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Local fluid velocities in the regions of interest were measured with a traverse-mounted two-component LDV. This LDV system uses Dantec FO probes with 111 mm beam spacing and 1600 mm focal length for an in-water probe volume 0.17 mm in diameter and 6.5 mm in length. Laser wavelengths of 514.5 and 488.0 nm were used for the two LDV channels. Both velocity channels were Bragg-cell shifted to allow measurement of flow velocities with either sign. The flow components measured were streamwise mean (U) and fluctuating (u) velocities (positive downstream), and vertical mean (V) and fluctuating (v) velocities (positive upward). Reynolds shear stress, , was also measured. The LDV data processors were burst signal analyzers from Dantec, so flow statistics were determined from tabulations of individual particle passages through the LDV focal volume. Continuous time histories of flow velocity were not analyzed. Thus, turbulence spectra are not presented here. The LDV system was calibrated with a reference velocity from a spinning disk attached to a Compumotor SM32 motor driven by a Compumotor TQ10X Servo Controller. Bias error in this calibration is introduced through uncertainty in (1) disk rotational speed, (2) disk radius, and (3) the linear regression fit of the calibration curve. Disk speed uncertainty based on the manufacturer's spec is ±0.040 revolutions per second. The disk radius is 100 mm with an uncertainty of ±0.35 mm arising from the need to locate the disk center. The uncertainty introduced by the linear regression fit is no greater than ±30 mm/s. Based on these values, the maximum calibration bias in the LDV velocity measurement ranges from ±1.4% at a flow speed of 3 m/s to ±0.4% at 18.3 m/s. This translates into a maximum fractional bias error in the normalized mean velocities (dimensionless) of ±0.02 at 3 m/s to ±0.006 at 18.3 m/s. Through the normalization velocity, bias also enters the normalized mean squares of the velocity fluctuations, but is limited to a fraction of ±0.002 or less. Precision error is also present in the calibration, but 1000 LDV samples are taken to produce each calibration curve point, rendering this error negligible. In order to make proper use of facility time, a nominal sampling period of 0.6 min per coordinate location was chosen for collection of all data sets. Since the LDV data rate changed with survey location and flow speed, the collection period was controlled (when necessary) by varying the number of samples per coordinate location between 500 and 12,000. At some coordinates, fewer samples were acquired than sought either due to low data rate (timeout) or failed data acceptance criteria for the LDV bursts. In such cases, measurements from coordinate locations with fewer than 100 samples were discarded in post-processing. The main impact of this test-timing constraint was felt in the separated and reverse-flow near the foil's trailing edge at the 3 m/s test speed. Otherwise the 0.6-minute data-point interval was well matched to the experiment. The vortex-shedding oscillation time scale for the foil's wake was calculated (Blake 1986) to be 60 ms at 3 m/s and 10 ms at 18.3 m/s. Hence, the data point collection interval represents 600 to 3600 fundamental wake oscillations, so statistical uncertainty in the measured mean velocities and turbulence quantities should be merely a few percent. This contention is backed up by the relatively smooth measured profiles shown on Figs. 8–24. In all cases, the LDV's random fluctuation level (approx. 1% of the freestream speed) were incoherently subtracted from the reported u and v fluctuation levels. Scatter in the plotted data points is probably the best measure of the extent of statistical convergence. The two-component LDV measurements were made at five stations for a total of over 3,100 coordinate locations within the flow. The five stations are shown on Fig. 3 and are referred to as follows: (1) the inflow plane, with its normal along the flow direction; (2) the leading edge line, a vertical line of data just upstream of the hydrofoil; (3) the trailing edge region, a plane with its normal perpendicular to the flow and centered near the point of separation, (4) the near wake plane, a plane slicing through the trailing edge region with its normal along the flow direction; and (5) the far wake line, a vertical line of data further downstream of the trailing edge. the authoritative version for attribution. Figure 3. Location of LDV Surveys

