ie IMAG cei Sit DTNSROC- Ba /022_ DAVID W. TAYLOR NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER Bethesda, Maryland 20084 DTNSRDC-82/022 STERN BOUNDARY-LAYER FLOW ON A THREE-DIMENSIONAL BODY OF 3:1 ELLIPTIC CROSS SECTION by Nancy C. Groves Garnell S. Belt Thomas T. Huang APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED SHIP PERFORMANCE DEPARTMENT RESEARCH AND DEVELOPMENT REPORT UNDARY-LAYER FLOW ON A THREE-DIMENSIONAL BODY iPTIC CROSS SECTION May 1982 DTNSRDC-82/022 MAJOR DTNSRDC ORGANIZATIONAL COMPONENTS DTNSRDC COMMANDER 00 TECHNICAL DI deuhent! OFFICER-IN-CHARGE ANNAPOLIS OFFICER-IN-CHARGE CARDEROCK SYSTEMS DEVELOPMENT DEPARTMENT AVIATION AND SURFACE EFFECTS DEPARTMENT SHIP PERFORMANCE DEPARTMENT 15 COMPUTATION, MATHEMATICS AND LOGISTICS pseu rats STRUCTURES DEPARTMENT PROPULSION AND AUXILIARY SYSTEMS DEPARTMENT SHIP ACOUSTICS DEPARTMENT SHIP MATERIALS CENTRAL ENGINEERING INSTRUMENTATION DEPARTMENT o DEPARTMENT ) H GPO 866 993 NDW-DTNSRDC 3960/43 (Rev. 2-80 UNCLASSIFIED @ 2 JUN SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) 4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED STERN BOUNDARY-LAYER FLOW ON A eee THREE-DIMENSIONAL BODY OF 3:1 6. PERFORMING ORG. REPORT NUMBER ELLIPTIC CROSS SECTION . AUTHOR(s) 8. CONTRACT OR GRANT NUMBER(s) Nancy C. Groves Garnell S. Belt Thomas T. Huang . PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK David W. Taylor Naval Ship Research Doge GREMeAe ALISO and Development Center Project ZR 000 O01 Bethesda, Maryland 20084 Work Unit 1542-103 . CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE May 1982 13. NUMBER OF PAGES 116 . MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) 15. SECURITY CLASS. (of thie report) UNCLASSIFIED 15a. DECLASSIFICATION/ DOWNGRADING SCHEDULE . DISTRIBUTION STATEMENT (of this Report) APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED . DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report) - SUPPLEMENTARY NOTES . KEY WORDS (Continue on reverse aside if necessary and identify by block number) Three-Dimensional Turbulent Boundary Layer Thick ,Stern Boundary Layer 3:1 Elliptic Cross Section . ABSTRACT (Continue on reverse side if necessary and identify by block number) A comprehensive set of experimental pressure, velocity, and turbulence data are presented across the stern of a three-dimensional model having 3:1 elliptic transverse cross sections. The axisymmetric displacement body con- cept is extended to three-dimensions and the pressure and velocity data are compared with the predictions of existing three-dimensional theoretical methods. The surface pressures for the displacement body are found to model, (Continued on reverse side) DD 6 on", 1473 EDITION OF 1 Nov 65 Is OBSOLETE UNCLASSIFIED S/N 0102-LF-014-6601 eee Oe SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) (Block 20 continued) satisfactorily, the measured pressure coefficients in all regions except over the aft 7 percent of body length. In this tail region, the boundary layer is much thicker than the cross section dimensions and the theory overpredicts the measured distributions of the mean velocity. Agreement is particularly poor in the inner region of the tail boundary layer, indicating a need to examine the eddy viscosity model currently used in computing the thick stern boundary layer of three-dimensional models. As was found in the axisymmetric case, the measured values of turbulence intensity, eddy viscosity, and mixing-length parameters in the stern region are much smaller than those of a thin boundary layer. UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) TABLE OF CONTENTS ILIESHE OI TIGUWIRIBSS "56 “ola o Ss IKIESHE OR MWNWHDS S56 60 MS OMG Nag. (5 NGQIVMIELON 5 5 6 66,0 0 0 0 0 6 0 6 ABSTRACT .. . ADMINISTRATIVE INFORMATION .... TON TONCIDWICMILONT 56g 5 6 6 6 4 6 0 8 WIND TUNNEL AND MODEL. ... INSTRUMENTATION. . . DISPLACEMENT BODY METHOD. . COMPARISON OF EXPERIMENTAL AND THEORETICAL RESULTS ....... PRESSURE DISTRIBUTION . MEAN VELOCITY PROFILES. MEASURED TURBULENCE CHARACTERISTICS. MEASURED REYNOLDS STRESSES. [RIDIN WATSCORIEING ZAINDD) WODONG IGINEWsl, 6G 5 6 6 56 6 6 6 6 oO CONCEUSEONS) Vener es ACKNOWLEDGMENTS. . . 2. 2 2 « « « « FH IINEMGES) 6 6 6 6 6 4 Ga 4 6 6 1 - Schematic of the Three-Dimensio 3:1 Elliptic Transverse Cross § 2 - Model Mounted in Anechoic Wind to | 4 - Schematic of a Typical Section LIST OF FIGURES nal Afterbody Having a QCiEMCMls 6 6 6605600 0 Tunnel. Schematic of the Pressure Tap Locations. ......... .» atex/E. garetts snelt + atatat 20 2a 23 23 Page Schematic of a Two-Element Sensor Alined 90 Degrees toyEach) Other andw45) Degrees to the Probe wAxise) ai ircr Men tiii tel enone tne 24 Computed and Measured Stern Pressure Distribution for Angular TOCatiOn ey wus je te sey ey ot on ateLIeIn tee enero ter tee a ssmareeet canen Soa vctan ene mnrs 25 Measured Mean Axial and Radial Velocity Distributions .......... 27 Computed and Measured Mean Axial Velocity Distributions ......... 30 Measured Distributions of Reynolds Stresses at Angular Location O,= 67 (Desrees! fo. a em ie) el see tie) tee eee eu is en ine rel te) 32 Measured Distributions of Reynolds Stresses at Angular LOCatdon sO = iSO! Desreesy ie ie ie, U5) vee: Vee pee elites ne ial ete, ste icy) Soukron Ucmmcomars 34 Measured Distributions of Reynolds Stresses at Angular LOcaEom @ = 83 WeeGSS 5 5656 65005006000000005000050000 36 Measured Distributions of Reynolds Stresses at Angular Locaiiom 9 = 86 DESEEGES o 656 056060650056006000000000 0 37 Measured Distributions of Reynolds Stresses at Angular Location @ = 87 Meaeree@S 6 6 6 66 6 660000 oP Gi nse aera ee oy ohare 38 Measured Distributions of Turbulent Structure Parameter ........ . 40 Measuneds Diltstr1butdon's sot sLddiyaVEtsiCosuityzmmemcni cine aieltcn nim Nit- nT imLcnn cH Nona InT® 44 Nensicea Dasineibmeioms Oi Whiralins Ieee 6 6 6 0 oo KO 48 Turbulence Area Representing the Square-Root of the hissing IEEE 5 g 50 00000 Do DODO ooo oO 52 Proposed Similarity Concept for Mixing Length of Thovebelleme Bowimndeiny eyes 6 5 0 6 6 0060 6 6 OO 8 53 LIST OF TABLES Model, ORFSGES Ginene@s)>o 6 6 0.06 6656006066500 060007 55 00 6 58 MeAgureedl Riresetise COEIrIeIEMES5 5 6006000000006000500 0160 4 64 Measured Mean and Turbulent Velocity Characteristics for Varying Axial Locations Along O-Degree Plane. . ....+.4-++-+-+ +6 65 iv Measured Mean and Turbulent Velocity Characteristics Axial Locations Along 6/7-—Degree Plane. Measured Mean and Turbulent Velocity Characteristics Axial Locations Along 80-Degree Plane. Measured Mean and Turbulent Velocity Characteristics Axial Locations Along 83-Degree Plane. . . Measured Mean and Turbulent Velocity Characteristics Axial Locations Along 86-Degree Plane. Measured Mean and Turbulent Velocity Characteristics Axial Locations Along 8/7-—Degree Plane. Measured Mean and Turbulent Velocity Characteristics Axial Locations Along 90-Degree Plane. for for Varying Varying Varying Varying Varying Varying Page 78 85 89 | 93 NY) NOTATION -1/2 A Van Driest's damping factor, A = 26v =) a Length of major elliptical axis at given x/L es) PP ay Turbulence structure parameter, a, = uv /q a* Effective displacement thickness, see Equation (3) b Length of minor elliptical axis at given x/L © Pressure coefficient, C_ = (p-p )/ (1/2002) =1- (UU _/U 2 3) oD) fo) fo) eo hy> hy Metric coefficients Ks Ky Geodesic curvatures of the curves z = constant and x = constant, respectively Kio Koy Functions of the geodesic curvatures and metric coefficients L Total body length My, Mixing length parameter: In the inner region & = 0.4 y [1-exp(-y/A) ] In the outer region DBs AyD —_ 2 |fem Py . (oe au Nee on on on e e e ny Coordinate measured normal to the body profile in the y-z plane P Measured local static pressure Pa Measured ambient pressure Ps Measured static pressure Py Measured dynamic total pressure Turbulence parameter, a =U 9 WY oF Ww Reynolds number based on model length, R, = ag Radius of curvature at major or minor axis of elliptic cross section Computed potential flow velocity on the displacement body Free-stream velocity Potential flow velocity at the edge of the boundary layer Mean velocity components in the x, y, and z directions, respectively Mean velocity components in the x, no» and § directions, respectively Turbulent fluctuations in the x, no» and @ directions, respectively Reynolds stresses Coordinates used to present measured boundary layer data Nonorthogonal boundary-layer coordinates, see Reference 6. Location of the thick stern boundary layer Angle between the body surface and the body axis Boundary-layer thickness at major and minor axis, respectively, elliptical cross section Boundary-layer thickness measured in n direction. Planar displacement thickness Eddy viscosity vii o| Ax tw Eddy viscosity in the inner and outer regions, respectively, see Equation (2) Angular coordinate measure in the y-z plane from the z-axis to the line joining the surface offset and elliptic center Angle between the x and z coordinates Effective displacement area Kinematic viscosity of the fluid Mass density of the fluid Shear stress at the wall viii ABSTRACT A comprehensive set of experimental pressure, velocity, and turbulence data are presented across the stern of a three- dimensional model having 3:1 elliptic transverse cross sections. The axisymmetric displacement body concept is extended to three- dimensions and the pressure and velocity data are compared with the predictions of existing three-dimensional theoretical methods. The surface pressures for the displacement body are found to model, satisfactorily, the measured pressure coefficients in all regions except over the aft 7 percent of body length. In this tail region, the boundary layer is much thicker than the cross section dimensions and the theory overpredicts the measured distributions of the mean velocity. Agreement is particularly poor in the inner region of the tail boundary layer, indicating a need to examine the eddy viscosity