λd=0.2, which is less than half the Kelvin-source value. Note that the largest difference among the Kelvin parameters is Clearly, there are many different amplitudes in the wave field with the highest values at the bow wave crest and fore-shoulder wave trough. The local region characteristics are similar to those of the wave profiles. In the global region, the complex interaction of the wave systems is apparent. The rapid rise of the free surface at the FP and the thin film of fluid in that region is indicated by a dense clustering of contours. Free-surface turbulence and general unsteadiness of the wave pattern is suggested by the irregularity of the contours at the afterbody and stern. ζx is in phase with the elevations and has similar patterns. For β=5° and 10° and Fr=0.316, the wave patterns become asymmetrical, i.e., for β=5° on the port and starboard sides αK=25° and αK=22°, respectively, and for β=10° on the port and starboard sides αK=30° and αK=21.5°, respectively. Note that αK increases on the port side and remains nearly constant on the starboard side with increasing β. Also, the total wave-envelope angle (αT) increases in comparison to that for β=0° by about β. On the port side, θ decreases with increasing β and λd increases with β, i.e., λd=0.31 and 0.34 for β=5° and 10°, respectively, such that λt/λd continuously decreases. On the starboard side, θ decreases at β=5° and then increases for β=10°. Also, λd increases and then decreases with increasing β, i.e., λd=0.35 and 0.18 for β=5° and 10°, respectively, such that λτ/λd decreases and then increases. The wave amplitudes clearly increase on the port side and decrease on the starboard side of the model with increasing β. In particular, note the increases and reductions of the bow wave on port and starboard sides, respectively, when β is increased from 0° to 10°. Observation of ζx indicates that the port bow-wave crestline curves back toward the model with increasing x. As per the wave profiles, the local regions are affected only for x<0.25, whereas in the global regions, the wave patterns are significantly changed with increasing β from bow to stern. Dense clustering of contours at the bow on the port side confirms the sharp increase of the bow wave at the FP and the existence of an attached thin film. For β=5° and 10°, the wave breaking in the patterns is difficult to distinguish in the contours because of the unsteadiness of this phenomenon. Free-surface turbulence and unsteadiness are mainly evident along the crestline of the port bow wave and at the stern and near wake as per β=0°. However, at the stern for the yaw cases, the merging of the port and starboard systems is a source of increased unsteadiness. These results are consistent with the photographs ( Figure 5). Similarly with β=0°, ζx is in phase with ζ and has similar patterns, but the magnitudes are significantly increased and decreased globally on the port and starboard sides, respectively.