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. at 3.0 and 18.3 m/s HYDROFOIL TURBULENT BOUNDARY LAYER SEPARATION AT HIGH REYNOLDS NUMBERS Figure 11. Trailing-edge LDV measurements of normalized vertical mean velocity V/Uref, taken at 3.0 and 18.3 m/s Figure 10. Trailing-edge LDV measurements of normalized streamwise mean velocity U/Uref, taken at 3.0 and 18.3 m/s Figure 12. Trailing-edge LDV measurements of normalized streamwise mean square velocity fluctuations /Uref2, taken 319

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. 3.0 and 18.3 m/s HYDROFOIL TURBULENT BOUNDARY LAYER SEPARATION AT HIGH REYNOLDS NUMBERS Figure 14. Trailing-edge LDV measurements of normalized Reynolds Stress /Uref2, taken at 3.0 and 18.3 m/s Figure 15. Trailing-edge LDV measurements of normalized streamwise mean velocity U/Uref, taken at 6.0 and 12.0 m/s Figure 13. Trailing-edge LDV measurements of normalized vertical mean square velocity fluctuations /Uref2, taken at 320

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. 6.0 and 12.0 m/s at 6.0 and 12.0 m/s HYDROFOIL TURBULENT BOUNDARY LAYER SEPARATION AT HIGH REYNOLDS NUMBERS Figure 16. Trailing-edge LDV measurements of normalized vertical mean velocity V/Uref, taken at 6.0 and 12.0 m/s Figure 18. Trailing-edge LDV measurements of normalized vertical mean square velocity fluctuations, /Uref2, taken at Figure 17. Trailing-edge LDV measurements of normalized streamwise mean square velocity fluctuations /Uref2, taken 321

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line HYDROFOIL TURBULENT BOUNDARY LAYER SEPARATION AT HIGH REYNOLDS NUMBERS 322 Figure 19. Trailing-edge LDV measurements of normalized Reynolds Stress /Uref2, taken at 6.0 and 12.0 m/s The unconventional grouping of the data was chosen to ease the comparisons between the lowest and highest Reynolds number measurements. In both figures, the foil's trailing edge appears as an outline within the left half of each frame. The vertical and horizontal axes show the spatial coordinates normalized by the foil chord. Here x/C= 1 lies at the tip of the foil's trailing edge, and y/C=0 corresponds to the vertical location of the flat pressure side of the foil. The vertical lines within each frame correspond to the location where the various flow quantities were measured. The horizontal distance from these vertical lines to the measured data points represents the measured field values. A scale that allows numerical determination of the field values is provided inside the foil outline of each figure. Negative measured values appear to the left of their vertical location lines while positive values appear to the right. For example, consider the measurements at x/C=1 on Figs. 10 and 11 for the normalized mean velocity components, U and V. On both frames, U and V go to zero at the tip of the foil and the measured data touch the vertical location line that passes through the tip of the foil. Using this as a reference point on Fig. 10, U above the foil tip is seen to have a weak reverse flow region and then a smoothly increasing value as y/C increases. Below the foil, U appears to change discontinuously to a positive value. The discontinuity occurs because the finite- size LDV focal volume can only be centered about one or two focal volume diameters from the foil's surface on closest approach because of optical constraints. In addition, the measurements near the topside (underside) of the foil must be made from above (below) with the LDV optics tilted slightly downward (upward). Thus, all the profiles for x/C≤1 (and some for x/C>1) are constructed from two or more separate traverses of the LDV system. Perhaps the most important feature of the data shown on Figs. 10 through 19, is the Reynolds number dependence. This is most clearly seen in the profiles of the turbulence quantities for the newly separated suction-side boundary layer (see Figs. 12, 13, and 14). Here, the peak values in the turbulence quantities are interpreted as lying at the center of the developing shear layer and the height of this shear layer above the foil surface is clearly different for the 3 m/s and 18.3 m/ s