model currently used in computing the thick stern boundary layer of three-dimensional models. As was found in the axisymmetric case, the measured values of turbulence intensity, eddy viscosity, and mixing-length parameters in the stern region are much smaller than those of a thin boundary layer. ADMINISTRATIVE INFORMATION The work described in this report was funded under the David W. Taylor Naval Ship Research and Development Center's Independent Research Program, Program Element 61152N, Project Number ZR 000 01, and Work Unit 1542-103. INTRODUCTION Many single-screw ship propellers operate inside of thick stern boundary layers. Satisfactory predictions of turbulent boundary-layer characteristics can be made for the forward portions of a body by solving the boundary-layer equations in either integral or differential forms. However, at the ship stern, the thickness of the boundary layer increases rapidly, mainly due to the diminishing cross-sectional area. The thickness of the stern boundary layer usually exceeds the thickness of the body. Detailed measurements of the turbulent boundary-layer characteristics in the thick stern boundary layers of axisymmetric bodies have been made by Huang et Hie eos in order to gain insight into the physics of thick stern boundary layers. These measurements have been used to validate the displacement body concept as suggested by ineeiea: and Lighthill* for solving viscid-inviscid flow interaction and an improved turbulence model has been obtained for computing thick axisymmetric boundary *A complete listing of references is given on page 105. layers on two convex sterns and one concave seen 2° The present work is an initial investigation into extending to three-dimensions the previous studies on axisymmetric bodies by Huang et a? Experiments have been made to measure the flow eee the thick stern boundary layer of a three-dimensional body having a 3:1 elliptical transverse cross section. A 10.06 ft (3.07 m) fiberglass model was tested in the Center's Anechoic Flow Facility at a speed of 100 ft/sec (30.48 m/s), resulting in an overall Reynolds number based on length of 6.5 x 10, Pressure taps, embedded in the model, were used to measure the pressure distribution on the surface. Velocity and turbulence characteristics were measured using a two-element hot-film sensor and were analyzed with an on-line computer. Measurements include mean velocity profiles, turbulence intensities, Reynolds stresses, eddy viscosity, and mixing length. Several experimental quantities are compared with data from existing theoreti- cal methods using an iterative scheme. The potential flow distribution on the body surface is computed using the XYZ Potential Flow (XYZPF) computer code of Dawson and Deeias® An initial boundary-layer computation, using the McDonnell Douglas Gseporatiion, © Cebeci, Chang, Kaups (7K) computer code, is made using the potential- flow pressure distribution on the body. Flow separation is predicted for this model by the c°K code at axial locations greater than 4 percent of the body length and angular locations greater than 75 degrees. Excessive boundary-layer growth in the separated region caused the boundary-layer calculation to abort prematurely at 81 percent of the body length. Predictions of the effective displacement thickness for the remaining 19 percent of the body length are obtained by extrapolation. The potential and boundary-layer flow calculations are repeated once for a modified body and wake geometry, formed by adding the computed effective displacement thickness. Comparison of predicted and measured results shows that this procedure predicts accurate values of pressure over the forward 93 percent of the body and accurate mean velocity profiles in locations where the boundary layer is thin compared with cross-— sectional area. The measured eddy viscosity distribution is compared with the thin boundary-layer model of Gapeet! and is found to be smaller than predictions. In the following sections, the experimental techniques and model geometry are given in detail. The experimental data are presented and compared with theoretical predictions. The raw data and derived results are given in tabular form for inde- pendent use by other investigators. WIND TUNNEL AND MODEL The experimental investigation was conducted in the DINSRDC Anechoic Wind Tunnel Facility. The wind tunnel has a closed jet test section that is 8 ft (2.4 m) square and 13.75 ft (4.19 m) long. The corners have fillets which are carried through the contraction. The test section is followed by an acoustically-lined large chamber 23.5 ft (7.16 m) long. It was found previously, by Huang et ple that the ambient free-stream turbulence levels, (V.-?)u, 100, are 0.075, 0.090, 0.100 and from 0.12 to 0.15 for free-stream velocities, Use OF Da .k, BS. S851, eine 4557 m/s, respectively. Integration of the measured noise spectrum levels in the test section from 10 to 10,000 Hz indicated that the typical background acoustic noise levels at 30.5 m/s were about 93 dB re 0.0002 dymne//en- (0.0002 Pa). These levels of ambient turbulence and acoustic noise were considered low enough so as not to unfavorably affect the measurement of boundary-layer characteristics. The maximum air speed that can be achieved is 200 ft/sec (61 m/s); in the present experiments the wind tunnel velocity was held constant at 100 ft/sec (30.48 m/s). A simple three-dimensional body, having a 3:1 elliptic transverse cross section with a bow entrance length of 6.23 ft (1.897 m), was used for the present experi- mental investigation. The total model length is 10.06 ft (3.07 m) with a maximum major axis of 1.588 ft (0.48 m) and a maximum minor axis of 6.35 in. (16512 em). A schematic of the three-dimensional afterbody with the 3:1 elliptic cross section is shown in Figure 1. The major and minor elliptic axes are shown in Figure 1 as a and b, respectively. The model is shown in the anechoic wind tunnel facility in Figure 2. The support struts shown in the figure are not the struts used for this experiment. Model offsets are presented in Table l. The model was supported by two streamlined struts separated by one-third of the model length. The struts are 0.5-in. (1.27-cm) thick with a 1.5-in. (3.81l-cm) chord upstream and 2.25-in. (5. 72-cm) thick with a 6.0-in. (15.24-cm) chord downstream. The model is designed to rotate 90 degrees radially about a center axis to permit vertical traversing normal to the surface pressure taps (see section on Instrumentation). The disturbances generated by the supporting struts were within the region below the horizontal centerplane. Therefore, all of the experimental data were taken above the model on the vertical centerplane along the upper meridian where there was no effect from the supporting struts. One-half of the model length protruded beyond the closed jet working section into the open-jet section. The ambient static pressure coefficients across and along the entire open-jet chamber (7.2 m xX 7.2 m X 6.4 m) were found to vary less than 0.3 percent of the dynamic pressure. Tunnel blockage and longitudinal pressure gradient effects along the tunnel length were almost completely removed by testing the afterbody in the open- jet section. The location of the boundary-layer transition from laminar to turbulent flow was artificially induced by a 0.024-in. (0.61-mm) diameter trip wire located at x/L = 0.05. Huang et ails found that the trip wire effectively moved the location of the virtual origin to x/L = 0.015 for axisymmetric models at a length Reynolds number @iF 5o°) X tobe The virtual oriipia® for the turbulent flow is defined such that the sum of the laminar frictional drag from the nose to the trip wire, the parasitic drag of the trip wire, and the turbulent frictional drag aft of the trip wire is equal to the sum of the laminar frictional drag from the nose to the virtual origin and the turbulent frictional drag from the virtual origin to the after end of the model. The virtual origin locations for the three-dimensional body are expected to be different for different streamlines. Due to the limited number of grid locations used in the present calculation, the location of the transition for the c?K boundary- layer calculation is set at a constant value of x/L = 0.030. The computed differ- ences in velocities using x/L = 0.01 and x/L = 0.03, for axisymmetric body get are found to be less than 0.1 percent of the free-stream velocities in the tail region. Thus, the error of using the constant transition location of x/L = 0.03 for the present c?K computation is expected to be negligible. INSTRUMENTATION A series of 0.03l-in. (0.8-mm) diameter pressure taps were embedded normal to the surface of the stern at nine x/L locations. When the model was rotated about its axis, the pressure taps were at the upper meridian location. Additional taps were added for model alinement; see Figures 3 and 4. The model was alined by balancing the surface static pressure about a line of symmetry. From Figure 3, the model is alined when symmetrically located pressure taps at c and d, and at e and f, give equal pressures, i.e., p (c) = p (d), p (e) = p (£). The model was rotated to eight test positions and the alinement was checked by the pressure balance technique. A Preston tube using a 0.072-in. (1.83-mm) inside diameter was attached and alined with the flow at the pressure taps to measure the shear stress. The Preston tube was calibrated in a l-in. (2.54~cm) diameter water-pipe flow facility described by Huang and von henezekae These pressure taps were connected to a multiple pres- Sure scanivalve system that takes one integral pressure transducer with its zeroing circuit and measures a single pressure in sequence along the stern upper meridian. The pressure transducer was designed for measuring low pressures of up to 1 psi (6.895 x 10°? Pa). The zero-drift linearity, scanivalve hysteresis, and pressure transducer zeroing circuit were carefully checked and the overall