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Twenty-First Symposium on NAVAL HYDRODYNAMICS Mean-velocity and pressure field Results for the mean-velocity and pressure field are presented next in Figures 9a–f. Contours of ωx, H, p, u, and v and w are shown at all crossplanes for high Fr only, although, the discussions for the low-Fr tests are retained. The overall trends are similar between the high- and low-Fr flow fields, although significant free-surface effects are also exhibited for the former. Equations for the total head and axial vorticity are: (9) (10) TECPLOT was used to numerically differentiate the mean crossplane velocities to obtain ωx. Results are presented at x=0, 0.1, 0.2, 0.4, 0.6, 0.8, 0.9, 1, 1.1, and 1.2, and the discussions begin with the forebody and proceed to the afterbody and wake. Also, discussions in (13) provide details of each crossplane and the wave-induced vortex. Most of the flow features in the yawed condition are significantly different than for the zero-yaw case. In many respects, the flow is completely altered. The boundary-layer and wake development is dominated by strong crossflow effects and vortices as opposed to px and weak crossflow effects observed in the β=0° case. The wave-induced effects at Fr=0.316 are explainable similarly as for β=0°, i.e., the Fr-related differences in the velocity and pressure correlate with the wave field, which, however, is significantly more complex for β=10° than for β=0° creating a more complex boundary-layer and wake response. Also, a wave-induced vortex is identified due to the breaking bow wave on the port side of the hull which significantly affects the boundary layer and wake. In Figure 9a, the extensive vorticity (ωx) in the flow field is evident. On the forebody at Fr=0.316, keel and bilge vortices are visible beginning at x=0.1. The keel vortex is relatively weak and not evident beyond x=0.4. The bilge vortex is relatively strong. The vortex core is off the body and moves further from the centerplane with increasing x. On the afterbody, the forebody-bilge vortex weakens, but is distinct with a trajectory off of the body toward the free surface at an angle of about 5.7° to the centerplane. An afterbody-bilge vortex develops as per β=0°. The vorticity has a core region that is off of the body and toward the free surface with a tail that extends toward the centerplane. There appears to be a weak interaction with the forebody-bilge vortex. At x=1 (AP), a counter rotating keel vortex is evident. In the wake, the forebody-bilge vortex dissipates and diffuses with trajectory towards the free surface. The afterbody-bilge vortex becomes oval shaped and dissipates and diffuses with a trajectory off the body towards the free surface at an angle of about 2.1° to the centerplane. The vorticity pattern appears to rotate counterclockwise and reorganizes its structure with increasing x, i.e., at x=1, the long axis is parallel with the stern stem and subsequently parallel with the free surface at x=1.2. The afterbody-keel vortex dissipates relatively fast with a trajectory as per the afterbody-bilge vortex but at an angle of about 2.6° to the centerplane. There is limited interaction between fore- and afterbody-bilge vortices. A wave-induced vortex is evident on the port-side forebody which initiates between 0.2<x<0.4 underneath the breaking bow wave and follows a trajectory near the free surface along the side of the hull. The vorticity in the breaking wave wake is also visible at x=0.4 near the free surface. At the afterbody and wake, the wave-induced vortex dissipates, diffuses, and mixes with the hull boundary layer and wake such that the overall wake pattern is largely increased on the port side. At Fr=0.16, the overall flow pattern is very similar to that of Fr=0.316 but with two important differences. First, the bilge and keel vortices appear weaker, and the trajectories are altered somewhat, i.e., the forebody-bilge vortex is at an angle of 5° to the centerplane (outward) and does not intersect the free surface at x=1.2 in contrast to Fr=0.316; the afterbody-bilge vortex is at an angle of 2° (outward); and the afterbody-keel vortex is at an angle of 2.8° to the centerplane (inward). Secondly, and due to the reduced wavefield, there is no wave-induced vorticity in the flowfield. In Figure 9b, the viscous regions (H) on the hull and in the wake are evident. At Fr=0.316, the patterns correlate with ωx and the boundary layer and wake losses but with stronger interactions between the loss regions of the keel and bilge vortices on the afterbody creating a somewhat more complex pattern. H displays similar Fr differences as per ωx for the wave-induced vortex. For Fr=0.16 and in contrast with ωx which decreased in magnitude with decreasing Fr, H is somewhat increased in magnitude which is a Reynolds number effect, i.e., the viscous regions are thicker for the lower Fr. In Figure 9c, the pressure (p) field around the hull is displayed. At the forebody for Fr=0.316, p correlates with u such that the trends are the same but the magnitudes are reversed especially in the bow and stern regions and at the midbody where the flow accelerates and the pressure is low. An asymmetric stagnation-type flow is exhibited at the FP. In general, high and low pressure regions exist on the port and starboard sides, respectively. The lowest pressures are in regions of high ωx with minimums in the core