tests. Thus, the vertical extent of the near-wake is slightly larger at the lower flow speed, and this increased wake width extends out past the right edge of each figure. The mean velocity profiles are consistent with this trend. On Fig. 10, the streamwise velocity profile is clearly inflected at x/C=0.986 (−30 mm) for the 3 m/s data while the same profile is fuller and not inflected for the 18.3 m/s measurements. Similarly, the vertical velocity at the same location (Fig. 11) is less negative close to the foil surface at 3 m/s than at 18.3 m/s. Taken together these mean-flow findings and those for the turbulence quantities, all suggest that the suction side boundary layer separates closer to the foil's trailing edge at the higher speed. Thus, a simple interpretation of the suction side flow emerges. For the fixed geometry of the foil, increasing the Reynolds number through increases in tunnel speed act to thin the suction side boundary layer. A thinner boundary layer better is able to resist separation in the adverse pressure gradient that exists on the aft half of the suction surface. Thus at higher Reynolds number, the suction side boundary layer makes it further past the knuckle and separates closer to the trailing edge and the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line HYDROFOIL TURBULENT BOUNDARY LAYER SEPARATION AT HIGH REYNOLDS NUMBERS 324 Figure 22. Station 5 (Far Wake), /Uref2 Figure 21. Station 5 (Far Wake), V/Uref Figure 24. Station 5 (Far Wake), /Uref2 Figure 23. Station 5 (Far Wake), /Uref2 As with the near-trailing edge measurements, these wake measurements can be compared to classical results. The measured peak Reynolds shear stress normalized by the wake deficit velocity is 0.06±0.01, and this compares well to the range of values (0.04 to 0.08) found by Narasimha and Prabhu (1972) although 0.06 is somewhat above their equilibrium peak value of 0.045. Likewise the measured peak streamwise and vertical velocity fluctuations are approximately the same and this matches the expectation for two-dimensional wake flows (Townsend 1976). However the measured fluctuations levels in the foil's wake are a factor of two higher than those in a classical two- the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line HYDROFOIL TURBULENT BOUNDARY LAYER SEPARATION AT HIGH REYNOLDS NUMBERS 325 dimensional wake flow. This difference may arise because the hydrofoil's wake has both wake-deficit and shear-flow characteristics that are produced by the drag and lift loads on the foil, respectively. PRELIMINARY COMPARISONS TO CALCULATIONS Computations of the flow over the foil were made using Mississippi State's UNCLE flow solver (UNsteady Computation of FieLd Equations) which is based on the Reynolds-averaged Navier-Stokes (RANS) equations (Arabshahi et al. 1999). Time and resource limitations only allowed calculations at the 3 m/s and 6 m/s to be completed. The computations were run as a two-dimensional model, and did not include the tunnel walls in the simulation. A C-type grid containing 170,000 points was used for the computation. As shown in Fig. 25, the point distribution was densely packed near the foil surface and in the trailing edge region; y+ values near the surface are less than 0.5. Figure 25. Computational Grid In order to correct the computed data for the effects of the test-section walls and the boundary layers developing on them, a free-stream velocity correction was computed in the following manner: at the location where the computed surface pressure is zero, measured velocities equidistantly located above and below the foil are averaged. The computed surface pressure was zero at x/C=0.98. This method is based on experimental results by Jiang et al., (1990) who determined that the axial location where the surface pressure is zero is least affected by tunnel walls, and best represents what the unbounded velocity should be. This value was computed to be 1.065 for the 3 m/s case and 1.064 for the 6 m/s case. Comparisons between the computed results and the experimental measurements were performed at x/C locations of 0.978, 0.992, 1.0, and 1.028 as shown in Figures 26–33. Computations were completed at Reynolds numbers of 9 and 16 million and the results match favorably with the experimental measurements of the mean flow. The q-ω turbulence model gave more favorable results than the k-ε model in matching velocity profiles at the trailing edge region. Figure 26. Normalized mean streamwise velocity at x/ Figure 27. Normalized mean streamwise velocity at x/ the authoritative version for attribution. C=0.978, 3.0 m/s C=0.992, 3.0 m/s