accuracy was found to be within 0.5 percent of the dynamic pressure. The mean axial and radial velocities and the turbulence intensities for the Reynolds stress calculations were measured by a TSI, Inc. Model 1241-20 'X" type hot-film probe. The probe elements are 0.002 in. (0.05 mm) in diameter with a sensing length of 0.04 in. (1.0 mm). The spacing between the two cross elements is 0.04 in. (1.0 mm). A typical schematic of the hot-film probe used is shown in Figure 5. A two-channel hot-wire and hot-film anemometer with linearizers was used to monitor the response of the hot-film probe. A temperature compensating sensor (probe) was used with each hot-film element to regulate the operating temperature of the sensor with changes in air temperature. The "X" hot film and its temperature- compensated sensor were calibrated together through the expected air temperature- range and supplied with their individual linearization polynomial coefficients at the factory. The frequency response of the anemometer system, for reliable measurements claimed by the manufacturer, is 0 to 100 kHz. Calibration of the "X" hot film was made before and after each set of measurements. It was found that the hot-film anemometer system had a +0.5 percent accuracy, +0.75 ft/sec (+0.23 m/s) accuracy at the free-stream velocity of 150 ft/sec (45.72 m/s), during the entire experiment. An estimate was made of the crossflow velocity by yawing the "X"' hot-film probe in the free stream. It was found that the crossflow velocities were about one percent of the free-stream velocity. The linearized signals were fed into a Time/Data Model 1923-C real-time analyzer. Both channels of the analog signal were digitized at a rate of 128 points per second for 8 sec. These data were immediately analyzed by a computer to obtain the individual components of mean velocity, turbulence fluctuation, and Reynolds stress on a real time basis. A traversing system with a streamlined strut was mounted on a guide plate that permitted the traverse to be locked in various stationary positions parallel to the longitudinal model axis. DISPLACEMENT BODY METHOD The theoretical method evaluated in this report is an initial attempt at ex- tending to three-dimensions the displacement body concept described by Wang and 1,2 Teenie” and by Huang et al., for axisymmetric bodies. The pressure distribution is calculated using the XYZ Potential Flow (XYZPF) computer code of Dawson and Mesine The input offsets to the XYZPF code are given in Table 1. The boundary- layer flow over the body is calculated by using the differential method of Cebeci, Chang, and Kaups (denoted c?x) .° The flow in the wake is modeled only in the near wake region of 0.93 < x/L < 1.05. The c*K method consists of using Keller's two-point finite difference metinad and Cebeci and Stewartson's meocedtres for computing flows in which the transverse velocity component contains regions of reverse flow to solve three-dimensional boundary-layer equations. The governing equations for three-dimensional incom- pressible laminar and turbulent flows are given by Continuity Equation P) x fe) = ro) aa ae ae (uh, sin 6) + 5, (why, sin 6) + Ba sin 9) = 0 (la) x-Momentum Equation u ou ,w ou du 2 =~ 2 =~ hy 9x * hy 82 aay, By - Kyu cot 6 + Kow esc 8 + K) uw 2—-— ae pe py SCH Oh ol(pi/o)) 4 cot 8 csc 8 3(p/p) mn coum (v ou -wv) (1b) hy Ox hy Oz Oy oy z—-Momentum Equation u ow ,w ow ow 2 = hy ax by Gat V ae Kaw Gore O Kyu esc 8 + Ky uw — = 2 = Beat z esc 8 (p/p) _ eset @ A@/o) fa om be Pe vw ) (1c) 1 x hy Oz oy fe) where u, v, and w = velocity components in the x, y, and z directions, respectively xX, y, and z = nonorthogonal boundary-layer coordinates, as given in Reference 6 0 = fluid density P = pressure on the body h> hy = metric coefficients Ki> K, = geodesic curvatures of the curves z = constant and Xx = constant, respectively Kio Roy = functions of the geodesic curvatures and metric coefficients 8 = angle between the coordinates x and z Vv = kinematic viscosity of the fluid (= < | ES ll Reynolds stresses The eddy—viscosity concept is used to relate the Reynolds stresses to the mean velocity profiles by u : ; —— < < é. a inner region 0 y ye uv = (2) gee outer region < dy? g Ve YW Tea 2 cos 8 (34) (32) | 1/2 which is the eddy viscosity in the inner region with & N ro) > << ee oe 1 © x ~ = Ph ——— SS! pewatls (3). i (=) i + 2 cos 0 =] @) | W WwW Ww 0.0168 | Cu, 4, dy E€ = ro) 0 u. = (u oo 28D w_ cos 6) 1/2 te e e ee Cg Gu eg Deeg cos 9) 1/2 y_ is the value of y at which Sree The displacement model presently used for this body adds the theoretical effec- tive displacement thickness (defined below) to the body surface along the major (y)- and minor (z)-axes of the elliptic cross section. . The surface profile along each of these axes is extended by hand-fairing from the location of separation, or 93 percent of body length (whichever occurs first), to 5-percent aft of the body, resulting in an open body. An elliptical cross section is defined between the off- sets of the major and minor axes. The cK computer program has been modified to compute the effective displace- ment thickness a* at the major and minor axes along the axial length of the body. The definition for a*, which is similar to the axisymmetric expression, is iP 4p yr 24 Ohx cos a c c a* = FF CMa en, (3) cos @ where alae radius of curvature at the particular axis of interest in the y-z plane, ThS als S Ig = c ; 2 y-axis diy and “Z-axis dz u (. =) rdy “te A* = effective displacement area, /|* -| 0 Q® = angle between the body surface and the body axis, fs -1 /Ay x -l (52) a = tan (5) or ® = tan = If = re =F W/ COS @! y = normal distance from the wall Unlike the procedure for an axisymmetric body, which uses an iterative procedure consisting of the calculation of pressure and boundary-layer flow over successive displacement bodies, the present scheme for three-dimensional bodies uses only one iteration. The uncertainties in defining the displacement body in the region between the major and minor axes and in the near-wake region lead one to question the use- fulness of an iterative procedure at present. It is anticipated, however, that once improvements are made in defining the displacement model over the entire body length and in the wake region, an iterative procedure will be adopted again. One further obstacle arose in defining the displacement body for the 3:1 trans- verse cross-sectional model. Excessive boundary-layer growth in the c?K boundary-— layer computation caused the computer program to abort prematurely. No values for the effective displacement thickness were computed along the major elliptic axis meridian for locations greater than 81 percent of the body length. A careful hand- fairing was used to define the effective displacement thickness along the major axis meridian. COMPARISON OF EXPERIMENTAL AND THEORETICAL RESULTS All data are presented in the coordinate system used to experimentally measure the boundary-layer flow. The coordinate system, denoted x - 1 ie 6, is given in Figures 1 and 4. The axial coordinate x is measured from the nose of the body and passes through the center of the elliptic profile. The coordinates n, and 9 are defined along an axial cut normal to the x-axis, i.e., in the y-z plane. The normal component ny is measured from the model surface and is normal to the elliptic surface. The angular coordinate 9 is defined as the angle, in degrees, measured from the z-axis to the line joining the surface offset and elliptic center. PRESSURE DISTRIBUTION The steady pressure was measured along the stern surface using pressure taps. These taps are located at nine axial and five radial positions, for a total of 45 measurements. The pressure coefficient o is computed from the measured pressures by the relationship SS (4) P PEP, = eu,” where p = measured local static pressure pan measured ambient pressure Deer measured dynamic total pressure Beth measured static pressure 0 = mass density of the fluid Ue = free-stream velocity The measured values of the pressure coefficients are given in Table 2 and com- pared in Figure 6 with two analytically-predicted distributions of pressure 10 coefficient. The dashed curve, denoted by potential flow theory, represents the predictions of the XYZ potential flow method of Dawson and nears before using the displacement body concept. The solid curve shows oe on the displacement body after one iteration of the displacement body procedure. The computed pressure coefficient (=) Gels |= (5) Pp Ww. where UE is the computed potential flow velocity on the displacement body and UR is is the free-stream velocity, 100 ft/sec (30.48 m/s). Two results are immediately apparent from the comparisons given in Figure 6. First, the theory was not able to predict accurately the values of the pressure coefficient for x/L > 0.93. At these locations, the boundary layer is much thicker than the body cross section and theoretical displacement thicknesses were not available due to premature abortion of the computer code calculation in the separa- tion region. Second, the predictions using the displacement body concept agree more closely with the measured values than do the data denoted as potential flow. After one iteration of the displacement procedure, overall agreement between theoretical and measured values of the pressure coefficient is considered good even though the predicted values are slightly lower than the measured values. No further iterations of the displacement method have been implemented at present. Further refinement of the three-dimensional wake and near wake region by the displacement body conception should improve the accuracy of the theoretical prediction. MEAN VELOCITY PROFILES Mean velocity measurements were taken with an "X'"' hot-film sensor which was stepped away from the body in the ny direction. Measurements of velocity in the axial x and normal n, directions, uy and Wao respectively, were taken with the probe elements alined vertically. The