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Twenty-First Symposium on NAVAL HYDRODYNAMICS regions. The bow wave stagnation effects are evident as increased pressures at x=0 and 0.1. The pressure differences (Δp) between port and starboard sides are reduced near the midbody at x=0.4. At the afterbody, there is a continued reduction in Δp up to x=0.9 when Δp increases with the lowest p in regions of high ωx and minimums in the core regions, i.e., for the forebody-bilge vortex and afterbody-bilge vortex. In the wake, the pressure recovery is evident. Lastly, apparently, most of the sideforce is generated at the bow and somewhat from the stern where Δp is largest. For Fr=0.16, the general patterns are similar except for some diminished features due to Fr and viscous effects. The pressure at the bow and stern is lower due to the reduced port-side bow wave system and wave effects at the AP and wake, respectively. The pressure field is higher over the midbody and stern especially in the regions of the vortex cores. In Figure 9d, for Fr=0.316 and 0.16, the axial velocity (u) contours are similar to H but also exhibit inviscid effects such as the stagnation flow at the FP and the acceleration around the midbody. For Fr=0.316, wave-induced effects are especially apparent at the bow on the port side and also at x=0.4 in association with the wave-induced vortex. By cross-referencing Figure 9d with Figure 9c, one can observe the correlation between u and p particularly at the bow and midbody and moderately at the stern. In Figures 9e and 9f, the combination of transverse (v) and vertical (w) components of velocity show the nature of the crossflow that is induced by the keel, bilge, and wave-induced vortices. w is particularly useful for observing wave effects near the free surface and charting the dissipation and diffusion of vortices in the flow field. Differences in v and w due to Fr are evident at every x-station and confirm increased wave effects and vorticity for Fr=0.316. CONCLUSIONS Yaw effects on model-scale ship flow is documented through towing-tank experiments for a 3.048 m Series 60 CB=0.6 model ship. The data includes: photographs and video; resistance, side force, and yaw moment; sinkage, trim, and heel angle; wave profiles along the hull and wave elevations; and mean-velocity and pressure fields for numerous crossplanes from the bow to the near wake. Detailed descriptions are provided of the experimental equipment, procedures, and uncertainty analysis. Comparison of results for low and high Fr with those from an earlier study for the without-yaw condition enables identification of the salient yaw- and wave-induced effects. When β increases, the forces, yaw moment, and displacements increase significantly. Increases in β produce asymmetric wave profiles and elevations with large changes in ζ locally at the bow and globally everywhere. In contrast to the zero-yaw case, the yaw condition is dominated by strong crossflow effects that drive the flow from the port to the starboard side and asymmetric vorticity development at the forebody bilge, forebody keel, afterbody bilge, and afterbody keel. The distinct vorticity in the flow field originates from strong yaw-induced crossflow and large crossplane pressure gradients. Most of the wave-induced effects on the boundary layer and wake are explained as per β=0°, by correlating the wave elevations and slopes of the free surface with the velocities and pressures in the underlying flow for both values of Fr. Some unresolved issues for the Series 60 CB=0.6 in yaw are hull surface-pressure distributions, turbulence measurements, unsteady flow, and wave breaking. These data would compliment the present study. With regard to future work and recommendations, a new modern combatant hull form (DTMB model 5415) shown in Figure 4b has been adopted for study by the US Navy in support of rapid advancements in CFD. Some of the crucial unresolved issues in experimental ship hydrodynamics and model testing are turbulence measurements, unsteady problems such as a ship in waves, bubble entrainment into the flow field, wave breaking, bow flow, effects of appendages, and rigorous uncertainty analysis. Currently, efforts are underway at the IIHR to equip the towing tank with PIV instrumentation and a wavemaker for unsteady experiments with model 5415. ACKNOWLEDGMENTS This research was sponsored by the Office of Naval Research under Contract N00014–92-J-1092 under the administration of Dr. E.P.Rood whose support is greatly appreciated. The first author is indebted to The Department of Mechanical Engineering of The University of Iowa who provided partial financial support over the course of this study. REFERENCES 1. Longo, J. and Stern, F., ( 1995), “Evaluation of surface-ship resistance and propulsion model-scale database for CFD validation,” J. Ship Research, Vol. 40, No. 2, pp. 112–116. 2. Day, W.G. and Hurwitz, R.B., ( 1980), “Propeller-disk wake survey data for model 4989 representing the FF 1052-class ship in a turn and with a bass dynamometer boat,” Report no. SPD-0011–21, David Taylor Naval Research and Development Center, Bethesda, MD.