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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. C=0.978, 6.0 m/s C=1.000, 3.0 m/s Figure 30. Normalized mean streamwise velocity at x/ Figure 28. Normalized mean streamwise velocity at x/ C=0.992, 6.0 m/s C=1.028, 3.0 m/s HYDROFOIL TURBULENT BOUNDARY LAYER SEPARATION AT HIGH REYNOLDS NUMBERS Figure 31. Normalized mean streamwise velocity at x/ Figure 29. Normalized mean streamwise velocity at x/ 326

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line HYDROFOIL TURBULENT BOUNDARY LAYER SEPARATION AT HIGH REYNOLDS NUMBERS 327 Figure 32. Normalized mean streamwise velocity at x/ Figure 33. Normalized mean streamwise velocity at x/ C=1.000, 6.0 m/s C=1.028, 6.0 m/s SUMMARY AND CONCLUSIONS Controlled tests of a two-dimensional hydrofoil at chord-based Reynolds numbers from 6 to 60 million have been performed. Two-component Laser-Doppler Velocimetry measurements have been made upstream of the foil, near the foil's leading and trailing edges, and in the wake of the foil. Spanwise uniformity of the flow over the foil was verified. Results for mean flow and turbulence quantities have been presented. Although this investigation will continue, the present results lead to three conclusions. First, the foil's near wake flow features appear to be Reynolds number dependent. This is shown most prominently in Figs. 12–14 where all of the profiles of the normalized Reynolds stress components on the suction side of the foil near its trailing edge show clear differences between test results at 3.0 m/s and 18.3 m/s. Second, the observed Reynolds number dependence is consistent with suction side boundary layer separation occurring closer to the trailing edge at higher Reynolds number. Support for this conclusion can be found in measurements of the mean flow and the turbulence quantities. And finally, classical RANS-based turbulence models appear to hold some promise for simulating the mean flow over this hydrofoil. However, the results presented here in Figs. 26 through 33 should be considered preliminary. ACKNOWLEDGEMENTS The authors of this paper wish to the acknowledge the contributions of Paul Tortora and Ronnie Bladh of the University of Michigan; William Blake, Ken Edens, Bob Etter, Ted Farabee, Jon Gershfeld, Joe Gorski, Tom Mathews, David Schwartzenberg, Jim Valentine, Phil Yarnall, and the LCC crew from the Naval Surface Warfare Center— Carderock Division; Lafe Taylor, Min-Yee Jaing, and David Whitfield from Mississippi State University; and Pat Purtell and Candace Wark from the Office of Naval Research. In addition, the authors wish to thank the Office of Naval Research for supporting this research effort under contract nos. N00014–99–1–0341, and N00014–99–1–0856. REFERENCES Arabshahi, A., Beddhu, M., Briley, W., Chen, J., Gaither, A., Janus, J., Jaing, M., Marcum, D., McGinley, J., Pankajakshan, R., Remotigue, M., Sheng, C., Sreenivas, K., Taylor, L., Whitfield, D. (1999) “A perspective on naval hydrodynamic flow simulation,” 22nd Symposium on Naval Hydrodynamics (National Academy Press, Washington DC), pp. 920–934. Blake, W.K. (1986) Mechanics of Flow Induced Sound and Vibration, Vol. II. (Academic Press, Orlando). Crighton, D.G. (1985) “The Kutta condition in unsteady flow,” Annual Review of Fluid Mechanics, Vol. 17, 411–445. the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line HYDROFOIL TURBULENT BOUNDARY LAYER SEPARATION AT HIGH REYNOLDS NUMBERS 328 Hinze, J.O. (1975) Turbulence, 2nd Ed. (McGraw Hill, New York), pp. 638–643. Jiang, C.W., Liu, H.L., and Huang, T.T. (1990) “Determination of wind tunnel wall effects and corrections”. In Proceedings of the 19th International Towing Tank Conference, held in Madrid, Spain, Vol. 2, PS-2.4, 310–317. McCormick, B.W. (1979) Aerodynamics, Aeronautics, and Flight Mechanics (John Wiley & Sons, New York), pp. 76–82. Narasimha, R., and Prabhu, A. (1972) “Equilibrium and relaxation in turbulent wakes,” J. Fluid Mech., Vol. 54, Pt. 1, 1–17. Simpson, R.L. (1989) “Turbulent boundary layer separation,” Annual Review of Fluid Mechanics, Vol. 21, 205–234. Townsend, A.A. (1976) The Structure of Turbulent Shear Flow (Cambridge University Press, Cambridge), pp. 202, 217. Wang, M., Lele, S.K., and Moin, P. (1996) “Computation of Quadrapole Noise Using Acoustic Analogy,” AIAA Journal, Vol. 34. 