sensor elements were rotated 90 degrees to the horizontal position to measure the mean velocity Wo in the 8 direction. An on-line computer was used to collect data at a sample rate of 1024 data values in 8 sec. ial The measured values of the mean velocity components are listed in Tables 3 through 9 along with other measured quantities. Tables 3 through 9 give the measured data along the 0, 67, 80, 83, 86, 87, and 90-degree planes, respectively, for various axial locations along the model. The velocity components are non- dimensionalized by the free-stream velocity US: As shown in the tables, the mean axial velocity is the largest of the three measured components. Measured mean velocity profiles in the x and ny directions are shown in Figures 7a through 7c. Each figure presents the profiles at various angular positions for a particular axial location. Figure 7a shows that the mean axial velocity profiles vary only slightly with angular position on the model at x/L = 0.719. Also, the boundary layer is thin, with an overall thickness of less than 1 in. Little variation in the normal velocity component is noted. Examining the profiles further aft on the model, the boundary layer thickens with increased angular position. Little varia- tion in profile occurs for angles less than or equal to 80 degrees. However, profiles between 80 and 90 degrees become increasingly fuller with increased angular location. From repeated measurements, the accuracies of the experimental measure- ments of u,/U, and vi/U5 are estimated to be about 0.5 percent and 1.0 percent, respectively. Comparisons of the measured and predicted mean axial velocity profiles are shown in Figure 8 at selected positions along the model. The circular symbols represent the "X'' hot-film measurements and the solid curves represent the theoreti- cal results of the coK meenode using the displacement body concept. Calculations using the c?K computer code were made using the initial velocity profiles generated within the computer code. Calculations were begun at 1.5 percent of the body length with the transition located at 3 percent of the body length. Use of a limited, discrete set of offsets to define the model for computational purposes forced the use of this transition location. As shown in Figure 8a, the c°K method, used with and without the displacement body, predicted the same profile at x/L = 0.719 and 0 degrees. For the basic body geometry, prior to using the displacement body concept, the °K method experienced excessive boundary-layer growth and aborted prematurely, giving no predictions for axial locations x/L > 0.81 and angles greater than 80 degrees. The agreement between the computed and measured mean axial velocity pro- files is good at x/L = 0.719 and O degrees where the boundary layer is thin. 12 Agreement is also fairly good at x/L = 0.954 and 0 degrees. However, at x/L = 0.954 and angular positions 83, 86, and 20 dleszeas. the measured axial velocity components are smaller than the predicted components, with flow reversal predicted at 83 and 86 degrees. Agreement inside the boundary layer is particularly poor. Since the eddy viscosity model plays an important role in this region, it is essential to examine the eddy viscosity model used for computing the thick three-dimensional stern boundary layer. MEASURED TURBULENCE CHARACTERISTICS The turbulence characteristics of the thick three-dimensional boundary layer were measured using an "X" hot-film probe. An on-line computer was used to collect data at a sample rate of 1024 data values in 8 sec. The root-mean-square values of turbulence velocity were recorded at each probe position and the eddy viscosity and mixing length values were computed from the measured Reynolds stresses and the measured mean velocity profiles. MEASURED REYNOLDS STRESSES . : * Zz = = Zz DD De ca The distribution of the Reynolds stresses -u'v_., -u_w,, u. , V_, and w xn x 0 x n tS) represent the turbulence characteristics in the thick boundary layer. The mean- < ; A P ; ; Bein we square turbulent velocity fluctuations u, in the axial direction and 5G og the ny direction, and the Reynolds stress Say were measured with the "X" hot-film probe elements alined vertically. The probe elements were rotated 90 degrees te the hori- Ake P 20) F ; zontal position to measure both the turbulent fluctuation We in the 6 direction and —— 5 ee. the Reynolds stress —U, Wo: Linear interpolation was used to approximate We and eee mae —-u_W x 8 All measured values of the turbulent fluctuations and the measured Reynolds stresses at the same off-body positions as the data measured in the vertical direction. are given in Tables 3 through 9. The nondimensionalized distributions of the measured turbulent fluctuations v ee) pew Waa e x. «0 ji 75) locations along the model, are shown in Figures 9 through 13. As can be seen and ¥ aa and Reynolds stress -100 u'v_/U_ at selected 9 fo) i O) =) 13 in Tables 4 through 8, the Reynolds stress “uWe is typically one order of magnitude less than the Reynolds stress LV oa: An exception to this trend occurs for the angular location of 80 degrees, where measured values of “Uwe exceed the values of TE wo This is the region of predicted separation by the °K computer code. The measured distributions of “u,Wy are not depicted graphically. The results given in Figures 9 through 13 and in Tables 4 through 8 indicate DP e s 2 that uy. /U, is the largest component of turbulent velocity fluctuation and that the normal component vi? /u, is the smallest component. In addition, the fluctuations are larger near the body's surface and reduce to values near zero as the edge of the boundary layer is approached. At the body's surface, the no-slip boundary con- dition requires the velocity and turbulent fluctuations to be zero, indicating that a sharp gradient exists in the turbulent fluctuations at the wall. This gradient, which becomes apparent in the measured data as the boundary layer thickens, is evident at all angular locations where x/L > 0.914. Similar trends have been noted by Huang et ails? for axisymmetric bodies. The measured distributions of the Reynolds stress - 100 uv /U, are also shown in Figures 9 through 13. The maximum value of this component of Reynolds stress generally occurs near the body wall showing little variation with location along the model. When the boundary layer is thin, the spatial resolution of the '"X" hot-film probe may not be fine enough to measure precisely the Reynolds stress distributions near the wall. The maximum value of the = Reynolds stress occurs near the wall for all locations measured except x/L = 0.914 and 8 = 86 degrees. A turbulence structure parameter ay where ay uve/q° and ar = wo 4 + cae was investigated by Huang et afl, 29° for axisymmetric bodies. Huang's results for axisymmetric bodies showed that this parameter has a value of 0.16 for 0 SOR < 0.6 6 and that the value of a, decreases toward the edge of the boundary layer. 1 The parameter Ss used to normalize the distance from the model no» is defined as the distance from the wall surface in the ny direction at which the measured tur- bulent fluctuation uw? /U, reaches the value 0.01. Figures 14a through 14d show the 14 range of values of the parameter a, for the three-dimensional body. At most axial I positions for the O- and 6/7-degree locations, the value of a, is, approximately, 1 OMlGe tor n,/ 6. < 0.8. The value of ay reduces to 0.07 at the 80-degree plane, the region of separation predicted by the cK computer code. For the remaining angular positions, the value ay fluctuates between 0.04 and 0.16. A reduction in the value of the parameter a, was also found by Shiloh et allo” maak separation for an airfoil type flow. i The free-stream turbulent velocity fluctuations were not removed from the measured values of ace The reduction in the values of a, near the edge of the boundary layer may be caused, in part, more by the larger contribution of the free- 2 —S>— stream turbulence to q than to Sue Ls EDDY VISCOSITY AND MIXING LENGTH The values of eddy viscosity and mixing length are not measured directly, but are obtained, as in the axisymmetric case, from the measured values of the Reynolds stress Suave and the mean velocity gradient du, /on,- The definitions used to compute these quantities are du, dW sl du, +2 Ba a, } Cos Is) au (6) e e e When the values of Wy/u,, are less than 0.1 and the value of ® is 90 degrees for the present measurements, Equation (6) may be approximated by du x on | on al Os aoe ee 2 uF (7) 15 A spline curve is used to fair the experimental data before the velocity gradient is obtained numerically. The nondimensional distributions of the eddy viscosity £/ (Wg0 determined from the data are shown in Figures 15a through 15d. The parameters Us and oa are defined as the potential flow velocity at the edge of the boundary layer and the planar displacement thickness, respectively, for the displacement body. The solid curve shown in these figures is the Cebeci and smith’ thin boundary layer formula, given by (8) All values of eddy viscosity for the 3:1 elliptic model are smaller than the experi- mentally-derived values recommended by Cebeci and Smith for thin boundary layers. The experimentally—-determined distributions of the nondimensional mixing length, B/S are shown in Figures l6a through 16d. The solid Faas in these figures represents the thin boundary-layer model of Bradshaw et al. Agreement between theory and measurements is, at best, fair for angular locations of O and 6/7 degrees; for angular locations greater than 6/7 degrees, the measured values of mixing length are much smaller than the predictions. For an axisymmetric turbulent boundary layer, Huang et Men proposed a turbulence model relating the mixing length to the square root of the entire tur- bulence annulus area between the body surface and the edge of the boundary layer. As seen in Figures 14 through 16, the values of measured turbulence intensity, eddy viscosity, and intermittency across a turbulent boundary layer decrease from a maximum value at 60 percent of the boundary-layer thickness to zero at the outside edge of the boundary layer. The effective gross turbulence area relevant to the mixing length parameter is [(at+0.65_) (b+0. 