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Twenty-First Symposium on NAVAL HYDRODYNAMICS 3. Stern, F., Hwang, W.S., and Jaw, S.Y., ( 1989), “Effects of waves on the boundary layer of a surface-piercing flat plate: experiment and theory,” J. Ship Research, Vol. 33, No. 1, pp. 63–80. 4. Stern, F., Choi, J.E., and Hwang, W.S., ( 1993), “Effects of waves on the wake of a surface-piercing flat plate: experiment and theory,” J. Ship Research, Vol. 37, No. 2, pp. 102–118. 5. Stern, F., Parthasarathy, R.N., Huang, H.P., and Longo, J., ( 1994), “Effects of waves and free surface on turbulence in the boundary layer of a surface-piercing flat plate,” ASME Symposium on Free-Surface Turbulence, Invited Speaker, Lake Tahoe, NV., pp. 37– 51. 6. Longo, J., and Huang, H.P., and Stern, F., ( 1996), “Solid/free-surface juncture boundary layer and wake,” Physics of Fluids, (in review). 7. Toda, Y., Stern, F., and Longo, J., ( 1992), “Mean-flow measurements in the boundary layer and wake and wave field of a Series 60 CB=0.6 ship model-part 1: Froude numbers 0.16 and 0.316,” J. Ship Research, Vol. 36, No. 4, pp. 360–377. 8. Longo, J., Stern, F., and Toda, Y., ( 1993), “Mean-flow measurements in the boundary layer and wake and wave field of a Series 60 CB=0.6 ship model-part 2: scale effects on near-field wave patterns and comparisons with inviscid theory,” J. Ship Research, Vol. 37, No. 1, pp. 16–24. 9. Stern, F., Longo, J., Zhang, Z.J., and Subramani, A., ( 1996), “Detailed bow-flow data and CFD for a Series 60 CB=0.6 ship model for Froude number 0.316,” J. Ship Research, Vol. 40, No. 3, pp. 61–67. 10. Tobak, M. and Peak, D.J., ( 1982), “Topology of three-dimensional separated flows,” Annual Review of Fluid Mechanics, Vol. 14, pp. 61–85. 11. Cointe, R. and Tulin, M.P., ( 1994), “A theory of steady breakers,” J. of Fluid Mechanics, Vol. 276, pp. 1–20. 12. Rood, E.P. and Katz, J., ( 1994), ASME symposium on free-surface turbulence, FED-Vol. 181, 169 pp. 13. Longo, J., ( 1996), “Yaw effects on model-scale ship flows,” Ph.D. Thesis, The Department of Mechanical Engineering, The University of Iowa, Iowa City, IA., 275pp. 14. ITTC, ( 1987), “Report of the resistance and flow committee,” 18th International Towing Tank Conference, Kobe, Japan, pp. 47–92. 15. Todd, F.H., ( 1963), “Series 60 methodical experiments with models of single-screw merchant ships,” David Taylor Model Basin Report 1712. 16. Houser, D., Toda, Y., and Stern, F., ( 1989), “High-resolution, low noise, capacitance-wire waveheight interface, ” Proc. IAHR Workshop on Instrumentation for Hydraulics Laboratories , Burlington, Canada. 17. Rood, E.P. and Telionis, D.P., ( 1991), “J. of fluids engineering policy on reporting uncertainties in experimental measurements and results,” ASME J. Fluids Eng., Vol. 113, pp. 313–314. 18. Coleman, H.W. and Steele, W.G., ( 1989), “Experimentation and uncertainty analysis for engineers,” John Wiley & Sons, New York. 19. Todd, F.H., ( 1967), “Resistance and propulsion,” Principles of Naval Architecture, The Society of Naval Architects and Marine Engineers, New York, New York, pp. 288–447. 20. Toda, Y., Stern, F., Tanaka, I., and Patel, V.C., ( 1988), “Mean-flow measurements in the boundary layer and wake of a Series 60 CB=0.6 model ship with and without propeller,” J. Ship Research, Vol. 34, No. 4, pp. 225–252. 21. Newman, J.N., ( 1977), “Marine hydrodynamics,” MIT Press, Cambridge, MA., 401 pp. Table 1. Uncertainties for the yaw experiments Result 0.316 0.16 Total resistance (CT) 0.6% 6.0% Sideforce (CS) 0.1% 1.8% Yaw moment (CM) 0.1% 4.1% Sinkage, trim, heel (σ, τ, η) 0.3% 1.8% Wave profiles (ζ) 1.3% 2.6% Global elevations (ζ) 1.1% 2.2% Local elevations (ζ) 1.3% 5.0% Mean velocities (u, v, w) 1.5% 1.5% Mean pressure (p) 3.0% 3.0% Vector direction (α, ) 1.3% 1.3% Table 2. Wave-pattern parameters (Fr=0.316)   † Kelvin Port/Starboard     0° 5° 10° αK 19°28' 22°/22° 25°/22° 30°/21.5° θ 35°16' 53°/53° 44°/36° 41.5/53.5° λt ‡ 0.63 0.55/0.55 0.6/0.53 0.6/0.5 λd ‡ 0.42 0.2/0.2 0.31/0.35 0.34/0.18 †: Kelvin wave pattern illustrated in Figure 4c ‡: nondimensionalized by ship length

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Twenty-First Symposium on NAVAL HYDRODYNAMICS Figure 1. Experimental coordinate system Figure 2. Carriage with Series 60 CB=0.6 at yaw Figure 3. Summary of experimental measurements Figure 4. Kelvin ship-wave patterns

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Twenty-First Symposium on NAVAL HYDRODYNAMICS Figure 5. Photographs of the wave field at Fr=0.316 Figure 6. Forces, moment, and displacements

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Twenty-First Symposium on NAVAL HYDRODYNAMICS Figure 7. Wave profiles Figure 8. Wave elevations

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Twenty-First Symposium on NAVAL HYDRODYNAMICS Figure 9. Mean-flow measurements for the Series 60 CB=0.6 at yaw: β=10°, Fr=0.316

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Twenty-First Symposium on NAVAL HYDRODYNAMICS Figure 9.