2247–2254. White, F.M. (1991) Viscous Fluid Flow, 2nd Ed. (McGraw Hill, Inc., New York), pp. 433–435. Wygnanski, I. And Fiedler, H.E. (1970) “The two-dimensional mixing region,” J. Fluid Mech., Vol. 41, 327–361. the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line HYDROFOIL TURBULENT BOUNDARY LAYER SEPARATION AT HIGH REYNOLDS NUMBERS 329 DISCUSSION S.Cordier Bassin d'Essais des Carenes, France Ship propeller finish tends to be degraded as the ships are in service. Could you discuss the influence of surface finish on the type of results you are presenting? AUTHOR'S REPLY Surface roughness can lead to premature transition to turbulence in the boundary layer and disrupt the flow within the boundary layer. The surface of the HIFOIL has been highly polished to an RMS surface roughness of 0.25 micrometers or less, and at the Reynolds numbers of interest, the hydrofoil can be considered hydrodynamically smooth (k +<0.5). Consequently, natural boundary layer transition is expected to occur near the leading edge, and the exact location of transition may vary somewhat with Reynolds number. However, the boundary layer has completely undergone transition by the time it reaches the trailing edge, and we do not expect that small variation in upstream transition location to influence the trailing edge flow. For fully rough flows (k+ >30), the surface roughness can significantly affect the developed boundary layer flow. It is possible that a propeller blade in service could exhibit such roughness (>25 micrometers RMS, say). Its effect on the flow would depend not only on the RMS level, but the topology of the roughness elements and their location on the blade surface. Since the focus or our experiments was scientific in nature we choose to study the simplest case (i.e. smooth) first. However, we are interested in investigating the effects of roughness in the future. DISCUSSION F.Di Felice Instituto Nazional per Studi ed Experienze di Architettura Navale (INSEAN), Italy LDV provides velocity measurement at a point and results are shown on a reference frame referred to the model. Did you take into account the model deformations? If yes, how? Measurements show find points near the surface. How do you solve the problem of the liquidation of the optical access when measuring the velocity component normal to the profile surface (vortical)? AUTHOR'S REPLY An experimental procedure was devised to measure velocities close to the surface of the hydrofoil. With the probe volume located in the area of interest, the LDV head was tilted to the minimum angle from horizontal at which all four LDV beams cleared the hydrofoil. Since this angle is known, its effect on the measured stream wise and transverse velocities could be taken into account. However, in all cases the angle effect was negligible in comparison with other sources of error and so was disregarded. The channel flow was then set on condition so that the hydrofoil assumed its lift-loaded shape. At this time the sharp tip of the trailing edge was located and used as the spatial reference point for the LDV measurements. (The point was located by observing the appearance and disappearance of the surface laser flare as the LDV probe volume was scanned vertically across the tip of the trailing edge.) Using this reference point with the hydrofoil's known angle of attack and surface contour, any given spatial coordinate for the LDV data may be related to any point on the hydrofoil surface. The LDV spatial coordinates were arranged in vertical columns rising from the hydrofoil surface. In order to ensure that data was acquired as close to the surface as conditions allowed, each column began with a coordinate just inside the hydrofoil surface and marched outward from the surface at increments of 0.2 mm. (Probe volume diameter was 0.17 mm.) Coordinates below the surface and so near to the surface as to be affected by surface flare timed-out before producing data. Such data dropouts were discarded in post-processing. the authoritative version for attribution.