66, )-(ate,) (bre, ) 15 where E. and €, are b the effective thicknesses of the separation bubble (low turbulence mixing) in the direction of the major and minor axes, a and b, respectively, of the elliptical cross-section, and Oo. and OF are the boundary-layer thicknesses along the a and b 16 axes. A new mixing length model is assumed to apply to a thick three-dimensional stern boundary layer. The schematic representation of effective turbulence areas, as determined by the areas between the body surfaces and the contours of 0.66. at x/L = 0.81 and 0.95, are shown in Figure 17. The outside edges of the effective DP turbulence areas are very close to the contours of uy /U, = 0.04. Further out- side of these edges, turbulence intensities reduce to 0.01 at the edge of the boundary layer. The mixing length parameter is assumed to be proportional to the square-root of these effective turbulence areas, e.g., UN V(at0. 65) (b+0. 65, )—ab = A(x) where the value of E is assumed to be small and will be neglected and the value of ey is zero since no separation occurs there. The values of Ss and ey may not be negligible if the separation region is so large that the effective turbulence area is reduced significantly. However, in the inner region, the conventional mixing length in the wall region, Equation (2), is assumed to apply. The mixing length & is assumed to be the same at the intersection of the inner and the outer region, Ney in Equation (2). Figures 18a through 18c show the aaeane paste length distributions for three axisymmetric bodies studied by Huang et al. ’ These figures show that the measured values for the three axisymmetric models agree reasonably well; each peaking at a value of approximately 0.05. The values of 2/A at various locations for the present three-dimensional model are shown in Figures 18d through 18j. With the exception of the 80-degree angular location, values of the non- dimensional mixing length remain fairly constant over the stern with respect to both angular and axial positions. The data in Figure 18 support the use of a revised mixing length formulation. The existing thin turbulent boundary-layer method can be applied to the axisymmetric or three-dimensional elliptical body at locations forward of where the boundary layer thickness reaches 20 percent of the major or minor axis value. Downstream of this location, the apparent mixing length &% may be approximated by the thin flat boundary layer of Bradshaw et Ase? as iL7/ = = 1 ope ©. < Oo enrl GO. < a7) a— b— (9) V[a(x)+0. 66. (x) ][b(x)+0. 65, (x) ]-a (x) b (x) a Se —_ for 6_ > 0.28 a © Ta Gg) #0. 68, rpg) ITP Gry) +0. 65), Gig) Ia GDP Oy) or 8, > 0.2b where x is the axial location downstream of the initial location of the thick stern boundary layer x The beginning of the thick stern boundary layer is selected as the axial Les where the local value of 6. or 5, grows to the value of 0.2a or 0.2b, respectively (whichever occurs first). The new formulation can be incorporated into existing axisymmetric and three-dimensional turbulent boundary-layer differ- ential methods and must be evaluated for a variety of stern boundary layers before its validity can be fully established. CONCLUSIONS The results of recent experimental investigations of the thick stern boundary layer on a three-dimensional body having 3:1 elliptic transverse cross sections are presented. Comprehensive boundary layer measurements, including mean and turbulence velocity profiles and static pressure distributions are given in detail. An initial attempt has been made at extending to three dimensions the Lighthill and Preston displacement body concept used to treat the viscid-inviscid stern flow interaction on axisymmetric bodies. The results of this initial investigation indi- cate that the use of the displacement model method significantly improves theoretical predictions of the measured pressure coefficients on the body surface. However, agreement between measured and predicted pressure coefficients remains poor in the thick stern boundary-layer region over the last 7 percent of the body. Theoretical predictions of the measured mean axial velocity profiles are satisfactory in the thin boundary-layer region, but are generally larger than the measured values when the boundary layer thickens. Refinements in the present displacement body modeling scheme to determine the effective displacement thickness accurately over the entire model surface and wake may improve the pressure distribution predictions in the thick stern boundary layer. 18 Measured values of eddy viscosity and mixing length in the thick stern boundary layer were found to be smaller than values which have been proposed for thin boundary layers. Because eddy viscosity and mixing length models play an important role in boundary-layer calculations, a modification of the theoretical mixing length model is proposed which may improve the prediction of the boundary layer. Further work in this area is needed. A larger data base of experimental results on a variety of three-dimensional geometries will aid in the development of improved theoretical models to predict the viscid-inviscid stern flow interaction. The pro- posed new mixing length formulation must be evaluated further. ACKNOWLEDGMENTS The authors would like to thank the staff at the DINSRDC Anechoic Flow Facility for their cooperation during the testing and to express their gratitude to Dr. K.C. : : ; 2 Chang for his many consultation sessions on the use of the C K computer program. 19 uoT}I9g SSOID eSteASUeIT, ITIdTTTG 1[:€ e ButAey Apoqierzy Teuotsusulrg—se1y4], ey. Jo oTJeWeYoS - | oaNnSty (WLOL'O) 4 62S5°0 = SIXV HONIIW XVW (Wy8p'0) 4 88S'L = SIXVY HOrTVW XVIN seeiBap Q ‘z | 20 Figure 2a - Stern View Figure 2b - Frontal View Figure 2 —- Model Mounted in Anechoic Wind Tunnel 21 ae AYLAIINAS JO ANIT _, — ‘ T/X 3e uotqoesg TeordAy e FO OTJeWEYDS —- 4» oANn3Ty 3} NOILVLOU A 006 Teor. FTONV NOILVI01 dvi 008 SS ee eS ewes / U e) om = fe) 2 Wes | NOILISOd 1VOILYAA OL GALVLOY SANVI1d 00 JDVAYNS OL IVINYON SdV1 SHNSS3Yd suotjeoo7] dey, einsseirg ayy JO OTAJeWeYIS - € seANsTyY ( ume: O fa) 8 O | e r© [es O—Q p rs < 3 | 9 vi be) m ” na—O = n = S | =| iS) m m | =a x ° O P | < a) ° = o LNAWANITVY 12GOW HOS YN =1x G3SN dVLADVSHNS e © gq dvL3ADvsHNS O 23 TURBULENCE IN az FLOW DIRECTION = 0.5 in. (12.7 mm) is SENSOR B oC eay Ta SS / \ | MEAN be OY Te u, SENSOR A rae ao ie 04 in. TYPICAL SPACING (1.0mm) 204 stem TURBULENCE IN DIRECTION PERPENDICULAR TO MEAN FLOW = FILM SENSOR: 0.002 in. DIA. (0.05 mm) Figure 5 - Schematic of a Two-Element Sensor Alined 90 Degrees to Each Other and 45 Degrees to the Probe Axis 24 0.3 02) 0.3 © MEASUREMENT —=- POTENTIAL FLOW THEORY —— DISPLACEMENT BODY CONCEPT ae Good 0 = Onl | | ! | aja 0.75 0.80 0.85 0.90 0.95 7.00 0.70 0.75 0.80 0.85 090 0.95 1.00 x/L Figure 6a - 9 = O Degrees x/L Figure 6b - 8 = 45 Degrees 0.3 0.2 -0.1 ! 0.70 0.75 0.80 0.85 0.90 x/L Figure 6c - 8 = 67 Degrees 0.3 | 0.2 |— ae /O ©) Cp On vo = iE | — | l l -0.11___ i 0.70 0.75 0.80 0.85 x/L 0.90 0.95 Figure 6d - 9 = 80 Degrees 1.00 0.70 0.75 0.80 0.85 0.90 0.95 1.00 x/L Figure 6e - § = 90 Degrees Figure 6 —- Computed and Measured Stern Pressure Distribution for Angular Location 25 1.00 \deddshen age. en rae PEGA sh RSIS PR mH, Bel WM Figure 7 - Measured Mean Axial and Radial Velocity Distributions 10 DATA PLANE ANGLE x/L=0.719 0 (degrees) 67 (degrees) 8 80 (degrees) 87 (degrees) 90 (degrees) 0 —0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 u,/U, or v,/U, x/L= 0.810 Ne (in) -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 u,/U, or v,/U, Figure 7a —- Nondimensional Axial Lengths, x/L = 0.719 and 0.810 27 Ng (in) Figure 7 (Continued) DATA PLANE ANGLE ° 0 (degrees) 67 (degrees) 80 (degrees) 83 (degrees) 86 (degrees) 87 (degrees) 90 (degrees) x/L = 0.854 0.4 0.6 0.8 1.0 1.2 -0.2 0.0 0.2 u,/U, or v,/U, x/L = 0.894 $+ gh Q 2) ‘ F+omrSto, < £ f ¥ £ ae Shehaxt 335595 5 6fcc ¢ 0.4 0.6 0.8 1.0 1.2 —0.2 0.0 0.2 u,/U, or v,/U, Figure 7b - Nondimensional Axial Lengths, x/L = 0.854 and 0.894 28 Figure 7 (Continued) DATA PLANE ANGLE 0 (degrees) 67 (degrees) 80 (degrees) 83 (degrees) 86 (degrees) 87 (degrees) 90 (degrees) Ne (in) Cc Me Re 16 < + —0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 u,/U, or v,/U, 0 ! -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 u,/U, or v_lU, Figure 7c - Nondimensional Axial Lengths, x/L = 0.934 and 0.954 29 Figure 8 - Computed and Measured Mean Axial Velocity Distributions x/L=0.719, 0=0 degrees © MEASUREMENT O ——_ THEORY’: POTENTIAL-FLOW & DISPLACED BODY Cp Ng (ft) x/L=0.954, 0=0 degrees Ng (ft) Ng (ft) Ng (ft) Ne (ft) Figure 8 (Continued) x/L=0.954, 0=86 degrees O MEASUREMENT THEORY: DISPLACED BODY Cp x/L=0.954, ©=90 degrees 31 4) oOL0 768°O pue “7¢8°0 ‘OT8‘°O ‘61Z°O = 1/X ‘Syq8uUeT TeTxy [eucTSuUeUTpuoN - eG oAN3Ty 80°0 90°0 70'0 c0'0 00 cL0 OL'0 80'0 90°0 0) c0'0 0 (ui) Pu v68'0=1/ vS8'0=1/ é o o = (6°n/) “aX n— ae Ei z CONS: S 2 M2. /\ 4 te} u NzALN x v OLs'0=1/* 6LL0=1/* ] Ss9ei3eq /9 = g UOTIeD0T APTNSUY Ie sesselqs sppToudey JO SuUCTINGTIISTG pernseewW - 6 saN3Tq 32 766°O pue “7€6°0 (4 L 0 "Ml 6°O = 0) 800 90°0 z0'0 0'0 clo OL vE60= 1/* S ai c0'0 90 (ut) Pu 0 (penutquo)) 6 24NnsTy 1/* ‘syuqSuey [eTXy TeuoTSUeUTpUON - 46 e4n3Ty (ut) 2u vL60=1/x 33) Figure 10 - Measured Distributions of Reynolds Stresses at Angular Location 9 = 80 Degrees 5 x/L=0.719 ° [a2 0, x/L=0.810 x SA FEM 4 # \/w' ZU, +, /100 (—u',v', /U,2) 3 £ £ “@ @ ce c 2 1 0 0.0 0.02 0.04 0.06 0.08 x/L=0.854 x/L=0.894 Ng (in) Ng (in) 0.0 0.02 0.04 0.06 0.08 0.0 0.02 0.04 0.06 0.08 Figure 10a - Nondimensional Axial Lengths, x/L = 0.719, 0.810, 0.854, and 0.894 34 80°0 B00 90" 7G6°O Pue “¥€6°O *716°O = 1/X ‘SyqBUeT TeTxy [TeuoTSUeUTpUuON - qo] eanSTy c0'0 (on! “A%n—) OOL (ui) 2u c0'0 (panutquo9) OT ean8tgq (ut) 9u 35) se0iseq €§ = Q UOTIeDOT Aep[NBuy ie Sesseiqig sprToudey Jo suOTINQTIISTG peansee_] - [TT oean3ty cL 0 OL'0 80°0 90'0 v0'0 c0'0 00 cL 0 OL0 800 90'0 yS6'0=1/ vL60=1/* 36 cL0 OL0 800 90'0 v0'0 c0'0 00 (ui) 2u v68'0=1/* S90i3eq 98 = 9 UOT}eDOT Ae[N3uy ie SOSS81}G Splousey Jo suoTjng}tAIstq peanseo, - ZI ean3ry clo oL0 800 90°0 00 c0'0 00 cL 0 oL0 80°0 90°0 00 c0'0 00 0 cL 0 0) 800 900 v0'0 c0'0 00 (2°A/ 2A“ —) OOL ¥68'0=1/x 37 768°0 pue “¥¢8°O “O18°G “61Z°0 = 1/X ‘Syasue] [eTxXy [TeUOTSUSUITpuoN - BE] oansTy zo 83) OL'0 80'0 90°0 0'0 z0'0 00 zo o1o 80'0 90°0 0 v0'0 c0'0 =} © ej v v68'0=1/ 7S8°0=1/ G l cL 0 OL‘0 80°0 90°0 ¥0'0 c0'0 00 0 L (4 3 2 te} u,Xx 5 (2 Al WA.) OOL € v OL8'0=1/x 6LL0=1/x i] Ssoviseq /g = g UOT}JeD0T AeTNSsuy Je sesseizqg sp[oudey Jo suoTINqTAISTq peanseay - €] eaN3Tg (ut) Pu (ul) 2u 38 Figure 13 (Continued) x/L=0.914 o Anti, ~ Miu, + Swi, + /i00 (-u',v', [U,2) Ng (in) 0.06 0.08 0.10 0.12 Ng (in) 0.0 0.02 0.06 0.08 0.10 0.12 Figure 13b —- Nondimensional Axial Lengths, x/L = 0.914 and 0.934 39 Figure 14 - Measured Distributions of Turbulent Structure Parameter 0.28 0 Degrees 0.24 Figure 14a - Angular Locations, 0= 0 and 67 Degrees 40 Figure 14 (Continued) 0.28 80 Degrees 0.24 nT 3 0.12 83 Degrees 0.24 0.16 0.12 0.08 0.04 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Ne/d, Figure 14b - Angular Locations 9 = 80 and 83 Degrees 41 0.28 TI lie ae ivi ae iL x 86 Degrees A 0.854 0.24 O 0.894 V 0.914 © 0.954 0.20 ~ (0.16 | % O onare 0.12 7 @) os a Ape A ‘A @) ] A Db 7 0.08 Y + oO i Oo oS © 0.04 Oo ma) A 0.0 | l CSF | aes esas) 0.0 0.2 0.4 0.6 0.8 1.2 1.4 Ne/5r O28) ngiearet ee hae RELL eT 87 Degrees 0.24 | 0.20 |- N 0.16 ferenal com oO O is 012} Bia cuss 08 o pO WO O 0.08)- A AO © Soo © fe) SS, GS © O aN 4 IX (e) (e) 0.04 oO o4 o Og 4 oO O 0.0 ei iL©@ | | 0.0 0.2 0.4 0.6 0.8 Ne/d, Figure 14 (Continued) Figure 14c - Angular Locations, 8 = 86 and 87 Degrees 42 Figure 14 (Continued) 90 Degrees 0.2 0.4 0.6 0.8 1.0 Ne/or Figure 14d - Angular Location, 8 = 9) Degrees 43 1.2 €/U55p* €/U55p* Figure 15 - Measured Distributions of Eddy Viscosity 0.020 0 Degrees 0.016 0.012 0.008 0.020 67 Degrees 0.016 oe 0.012 0.008 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Ne/d, Figure 15a - Angular Locations, 8 = 0 and 67 Degrees 44 €/U55p* €/U55p* Figure 15 (Continued) 0.020 80 Degrees 0.016 fs 0.0 0.2 0.4 0.6 0.8 1.0 Ue 1.4 0.020 0.016 Ne/d, Figure 15b - Angular Locations, 8 = 80 and 83 Degrees 45 Figure 15 (Continued) 0.020 86 Degrees 0.016 0.012 €/U55p* 0.020 87 Degrees 0.016 a 0.004 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Figure 15c - Angular Locations, 8 = 86 and 87 Degrees 46 0.020 90 Degrees Figure 15d - Angular Location, 9 = 90 Degrees Figure 15 (Continued) 47 Figure 16 - Measured Distributions of Mixing Length 0.14 =| ae 1a alow x/L 0 Degrees O 0.719 0 0.810 O 0.12 0.08 Loo 0.06 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 67 Degrees 0.10 0.08 hols 0.06 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Ne/ or Figure 16a - Angular Locations, ® = O and 67 Degrees 48 Figure 16 (Continued) 0.14 80 Degrees 0.12 0.10 0.08 ses 0.06 0.04 0.02 83 Degrees hy/ 5, 0.06 0.04 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Ne/d, Figure 16b - Angular Locations, 9 = 80 and 83 Degrees 49 Figure 16 (Continued) 86 Degrees LONE, 87 Degrees Ao 6, 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Ne/d, Figure 16c - Angular Locations, 8 = 86 and 87 Degrees 50 Ae/ 0.10 0.06 0.04 0.02 0.0 Figure 16 (Continued) le x/L 90 Degrees 0.0 ; y 0.6 Ne/dr Figure 16d - Angular Location, 8 = 90 Degrees 51 Yasue] BUTXTW eY42 FO JOOY-eAenbs sy, BuTqueseidey eezy soUeTNqIAN] - /] ean3tTy G6°O = 1/X ‘UOTIeD0T TeTXY - G/T eansTy 8'0 ; v'0 0 v0- 80- 52 Figure 18 - Proposed Similarity Concept for Mixing Length 0.10 0.08 0.06 0.04 R/ (to + 0.64)? — ro? 0.02 0.10 0.08 0.06 (ro + 0.64, = To” 0.04 0.02 of Turbulent Boundary Layer AFTERBODY 1 FROM 12th SYMPOSIUM ON NAVAL HYDRODYNAMICS JUNE 1978 0.2 0.4 0.6 0.8 1.0 1.2 (r—r,)/5, Figure 18a - Afterbody 1 AFTERBODY 2 FROM 12th SYMPOSIUM ON NAVAL HYDRODYNAMICS JUNE 1978 0.2 0.4 0.6 0.8 1.0 1.2 (r—r)/5 . Figure 18b — Afterbody 2 53 1.4 1.4 Figure 18 (Continued) 0.10 AFTERBODY 5 FROM DTNSRDC - 80/064 AUGUST 1980 0.08 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 (r—r)/5, Figure 18c - Afterbody 5 0.06 0.04 Figure 18d - Present Model, O Degree Plane +54 Figure 18 (Continued) 0.06 0.04 (a+0.6dq)(b +0.6dp) - ab 0.02 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Ne/ or Figure 18e - Present Model, 6/7 Degree Plane (a+0.6daib+0.6dp) - ab Ne/dr Figure 18f - Present Model, 80 Degree Plane 55 Figure 18 (Continued) 0.06 ab 0.04 (a+0.6da)(b+0.6dp) - 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Ne/dr Figure 18g - Present Model, 83 Degree Plane ab 0.04 (a+ 0.6dqilb+0.6dp) - 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Ne/dr Figure 18h - Present Model, 86 Degree Plane 56 Figure 18 (Continued) £/\ha+0.65aNb +0.65p) - ab Ne/dr Figue 18i - Present Model, 8/7 Degree Plane £/\Ja+0.65aNb + 0.605) - ab Ne/5, Figure 18} - Present Model, 90 Degree Plane 5)7/ TABLE 1 - MODEL OFFSETS (INCHES) Re y A x ¥ Z 0.00000 0.00000 0.00000 -2.40000 3.38476 -0.84619 0.00000 0.00000 0.00000 -2.40000 3.80785 -0.61474 0.00000 0.00000 0.00000 -2.40000 4.08710 -0,36463 0.00000 0.00000 0.00000 -2.40000 4.23095 0.00006 0.00000 0.00000 0.00000 -3.00000 0.00000 -1.57974 0.00000 0.00000 0.00000 -3,00000 0.42453 -1.57333 0.00000 0.00000 0.00000 -3.00000 0.85306 -1.55394 0.00000 0.00000 0.00000 -3.00000 1.42177 -1.50698 0.00000 0.00000 0.00000 -3.00000 1.99047 -1.43365 0.00000 0.00000 0.00000 -3.00000 2.60657 -1.31934 0.00000 0.00000 0.00000 -3.00000 3.22267 -1.15828 -0.40000 0.00000 -0.69647 -3.00000 3.79138 -0.94784 -0.460000 0.18805 -0.469365 -3.00000 4.26530 -0.468859 -0.60000 0.37610 -0.468510 -3.00000 4.57809 -0.40843 -0.60000 0.62683 -0.66439 -3.00000 4.73922 0.00000 -0.60000 0.87756 -0.63207 -4,20000 0.00000 -1.87052 -0.40000 1.14918 -0.58167 -4.20000 0.50504 -1.86293 -0.60000 1.42081 -0.51066 -4.20000 1.01008 -1.83996 -0.60000 1.67154 -0.41788 -4,20000 1.68347 -1.78436 -0.60000 1.88048 -0.30359 -4.20000 2.35685 -1.69754 -0.60000 2.01838 -0.18007 -4,20000 3.08635 -1.56219 -0.60000 2.08942 9.00000 -4,20000 3.81585 -1.37149 -1.20000 0.00000 -0.99022 -4.20000 4.48924 -1.12231 -1.20000 0.26736 -0.98621 -4.20000 5.05040 -0.81534 -1.20000 0.53472 -0.97405 -4.20000 5.42076 -0.48361 -1.20000 0.89120 -0.94461 -4.20000 5.61155 0.00000 -1.20000 1.24768 -0.89865 -5.40000 0.00000 -2.11461 -1.20000 1.63387 -0.82700 -5.40000 0.57095 -2.10603 -1.20000 2.02006 -0.72604 -5.40000 1.14189 -2.08008 -1.20000 2.37654 -0.59413 -5.40000 1.90315 -2.01721 -1.20000 2.67361 -0.43163 -5.40000 2.66441 -1.91906 -1.20000 2.86967 -0.25601 -5.40000 3.48911 -1.76605 -1.20000 2.97067 0.00000 -5.40000 4.31381 -1.55046 -1.80000 0.00000 -1.21776 -5.40000 5.07508 - 1.26877 -1.80000 0.32880 -1.21282 -5.40000 5.70946 -0.92174 -1.80000 0.65759 -1.19787 -5.40000 6.12815 -0.54672 -1.80000 1.09599 -1.16167 -5.40000 6.34384 0.00000 -1.80000 1.53438 -1.10515 -§.60000 0.00000 -2.32285 -1.80000 2.00931 -1.01703 -6.60000 0.62717 -2,31343 -1.80000 2.48424 -0.89288 -$.60000 1.25434 -2.28491 -1.80000 2.92263 -0.73066 -6.60000 2.09057 -2.21586 -1.80000 3.28796 -0.53081 -6.60000 2.92680 -2.10805 -1.80000 3.52908 -0.31484 -6.60000 3.83271 -1.93997 -1.80000 3.65329 0.00000 -6.60000 4.73862 -1.70314 -2.40000 0.00000 -1.41032 -6.60000 5.57485 -1.39371 -2.40000 0.38079 -1.40459 -6.60000 6.27170 -1.01251 -2.40000 0.76157 -1.38728 -6.60000 6.73163 -0.60056 -2.40000 1.26928 -1.34536 -6.60000 6.96856 0.00000 -2.40000 1.77700 -1.27990 -7.80000 0.00000 -2.50114 -2.40000 2.32702 -1.17785 -7,80000 0.67531 -2.49099 -2.40000 2.87705 -1.03406 -7,80000 1.35062 -2.46029 58 x -7.80000 -7.80000 -7.80000 -7.80000 -7.80000 -7.80000 -7.80000 -7.80000 -9.00000 -9.00000 -9.00000 -97.00000 -9.00000 -97.00000 -97.00000 -9.00000 -9.00000 -9.00000 -9.00000 -10.80000 -10.80000 -10.80000 -10.80000 -10.80000 -10.80000 -10.80000 -10.80000 -10.80000 -10.80000 -10.80000 -13.20000 -13.20000 -13.20000 -13.20000 -13.20000 -13.20000 -13.20000 -13.20000 -13.20000 -13.20000 -135.20000 -14.40000 -14.40000 -14.40000 -14.40000 -14.40000 -14.40000 -14.40000 -14.40000 -14.40000 -14.40000 Yi 2.25103 3.15144 4.12689 5.10234 6.00275 6.75309 7.24832 7.50343 0.00000 0.71636 1.43273 2.38788 3.34303 4.37778 5.41253 6.36768 7.16364 7.68898 7.95960 0.00000 0.76617 1.53234 2.55390 3.57546 4.68215 5.78884 6.81040 7.66170 8.22355 8.51300 0.00000 0.81327 1.62654 2.71091 3.79527 4.96999 6.14472 7.22908 8.13272 B.72912 9.03635 - 0.00000 0.82955 1.65910 2.765916 3.87122 5.06946 6.26769 7.37376 8.297548 8.903581 TABLE 1 (Continued) Zz -2.358594 -2.26985 -2.08887 -1.83387 -1.50069 -1.09022 -0.64665 0.00000 -2.65320 -2.64243 -2.60987 =2.53099 -2.40784 -2.21386 -1.94556 -1.59192 -1.15450 -0.468596 0.00000 -2.85767 -2.82615 -2.79132 -2.70696 -2.97925 -2.36992 -2.08061 -1.70260 -1.23691 -0.73366 0.00000 -3.01212 -2.99989 -2.96292 -2.87338 =2.75307 -2.91561 -2.20852 -1.80727 =1.31295 -0.77876 0.00000 -3.07240 -3.05993 -3.02222 -2.93088 -2.78828 -2.96596 =2.25272 -1.84344 -1.33923 -0.79434 59 x -14.40000 -16.20000 -16.20000 -16.20000 -16.20000 -16.20000 -16.20000 -16.20000 -16.20000 -16.20000 -16.20000 -16.20000 -18.00000 -18.00000 -18.00000 -18.00000 -18.00000 -18.00000 -18.00000 -18.00000 -18.00000 -18.00000 -18.00000 -20.40000 -20.40000 -20.40000 -20.40000 -20.40000 -20.40000 -20.40000 -20.40000 -20.40000 -20.40000 -20.40000 -22.80000 -22.80000 -22.80000 -22.80000 -22.80000 -22.80000 -22.80000 -22.80000 -22.80000 -22.80000 -22.80000 -27.60000 -27.60000 -27.60000 -27.60000 -27.60000 -27.60000 y 9.21719 0.00000 0.84611 1.69222 2.82036 3.94851 3.17066 6.59282 7292097 8.46109 9.08197 9.40121 0.00000 0.85472 1.70944 2.84907 3.98870 5.22330 6.45789 7.997952 8.94721 9.17401 9.49690 0.00000 0.85737 1.71473 2.85788 4.00104 3.25945 6.47787 7.62102 8.97345 9.20239 9.92628 0.00000 0.85737 1.71473 2.85788 4.00104 9.235945 6.47787 7.62102 8.97365 9.20239 9.92628 0.00000 0.85737 1.71473 2.89788 4.00104 3225945 Zz 0.00000 -3.13374 -3.12102 -3.08255 -2.98939 -2.84394 -2.61719 -2.29769 -1.88024 -1.365%6 -0.81020 0.00000 -3.16563 -3.15279 -3.11393 -3.01982 -2.87289 -2,64383 -2.52108 -1.89938 -1.37987 -0.81845 0.00000 -3.17543 -3.16254 -3.12556 -3.02916 -2.88178 -2.65200 -2.52826 -1.90526 -1.58414 -0.82098 0.00000 3.17543 -3.16254 -3.12356 -3.02916 -2.88178 -2.69200 -2.32826 -1.90526 -1.58414 -0.82098 0.00000 -3.17943 -3.16254 -3.12356 -3.02916 -2.88178 -2.69200 x -27.60000 -27.60000 -27.60000 -27.60000 -27.60000 -31.20000 -31.20000 -31.20000 -31.20000 -31.20000 -31.20000 -31.20000 -31.20000 -31.20000 -31.20000 -31.20000 -36.00000 -36.00000 -36.00000 -36.00000 -36.00000 -36.00000 -34.00000 -36.00000 -36.00000 -36.00000 -36.00000 -39.60000 -39.60000 -39.60000 -59.60000 -39.60000 -39.60000 -39.60000 -39.60000 -39.60000 -39.60000 -397.60000 -44.40000 -44.40000 -44.40000 -44.40000 -44.40000 -44.40000 -44.40000 -44.40000 ~44.40000 -44.40000 -44.40000 -50.40000 -30.40000 y 6.47787 7.62102 8.573465 9.20239 9.92628 0.00000 0.85737 1.71473 2.85788 4.00104 3.235945 6.47787 7.62102 8.97365 9.20239 9.92628 0.00000 0.85737 1.71473 2.85788 4.00104 9.23945 6.47787 7.62102 8.57365 9.20239 9.92628 0.00000 0.85737 1.71473 2.85788 4.00104 5023945 6.47787 7.62102 8.573465 9.20239 9.92628 0.00000 0.85737 1.71473 2.85788 4.00104 3.23945 6.47787 7.62102 8.57365 9.20239 9.92628 0.00000 0.85737 TABLE 1 (Continued) Z -2.52826 -1.90526 -1.58414 -0.82098 0.00000 -3.17543 -3.16254 -3.12356 -3.02916 -2.88178 -2.65200 -2,32826 -1.90526 -1.58414 -0.82098 0.00000 -3.17543 -3.16254 -3.12356 5.02916 -2.88178 -2.65200 -2.52826 -1.90526 -1.38414 -0.82098 0.00000 -3.17543 -3.16254 -3.12356 -3.02916 -2.88178 -2.65200 -2.52826 -1.90526 -1.38414 -0.82098 0.00000 -3.17543 -3.16254 -3.12356 -3.02916 -2.88178 -2.65200 -2.52826 -1.90526 -1.58414 -0.82098 0.00000 -3.17543 -3.16254 60 x -30.40000 -30.40000 -50.40000 -50.40000 -30.40000 -50.40000 -30.40000 -50. 40000 -30.40000 -36.40000 -56.40000 -36.40000 -36.40000 -56.40000 -56.40000 -36.40000 -36.40000 -36.40000 -36.40000 -36.40000 -60.00000 -60.00000 -60.00000 -60.00000 -60.00000 -60.00000 -60.00000 -60.00000 -60.00000 -60.00000 -60.00000 -63.60000 -63.60000 -63.60000 -63.60000 -63.60000 -63.60000 -63.60000 -63.60000 -63.60000 -63.60000 -63.60000 -68.40000 -68.40000 -68.40000 -68.40000 -68. 40000 -68.40000 -68.40000 -68.40000 -68.40000 y 1.71473 2.85788 4.00104 3223945 6.47787 7.62102 8.975469 9.20239 9.92628 0.00000 0.85737 1.71473 2.85788 4.00104 9025945 6.47787 7.62102 8.57365 9.20239 9.02628 0.00000 0.85737 1.71473 2.85788 4.00104 3225949 6.47787 7.62102 8.97365 9.20239 9.52628 0.00000 0.85737 1.71473 2.85788 4.00104 52235945 6.47787 7.62102 8.973565 9.20239 9.92628 0.00000 0.85737 1.71473 2.85788 4.00104 3623945 6.47787 7.62102 8.973565 Zz -3.12356 -3.02916 -2.88178 -2.49200 -2.52826 -1.90526 -1.358414 -0.82098 0.00000 -3.17543 -3.16254 -3.12356 -3.02916 -2.88178 -2.653200 -2.52826 -1.90526 -1.58414 -0.82098 0.00000 -3.17543 -3.16254 -3.12356 -3.02916 -2.88178 -2.69200 -2.32826 -1.90526 -1.58414 -0.82098 0.00000 -3.17543 -3.16254 9.12956 -3.02916 -2.88178 -2.69200 -2.32826 -1.90526 -1.358414 -0.82098 0.00000 -3.17543 3.16254 -3.12356 -3.02916 -2.88178 -2.65200 -2.52826 -1.90526 -1.38414 x -68.40000 -68.40000 -73.20000 -73.20000 -73.20000 -73.20000 -73.20000 -73.20000 -73.20000 -73.20000 -73.20000 -73.20000 -73.20000 -76.99344 -76.99344 -76.993544 -76.993544 -76.99344 -76.99344 -76.99344 -76.99344 -76.99344 -76.99344 -76.99344 -81.56868 -81.56868 -81.56868 -81.56868 -81.56868 -81.56868 -81.56868 -81.56868 -81.56868 -81.56868 -81.56868 -85.22880 -85.22880 -85.22880 -85.22880 -85.22880 -85.22880 -85.22880 -85.22880 -85.22880 -85.22880 -85.22880 -88.88892 -88.88892 -88.88892 -88.88892 -88.88892 y 9.20239 9.52628 0.00000 0.85737 1.71473 2.85788 4.00104 5.25945 6.47787 7.62102 8.573465 9.20239 9.92628 0.00000 0.85633 1.71265 2.85442 3.99619 5223311 6.47003 7.61180 8.56327 9.19124 9.01474 0.00000 0.84736 1.69471 2.82452 3.95433 3.17829 6.40226 72935207 8.47357 9.09497 9.41508 0.00000 0.83075 1.66149 2.76915 3.87682 9.07678 6.27675 7.38441 8.30746 8.91668 9.235051 0.00000 0.80434 1.60869 2.68114 3.75360 TABLE 1 (Continued) Zz -0.82098 0.00000 -3.17543 -3.16254 -3.12356 -3.02916 -2.88178 -2.65200 -2.32826 =1.90526 -1.38414 -0.82098 0.00000 -3.17158 -3.15871 3.11978 ~3.02550 -2.87829 -2.64879 -2.32544 -1.90295 -1.58246 -0.81999 0.00000 -3.13836 -3.12562 -35.08710 -2.993581 -2.84814 -2.62105 -2.30108 -1.88302 -1.36798 -0.81140 0.00000 -35.07684 -3.06435 -3.02658 =2.93512 -2.79230 -2.96967 =2.29997 -1.84610 -1.34116 -0.79549 0.00000 -2.97905 -2.96696 -2.930359 -2.84183 -2.70356 61 x -88.88892 -88.88892 -88.88892 -88.88892 -88.88892 -88.88892 “91417648 -91.17648 -91.17648 -91.1764€ -91.17646 -91.17648 -91.17648 -91.17648 -91.17648 -91.17648 -91.17648 -94.83660 -94.83640 -94.83660 -9 4.83660 -94.83660 -94 83660 -94.83660 -94 83660 -9 4.83660 -9 4.83660 -9 4.83660 -97.12416 -97.12416 -97.12416 -97.12416 -97.12416 -97.12416 -97.12416 -97.12416 -97.12416 -97.12416 -97.12416 -100.78428 -100.78428 -100.78428 -100.78428 -100.78428 -100.78428 -100.78428 -100.78428 -100.78428 -100.78428 -100.78428 -103.52940 y 4.91543 6.07725 7.14971 8.04343 8.63328 8.93714 0.00000 0.78253 1.96505 2.60842 3.69179 4.78211 5.91242 6.95579 7.82527 8.39912 B.69474 0.00000 0.73860 1.47720 2.46199 3.44679 4.91366 5.98052 6.965352 7.58598 7.92762 8.20645 0.00000 0.70514 1.41029 2.35048 3.29067 4.30921 9.32775 6.26795 7.05144 7.96854 7.83493 0.00000 0.64108 1.28215 2.13692 2.99168 3.971768 4.84368 3.69845 6.41075 6.88087 7.12306 0.00000 Z -2.48799 -2.18427 -1.78743 -1.29854 -0.77021 0.00000 -2.89825 -2.88648 -2.85091 -2.76475 -2.65023 -2.42051 -2.12503 -1.73895 -1.26332 -0.74932 0.00000 -2.73595 72.72445 -2.69087 -2.60955 -2.48258 -2.28463 -2.00574 -1.64133 -1.19240 -0.70725 0.00000 -2.61164 -2.60105 =2.96899 -2.49135 -2.37013 -2.18115 -1.91489 -1.96699 =1.13839 -0.67522 0.00000 =2.37435 -2.56472 -2.53557 -2.26499 -2.15478 -1.98298 -1.74090 -1.42461 -1.05496 -0.61387 0.00000 -2.16142 x -103.52940 -103.52940 -103.52940 -103.52940 -103.92940 -103.52940 -103.52940 -103.52940 -103.92940 -103.52940 -105.81696 -105.81696 -105.81696 -105.81696 -105.81696 -105.81696 -105.81696 -105.81696 -105.81694 -105.81696 -105.81696 -108.10452 -108.10452 -108.10452 -108.10452 -108.10452 -108.10452 -108.10452 -108.10452 -108.10452 -108.10452 -108.10452 -110.39220 -110.39220 -110.39220 -110.39220 -110.39220 -110.39220 -110.39220 -110.59220 -110.39220 -110.39220 -110.39220 -111.60000 -111.60000 -111.60000 -111.60000 -111.60000 -111.60000 -111.60000 -111.60000 yi 0.58358 1.16717 1.94528 2.72539 3.96634 4.40930 9.18741 9.83584 6.26380 6.48426 0.00000 0.92873 1.09746 1.76244 2.46741 3.235113 3.99486 4.69983 5.28731 3.67004 3.87479 0.00000 0.46686 0.93373 1.59621 2.17870 2.85305 3.02741 4.14990 4.66863 5.01100 5.18737 0.00000 0.39723 0.79447 1.52411 1.85375 2.42753 3.00131 3.593076 3.97233 4.26363 4.41370 0.00000 0.35597 0.71115 1.18524 1.65934 2.17294 2.68655 3.16065 TABLE 1 (Continued) Z -2.15265 52112612 -2.06186 -1.96154 -1.80514 -1.98478 -1.29685 -0.974214 -0.59882 0.00000 -1.95826 -1.95052 -1.92628 -1.846806 sal raya a7, -1.63547 -1.433582 ~1.17496 -0.85359 -0.90629 0.00000 -1.72912 -1.72211 -1.70088 -1.64948 -1.96922 -1.44410 -1.26781 -1.03747 -0.75371 -0.44705 0.00000 -1.47123 -1.46526 -1.44720 -1.40347 -1.53518 -1.22872 -1.07873 -0.88274 -0.64150 -0.58038 0.90000 -1.51694 -1.31159 -1.29543 -1.25628 -1.19515 -1.09986 -0.96559 -0.79016 62 x -111.60000 -111.60000 -111.60000 -114.00000 -114.00000 -114.00000 -114.000090 -114.00000 -114.00000 -114.00000 -114.00000 -114.00000 -114.00000 -114.00000 -114.84000 -114.84000 -114.84000 -114.84000 -114.84000 -114.84000 -114.84000 -114.84000 -114.84000 -114.84000 -114.84000 -116.04000 -116.04000 -116.04000 -116.04000 -116.04000 -116.04000 -116.04000 -116.04000 -116.04000 -116.04000 -116.04000 -116.88000 -116.88000 -116.88000 -114.88000 -116.88000 -116.88000 -116.88000 -114.88000 -116.88000 -116.88000 -116.88000 -117.36000 -117.36000 -117.356000 -117.36000 y 3.55573 3.81648 3.95081 0.00000 0.26391 0.92783 0.87971 1.23159 1.61280 1.99401 2.34989 2.63913 2.83266 2.93236 0.00000 0.22915 0.45830 0.76383 1.06937 1.40036 1.73136 2.03689 2.29190 2.49995 2.04611 0.00000 0.17833 0.55666 0.59444 0.83222 1.08981 1.34740 1.98517 1.78332 1.91410 1.98147 0.00000 » 14310 228620 247701 ~66781 287451 208122 227202 1.43102 1.53596 1.59002 0.00000 0.12767 0.259534 0.42556 —_———-occo & Zz -0.57404 -0.54048 0.00000 -0.97745 -0.97349 -0.96149 -0.93243 -0.88706 -0.81634 -0.71668 -0.98647 -0.42606 -0.25271 0.00000 -0.84870 -0.84526 -0.83484 -0.80961 -0.77022 -0.70881 -0.62228 -0.50922 -0.56994 -0.21943 0.00000 -0.66049 -0.65781 -0.64970 -0.65007 -0.99941 =0.99162 -0.48428 -0.39629 -0.28790 -0.17076 0.00000 -0.953001 -0.92786 -0.92135 -0.00559 -0.48099 -0.44264 -0.58861 -0.351800 -0.23102 -0.13703 0.00000 -0.47285 -0.47093 -0.46513 -0.45107 x -117.36000 -117.36000 -117.34000 -117.36000 -117.36000 -117.36000 -117.36000 -117.72000 -117.72000 -117.72000 -117.72000 -117.72000 -117.72000 -117.72000 -117.72000 -117.72000 -117.72000 -117.72000 -118.44000 -118.44000 -118.44000 -118.44000 -118.44000 -118.44000 -118.44000 -118.44000 -118.44000 -118.44000 -118.44000 -118.92000 -118.92000 -118.92000 -118.92000 -118.92000 -118.92000 -118.92000 -118.92000 -118.92000 -118.92000 -118.92000 -119.52000 -119.52000 -1197.52000 -119.52000 -119.52000 -119.52000 -119.52000 -119.52000 -119.52000 -119.52000 -119.52000 y/ 0.59579 0.78020 0.96461 1.13484 1.27669 1.37032 1.41855 0.00000 0.11941 0.23882 0.39803 0.55724 0.72971 0.90219 1.06140 1.19408 1.28164 1.32675 0.00000 0.10475 0.20951 0.34918 0.48885 0.64017 0.79148 0.93115 1.04754 1.12436 1.16394 0.00000 0.09493 0.18987 0.31645 0.44302 0.58015 0.71728 0.84386 0.94934 1.01896 1.05482 0.00000 0.08153 0.16306 0.27176 0.38044 0.49822 0.61599 0.72469 0.81528 0.87506 0.90586 TABLE 1 (Continued) Zz -0.42912 -0.59491 -0.34670 -0.28371 -0.20611 -0.12225 0.00000 -0.44225 -0.44046 -0.45503 0.42188 -0.40135 -0.36935 -0.32426 -0.265355 SON 19277 -0.11434 0.00000 -0.38798 -0.38640 -0.38164 -0.57011 -0.35210 -0.32403 -0.28447 =0.2327 -0.16912 -0.10031 0.00000 -0.35161 -0.35018 -0.54586 -0.335541 -0.31909 -0.29365 -0.25780 -0.21096 -0.15326 -0.09091 0.00000 -0.50195 -0.50073 -0.29702 -0.28805 -0.27403 -0.25218 -0.22140 -0.18117 -0.13162 -0.07807 0.00000 63 x -119.76000 -119.76000 -119.76000 -119.76000 -119.76000 -119.76000 -119.76000 -119.76000 -119.76000 -119.76000 -119.76000 -120.24000 -120.24000 -120.24000 -120.24000 -120.24000 -120.24000 -120.24000 -120.24000 -120.24000 -120.24000 -120.24000 -120.48000 -120.48000 -120.48000 -120.48000 -120.48000 -120.48000 -120.48000 -120.48000 -120. 48000 -120.48000 -120.48000 -120.72000 -120.72000 -120.72000 -120.72000 -120.72000 -120.72000 -120.72000 -120.72000 -120.72000 -120.72000 -120.72000 Vy 0.00000 0.07545 0.15090 0.25149 0.35209 0.46107 0.57005 0.67065 0.75448 0.80981 0.83831 0.00000 0.05362 0.10725 0.17875 0.259025 0.32770 0.40516 0.47664 0.53624 0.97557 0.99583 0.00000 0.03539 0.07077 0.11795 0.16513 0.214625 0.26736 0.31454 0.35386 0.37981 0.39318 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 Zz -0.27944 -0.27830 -0.27487 -0..26657 -0..25360 -0.23338 -0.20489 -0.16746 -0.12180 -0.07225 0.00000 -0.19861 -0.19780 -0.19536 -0.18946 -0.18024 -0.16587 -0.14562 0.11917 -0.08657 -0.05135 0.00000 -0.13106 -0. 13053 -0.12892 -0.12502 -0.11894 -0.10946 -0.09609 -0.07864 -0.05713 -0.03388 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 TABLE 2 — MEASURED PRESSURE COEFFICIENTS 0, Angular Position (deg) fiaslo| 2 saOmsc oberees, _[nakOdns, ORME" hoy AACE 719 oO Oo So =o 2&2 So Oo Oo OS 64 76€0°0 90470 °0 $6200 8810 °0 ¢¢c0 0 0€¢20°O 7920 0 8cc0 0 €2720°0 6020°0 7810 °0 1900 °0 ae-(4199*0+4) (99 0+")! % £46B0°O O45 0 * O M89)? oO 002070 MOZO°O EHh9O*O te FAQ)? ONIAUVA YO SOLLSTMALOVYVHO ALIOOTHA INA INGYNL CNV NVAWN GHYNSVAN - € ATAVL a3 TYCT°O = 49 EEQO*O OEBO? T EHO? O POO * T ByOO?rO PY Aw O LEO0%O LOL? O HLO0%O DIO? O Tee * 0 £H00°O LOOT * O 6490°O Gav *O TBO? O O60¥ °O EBOO *O POs * O OBOO *O : LEQO?O aF €€Sc°0 = 4 O40°O 401% O 6E1°O HOT? O Bet? O ek © cs aa) fig. LO GET? O Moa tb SQ) Ov Tt *O wit? O (oe) G/Z0"T = = a 100°0 Z00°O T00°0 fOO*O Z1Q0°0 = 6000 6&00°O 0° 9yTO?O 9tO%O BO80°O 61O°O E2200 EeOrd Tepo'O v&0°O 890° 680°O 8P90'O 660°O 1TeZ0°0 TEO*O EHO COO OLOT*O E00 Oro Sore 61Z°O0 = 1/¥ - VE ATAVL HUNVId AXXOHC-O ONOTV SNOILVOOT IVIXV 43 88110 43 €60°0 ¥O0°O OO *%O &00°O BLOrO fvQ?O BYO*O P9O*O O40°O fo) ig x . n\ = ia BS 0 o r@) seer SAO} (Qe PEO? Om VEO*O~ ODEO Ole FOO: He (LOVE Ole PO) vtO'O= OTO*O= 33 86S2°0 il o 43 20200 = x 9 £66°O OOO * TF OO0' T O00' T OOO? T 266'O £B86*O 696° O 16° O Lb *O 0060 6909 *O BEB *O B88ecr*O ZEB T*O TOOT*TO 8cé0*°0O BE8o*oO Eco? O EL90°O 090*O veEsSoO*o fav 0 * 0 B2E0*O POLO *O hEEO*O Std 'O (a3) °u 65 c090°0 77S0 0 00€0°0 Gce0 0 7920 °0 09¢0°0 49¢0°0 6€c0°0 7770 “0 0610 °0 88T0°O cLTO°O 1700 °0 qe- (499 °0+4) ("99 70+8) 12¥0°O Oe PO? O FOTO’ O ay €¢yT 0 = 9 O0°O 69O00°O OO* COU PO PLZO0°O BEQO*O 8400? O EhOO%O 6460050 BoOO*O vQO* BOO? OOO Bhd * O 40 18 mo) Th68T°O OT8°0 a5 €71C°O i] Qo O€cO°T £00079 4.800% O EETLOPRO BPO eO LEvaro ZB50°O . [0080 T/* - d€ wTdvL (penutquo)) € A1aVL LQO0*9O TQ0*%O &QQ?O 9OO°9 LO?) TEO*9O LEO? O 435 67ST°O a5 STT'O LOO%O £00? O ¥OO? O ZQ0°O é 10% 0) OF OQ? O GIO? QO 020°O 040° 9 &20°O x) > 990? O-= O&O * O- & “EQ * O- £8 * Om tin) ® Qo 0'0~ 6v0°O~ OG0* O~ 9 O° Q- byO*O= v0? O~ Boo" oO- EEO? Oe EO * Om EO! O= 0° 0 H H ai 62790 = # d 33 0020°0 = x 9 OO0r T 000° T LOoO?°T POO" T 5666°0O t65°0O E46°O vd *O BL6°O 068°O T9B? 0 9TB°O OL6°O Bb 2° O BOLT O BY9"O SA 92908 iSO? O BYvo?O KeLO PO SEO) JLEO*O G9TO*O vOTO'O (a)°u 66 9850 °0 €S470°0 8c£0°0 6S€0°0 GO€0°0 9120°0 68¢0°0 7620 °0 90€0°0 7€720°0 c610°0 €0c0°0 8810°0 791T0°0O ¢700°0 qe-(199°0+4) ("99 °0+8) OBY TT O 6éZOT*O EGLOTO LE B0°0 9EEO*O DEED * © BEP9O*0O BAO? O sh AON) Byro* dO COL0°O LOTO*O et 4 2 |co dF TESTO 9800°O 00°0 £900°0 b800°0 vZ00°0 1800°O 9600°O 8400°0 BOTO*O 1400°0 8£00°0 8400°0 £00*O 90080 6100°O = 9 O90 * 1 6¥06°O ELLY? O POLE? O mo) o@ PG ©) 2 bo} Ov Q B9HO*O 768°0 = '1/X — 0€ FIEVL 13 O€8T'O = 4 vGT? Q 74707 = &LOO*O ¥BOO*O B70 °O H9L0° 0 MOZO° O EBLO°O ZTOLEO 9LH0°O Ph6460°O vLOT YO (penutquoj) € ATAVL £000 100*O E000 Z00*0 1O°O 080°0 TEO*O E00 ZE0°0 TeO*O 1E0*0 eLO%O E00 E00 vE0°0 E00 “O°0 EO°0 3F CO6TO a5 0CT0 ¥O0*O 9O0°O vOO0*O 800%O 61O°O BaO*O fe0*O BL0°O Seo'o 8vO*O TO *O 450 *O fO*O BO? Q &9O*O FO *O 020°O 1Z0°O £v0*O~ 950% O- E9O*O- 90 *O~ 90° Om B9O*O= E90 O~ E90*O= 090*O- 650° O~ 250° O- GG0%O~ 0° O- MELO)O =p AF T67S 50) = 2 d 33 97Z0°0 = x 9 LOO'T £00°*T Z00*T Z00°T c00°T 986°O 026°O e56°O vE6*O 848° 0 £28*°O £78°O O£8°O £08*0O 9LL°0 Br2° 0 TTZ*O FAAS 0) s Gécr*o v08E*0 vOZt*o B9ET*O 980T*0 EL40*O 9580°0O TSZ0°0 0S90°0 69S0°0 96¥0°O 80%0*O EPEO*O BTELO*O 99E0*O 8tzoro T9TO*O rOTO'O (az)°u 67 L0€0°0 6770 °0 Tc20°0 LTZO0°O 6920°0 99720°0 €S20°0 OTZO0°O L120 °0O c810°0 €ZT0°O 6£00°0 £700°0O qe-(199°0+4) ("99 °0+8) E90 °O 160O°O 6b 70° O lyyo*o Or7O*dO TZ£0°O EO? O L9TO*O ¥A00°O a uu so |co 433 COLT 0) = 59 E00 %O O98? O EOTO*O 0E2L9°O Mh OO * O 6225 °O 0900°0O POLS SO ¥B00°O EOE? oO £800°O LU62°O 6800°O O98 ° 0 8400*O TEBe * 7800 °O rea ae ~ 0) 9L00°O OTBE*O ¥600°O TET? oO £EOO?O Z140°O 6TOO0*O v¥Z0O°O 768°0 = I/* - G€ AIGVL mer (ASv/G (0) Gl 0666°0 = 6¢T°O £200 £BT°O TSO L457 °O STtvO 69T°O 9ES0 £aT eo L390 OST *O 9TLO OSE*O 6280 eat? oO vdO TST*O SEOT £ST*O TOT ¥9T?O Tet 99-0 69TT CLT O TOTT (0) i om (0) “@ °@) %@) °@) 0) °@) °@ m0) m0) m0) °@) (0) (penufjuo)) € ATAVL €00°O c00°O €00°O £00°O 500°0O £TO°O c&Q*O Sc0°O 4EO*O O£0°O T£O*O ££0°O veO*O S£O*O S£0°O SEO *O S£0°O S20 *O 435 89472 °0 43 OVT°O ~700°O 700°O ¥00°O 500°O 500°O ZTO0°O 9£0°O Tv0°O £vO°O 050°0O &&0°O 2£50°O 090°O ¥90°O 690°O cZ£0°O £20°O cZ0°O 33 69€7°0 = & d 3F 7E70°O = x 9 000'T TOO’T TOO°T TOO*T c00°T 566°0O €56°O £L6°O 806°0 Z£28°0O ¥v8°O ZTB°O ¥82°O £v2°O 602°O &S9°O BT9°O éZ35°QO a n * ZOLS*O vTB£"0 695c°0 Z06T*O LEST *O B0ET*0 Ty60'O £080°0 cvZ0*0 EV¥90*O 850°0 cZvo*o 66£0°0 ZE£0°0 £Sz0°0 S8T0*0 8zT0°0 voto*o (a3)°u 68 ¢€L40°0 6€£0°0 T9€0°0 LO€0°0O S€€0°0 6TE0°0O 9TEO*O 7970 °0O 1920 °0O T€20°0 6020°0 LLT0°0 0ZTO°O cETO"O 6S00°0 qe- (499044) ("99 "0+8) T0460°0 9¥90°O 0690°O 9850°0O 62:90°O OT90*O vO9IO'O vOSEO*) 8670°O TvvO*O 00790°O BELO *O Aop Oa) TSEEO%O ETLO*O) q 33 T00Z°0 = 9 €T00°O 8c00°0 £900°O ¥Z00°O 9600°O €600°O vOTO*O T600°O0 T600°O 7800°0O B400°O 8900'0O 6900°O OG00°O VEOO'O 50° T 68680 BEL4°0 E4590 0095 ' 68240 °0 voto ELIL*O ELTE*O T99°0 6EER0 AWE O@ oEL*O 1960°O 76900 IF 8960°0 = 4 6v70°0O TET *O £6T*o O6T°O vVéT*O v2ZT*O vLT°O OZT*O BET" O Sat *O ba aa) £41T°O 5000°O £500°O exLeo*oO B8£70*O 6190°O EL90'O SéB0"O 6040°O 0260°O OOOT*O OvOT*O BET? oO PATO 6OT*O 6B8ET*O 7€6°O = 1/X - HE AIGVL (penut3uo0D) € FTAVL TO0*O TOO*O €00°O £00°O 900°O clO*O 610°O veEO*O ZE0*O 6€0°O T£0°O £.0°O ve0°O £L0*O 9Z0*O BeO *O 4EO°O veo*O BLO °O 33 SC9IE*O = 33 OST*O = £00°O 700°O £00*O 500°O Z00°*O 9T0*O 6c0°O Ov0*O 270°O 470°O S0*O PEO" O B8E0°O £9O°O ¥9O'"O £90" O 020°O 020°O UASNTO) eB 9 a 9 E20 °O- 620° O- 870*°0- T90*O- PRO? 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O- Ort? o- 9E 15 + On i i=] 43 S66T 33 TOCT*O T65°O 166°O 066°9 086*0 6° O cEb"O £8B°9O 9¥B°O 8462°0 6£E2°0 69 *O 989" O 609° 0 B25 °O vo* O 660°O S9¥'O ayy? o OT’ to OLZ£L°O iene) A ‘9 =e d =¥ 0 ZB99°O SIOP*O 1605°O BLO 1a *Q LYb6L?O BEIL?O Seek oO ELGE?O EEG" O ba Y 0) ew a 0) ¥98O°O LEIO SO &2£¥0°O Z90°O ¥Z20°O ZL1.0°O vOTO*O (a3)°u 104 REFERENCES 1. Huang, T.T. et al., "Stern Boundary-Layer Flow on Axisymmetric Bodies," 12th Symposium on Naval Hydrodynamics, Washington, D.C. (5-9 Jun 1978). Available from National Academy of Sciences, Wash., D.C., pp. 127-147 (1979). 2. Huang, T.T. et al., '"Boundary-Layer Flow on an Axisymmetric Body with an Inflected Stern," DINSRDC Report 80/064 (1980). 3. Preston, J.H., ''The Effect of the Boundary Layer and Wake on the Flow Past a Symmetrical Aerofoil at Zero Incidence; Part I, The Velocity Distribution at the Edge of and Outside the Boundary Layer and Wake,'’ ARC R&M 2107 (1945). 4. Lighthill, M.J., "On Displacement Thickness," Journal of Fluid Mechanics, Vol. 4, pp. 383-392 (1958). 5. Dawson, C. and J. Dean, "The XYZ Potential Flow Program,'' NSRDC Report 3892 (1972). 6. Cebeci, T. et al., "A General Method for Calculating Three-Dimensional Laminar and Turbulent Boundary Layers on Ship Hulls," McDonnell Douglas Corporation Report J7998 (1978). Also Ocean Engineering, Vol. 7, pp. 229-289, Pergamon Press, Great Britain (1980). 7. Cebeci, T. and A.M.O. Smith, "Analysis of Turbulent Boundary Layers," Academic Press, New York (1974). 8. McCarthy, J.H. et al., "The Roles of Transition, Laminar Separation, and Turbulence Stimulation in the Analysis of Axisymmetric Body Drag," 11th Office of Naval Research Symposium on Naval Hydrodynamics, London (1976). 9. Huang, T.T. and C.H. von Kerczek, "Shear Stress and Pressure Distribution on a Surface Ship Model: Theory and Experiment," 9th Office of Naval Research Symposium on Naval Hydrodynamics, Paris (1972). Proceedings are available in U.S. Government Printing Office as ACR-203, Vol. 2. 10. Wang, H.T. and T.T. Huang, "Calculation of Potential Flow/Boundary Layer Interaction on Axisymmetric Bodies," The American Society of Mechanical Engineers Symposium on Turbulent Boundary Layers, Niagara Falls, N.Y., pp. 47-57 (18-20 Jun 1197.9)" 105 11. Keller, H.B., "A New Difference Scheme for Parabolic Problems," Numerical Solution of Partial Differential Equations, II, J. Bramble (ed.), Academic Press, New York (1970). 12. Shiloh, K. et al., "The Structure of a Separating Turbulent Boundary Layer. Part 3: Transverse Velocity Measurements," Journal of Fluid Mechanics, Wola ils} (QIEKSID) 13. 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